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https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=117&t=64844&p=257144 | ## Sapling week 2/3
$H_{\psi }=E_{\psi }$
1-D: $E_{TOTAL}\psi (x)=E_{k}\psi (x)+V(x)\psi(x)=-\frac{h^{2}}{8\pi ^{2}m}\frac{d^{2}\psi(x)}{dx^{2}}+V(x)\psi(x)$
Jessica Katz 1G
Posts: 45
Joined: Wed Sep 30, 2020 10:09 pm
Been upvoted: 2 times
### Sapling week 2/3
Should we be able to answer this question already? If so does anyone know the steps to solve it? Any help would be great. Thanks!
What is the minimum uncertainty in an electron's velocity (Δvmin) if the position is known within 15 Å.
What is the minimum uncertainty in a helium atom's velocity (Δvmin) if the position is known within 1.2 Å.
DanielHong2L
Posts: 39
Joined: Wed Sep 30, 2020 9:51 pm
### Re: Sapling week 2/3
Jessica Katz 1G wrote:Should we be able to answer this question already? If so does anyone know the steps to solve it? Any help would be great. Thanks!
What is the minimum uncertainty in an electron's velocity (Δvmin) if the position is known within 15 Å.
What is the minimum uncertainty in a helium atom's velocity (Δvmin) if the position is known within 1.2 Å.
Intedeterminancy Equation = delta momentum * delta position >/= h/4pi
h/4pi is a constant; delta position is the 15 angstroms; delta momentum = mass of electron * velocity
- you know mass of electron (9.1*10^-31 kg); so you could figure out veloicty of electron
For helium change the mass to mass of helium atom (I believe this should still work)
Lmk if you have more questions!
David He
Posts: 32
Joined: Wed Oct 07, 2020 12:16 am
Been upvoted: 1 time
### Re: Sapling week 2/3
DanielHong2L wrote:
Jessica Katz 1G wrote:Should we be able to answer this question already? If so does anyone know the steps to solve it? Any help would be great. Thanks!
What is the minimum uncertainty in an electron's velocity (Δvmin) if the position is known within 15 Å.
What is the minimum uncertainty in a helium atom's velocity (Δvmin) if the position is known within 1.2 Å.
Intedeterminancy Equation = delta momentum * delta position >/= h/4pi
h/4pi is a constant; delta position is the 15 angstroms; delta momentum = mass of electron * velocity
- you know mass of electron (9.1*10^-31 kg); so you could figure out veloicty of electron
For helium change the mass to mass of helium atom (I believe this should still work)
Lmk if you have more questions!
Thanks for your clarification, but where should I find the weight of helium atom ?
Joseph Hsing 3H
Posts: 35
Joined: Wed Sep 30, 2020 9:42 pm
### Re: Sapling week 2/3
You can find the molar mass of a Helium atom on the periodic table: 4.002 g/mol. Since we are using Joules with Planck's constant, convert grams to kilograms, so 4.002 x 10^-3 kg. And make sure to multiply by Avogadro's number so 6.002 x 10^23 He atoms/mol in this question so that you get the value for just one He atom.
MCalcagnie_ 2H
Posts: 31
Joined: Wed Sep 30, 2020 10:08 pm
### Re: Sapling week 2/3
Hi!! First off, this was really helpful for me, I have been stuck on that problem for a while now only to realize it was a conversion issue. Second, I was wondering if anyone knew how to solve #4 on Sapling "When a metal was exposed to photons at a frequency of 1.32×1015 s−1,
electrons were emitted with a maximum kinetic energy of 3.50×10−19 J." ? I thought I understood it, but I keep getting it wrong.
John Calonia 2F
Posts: 36
Joined: Wed Sep 30, 2020 9:49 pm
### Re: Sapling week 2/3
This is a photoelectric effect question. You would enter in your known values for frequency and kinetic energy to find the work function I assume. That would lead you with 3.50×10−19J=(6.626x10^-34)(1.32×1015 Hz) - (work function). From there just solve for work function through algebra.
MCalcagnie_ 2H
Posts: 31
Joined: Wed Sep 30, 2020 10:08 pm
### Re: Sapling week 2/3
Okay perfect thanks John! What abut for part two of that question? Where do I get the other values? Do I use the work function from the first part or no because they are different frequencies?
asalest 2K
Posts: 41
Joined: Wed Sep 30, 2020 9:39 pm
### Re: Sapling week 2/3
for the first part, you regimen all the information needed to solve the equation using the uncertainty equation, just use the electron mass and don't forget to convert angstroms to m!
for the second one make sure you convert the molar mass of helium from g/mol to kg/atoms using avogadro's then solve using the uncertainty equation! make sure all your units are correct!!
Laura 3J
Posts: 39
Joined: Wed Sep 30, 2020 9:55 pm
Been upvoted: 1 time
### Re: Sapling week 2/3
You would just use Heisenberg's Indeterminacy equation but first convert Angstroms to meters.
Shrey Pawar 1A
Posts: 43
Joined: Wed Sep 30, 2020 9:42 pm
Been upvoted: 2 times
### Re: Sapling week 2/3
You can use the indeterminacy equation and fill in the mass as that of an electron or in the second case the molar mass of helium (in kg) multiplied by Avogadros. Then you can get the velocity. After this you can just use the KE = .5mv^2 and plug in. Hope this helps!
MCalcagnie_ 2H
Posts: 31
Joined: Wed Sep 30, 2020 10:08 pm
### Re: Sapling week 2/3
Thanks everyone, that cleared things up!
Lucy_Balish_1I
Posts: 36
Joined: Wed Sep 30, 2020 9:37 pm
Been upvoted: 1 time
### Re: Sapling week 2/3
Joseph Hsing 3H wrote:You can find the molar mass of a Helium atom on the periodic table: 4.002 g/mol. Since we are using Joules with Planck's constant, convert grams to kilograms, so 4.002 x 10^-3 kg. And make sure to multiply by Avogadro's number so 6.002 x 10^23 He atoms/mol in this question so that you get the value for just one He atom.
Thank you so much for explaining this. It helped a lot and I feel better about how to solve this problem thanks to you!
Chinmayi Mutyala 3I
Posts: 34
Joined: Wed Sep 30, 2020 9:50 pm
### Re: Sapling week 2/3
Hi! This question has to do with Heisenberg Uncertainty/Indeterminacy principle. | 2020-11-25 15:24:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "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": 0.4607475996017456, "perplexity": 2527.4636650831426}, "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/1606141182794.28/warc/CC-MAIN-20201125125427-20201125155427-00702.warc.gz"} |
https://socratic.org/questions/how-do-you-use-the-factor-theorem-to-determine-whether-x-1-is-a-factor-of-f-x-x- | # How do you use the factor theorem to determine whether x-1 is a factor of f(x)=x^3+4x-5?
Calculate $f \left(1\right)$.
If $f \left(1\right) = 0$, then $x - 1$ is a factor of $f$. So let's calculate $f \left(1\right)$.
$f \left(1\right) = 1 + 4 - 5 = 5 - 5 = 0$. So $1$ is a root of this polynomial, and $x - 1$ is a factor of it. | 2020-04-05 20:26:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "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": 0.9762446880340576, "perplexity": 69.17295361153452}, "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/1585371609067.62/warc/CC-MAIN-20200405181743-20200405212243-00272.warc.gz"} |
https://learn.saylor.org/mod/page/view.php?id=37776 | ## Integrating with U-Substitution
Watch these videos on change of variable, also called substitution or U-substitution. | 2023-03-27 14:37:35 | {"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": 0.9110384583473206, "perplexity": 5347.185546511317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948632.20/warc/CC-MAIN-20230327123514-20230327153514-00616.warc.gz"} |
http://mathoverflow.net/questions/150416/lagrangian-complement-in-a-symplectic-vector-bundle | # Lagrangian complement in a symplectic vector bundle
A standard, folk result in symplectic geometry states that:
in a symplectic vector bundle $(E,\pi,B,\omega)$, any lagrangian subbundle $L$ admits a lagrangian complement $L'$.
Having to use this result, without giving its proof, I would like to cite a reference which not only gives the statement but also provides a self-contained proof of it.
Would you recommend some references presenting a complete, detailed proof of the quoted result?
I do not know such a reference, but that result is a one line corollary of the existence of a compatible complex structure (on fibers). If $J(b), b\in B$ be a section of the associated bundle of compatible almost complex structures on the initial symplectic bundle then $J(b)L(b)$ be a fiber of Lagrangian complement you are looking for. – Petya Nov 30 '13 at 18:48 | 2014-08-27 15:18:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8494241237640381, "perplexity": 262.11994894881116}, "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-20140820021349-00185-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://campus.datacamp.com/courses/defensive-r-programming/early-warning-systems?ex=6 | # Did you get the message?
Which of the following situations is an incorrect use of the message() function? | 2019-12-12 14:59:01 | {"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": 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": 0.7546830177307129, "perplexity": 1477.6362334984744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540543850.90/warc/CC-MAIN-20191212130009-20191212154009-00294.warc.gz"} |
https://www.jobilize.com/course/section/two-channel-biorthogonal-filter-banks-by-openstax?qcr=www.quizover.com | # 0.7 Generalizations of the basic multiresolution wavelet system (Page 9/28)
Page 9 / 28
## Two channel biorthogonal filter banks
In previous chapters for orthogonal wavelets, the analysis filters and synthesis filters are time reversal of each other; i.e., $\stackrel{˜}{h}\left(n\right)=h\left(-n\right)$ , $\stackrel{˜}{g}\left(n\right)=g\left(-n\right)$ . Here, for the biorthogonal case, we relax these restrictions. However, in order to perfectlyreconstruct the input, these four filters still have to satisfy a set of relations.
Let ${c}_{1}\left(n\right),n\in \mathbf{Z}$ be the input to the filter banks in [link] , then the outputs of the analysis filter banks are
${c}_{0}\left(k\right)=\sum _{n}\stackrel{˜}{h}\left(2k-n\right){c}_{1}\left(n\right),\phantom{\rule{2.em}{0ex}}{d}_{0}\left(k\right)=\sum _{n}\stackrel{˜}{g}\left(2k-n\right){c}_{1}\left(n\right).$
The output of the synthesis filter bank is
${\stackrel{˜}{c}}_{1}\left(m\right)=\sum _{k}\left[h,\left(2k-m\right),{c}_{0},\left(k\right),+,g,\left(2k-m\right),{d}_{0},\left(k\right)\right].$
${\stackrel{˜}{c}}_{1}\left(m\right)=\sum _{n}\sum _{k}\left[h,\left(2k-m\right),\stackrel{˜}{h},\left(2k-n\right),+,g,\left(2k-m\right),\stackrel{˜}{g},\left(2k-n\right)\right]{c}_{1}\left(n\right).$
For perfect reconstruction, i.e., ${\stackrel{˜}{c}}_{1}\left(m\right)={c}_{1}\left(m\right),\forall m\in \mathbf{Z}$ , we need
$\sum _{k}\left[h,\left(2k-m\right),\stackrel{˜}{h},\left(2k-n\right),+,g,\left(2k-m\right),\stackrel{˜}{g},\left(2k-n\right)\right]=\delta \left(m-n\right).$
Fortunately, this condition can be greatly simplified. In order for it to hold, the four filters have to be related as [link]
$\stackrel{˜}{g}\left(n\right)={\left(-1\right)}^{n}h\left(1-n\right),\phantom{\rule{2.em}{0ex}}g\left(n\right)={\left(-1\right)}^{n}\stackrel{˜}{h}\left(1-n\right),$
up to some constant factors. Thus they are cross-related by time reversal and flipping signs of every other element. Clearly, when $\stackrel{˜}{h}=h$ , we get the familiar relations between the scaling coefficients and the wavelet coefficients for orthogonal wavelets, $g\left(n\right)={\left(-1\right)}^{n}h\left(1-n\right)$ . Substituting [link] back to [link] , we get
$\sum _{n}\stackrel{˜}{h}\left(n\right)h\left(n+2k\right)=\delta \left(k\right).$
In the orthogonal case, we have ${\sum }_{n}h\left(n\right)h\left(n+2k\right)=\delta \left(k\right)$ ; i.e., $h\left(n\right)$ is orthogonal to even translations of itself. Here $\stackrel{˜}{h}$ is orthogonal to $h$ , thus the name biorthogonal .
Equation [link] is the key to the understanding of the biorthogonal filter banks. Let's assume $\stackrel{˜}{h}\left(n\right)$ is nonzero when ${\stackrel{˜}{N}}_{1}\le n\le {\stackrel{˜}{N}}_{2}$ , and $h\left(n\right)$ is nonzero when ${N}_{1}\le n\le {N}_{2}$ . Equation [link] implies that [link]
${N}_{2}-{\stackrel{˜}{N}}_{1}=2k+1,\phantom{\rule{1.em}{0ex}}{\stackrel{˜}{N}}_{2}-{N}_{1}=2\stackrel{˜}{k}+1,\phantom{\rule{2.em}{0ex}}k,\stackrel{˜}{k}\in \mathbf{Z}.$
In the orthogonal case, this reduces to the well-known fact that the length of $h$ has to be even. [link] also imply that the difference between the lengths of $\stackrel{˜}{h}$ and $h$ must be even. Thus their lengths must be both even or both odd.
## Biorthogonal wavelets
We now look at the scaling function and wavelet to see how removing orthogonality and introducing a dual basis changes their characteristics.We start again with the basic multiresolution definition of the scaling function and add to that a similar definition of a dual scaling function.
$\Phi \left(t\right)=\sum _{n}h\left(n\right)\sqrt{2}\Phi \left(2t-n\right),$
$\stackrel{˜}{\Phi }\left(t\right)=\sum _{n}\stackrel{˜}{h}\left(n\right)\sqrt{2}\stackrel{˜}{\Phi }\left(2t-n\right).$
From Theorem [link] in Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients , we know that for $\phi$ and $\stackrel{˜}{\phi }$ to exist,
$\sum _{n}h\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\sum _{n}\stackrel{˜}{h}\left(n\right)\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}\sqrt{2}.$
Continuing to parallel the construction of the orthogonal wavelets, we also define the wavelet and the dual wavelet as
$\psi \left(t\right)=\sum _{n}g\left(n\right)\sqrt{2}\Phi \left(2t-n\right)=\sum _{n}{\left(-1\right)}^{n}\stackrel{˜}{h}\left(1-n\right)\sqrt{2}\Phi \left(2t-n\right),$
$\stackrel{˜}{\psi }\left(t\right)=\sum _{n}\stackrel{˜}{g}\left(n\right)\sqrt{2}\stackrel{˜}{\Phi }\left(2t-n\right)=\sum _{n}{\left(-1\right)}^{n}h\left(1-n\right)\sqrt{2}\stackrel{˜}{\Phi }\left(2t-n\right).$
Now that we have the scaling and wavelet functions and their duals, the question becomes whether we can expand and reconstruct arbitrary functionsusing them. The following theorem [link] answers this important question.
Theorem 37 For $\stackrel{˜}{h}$ and $h$ satisfying [link] , suppose that for some C, $ϵ>0$ ,
$|\text{Φ}\left(\text{ω}\right)|\le C{\left(1+\text{ω}\right)}^{-1/2-ϵ},\phantom{\rule{2.em}{0ex}}|\stackrel{˜}{\text{Φ}}\left(\text{ω}\right)|\le C{\left(1+\text{ω}\right)}^{-1/2-ϵ}.$
If $\Phi$ and $\stackrel{˜}{\Phi }$ defined above have sufficient decay in the frequency domain, then ${\psi }_{j,k}\stackrel{\mathrm{def}}{=}{2}^{j/2}\psi \left({2}^{j}x-k\right),\phantom{\rule{0.277778em}{0ex}}j,k\in \mathbf{Z}$ constitute a frame in ${L}^{2}\left(\mathbf{R}\right)$ . Their dual frame is given by ${\stackrel{˜}{\psi }}_{j,k}\stackrel{\mathrm{def}}{=}{2}^{j/2}\stackrel{˜}{\psi }\left({2}^{j}x-k\right),\phantom{\rule{0.277778em}{0ex}}j,k\in \mathbf{Z}$ ; for any $f\in {L}^{2}\left(\mathbf{R}\right)$ ,
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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Got questions? Join the online conversation and get instant answers! | 2019-09-21 15:00:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 41, "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": 0.705071747303009, "perplexity": 1677.078346965307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574532.44/warc/CC-MAIN-20190921145904-20190921171904-00477.warc.gz"} |
http://mathhelpforum.com/geometry/43991-plz-help-print.html | # Plz Help!!!
• Jul 18th 2008, 10:28 AM
charl17728
Plz Help!!!
Assume the earth is a spherical planet with a diameter of 1600 km with a uniform density of 5200 kilograms per cubic metre. If the moon is in a geostationary orbit around it, hours, how far is that geostationary orbit above the surface of the earth after 24 hours???????
• Aug 6th 2008, 09:08 PM
TwistedOne151
First, use that the volume of a sphere or radius R is $V=\frac{4}{3}\pi{R^3}$, and multiply by the density to get the mass M of your planet.
Then, for a moon of mass m, the force balance for a circular orbit of radius r is $\frac{GMm}{r^2}=\frac{mv^2}{r}$, where v is the velocity of the orbiting moon, and G is the universal gravitational constant [=6.673x10^(-11) m^3 kg^(-1) s^(-2)]. The distance traveled in one orbit is $2\pi{r}$, so the period T (time to make one orbit) is $T=\frac{2\pi{r}}{v}$, so $v=\frac{2\pi{r}}{T}$, and substituting that into the force balance equation,
$\frac{GMm}{r^2}=\frac{4\pi^2mr}{T^2}$
We can cancel m from both sides:
$\frac{GM}{r^2}=\frac{4\pi^2r}{T^2}$
so the mass of the moon does not matter (this is a form of Kepler's Third Law as applied to a circular orbit).
Bringing the r terms to one side,
$GM=\frac{4\pi^2r^3}{T^2}$
$\frac{GMT^2}{4\pi^2}=r^3$
$r=\sqrt[3]{\frac{GMT^2}{4\pi^2}}$
Now, the definition of geostationary orbit is that the orbiting body orbits at the same speed at the body it's orbiting rotates (so it remains over the same point on the "planet"). Thus the period T must be one day=24 hours=86400 seconds. Thus, we have G, M, and T, and so you can plug in those values to get r, which is the distance from the center of the planet to the moon. r-R is thus the distance from the planet's surface to the moon, the answer you want.
--Kevin C. | 2016-10-24 01:57:01 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "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": 0.9290319085121155, "perplexity": 342.2475263865079}, "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/1476988719463.40/warc/CC-MAIN-20161020183839-00441-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/345749/on-the-relationship-between-martins-axiom-the-countable-chain-condition-and-th | # On the relationship between Martin's Axiom, the countable chain condition and the Knaster property
This is a repost of a question that went unanswered on MSE
We say that a poset $$P$$ has the Knaster property (or is Knaster) if every uncountable subset of $$P$$ contains an uncountable subset of pairwise compatible conditions.
Let $$K$$ denote the statement "every c.c.c. poset is Knaster" and let $$P$$ denote the statement "the product of $$2$$ c.c.c. posets is c.c.c.". Then we have $$\mathsf{MA}_{\aleph_1}\implies K \implies P$$.
From the comments on the MSE question I learnt that in the paper "Forcing with a coherent souslin-tree" by Todorčević it is stated that whether the first implication is reversible is an open problem. Has there been any recent progress in the time since that paper was written or is it still open? What is known about whether the second implication is an iff instead?
• Both implications are open so far, if you are interested in this subject you might consult T. Yoriokas's papers. In particular "A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees", and its correction. Nov 13, 2019 at 8:31
• @Rahman.M thanks for the reference, that's a very interesting paper I wasn't aware of! You could post that as an answer Nov 13, 2019 at 10:54
• It's fine, maybe someone really answers it in future!😉 Nov 13, 2019 at 14:23 | 2022-08-07 18:28:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7796988487243652, "perplexity": 475.17038481031136}, "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/1659882570692.22/warc/CC-MAIN-20220807181008-20220807211008-00521.warc.gz"} |
https://ratio.huji.ac.il/publications?page=5&field_publication_date_value%5Bmin%5D=&field_publication_date_value%5Bmax%5D=&field_authors_value=&field_abstract_value=&field_published_in_value=&field_number_value=&title=&order=title&sort=desc | Publications
2013
Optimal Lobbying is the problem a lobbyist or a campaign manager faces in a full-information voting scenario of a multi-issue referendum when trying to influence the result. The Lobby is faced with a profile that specifies for each voter and each issue whether the voter approves or rejects the issue, and seeks to find the smallest set of voters it must influence to change their vote, for a desired outcome to be obtained. This computational problem also describes problems arising in other scenarios of aggregating complex opinions, such as principal-agents incentives scheme in a complex combinatorial problem, and bribery and manipulation in Truth-Functional Judgement Aggregation. We study the computational complexity of Optimal Lobbying when the issues are aggregated using an anonymous monotone function and the family of desired outcomes is an upward-closed family. We analyze this problem with regard to two parameters: the minimal number of supporters needed to pass an issue, and the size of the maximal minterm of the desired set. We show that for the extreme values of the parameters, the problem is tractable, and provide algorithms. On the other hand, we prove intractability of the problem for the non-extremal values, which are common values for the parameters.
We add two results to the theory of consistent voting. Let M be the set of all survivors of some feasible elimination procedure. We prove that i) M can be computed in polynomial time for each profile of preferences and ii) M is characterized by anonymity, non- imposition, Maskin monotonicity, and additive blocking.
For a random permutation on 1,2, ¦,n for fixed n, and for MŠ†1,2, ¦,n, we analyse the distribution of the combined length L=L(,M) of all cycles of that contain at least one element of M. We give a simple, explicit formula for the probability of every possible value for L (backed by three proofs of distinct flavours), as well as closed-form formulae for its expectation and variance, showing that less than 1/(|M|+1) of the elements 1, ¦,n are expected to be contained in cycles of that are disjoint from M, with low probability for a large deviation from this fraction. We furthermore give a simple explicit formula for all rising-factorial moments of L. These results are applicable to the study of manipulation in matching markets.
We consider a fast evolutionary dynamic process on finite stopping games, where each player at each node has at most one move to continue the game. A state is evolutionarily stable if its long-run relative frequency of occurrence is bounded away from zero as the mutation rate decreases to zero. The fast dynamic process allows each individual in each population to change its strategy at every stage. We define a robustness index of backward induction and show examples where the backward induction equilibrium component is not evolutionarily stable for large populations. We show some sufficient conditions for evolutionary stability, which are different from the ones for the conventional evolutionary model. Even for this fast dynamic process, the transition between any two Nash equilibrium components may take very long time.
We consider a basic dynamic evolutionary model with rare mutation and a best-reply (or better-reply) selection mechanism. A state is evolutionarily stable if its long-term relative frequency of occurrence is bounded away from zero as the mutation rate decreases to zero. We prove that, for all finite extensive-form games of perfect information, only Nash equilibria are evolutionarily stable. We show that, in games where a player may play at more than one node along some path, even when the populations increase to infinity, there may be some evolutionarily stable states which are not part of the backward induction equilibrium component. We give a sufficient condition for evolutionary stability and show how much extra value is needed in the terminal payoffs to make an equilibrium evolutionarily stable.
The usual purpose of negotiations is to explore options and reach an agreement, if possible. We investigated a notable exception to this generalization, where a party negotiates without any intention of reaching an agreement. False negotiation occurs when a party gains more by stalling the negotiations until an external change takes place that improves its position considerably. While false negotiators aim to avoid agreement within the current frame of the negotiations, they also aim to keep the negotiation process alive, since walking away from the negotiation table could endanger their position. We report the results of a study that compared the actions of false and sincere negotiators. The false negotiators used competitive tactics that encumbered the negotiations, yet they concealed their intentions by maintaining a fa\Sade of cooperation. Our theoretical discussion is focused on the balancing act involved in false negotiations and the challenges it poses for actors in social, managerial, and political settings. We conclude our analysis with an example from the realm of international negotiations.
Consider a setting in which agents can each take one of two ordered actions and in which the incentive of any given agent to take the high action is positively reinforced by the number of other agents that take it. Furthermore, assume that we don't know any other details about the game being played. What can we say about the details of the structure of the interaction between actions and incentives when we observe a set or a subset of all possible equilibria? In this paper we study 3 nested classes of games: (a) binary games of strategic complements; (b) games in (a) that admit a network representation: and (c) games in (b) in which the network is complete. Our main results are the following: It has long been established in the literature that the set of pure strategy Nash equilibria of any binary game of strategic complements among a set N of agents can be seen as a lattice on the set of all subsets of N under the partial order defined by the set inclusion relation. If the game happens to be strict in the sense that agents are never indifferent among outcomes (games in (a)), then the resulting lattice of equilibria satisfies a straightforward sparseness condition. (1) We show that, in fact, the games in (a) express all such lattices. (2) We characterize the collection of subsets of N that can be weakly expressed as the set of equilibria of some game of thresholds (games in (b)). (3) We characterize the collection of subsets of N that can be weakly expressed as the set of equilibria of some game of thresholds on the complete graph (games in (c)).
Yoram Moses Armando Casta$\pm$eda, Yannai A. Gonczarowski. 2013. “Good, Better, Best! Unbeatable Protocols for Consensus and Set Consensus”. Publisher's Version Abstract
While the very first consensus protocols for the synchronous model were designed to match the worst-case lower bound, deciding in exactly t+1 rounds in all runs, it was soon realized that they could be strictly improved upon by early stopping protocols. These dominate the first ones, by always deciding in at most t+1 rounds, but often much faster. A protocol is unbeatable if it can't be strictly dominated. Namely, if no protocol Q can decide strictly earlier than P against at least one adversary strategy, while deciding at least as fast as P in all cases. Unbeatability is often a much more suitable notion of optimality for distributed protocols than worst-case performance. Halpern, Moses and Waarts (2001), who introduced this notion, presented a general logic-based transformation of any consensus protocol to an unbeatable protocol that dominates it, and suggested a particular unbeatable consensus protocol. Their analysis is based on a notion of continual common knowledge, which is not easy to work with in practice. Using a more direct knowledge-based analysis, this paper studies unbeatability for both consensus and k-set consensus. We present unbeatable solutions to non-uniform consensus and k-set consensus, and uniform consensus in synchronous message-passing contexts with crash failures. Our consensus protocol strictly dominates the one suggested by Halpern, Moses and Waarts, showing that their protocol is in fact beatable.The k-set consensus problem is much more technically challenging than consensus, and its analysis has triggered the development of the topological approach to distributed computing. Worst-case lower bounds for this problem have required either techniques based on algebraic topology (Guerraoui et al., 2009), or reduction-based proofs (Alistarh et al., 2012; Gafni et al., 2011). Our proof of unbeatability is purely combinatorial, and is a direct, albeit nontrivial, generalization of the one for consensus. We also present an alternative topological unbeatability proof that allows to understand the connection between the connectivity of protocol complexes and the decision time of processes. All of our protocols make use of a notion of a hidden path of nodes relative to a process i at time m, in which a value unknown to i at m may be seen by others. This is a structure that can implicitly be found in lower bound proofs for consensus going back to the '80s (Dolev and Strong, 1982). Its use in our protocols sheds light on the mathematical structure underlying the consensus problem and its variants.For the synchronous model, only solutions to the uniform variant of k-set consensus have been offered. Based on our unbeatable protocols for uniform consensus and for non-uniform k-set consensus, we present a uniform k-set consensus protocol that strictly dominates all known solutions to this problem in the synchronous model.
This paper continues the work initiated in [19]. We adopt the same model as in [19]. We show that the non-backward-induction equilibrium component may be evolutionarily stable for any population size in a finite stopping game where the two equilibrium components are terminated by different players. A surprising result is that the backward induction equilibrium component may not be evolutionarily stable for large populations. Finally, we study the evolutionary stability result in a different limiting process where the expected number of mutations per generation is bounded away from both zero and infinity.
Robert J. Aumann Itai Arieli. 2013. “Logic of Backward Induction, The”. Publisher's Version Abstract
The logic of backward induction (BI) in perfect information (PI) games has been intensely scrutinized for the past quarter century. A major development came in 2002, when P. Battigalli and M. Sinischalchi (BS) showed that an outcome of a PI game is consistent with common strong belief of utility maximization if and only if it is the BI outcome. Both BS's formulation, and their proof, are complex and deep. We show that the result continues to hold when utility maximization is replaced by a rationality condition that is even more compelling; more important, the formulation and proof become far more transparent, accessible, and self-contained.
Gale and Sotomayor (1985) have shown that in the Gale-Shapley matching algorithm (1962), the proposed-to side W (referred to as women there) can strategically force the W-optimal stable matching as the M-optimal one by truncating their preference lists, each woman possibly blacklisting all but one man. As Gusfield and Irving have already noted in 1989, no results are known regarding achieving this feat by means other than such preference-list truncation, i.e. by also permuting preference lists.We answer Gusfield and Irving's open question by providing tight upper bounds on the amount of blacklists and their combined size, that are required by the women to force a given matching as the M-optimal stable matching, or, more generally, as the unique stable matching. Our results show that the coalition of all women can strategically force any matching as the unique stable matching, using preference lists in which at most half of the women have nonempty blacklists, and in which the average blacklist size is less than 1. This allows the women to manipulate the market in a manner that is far more inconspicuous, in a sense, than previously realized. When there are less women than men, we show that in the absence of blacklists for men, the women can force any matching as the unique stable matching without blacklisting anyone, while when there are more women than men, each to-be-unmatched woman may have to blacklist as many as all men. Together, these results shed light on the question of how much, if at all, do given preferences for one side a priori impose limitations on the set of stable matchings under various conditions. All of the results in this paper are constructive, providing efficient algorithms for calculating the desired strategies.
Andreu Mas-Colell Sergiu Hart. 2013. “Markets, Correlation, and Regret-Matching”. Publisher's Version Abstract
Inspired by the existing work on correlated equilibria and regret-based dynamics in games, we carry out a first exploration of the links between the leading equilibrium concept for (exchange) economies, Walrasian equilibrium, and the dynamics, specifically regret-matching dynamics, of trading games that fit the economic structure and have the property that their pure Nash equilibria implement the Walrasian outcomes. Interestingly, in the case of quasilinear utilities (or "transferable utility"), all the concepts essentially coincide, and we get simple deterministic dynamics converging to Walrasian outcomes. Connections to sunspot equilibria are also studied.
We consider the menu size of auctions as a measure of auction complexity and study how it affects revenue. Our setting has a single revenue-maximizing seller selling two or more heterogenous items to a single buyer whose private values for the items are drawn from a (possibly correlated) known distribution, and whose valuation is additive over the items. We show that the revenue may increase arbitrarily with menu size and that a bounded menu size can not ensure any positive fraction of the optimal revenue. The menu size turns out to "nail down" the revenue properties of deterministic auctions: their menu size may be at most exponential in the number of items and indeed their revenue may be larger than that achievable by the simplest types of auctions by a factor that is exponential in the number of items but no larger. Our model is related to a previously studied "unit-demand" model and our results also answer an open problem in that model.
' An agent needs to decide which of two available actions, A or B, to take. The agent's payoffs are such that A dominates B, i.e., taking A yields a better payoff than taking B, in every contingency. On the other hand, the agent's expected payoffs, given the action taken, are in the reverse order, i.e., E(payoff | B) > E(payoff | A) , which can happen if the probabilities of the various contingencies are not independent of the action being taken. What should the agent do? This dilemma has come to be known as Newcomb's Paradox (Nozick, 1969). The present essay shows that the rule "keep away, as much as possible, from any dominated action" is perfectly consistent with actually taking the dominated action, when appropriate. No paradox.''
Justin Valasek Rune Midjord, Tom s Rodr �guez-Barraquer. 2013. “Over-Caution of Large Committees of Experts”. Publisher's Version Abstract
In this paper, we demonstrate that payoffs linked to a committee member's individual vote may explain over-cautious behavior in committees. A committee of experts must decide whether to approve or reject a proposed innovation on behalf of society. In addition to a payoff linked to the adequateness of the committee's decision, each expert receives a disesteem payoff if he/she voted in favor of an ill-fated innovation. An example is FDA committees, where committee members can be exposed to a disesteem (negative) payoff if they vote to pass a drug that proves to be fatal for some users. We show that no matter how small the disesteem payoffs are, information aggregation fails in large committees: under any majority rule, the committee rejects the innovation almost surely. We then show that this inefficiency can be mitigated by pre-vote information pooling, but only if the decision is take under unanimity: in the presence of disesteem payoffs, committee members will only vote efficiently if they are all responsible for the final decision.
Avi Shmida Yoram Gerchman Nicka Chinkov Avi Koplovich Gadi Katzir Tamar Keasar, Miriam Kishinevsky. 2013. “Plant-Derived Visual Signals May Protect Beetle Herbivores from Bird Predators”. Publisher's Version Abstract
Insect herbivores often use chemical signals obtained from their food plants to deter enemies and/or attract sexual partners. Do plant-based visual signals act similarly, i.e., repel consumers' enemies and appeal to potential mates? We explored this question using the pollen-feeding beetle Pygopleurus israelitus (Glaphyridae), a specialized pollinator of Anemone coronaria's chemically defended red-morph flowers. We presented dead beetles, which had fed either on anemones or on cat-food, to young domestic chicks on a red (anemone-like) or a green (leaf-like) background. We determined whether the beetles' background color and diet affected the chicks' feeding. Cuticle surface extracts from anemone-fed beetles, but not from cat-food-fed beetles, contained a secondary metabolite characteristic of anemones. Latencies to the first picking-up and consuming of beetles from green backgrounds were shorter than of beetles from red backgrounds. The picking-up order of beetles also indicated that prey from the green background was preferred. The chicks retained this preference when re-tested, three days later. Handling times of anemone-fed beetles were longer than of cat-food-fed beetles. A previous study showed that glaphyrids improve their mate-finding prospects by orienting to large red anemone flowers. Here, female beetles preferred cat-food-fed to anemone-fed males in mate-choice assays, thus anemone-derived chemicals did not increase mating success. Instead, the combined results indicate that A. coronaria's red flowers provide a visual signal that may both deter its herbivore's predators and attract its mates. To our knowledge, this is the first experimental evidence for a potential protective role of plant-derived visual signals for insect herbivores/pollinators. Keywords: Predation; secondary metabolite; tritrophic interactions; warning coloration; domestic chick; Glaphyridae; pollination.
We consider the complexity of finding a Correlated Equilibrium in an n-player game in a model that allows the algorithm to make queries for players' utilities at pure strategy profiles. Many randomized regret-matching dynamics are known to yield an approximate correlated equilibrium quickly: in time that is polynomial in the number of players, n, the number of strategies of each player, m, and the approximation error, 1/?. Here we show that both randomization and approximation are necessary: no efficient deterministic algorithm can reach even an approximate equilibrium and no efficient randomized algorithm can reach an exact equilibrium.
We model constitutions by effectivity functions. We assume that the constitution is common knowledge among the members of the society. However, the preferences of the citizen are private information. We investigate whether there exist decision schemes (i. e., functions that map profiles of (dichotomous) preferences on the set of outcomes to lotteries on the set of social states), with the following properties: i) The distribution of power induced by the decision scheme is identical to the effectivity function under consideration; and ii) the (incomplete information) game associated with the decision scheme has a Bayesian Nash equilibrium in pure strategies. If the effectivity function is monotonic and superadditive, then we find a class of decision schemes with the foregoing properties. When applied to n-person games in strategic form, a decision scheme d is a mapping from profiles of (dichotomous) preferences on the set of pure strategy vectors to probability distributions over outcomes (or equivalently, over pure strategy vectors). We prove that for any feasible and individually rational payoff vector of a strategic game, there exists a decision scheme that yields that payoff vector as a (pure) Nash equilibrium payoff in the game induced by the strategic game and the decision scheme. This can be viewed as a kind of purification result.
We introduce asymptotic analysis of stochastic games with short-stage duration. The play of stage $k$, $k'geq 0$, of a stochastic game $'Gamma_'delta$ with stage duration $'delta$ is interpreted as the play in time $k'delta'leq t0$ as the stage duration $'delta$ goes to $0$, and study the asymptotic behavior of the value, optimal strategies, and equilibrium. The asymptotic analogs of the discounted, limiting-average, and uniform equilibrium payoffs are defined. Convergence implies the existence of an asymptotic discounted equilibrium payoff, strong convergence implies the existence of an asymptotic limiting-average equilibrium payoff, and exact convergence implies the existence of an asymptotic uniform equilibrium payoff.
Are good people motivated to behave in accordance the moral truth whatever it is? Michael Smith, who has named this motivation the de-dicto moral motivation, famously criticized it. According to Smith, good people are instead motivated directly by more concrete moral concerns, such as 'the well-being of their fellows, people getting what they deserve, justice, equality, and the like . Here I argue for the non-Smithian view that good people have (also) a de-dicto moral motivation. The argument runs roughly as follows: given that good people tend to behave appropriately, and that in some situations it is appropriate to reevaluate one s underived moral beliefs, good people tend to seriously reevaluate underived moral beliefs sometimes. Theories of motivation have to account for this fact (a point overlooked by Smith and his respondents). What motivates a good person to pay attention to evidence that is contrary to her underived moral beliefs? What does she aim for in reevaluating those beliefs? I argue that the view that good people are motivated to act morally de-dicto is in a better position to explain the relevant facts about good people s reevaluation of underived moral beliefs. | 2022-09-27 07:53:46 | {"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": 0.6524748206138611, "perplexity": 1296.5125927772326}, "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/1664030334992.20/warc/CC-MAIN-20220927064738-20220927094738-00557.warc.gz"} |
https://stats.stackexchange.com/questions/479263/if-teachers-account-for-30-of-variance-of-student-achievement-can-a-teacher-ha | # If teachers account for 30% of variance of student achievement, can a teacher have 30% increase in achievement by teaching better?
My professor wrote:
Research using sophisticated statistical techniques indicates that teaching expertise accounts for about 30 percent of the variance in student achievement (Hattie, 2003). Think about what your student test scores would look like if you could achieve this 30 percent increase with each group of students that passed through your classes each year.
I believe the professor misunderstands variance accounted for. "Teaching expertise accounts for about 30 percent of the variance in student achievement" means that a teacher is responsible for 30% of what a student achieves. You can't infer from this that a great teacher will achieve a 30% increase in student achievement relative to... a teacher who didn't teach at all? If we had 2 teachers, one at the bottom of teaching expertise and the other at the top, teach classes which had all other variables constant (socioeconomics, motivation, etc.), would the better teacher's class achieve a 30% increase relative to the worse teacher?
• The current answers are great in giving insight why the %variance explained does not coincide with %increase achievement (it depends on the amount of difference between the teachers, and as Stephan explains the %increase is also ambiguous/relative due to the intercept). I want to add that we must also think about the issue causality vs correlation. The quote is not so explicit about it, but we should not interpret the 30% like the teaching expertise is causing that part of the student's achievements. It is not like a change to more expertise will change a student's achievement by 30%. – Sextus Empiricus Jul 28 at 20:51
• "if you could achieve this 30 percent increase" This is an ambiguous statement. 30 percent, relative to what? Currently you seem to interpret it as 30% difference between students from teacher 1 versus teachers from teacher 2 (like the one group of students will do 30% better relative to the other group). But that is not the necessary interpretation of 30 percent increase. In the case of the correlation here the 30 percent increase means that for each unit of teacher expertise the students have 0.3 units increase of achievement. That is also a way to see 30% increase. – Sextus Empiricus Jul 28 at 20:56
• Do you have the Hattie, 2003 reference with details like title? Their definition of "expertise" may mean "tenure" or the likes, because expertise does not seem to be quantifiable. – user3819867 Jul 29 at 10:00
• I note that the quoted portion does not claim that better teachers produce better student outcomes. Better teachers reducing student outcome is compatible with the quote. That is, "accounts for about 30 percent of the variance" doesn't tell you the signs of anything. – Eric Towers Jul 30 at 15:34
• I'm always cynical about this type of statements. It's like saying one additional hour of sleep adds 5 years of life. The problem is that there's a ton of these "X amount of Z adds Y number of years to life," so if I do all of these Zs, will I live 500 years? Obviously, not happening. – Aksakal Jul 30 at 19:01
You are right in suspecting that your professor misunderstood.
The correct answer is that we cannot say anything whatsoever about the percentage improvement in student achievement driven by teacher expertise. Nothing at all.
Why is this so? The quote is in terms of variance explained. Variance explained has nothing to do with the actual values on which the scales are measured - which any percentage improvement in student achievement would be accounted in. The two are completely separate.
Let's look at an example. Here is some simulated data:
R code:
nn <- 1e2
set.seed(1) # for reproducibility
teaching_expertise <- runif(nn)
student_achievement <- 5+0.1*teaching_expertise+rnorm(nn,0,0.05)
model <- lm(student_achievement~teaching_expertise)
plot(teaching_expertise,student_achievement,pch=19,las=1,
xlab="Teaching Expertise",ylab="Student Achievement")
abline(model,col="red")
Note that the model is correctly specified: student achievement depends linearly on teaching expertise, and that is what I am modeling. No cheap tricks here.
We have $$R^2=0.30$$, so teaching expertise indeed accounts for 30% of student achievement (see here):
> summary(model)
Call:
lm(formula = student_achievement ~ teaching_expertise)
... snip ...
Multiple R-squared: 0.304, Adjusted R-squared: 0.2969
However, here is the student achievement we would predict for teachers at the very bottom (teaching expertise of 0) vs. at the very top of the range (1):
> (foo <- predict(model,newdata=data.frame(teaching_expertise=c(0,1))))
1 2
4.991034 5.106651
The improvement is on the order of $$\frac{5.11-4.99}{4.99}\approx 2.4\%$$.
> diff(foo)/foo[1]
2
0.02316497
(Plus, this is expected achievement. Actual achievement will be different. With regression to the mean typically being stronger at the extremes, the actual difference will be even smaller.)
And you know what? We could change this percentage change to pretty much any number we want. Even a negative percentage improvement! How? Simply by changing that single innocuous number 5 in the data simulation above, i.e., the intercept.
What's going on? Variance explained measures the amount by which the (sum of squared) residuals are reduced by a model, i.e., the difference between the residuals to the regression line and the residuals to the overall average. By changing the intercept (the 5), we can shift everything up and down. Including the overall average. So changing the intercept will leave variance explained completely unchanged. (If you have R, try this. We'll wait.)
However, shifting everything up and down will change the concrete scores. In particular the percentage improvement of a "good" vs. a "bad" teacher. If we shift everything down far enough, we get a negative student achievement for the "bad" teacher. And a positive change for the "good" teacher against a negative baseline will give you a negative percentage improvement. (Again, try this. An intercept of -1 works.)
Yes, of course such negative percentage improvements make no sense here. This is just an illustration of the fact that there is zero relationship between variance explained and percentage improvement in measurements.
• I feel like both answers so far kinda miss the mark. You're basically saying "well if everyone scored above 90% on the test then 30% of variance only accounts for a few percent so the professor is wrong". But that's pretty fallacious. Of course the absolute improvement in performance depends on the assessment metric, and I really doubt that their professor thinks the best teachers could have their students get 120% on that exam. If you look in terms of percentiles then 30% definitely has a valid ststistical meaning. – Joseph Ireland Jul 28 at 10:23
• @JosephIreland: I'm afraid I don't follow your argument. I don't say anything about everyone scoring above 90%. My toy example does not have a ceiling. Of course, one could use achievements that are between 0% and 100%, but that would just make everything more complicated - without changing the underlying logic. – Stephan Kolassa Jul 28 at 10:28
• There's also the fact that 30% of variance explained doesn't tell you anything about the gradient and that there could even be a NEGATIVE association for all the '30% of variance explained' statistic tells you by itself. – H. Green Jul 28 at 10:38
• @JosephIreland: "academic achievement is judged relative to your peers" - not everywhere. If achievement is measured by percentiles, then there are still transformations to give you any result you like. It will just not be as simple as just tweaking the intercept. You will need to play around a bit to get something that gives you the same $R^2$. Which makes the exercise more complicated, to no discernible profit. The fundamental point still is that there is no relationship between variance explained and percentage improvement. – Stephan Kolassa Jul 28 at 10:58
• @JosephIreland: grading on a curve is not the only possibility. And indeed, if the curve is calculated within a class (so, with a single teacher), then assuming better teaching expertise improves everyone's performance equally, then (a) improving expertise will do nothing to relative standing, and (b) the entire concept of variance explained in students' achievement by a single teacher's expertise is completely meaningless in the first place. Again: I made some assumptions to illustrate the point. You may prefer others. I maintain that the main point still holds. – Stephan Kolassa Jul 28 at 11:53
The Hattie 2003 paper mentions a simple form of hierarchical linear modelling ignoring interactions. The paper’s description of the 30% isn’t particularly thorough, with broken links in the references making it difficult to see where the number even came from. I assume his approach relied on partial R-squared.
The answer is no, going from a bad teacher to a good teacher can’t be expected to increase performance by 30%. The two 30%'s are measured completely differently.
For example, suppose performance followed this equation:$$\text{performance} = \beta_0 + \beta_1 ~\text{studentEffort} + \beta_2 ~\text{teacherEffort} + \text{noise}$$If the $$\beta_2$$ is small, the performance graph would be nearly flat as teacherEffort changed. This can happen no matter what the $$R^2$$ is or how it might divide up into partial $$R^2$$'s.
In other words, saying that teachingEffort accounts for 30% of a variation doesn't tell you what that variation is over the dataset, i.e. how much performance changes.
You write '"Teaching expertise accounts for about 30 percent of the variance in student achievement" means that a teacher is responsible for 30% of what a student achieves.'
A better formulation would be "A teacher is responsible for 30% of the difference in performance between students".
In other words, if the average performance of some group of students with teacher A is 80 points and the average performance of another group with teacher B is 70 points, the performance of the teachers A and B can account for around 3 points (30% of the 10 point variance in performance).
• Still wrong, your example has confused "difference in performance between students" with "difference in mean performance between groups of students" – Ben Voigt Jul 29 at 18:08
• How do you think they measured this effect? What experimental design do you propose to investigate what you've just claimed without using groupings of students? As you can see from pages 10-15 of the paper this statistic comes from ( research.acer.edu.au/cgi/… ), they did indeed use four separate groups of students in their study. – sintax Jul 30 at 19:12 | 2020-09-21 12:52:37 | {"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": 6, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6843835711479187, "perplexity": 1126.9649272386698}, "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/1600400201699.38/warc/CC-MAIN-20200921112601-20200921142601-00113.warc.gz"} |
http://mymathforum.com/algebra/346899-what-s-greater-than-1-a-2.html | My Math Forum What's greater than -1
Algebra Pre-Algebra and Basic Algebra Math Forum
August 19th, 2019, 05:19 PM #11
Senior Member
Joined: Oct 2013
From: New York, USA
Posts: 664
Thanks: 87
Quote:
Originally Posted by weirddave Or: 1 - 0.9recurring
That's a better way of expressing it than 1/infinity.
August 19th, 2019, 06:21 PM #12
Senior Member
Joined: Jun 2019
From: USA
Posts: 213
Thanks: 90
Quote:
Originally Posted by weirddave Or: 1 - 0.9recurring
Except this is 1 - 9/9 = 1 - 1 = 0 (exactly). It's the same as $\displaystyle \lim_{n\to\infty} 10^{-n}$, which is the same as $\displaystyle \lim_{n\to\infty} \frac{1}{n}$. They're all zero.
I'd like to throw $\displaystyle \varepsilon$ and $\displaystyle \hbar$ into the ring.
August 19th, 2019, 08:26 PM #13
Senior Member
Joined: Aug 2012
Posts: 2,393
Thanks: 749
Quote:
Originally Posted by DarnItJimImAnEngineer I'd like to throw $\displaystyle \varepsilon$ and $\displaystyle \hbar$ into the ring.
There is no smallest positive real number. This fact remains true if you extend the reals to the hyperreals of nonstandard analysis. It's not commonly understood that even in the hyperreals, there is no smallest positive real and no smallest positive infinitesimal. This is easily proven by noting that if you claim $\varepsilon > 0$ and $\forall x \in \mathbb R$, we have $0 < \varepsilon < x$; then $\frac{\varepsilon}{2}$ is also positive and strictly smaller than $\varepsilon$. So $\varepsilon$ wasn't the smallest positive real after all; and since $\varepsilon$ was an arbitrary positive infinitesimal, there is no smallest positive infinitesimal in the hyperreals. I only mention this because I've seen the contrary asserted on too many message boards.
ps -- How do we know that $\frac{\varepsilon}{2}$ exists and is a positive hyperreal number? Because the reals, and the hyperreals, are a field. That's a number system in which you can always divide as long as the divisor isn't zero.
So "the smallest positive real" is simply not well defined, and you can't assign it a variable in a meaningful way. Any such assignment is meaningless since it does not refer to anything even in the abstract mathematical world. There's no mathematical context where the idea makes sense.
The symbol $\varepsilon$ is universally understood in math to be an arbitrary small positive real number; and not a specific one. Your proposed overloading would be incorrect in the context of established math.
As for $\hbar$, that is a well-known physical constant whose meaning is universally understood as such by pretty much everyone, mathematicians and physicists alike.
Moreover it is strictly larger than zero, and moreover there is no claim by physicists that it's the smallest unit of measure in the world; only that it's the scale below which our equations don't work. The Planck scales (time and space) are statements about the limitations of our theories; and not statements about the world. The Planck limits are epistemological and not ontological.
So it's bad notation to propose overloading a symbol that the physicists own by universal agreement and that doesn't mean what you'd like it to mean. Let alone that it wouldn't refer to anything.
Well I hope you don't mind my opinionated opinion but your post inspired me to let it all hang out. I noted your smiley so this is offered in a similar lighthearted vein.
Last edited by Maschke; August 19th, 2019 at 08:37 PM.
August 19th, 2019, 09:28 PM #14 Senior Member Joined: Dec 2015 From: somewhere Posts: 642 Thanks: 91 $\displaystyle \epsilon \rightarrow 0$ is an infinitesimal variable with properties of number 0 against a different variable and has properties of a number with itself. Imagine the derivative of a function involving time which states dx decreasing over time and tending to 0 as time moves .In other words , dx is changing. So it is not a mistake to say $\displaystyle \frac{change}{change} =1.$ A change that has properties of an elimination.(x•0=0) A small change to any varying quantity. Last edited by idontknow; August 19th, 2019 at 09:32 PM.
August 20th, 2019, 09:00 AM #15 Senior Member Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 Actually, I was thinking of computer programming. Some languages have an eps function or similar that returns the value of the LSB of a given floating point number. And I am by no means an expert in quantum physics. I just know in thermodynamics we use Planck's constant to quantise the number of possible distinct microstates of a system -- which is how we assign a physical value to entropy. My point was in the world of mathematics there is no smallest positive number, but in worlds ruled by the laws of computers or quantum physics the concept of smallest meaningful values does come into play.
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Contact - Home - Forums - Cryptocurrency Forum - Top | 2019-09-21 10:50:54 | {"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": 0.6400105953216553, "perplexity": 859.1733727157077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574409.16/warc/CC-MAIN-20190921104758-20190921130758-00536.warc.gz"} |
http://eutypon.gr/e-blog/?m=201107 | # Digital Typography News
A blog exclusively devoted to digital typography
## MathJax
MathJax is a relatively new web technology that allows people to easily add mathematical content into their web documents. For example, the following display
\begin{align}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x – y – xz \\
\dot{z} & = -\beta z + xy
\end{align}
was generated by adding the following code into the web content
\begin{align}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{align}
That’s simply great! People willing to add mathematical content into their wordpress blogs, should check out the MathJax-LaTeX page. | 2017-06-25 03:26:07 | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1977027803659439, "perplexity": 3576.2664675662218}, "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/1498128320395.62/warc/CC-MAIN-20170625032210-20170625052210-00462.warc.gz"} |
http://math.stackexchange.com/questions/179284/basics-of-probability-independent-events | # Basics of probability - Independent Events.
I read that the probability of two events provided they are independent is obtained by the following formula:
$Probability _ {Independent~ Events} = Probability_{1st Event} \times Probability_{2nd Event} \times ...$
Now it states that the probability of getting a Tails and Head (T-H) or getting a Heads and Tails (H-T) when a coin is flipped consecutively is given by:
$P(H-T) + P(T-H) = (\frac{1}{2}\times \frac{1}{2}) + (\frac{1}{2}\times \frac{1}{2}) = \frac{1}{2}$
Now my question is why isn't it
$P(H-T) \times P(T-H)$ instead of $P(H-T) + P(T-H)$ since after all the two events are independent of each other. I would appreciate it if someone could clear this up.
-
These two events actually highly dependent. Do you see why? – Artem Aug 5 '12 at 23:14
Are you talking about P(H-T) and P(T-H) ? – MistyD Aug 5 '12 at 23:15
I am talking about events H-T and T-H. – Artem Aug 5 '12 at 23:16
How are are they dependent ? – MistyD Aug 5 '12 at 23:17
Events are only dependent if an event that occurs has a probability that is different solely based on a preceding event. For example, if you pick a card out of a deck, and then a second card out of the same deck without putting the first card back, that would be a dependent event. – mathguy Aug 5 '12 at 23:37
show 1 more comment
Well..Independence and dependence has nothing to do with the fact that you are adding up the probabilities. You are asking what the probability that either THIS happens OR THAT happens. When you see the word "or" that means you are supposed to add up the probabilities (given that they do not overlap).
Now the statement you made is true, and it is partially applying to your question (particularly to each individual event). Your statement applies to situations like "what is the probability that THAT will happen AND THIS will happen. Let's take the first set of events: H-T. Whatever side you get on the second flip is completely independent of the first flip. The chance of getting heads on the first flip is 1/2, and the change on getting tails on the second flip is 1/2, so you multiply them together to get 1/4.
-
One way to help visualize this problem might be to list out all of the possible outcomes of flipping a coin twice. There is a much more rigorous statement to be made which is hinted at above about the independence and dependence, but think of it this way.
If you are trying to assess $P(Heads, then\;Tails)$ or the reverse, think about this:
Your probability space $\Omega$ consists of all possible outcomes of 2 consecutive coin tosses. Since there are only 4 possible outcomes in this example, we list them explicitly:
$\Omega=\{x,y\}:x\in H,T\;\;y\in H,T$
$\Omega= \{\{H,T\},\{H,H\},\{T,H\},\{T,T\}\}$
The above lists each possible outcome, and the associated probability for each is $\frac14$. Now if you want to know the combined probability that you flip Heads, then Tails -OR- Tails then Heads, it is simply the addition of each probability.
But if you wanted to know the probability of flipping heads, and then flipping tails, since with a fair coin the events are independent, we just apply the formula you mentioned and multiply each event's probability together $\frac12 \times\frac12=\frac14$
The reason you might be getting confused on this is because the odds of flipping Heads then Tails is highly dependent upon whether or not you flip heads on the first coin.
Does that make sense?
-
Yes it does. However I am currently sticking to the fact that if I need an "or" then I'll add otherwise I''ll multiply – MistyD Aug 6 '12 at 1:20
nothing wrong with that for basic problems, but in case things get a little more complicated hopefully this is of some value :) GOod luck – Justin Aug 6 '12 at 1:38
You use the term $\it{or}$, so the question you are trying to answer has nothing to do with whether the two events are independent or not. The formula you gave means that the probability that two events occur $\it{simultaneouly}$ can be written as the product of the probabilities of the one event and of the other event if these events are independent. | 2013-12-20 19:34:10 | {"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": 0.8055877089500427, "perplexity": 244.6145092644991}, "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/1387345773090/warc/CC-MAIN-20131218054933-00018-ip-10-33-133-15.ec2.internal.warc.gz"} |
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### Custom Intellisense Presenter problems
I am creating my own intellisense presenter these days, and I have posted this thread and downloaded this editor sample. After run the sample, I found some problems, but didn't find how to solve them. ... | 2013-05-20 17:37:07 | {"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": 0.3773081302642822, "perplexity": 8462.561617010722}, "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/1368699138006/warc/CC-MAIN-20130516101218-00052-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1143134/how-does-author-reach-step-of-sa-tm-equiv-1-pmod-m | How does author reach step of $sa + tm \equiv 1 \pmod m$?
This is a proof of a theorem from my book, Discrete Mathematics and its Applications
Theorem 1
If $a$ and $m$ are relatively prime integers and $m>1$, then an inverse of $a$ modulo $m$ exists. Furthermore, this inverse is unique modulo $m$. (That is, there is a unique positive integer $\overline a$ less than m that is an inverse of $a$ modulo $m$ and every other inverse of $a$ modulo $m$ is congruent to $\overline a$ modulo $m$.
Proof: By Theorem 6 of Section 4.3, because $\text{gcd}(a,m)=1$, there are integers $s$ and $t$ such that $$sa+tm=1$$
This implies that $$sa+tm\equiv1 \pmod m$$
Here is theorem 6 of Section 4.3:
Theorem 6
Beloit's Theorem: If $a$ and $b$ are positive integers, then there exist integers $s$ and $t$ such that $\text{gcd}(a,b)\equiv sa+tb$.
The first part of the proof, "because $\text{gcd}(a,m)=1$" makes sense because the conditional statement includes the statement that "$a$ and $m$ are relatively prime integers", meaning that their gcd is $1$. I don't get the step that the author uses to get from
$$sa + tm = 1$$
to
$$sa+tm\equiv1 \pmod m$$
Can someone explain how the author got to that step? Can someone give a general overview of inverse of modulo as well? I don't really understand it from my book. I understand modulus: $7 \bmod 3$ is $1$ but what would inverse of $7 \bmod 3$ get you?
• I would prefer to say and therefore $sa\equiv 1\pmod{m}$. This follows from $sa+tm=1$, since that says that $sa-1$ is a multiple of $m$. – André Nicolas Feb 11 '15 at 5:57
• Where did tm go? – committedandroider Feb 11 '15 at 6:01
• It was explained above. We have $sa+tm=1$ and therefore $sa-1=-tm$, so $sa-1$ is a multiple of $m$, so $sa-1\equiv 0\pmod{m}$, so $sa\equiv 1\pmod{m}$. – André Nicolas Feb 11 '15 at 6:07
• @AndréNicolas That comment should be the answer :) Thanks for making it clear – committedandroider Feb 12 '15 at 0:37
• As long as comments and answer together make everything clear, task accomplished. – André Nicolas Feb 12 '15 at 1:14
Hint: $1 = 1 \pmod m$ for any $m \in \mathbb{N}$. | 2020-01-21 21:29:03 | {"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": 0.8344668745994568, "perplexity": 149.347550346204}, "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-2020-05/segments/1579250605075.24/warc/CC-MAIN-20200121192553-20200121221553-00153.warc.gz"} |
http://math.stackexchange.com/questions/5827/measurable-function-remaining-constant | # Measurable function remaining constant
This is a problem which appeared in one of my tests, which i wasn't able to solve.
Let $\Omega$ be a uncountable set. Let $S$ be the collection of subsets of $\Omega$ given by: $A \in S$ if and only if $A$ is countable or $A^{c}$ is countable. Suppose $f: \Omega \to \mathbb{R}$ is a real measurable function. Prove that there exists a $y \in \mathbb{R}$ and a countable set $B$ such that the $f(x)=y$ is on $B^{c}$.
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You mean to say that $S$ is a collection of subsets of $\Omega$ and not all of the subsets. – Asaf Karagila Oct 1 '10 at 16:16
I guess you mean $S$ is the collection of all subsets $A$ such that $A$ or $A^c$ is infinite countable (many authors say that a set is countable if it is finite or infinite countable). – AD. Oct 1 '10 at 18:14
What about $f:\mathbb{R}\ni x\mapsto x\in\mathbb{R}$ (the identity) - $f$ is not constant on any subset except singleton sets. – AD. Oct 1 '10 at 18:17
@AD $f^{-1}(0,1) = (0,1)$ which is neither countable not co-countable, so the identity function is not measurable. – ACARCHAU Oct 1 '10 at 18:22
@A.D. Given a set of infinite cardinality $\kappa$ and an infinite cardinal $\lambda\lt\kappa$, the collection of all subsets that have cardinalty at most $\lambda$ or are complements of subsets of cardinality at most $\lambda$ is a $\sigma$-algebra. So here, $S$ is a $\sigma$-algebra. – Arturo Magidin Oct 1 '10 at 22:26
Note, that the countable intersection of co-countable sets (i.e., sets whose complement is countable) is co-countable ( since its complement is a countable union of countable sets).
Now, the union of the inverse image of the sets $[n,n+1]$ under as $n$ varies over all integers is all of $\Omega$. Since the inverse image of each $[n,n+1]$ is either countable or co-countable, so at least one of them must be co-countable (since $\Omega$ is uncountable).
Say $[n_1,n_1+1] = [a_1,b_1]$ is a set with co-countable (and hence uncountable) inverse image. Clearly, one of $[n_1, n_1 + 1/2]$ and $[n_1+1/2,n_1+1]$ must have a co-countable inverse image, call it $[a_2,b_2]$, ... proceeding in this manner we get a sequence of nested intervals $[a_n,b_n]$ each of whose inverse image is co-countable and $\lim_{n\to\infty} b_n - a_n = 0$, their intersection consists of a single point, say $y$, and $f^{-1}(\{y\}) = \cap f^{-1}( [a_n,b_n] )$ being an intersection of co-countable sets is co-countable. Call $B^{c} = f^{-1}(\{y\})$, we are done.
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Nice :) – AD. Oct 2 '10 at 5:39
Why can you say that R is the union of $[n,n+1]$ with $n$ an integer? – user54297 Mar 23 '14 at 21:01
Here's a fun way to write the solution.
Define on $(\Omega, \mathcal{S})$ the probability measure $P(A) = 0$ if $A$ is countable, $P(A)=1$ if $A^c$ is countable. Then $f$ can be seen as a random variable $X$. Since all events have probability $0$ or $1$, all events, and hence all random variables, are independent. Now there must be some $N$ with $P(|X| \le N) > 0$ (since $\Omega = \bigcup_{N=1}^\infty \{|X| \le N\}$ and $P$ is countably additive), hence $P(|X| \le N) = 1$. So $X$ is a.s. bounded and in particular is $L^2$. But $Var(X) = Cov(X,X) = 0$ since $X$ is independent of itself! So $X = EX$ a.s., i.e. except on a countable set.
- | 2016-02-05 22:43:14 | {"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": 0.9498435854911804, "perplexity": 129.20085297768748}, "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-2016-07/segments/1454701145519.33/warc/CC-MAIN-20160205193905-00191-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://legendtechz.blogspot.com/2014/02/explain-about-ideal-low-pass-filter-frequency-domain.html | 35. Explain about Ideal Low Pass Filter (ILPF) in frequency domain.
35. Explain about Ideal Low Pass Filter (ILPF) in frequency domain.
Lowpass Filter:
The edges and other sharp transitions (such as noise) in the gray levels of an image contribute significantly to the high-frequency content of its Fourier transform. Hence blurring (smoothing) is achieved in the frequency domain by attenuating us the transform of a given image.
G (u, v) = H (u, v) F(u, v)
where F (u, v) is the Fourier transform of an image to be smoothed. The problem is to select a filter transfer function H (u, v) that yields G (u, v) by attenuating the high-frequency components of F (u, v). The inverse transform then will yield the desired smoothed image g (x, y).
Ideal Filter:
A 2-D ideal lowpass filter (ILPF) is one whose transfer function satisfies the relation
$H(u,v)=\begin{cases}1 & D(u,v) \leq D_{0}\\0 & D(u,v) > D_{0}\end{cases}$
where D is a specified non negative quantity, and D(u, v) is the distance from point (u, v) to the
origin of the frequency plane; that is,
$D(u,v)=\sqrt{(u^{2}+v^{2})}$
Figure 3 (a) shows a 3-D perspective plot of H (u, v) u a function of u and v. The name ideal filter indicates that oil frequencies inside a circle of radius D0 are passed with no attenuation, whereas all frequencies outside this circle are completely attenuated.
Fig.3 (a) Perspective plot of an ideal lowpass filter transfer function; (b) filter cross section.
The lowpass filters are radially symmetric about the origin. For this type of filter, specifying a cross section extending as a function of distance from the origin along a radial line is sufficient, as Fig. 3 (b) shows. The complete filter transfer function can then be generated by rotating the cross section 360 about the origin. Specification of radially symmetric filters centered on the N x N frequency square is based on the assumption that the origin of the Fourier transform has been centered on the square.
For an ideal lowpass filter cross section, the point of transition between H(u, v) = 1 and H(u, v) = 0 is often called the cutoff frequency. In the case of Fig.3 (b), for example, the cutoff frequency is Do. As the cross section is rotated about the origin, the point Do traces a circle giving a locus of cutoff frequencies, all of which are a distance Do from the origin. The cutoff frequency concept is quite useful in specifying filter characteristics. It also serves as a common base for comparing the behavior of different types of filters.
The sharp cutoff frequencies of an ideal lowpass filter cannot be realized with electronic components, although they can certainly be simulated in a computer. | 2018-12-19 01:25:40 | {"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": 0.751014232635498, "perplexity": 605.4810104548203}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376830305.92/warc/CC-MAIN-20181219005231-20181219031231-00149.warc.gz"} |
http://ncatlab.org/nlab/show/simple+ring | # nLab simple ring
A ring $R$ is simple if it is it is a simple object in the category of $R$-$R$-bimodules.
This can be stated in more elementary terms in any of the following equivalent ways:
• $R$ is nontrivial and has no nontrivial two-sided ideals.
• $R$ has exactly two two-sided ideals (which must be $R$ itself and its zero ideal).
In constructive algebra, this is too strong; we must say:
• For each two-sided ideal $I$, $I$ is the zero ideal if and only if $I$ is proper (not equal to $R$).
Revised on November 27, 2009 17:06:30 by Toby Bartels (173.60.119.197) | 2014-08-02 08:37:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8078218102455139, "perplexity": 309.4272420341335}, "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/1406510280868.21/warc/CC-MAIN-20140728011800-00396-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://brilliant.org/practice/cryptograms-2/ | ### Mathematical Fundamentals
$\Large \begin{array} { c c c } & 1 & \color{#EC7300}{B} \\ + & \color{#EC7300}{B} & 6 \\ \hline & 7 & 1 \\ \end{array}$
Cryptograms are puzzles where capital letters stand in for the digits of a number. If the same letter is used twice, it’s the same digit in both places. And, in all of our puzzles, different letters represent different digits.
Solve enough cryptograms, and you’ll start to see patterns — tricks that will help you solve puzzles like these faster and more skillfully.
# Cryptograms
$\Large \begin{array} { c c c } & 1 & \color{#EC7300}{B} \\ + & \color{#EC7300}{B} & 6 \\ \hline & 7 & 1 \\ \end{array}$
What digit in place of $\color{#EC7300}{B}$ would make this sum true?
# Cryptograms
$\large \begin{array} { l l l l } & & \color{#20A900}{Z} & \color{#20A900}{Z} \\ & & \color{#20A900}{Z} & \color{#20A900}{Z} \\ & & & \color{#20A900}{Z} \\ & & & \color{#20A900}{Z} \\ +& & &\color{#20A900}{Z} \\ \hline & 1 & 0 & 0 \end{array}$ What digit is ${\color{#20A900}{Z}}?$
# Cryptograms
The remainder of the problems in this quiz get increasingly difficult as you go along. You will be called on to use all of the strategies below methodically and deliberately to work out solutions:
• Converting the problem into equations that take the place values of the letters into account. For example, $R2D2 = 1000R + 200 + 10D + 2.$
• Keeping track of how carrying the overflow from one place value to the next if the sum is greater than or equal to 10 impacts the cryptogram.
• Keeping track of all of the possibilities carefully and in an organized way so that you remember what you've ruled out and what's still possible.
Which of these techniques did you already use while solving the two previous warmup puzzles?
# Cryptograms
$\Large \begin{array} {c c c c } & & \color{#D61F06}{A} & \color{#3D99F6}{B} \\ + & & \color{#D61F06}{A} & \color{#3D99F6}{B} \\ \hline & 1 & 3 & 8 \\ \end{array}$
What is the value of $\color{#D61F06}{A} \color{#333333}{?}$
# Cryptograms
$\Large \begin{array} {c c c } & 1 & \color{#3D99F6}{E} \\ \times & & \color{#3D99F6}{E} \\ \hline & 9 & \color{#3D99F6}{E} \\ \end{array}$
What digit in place of $\color{#3D99F6}{E}$ would make this multiplication true?
# Cryptograms
$\def\arraystretch{1.5} \begin{array} {ccccc} \large & & & & \large \color{#69047E}{X}& \large \color{#69047E}{X}\\ \large & & & & \large \color{#D61F06}{Y}& \large \color{#D61F06}{Y}\\ \large + & & & & \large \color{#3D99F6}{Z}& \large \color{#3D99F6}{Z}\\ \hline \large& & & \large \color{#69047E}{X} & \large \color{#D61F06}{Y} & \large \color{#3D99F6}{Z} \end{array}$
If each letter represents a different nonzero digit, what must $\large \color{#3D99F6}{Z}$ be?
× | 2020-04-03 04:59:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 34, "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": 0.5278012156486511, "perplexity": 799.7870292851167}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370510287.30/warc/CC-MAIN-20200403030659-20200403060659-00081.warc.gz"} |
https://kerodon.net/tag/00CL | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Exercise 2.1.2.20. Let $\operatorname{\mathcal{C}}$ be a monoidal category with unit object $\mathbf{1}$. Show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagrams
$\xymatrix { X \otimes (Y \otimes \mathbf{1}) \ar [rr]^{\alpha _{X,Y,\mathbf{1}} } \ar [dr]_{\operatorname{id}_ X \otimes \rho _{Y}} & & (X \otimes Y) \otimes \mathbf{1} \ar [dl]^{ \rho _{ X \otimes Y} } \\ & X \otimes Y & }$
$\xymatrix { \mathbf{1} \otimes (X \otimes Y) \ar [dr]_{\lambda _{X \otimes Y}} \ar [rr]^{ \alpha _{ \mathbf{1}, X, Y} } & & ( \mathbf{1} \otimes X) \otimes Y \ar [dl]^{ \lambda _ X \otimes \operatorname{id}_ Y} \\ & X \otimes Y & }$
are commutative (for a more general statement, see Proposition 2.2.1.16). | 2021-12-03 16:16:29 | {"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": 0.9480682611465454, "perplexity": 319.9486278601585}, "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/1637964362891.54/warc/CC-MAIN-20211203151849-20211203181849-00546.warc.gz"} |
https://www.mersenneforum.org/showthread.php?s=05ff23596269c4c967b2a7ef77a138d0&t=12004 | mersenneforum.org Predicting the needed time for high n-values?
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2009-06-07, 19:45 #1 Rincewind Oct 2006 103 Posts Predicting the needed time for high n-values? Is there a way to predict/calculate the time a LLRnet-Test with a higher n-value would take? I know that it depenends on the cpu, but is there a way to calculate it in relation to a lower n? Like: Code: n=300.000 : n=1.500.000 5h : xh
2009-06-07, 20:27 #2 Mini-Geek Account Deleted "Tim Sorbera" Aug 2006 San Antonio, TX USA 17·251 Posts Maybe this is different with base 5 numbers than base 2 numbers, but a doubling of n produces roughly a quadrupling of testing time, so e.g. if n=300K is 5 hours (from your example), n=600K would be 20 hours, n=1200K would be 80 hours, so I'd guesstimate that n=1500K would be around 100 hours. Of course, I could just be completely wrong. A better way would be to run some iterations and multiply appropriately.
2009-06-07, 20:53 #3
masser
Jul 2003
1,433 Posts
Quote:
Originally Posted by Mini-Geek Maybe this is different with base 5 numbers than base 2 numbers, but a doubling of n produces roughly a quadrupling of testing time, so e.g. if n=300K is 5 hours (from your example), n=600K would be 20 hours, n=1200K would be 80 hours, so I'd guesstimate that n=1500K would be around 100 hours. Of course, I could just be completely wrong. A better way would be to run some iterations and multiply appropriately.
Ditto.
2009-06-09, 03:19 #4 gd_barnes May 2007 Kansas; USA 5×13×157 Posts Ditto here also from my experience with the numerous different bases on the CRUS project. To be more exact, if an n=300K test takes 5 hours, an n=1.5M test, since the exponent is 5 times as high, would take 25 times longer, i.e. 125 hours.
2009-06-11, 12:24 #5 Rincewind Oct 2006 103 Posts I started the following three pairs in llrnet, and noted the time and the percentage after restarting llrnet and calculated the time a certain pair would need to finish. Code: Pairs: 64258 328758 64258 657558 64258 1972518 (I got the first pair from the server after starting llrnet. The other two n-values are generated with sr5sieve [sr5sieve -a 64258 657516 2000000] and are about 2 times // 6 times as big as the n-value from the server) And the results are: If you increase the n-value with the factor 2, the needed time is increased by factor ~4 (2^2) Increasing the n-value with the factor 6 causes the needed time to be increased by the factor ~36 (6^2) So, your answers were correct. Thanks. Last fiddled with by Rincewind on 2009-06-11 at 13:08 Reason: wrong parameter (1972548 -> 2000000)
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http://sondavoinemaigrir.com/torani-syrup-icbste/0c4bcf-a-good-estimator-is-consistent | In the above example, if we choose $\hat{\Theta}_1=X_1$, then $\hat{\Theta}_1$ is also an unbiased estimator of $\theta$: \begin{align}%\label{} B(\hat{\Theta}_1)&=E[\hat{\Theta}_1]-\theta\\ &=EX_1-\theta\\ &=0. There is a random sampling of observations.A3. E ( α ^) = α . These are: Unbiasedness; Efficiency; Consistency; Let’s now look at each property in detail: Unbiasedness. Find the asymptotic joint distribution of the MLE of $\alpha, \beta$ and $\sigma^2$ Hot Network Questions Why do the Pern novels use regular words as profanity? We have already seen in the previous example that $$\overline X$$ is an unbiased estimator of population mean $$\mu$$. Formal Definition: The estimator is a consistent estimator of the population parameter βj if the probability limit of is βj, ⦠In class, we mentioned that Consistency is an ideal property of a good estimator. Other Properties of Good Estimators •An estimator is efficient if it has a small standard deviation compared to other unbiased estimators ... –That is, robust estimators work reasonably well under a wide variety of conditions •An estimator is consistent if For more detail, see Chapter 9.1-9.5 T n Ö P TÖ n T ! Point estimation, in statistics, the process of finding an approximate value of some parameter—such as the mean (average)—of a population from random samples of the population. There are three desirable properties every good estimator should possess. On the other hand, a good state-of-charge estimator is consistent and it is dependable for any driving profile and this enhances the overall power system reliability. In statistics, a consistent estimator or asymptotically consistent estimator is an estimatorâa rule for computing estimates of a parameter θ 0 âhaving the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ 0. See the answer. A consistent estimator in statistics is such an estimate which hones in on the true value of the parameter being estimated more and more accurately as the sample size increases. Consider the following example. It produces a single value while the latter produces a range of values. Thus estimators with small variances are more concentrated, they estimate the parameters more precisely. It is satisfactory to know that an estimator θËwill perform better and better as we obtain more examples. An Unbiased Estimator, ê, Is Consistent If, Among Other Assumptions) Lim Var(Ô) = 0 N- (a) (4 Pts) In Your Own Words, Interpret What It Means To Be A Consistent Estimator. Hi there! Required fields are marked *. If an estimator is not an unbiased estimator, then it is a biased estimator. Similarly we deal with point estimation of p. ⦠A Bivariate IV model Letâs consider a simple bivariate model: y 1 =β 0 +β 1 y 2 +u We suspect that y 2 is an endogenous variable, cov(y 2, u) â 0. by Marco Taboga, PhD. The variance of $$\widehat \alpha$$ approaches zero as $$n$$ becomes very large, i.e., $$\mathop {\lim }\limits_{n \to \infty } Var\left( {\widehat \alpha } \right) = 0$$. Consistent estimators: De nition: The estimator ^ of a parameter is said to be consistent estimator if for any positive lim n!1 P(j ^ j ) = 1 or lim n!1 P(j ^ j> ) = 0 We say that ^converges in probability to (also known as the weak law of large numbers). An unbiased estimator which is a linear function of the random variable and possess the least variance may be called a BLUE. Example: Let be a random sample of size n from a population with mean µ and variance . That is if θ is an unbiased estimate of θ, then we must have E (θ) = θ⦠parameter with many samples, do not vary much with each sample) Sample mean (AKA mean/average) - one of the simplest estimators - can act as an estimator for the population expectation. Therefore, the IV estimator is consistent when IVs satisfy the two requirements. If an estimator converges to the true value only with a given probability, it is weakly consistent. A good example of an estimator is the sample mean x, which helps statisticians to estimate the population mean, μ. An estimator is said to be consistent if: a. it is an unbiased estimator. Estimators are essential for companies to capitalize on the growth in construction. The Gauss-Markov theorem states that if your linear regression model satisfies the first six classical assumptions, then ordinary least squares regression produces unbiased estimates that have the smallest variance of all possible linear estimators.. Estimating is one of the most important jobs in construction. You will often read that a given estimator is not only consistent but also asymptotically normal, that is, its distribution converges to a normal distribution as the sample size increases. In my opinion, when we have good predictive estimators, we should . The simplest way of showing consistency consists of proving two sufficient conditions: i) the estimator ⦠Definition of Consistent Estimator in the context of A/B testing (online controlled experiments). An estimator, $$t_n$$, is consistent if it converges to the true parameter value $$\theta$$ as we get more and more observations. An estimator ⦠In developing this article I came up with three areas in regard to what I think makes up a good estimator. Consistent Estimator. The obvi-ous way to estimate dy=dz is by OLS regression of y on z with slope estimate (z0z) 1z0y. Properties of Good Estimators ¥In the Frequentist world view parameters are Þxed, statistics are rv and vary from sample to sample (i.e., have an associated sampling distribution) ¥In theory, there are many potential estimators for a population parameter ¥What are characteristics of good estimators? Inconsistent estimator. - good estimators give good indication of pop. Although a biased estimator does not have a good alignment of its expected value with its parameter, there are many practical instances when a biased estimator can be useful. All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. b. Deï¬nition 1. All that remains is consistent estimation of dy=dz and dx=dz. A BLUE therefore possesses all the three properties mentioned above, and is also a linear function of the random variable. An estimator is said to be consistent if its value approaches the actual, true parameter (population) value as the sample size increases. The two main types of estimators in statistics are point estimators and interval estimators. We already made an argument that IV estimators are consistent, provided some limiting conditions are met. The variance of must approach to Zero as n tends to infinity. The estimator is a consistent estimator of the population parameter βj if its sampling distribution collapses on, or converges to, the value of the population parameter βj as ˆ (N) βj ˆ (N) βj N →∞. Being unbiased. Therefore, the IV estimator is consistent when IVs satisfy the two requirements. c. an estimator whose expected value is equal to zero. From the second condition of consistency we have, $\begin{gathered} \mathop {\lim }\limits_{n \to \infty } Var\left( {\overline X } \right) = \mathop {\lim }\limits_{n \to \infty } \frac{{{\sigma ^2}}}{n} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\sigma ^2}\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{n}} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\sigma ^2}\left( 0 \right) = 0 \\ \end{gathered}$. The variable z is called a(n) _____ variable. A notable consistent estimator in A/B testing is the sample mean (with proportion being the mean in the case of a rate). This sounds so simple, but it is a critical part of their ability to do their job. Consistent and asymptotically normal. consistent theme I hear is that âa good estimator should be able to write a good scope.â I have to confess: I donât know what that means, and I believe the people telling me that are not really sure what it means either. An estimator is a random variable and an estimate is a number (that is the computed value of the estimator). Definition: An estimator Ì is a consistent estimator of θ, if Ì â , i.e., if Ì converges in probability to θ. ANS: A PTS: 1 REF: SECTION 10.1 4. As we have ⦠$$\mathop {\lim }\limits_{n \to \infty } E\left( {\widehat \alpha } \right) = \alpha$$. Asymptotic (infinite-sample) consistency is a guarantee that the larger the sample size we can achieve the more accurate our estimation becomes. A consistent estimator in statistics is such an estimate which hones in on the true value of the parameter being estimated more and more accurately as the sample size increases. Example: Let be a random sample of size n from a population with mean µ and variance . Question: 5. Also, by the weak law of large numbers, $\hat{\sigma}^2$ is also a consistent estimator of $\sigma^2$. An estimator is Fisher consistent if the estimator is the same functional of the empirical distribution function as the parameter of the true distribution function: θË= h(F n), θ = h(F θ) where F n and F θ are the empirical and theoretical distribution functions: F n(t) = 1 n Xn 1 1{X i ⤠t), F θ(t) = P θ{X ⤠t}. an estimator whose variance is equal to one. Show that Ì
â is a consistent estimator of µ. Consistent estimators •We can build a sequence of estimators by progressively increasing the sample size •If the probability that the estimates deviate from the population value by more than ε«1 tends to zero as the sample size tends to infinity, we say that the estimator is consistent. Definition of consistent estimator in the Definitions.net dictionary. Consistency: An estimator is said to be "consistent" if increasing the sample size produces an estimate with smaller standard error. Consistency : An estimators called consistent when it fulfils following two conditions must be Asymptotic Unbiased. ð Below is a list of consistent estimator words - that is, words related to consistent estimator. Like this glossary entry? Note that if an estimator is unbiased, it is not necessarily a good estimator. "Statistical Methods in Online A/B Testing". Similarly estimate dx=dz by OLS regression of x on z with slope estimate (z0z) 1z0x. No, not all unbiased estimators are consistent. In econometrics, Ordinary Least Squares (OLS) method is widely used to estimate the parameters of a linear regression model. An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter.. The proof for this theorem goes way beyond the scope of this blog post. 5. Consistent estimators: De nition: The estimator ^ of a parameter is said to be consistent estimator if for any positive lim n!1 P(j ^ j ) = 1 or lim n!1 P(j ^ j> ) = 0 We say that ^converges in probability to (also known as the weak law of large numbers). The variance of $$\overline X$$ is known to be $$\frac{{{\sigma ^2}}}{n}$$. lim n â â. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. Without the solid background in construction, they cannot do a fair or accurate estimate. An estimator has this property if a statistic is a linear function of the sample observations. Consistent . Question: What Are Three Properties Of A Good Estimator? In others there may be many different transformations of x into (y,z) for which y is sufficient. In the absence of an experiment, researchers rely on a variety of statistical control strategies and/or natural experiments to reduce omitted variables bias. An exception where bIV is unbiased is if the original regression equation actually satisfies Gauss-Markov assumptions. This refers to a ⦠Demand for well-qualified estimators continues to grow because construction is on an upswing. We did not show that IV estimators are unbiased, and in fact they usually are not. Now, consider a variable, z, which is correlated y 2 but not correlated with u: cov(z, y 2) ≠0 but cov(z, u) = 0. For this reason, consistency is known as an asymptotic property for an estimator; that is, it gradually approaches the true parameter value as the sample size approaches infinity. 4, Regression and matching Although it is increasingly common for randomized trials to be used to estimate treatment effects, most economic research still uses observational data. Let Z 1,Z Being consistent. We say that the PE βâ j is an unbiased estimator ⦠The accuracy of any particular approximation is not known precisely, though probabilistic statements concerning the accuracy of such numbers as found over many experiments can be constructed. This seems sensible - weâd like our estimator to be estimating the right thing, although weâre sometimes willing to make a tradeoff between bias and variance. The attractiveness of different ⦠Thus, if we have two estimators $$\widehat {{\alpha _1}}$$ and \widehat {{\a â¡. What is standard error? If at the limit n â â the estimator tend to be always right (or at least arbitrarily close to the target), it is said to be consistent. A fourth benefit of a good state of charge estimator has to do with increasing the density of your energy storage of your battery pack. However, even without any analysis, it seems pretty clear that the sample mean is not going to be a very good choice of estimator of the population minimum. use them in stead of unbiased estimator. These are: Unbiasedness; Efficiency; Consistency; Letâs now look at each property in detail: Unbiasedness. Example 1: The variance of the sample mean X¯ is σ2/n, which decreases to zero as we increase the sample size n. Hence, the sample mean is a consistent estimator for µ. An estimator that has the minimum variance but is biased is not good; An estimator that is unbiased and has the minimum variance of all other estimators is the best (efficient). For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev. Your email address will not be published. This problem has been solved! BLUE stands for Best Linear Unbiased Estimator. \end{align} By linearity of expectation, \hat{\sigma}^2 is an unbiased estimator of \sigma^2. Linear regression models have several applications in real life. For there to be a consistent estimator the parameter variance should be a decreasing function as the sample size increases. Proof: omitted. B. Theorem: An unbiased estimator Ì for is consistent, if â ( Ì ) . A point estimator is defined as: a single value that estimates an unknown population parameter. An estimator is consistent if it approaches the true parameter value as the sample size gets larger and larger. The estimator is a consistent estimator of the population parameter βj if its sampling distribution collapses on, or converges to, the value of the population parameter βj as Ë (N) βj Ë (N) βj N ââ. \end{align} Nevertheless, we suspect that \hat{\Theta}_1 is probably not as good ⦠An estimator\widehat \alpha $$is said to be a consistent estimator of the parameter$$\widehat \alpha $$if it holds the following conditions: Example: Show that the sample mean is a consistent estimator of the population mean. One such case is when a plus four confidence interval is used to construct a confidence interval for a population proportion. of which a consistent estimate is avar[(ˆδ(Sˆ−1)) = (S0 xz ˆS−1S )−1 (1.11) The efficient GMM estimator is defined as ˆδ(Sˆ−1)=argmin δ ngn(δ) 0ˆS−1g n(δ) which requires a consistent estimate of S.However, consistent estimation of S, in turn, requires a consistent estimate of … It uses sample data when calculating a single statistic that will be the best estimate of the unknown para⦠Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more. C. Having relative efficiency. Its quality is to be evaluated in terms of the following properties: 1. δ is an unbiased estimator of For fun δ is a consistent estimator of δ is a from STAT 410 at University of Illinois, Urbana Champaign sample analog provides a consistent estimate of ATE.$$\widehat \alpha $$is an unbiased estimator of$$\alpha $$, so if$$\widehat \alpha $$is biased, it should be unbiased for large values of$$n$$(in the limit sense), i.e. An estimator is consistent if it satisfies two conditions: a. characteristic interested in (ideally provide a value close to true value of the population parameter, average out to true pop. An estimator is said to be unbiased if its expected value is identical with the population parameter being estimated. The linear regression model is âlinear in parameters.âA2. These are: 1) Unbiasedness: the expected value of the estimator (or the mean of the estimator) is simply the figure being estimated. The OLS estimator is an efficient estimator. Its variance converges to 0 as the sample size increases. There are 20 consistent estimator-related words in total, with the top 5 most semantically related being estimator, convergence in probability, statistics, sample size and almost sure convergence.You can get the definition(s) ⦠d. an estimator whose variance goes to zero as the sample size goes to infinity. This property isn’t present for all estimators, and certainly some estimators are desirable (efficient and either unbiased or consistent) without being linear. There are four main properties associated with a "good" estimator. Among a number of estimators of the same class, the estimator having the least variance is called an efficient estimator. Unbiased estimator. When a biased estimator is used, bounds of the bias are calculated. A consistent estimator is one which approaches the real value of the parameter in the population as the size of the sample, n, increases. MLE for a regression with alpha = 0. An implication of sufficiency is that the search for a good estimator can be restricted to estimators T(y) that depend only on sufficient statistics y. Good estimators bend over backwards, at times at their own loss, to do the right thing. An estimator that converges to a multiple of a parameter can be made into a consistent estimator by multiplying the estimator by a scale factor, namely the true value divided by the asymptotic Therefore, your estimate is consistent with the sample size. A good estimator, as common sense dictates, is close to the parameter being estimated. If there are two unbiased estimators of a population parameter available, the one that has the smallest variance is said to be: An efficient estimator is the "best possible" or "optimal" estimator of a parameter of interest. Typically, estimators that are consistent begin to converge steadily. Which of the following is not a characteristic for a good estimator? But the sample mean Y is also an estimator of the popu-lation minimum. Note that being unbiased is a precondition for an estima-tor to be consistent. In other words: the average of many independent random variables should be very ⦠So for any n0, n1, ... , nx, if nx2 > nx1 then the estimator's error decreases: εx2 < &epsilonx1. The sequence is strongly consistent, if it converges almost surely to the true value. In some problems, only the full sample x is a sufficient statistic, and you obtain no useful restriction from sufficiency. An estimator is said to be consistent if it converges in probability to the unknown parameter, that is to say: (2.99) which, in view of , means that a consistent estimator satisfies the convergence in probability to a constant, with the unknown parameter being such a constant. Unbiasedness. Both these hold true for OLS estimators and, hence, they are consistent estimators. This satisfies the first condition of consistency. Select a letter to see all A/B testing terms starting with that letter or visit the Glossary homepage to see all. When one compares between a given procedure and a notional "best ⦠So for any n 0, n 1,..., n x, if n x2 > n x1 then the estimator's error decreases: ε x2 < &epsilon x1. Consistency. For the point estimator to be consistent, the expected value should move toward the true value of the parameter. (William Saroyan) ... meaning that it is consistent, since when we increase the number of observation the estimate we will get is very close to the parameter (or the chance that the difference between the estimate and the parameter is large (larger than epsilon) is zero). Now, consider a variable, z, which is correlated y 2 but not correlated with u: cov(z, y 2) â 0 but cov(z, u) = 0. If convergence is almost certain then the estimator is said to be strongly consistent (as the sample size reaches infinity, the probability of the estimator being equal to the true value becomes 1). But in practice, that is not typically how such things behave. In other words, an estimator is unbiased if it produces parameter estimates that are on average correct. The estimator needs to have a solid background in construction. Let us show this using an example. b. Your email address will not be published. Most efficient or unbiased. A mind boggling venture is to find an estimator ⦠Information and translations of consistent estimator in the most comprehensive dictionary definitions resource on the web. Hence,$$\overline X $$is also a consistent estimator of$$\mu . - good estimators give good indication of pop. Proof: omitted. An estimator which is not consistent is said to be inconsistent. An estimator is said to be consistent if it converges in probability to the unknown parameter, that is to say: (2.99) which, in view of , means that a consistent estimator satisfies the convergence in probability to a constant, with the unknown parameter being such a constant. In other words: the average of many independent random variables should be very close to the true mean with high probability. The most efficient point estimator is the one with the smallest variance of all the unbiased and consistent estimators. It is asymptotically unbiased. Both weak and strong consistency are extensions of the Law of Large Numbers (LLN). An estimator is said to be consistent if: the difference between the estimator and the population parameter grows smaller as the sample size grows larger. This suggests the following estimator for the variance \begin{align}%\label{} \hat{\sigma}^2=\frac{1}{n} \sum_{k=1}^n (X_k-\mu)^2. Good people are good because they've come to wisdom through failure. characteristic interested in (ideally provide a value close to true value of the population parameter, average out to true pop. said to be consistent if V(ˆµ) approaches zero as n → ∞. We say that the estimator is a finite-sample efficient estimator (in the class of unbiased estimators) if it reaches the lower bound in the Cramér–Rao inequality above, for all θ ∈ Θ. parameter with many samples, do not vary much with each sample) Sample mean (AKA mean/average) - one of the simplest estimators - can act as an estimator ⦠The con⦠can we say for certain if it is a good estimator or not, but it is certainly a natural first choice. There are three desirable properties every good estimator should possess. What does consistent estimator mean? From the last example we can conclude that the sample mean\overline X is a BLUE. 1. A Bivariate IV model Let’s consider a simple bivariate model: y 1 =β 0 +β 1 y 2 +u We suspect that y 2 is an endogenous variable, cov(y 2, u) ≠0. Point estimation is the opposite of interval estimation. Use MGF to show $\hat\beta$ is a consistent estimator of $\beta$ 1. Indeed, any statistic is an estimator. 3. An estimator α ^ is said to be a consistent estimator of the parameter α ^ if it holds the following conditions: α ^ is an unbiased estimator of α , so if α ^ is biased, it should be unbiased for large values of n (in the limit sense), i.e. A. In Class, We Mentioned That Consistency Is An Ideal Property Of A Good Estimator. The linearity property, however, can … This notion is equivalent to convergence in probability deï¬ned below. Show that ̅ ∑ is a consistent estimator … Unbiased, Consistent, And Relatively Efficient Consistent, Confident, And Accurate Even With A Small Sample Robust, Confident, And Practical OOOO Unbiased, Robust, And Confident Relatively Efficient, Accurate Even With A Small Sample, And Practical None Of The Above . Consistency is a property involving limits, and mathematics allows things to be arbitrarily far away from the limiting value even after "a long time." A point estimator is a statistic used to estimate the value of an unknown parameter of a population. An unbiased estimator, 0, is consistent if, among other assumptions) lim Var(0) = 0 (a) (4 pts) In your own words, interpret what it means to be a consistent estimator. Suppose we are trying to estimate $1$ by the following procedure: $X_i$s are drawn from the set $\{-1, 1\}$. A good example of an estimator is the sample mean x, which helps statisticians to estimate the population mean, μ. In order to obtain consistent estimators of 0 and 1 , when x and u are correlated, a new variable z is introduced into the model which satisfies the following two conditions: Cov(z,x) 0 and Cov (z,u) = 0. An unbiased estimator of a population parameter is defined as: an estimator whose expected value is equal to the parameter. You might think that ⦠The definition of "best possible" depends on one's choice of a loss function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. Definition: An estimator ̂ is a consistent estimator of θ, if ̂ → , i.e., if ̂ converges in probability to θ. Theorem: An unbiased estimator ̂ for is consistent, if → ( ̂ ) . Meaning of consistent estimator. Unbiased is if the original regression equation actually satisfies Gauss-Markov assumptions in problems... 'Ve come to wisdom through failure a good estimator should possess in there... Estimator or not, but it is certainly a natural first choice: what are three properties mentioned,... Then it is satisfactory to know that an estimator is a linear function of the sample produces... For companies to capitalize on the web needs to have a solid background in construction variables bias for OLS and! ( OLS ) method is widely used to estimate the population parameter growth in construction through failure µ variance! Without the solid background in construction there are four main properties associated with given...: 1 strong Consistency are extensions of the most efficient point estimator is unbiased is a linear function the! Definitions resource on the web it converges almost surely to the true mean with high probability two requirements properties 1. With a given parameter is said to be inconsistent to converge steadily consistent. In statistics are point estimators and, hence, $\hat { \sigma ^2! By OLS regression of x into ( y, z ) for which y is sufficient and... All the three properties mentioned above, and is also a consistent estimator of linear... Detail: Unbiasedness ; Efficiency ; Consistency ; Let ’ s now look each! \To \infty } E\left ( { \widehat \alpha } \right ) = \alpha$ ${. Estimators bend over backwards, at times at their own loss, to the! Almost surely to the parameter is not an unbiased estimator already made an argument that IV estimators essential. Then it is an ideal property of a population parameter, average out to true value the! We did not show that IV estimators are consistent begin to converge steadily with the smallest variance of the! Econometrics, Ordinary Least Squares ( OLS ) method is widely used to estimate the value of an population. Case is when a biased estimator \mathop { \lim } \limits_ { n \to \infty } E\left ( \widehat. Bounds of the random variable and possess the Least variance may be different! Are met the estimator needs to have a solid background in construction the last example we conclude!$ is an ideal property of a good estimator or visit the Glossary homepage to see A/B. Glossary homepage to see all A/B testing is the one with the smallest variance of must to... Satisfies two conditions: a PTS: 1 ( with proportion being the mean in the case of population! Tends to infinity provide a value close to the true value of the most comprehensive dictionary definitions resource on growth! Terms of the bias are calculated average of many independent random variables should be very close to true.... ; Letâs now look at each property in detail: Unbiasedness jobs in construction, they can not do fair! From a population parameter, average out to true value only with a good estimator... The growth in construction the two requirements homepage to see all limiting conditions are.. Size we can conclude that the sample observations above, and you No! Section 10.1 4: 1 REF: SECTION 10.1 4 unknown population parameter being estimated weakly! Property, however, can … No, not all unbiased estimators are,! Note that being unbiased is a consistent estimator in the absence of an unknown parameter of a good should! Widely used to estimate the population parameter, an estimator is consistent when IVs satisfy the two.! Consistent if: a. it is a statistic used to estimate the value of an estimator whose expected is. They are consistent estimators Note a good estimator is consistent being unbiased is if the original equation... Variance may be called a ( n ) _____ variable a PTS: 1 good estimators bend over,! Is also a consistent estimator words - that is, words related consistent... Definitions resource on the web linear regression models have several applications in life. Is on an upswing a ( n ) _____ variable and strong Consistency are extensions the. Ivs satisfy the two requirements the estimator needs to have a solid background construction... ( OLS ) method is widely used to construct a confidence interval for a good a good estimator is consistent... A statistic is a BLUE therefore possesses all the three properties mentioned,... Also a consistent estimator the parameter a letter to see all A/B testing terms starting that. Words - that is, words related to consistent estimator words - that is not a for. Therefore possesses all the three properties of a rate ) produces an estimate with standard... Up with three areas in regard to what I think makes up good... Are point estimators and interval estimators ˆµ ) approaches zero as the sample size goes to infinity Ordinary... { n \to \infty } E\left ( { \widehat \alpha } \right ) = \alpha $\overline... Regression models have several applications in real life types of estimators in statistics are point estimators a good estimator is consistent estimators... No useful restriction from sufficiency is defined as: a single value that estimates an unknown parameter of a example! True pop be consistent if it is certainly a natural first choice deal with point estimation of and. Asymptotic unbiased used, bounds of the following properties: 1 such case is when plus!: a single value that estimates an unknown population parameter being estimated produces parameter that... And strong Consistency are extensions of the random variable see all you obtain No useful restriction from sufficiency =. Interval for a good estimator an exception where bIV is unbiased is a linear regression models.A1 requirements. The con⦠therefore, the IV estimator is said to be a decreasing function as the sample size we achieve... And possess the Least variance may be called a ( n ) _____ variable growth in construction probability... That estimates an unknown parameter of a good estimator should possess \beta$.. ( y, z ) for which y is also a consistent estimator of a good estimator should possess is! And/Or natural experiments to reduce omitted variables bias \hat\beta $is also estimator... For well-qualified estimators continues to grow because construction is on an upswing there may many... Or visit the Glossary homepage to see all A/B testing is the sample size produces an estimate smaller! A linear function of the following is not necessarily a good estimator mean... 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https://support.bioconductor.org/p/59337/#59339 | granges() method for GenomicRanges objects akin to ranges()...
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Tim Triche ★ 4.2k
@tim-triche-3561
Last seen 23 months ago
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I wanted something to extract @ranges from a GRanges object along with its @seqnames, @strand, and @seqinfo. Essentially, everything but the mcols. Does this make sense? Is there a lighter-weight way to avoid any copying in-flight? setMethod("granges", "GRanges", function(x) { GRanges(seqnames=seqnames(x), ranges=ranges(x), strand=strand(x), seqinfo=seqinfo(x)) }) The fact that I'm constructing an entire new GRanges makes me a little queasy... that said, it has turned out to be useful when I just want a short list of locations, as for debugging plotting functions, profile plots, or what have you. Statistics is the grammar of science. Karl Pearson <http: en.wikipedia.org="" wiki="" the_grammar_of_science=""> [[alternative HTML version deleted]]
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@jeff-johnston-6497
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On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> wrote: > I wanted something to extract @ranges from a GRanges object along with its > @seqnames, @strand, and @seqinfo. Essentially, everything but the mcols. > > Does this make sense? Is there a lighter-weight way to avoid any copying > in-flight? > > > setMethod("granges", "GRanges", function(x) { > GRanges(seqnames=seqnames(x), > ranges=ranges(x), > strand=strand(x), > seqinfo=seqinfo(x)) > }) > > > The fact that I'm constructing an entire new GRanges makes me a little > queasy... that said, it has turned out to be useful when I just want a > short list of locations, as for debugging plotting functions, profile > plots, or what have you. > Perhaps just this: setMethod("granges", "GRanges", function(x) { mcols(x) <- NULL x })
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Right, what I was wondering however is whether it's possible not to create or modify the object at all, but rather access only the necessary bits. It seems like a slightly different structure that puts all the location in one place (say @granges) and the metadata in another (as it presently is) might be handy to avoid this hoohah. That's rather a larger change. --t > On May 4, 2014, at 3:23 PM, "Johnston, Jeffrey" <jjj at="" stowers.org=""> wrote: > > >> On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> wrote: >> >> I wanted something to extract @ranges from a GRanges object along with its >> @seqnames, @strand, and @seqinfo. Essentially, everything but the mcols. >> >> Does this make sense? Is there a lighter-weight way to avoid any copying >> in-flight? >> >> >> setMethod("granges", "GRanges", function(x) { >> GRanges(seqnames=seqnames(x), >> ranges=ranges(x), >> strand=strand(x), >> seqinfo=seqinfo(x)) >> }) >> >> >> The fact that I'm constructing an entire new GRanges makes me a little >> queasy... that said, it has turned out to be useful when I just want a >> short list of locations, as for debugging plotting functions, profile >> plots, or what have you. > > > Perhaps just this: > > setMethod("granges", "GRanges", function(x) { > mcols(x) <- NULL > x > }) > > >
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Why not just do mcols(gr) <- NULL It's way more obvious than granges(gr). And that should happen virtually instantaneously in R 3.1, regardless of the length. On Sun, May 4, 2014 at 3:55 PM, Tim Triche, Jr. <tim.triche@gmail.com>wrote: > Right, what I was wondering however is whether it's possible not to create > or modify the object at all, but rather access only the necessary bits. > > It seems like a slightly different structure that puts all the location in > one place (say @granges) and the metadata in another (as it presently is) > might be handy to avoid this hoohah. That's rather a larger change. > > --t > > > On May 4, 2014, at 3:23 PM, "Johnston, Jeffrey" <jjj@stowers.org> wrote: > > > > > >> On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche@gmail.com> > wrote: > >> > >> I wanted something to extract @ranges from a GRanges object along with > its > >> @seqnames, @strand, and @seqinfo. Essentially, everything but the > mcols. > >> > >> Does this make sense? Is there a lighter-weight way to avoid any > copying > >> in-flight? > >> > >> > >> setMethod("granges", "GRanges", function(x) { > >> GRanges(seqnames=seqnames(x), > >> ranges=ranges(x), > >> strand=strand(x), > >> seqinfo=seqinfo(x)) > >> }) > >> > >> > >> The fact that I'm constructing an entire new GRanges makes me a little > >> queasy... that said, it has turned out to be useful when I just want a > >> short list of locations, as for debugging plotting functions, profile > >> plots, or what have you. > > > > > > Perhaps just this: > > > > setMethod("granges", "GRanges", function(x) { > > mcols(x) <- NULL > > x > > }) > > > > > > > > _______________________________________________ > Bioc-devel@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/bioc-devel > [[alternative HTML version deleted]]
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Hi Tim, I was looking for a similar function a while ago, and created the 'grangesPlain' function in 'SomaticSignatures': grangesPlain <- function (x) { mcols(x) = NULL x = as(x, "GRanges") return(x) } It removes the metadata columns, as Michael described. Further, it performs an explicit conversion to a 'GRanges' object - in case that 'x' has a derived class like a 'VRanges'. The main motivation for an extra function is that you can use it inline, e.g resize(sort(grangesPlain(x)), ...) works. It would be great to have such functionality as part of the bioc core. Best wishes Julian On 05.05.2014 01:56, Michael Lawrence wrote: > Why not just do > > mcols(gr) <- NULL > > It's way more obvious than granges(gr). And that should happen virtually > instantaneously in R 3.1, regardless of the length. > > > > > On Sun, May 4, 2014 at 3:55 PM, Tim Triche, Jr. <tim.triche at="" gmail.com="">wrote: > >> Right, what I was wondering however is whether it's possible not to create >> or modify the object at all, but rather access only the necessary bits. >> >> It seems like a slightly different structure that puts all the location in >> one place (say @granges) and the metadata in another (as it presently is) >> might be handy to avoid this hoohah. That's rather a larger change. >> >> --t >> >>> On May 4, 2014, at 3:23 PM, "Johnston, Jeffrey" <jjj at="" stowers.org=""> wrote: >>> >>> >>>> On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> >> wrote: >>>> >>>> I wanted something to extract @ranges from a GRanges object along with >> its >>>> @seqnames, @strand, and @seqinfo. Essentially, everything but the >> mcols. >>>> >>>> Does this make sense? Is there a lighter-weight way to avoid any >> copying >>>> in-flight? >>>> >>>> >>>> setMethod("granges", "GRanges", function(x) { >>>> GRanges(seqnames=seqnames(x), >>>> ranges=ranges(x), >>>> strand=strand(x), >>>> seqinfo=seqinfo(x)) >>>> }) >>>> >>>> >>>> The fact that I'm constructing an entire new GRanges makes me a little >>>> queasy... that said, it has turned out to be useful when I just want a >>>> short list of locations, as for debugging plotting functions, profile >>>> plots, or what have you. >>> >>> >>> Perhaps just this: >>> >>> setMethod("granges", "GRanges", function(x) { >>> mcols(x) <- NULL >>> x >>> }) >>> >>> >>> >> >> _______________________________________________ >> Bioc-devel at r-project.org mailing list >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> > > [[alternative HTML version deleted]] > > _______________________________________________ > Bioc-devel at r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/bioc-devel >
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Lovely. Hadn't even thought about the VRanges aspect in a while, but this is essentially the use case I had as well. Thanks! --t > On May 5, 2014, at 1:25 AM, Julian Gehring <julian.gehring at="" embl.de=""> wrote: > > Hi Tim, > > I was looking for a similar function a while ago, and created the 'grangesPlain' function in 'SomaticSignatures': > > grangesPlain <- > function (x) > { > mcols(x) = NULL > x = as(x, "GRanges") > return(x) > } > > It removes the metadata columns, as Michael described. Further, it performs an explicit conversion to a 'GRanges' object - in case that 'x' has a derived class like a 'VRanges'. > > The main motivation for an extra function is that you can use it inline, e.g > > resize(sort(grangesPlain(x)), ...) > > works. It would be great to have such functionality as part of the bioc core. > > Best wishes > Julian > > >> On 05.05.2014 01:56, Michael Lawrence wrote: >> Why not just do >> >> mcols(gr) <- NULL >> >> It's way more obvious than granges(gr). And that should happen virtually >> instantaneously in R 3.1, regardless of the length. >> >> >> >> >> On Sun, May 4, 2014 at 3:55 PM, Tim Triche, Jr. <tim.triche at="" gmail.com="">wrote: >> >>> Right, what I was wondering however is whether it's possible not to create >>> or modify the object at all, but rather access only the necessary bits. >>> >>> It seems like a slightly different structure that puts all the location in >>> one place (say @granges) and the metadata in another (as it presently is) >>> might be handy to avoid this hoohah. That's rather a larger change. >>> >>> --t >>> >>>> On May 4, 2014, at 3:23 PM, "Johnston, Jeffrey" <jjj at="" stowers.org=""> wrote: >>>> >>>> >>>>> On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> >>> wrote: >>>>> >>>>> I wanted something to extract @ranges from a GRanges object along with >>> its >>>>> @seqnames, @strand, and @seqinfo. Essentially, everything but the >>> mcols. >>>>> >>>>> Does this make sense? Is there a lighter-weight way to avoid any >>> copying >>>>> in-flight? >>>>> >>>>> >>>>> setMethod("granges", "GRanges", function(x) { >>>>> GRanges(seqnames=seqnames(x), >>>>> ranges=ranges(x), >>>>> strand=strand(x), >>>>> seqinfo=seqinfo(x)) >>>>> }) >>>>> >>>>> >>>>> The fact that I'm constructing an entire new GRanges makes me a little >>>>> queasy... that said, it has turned out to be useful when I just want a >>>>> short list of locations, as for debugging plotting functions, profile >>>>> plots, or what have you. >>>> >>>> >>>> Perhaps just this: >>>> >>>> setMethod("granges", "GRanges", function(x) { >>>> mcols(x) <- NULL >>>> x >>>> }) >>> >>> _______________________________________________ >>> Bioc-devel at r-project.org mailing list >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >> [[alternative HTML version deleted]] >> >> _______________________________________________ >> Bioc-devel at r-project.org mailing list >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >>
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The in-line usage makes sense. How about dropmcols() or something, at least for removing the mcols? On Mon, May 5, 2014 at 1:25 AM, Julian Gehring <julian.gehring@embl.de>wrote: > Hi Tim, > > I was looking for a similar function a while ago, and created the > 'grangesPlain' function in 'SomaticSignatures': > > grangesPlain <- > function (x) > { > mcols(x) = NULL > x = as(x, "GRanges") > return(x) > } > > It removes the metadata columns, as Michael described. Further, it > performs an explicit conversion to a 'GRanges' object - in case that 'x' > has a derived class like a 'VRanges'. > > The main motivation for an extra function is that you can use it inline, > e.g > > resize(sort(grangesPlain(x)), ...) > > works. It would be great to have such functionality as part of the bioc > core. > > Best wishes > Julian > > > > On 05.05.2014 01:56, Michael Lawrence wrote: > >> Why not just do >> >> mcols(gr) <- NULL >> >> It's way more obvious than granges(gr). And that should happen virtually >> instantaneously in R 3.1, regardless of the length. >> >> >> >> >> On Sun, May 4, 2014 at 3:55 PM, Tim Triche, Jr. <tim.triche@gmail.com>> >wrote: >> >> Right, what I was wondering however is whether it's possible not to >>> create >>> or modify the object at all, but rather access only the necessary bits. >>> >>> It seems like a slightly different structure that puts all the location >>> in >>> one place (say @granges) and the metadata in another (as it presently is) >>> might be handy to avoid this hoohah. That's rather a larger change. >>> >>> --t >>> >>> On May 4, 2014, at 3:23 PM, "Johnston, Jeffrey" <jjj@stowers.org> >>>> wrote: >>>> >>>> >>>> On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche@gmail.com> >>>>> >>>> wrote: >>> >>>> >>>>> I wanted something to extract @ranges from a GRanges object along with >>>>> >>>> its >>> >>>> @seqnames, @strand, and @seqinfo. Essentially, everything but the >>>>> >>>> mcols. >>> >>>> >>>>> Does this make sense? Is there a lighter-weight way to avoid any >>>>> >>>> copying >>> >>>> in-flight? >>>>> >>>>> >>>>> setMethod("granges", "GRanges", function(x) { >>>>> GRanges(seqnames=seqnames(x), >>>>> ranges=ranges(x), >>>>> strand=strand(x), >>>>> seqinfo=seqinfo(x)) >>>>> }) >>>>> >>>>> >>>>> The fact that I'm constructing an entire new GRanges makes me a little >>>>> queasy... that said, it has turned out to be useful when I just want a >>>>> short list of locations, as for debugging plotting functions, profile >>>>> plots, or what have you. >>>>> >>>> >>>> >>>> Perhaps just this: >>>> >>>> setMethod("granges", "GRanges", function(x) { >>>> mcols(x) <- NULL >>>> x >>>> }) >>>> >>>> >>>> >>>> >>> _______________________________________________ >>> Bioc-devel@r-project.org mailing list >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >>> >>> >> [[alternative HTML version deleted]] >> >> >> _______________________________________________ >> Bioc-devel@r-project.org mailing list >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >> [[alternative HTML version deleted]]
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On 05/05/2014 06:55 AM, Michael Lawrence wrote: > The in-line usage makes sense. How about dropmcols() or something, at least > for removing the mcols? generalize as setMcols, like setNames? setMcols(x, NULL) > > > On Mon, May 5, 2014 at 1:25 AM, Julian Gehring <julian.gehring at="" embl.de="">wrote: > >> Hi Tim, >> >> I was looking for a similar function a while ago, and created the >> 'grangesPlain' function in 'SomaticSignatures': >> >> grangesPlain <- >> function (x) >> { >> mcols(x) = NULL >> x = as(x, "GRanges") >> return(x) >> } >> >> It removes the metadata columns, as Michael described. Further, it >> performs an explicit conversion to a 'GRanges' object - in case that 'x' >> has a derived class like a 'VRanges'. >> >> The main motivation for an extra function is that you can use it inline, >> e.g >> >> resize(sort(grangesPlain(x)), ...) >> >> works. It would be great to have such functionality as part of the bioc >> core. >> >> Best wishes >> Julian >> >> >> >> On 05.05.2014 01:56, Michael Lawrence wrote: >> >>> Why not just do >>> >>> mcols(gr) <- NULL >>> >>> It's way more obvious than granges(gr). And that should happen virtually >>> instantaneously in R 3.1, regardless of the length. >>> >>> >>> >>> >>> On Sun, May 4, 2014 at 3:55 PM, Tim Triche, Jr. <tim.triche at="" gmail.com="">>>> wrote: >>> >>> Right, what I was wondering however is whether it's possible not to >>>> create >>>> or modify the object at all, but rather access only the necessary bits. >>>> >>>> It seems like a slightly different structure that puts all the location >>>> in >>>> one place (say @granges) and the metadata in another (as it presently is) >>>> might be handy to avoid this hoohah. That's rather a larger change. >>>> >>>> --t >>>> >>>> On May 4, 2014, at 3:23 PM, "Johnston, Jeffrey" <jjj at="" stowers.org=""> >>>>> wrote: >>>>> >>>>> >>>>> On May 4, 2014, at 3:50 PM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> >>>>>> >>>>> wrote: >>>> >>>>> >>>>>> I wanted something to extract @ranges from a GRanges object along with >>>>>> >>>>> its >>>> >>>>> @seqnames, @strand, and @seqinfo. Essentially, everything but the >>>>>> >>>>> mcols. >>>> >>>>> >>>>>> Does this make sense? Is there a lighter-weight way to avoid any >>>>>> >>>>> copying >>>> >>>>> in-flight? >>>>>> >>>>>> >>>>>> setMethod("granges", "GRanges", function(x) { >>>>>> GRanges(seqnames=seqnames(x), >>>>>> ranges=ranges(x), >>>>>> strand=strand(x), >>>>>> seqinfo=seqinfo(x)) >>>>>> }) >>>>>> >>>>>> >>>>>> The fact that I'm constructing an entire new GRanges makes me a little >>>>>> queasy... that said, it has turned out to be useful when I just want a >>>>>> short list of locations, as for debugging plotting functions, profile >>>>>> plots, or what have you. >>>>>> >>>>> >>>>> >>>>> Perhaps just this: >>>>> >>>>> setMethod("granges", "GRanges", function(x) { >>>>> mcols(x) <- NULL >>>>> x >>>>> }) >>>>> >>>>> >>>>> >>>>> >>>> _______________________________________________ >>>> Bioc-devel at r-project.org mailing list >>>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >>>> >>>> >>> [[alternative HTML version deleted]] >>> >>> >>> _______________________________________________ >>> Bioc-devel at r-project.org mailing list >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >>> >>> > > [[alternative HTML version deleted]] > > _______________________________________________ > Bioc-devel at r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/bioc-devel > -- Computational Biology / Fred Hutchinson Cancer Research Center 1100 Fairview Ave. N. PO Box 19024 Seattle, WA 98109 Location: Arnold Building M1 B861 Phone: (206) 667-2793
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Hi, On 05.05.2014 16:22, Martin Morgan wrote: > generalize as setMcols, like setNames? setMcols(x, NULL) How about Tim's original suggestion, to add a 'granges' method that works on a 'GRanges' input? The current definition granges(x, use.mcols=FALSE, ...) seem suited for this. Best wishes Julian
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That's exactly what I was after -- the generic is already defined, so why not use it? --t > On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring at="" embl.de=""> wrote: > > Hi, > >> On 05.05.2014 16:22, Martin Morgan wrote: >> generalize as setMcols, like setNames? setMcols(x, NULL) > > How about Tim's original suggestion, to add a 'granges' method that works on a 'GRanges' input? The current definition > > granges(x, use.mcols=FALSE, ...) > > seem suited for this. > > Best wishes > Julian
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In my opinion, granges() is not very clear as to the intent. The mcols are part of the GRanges, so why would calling granges() drop them? I think we want something similar to unclass(), unname(), etc. This why I suggested dropmcols(). On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche@gmail.com>wrote: > That's exactly what I was after -- the generic is already defined, so why > not use it? > > --t > > > On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring@embl.de> > wrote: > > > > Hi, > > > >> On 05.05.2014 16:22, Martin Morgan wrote: > >> generalize as setMcols, like setNames? setMcols(x, NULL) > > > > How about Tim's original suggestion, to add a 'granges' method that > works on a 'GRanges' input? The current definition > > > > granges(x, use.mcols=FALSE, ...) > > > > seem suited for this. > > > > Best wishes > > Julian > [[alternative HTML version deleted]]
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I agree with Michael on this. I can see why, in some usage cases, granges() is convenient to have with use.mcols=FALSE (which seems to have been added in the latest release). But in my usage of granges(), where I call granges() on objects like SummarizedExperiments and friends, I have been expecting granges() to give me the GRange component of the object. Not a crippled version of the GRange component. This is - to me - very counter intuitive and I wish I had seen this earlier. It is particular frustrating that this default is part of the generic. Best, Kasper On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence <lawrence.michael@gene.com> wrote: > In my opinion, granges() is not very clear as to the intent. The mcols are > part of the GRanges, so why would calling granges() drop them? I think we > want something similar to unclass(), unname(), etc. This why I suggested > dropmcols(). > > > > > On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche@gmail.com> >wrote: > > > That's exactly what I was after -- the generic is already defined, so why > > not use it? > > > > --t > > > > > On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring@embl.de> > > wrote: > > > > > > Hi, > > > > > >> On 05.05.2014 16:22, Martin Morgan wrote: > > >> generalize as setMcols, like setNames? setMcols(x, NULL) > > > > > > How about Tim's original suggestion, to add a 'granges' method that > > works on a 'GRanges' input? The current definition > > > > > > granges(x, use.mcols=FALSE, ...) > > > > > > seem suited for this. > > > > > > Best wishes > > > Julian > > > > [[alternative HTML version deleted]] > > _______________________________________________ > Bioc-devel@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/bioc-devel > [[alternative HTML version deleted]]
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Hi, I have no problem using granges() for that. Just to clarify: (a) it would propagate the names() (b) it would drop the metadata() (c) the mcols() would propagate only if 'use.mcols=TRUE' was specified ('use.mcols' is FALSE by default) (d) it would return a GRanges *instance* i.e. input object 'x' would be downgraded to GRanges if it extends GRanges @Kasper: granges() on SummarizedExperiment ignores the 'use.mcols' arg and always propagates the mcols. Alternatively you can use rowData() which also propagates the mcols. granges() is actually just an alias for rowData() on SummarizedExperiment objects. H. On 05/05/2014 10:31 AM, Kasper Daniel Hansen wrote: > I agree with Michael on this. > > I can see why, in some usage cases, granges() is convenient to have with > use.mcols=FALSE (which seems to have been added in the latest release). > But in my usage of granges(), where I call granges() on objects like > SummarizedExperiments and friends, I have been expecting granges() to give > me the GRange component of the object. Not a crippled version of the > GRange component. > > This is - to me - very counter intuitive and I wish I had seen this > earlier. It is particular frustrating that this default is part of the > generic. > > Best, > Kasper > > > On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence <lawrence.michael at="" gene.com="">> wrote: > >> In my opinion, granges() is not very clear as to the intent. The mcols are >> part of the GRanges, so why would calling granges() drop them? I think we >> want something similar to unclass(), unname(), etc. This why I suggested >> dropmcols(). >> >> >> >> >> On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche at="" gmail.com="">>> wrote: >> >>> That's exactly what I was after -- the generic is already defined, so why >>> not use it? >>> >>> --t >>> >>>> On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring at="" embl.de=""> >>> wrote: >>>> >>>> Hi, >>>> >>>>> On 05.05.2014 16:22, Martin Morgan wrote: >>>>> generalize as setMcols, like setNames? setMcols(x, NULL) >>>> >>>> How about Tim's original suggestion, to add a 'granges' method that >>> works on a 'GRanges' input? The current definition >>>> >>>> granges(x, use.mcols=FALSE, ...) >>>> >>>> seem suited for this. >>>> >>>> Best wishes >>>> Julian >>> >> >> [[alternative HTML version deleted]] >> >> _______________________________________________ >> Bioc-devel at r-project.org mailing list >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> > > [[alternative HTML version deleted]] > > _______________________________________________ > Bioc-devel at r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/bioc-devel > -- Hervé Pagès Program in Computational Biology Division of Public Health Sciences Fred Hutchinson Cancer Research Center 1100 Fairview Ave. N, M1-B514 P.O. Box 19024 Seattle, WA 98109-1024 E-mail: hpages at fhcrc.org Phone: (206) 667-5791 Fax: (206) 667-1319
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Wondering, Is it too off the beaten track to expect mcols<-(x,NULL) to work? hint: it does not >-----Original Message----- >From: bioc-devel-bounces at r-project.org [mailto:bioc-devel-bounces at r-project.org] On Behalf Of Hervé Pagès >Sent: Monday, May 05, 2014 1:28 PM >To: Kasper Daniel Hansen; Michael Lawrence >Cc: Johnston, Jeffrey; ttriche at usc.edu; bioc-devel at r-project.org; bioconductor at r-project.org >Subject: Re: [Bioc-devel] [BioC] granges() method for GenomicRanges objects akin to ranges()... > >Hi, > >I have no problem using granges() for that. Just to clarify: > (a) it would propagate the names() > (b) it would drop the metadata() > (c) the mcols() would propagate only if 'use.mcols=TRUE' was > specified ('use.mcols' is FALSE by default) > (d) it would return a GRanges *instance* i.e. input object 'x' > would be downgraded to GRanges if it extends GRanges > >@Kasper: granges() on SummarizedExperiment ignores the 'use.mcols' >arg and always propagates the mcols. Alternatively you can use rowData() >which also propagates the mcols. granges() is actually just an alias >for rowData() on SummarizedExperiment objects. > >H. > > >On 05/05/2014 10:31 AM, Kasper Daniel Hansen wrote: >> I agree with Michael on this. >> >> I can see why, in some usage cases, granges() is convenient to have with >> use.mcols=FALSE (which seems to have been added in the latest release). >> But in my usage of granges(), where I call granges() on objects like >> SummarizedExperiments and friends, I have been expecting granges() to give >> me the GRange component of the object. Not a crippled version of the >> GRange component. >> >> This is - to me - very counter intuitive and I wish I had seen this >> earlier. It is particular frustrating that this default is part of the >> generic. >> >> Best, >> Kasper >> >> >> On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence <lawrence.michael at="" gene.com="">>> wrote: >> >>> In my opinion, granges() is not very clear as to the intent. The mcols are >>> part of the GRanges, so why would calling granges() drop them? I think we >>> want something similar to unclass(), unname(), etc. This why I suggested >>> dropmcols(). >>> >>> >>> >>> >>> On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche at="" gmail.com="">>>> wrote: >>> >>>> That's exactly what I was after -- the generic is already defined, so why >>>> not use it? >>>> >>>> --t >>>> >>>>> On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring at="" embl.de=""> >>>> wrote: >>>>> >>>>> Hi, >>>>> >>>>>> On 05.05.2014 16:22, Martin Morgan wrote: >>>>>> generalize as setMcols, like setNames? setMcols(x, NULL) >>>>> >>>>> How about Tim's original suggestion, to add a 'granges' method that >>>> works on a 'GRanges' input? The current definition >>>>> >>>>> granges(x, use.mcols=FALSE, ...) >>>>> >>>>> seem suited for this. >>>>> >>>>> Best wishes >>>>> Julian >>>> >>> >>> [[alternative HTML version deleted]] >>> >>> _______________________________________________ >>> Bioc-devel at r-project.org mailing list >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >>> >> >> [[alternative HTML version deleted]] >> >> _______________________________________________ >> Bioc-devel at r-project.org mailing list >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> > >-- >Hervé Pagès > >Program in Computational Biology >Division of Public Health Sciences >Fred Hutchinson Cancer Research Center >1100 Fairview Ave. N, M1-B514 >P.O. Box 19024 >Seattle, WA 98109-1024 > >E-mail: hpages at fhcrc.org >Phone: (206) 667-5791 >Fax: (206) 667-1319 > >_______________________________________________ >Bioc-devel at r-project.org mailing list >https://stat.ethz.ch/mailman/listinfo/bioc-devel
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Hi Malcolm, On 05/05/2014 01:00 PM, Cook, Malcolm wrote: > Wondering, > > Is it too off the beaten track to expect > > mcols<-(x,NULL) > args(mcols<-) function (x, ..., value) Arguments after the ellipsis must be named: mcols<-(x, value=NULL) Nothing we can do about this. Cheers, H. > > to work? > > hint: it does not > > >-----Original Message----- > >From: bioc-devel-bounces at r-project.org [mailto:bioc-devel- bounces at r-project.org] On Behalf Of Hervé Pagès > >Sent: Monday, May 05, 2014 1:28 PM > >To: Kasper Daniel Hansen; Michael Lawrence > >Cc: Johnston, Jeffrey; ttriche at usc.edu; bioc-devel at r-project.org; bioconductor at r-project.org > >Subject: Re: [Bioc-devel] [BioC] granges() method for GenomicRanges objects akin to ranges()... > > > >Hi, > > > >I have no problem using granges() for that. Just to clarify: > > (a) it would propagate the names() > > (b) it would drop the metadata() > > (c) the mcols() would propagate only if 'use.mcols=TRUE' was > > specified ('use.mcols' is FALSE by default) > > (d) it would return a GRanges *instance* i.e. input object 'x' > > would be downgraded to GRanges if it extends GRanges > > > >@Kasper: granges() on SummarizedExperiment ignores the 'use.mcols' > >arg and always propagates the mcols. Alternatively you can use rowData() > >which also propagates the mcols. granges() is actually just an alias > >for rowData() on SummarizedExperiment objects. > > > >H. > > > > > >On 05/05/2014 10:31 AM, Kasper Daniel Hansen wrote: > >> I agree with Michael on this. > >> > >> I can see why, in some usage cases, granges() is convenient to have with > >> use.mcols=FALSE (which seems to have been added in the latest release). > >> But in my usage of granges(), where I call granges() on objects like > >> SummarizedExperiments and friends, I have been expecting granges() to give > >> me the GRange component of the object. Not a crippled version of the > >> GRange component. > >> > >> This is - to me - very counter intuitive and I wish I had seen this > >> earlier. It is particular frustrating that this default is part of the > >> generic. > >> > >> Best, > >> Kasper > >> > >> > >> On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence <lawrence.michael at="" gene.com=""> >>> wrote: > >> > >>> In my opinion, granges() is not very clear as to the intent. The mcols are > >>> part of the GRanges, so why would calling granges() drop them? I think we > >>> want something similar to unclass(), unname(), etc. This why I suggested > >>> dropmcols(). > >>> > >>> > >>> > >>> > >>> On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> >>>> wrote: > >>> > >>>> That's exactly what I was after -- the generic is already defined, so why > >>>> not use it? > >>>> > >>>> --t > >>>> > >>>>> On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring at="" embl.de=""> > >>>> wrote: > >>>>> > >>>>> Hi, > >>>>> > >>>>>> On 05.05.2014 16:22, Martin Morgan wrote: > >>>>>> generalize as setMcols, like setNames? setMcols(x, NULL) > >>>>> > >>>>> How about Tim's original suggestion, to add a 'granges' method that > >>>> works on a 'GRanges' input? The current definition > >>>>> > >>>>> granges(x, use.mcols=FALSE, ...) > >>>>> > >>>>> seem suited for this. > >>>>> > >>>>> Best wishes > >>>>> Julian > >>>> > >>> > >>> [[alternative HTML version deleted]] > >>> > >>> _______________________________________________ > >>> Bioc-devel at r-project.org mailing list > >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel > >>> > >> > >> [[alternative HTML version deleted]] > >> > >> _______________________________________________ > >> Bioc-devel at r-project.org mailing list > >> https://stat.ethz.ch/mailman/listinfo/bioc-devel > >> > > > >-- > >Hervé Pagès > > > >Program in Computational Biology > >Division of Public Health Sciences > >Fred Hutchinson Cancer Research Center > >1100 Fairview Ave. N, M1-B514 > >P.O. Box 19024 > >Seattle, WA 98109-1024 > > > >E-mail: hpages at fhcrc.org > >Phone: (206) 667-5791 > >Fax: (206) 667-1319 > > > >_______________________________________________ > >Bioc-devel at r-project.org mailing list > >https://stat.ethz.ch/mailman/listinfo/bioc-devel > -- Hervé Pagès Program in Computational Biology Division of Public Health Sciences Fred Hutchinson Cancer Research Center 1100 Fairview Ave. N, M1-B514 P.O. Box 19024 Seattle, WA 98109-1024 E-mail: hpages at fhcrc.org Phone: (206) 667-5791 Fax: (206) 667-1319
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>On 05/05/2014 01:00 PM, Cook, Malcolm wrote: >> Wondering, >> >> Is it too off the beaten track to expect >> >> mcols<-(x,NULL) > > > args(mcols<-) > function (x, ..., value) > >Arguments after the ellipsis must be named: > > mcols<-(x, value=NULL) Herve - Great - of course - so - does this not provide the means requested by the original poster? > >Nothing we can do about this. > >Cheers, >H. > >> >> to work? >> >> hint: it does not >> >> >-----Original Message----- >> >From: bioc-devel-bounces at r-project.org [mailto:bioc-devel- bounces at r-project.org] On Behalf Of Hervé Pagès >> >Sent: Monday, May 05, 2014 1:28 PM >> >To: Kasper Daniel Hansen; Michael Lawrence >> >Cc: Johnston, Jeffrey; ttriche at usc.edu; bioc-devel at r-project.org; bioconductor at r-project.org >> >Subject: Re: [Bioc-devel] [BioC] granges() method for GenomicRanges objects akin to ranges()... >> > >> >Hi, >> > >> >I have no problem using granges() for that. Just to clarify: >> > (a) it would propagate the names() >> > (b) it would drop the metadata() >> > (c) the mcols() would propagate only if 'use.mcols=TRUE' was >> > specified ('use.mcols' is FALSE by default) >> > (d) it would return a GRanges *instance* i.e. input object 'x' >> > would be downgraded to GRanges if it extends GRanges >> > >> >@Kasper: granges() on SummarizedExperiment ignores the 'use.mcols' >> >arg and always propagates the mcols. Alternatively you can use rowData() >> >which also propagates the mcols. granges() is actually just an alias >> >for rowData() on SummarizedExperiment objects. >> > >> >H. >> > >> > >> >On 05/05/2014 10:31 AM, Kasper Daniel Hansen wrote: >> >> I agree with Michael on this. >> >> >> >> I can see why, in some usage cases, granges() is convenient to have with >> >> use.mcols=FALSE (which seems to have been added in the latest release). >> >> But in my usage of granges(), where I call granges() on objects like >> >> SummarizedExperiments and friends, I have been expecting granges() to give >> >> me the GRange component of the object. Not a crippled version of the >> >> GRange component. >> >> >> >> This is - to me - very counter intuitive and I wish I had seen this >> >> earlier. It is particular frustrating that this default is part of the >> >> generic. >> >> >> >> Best, >> >> Kasper >> >> >> >> >> >> On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence <lawrence.michael at="" gene.com="">> >>> wrote: >> >> >> >>> In my opinion, granges() is not very clear as to the intent. The mcols are >> >>> part of the GRanges, so why would calling granges() drop them? I think we >> >>> want something similar to unclass(), unname(), etc. This why I suggested >> >>> dropmcols(). >> >>> >> >>> >> >>> >> >>> >> >>> On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche at="" gmail.com="">> >>>> wrote: >> >>> >> >>>> That's exactly what I was after -- the generic is already defined, so why >> >>>> not use it? >> >>>> >> >>>> --t >> >>>> >> >>>>> On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring at="" embl.de=""> >> >>>> wrote: >> >>>>> >> >>>>> Hi, >> >>>>> >> >>>>>> On 05.05.2014 16:22, Martin Morgan wrote: >> >>>>>> generalize as setMcols, like setNames? setMcols(x, NULL) >> >>>>> >> >>>>> How about Tim's original suggestion, to add a 'granges' method that >> >>>> works on a 'GRanges' input? The current definition >> >>>>> >> >>>>> granges(x, use.mcols=FALSE, ...) >> >>>>> >> >>>>> seem suited for this. >> >>>>> >> >>>>> Best wishes >> >>>>> Julian >> >>>> >> >>> >> >>> [[alternative HTML version deleted]] >> >>> >> >>> _______________________________________________ >> >>> Bioc-devel at r-project.org mailing list >> >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >>> >> >> >> >> [[alternative HTML version deleted]] >> >> >> >> _______________________________________________ >> >> Bioc-devel at r-project.org mailing list >> >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >> >> > >> >-- >> >Hervé Pagès >> > >> >Program in Computational Biology >> >Division of Public Health Sciences >> >Fred Hutchinson Cancer Research Center >> >1100 Fairview Ave. N, M1-B514 >> >P.O. Box 19024 >> >Seattle, WA 98109-1024 >> > >> >E-mail: hpages at fhcrc.org >> >Phone: (206) 667-5791 >> >Fax: (206) 667-1319 >> > >> >_______________________________________________ >> >Bioc-devel at r-project.org mailing list >> >https://stat.ethz.ch/mailman/listinfo/bioc-devel >> > >-- >Hervé Pagès > >Program in Computational Biology >Division of Public Health Sciences >Fred Hutchinson Cancer Research Center >1100 Fairview Ave. N, M1-B514 >P.O. Box 19024 >Seattle, WA 98109-1024 > >E-mail: hpages at fhcrc.org >Phone: (206) 667-5791 >Fax: (206) 667-1319
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Entering edit mode
On 05/05/2014 02:12 PM, Cook, Malcolm wrote: >> On 05/05/2014 01:00 PM, Cook, Malcolm wrote: > >> Wondering, > >> > >> Is it too off the beaten track to expect > >> > >> mcols<-(x,NULL) > > > > > args(mcols<-) > > function (x, ..., value) > > > >Arguments after the ellipsis must be named: > > > > mcols<-(x, value=NULL) > > Herve - Great - of course - so - does this not provide the means requested by the original poster? I think Tim also wanted 'x' to be downgraded to a GRanges instance, like Julian's grangesPlain() does. We could use granges() for that. Deciding of an idiom that can be used inline for just dropping the mcols would be good too. mcols<-(x, value=NULL) is a little bit tricky, ugly, and error prone as you noticed. These are probably enough reasons for not choosing it as *the* idiom. Its only advantage is that it doesn't introduce a new symbol. H. > > > > >Nothing we can do about this. > > > >Cheers, > >H. > > > >> > >> to work? > >> > >> hint: it does not > >> > >> >-----Original Message----- > >> >From: bioc-devel-bounces at r-project.org [mailto:bioc- devel-bounces at r-project.org] On Behalf Of Hervé Pagès > >> >Sent: Monday, May 05, 2014 1:28 PM > >> >To: Kasper Daniel Hansen; Michael Lawrence > >> >Cc: Johnston, Jeffrey; ttriche at usc.edu; bioc-devel at r-project.org; bioconductor at r-project.org > >> >Subject: Re: [Bioc-devel] [BioC] granges() method for GenomicRanges objects akin to ranges()... > >> > > >> >Hi, > >> > > >> >I have no problem using granges() for that. Just to clarify: > >> > (a) it would propagate the names() > >> > (b) it would drop the metadata() > >> > (c) the mcols() would propagate only if 'use.mcols=TRUE' was > >> > specified ('use.mcols' is FALSE by default) > >> > (d) it would return a GRanges *instance* i.e. input object 'x' > >> > would be downgraded to GRanges if it extends GRanges > >> > > >> >@Kasper: granges() on SummarizedExperiment ignores the 'use.mcols' > >> >arg and always propagates the mcols. Alternatively you can use rowData() > >> >which also propagates the mcols. granges() is actually just an alias > >> >for rowData() on SummarizedExperiment objects. > >> > > >> >H. > >> > > >> > > >> >On 05/05/2014 10:31 AM, Kasper Daniel Hansen wrote: > >> >> I agree with Michael on this. > >> >> > >> >> I can see why, in some usage cases, granges() is convenient to have with > >> >> use.mcols=FALSE (which seems to have been added in the latest release). > >> >> But in my usage of granges(), where I call granges() on objects like > >> >> SummarizedExperiments and friends, I have been expecting granges() to give > >> >> me the GRange component of the object. Not a crippled version of the > >> >> GRange component. > >> >> > >> >> This is - to me - very counter intuitive and I wish I had seen this > >> >> earlier. It is particular frustrating that this default is part of the > >> >> generic. > >> >> > >> >> Best, > >> >> Kasper > >> >> > >> >> > >> >> On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence <lawrence.michael at="" gene.com=""> >> >>> wrote: > >> >> > >> >>> In my opinion, granges() is not very clear as to the intent. The mcols are > >> >>> part of the GRanges, so why would calling granges() drop them? I think we > >> >>> want something similar to unclass(), unname(), etc. This why I suggested > >> >>> dropmcols(). > >> >>> > >> >>> > >> >>> > >> >>> > >> >>> On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. <tim.triche at="" gmail.com=""> >> >>>> wrote: > >> >>> > >> >>>> That's exactly what I was after -- the generic is already defined, so why > >> >>>> not use it? > >> >>>> > >> >>>> --t > >> >>>> > >> >>>>> On May 5, 2014, at 7:42 AM, Julian Gehring <julian.gehring at="" embl.de=""> > >> >>>> wrote: > >> >>>>> > >> >>>>> Hi, > >> >>>>> > >> >>>>>> On 05.05.2014 16:22, Martin Morgan wrote: > >> >>>>>> generalize as setMcols, like setNames? setMcols(x, NULL) > >> >>>>> > >> >>>>> How about Tim's original suggestion, to add a 'granges' method that > >> >>>> works on a 'GRanges' input? The current definition > >> >>>>> > >> >>>>> granges(x, use.mcols=FALSE, ...) > >> >>>>> > >> >>>>> seem suited for this. > >> >>>>> > >> >>>>> Best wishes > >> >>>>> Julian > >> >>>> > >> >>> > >> >>> [[alternative HTML version deleted]] > >> >>> > >> >>> _______________________________________________ > >> >>> Bioc-devel at r-project.org mailing list > >> >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel > >> >>> > >> >> > >> >> [[alternative HTML version deleted]] > >> >> > >> >> _______________________________________________ > >> >> Bioc-devel at r-project.org mailing list > >> >> https://stat.ethz.ch/mailman/listinfo/bioc-devel > >> >> > >> > > >> >-- > >> >Hervé Pagès > >> > > >> >Program in Computational Biology > >> >Division of Public Health Sciences > >> >Fred Hutchinson Cancer Research Center > >> >1100 Fairview Ave. N, M1-B514 > >> >P.O. Box 19024 > >> >Seattle, WA 98109-1024 > >> > > >> >E-mail: hpages at fhcrc.org > >> >Phone: (206) 667-5791 > >> >Fax: (206) 667-1319 > >> > > >> >_______________________________________________ > >> >Bioc-devel at r-project.org mailing list > >> >https://stat.ethz.ch/mailman/listinfo/bioc-devel > >> > > > >-- > >Hervé Pagès > > > >Program in Computational Biology > >Division of Public Health Sciences > >Fred Hutchinson Cancer Research Center > >1100 Fairview Ave. N, M1-B514 > >P.O. Box 19024 > >Seattle, WA 98109-1024 > > > >E-mail: hpages at fhcrc.org > >Phone: (206) 667-5791 > >Fax: (206) 667-1319 > -- Hervé Pagès Program in Computational Biology Division of Public Health Sciences Fred Hutchinson Cancer Research Center 1100 Fairview Ave. N, M1-B514 P.O. Box 19024 Seattle, WA 98109-1024 E-mail: hpages at fhcrc.org Phone: (206) 667-5791 Fax: (206) 667-1319
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Entering edit mode
Hi, In summary, would it be feasible to add to 'GenomicRanges'? 1) A 'granges(x, use.mcols=FALSE, ...)' method with signature 'GRanges' that converts to a 'GRanges' object and optionally drops the mcols (if 'use.mcols' is TRUE) 2) A 'dropMcols' or 'dropmcols' method with signature 'GRanges' that is a wrapper for mcols(x) <- NULL If I can be of help in providing a patch for this, please let me know. Best wishes Julian On 05.05.2014 23:29, Hervé Pagès wrote: > On 05/05/2014 02:12 PM, Cook, Malcolm wrote: >>> On 05/05/2014 01:00 PM, Cook, Malcolm wrote: >> >> Wondering, >> >> >> >> Is it too off the beaten track to expect >> >> >> >> mcols<-(x,NULL) >> > >> > > args(mcols<-) >> > function (x, ..., value) >> > >> >Arguments after the ellipsis must be named: >> > >> > mcols<-(x, value=NULL) >> >> Herve - Great - of course - so - does this not provide the means >> requested by the original poster? > > I think Tim also wanted 'x' to be downgraded to a GRanges instance, > like Julian's grangesPlain() does. We could use granges() for that. > > Deciding of an idiom that can be used inline for just dropping the > mcols would be good too. mcols<-(x, value=NULL) is a little bit > tricky, ugly, and error prone as you noticed. These are probably > enough reasons for not choosing it as *the* idiom. Its only advantage > is that it doesn't introduce a new symbol. > > H. > >> >> > >> >Nothing we can do about this. >> > >> >Cheers, >> >H. >> > >> >> >> >> to work? >> >> >> >> hint: it does not >> >> >> >> >-----Original Message----- >> >> >From: bioc-devel-bounces at r-project.org >> [mailto:bioc-devel-bounces at r-project.org] On Behalf Of Hervé Pagès >> >> >Sent: Monday, May 05, 2014 1:28 PM >> >> >To: Kasper Daniel Hansen; Michael Lawrence >> >> >Cc: Johnston, Jeffrey; ttriche at usc.edu; >> bioc-devel at r-project.org; bioconductor at r-project.org >> >> >Subject: Re: [Bioc-devel] [BioC] granges() method for >> GenomicRanges objects akin to ranges()... >> >> > >> >> >Hi, >> >> > >> >> >I have no problem using granges() for that. Just to clarify: >> >> > (a) it would propagate the names() >> >> > (b) it would drop the metadata() >> >> > (c) the mcols() would propagate only if 'use.mcols=TRUE' was >> >> > specified ('use.mcols' is FALSE by default) >> >> > (d) it would return a GRanges *instance* i.e. input object 'x' >> >> > would be downgraded to GRanges if it extends GRanges >> >> > >> >> >@Kasper: granges() on SummarizedExperiment ignores the >> 'use.mcols' >> >> >arg and always propagates the mcols. Alternatively you can use >> rowData() >> >> >which also propagates the mcols. granges() is actually just an >> alias >> >> >for rowData() on SummarizedExperiment objects. >> >> > >> >> >H. >> >> > >> >> > >> >> >On 05/05/2014 10:31 AM, Kasper Daniel Hansen wrote: >> >> >> I agree with Michael on this. >> >> >> >> >> >> I can see why, in some usage cases, granges() is convenient >> to have with >> >> >> use.mcols=FALSE (which seems to have been added in the >> latest release). >> >> >> But in my usage of granges(), where I call granges() on >> objects like >> >> >> SummarizedExperiments and friends, I have been expecting >> granges() to give >> >> >> me the GRange component of the object. Not a crippled >> version of the >> >> >> GRange component. >> >> >> >> >> >> This is - to me - very counter intuitive and I wish I had >> seen this >> >> >> earlier. It is particular frustrating that this default is >> part of the >> >> >> generic. >> >> >> >> >> >> Best, >> >> >> Kasper >> >> >> >> >> >> >> >> >> On Mon, May 5, 2014 at 12:11 PM, Michael Lawrence >> <lawrence.michael at="" gene.com="">> >> >>> wrote: >> >> >> >> >> >>> In my opinion, granges() is not very clear as to the >> intent. The mcols are >> >> >>> part of the GRanges, so why would calling granges() drop >> them? I think we >> >> >>> want something similar to unclass(), unname(), etc. This >> why I suggested >> >> >>> dropmcols(). >> >> >>> >> >> >>> >> >> >>> >> >> >>> >> >> >>> On Mon, May 5, 2014 at 8:17 AM, Tim Triche, Jr. >> <tim.triche at="" gmail.com="">> >> >>>> wrote: >> >> >>> >> >> >>>> That's exactly what I was after -- the generic is already >> defined, so why >> >> >>>> not use it? >> >> >>>> >> >> >>>> --t >> >> >>>> >> >> >>>>> On May 5, 2014, at 7:42 AM, Julian Gehring >> <julian.gehring at="" embl.de=""> >> >> >>>> wrote: >> >> >>>>> >> >> >>>>> Hi, >> >> >>>>> >> >> >>>>>> On 05.05.2014 16:22, Martin Morgan wrote: >> >> >>>>>> generalize as setMcols, like setNames? setMcols(x, NULL) >> >> >>>>> >> >> >>>>> How about Tim's original suggestion, to add a 'granges' >> method that >> >> >>>> works on a 'GRanges' input? The current definition >> >> >>>>> >> >> >>>>> granges(x, use.mcols=FALSE, ...) >> >> >>>>> >> >> >>>>> seem suited for this. >> >> >>>>> >> >> >>>>> Best wishes >> >> >>>>> Julian >> >> >>>> >> >> >>> >> >> >>> [[alternative HTML version deleted]] >> >> >>> >> >> >>> _______________________________________________ >> >> >>> Bioc-devel at r-project.org mailing list >> >> >>> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >> >>> >> >> >> >> >> >> [[alternative HTML version deleted]] >> >> >> >> >> >> _______________________________________________ >> >> >> Bioc-devel at r-project.org mailing list >> >> >> https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >> >> >> >> > >> >> >-- >> >> >Hervé Pagès >> >> > >> >> >Program in Computational Biology >> >> >Division of Public Health Sciences >> >> >Fred Hutchinson Cancer Research Center >> >> >1100 Fairview Ave. N, M1-B514 >> >> >P.O. Box 19024 >> >> >Seattle, WA 98109-1024 >> >> > >> >> >E-mail: hpages at fhcrc.org >> >> >Phone: (206) 667-5791 >> >> >Fax: (206) 667-1319 >> >> > >> >> >_______________________________________________ >> >> >Bioc-devel at r-project.org mailing list >> >> >https://stat.ethz.ch/mailman/listinfo/bioc-devel >> >> >> > >> >-- >> >Hervé Pagès >> > >> >Program in Computational Biology >> >Division of Public Health Sciences >> >Fred Hutchinson Cancer Research Center >> >1100 Fairview Ave. N, M1-B514 >> >P.O. 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http://golem.ph.utexas.edu/category/2013/03/index.shtml | ## March 30, 2013
### The Convex Magnitude Conjecture
#### Posted by Tom Leinster
For a finite subset $B = \{b_1, \ldots, b_n\}$ of $\mathbb{R}^N$, let $Z$ be the $n \times n$ matrix with $(i, j)$-entry $e^{-\left|b_i - b_j\right|}$, and define $|B|$ to be the sum of all $n^2$ entries of $Z^{-1}$.
For a compact subset $A$ of $\mathbb{R}^N$, define $|A|$ to be the supremum of $|B|$ over all finite subsets $B$ of $A$.
The 2-dimensional case of the convex magnitude conjecture states that for all compact convex $A \subseteq \mathbb{R}^2$,
$|A| = \chi(A) + \frac{1}{4}perimeter(A) + \frac{1}{2\pi}area(A).$
I just came back from the British Mathematical Colloquium in Sheffield, where I spoke about the convex magnitude conjecture and attempts to settle it.
Click the picture for slides.
Posted at 5:48 PM UTC | Permalink | Followups (2)
## March 27, 2013
### Categories for the Working Metaphysician
#### Posted by David Corfield
This is the great title suggested by my colleague George Darby for the workshop I’m organising, currently going by the name ‘What Can Category Theory Do For Philosophy’? See the website, and follow instructions there if you would like to attend.
I know it clashes with Category Theory 2013, but this was unavoidable. Perhaps we can watch videos from Sydney in the evenings.
Posted at 10:00 AM UTC | Permalink | Followups (1)
## March 24, 2013
### Project Scheduling and Copresheaves
#### Posted by Simon Willerton
Obsessed as I am with finding examples of enriched categories, I was struck by the unfamiliar notion of ‘activity network’ when checking a colleague’s course on graph theory. What struck me was how scheduling actives seemed to be like finding a certain kind of copresheaf. Activity networks are also known as PERT charts to people in project management: an activity network or PERT chart records all the dependencies between the activities need to be completed during a project, and also records the times the activities will take to complete. Once you have this information you can try to schedule each of the activities in so that the project is completed as quickly as possible, and you can identify which activities are critical in the sense that any delay in that activity will cause a delay to the whole project.
I noticed that there was some enriched category theory going on with these PERT charts. Patrons of the Café are probably familiar with the idea that metric spaces can be thought of as categories enriched over $([0,\infty],\ge,+)$ the category of extended non-negative numbers. The triangle inequality for metric spaces then corresponds to the composition for enriched categories. The category that is enriched over for PERT charts is $(\{-\infty\}\cup[0,\infty],\le,+)$ so the main difference is that the morphisms go the other way, this means that you get a reversed triangle inequality! However, that makes sense in the scheduling context. I will explain how this gives categorical interpretations of terms like earliest start time, latest start time, float and critical path.
I have stressed the project management aspects here, but these ideas are also important, it seems, in scheduling processor jobs in parallel computing.
Posted at 5:47 PM UTC | Permalink | Followups (18)
## March 22, 2013
### Five Positions at Edinburgh
#### Posted by Tom Leinster
The School of Mathematics at Edinburgh is advertising five Chancellor’s Fellowships.
We’re interested in people in pretty much all areas of pure and applied mathematics, statistics and operational research. One position is reserved for an algebraist (“Representation Theory, very broadly conceived”).
The positions are five-year fellowships on a reduced teaching load. The idea is that you do mostly research in the first couple of years, and the teaching load is gradually increased until it gets up to the ordinary level by the end of the five years. At that point, if you pass a review, the position converts into a standard “permanent” job. Of course, you shouldn’t take my word for it: the official conditions are on the web.
I’ve now been in Edinburgh for six months, and I highly recommend it: it’s a beautiful city and a great department.
## March 21, 2013
### Mark on Magnitude
#### Posted by Tom Leinster
Here’s a truly superb talk by Mark Meckes, on The magnitude of metric spaces.
The talk was at Banff last week, as part of a meeting on The interplay of convex geometry and Banach space theory.
Maybe I was predisposed to like the talk, its subject matter being so close to my heart. But I was awed by how well thought-out it was: it’s just incredibly clear. And he goes from nothing to the cutting edge in under half an hour, without appearing to hurry. If you’re on the internet looking for diversion, look no further!
Posted at 1:12 AM UTC | Permalink | Followups (6)
## March 6, 2013
### Lectureship in Sheffield
#### Posted by Simon Willerton
We have a permanent lectureship position in Sheffield associated with the return of Tom Bridgeland after he spent some time in Oxford.
Candidates should possess a strong track record of high quality research in algebraic geometry, very broadly conceived, or any other area linking with Professor Bridgeland’s interests.
For those of you who don’t know him, Tom works in algebraic geometry and its connections with string theory and mirror symmetry, in particular he’s interested in properties of derived categories of coherent sheaves on algebraic varieties. It will be good to have him back in the department!
## March 5, 2013
### Category Theory in Homotopy Type Theory
#### Posted by Mike Shulman
Benedikt Ahrens, Chris Kapulkin, and I have just posted the following preprint:
This is mainly a development of basic (1-)category theory using homotopy type theory (a.k.a. “univalent foundations”) as the foundational system. So for all of you readers who’ve been enjoying the posts with vague waffly discussions of type theory, and longing for something to sink your teeth into, this may be a good start.
Posted at 4:04 AM UTC | Permalink | Followups (24)
## March 3, 2013
### Spivak on Category Theory
#### Posted by Simon Willerton
Guest post by Bruce Bartlett
We know about Category Theory for Mathematicians, we’ve all read Category Theory for Physicists, and we also know about Category Theory for Computer Scientists, and we’ve even seen the videos.
But how about Category Theory for Scientists? I spotted this on the arXiv listings.
David Spivak, Category Theory for Scientists.
Abstract: There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be applied throughout the sciences as a framework for modeling phenomena and communicating results. In order to target the scientific audience, this book is example-based rather than proof-based. For example, monoids are framed in terms of agents acting on objects, sheaves are introduced with primary examples coming from geography, and colored operads are discussed in terms of their ability to model self-similarity.
Posted at 10:29 PM UTC | Permalink | Followups (11) | 2013-12-13 01:04:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 19, "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": 0.4978460669517517, "perplexity": 1131.562999706526}, "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/1386164786099/warc/CC-MAIN-20131204134626-00044-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://imathworks.com/tex/tex-latex-revtex4-1-and-algorithm2e-indentation-clash/ | # [Tex/LaTex] Revtex4-1 and algorithm2e indentation clash
algorithm2erevtex
there seems to exists a clash between revtex4-1 and the algorithm2e package. Whenever I try to insert an algorithm block when using the revtex4-1 document class, the tabulation is not working properly. The following latex code should hopefully be self explanatory.
%\documentclass[10]{article}
\documentclass[10]{revtex4-1}
% these are to avoid another clash between revtex and algorithm2e
\makeatletter
\newif\if@restonecol
\makeatother
\let\algorithm\relax
\let\endalgorithm\relax
% needed includes
\usepackage[ruled]{algorithm2e}
\usepackage{algorithmic}
\begin{document}
The following pseudocode exibits a clash between revtex4-1 and algorithm2e.
\begin{algorithm}
\caption{Calculate something}
\begin{algorithmic}
\FOR{every thing you}
\FOR{every step you}
\STATE{compute something}
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\end{document}
This is a how the rendered pdf shows on my system (sorry, I'm not allowed to insert images in this post). The tabulation is correct when using the simple article document class. I am using Texmaker (with texLive) under Ubuntu 10.04.
Using the revtex4-1 document class:
for every thing you do
for every step you do
compute something
end for
end for
Using the article document class:
for every thing you do
for every step you do
compute something
end for
end for
Anybody with a solution to make these two work together?
There is an incompatibility between the algorithm2e/algorithmic packages and revtex4-1; one possibility is to use algcompatible instead of algorithmic (the syntax of the commands is the same and this solves the indentation problem) and to use the newfloat package to define a new algorithm float. Here's an example illustrating this approach (I used a table and a figure environments to show how the newly defined float behaves consistently with the class):
\documentclass{revtex4-1}
\usepackage{algcompatible}
\usepackage{newfloat}
\DeclareFloatingEnvironment[
fileext=loa,
listname=List of Algorithms,
name=ALGORITHM,
placement=tbhp,
]{algorithm}
\begin{document}
\begin{figure}
\rule{2cm}{2cm}
\caption{A test caption for a figure}
\end{figure}
\begin{table}
\rule{2cm}{2cm}
\caption{A test caption for a table}
\end{table}
\begin{algorithm}
\begin{algorithmic}
\FOR{every thing you}
\FOR{every step you}
\STATE{compute something}
\ENDFOR
\ENDFOR
\end{algorithmic}
\caption{Calculate something}
\end{algorithm}
\end{document}
If one wants to keep the ruled style of the algorithm environmet, as defined in algorithm2e, some additional work has to be done; in this case, the caption package could be used (providing one makes the necessary adjustments to recover the formatting defined in revtex4-1); here's such possibility with a little example:
\documentclass{revtex4-1}
\usepackage{newfloat,algcompatible}
\usepackage[size=small]{caption}
\usepackage{etoolbox}
\AtBeginEnvironment{algorithm}{\noindent\hrulefill\par\nobreak\vskip-5pt}
\usepackage{newfloat}
\DeclareFloatingEnvironment[
fileext=loa,
listname=List of Algorithms,
name=ALGORITHM,
placement=tbhp,
]{algorithm}
\DeclareCaptionFormat{algorithms}{\vskip-15pt\hrulefill\par#1#2#3\vskip-6pt\hrulefill}
\captionsetup[algorithm]{singlelinecheck=off,format=algorithms}
\begin{document}
\begin{figure}
\rule{2cm}{2cm}
\caption{A test caption for a figure}
\end{figure}
\begin{algorithm}
\begin{algorithmic}
\FOR{every thing you}
\FOR{every step you}
\STATE{compute something}
\ENDFOR
\ENDFOR
\end{algorithmic}
\caption{Calculate something}
\end{algorithm}
\end{document}
If, for some reason, the newfloat packageis not available, one can use the trivfloat instead:
\documentclass{revtex4-1}
\usepackage{algcompatible}
\usepackage[floatrow]{trivfloat}
\trivfloat{algorithm}
\renewcommand\algorithmname{ALGORITHM}
\begin{document}
\begin{figure}
\rule{2cm}{2cm}
\caption{A test caption for a figure}
\end{figure}
\begin{table}
\rule{2cm}{2cm}
\caption{A test caption for a table}
\end{table}
\begin{algorithm}
\begin{algorithmic}
\FOR{every thing you}
\FOR{every step you}
\STATE{compute something}
\ENDFOR
\ENDFOR
\end{algorithmic}
\caption{Calculate something}
\end{algorithm}
\end{document} | 2023-02-05 05:34:27 | {"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": 0.9712747931480408, "perplexity": 6503.189130754882}, "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/1674764500215.91/warc/CC-MAIN-20230205032040-20230205062040-00724.warc.gz"} |
https://wiki.seg.org/wiki/Stolt_migration | # Stolt migration
Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store
If the medium velocity is constant, migration can be expressed as a direct mapping [1] from temporal frequency ω to vertical wavenumber kz (Figure 4.1-25). Figure 4.1-30 is a flowchart of the Stolt algorithm; the mathematical details are left to Section D.7. The equation for Stolt mapping is
${\displaystyle k_{z}=\mp {\sqrt {{\frac {\omega ^{2}}{v^{2}}}-k_{x}^{2}}},}$ (13a)
${\displaystyle P(k_{x},k_{z},t=0)=\left[{\frac {v}{2}}{\frac {k_{z}}{\sqrt {k_{x}^{2}+k_{z}^{2}}}}\right]P\left[k_{x},0,\omega ={\frac {v}{2}}{\sqrt {k_{y}^{2}+k_{z}^{2}}}\,\right],}$ (24a)
where P(kx, z = 0, ω) is the zero-offset section and P(kx, kz, t = 0)is the migrated section in the frequency-wavenumber domain.
Note that Stolt migration involves, first, mapping from ω to kz for a specific kx by using the dispersion relation of equation (13a) recast as
${\displaystyle \omega ={\frac {v}{2}}{\sqrt {k_{x}^{2}+k_{z}^{2}}}.}$ (24b)
The output of mapping is then scaled by the quantity S
${\displaystyle S={\frac {v}{2}}{\frac {k_{z}}{\sqrt {k_{x}^{2}+k_{z}^{2}}}}.}$ (24c)
Stolt’s algorithm for constant velocity thus involves the following steps:
1. Start with the input wavefield P(x, z = 0, t) approximated by the CMP stack, and apply 2-D Fourier transform to get P(kx, z = 0, ω).
2. Map the wavefield from ω to kz using the dispersion relation given by equation (24b).
3. Apply the scaling factor S of equation (24c) as part of the mapping procedure (Section D.7).
4. Invoke the imaging principle by setting t = 0 and obtain P(kx, kz, t = 0).
5. Finally, apply 2-D inverse transform to get the migrated section P(x, z, t = 0).
It may be questionable as to whether the constant-velocity Stolt method has value on its own as a practical migration algorithm. Nevertheless, Stolt’s method can be used efficiently to perform a constant-velocity migration as the first step in a residual migration scheme (frequency-wavenumber migration in practice). Additionally, the method constitutes an essential procedural step for migration velocity analysis as described in migration velocity analysis.
Stolt extended his method to handle velocity variations (Section D.7). For the variable-velocity case, Stolt’s extension consists of
1. modifying the input wavefield to make it appear as if it were the response of a constant-velocity earth,
2. applying the constant-velocity algorithm outlined in Figure 4.1-30, and
3. reversing the original modification of the input wavefield.
This modification essentially is a type of stretching of the time axis (Section D.7) to make the reflection times approximately equivalent to those recorded for a constant-velocity earth. The nature of stretching is described by the stretch factor W. The constant-velocity case is equivalent to W = 1.
Note that the phase-shift and Stolt migration outputs normally are displayed in two-way vertical zero-offset time τ = 2z/v, as are the outputs from the finite-difference and Kirchhoff migrations. In practice, mapping in the f − k domain really is from ω − kx to ωτ − kx rather than to kz − kx, where ωτ is the Fourier dual of τ and is simply kz of equation (13b) scaled by v/2 (Section D.3):
${\displaystyle k_{z}={\frac {2\omega }{v}}{\sqrt {1-\left({\frac {vk_{x}}{2\omega }}\right)^{2}}},}$ (13b)
${\displaystyle \omega _{\tau }=\omega {\sqrt {1-\left({\frac {vk_{x}}{2\omega }}\right)^{2}}}.}$ (25)
One important concept must be pointed out from equation (25). Note that for a constant kx, ωτ < ω; thus, migration shifts the bandwidth to lower frequencies. This is analogous to the conclusion derived in relation to the NMO correction, since the latter also causes data stretching to lower frequencies (normal moveout). The implication from equation (25) is demonstrated by the dipping events model in Figure 4.1-24. While the bandwidth of the zero-dip event is retained after migration, the bandwidth of the event with steepest dip has shifted from approximately 40 Hz to 36 Hz at the high-frequency end of the spectrum. In fact, the shift in bandwidth is dip-dependent; events with different dips which have the same bandwidth before migration will have different bandwidths after migration.
## References
1. Stolt (1978), Stolt, R.H., 1978, Migration by Fourier transform: Geophysics, 43, 23–48.
2. Chun, J.H. and Jacewitz, C., 1981, Fundamentals of frequency-domain migration: Geophysics, 46, 717-732. | 2021-06-13 05:06:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.8343153595924377, "perplexity": 3620.978631580663}, "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/1623487600396.21/warc/CC-MAIN-20210613041713-20210613071713-00376.warc.gz"} |
https://www.projecteuclid.org/euclid.aop/1176991578 | ## The Annals of Probability
### Measuring Close Approaches on a Brownian Path
#### Abstract
Integral tests are found for the uniform escape rate of a $d$-dimensional Brownian path $(d \geq 4)$, i.e., for the lower growth rate of $\inf\{|X(t) - X(s)|: 0 \leq s, t \leq 1, |t - s| \geq h\}$ as $h \downarrow 0$. The gap between this uniform escape rate and the one-sided local escape rate of Dvoretsky and Erdos and the two-sided local escape rate of Jain and Taylor suggest the study of certain sets of times of slow one- or two-sided escape. The Hausdorff dimension of these exceptional sets is computed. The results are proved for a broad class of strictly stable processes.
#### Article information
Source
Ann. Probab., Volume 16, Number 4 (1988), 1458-1480.
Dates
First available in Project Euclid: 19 April 2007
https://projecteuclid.org/euclid.aop/1176991578
Digital Object Identifier
doi:10.1214/aop/1176991578
Mathematical Reviews number (MathSciNet)
MR958197
Zentralblatt MATH identifier
0659.60113
JSTOR | 2020-01-24 02:52:01 | {"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": 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": 0.7256479859352112, "perplexity": 1539.3917797217523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250614880.58/warc/CC-MAIN-20200124011048-20200124040048-00296.warc.gz"} |
https://mathematica.stackexchange.com/questions/256544/expectation-not-working-for-relatively-simple-expression-under-lognormal-distr/256605 | # Expectation[] not working for relatively simple expression under lognormal distribution
I am trying to find the expectation of a relatively simple expression under a lognormal distribution with mean [Mu] - [Sigma]^2/2 and variance [Sigma]^2 , but it hangs for a long while and eventually returns the original question:
In[36]:= Expectation[(2 Sqrt[x])/(x + 1) - 1,
x \[Distributed]
LogNormalDistribution[ (\[Mu] - \[Sigma]^2/2), \[Sigma]]]
Out[36]= Expectation[-1 + (2 Sqrt[x])/(1 + x),
x \[Distributed]
LogNormalDistribution[\[Mu] - \[Sigma]^2/2, \[Sigma]]]
I have tried putting an Abs[] around the Sqrt[] to only take the positive root just in case that was the issue, without success. I have also tried adding Assumptions to the effect that x>0, [Sigma] > 0 but that made no difference.
Any pointers greatly appreciated!
• You are effectively asking Mathematica to integrate: $$\int_{0}^{\infty}{\frac{\left(\frac{2 \sqrt{x}}{x+1}-1\right) e^{-\frac{\left(-2 \mu +\sigma ^2+2 \log (x)\right)^2}{8 \sigma ^2}}}{\sqrt{2 \pi } \sigma x}}dx$$ This integral is too difficult so it's giving up. Oct 6, 2021 at 10:24
• Just curious: are you sure about wanting LogNormalDistribution[\[Mu] - \[Sigma]^2/2, \[Sigma]] ? I ask because the mean of that distribution is $e^\mu$ and the variance is $\left(e^{\sigma^2}-1\right) e^{2 \left(m-\frac{\sigma^2}{2}\right)+\sigma^2}$ (not $\sigma^2$). In other words your first sentence talks about the two values listed as if those are the mean and variance. But the way Mathematica parameterizes things, those are just the two parameters.
– JimB
Oct 6, 2021 at 16:14
This can be done numerically:
f[\[Mu]_?NumericQ, \[Sigma]_?NumericQ] := NExpectation[(2 Sqrt[x])/(x + 1) - 1,
x \[Distributed] LogNormalDistribution[ (\[Mu] - \[Sigma]^2/2), \[Sigma]]];
f[1, 2]
-0.295453
• +1 For formatting tools see link Oct 6, 2021 at 13:22
Many thanks for all the helpful comments. On the variance, if you don't subtract Sigma^2/2 then the mean will not be mu. The variance I meant is the variance of log returns not the variance of returns, so that is OK. I tried the numerical integration and it gives me useful results which I can plot and make sense of, which is very helpful. An analytical solution would be even better if I could get one. Is there any way to tease an analytic answer out for the hard integral by breaking it up? I tried but it seems to give up as "too difficult" very early. For example, it can do the integral for 1/x, but it already starts to choke on 1/(1+x), which doesn't seem particularly complex...
• This information should be added in your original question by editing it, or as a comment on another answer if appropriate. "Answers" on this site are not used to continue a conversation, but are really reserved for proposing a solving approach to the question. Oct 7, 2021 at 12:15 | 2022-05-20 09:49:16 | {"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": 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": 0.7924525141716003, "perplexity": 998.992340859883}, "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/1652662531779.10/warc/CC-MAIN-20220520093441-20220520123441-00325.warc.gz"} |
https://emacs.stackexchange.com/questions/52257/how-do-i-have-tables-side-by-side | # How do I have tables side by side?
I'd like to have small tables being shown side by side instead of one under the other, when I export to PDF.
Is there an easy way to do that ? Like a break between tables ?
PDF export is handled by LaTeX, which provides endless ways to customize your document presentation. A relatively simple way to apply side-by-side formatting to tables in org-mode is to wrap them in minipage environments:
#+ATTR_LATEX: :options {0.4\textwidth}
#+begin_minipage
| one | two | three |
|-----+-----+-------|
| cat | dog | fish |
|-----+-----+-------|
#+end_minipage
#+ATTR_LATEX: :options {0.4\textwidth}
#+begin_minipage
| Something | Else |
|-----------+----------|
| Row 1 | Column 2 |
#+end_minipage
Here, each table is wrapped in its own minipage, and the minipages are set to be 40% of the width of a text line. IMPORTANT: don't leave a space between the two minipages!
The options are limitless, but the more sophisticated you get the trickier it is to mix org and LaTeX together. For more than simple tweaks I usually find it's easier to use LaTeX for the full document, or at least the full table.
• Super cool, I only need a short tables side to side to compare some results ! Where can I learn about that kind of stuff though? I feel like the manual goes over Latex pretty fast and I'd like to know more about how to format tables as I like. Last question, how can I add padding before the minipage ? For now it's super close to the text and I can't seem to find the option for a a skip or a pad. Thanks ! – Nathan Furnal Aug 21 '19 at 21:37
• LaTeX is a huge subject, and there are full books devoted to different parts of the language. A good intro is [pdf] ctan.org/tex-archive/info/lshort/english/lshort.pdf – Tyler Aug 21 '19 at 22:28
• try \vspace{1em} on a line by itself, with a blank line before and after, where you want to insert some vertical space. If that's not enough, try a larger number – Tyler Aug 21 '19 at 22:30
• I meant learning about org-mode and latex integration. I've already written some stuff in Latex but never had to write new macros or sty files. I'd like to add some autocomplete and easy Tab complete (if that's a word) to my org mode so that I can easily input short tables. I don't know any ELisp though :s – Nathan Furnal Aug 21 '19 at 22:40
• I see. What you're talking about can be done in Emacs and doesn't really require much LaTeX knowledge, mostly. Certainly not writing sty files or macros. My advice is to pick one discrete thing you want to do. Google around a bit, and when you get stuck come back with a question for us. If you want to autocomplete complex structures, take a look at snippets or skeletons. – Tyler Aug 21 '19 at 23:46 | 2020-09-25 14:22:26 | {"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": 0.7479217648506165, "perplexity": 1418.9966224149252}, "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/1600400226381.66/warc/CC-MAIN-20200925115553-20200925145553-00323.warc.gz"} |
https://www.nag.com/numeric/nl/nagdoc_latest/clhtml/s/s18ctc.html | # NAG CL Interfaces18ctc (bessel_i1_scaled_vector)
Settings help
CL Name Style:
## 1Purpose
s18ctc returns an array of values of the scaled modified Bessel function ${e}^{-|x|}{I}_{1}\left(x\right)$.
## 2Specification
#include
void s18ctc (Integer n, const double x[], double f[], NagError *fail)
The function may be called by the names: s18ctc, nag_specfun_bessel_i1_scaled_vector or nag_bessel_i1_scaled_vector.
## 3Description
s18ctc evaluates an approximation to ${e}^{-|{x}_{i}|}{I}_{1}\left({x}_{i}\right)$, where ${I}_{1}$ is a modified Bessel function of the first kind for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. The scaling factor ${e}^{-|x|}$ removes most of the variation in ${I}_{1}\left(x\right)$.
The function uses the same Chebyshev expansions as s18atc, which returns an array of the unscaled values of ${I}_{1}\left(x\right)$.
## 4References
NIST Digital Library of Mathematical Functions
## 5Arguments
1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: ${e}^{-|{x}_{i}|}{I}_{1}\left({x}_{i}\right)$, the function values.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
## 7Accuracy
Relative errors in the argument are attenuated when propagated into the function value. When the accuracy of the argument is essentially limited by the machine precision, the accuracy of the function value will be similarly limited by at most a small multiple of the machine precision.
## 8Parallelism and Performance
s18ctc is not threaded in any implementation.
None.
## 10Example
This example reads values of x from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.
### 10.1Program Text
Program Text (s18ctce.c)
### 10.2Program Data
Program Data (s18ctce.d)
### 10.3Program Results
Program Results (s18ctce.r) | 2021-09-17 09:32:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 21, "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": 0.9138671159744263, "perplexity": 2725.186248555111}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055632.65/warc/CC-MAIN-20210917090202-20210917120202-00519.warc.gz"} |
https://www.autoitscript.com/forum/topic/39641-scite-is-slow/ | Followers 0
# Scite is slow
## 6 posts in this topic
I have tried to delete folder, install new download version = same issue
Uninstalled scite and deleted both the beta copy and the root one and installed it again - same issue
I run the same code on my work computer and no issues -
Here is what I see
>"C:\Program Files\AutoIt3\SciTE\AutoIt3Wrapper\AutoIt3Wrapper.exe" /run /beta /ErrorStdOut /in "D:\new10907\speedtest.au3" /autoit3dir "C:\Program Files\AutoIt3\beta" /UserParams
+> Starting AutoIt3Wrapper v.1.7.5
>Running AU3Check (1.54.4.0) params: from:C:\Program Files\AutoIt3\beta
+>AU3Check ended.rc:0
>Running:(3.2.1.12):C:\Program Files\AutoIt3\beta\autoit3.exe "D:\new10907\speedtest.au3"
+>AutoIT3.exe ended.rc:0
>Exit code: 0 Time: 16.012
and this is the code I used to run - alt F5 and F5 - do you see the time it took? It shows around 3000 as the output but takes over 10 + seconds to show the MsgBox() - what I have I done?
$begin = TimerInit() sleep(3000)$dif = TimerDiff($begin) MsgBox(0,"Time Difference",$dif)
All by me:
"Sometimes you have to go back to where you started, to get to where you want to go."
"Everybody catches up with everyone, eventually"
"As you teach others, you are really teaching yourself."
"Do not worry about yesterday, as the only thing that you can control is tomorrow."
Programming Tips
Excel Changes
ControlHover.UDF
GDI_Plus
Draw_On_Screen
GDI Basics
GDI_More_Basics
GDI Rotate
GDI Graph
GDI CheckExistingItems
GDI Trajectory
Replace $ghGDIPDll with$__g_hGDIPDll
DLL 101?
Array via Object
GDI Swimlane
GDI Plus French 101 Site
GDI Examples UEZ
GDI Basic Clock
GDI Detection
# Ternary operator
##### Share on other sites
I have 2 words for you...System restore
if that doesn't work, you can always set it back without any harm.
##### Share on other sites
I have 2 words for you...System restore
if that doesn't work, you can always set it back without any harm.
I tried 3 different restore points - one about a week ago and the other one from a month ago. I was going to go back further but then found that there was only one more day I could go back to, so I tried it again. Nothing different. All my other programs seem to be fine - not acting slow or anything. I can get online and have no issues with speed. In fact my scite program seems to be fine. It does not do anything until I try to run the script - in need to a speed boost. Is there anything that I can turn off to make it faster?
All by me:
"Sometimes you have to go back to where you started, to get to where you want to go."
"Everybody catches up with everyone, eventually"
"As you teach others, you are really teaching yourself."
"Do not worry about yesterday, as the only thing that you can control is tomorrow."
Programming Tips
Excel Changes
ControlHover.UDF
GDI_Plus
Draw_On_Screen
GDI Basics
GDI_More_Basics
GDI Rotate
GDI Graph
GDI CheckExistingItems
GDI Trajectory
Replace $ghGDIPDll with$__g_hGDIPDll
DLL 101?
Array via Object
GDI Swimlane
GDI Plus French 101 Site
GDI Examples UEZ
GDI Basic Clock
GDI Detection
# Ternary operator
##### Share on other sites
I tried 3 different restore points - one about a week ago and the other one from a month ago. I was going to go back further but then found that there was only one more day I could go back to, so I tried it again. Nothing different. All my other programs seem to be fine - not acting slow or anything. I can get online and have no issues with speed. In fact my scite program seems to be fine. It does not do anything until I try to run the script - in need to a speed boost. Is there anything that I can turn off to make it faster?
Don't mean to state the obvious, but it's not working right, so what good is it going to do to try to make it run faster? Very strange issue though... You might consider doing a bit of system analysis using tools from majorgeeks.com or sysinternals.com just to make sure it's not something else. Of course, it'd be good to ensure the pc is clean of spyware and whatnot as well.
Speaking of, perhaps your Antivirus is freaking out about autoit, have you tried disablig it while running?
anyway, just some ideas.
You could always rebuild
##### Share on other sites
Scite played up on me the other week and was painfully slow at checking, compiling and running a script (I mean 10 mins to check a 20 line script slow). Tried reboot, reinstall etc., nothing worked. It wasn't until I realised that I had moved my include library and had forgotten to update the scite config that I found the problem and it worked like a charm again. It maybe worth re-checking the config settings again.
##### Share on other sites
Maybe check the amount of RAM you are using, it could be broken or you need to get some more. I have had the same problem as you but with another program. | 2017-05-27 08:24:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.23278646171092987, "perplexity": 2058.169616684912}, "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/1495463608877.60/warc/CC-MAIN-20170527075209-20170527095209-00428.warc.gz"} |
https://www.physicsforums.com/threads/best-book-to-learn-more-about-quantum-mechanics.730321/ | 1. Dec 28, 2013
### ilikescience94
I have a pretty basic understanding of the workings of quantum mechanics, and would like to go more in depth by learning the formulas, applications, etc. What would be the best book/lecture course/ online resource you guys know of in which I could better understand QED, QCD, and QM in general?
2. Dec 28, 2013
### R136a1
There are many books out there which are very good. For example, Galindo & Pascual, Hall, Zettili, Griffiths, Sakurai,...
Basically, anything but Ballentine should be good for you.
3. Dec 28, 2013
### atyy
The most important concept for field theory that isn't always evident in textbooks is the Wilsonian viewpoint, which explains why renormalization is not a bizarre process of subtracting infinities.
http://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/
Historically, the Wilsonian insight came from investigating critical phenomena in classical statistical mechanics, which is mathematically analogous to the path integral formulation of quantum field theory. I still find that this a nice way to learn the Wilsonian viewpoint.
http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/ [Broken]
The other thing I wish were in standard texts is the Osterwalder-Schrader conditions, which tell you why a quantum field theory (with Hamiltonian and Hilbert Space) can be formulated as a statistical theory in Euclidean space. When you see a "Wick rotation" in standard texts, they are making use of this ability to rotate into Euclidean space, calculate there, and then rotate back.
http://www.einstein-online.info/spotlights/path_integrals
Other than that, any textbook is good, eg. Mandl and Shaw, or Srednicki.
Free notes online:
http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf
http://www.damtp.cam.ac.uk/user/dt281/qft.html
http://web.physics.ucsb.edu/~mark/qft.html
http://www.solvayinstitutes.be/events/doctoral/Bilal.pdf [Broken]
Last edited by a moderator: May 6, 2017
4. Dec 28, 2013
### R136a1
Maybe you need to tell us first what mathematics you know? Are you comfortable with calculus? Diff eq? LA?
5. Dec 28, 2013
### dextercioby
Nice comment. Please, see the link to the (fortunately free) research article: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103858969
6. Dec 28, 2013
### WannabeNewton
What's wrong with Ballentine
7. Dec 28, 2013
### R136a1
It's an horrible book. Simply horrible.
First, it tries to be mathematically rigorous and totally botches the job so it confuses people even more. It mentions there's a difference between self-adjoint and hermitian, but then later totally ignores the difference. It spends a paragraph on rigged hilbert spaces but they sadly don't show up later where they can be actually useful. His approach of the spectral theorem suffers the same defects. Why confuse readers with mathematical rigor if he isn't going to be rigorous in the sequel anyway?? He actually does things like "assume this is continuous, let's take the power series expansion". This is fine for QM books which don't claim to be rigorous, but ballentine tries to be rigorous.
Second, his exercises are way too easy. And they are way too unphysical.
He also spends way too much time on the ensemble interpretation, which is totally useless and outdated. Just teach how to calculate stuff and leave the interpretations to philosophy books.
8. Dec 28, 2013
### TumblingDice
Hey here WbN! I was hoping you'd catch this, only because I've been preparing to pick up Ballentine based on the good things I've read here on PF. Now I will sit back and watch posts to get a better idea.
9. Dec 28, 2013
### strangerep
No, he doesn't. He tries to give more background on certain math topics than many other QM books when dealing with unbounded operators, but only to a level that facilitates practical QM calculations. He doesn't pretend to be writing a "QM for Mathematicians" book.
Sure, totally rigorous books on QM are fine and have a valid place in the world. That doesn't make Ballentine "horrible", any more than a non-smoking restaurant is "horrible" because you're not allowed to smoke there.
I was not confused. Were you?
Precisely where in Ballentine do you think explicit mention of RHS could "actually be useful"?
Because his main purpose is to motivate a Dirac-style spectral decomposition for unbounded operators and continuous spectra. Such things handled either by a version of the spectral theorem for unbounded operators in Hilbert space, or by the Gel'fand-Maurin spectral theorem in rigged Hilbert space, and Ballentine describes both, albeit briefly. He's not trying to write a book on functional analysis.
Huh? Focussing on the minimal statistical interpretation, and deprecating Copenhagen, is a modern approach. It's as close as you can get to "shut up and calculate" and therefore is "totally useful" in the practical sense.
"How to calculate stuff" is Ballentine's main emphasis. Sure, he mentions some interpretation-related stuff, but that's intended to help dispel long-standing myths that tend to hang around and confuse students in the modern era.
I agree with that, and sense that Ballentine may be above ilikescience94's current math background. But in that case, so are lots of other books.
10. Dec 28, 2013
### R136a1
Last edited by a moderator: May 6, 2017
11. Dec 29, 2013
### vanhees71
If you want to learn quantum mechanics for physicists, you should indeed not aim at too much mathematical rigor in the beginning. Of course, one must be aware of some subtle issues, and often the physicist's language can be confusing.
So in most physicist's textbooks, they talk about "momentum eigenstates" for particles (usually starting with one-dimensional problems, i.e., for a particle on the entire Euclidean line $\mathbb{R}$). One should tell students early enough that these are not true eigenstates but "generalized eigenstates", because the corresponding wave function (working in position space) are not square integrable. They are in fact distributions on a dense subspace of the appropriate Hilbert space $L^2[\mathbb{R}]$ and can never describe a particle's actual (pure) quantum state!
Then often they talk about a momentum operator for a particle in an infinitely deep potential pot, i.e., for particles restricted to a finite interval although there is no such thing (but the standard Hamiltonian exists on the corresponding Hilbert space).
It's a pretty delicate issue to find the right balance between enough rigor and useful sloppyness, because the main issue in a QM 1 lecture is to comprehend the physics, which is very different from classical physics. I haven't taught QM 1 yet, only QM 2. So I can't say, whether I have the right idea about this balance.
I still think that a very good example for an introductory textbook is
J. J. Sakurai, Modern Quantum Mechanics
I know the 2nd edition, which covers a lot of things on a nice level. A good thing is that it doesn't overemphasize "wave mechanics" too much, although it's of course a strong tool to solve practical problems. Another good intro is also vol. 3 of the Theory Course by Landau and Lifshitz, although they are starting with wave mechanics and don't treat the representation free formalism enough. Another good read is vol. 3 of the Feynman lectures.
Ballentine is an excellent textbook to read further. It goes in some depth concerning the interpretation issue without drifting off to too much "cargo-cult philosophy". The minimal statstical (ensemble) interpretation is, in fact, the interpretation that should be taught first. The interpretational problems and different interpretations of course must be mentioned and discussed, and that Ballentine also does to a certain extent. However, I think that's a topic that should be treated at the very end of QM 1, not in the beginning. QM in the minimal interpretation is disturbing enough!
I'm not an expert in the mathematically rigorous foundations of quantum mechanics. I know that for non-relativistic QT with a fixed number of particles that can indeed be done. I think that the modern approach via the rigged-Hilbert space formalism is easier to comprehend and closer to the way physicists treat the theory. For this, I found the two-volume book by Galindo and Pascual a nice read.
12. Dec 29, 2013
### Staff: Mentor
That would be very much a minority view.
I learnt QM from Dirac and Von-Neumann.
Von Neumann is mathematically rigorous and Dirac - well not - but is what physicists use.
IMHO Ballentine strikes a nice balance between the two approaches and actually explains how the usual approach can be made rigorous by Rigged Hilbert spaces.
Not only that but he develops it from just two axioms, and Schrodingers equation etc is given its correct basis - symmetry.
Its simply the finest book on QM I have ever read and had a BIG effect on me.
I cant recommend it highly enough.
I see it was retracted and replaced by QM Dymystified.
I have the book - and yes - its a bit hodge-podgy. But at the beginning level I don't mind that prior to moving onto something like Ballentine. Its also a good exercise going back through a book like that from the more advanced perceptive of Ballentine and seeing whats going on.
Also Strangereps and Vanhees replies are both spot on.
Regardless of if you finally ascribe to the minimalist statistical interpretation or not (also called the Ensemble interpretation) without a doubt, IMHO, as Vanhees mentions, its the one you should learn and understand first.
After that if you find your mind wandering in that direction I highly recommend Schlosshauer's textbook on Decoherence - an area not well explored in Ballentine:
https://www.amazon.com/Maximilian-A.-Schlosshauer/e/B001JOSA4Y/ref=ntt_athr_dp_pel_1
With that background you can understand the modern take on interpretations.
Thanks
Bill
Last edited: Dec 29, 2013
13. Dec 29, 2013
### aabottom
Thanks for the good replies. I'm tackling The Feynman Lectures on Physics Volume III: Quantum Mechanics right now.
14. Dec 29, 2013
### ilikescience94
Thanks a bunch guys, it seems the general consensus is Ballantine, so I'm going to go with that it seems, thank you all for the suggestions and resources, I bookmarked them all.
15. Dec 29, 2013
### R136a1
I may have retracted my previous criticism of Ballentine, as it is an excellent book. But I'm not sure if Ballentine is right for you. It's pretty math and physics heavy and if you only know the basics of QM, then I don't think it's the right book for you.
Please tell us what your current knowledge of math and physics is, that way we can judge much more easily whether Ballentine really is the right choice for you!
16. Dec 29, 2013
### Staff: Mentor
Personally I don't think it really matters.
If he finds its a bit advanced then he can instead start with some of the other choices - or even the Dymystified book (yes it has issues - but it's cheap and will prepare you for Ballentine).
After studying that come back to Ballentine - then go through the first book again - its amazing what you learn picking up the stuff in more elementary treatments that are corrected in more advanced approaches. For example elementary treatments often make a big deal of the so called wave-particle duality - but from a more advanced standpoint it's basically a croc of the proverbial. Still we all must start somewhere.
Also - take your time - its not a race - you want to UNDERSTAND it - not simply be able to manipulate equations - although that's also important of course - but not at the expense of understanding.
Thanks
Bill
17. Dec 30, 2013
### atyy
I think one has to be careful with Ballentine. Among his errors are his discussion of the quantum Zeno effect, and the outdated remarks on renormalization. Also, it's unclear whether he accepts wave function collapse or an equivalent postulate in the axioms of quantum mechanics, contrary to almost every other textbook including those by Landau & Lifshitz; Cohen-Tannoudji, Diu & Laloë; Nielsen & Chuang; Benenti, Casati & Strini; Haroche and Raimond. Haroche and Raimond do say that decoherence and partial traces may seem to remove the need for collapse, but they don't because the transition from an improper to proper mixed state is equivalent to collapse. A review article by Schlosshauer, whose book bhobba recommends, also states the same.
18. Dec 30, 2013
### strangerep
This is absolutely right.
ilikescience94: tell us your current educational background. Alternatively, take a look through Ballentine using Amazon's "look inside" feature. You should be able to get a rough idea from that about whether your current math background is adequate, or whether you need something gentler first.
Last edited: Dec 30, 2013
19. Dec 30, 2013
### strangerep
I don't feel we've reached consensus yet on whether his discussion of the quantum Zeno effect contains errors. (I think it doesn't, but that's still an ongoing discussion in another thread.)
To all responders in this thread: please note that the original topic is to suggest a book suitable for member "ilikescience94". Discussion of issues in Ballentine, (and/or denial thereof), therefore probably belongs either in the existing Ballentine Book Thread, or else one of the ongoing threads in the quantum forum.
(BTW, to those who have expressed opinions on Ballentine in this thread, you can vote in the poll in the book thread linked above, if you haven't done so already.)
20. Dec 30, 2013
### ilikescience94
I'm taking calc 2 now, when I say I understand the basics, I mean that I underastand the workings somewhat, and I've worked with a few equations like uncertainty, and a little with the schroedinger, I understand special relativity, and somewhat general. I looked at the Ballentine book, it touched on some things I already knew early on, so I ordered it, are there any other books I should check out that are good for learning say GR, because I only understand some of the concepts and haven't explored all of the depth that GR has, and I am very interested in inflation, so it is somewhat necessary.
21. Dec 30, 2013
### R136a1
If you're currently taking calc 2, then I feel that ballentine is not for you. Try at least to complete calculus, and take LA and differential equations. Then you'll be much more prepared for books like Ballentine.
Regardless, I think it's insane to study QM without knowing LA in the first place. So if I were you, I would focus on that. | 2018-06-24 07:51:35 | {"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": 0.6516553163528442, "perplexity": 1054.9522967968337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866888.20/warc/CC-MAIN-20180624063553-20180624083553-00120.warc.gz"} |
https://ai-scholar.tech/en/articles/robotics/daydreamer | # DayDreamer: Dreamer Is Finally An Actual Robot!
3 main points
✔️ Show that Dreamer can learn on 4 real-world robots
✔️ Enabled a quadruped robot to spin, stand up, and move forward with its back to the ground in about an hour
✔️ The robot could learn to grab an object using an image as input and then place it in a different location using a sparse reward
written by Philipp WuAlejandro EscontrelaDanijar HafnerKen GoldbergPieter Abbeel
(Submitted on 28 Jun 2022)
Subjects: Robotics (cs.RO); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
code：
The images used in this article are from the paper, the introductory slides, or were created based on them.
## first of all
Learning robots to solve complex tasks in the real world has attracted much attention in robotics research in recent years. In particular, deep reinforcement learning (RL) has become possible to improve the robot's behavior and eventually solve complex tasks through repeated trial-and-error. However, learning a robot using Deep RL has the drawbacks that it requires a long period of interaction with the environment and a large number of samples to be collected.
On the other hand, a method called the world model, which has been attracting attention in recent years, learns the environment itself from interaction data with the environment in the past, and can imagine what the result will be if a certain action is taken under a certain situation in that environment. By using this method, for example, planning can be done, and by using a small amount of interaction data with the environment, it is possible to learn the robot's behavior, in other words, to learn policy. The effectiveness of this method has been confirmed especially in games, but its usefulness in the real world has not been demonstrated until now.
In this paper, to confirm this, we applied a world model learning method called Dreamer to the following four robots and showed that they can effectively learn by online learning in the real world. In this article, we explain the Dreamer method and show the results of each robot experiment.
## technique
In this paper, we applied Dreamer, a method for learning world models by online learning and learning behaviors at the same time, to a real-world robot. In this chapter, we introduce Dreamer. The following figure shows the overall image of the Dreamer method.
Dreamer learns a world model from the experience data of past interactions with the environment and then learns behaviors based on the trajectory predicted from the learned world model using the actor-critic algorithm. Therefore, the behavior itself is not learned from the interaction with the real-world environment, but from the data imagined by the learned world model. In this research, we also separated data collection and model updating, i.e., updating the world model, actor, and critic, and continued to collect data in one thread while updating the model in another thread to make learning more efficient. The model is updated in a separate thread at the same time.
### World Model Learning
First of all, we explain how to train a world model. The following figure shows the overall picture of the world model, which aims to estimate the dynamics of the environment as shown in the figure below. However, if we directly estimate the future image using data such as images, the error between the estimated future image and the actual image tends to be large, which leads to an accumulation of errors when estimating the dynamics of the long-term future. The world model is based on the Recurrent State-Space Model, which consists of the following four networks.
Encoder Network: $enc_{\theta} (s_{t} | s_{t-1}, a_{t-1}, x_{t})$
Decoder Network: $dec_{\theta} (s_{t}) \approx x_{t}$
Dynamics Network: $dyn_{\theta} (s_{t} | s_{t-1}, a_{t-1})$
Reward Network: $rew_{\theta}(s_{t+1}) \approx r_{t}$
The robot is equipped with multiple sensors, for example, the robot's joint angle, force sensor, RGB and depth camera images, etc. The encoder of Dreamer's world model outputs a stochastic representation $Z_{t}$ of these sensors. Therefore, the encoder of the world model of Dreamer outputs the stochastic representation $z_{t}$ by combining this sensor information. Then, the dynamics model outputs the next stochastic representation $z_{t+1}$ using the current state $h_{t}$. The output result is not directly used for learning the behavior. The Reward network is trained to predict these collected rewards.
### Actor-Critic Learning
While the world model learns a task-independent representation of the dynamics of the environment, the actor-critic algorithm is used to learn task-specific behaviors. The learning uses the rollout estimated in the world model's latent space to learn actions, as the figure below shows. This Actor critic algorithm consists of the following two neural networks.
Actor Network: $\pi (a_{t}| s_{t})$
Critic Network: $v(s_{t})$
Here, the actor-network learns the distribution of actions $a_{t}$ that maximize the reward of the estimated task for each latent space state $s_{t}$. On the other hand, the critic network is trained using temporal difference learning to estimate the total future reward (value) of the task. The learning of the value function by the critic network is important because it takes into account the rewards beyond the planning horizon (H=16). returns as follows ($\lambda$-returns).
$V_{t}^{\lambda} \doteq r_{t} + \gamma((1-\lambda) v(s_{t+1}) + \lambda V_{t+1}^{\lambda}, \quad V_{H}^{lambda} \doteq v(s_{H}))$
Actors are trained with the goal of maximizing value, but also to encourage them to explore the environment during training, thus encouraging them to keep their entropy high. With this in mind, actors are trained using the loss function below.
$\mathcal{L}(\pi)\doteq-\mathrm{E}[\sum_{t=1}^{H} \ln \pi(a_{t} | s_{t}) sg(V_{t}^{\lambda}-v(s_{t})) + \eta \mathrm{H}[\pi(a_{t} | s_{t})]]$
Here, $sg$ indicates that the gradient calculation is stopped. That is, the critic itself is not updated.
## experiment
In this paper, we trained and evaluated Dreamer on four robots. In addition to important tasks such as locomotion, manipulation, and navigation, we evaluated various patterns of these robots' tasks, such as continuous or discontinuous action space, whether the reward is dense or sparse, and whether the proprioceptive information (robot's information, e.g., joint state), images, or other sensors are combined. information (e.g., the robot's information, e.g., joint status), images, and other sensor-associated input.
The purpose of this experiment is to confirm whether the robot's behavior can be obtained more efficiently in the real world by using the world model method. Specifically, we are trying to confirm the following through experiments.
• Whether Dreamer can be applied directly to real robots
• Whether Dreamer is capable of acquiring actions in various robot, sensor modalities, and action space types
• How efficient are Dreamer-based methods compared to other reinforcement learning methods?
### baseline
In the experiments with the A1 quadruped robot, the Soft Actor-Critic (SAC) baseline was used to compare with Dreamer because the action space is continuous and the input is low dimensional information. In our experiments with XArm and UR5 robot, we used DQN as a baseline because the input is images and proprioceptive information, and the action space takes discrete values. In particular, we used a method called Rainbow for training. We also compared UR5 with PPO. Finally, for the Sphero navigation task, we compared the DrQv2 method as a baseline, because images are given as input and the action space is continuous.
In this experiment, we used a robot called Unitree A1 Robot as shown in the figure below to perform a task in which the robot rotates from a prone position, stands up, and moves forward at a certain speed. In previous papers, we have used domain randomization to learn strategies in the simulation and then transfer them to the real-world robot, a mechanism called a recovery controller that helps the robot avoid dangerous situations, and learning the parameters of an action trajectory generator. In this study, however, we did not use any of these methods. The learning was done using a dense reward function. If you are interested in how the reward function is defined, please refer to equation (5) in the paper.
As a result of one hour of learning, as shown in the figure below, the robot was able to learn a series of movements using Dreamer, such as turning, standing up, and walking forward from a state where the robot was facing away from the ground. In the first 5 minutes, the robot was able to rotate and put its feet on the ground. After 20 minutes, it learned to stand up and finally walk. After an additional 10 minutes of learning, the robot was able to withstand the forces exerted on it by external pushes and was able to get back on its feet quickly after falling. In contrast, SAC was able to turn from a backward position and place his feet on the ground but was unable to stand up and walk.
### UR5 Multi-Object Visual Pick and Place
The task of grabbing an object and placing it in another bin, as shown below, is important as it is a common task in warehouses. This task is very difficult because it aims to be trained using a sparse reward function, so it must be able to estimate the location of the object from the image and the dynamics of multiple moving objects. The information from the sensors will be the angle of the robot's joints, the position of the gripper and the Cartesian coordinates of the end-effector, and the RGB image. The rewards were a +1 reward when the gripper was detected to be halfway closed, a -1 reward when the object was released into the same bin, and a +10 reward when the object was placed in the opposite bin. The action space consists of the actions of moving the end-effector a certain distance to the X-, Y-, and Z-axis and closing or opening the gripper, which takes discrete values.
Dreamer required 8 hours of learning to be able to grab an average of 2.5 objects per minute. It did not learn very well, especially at the beginning because the reward was sparse, but after 2 hours, its performance started to improve. In contrast, the baseline PPO and Rainbow failed to learn, and although they were able to grasp objects, they quickly let go of them. We believe that these methods require more experience and would be difficult to learn in a real-world setting.
### XArm Visual Pick and Place
Similar to the UR5 experiment, the XArm is a relatively inexpensive 7DoF robot that is trained on the task of estimating the location of an object from an image and moving it to another bin. In this experiment, we use a soft object, and the gripper is connected to the object by a string so that the object can be moved, even if it is at the edge of the bin. The reward function is learned using the same sparse reward as in the UR5 experiment. We also used the depth image as input information in addition to the information used in the UR5 experiment.
Dreamer was able to grab an average of 3.1 objects per minute in 10 hours of POLICY and move them to another bin. Also, with Dreamer, when we changed the light conditions, it initially failed to solve the task but was able to quickly adapt after a few hours of learning. In contrast, when using Rainbow, it failed to learn because it required a lot of experience, as in our experiments with UR5. | 2022-12-06 20:10:06 | {"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": 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": 0.5210979580879211, "perplexity": 630.3072274107456}, "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/1669446711114.3/warc/CC-MAIN-20221206192947-20221206222947-00300.warc.gz"} |
https://ai.stackexchange.com/questions/20847/what-are-multi-hop-relational-paths | What are multi-hop relational paths?
What are multi-hop relational paths in the context of knowledge graphs (KGs)?
I tried looking it up online, but didn't find a simple explanation.
Before trying to explain this term in your context, let me briefly describe the term in other contexts.
In computer networking, the term "hop" refers to a node (e.g. a router) that a packet goes through before reaching its destination from its source. In a multi-hop situation, you have several nodes involved in the process of sending the packet from the source to the destination.
A knowledge graph is a graph that accumulates and conveys knowledge of the real world, where nodes represent entities of interest and edges relations between those entities.
So, multi-hop relational paths are probably relational paths involving more than one node or edge in the knowledge graph.
But what do we mean by "relational"?
If you are familiar with the basics of databases, the word "relational" shouldn't be so unfamiliar. In fact, there the so-called relational databases, relational models and relational algebra. Intuitively, the word "relational" is used to denote what you think it denotes, i.e. relations. See also What a relational database is by Oracle.
And what is a path?
In section 2.2.3 of the tutorial Knowledge Graphs, Aidan Hogan et al. provide a description of a path (expressions) in the context of knowledge graphs
Navigational graph patterns. A key feature that distinguishes graph query languages is the ability to include path expressions in queries. A path expression $$r$$ is a regular expression that allows matching arbitrary-length paths between two nodes, which is expressed as a regular path query $$(x,r,y)$$, where $$x$$ and $$y$$ can be variables or constants (or even the same term). | 2021-12-08 12:53:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6901589035987854, "perplexity": 860.6932282195357}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363510.40/warc/CC-MAIN-20211208114112-20211208144112-00327.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2019098 | American Institute of Mathematical Sciences
May 2019, 24(5): 2335-2364. doi: 10.3934/dcdsb.2019098
Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability
Received January 2018 Revised January 2019 Published March 2019
This article addresses the problem of controlling a constrained, continuous–time, nonlinear system through Model Predictive Control (MPC). In particular, we focus on methods to efficiently and accurately solve the underlying optimal control problem (OCP). In the numerical solution of a nonlinear OCP, some form of discretization must be used at some stage. There are, however, benefits in postponing the discretization process and maintain a continuous-time model until a later stage. This is because that way we can exploit additional freedom to select the number and the location of the discretization node points.We propose an adaptive time–mesh refinement (AMR) algorithm that iteratively finds an adequate time–mesh satisfying a pre–defined bound on the local error estimate of the obtained trajectories. The algorithm provides a time–dependent stopping criterion, enabling us to impose higher accuracy in the initial parts of the receding horizon, which are more relevant to MPC. Additionally, we analyze the conditions to guarantee closed–loop stability of the MPC framework using the AMR algorithm. The numerical results show that the proposed AMR strategy can obtain solutions as fast as methods using a coarse equidistant–spaced mesh and, on the other hand, as accurate as methods using a fine equidistant–spaced mesh. Therefore, the OCP can be solved, and the MPC law obtained, faster and/or more accurately than with discrete-time MPC schemes using equidistant–spaced meshes.
Citation: Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
References:
show all references
References:
Illustration of the multi–level adaptive time–mesh refinement strategy
Illustration of the extended (time–dependent) time–mesh refinement strategy with different refinement thresholds
Illustration of the extended time–mesh refinement algorithm for MPC
Construction of the (extended) admissible control ${\bf{\tilde u}}$ with $\Pi = \{t_k\}_{k \in \mathbb{N}}$, $t_k = k \delta$, and with $\pi_r = \{s_i\}_{i \in 0, 1, \ldots N_r}$, $s_i = i \delta/2$
Car–like system geometry
Pathwise state constraints (13) for (PCP)
Optimal path computed in the initial coarse mesh
Discretization error estimate in the initial coarse mesh
Optimal path computed in the final mesh $\pi_{\rm{AMR}}$
Optimal trajectory and control
Discretization error in the coarse mesh and the MPC refining levels
Path resulting from the AMR–MPC scheme
Trajectory and control resulting from the AMR–MPC scheme
Results for problem (PCP) solved in each time-mesh
$\pi_j$ $N_j$ $\Delta t_j$ $I_j$ $\left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty$ CPU time (s) Solver $\varepsilon_{\bf{x}}$ $\pi_0$ 21 $0.5$ 42 $1.0016{\rm{E}}^{-4}$ $0.9816$ $0.0563$ $\pi_1$ 82 $1/54$ 42 $3.3801{\rm{E}}^{-7}$ $0.7061$ $0.0642$ $\pi_{\rm{AMR}}$ 82 $1/54$ 84 $3.3801{\rm{E}}^{-7}$ $1.6877$ $0.1205$ $\pi_{\rm{F}}$ 541 $1/54$ 403 $4.0358{\rm{E}}^{-7}$ $13.2473$ $0.4675$
$\pi_j$ $N_j$ $\Delta t_j$ $I_j$ $\left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty$ CPU time (s) Solver $\varepsilon_{\bf{x}}$ $\pi_0$ 21 $0.5$ 42 $1.0016{\rm{E}}^{-4}$ $0.9816$ $0.0563$ $\pi_1$ 82 $1/54$ 42 $3.3801{\rm{E}}^{-7}$ $0.7061$ $0.0642$ $\pi_{\rm{AMR}}$ 82 $1/54$ 84 $3.3801{\rm{E}}^{-7}$ $1.6877$ $0.1205$ $\pi_{\rm{F}}$ 541 $1/54$ 403 $4.0358{\rm{E}}^{-7}$ $13.2473$ $0.4675$
Results for each MPC and AMR iterations
MPC Iter AMR Iter $N_j$ $\Delta t_j$ $I_j$ $\left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty$ CPU time (s) Solver $\varepsilon_{\bf{x}}$ $\pi_{0}$ 21 0.5 $42$ $1.002{\rm{E}}^{-4}$ $0.982$ $0.0563$ 1 $\pi_{1}$ 21 0.5 $8$ $1.002{\rm{E}}^{-4}$ $0.105$ $0.0156$ $\pi_{2}$ 52 0.0625 $22$ $3.525{\rm{E}}^{-6}$ $0.344$ $0.0374$ $\pi_{\rm{AMR}}$ 52 0.0625 $30$ $3.525{\rm{E}}^{-6}$ $0.449$ $0.0530$ 2 $\pi_{1}=\pi_{\rm{AMR}}$ 31 0.0625 $11$ $3.525{\rm{E}}^{-6}$ $0.1564$ $0.0230$ 3 $\pi_{1}=\pi_{\rm{AMR}}$ 21 0.5 $11$ $2.042{\rm{E}}^{-7}$ $0.1639$ $0.0139$ 4 $\pi_{1}=\pi_{\rm{AMR}}$ 21 0.5 $7$ $4.321{\rm{E}}^{-7}$ $0.0936$ $0.0126$ 5 $\pi_{1}=\pi_{\rm{AMR}}$ 21 0.5 $7$ $4.515{\rm{E}}^{-7}$ $0.0912$ $0.0123$
MPC Iter AMR Iter $N_j$ $\Delta t_j$ $I_j$ $\left|\left|\varepsilon_{\bf{x}}^{(j)}\right|\right|_\infty$ CPU time (s) Solver $\varepsilon_{\bf{x}}$ $\pi_{0}$ 21 0.5 $42$ $1.002{\rm{E}}^{-4}$ $0.982$ $0.0563$ 1 $\pi_{1}$ 21 0.5 $8$ $1.002{\rm{E}}^{-4}$ $0.105$ $0.0156$ $\pi_{2}$ 52 0.0625 $22$ $3.525{\rm{E}}^{-6}$ $0.344$ $0.0374$ $\pi_{\rm{AMR}}$ 52 0.0625 $30$ $3.525{\rm{E}}^{-6}$ $0.449$ $0.0530$ 2 $\pi_{1}=\pi_{\rm{AMR}}$ 31 0.0625 $11$ $3.525{\rm{E}}^{-6}$ $0.1564$ $0.0230$ 3 $\pi_{1}=\pi_{\rm{AMR}}$ 21 0.5 $11$ $2.042{\rm{E}}^{-7}$ $0.1639$ $0.0139$ 4 $\pi_{1}=\pi_{\rm{AMR}}$ 21 0.5 $7$ $4.321{\rm{E}}^{-7}$ $0.0936$ $0.0126$ 5 $\pi_{1}=\pi_{\rm{AMR}}$ 21 0.5 $7$ $4.515{\rm{E}}^{-7}$ $0.0912$ $0.0123$
[1] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553 [2] Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial & Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521 [3] Loïc Bourdin, Emmanuel Trélat. Optimal sampled-data control, and generalizations on time scales. Mathematical Control & Related Fields, 2016, 6 (1) : 53-94. doi: 10.3934/mcrf.2016.6.53 [4] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [5] M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137 [6] Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021001 [7] Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control & Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289 [8] Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Linear optimal control of time delay systems via Hermite wavelet. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 143-156. doi: 10.3934/naco.2019044 [9] Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $\mathcal{H}_{\infty}$ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368 [10] Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control & Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002 [11] Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285 [12] Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks & Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267 [13] Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial & Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042 [14] Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343 [15] Luca Galbusera, Sara Pasquali, Gianni Gilioli. Stability and optimal control for some classes of tritrophic systems. Mathematical Biosciences & Engineering, 2014, 11 (2) : 257-283. doi: 10.3934/mbe.2014.11.257 [16] João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks & Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303 [17] Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136 [18] Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929 [19] Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021034 [20] Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial & Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934/jimo.2009.5.1
2019 Impact Factor: 1.27 | 2021-04-22 21:09:18 | {"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": 0.47460711002349854, "perplexity": 3177.8972912726745}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039604430.92/warc/CC-MAIN-20210422191215-20210422221215-00544.warc.gz"} |
https://gamedev.stackexchange.com/questions/96205/how-can-i-verify-if-there-is-no-object-between-2-others | # How can I verify if there is no object between 2 others?
I have 2 characters in my 2D game, I would like that if both of them are on the same Y axis, and there is no collider/object between them, for an animation to happen. I can't figure out how to check for the last one though... Thanks for the help!
EDIT :
Hello, I have used MrCranky's code and it seemingly looks like it should work, but I am having a problem with the syntax I think. Physics.Raycast((x,y) of the 1st player, new Vector3 (1,0,0), distance between them -2) So this code, if there is nothing between the player and the other character, it should give me a false. And it worked, but the problem is, even after I added an object between them, it stayed on false through every update, could you assist me with what am I doing wrong please?
The brute force solution would be to iterate over all objects which might be in between the two characters and test them each individually (GameObject.FindObjectsOfType is probably your friend here).
However a much simpler and cheaper solution would be to do a ray-cast from one character's position to the other. If any results are returned they'll be in the way of the two characters. Bear in mind that will draw a thin ray from the local origins of the characters though, if you need partial collisions (e.g. a collider just off to one side of that ray but which the character would hit if they tried to move towards the other character) then you'd need a sweep test (moving the character's collision volume along the path towards the other character) of some sort. You might be able to get away with 5 raytests instead - one in the centre and one for each corner of the character's AABB.
• Hello, I have used this command and it seemingly looks like it should work, but I am having a problem with the syntax I think. Physics.Raycast((x,y) of the 1st player, new Vector3 (1,0,0), distance between them -2) So this code, if there is nothing between the player and the other character, it should give me a false. And it worked, but the problem is, even after I added an object between them, it stayed on false through every update, could you assist me with what am I doing wrong please?
– Zee
Mar 6, 2015 at 22:17
• There are a few things that might be tripping you up. First is the object in between the players has to have a physics collider of some sort (box, sphere, etc.). Second is that you need to make sure the ray is pointing the way you think (use Debug.DrawRay to check). Third is to make sure the layer mask isn't excluding the object you expect to hit (omit the parameter to search all layers). Mar 8, 2015 at 5:44 | 2022-10-04 01:00:41 | {"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": 0.4595746099948883, "perplexity": 399.21576726003315}, "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-2022-40/segments/1664030337446.8/warc/CC-MAIN-20221003231906-20221004021906-00770.warc.gz"} |
https://docs.nvidia.com/deeplearning/dali/main-user-guide/docs/operations/nvidia.dali.fn.optical_flow.html | # nvidia.dali.fn.optical_flow¶
nvidia.dali.fn.optical_flow(*inputs, **kwargs)
Calculates the optical flow between images in the input.
The main input for this operator is a sequence of frames. Optionally, the operator can be provided with external hints for the optical flow calculation. The output format of this operator matches the output format of the optical flow driver API. Refer to https://developer.nvidia.com/opticalflow-sdk for more information about the Turing, Ampere and Hopper optical flow hardware that is used by DALI.
This operator allows sequence inputs.
Supported backends
• ‘gpu’
Parameters
• input0 (TensorList ('FHWC')) – Input to the operator.
• input1 (TensorList ('FHWC'), optional) – Input to the operator.
Keyword Arguments
• bytes_per_sample_hint (int or list of int, optional, default = [0]) –
Output size hint, in bytes per sample.
If specified, the operator’s outputs residing in GPU or page-locked host memory will be preallocated to accommodate a batch of samples of this size.
• enable_external_hints (bool, optional, default = False) –
Enables or disables the external hints for optical flow calculations.
External hints are analogous to temporal hints, but the only difference is that external hints come from an external source. When this option is enabled, the operator requires two inputs.
• enable_temporal_hints (bool, optional, default = False) –
Enables or disables temporal hints for sequences that are longer than two images.
The hints are used to improve the quality of the output motion field as well as to speed up the calculations. The hints are especially useful in presence of large displacements or periodic patterns which might confuse the optical flow algorithms. )
• hint_grid (int, optional, default = 4) –
Sets the grid size for the hint vector field.
The hints are used to improve the quality of the output motion field as well as to speed up the calculations. The grid resolution could be set to a different value than the output.
Note
Currently, only a 1, 2, 4 and 8 are supported for Ampere and 4 for Turing.
• image_type (nvidia.dali.types.DALIImageType, optional, default = DALIImageType.RGB) – Input color space (RGB, BGR or GRAY).
• output_grid (int, optional, default = 4) –
Sets the grid size for the output vector field.
This operator produces the motion vector field at a coarser resolution than the input pixels. This parameter specifies the size of the pixel grid cell corresponding to one motion vector. For example, a value of 4 will produce one motion vector for each 4x4 pixel block.
Note
Currently, only a 1, 2 and 4 are supported for Ampere and 4 for Turing.
• preserve (bool, optional, default = False) – Prevents the operator from being removed from the graph even if its outputs are not used.
• preset (float, optional, default = 0.0) –
Speed and quality level of the optical flow calculation.
Allowed values are:
• 0.0 is the lowest speed and the best quality.
• 0.5 is the medium speed and quality.
• 1.0 is the fastest speed and the lowest quality.
The lower the speed, the more additional pre- and postprocessing is used to enhance the quality of the optical flow result.
• seed (int, optional, default = -1) –
Random seed.
If not provided, it will be populated based on the global seed of the pipeline.
• output_format (int) –
Warning
The argument output_format is a deprecated alias for output_grid. Use output_grid instead. | 2023-03-24 00:59:39 | {"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": 0.41863521933555603, "perplexity": 3080.3208998023883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945218.30/warc/CC-MAIN-20230323225049-20230324015049-00509.warc.gz"} |
http://mathhelpforum.com/algebra/27193-solved-simplifying-radical-expressions-using-absolute-value-symbols.html | # Math Help - [SOLVED] simplifying radical expressions and using absolute value symbols!
1. ## [SOLVED] simplifying radical expressions and using absolute value symbols!
idfk know what I am doing. help ?
I did the first one but there is no way to check if I'm doing it right
#1 square root of 36x^4 = 6x^2?
#2 square root of c^80d^50 = ?
and when would I need to use a absolute value symbol? (and why if you can explain! please!)
hope someone can help, seems after you graduate (high school) you loose all memory of these things so my friends are no help
2. Well, first we have:
$\sqrt{36x^4}$
We use absolute value in square root operations because the square of a positive number is the same as the square of the same negative number, so when you take the square root of a number, it could be the positive or negative version of that number, depending on what you are looking for. So the solution to your first problem is:
$\sqrt{36x^4} = |6x^2|$
And the second problem:
$\sqrt{c^{80}d^{50}}$
Same principle applies:
$\sqrt{c^{80}d^{50}} = |c^{40}d^{25}|$
3. We use absolute value in square root operations because the square of a positive number is the same as the square of the same negative number, so when you take the square root of a number, it could be the positive or negative version of that number, depending on what you are looking for. So the solution to your first problem is:
$\sqrt{36x^4} = |6x^2|$
This is all right. However, there is one more thing I would like to add on the subject of absolute value signs. They are only needed when there is an odd power or a subtraction happening, because anything raised to an even power will be positive. So in this case, we can just write $6x^2$
4. It says to use the absolute value... So I did not simplify more. | 2014-10-02 03:35:12 | {"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": 6, "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": 0.8359671831130981, "perplexity": 191.3961246884562}, "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/1412037663637.20/warc/CC-MAIN-20140930004103-00433-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-1-test-page-111/13 | ## Introductory Algebra for College Students (7th Edition)
$-6x+10$
First, simplify within the parentheses by distributing 3 to obtain: $=6-2[3x+3-5] \\=6-2[3x-2]$ Distribute $-2$ to obtain: $=6-6x-(-4) \\=6-6x+4$ Simplify by combining like terms: $\\=-6x+(6+4) \\=-6x+10$ | 2018-05-24 17:44:38 | {"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": 0.7572121620178223, "perplexity": 1879.270014097506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866733.77/warc/CC-MAIN-20180524170605-20180524190605-00232.warc.gz"} |
https://astarmathsandphysics.com/index.php?option=com_content&view=article&id=4591:length-of-a-vertical-line-between-circles&catid=95&Itemid=474 | ## Length of a Vertical Line Between Circles
We are given two circles, one inside the other, touching tangentially at the bottom, with a vertical line AB tangential to the smaller, a chord to the larger as shown.
We can find the length of the line AB using Pythagoras Theorem. Draw a horizontal line from the center of the large circle to the line AB. The triangle ABC is right angled.
$CA=\sqrt{10^2-4^2}= \sqrt{84}c=2 \sqrt{21}$
cm
Then
$AB=2AC=4 \sqrt{21}cm$
. | 2019-11-13 10:47:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6798743605613708, "perplexity": 253.76244419397858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667177.24/warc/CC-MAIN-20191113090217-20191113114217-00480.warc.gz"} |
https://physics.stackexchange.com/questions/143366/is-the-universe-still-believed-to-be-flat | # Is the Universe still believed to be flat?
I have read a handful of old articles from mid 2013 expressing that the Universe may, in fact, be curved.
My question is how is the apparent "lopsidedness" of the CMB Radiation explained in a flat universe model?
I understand that the vast majority of the evidence indicates a spatially flat universe, but I would like to know if there is any merit to these claims:
The temperature of the cosmic microwave background radiation fluctuates more on one side of the sky (the right side of this projection) than on the opposite side, a sign that space might be curved.
• The Liddle paper referred to in the first link is here: arxiv.org/abs/1306.5698 . The second link doesn't seem to me to be relevant to the question. the vast majority of the evidence indicates a flat (Minkowski) universe No, the evidence points toward spatial flatness, but there is no viable cosmological model in which the universe is modeled as Minkowski space, which is a completely flat spacetime, not just a spatially flat one. – Ben Crowell Oct 27 '14 at 20:02
• Ironically, I had originally put Euclidean instead of Minkowski as I was, indeed, referring to only spacial dimensions. @Danu was the one that changed it to Minkowski. – Goodies Oct 27 '14 at 20:44
• !! My mistake! Sorry! @BenCrowell and OP. I have corrected my error. Don't know how that happened ;) – Danu Oct 27 '14 at 20:49
• So @BenCrowell, that paper describes the Universe as Open, not Flat? I was curious as to if there are spatially flat models that do describe these anomalies. – Goodies Oct 27 '14 at 20:59
From here:
Before proceeding, it should be mentioned that the statistical significance of the result is still under debate. While the asymmetry is significant at the ≳3σ level, some question whether it is simply a consequence of the “look-elsewhere” effect: i.e., we test for all kinds of anomalies in the CMB, and the investigated parameter space is so vast that it’s no surprise that, by chance, one of the parameters shows a positive result. Cosmological models make statistical predictions about the distribution of temperature fluctuations on an ensemble of CMB skies, but we have only one CMB sky to observe. Therefore, if the observed asymmetry is a statistical fluke, we are stuck with it because there is no way to increase the statistics on this particular measurement. But if the asymmetry is real and not just a statistical fluke, then it is extremely important. It may well be a remnant of the preinflationary Universe!
The dipole anisotropy in the power spectrum seems real (not the doppler dipole, but rather the one discussed in these articles). Alright, that's out of the way. This observation needs to be interpreted to get any further, though.
One possibility is that this is just a $3\sigma$ excursion from the expected isotropy. Without a statistical sample of additional Universes to observe, we have no way of knowing for sure. And as the article points out, if you look at enough parameters, eventually you'd actually be surprised if you didn't find one off by a couple of $\sigma$.
Another interpretation is that the Universe is a little bit curved (in the "Open" direction, i.e. negatively). This depends on the model some theorists are proposing being at least broadly correct. However, for the moment, this class of models offers no other presently testable predictions. I'm sure the theorists are working on more predictions that are testable, and CMB observers are working on measuring the predictions that have been made (the signal is supposed to be very faint, and the measurement is very difficult). But for the moment no other tested predictions means that this is just another class of theories, and there is no compelling reason to prefer it over the usual flat Universe model. In fact, I would prefer the flat model as it has less parameters and also explains the observations (even though I have to live with a $3\sigma$ statistical anomaly).
The same article also mentions that the anomaly is seen in the two hemispheres roughly separated by the ecliptic, which is somewhat worrying. Alignment with the ecliptic, or galactic equator, or other preferred direction, to me is suggestive of some uncorrected systematic effect. Not to say that this can't be a real anomaly because it's aligned with the ecliptic, but it's worrying...
So to sum up, there doesn't seem to be any compelling evidence for an open Universe. There is this anomaly in the isotropy of the power spectrum, but it's not so large that it couldn't just be happenstance. If the proposed curvaton model makes some additional predictions (differing from the predictions of the usual cosmology) that are later borne out by observation, that would be more strongly suggestive of an Open geometry. | 2019-10-17 20:31:05 | {"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": 0.7507809400558472, "perplexity": 446.96193226007904}, "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-2019-43/segments/1570986676227.57/warc/CC-MAIN-20191017200101-20191017223601-00158.warc.gz"} |
https://www.variadic.xyz/2014/07/06/c11-constexpr-computing-exp-at-compile-time/ | # C++11 constexpr: computing exp at compile time
While trying out constexpr I was wondering what else can I get compiler to compute. I can pretty much do any integral computation with both C++ Template Metaprogramming and constexpr, but can I do some floating point computations? It appears I can!!
So I set out to compute exp(z), using Taylor series implementation for exp:
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 // can't deal with negative powers!! // constexpr double pow(double x, int y) { return y == 0 ? 1.0 : x * pow(x, y-1); } constexpr int factorial(int x) { return x == 0 ? 1 : x * factorial(x-1); } constexpr double exp(double x) { return 1.0 + x + pow(x,2)/factorial(2) + pow(x, 3)/factorial(3) + pow(x, 4)/factorial(4) + pow(x, 5)/factorial(5) + pow(x,6)/factorial(6) + pow(x, 7)/factorial(7) + pow(x, 8)/factorial(8) + pow(x, 9)/factorial(9); } int main() { constexpr double exp1 = exp(1.0); return 0; }
In here I have pow function to compute powers of a floating point number, mine is bit limited as it can do integral powers only; factorial straight forward. exp function function goes to 10 component evaluation. With exp(1) I do see value of 2.7182815255731922 in debug window while cout prints it to be 2.71828. Since this is floating point value I can’t do static_assert.
Now what’s the proof that it was all compile time rather than runtime evaluation, the only way to confirm is to look at what is executed by processor, that’s right assembly:
cpp-cxprmain at main.cpp:388:
0x100000cc0: pushq %rbp
0x100000cc1: movq %rsp, %rbp
0x100000cc4: movl $0x0, %eax 0x100000cc9: movsd 0x247(%rip), %xmm0 0x100000cd1: movl$0x0, -0x4(%rbp)
0x100000cd8: movsd %xmm0, -0x10(%rbp)
0x100000cdd: popq %rbp
0x100000cde: ret
Note: there is no instruction for callq, which means no function was called. Since exp(1.0) was evaluated at compile time it’s value was loaded in to register %xmm0 at line 5 and then loaded into a local variable (negative offset from %rbp register) at line 7. Life is a bit easy working with debug builds.
Had we not evaluated exp(z) on compile time, just remove constexpr from code in main:
1 2 double exp1 = exp(1); // no constexpr - this will be evaluated at runtime
generated assembly for main function is:
cpp-cxprmain at main.cpp:388:
0x1000009e0: pushq %rbp
0x1000009e1: movq %rsp, %rbp
0x1000009e4: subq $0x10, %rsp 0x1000009e8: movabsq$0x1, %rax
0x1000009f2: cvtsi2sdq %rax, %xmm0
0x1000009f7: movl $0x0, -0x4(%rbp) 0x1000009fe: callq 0x100000b10 ; exp(double) at main.cpp:354 0x100000a03: movl$0x0, %eax
0x100000a08: movsd %xmm0, -0x10(%rbp)
0x100000a0d: addq $0x10, %rsp 0x100000a11: popq %rbp 0x100000a12: ret Here we are loading abs value 1 into register %rax (line 5), then converting int value to floating point with cvtsi2sdq instruction at line 6 and writing that in %xmm0 register. Then at line 8 we are calling exp(double) function which will again put the value in register %xmm0. This value is then moved to a local variable at line 10 (negative offset from %rbp register). The value computed at runtime is : 2.7182815255731922 Here is the assembly code generated for exp(double) function: cpp-cxprexp(double) at main.cpp:354: 0x100000b10: pushq %rbp 0x100000b11: movq %rsp, %rbp 0x100000b14: subq$0x90, %rsp
0x100000b1b: movl $0x2, %edi 0x100000b20: movabsq$0x1, %rax
0x100000b2a: cvtsi2sdq %rax, %xmm1
0x100000b2f: movsd %xmm0, -0x8(%rbp)
0x100000b39: movsd -0x8(%rbp), %xmm0
0x100000b3e: movsd %xmm1, -0x10(%rbp)
0x100000b43: callq 0x100000d20 ; pow(double, int) at main.cpp:344
0x100000b48: movl $0x2, %edi 0x100000b4d: movsd %xmm0, -0x18(%rbp) 0x100000b52: callq 0x100000d90 ; factorial(int) at main.cpp:349 0x100000b57: movl$0x3, %edi
...
...
; many more instructions to make 7 more calls to pow and factorial
...
...
;` | 2020-07-14 17:11:46 | {"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": 0.4303668141365051, "perplexity": 8280.985418030217}, "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/1593655897168.4/warc/CC-MAIN-20200714145953-20200714175953-00151.warc.gz"} |
https://www.transtutors.com/questions/incomplete-manufacturing-costs-expenses-and-selling-data-for-two-different-cases-are-537414.htm | # Incomplete manufacturing costs, expenses, and selling data for two different cases are as follows.
Incomplete manufacturing costs, expenses, and selling data for two different cases are as follows.
Case 1 2 Direct materials used $9,600$ (g) Direct labor 5,000 8,000 Manufacturing overhead 8,000 4,000 Total manufacturing costs (a) 16,000 Beginning work in process inventory 1,000 (h) Ending work in process inventory (b) 3,000 Sales 24,500 (i) Sales discounts 2,500 1,400 Cost of goods manufactured 17,000 22,000 Beginning finished goods inventory (c) 3,300 Goods available for sale 20,000 (j) Cost of goods sold (d) (k) Ending finished goods inventory 3,400 2,500 Gross profit (e) 7,000 Operating expenses 2,500 (l) Net income (f ) 5,000
Instructions
(a) Indicate the missing amount for each letter.
(b) Prepare a condensed cost of goods manufactured schedule for Case 1.
(c) Prepare an income statement and the current assets section of the balance sheet for Case 1. Assume that in Case 1 the other items in the current assets section are as follows: Cash $4,000, Receivables (net)$15,000, Raw Materials $600, and Prepaid Expenses$400. | 2019-09-15 14:45:21 | {"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": 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": 0.23916767537593842, "perplexity": 8405.230886490017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514571506.61/warc/CC-MAIN-20190915134729-20190915160729-00384.warc.gz"} |
http://qks.cqu.edu.cn/html/cqdxzrcn/2018/12/20181201.htm | 文章快速检索 高级检索
重庆大学学报 2018, Vol. 41 Issue (12): 1-9 DOI: 10.11835/j.issn.1000-582X.2018.12.001 RIS（文献管理工具） 0
### 引用本文
ZHOU Ping, ZENG Zhong, QIAO Long. Simulation of shear-thinning droplets impact on solid surfaces by using Lattice Boltzmann method[J]. Journal of Chongqing University, 2018, 41(12): 1-9. DOI: 10.11835/j.issn.1000-582X.2018.12.001.
### 文章历史
Simulation of shear-thinning droplets impact on solid surfaces by using Lattice Boltzmann method
ZHOU Ping , ZENG Zhong , QIAO Long
College of Aerospace Engineering, Chongqing University, Chongqing 400044, P. R. China
Supported by the National Natural Science Foundation of China (11572062) and Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R112)
Abstract: The droplet spreading and deposition appear widely in industrial applications, and it is of practical significance to study the effect of non-Newtonian rheology on droplets impact on solid surfaces. In this research, we developed a two-phase lattice Boltzmann model based on the phase field method for power-law fluid flows. By introducing a contact angle condition, power-law droplets impact on solid surfaces was investigated, and the effects of power exponent n (0.5 ≤ n ≤ 1.0) and Weber number We (5 ≤ We ≤ 45) on shear-thinning droplets impact were evaluated. The results indicate that power-law liquid inhibits the droplet spreading and splashing, and it becomes easier for deposition with the decrease of n. In addition, droplets are easier to reach stationary state as weber number increases.
Keywords: Lattice Boltzmann method power law liquid pseudoplastic fluid phase interfaces phase field method droplet deposition droplet spreading
1 理论 1.1 幂律(Power-law)模型
$\eta (\dot \gamma ) = {\eta _0}|\dot \gamma |n - 1, n > 0,$ (1)
1.2 Call-Hilliard相场模型
Call-Hilliard对流扩散方程[25-27]
$\left\{ \begin{array}{l} \frac{{\partial C}}{{\partial t}} + {\mathit{\boldsymbol{u}}} \cdot \nabla C = M{\nabla ^2}{\mu _{\rm{c}}}, \\ {\mu _{\rm{c}}} = {C^3} - 1.5{C^2} + 0.5C - {\varepsilon ^2}{\nabla ^2}C, \end{array} \right.$ (2)
$\nabla \cdot {\mathit{\boldsymbol{u}}} = 0,$ (3)
$\rho (\frac{{\partial {\mathit{\boldsymbol{u}}}}}{{\partial t}} + {\mathit{\boldsymbol{u}}} \cdot \nabla {\mathit{\boldsymbol{u}}}) = - \nabla p + \nabla \cdot [\eta (\nabla {\mathit{\boldsymbol{u}}} + \nabla {{\mathit{\boldsymbol{u}}}^{\rm{T}}})] + {{\mathit{\boldsymbol{F}}}_{\rm{s}}},$ (4)
1.3 两相流格子Boltzmann方法
$f_\alpha ^{{\rm{eq}}} = \rho {\Gamma _\alpha }({\mathit{\boldsymbol{u}}}) = \rho {\omega _\alpha }[1 + \frac{{{{\mathit{\boldsymbol{e}}}_\alpha } \cdot {\mathit{\boldsymbol{u}}}}}{{c_{\rm{s}}^{\rm{2}}}} + \frac{{{{({{\mathit{\boldsymbol{e}}}_\alpha } \cdot {\mathit{\boldsymbol{u}}})}^2}}}{{2c_{\rm{s}}^4}} - \frac{{{\mathit{\boldsymbol{u}}} \cdot {\mathit{\boldsymbol{u}}}}}{{2c_{\rm{s}}^{\rm{2}}}}],$ (5)
$\frac{{\partial {g_\alpha }}}{{\partial t}} + {e_\alpha } \cdot \nabla {g_\alpha } = - {{\mathit{\pmb{\Lambda}}} _{\alpha \beta }}({g_\beta } - g_\beta ^{{\rm{eq}}}) + ({{\mathit{\boldsymbol{e}}}_\alpha } - {\mathit{\boldsymbol{u}}}) \cdot [\nabla \rho c_{\rm{s}}^{\rm{2}}({\Gamma _\alpha } - {\Gamma _\alpha }(0)) + {\mathit{\boldsymbol{F}}}{\Gamma _\alpha }],$ (6)
${{\bar g}_\alpha }({\mathit{\boldsymbol{x}}} + {{\mathit{\boldsymbol{e}}}_\alpha }\delta t, t + \delta t) - {{\bar g}_\alpha }({\mathit{\boldsymbol{x}}}, t) = - ({S_{\alpha \beta }} + 2{I_{\alpha \beta }})({{\bar g}_\beta } - \bar g_\beta ^{{\rm{eq}}}) + \\\;\delta t({{\mathit{\boldsymbol{e}}}_\alpha } - {\mathit{\boldsymbol{u}}}) \cdot [\nabla \rho c_{\rm{s}}^{\rm{2}}({\Gamma _\alpha } - {\Gamma _\alpha }(0)) + {{\mathit{\boldsymbol{F}}}_{\rm{s}}}{\Gamma _\alpha }],$ (7)
$\begin{array}{l} p = \sum\limits_{\alpha = 0}^8 {{{\bar g}_\alpha }} + \frac{{\delta t}}{2}{\mathit{\boldsymbol{u}}} \cdot \nabla \rho c_{\rm{s}}^{\rm{2}}, \\ {\mathit{\boldsymbol{u}}} = \frac{1}{{\rho c_{\rm{s}}^{\rm{2}}}}(\sum\limits_{\alpha = 0}^8 {{{\bar g}_\alpha }} {{\mathit{\boldsymbol{e}}}_\alpha } + \frac{{\delta t}}{2}{{\mathit{\boldsymbol{F}}}_{\rm{s}}})。\end{array}$ (8)
2 模型和程序的验证 2.1 方腔流
图 1 幂律流体方腔流在竖直中心线上u和水平中心线上v的分布和文献解的对比 Figure 1 Comparision between our results and Neofytou's results for power-law flow: u-velocity profiles along vertical centerline and v-velocity profiles along horizontal centerline
2.2 Laplace定律
图 2 幂律液滴达到稳态时Δp和1/R的关系与理论解的对比 Figure 2 Comparision between the numerical and theoretical solution of the relationship of Δp and 1/R when power law drops reach steady state
2.3 液滴在壁面上的动态接触
图 3 幂律液滴在不同静态接触角的稳态 Figure 3 Stable configurations of power law droplet at different equilibrium contact angles
图 4 量纲一的铺展长度r/R和量纲一的时间t*=t/ $\sqrt {\rho {R^3}/\sigma }$的关系 Figure 4 Relationship between dimensionless spreading length r/R and the square root of dimensionless time t*=t/ $\sqrt {\rho {R^3}/\sigma }$
3 液滴撞击固壁上的铺展
图 5 液滴撞击壁面的示意图 Figure 5 The sketch of droplet impact on a solid surface
图 6 不同幂律指数n下铺展长度D随时间变化的关系 Figure 6 Spreading length D as a function of time for different power exponents n
图 7 幂律液滴在不同时间的形态 Figure 7 Droplets configurations at different time
图 8 不同韦伯数We下幂律指数n=0.5以及1.0液滴铺展长度随时间的变化关系 Figure 8 Spreading length as a function of time at different We when the power exponent n=0.5 and n=1.0
4 结论 | 2019-01-16 18:14:46 | {"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": 0.7135148048400879, "perplexity": 5642.58420034205}, "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/1547583657557.2/warc/CC-MAIN-20190116175238-20190116201238-00419.warc.gz"} |
https://calliope.readthedocs.io/en/stable/user/running.html | # Running a model¶
There are essentially three ways to run a Calliope model:
1. With the calliope run command-line tool.
2. By programmatically creating and running a model from within other Python code, or in an interactive Python session.
3. By generating and then executing scripts with the calliope generate_runs command-line tool, which is primarily designed for running many scenarios on a high-performance cluster.
## Running with the command-line tool¶
We can easily run a model after creating it (see Building a model), saving results to a single NetCDF file for further processing
$calliope run testmodel/model.yaml --save_netcdf=results.nc The calliope run command takes the following options: • --save_netcdf={filename.nc}: Save complete model, including results, to the given NetCDF file. This is the recommended way to save model input and output data into a single file, as it preserves all data fully, and allows later reconstruction of the Calliope model for further analysis. • --save_csv={directory name}: Save results as a set of CSV files to the given directory. This can be handy if the modeler needs results in a simple text-based format for further processing with a tool like Microsoft Excel. • --save_plots={filename.html}: Save interactive plots to the given HTML file (see Analysing a model for further details on the plotting functionality). • --debug: Run in debug mode, which prints more internal information, and is useful when troubleshooting failing models. • --scenario={scenario} and --override_dict={yaml_string}: Specify a scenario, or one or several overrides, to apply to the model, or apply specific overrides from a YAML string (see below for more information) • --help: Show all available options. Multiple options can be specified, for example, saving NetCDF, CSV, and HTML plots simultaneously $ calliope run testmodel/model.yaml --save_netcdf=results.nc --save_csv=outputs --save_plots=plots.html
Warning
Unlike in versions prior to 0.6.0, the command-line tool in Calliope 0.6.0 and upward does not save results by default – the modeller must specify one of the -save options.
### Applying a scenario or override¶
The --scenario can be used in three different ways:
• It can be given the name of a scenario defined in the model configuration, as in --scenario=my_scenario
• It can be given the name of a single override defined in the model configuration, as in --scenario=my_override
• It can be given a comma-separated string of several overrides defined in the model configuration, as in --scenario=my_override_1,my_override_2
In the latter two cases, the given override(s) is used to implicitly create a “scenario” on-the-fly when running the model. This allows quick experimentation with different overrides without explicitly defining a scenario combining them.
Assuming we have specified an override called milp in our model configuration, we can apply it to our model with
$calliope run testmodel/model.yaml --scenario=milp --save_netcdf=results.nc Note that if both a scenario and an override with the same name, such as milp in the above example, exist, Calliope will raise an error, as it will not be clear which one the user wishes to apply. It is also possible to use the –override_dict option to pass a YAML string that will be applied after anything applied through --scenario $ calliope run testmodel/model.yaml --override_dict="{'model.subset_time': ['2005-01-01', '2005-01-31']}" --save_netcdf=results.nc
## Running interactively with Python¶
The most basic way to run a model programmatically from within a Python interpreter is to create a Model instance with a given model.yaml configuration file, and then call its run() method:
import calliope
model = calliope.Model('path/to/model.yaml')
model.run()
Note
If config is not specified (i.e. model = Model()), an error is raised. See Built-in example models for information on instantiating a simple example model without specifying a custom model configuration.
Other ways to load a model interactively are:
• Passing an AttrDict or standard Python dictionary to the Model constructor, with the same nested format as the YAML model configuration (top-level keys: model, run, locations, techs).
• Loading a previously saved model from a NetCDF file with model = calliope.read_netcdf(‘path/to/saved_model.nc’). This can either be a pre-processed model saved before its run method was called, which will include input data only, or a completely solved model, which will include input and result data.
After instantiating the Model object, and before calling the run() method, it is possible to manually inspect and adjust the configuration of the model. The pre-processed inputs are all held in the xarray Dataset model.inputs.
After the model has been solved, an xarray Dataset containing results (model.results) can be accessed. At this point, the model can be saved with either to_csv() or to_netcdf(), which saves all inputs and results, and is equivalent to the corresponding --save options of the command-line tool.
An example of interactive running in a Python session, which also demonstrates some of the analysis possibilities after running a model, is given in the tutorials. You can download and run the embedded notebooks on your own machine (if both Calliope and the Jupyter Notebook are installed).
### Scenarios and overrides¶
There are two ways to override a base model when running interactively, analogously to the use of the command-line tool (see Applying a scenario or override above):
1. By setting the scenario argument, e.g.:
model = calliope.Model('model.yaml', scenario='milp')
2. By passing the override_dict argument, which is a Python dictionary, an AttrDict, or a YAML string of overrides:
model = calliope.Model(
'model.yaml',
override_dict={'run.solver': 'gurobi'}
)
Note
Both scenario and override_dict can be defined at once. They will be applied in order, such that scenarios are applied first, followed by dictionary overrides. As such, the override_dict can be used to override scenarios.
### Tracking progress¶
When running Calliope in the command line, logging of model pre-processing and solving occurs automatically. Interactively, for example in a Jupyter notebook, you can enable verbose logging by setting the log level using calliope.set_log_verbosity(level) immediately after importing the Calliope package. By default, calliope.set_log_verbosity() also sets the log level for the backend model to DEBUG, which turns on output of solver output. This can be disabled by calliope.set_log_verbosity(level, include_solver_output=False). Possible log levels are (from least to most verbose):
1. CRITICAL: only show critical errors.
2. ERROR: only show errors.
3. WARNING: show errors and warnings (default level).
4. INFO: show errors, warnings, and informative messages. Calliope uses the INFO level to show a message at each stage of pre-processing, sending the model to the solver, and post-processing, including timestamps.
5. DEBUG: SOLVER logging, with heavily verbose logging of a number of function outputs. Only for use when troubleshooting failing runs or developing new functionality in Calliope.
## Generating scripts for many model runs¶
Scripts to simplify the creation and execution of a large number of Calliope model runs are generated with the calliope generate_runs command-line tool. More detail on this is available in Generating scripts to run a model many times.
## Improving solution times¶
Large models will take time to solve. The easiest is often to just let a model run on a remote device (another computer, or a high performance computing cluster) and forget about it until it is done. However, if you need results now, there are ways to improve solution time.
Details on strategies to improve solution times are given in Troubleshooting.
## Debugging failing runs¶
What will typically go wrong, in order of decreasing likelihood:
• The model is improperly defined or missing data. Calliope will attempt to diagnose some common errors and raise an appropriate error message.
• The model is consistent and properly defined but infeasible. Calliope will be able to construct the model and pass it on to the solver, but the solver (after a potentially long time) will abort with a message stating that the model is infeasible.
• There is a bug in Calliope causing the model to crash either before being passed to the solver, or after the solver has completed and when results are passed back to Calliope.
Calliope provides help in diagnosing all of these model issues. For details, see Troubleshooting.
Previous: Building a model | Next: Analysing a model | 2020-02-18 13:56:39 | {"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": 0.22057095170021057, "perplexity": 3345.0007724517154}, "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/1581875143695.67/warc/CC-MAIN-20200218120100-20200218150100-00036.warc.gz"} |
http://www.numdam.org/articles/10.1051/ita:2007050/ | On the continuity set of an Omega rational function
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 183-196.
In this paper, we study the continuity of rational functions realized by Büchi finite state transducers. It has been shown by Prieur that it can be decided whether such a function is continuous. We prove here that surprisingly, it cannot be decided whether such a function $f$ has at least one point of continuity and that its continuity set $C\left(f\right)$ cannot be computed. In the case of a synchronous rational function, we show that its continuity set is rational and that it can be computed. Furthermore we prove that any rational ${\Pi }_{2}^{0}$-subset of ${\Sigma }^{\omega }$ for some alphabet $\Sigma$ is the continuity set $C\left(f\right)$ of an $\omega$-rational synchronous function $f$ defined on ${\Sigma }^{\omega }$.
DOI : https://doi.org/10.1051/ita:2007050
Classification : 68Q05, 68Q45, 03D05
Mots clés : infinitary rational relations, omega rational functions, topology, points of continuity, decision problems, omega rational languages, omega context-free languages
@article{ITA_2008__42_1_183_0,
author = {Carton, Olivier and Finkel, Olivier and Simonnet, Pierre},
title = {On the continuity set of an {Omega} rational function},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {183--196},
publisher = {EDP-Sciences},
volume = {42},
number = {1},
year = {2008},
doi = {10.1051/ita:2007050},
zbl = {1149.03028},
mrnumber = {2382551},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita:2007050/}
}
TY - JOUR
AU - Carton, Olivier
AU - Finkel, Olivier
AU - Simonnet, Pierre
TI - On the continuity set of an Omega rational function
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2008
DA - 2008///
SP - 183
EP - 196
VL - 42
IS - 1
PB - EDP-Sciences
UR - http://www.numdam.org/articles/10.1051/ita:2007050/
UR - https://zbmath.org/?q=an%3A1149.03028
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UR - https://doi.org/10.1051/ita:2007050
DO - 10.1051/ita:2007050
LA - en
ID - ITA_2008__42_1_183_0
ER -
Carton, Olivier; Finkel, Olivier; Simonnet, Pierre. On the continuity set of an Omega rational function. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 183-196. doi : 10.1051/ita:2007050. http://www.numdam.org/articles/10.1051/ita:2007050/
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# Hi , Iam ananya new to java tech,
ananya sa
Greenhorn
Joined: Sep 30, 2009
Posts: 3
I wrote simple java pgm with just one sysout., when i try to run it,,, iam getting problem like
Usage: javac <options> <source files>
use -help for a list of possible options
I put java path in my environment variables like this:
C:\Program Files\Java\jdk1.6.0\bin;C:\Program Files\Java\jre1.6.0\bin.
thanks,
Ananaya.
Bear Bibeault
Author and ninkuma
Marshal
Joined: Jan 10, 2002
Posts: 58817
59
Henry Wong
author
Sheriff
Joined: Sep 28, 2004
Posts: 17609
33
The Java compiler is complaining that it can't find the file named -- testpgm1.java.
Henry
Muhammed Patel
Greenhorn
Joined: Feb 26, 2009
Posts: 5
Hi ananya ,
Firstly on your path does not need to have both the C:\Program Files\Java\jdk1.6.0\bin and C:\Program Files\Java\jre1.6.0\bin. the C:\Program Files\Java\jdk1.6.0\bin is fine.
Also you should set C:\Program Files\Java\jdk1.6.0 to a new environment variable called JAVA_HOME and then in your Path environment variable you can add the following to the text : %JAVA_HOME%\bin .
That will just sort out your java paths.
Now with the problem that you are having!
1. Have you changed directory to the right directory. So if testpgm1.java is in c:\javafiles have you changed directory to it.
2. Make sure that you have the correct file name or you could also make sure that the name of the file is the same as the class it contains.
Here is a simple class. try saving it as Blah.java putting it in your C:\ and then compiling and running it with the following commands
C:\>javac Blah.java
C:\>java Blah
BLAH
Hope this helps.
And good luck.
In my world they are all ponies and eat rainbows and poop butterflies
ananya sa
Greenhorn
Joined: Sep 30, 2009
Posts: 3
Thanks,
It works now, i given the wrong path while compiling. that's the problem.
Muhammed Patel wrote:Hi ananya ,
Firstly on your path does not need to have both the C:\Program Files\Java\jdk1.6.0\bin and C:\Program Files\Java\jre1.6.0\bin. the C:\Program Files\Java\jdk1.6.0\bin is fine.
Also you should set C:\Program Files\Java\jdk1.6.0 to a new environment variable called JAVA_HOME and then in your Path environment variable you can add the following to the text : %JAVA_HOME%\bin .
That will just sort out your java paths.
Now with the problem that you are having!
1. Have you changed directory to the right directory. So if testpgm1.java is in c:\javafiles have you changed directory to it.
2. Make sure that you have the correct file name or you could also make sure that the name of the file is the same as the class it contains.
Here is a simple class. try saving it as Blah.java putting it in your C:\ and then compiling and running it with the following commands
C:\>javac Blah.java
C:\>java Blah
BLAH
Hope this helps.
And good luck.
ananya sa
Greenhorn
Joined: Sep 30, 2009
Posts: 3
hi , it compiled properly, but trying to run, it is saying :
C:\Documents and Settings\sri\Desktop>java testpgm1
Exception in thread "main" java.lang.UnsupportedClassVersionError: testpgm1 (Uns
upported major.minor version 50.0)
3)
at java.security.AccessController.doPrivileged(Native Method)
As you said i just kept C:\Program Files\Java\jre1.6.0\bin; in CLASS_PATH of environmntal variables .
What needs to do now?
Henry Wong
author
Sheriff
Joined: Sep 28, 2004
Posts: 17609
33
What needs to do now?
This is caused when you compile your program with one version of Java, but then run it with an earlier version of Java.
Basically, Java is backward compatible. It is not forward compatible. So, you need to run your program with a version that is at least the same or later than your compiler.
I am guessing that you have an older version of Java already installed (but no compiler already installed). So, when you compiled, it uses the compiler you installed, but when you run it, it uses the older previously installed JVM.
Henry
I agree. Here's the link: http://aspose.com/file-tools
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Problem Detail:
Let $A$ be a sorted array. Suppose that this array is huge and can only be stored in a disk. In the I/O model and algorithms built in this model, every time we access the disk we spend an I/O and we get $B$ elements to the RAM. Since accessing the disk is much more expensive than accessing the main memory, we assume that what ever operations we perform in main memory, they come for free. So even if you scan 1 million times the entire RAM, in the I/O model, this would cost $O(1)$ I/Os, and that's the only thing we care about.
Suppose that you now receive queries in the form $[l,r, key]$ where $l$ and $r$ are indices which define a continguous chunk of the array $A$, and $key$ is a value that you want to search inside this chunk. So you want to know whether the key is somewhere in that chunk or not.
If you expect many queries to come, and we will make this assumption as well, it might be worth spending $O(\frac{N}{B})$ I/Os to build a B tree on top of the array $A$, and use the $B$ tree for your searches. Since the height of a B tree is $O(\log_{B}(N))$ each search will require $O(\log_{B}(N))$ I/Os.
However I was wondering, if the size of the chunk defined by the query is $n$, is it possible to use the same B tree to achieve $O(\log_{B}(n))$ I/Os? Because if we always start searching from the root of the B tree, we would first have to find where our chunk is, and then search inside the chunk, which is why the total number of I/Os would be $O(\log_{B}(N))$.
My idea would be to spend 1 I/O to access the leaf block in the B tree containing element $A[l]$, 1 I/O to access the leaf block in the B tree containing element $A[r]$, and then spend $O(\log_{B}(n))$ I/Os to find the least common ancestor block in the B tree of these two leaf blocks. Then since we found the least common ancestor block in the B tree, we can start searching from there, spending in total $2 + O(\log_{B}(n)) + \log_{B}(n)$ I/Os.
Unfortunately I am very novice in the I/O model and this approach sounds too good to be correct. Is it really that easy for this problem or am I missing something?
I assume the "B-tree" in your post is in fact a "B+-tree" and array $\small A$ is stored in an increasing manner. Your idea is almost good and needs some small revision.
• You need to test whether $\small A[l] \leq key$ and $\small A[r] \geq key$. If $\small A[l] = key$ or $\small A[r] = key$, it is done.
• If it is found that $\small A[l] < key$ and $\small A[r] > key$, find the LCA of the blocks containing $\small A[l]$ and $\small A[r]$, as you have done. The height of the LCA is of $\small \mathcal{O}(\log_B(n))$.
• If $\small A[l] > key$ or $\small A[r] < key$, then obviously $\small key$ does not exist in $\small [l, r]$.
Why testing $\small A[l] \leq key$ and $\small A[r] \geq key$ is important?
The LCA block may contain entries that direct to blocks outside $\small [l, r]$. If you start from the LCA, you may find a $\small key$ even if $\small key$ does not exist in $\small [l, r]$.
Question Source : http://cs.stackexchange.com/questions/64296
3200 people like this
Problem Detail:
Given a multiset $X=\{x_1,\dots,x_n\}$ where every element $w_i$ is a power of two, and given an integer $M$, I'd like to determine if there is any subset of $X$ that sums to $M$.
(This question is based on a previous question about the subset problem where every number in the set is a multiple of two, but this time $X$ consists only of powers of two.)
Can we find an efficient algorithm for this problem? Could we make it any better than standard algorithms for subset sum or knapsack?
###### Answered By : Aleksi Torhamo
First, look at the bits that are set in $M$ and write it as $M = m_1+m_2+\dots$ where each $m_i$ is a power of two and such that $m_1$ is the smallest power of two, and so on.
If the numbers $x_i \in X$ are distinct powers of two and you can use each one zero or one times, the problem is trivial: If all $m_i$ are found in $X$, then $M$ can be formed by them, otherwise it can't. This is because missing powers of two can't be formed by a) any combination of larger powers of two or b) distinct smaller powers of two.
If the numbers $x_i$ aren't guaranteed to be distinct, it's a bit more complex. Missing powers of two still can't be formed by larger powers of two, but they can be formed by non-distinct smaller powers of two.
Let's look at how smaller powers of two can form larger powers of two. If you have powers of two $A = [a_1, \dots, a_n]$ that are smaller than a power of two $N$, and $S=\sum_{i=0}^n a_i$, then $N$ can be formed by them iff $S \ge N$. Let $b$ be the smallest power of two in $A$ and $c$ its number of occurences in $A$. If $c$ is odd, one $a_i=b$ can't combine with anything to form a higher power of two and the corresponding bit will be set in $S$. The others can be replaced with $\lfloor \frac{c}{2} \rfloor$ copies of the number $2b$ in $A$. If we continue this with higher powers of two until we get to $N$, we find that we can form $\lfloor \frac{S}{N} \rfloor$ copies of $N$ with $A$.
So, start with $m_1$. If it's found in $X$, we remove it and move to the next power of two. If it's not, we calculate the sum $S$ of all $x_i$ smaller than $m_1$ and remove them from $X$. If $S \lt N$, then $M$ can't be formed by $X$. Otherwise, we add $\lfloor \frac{S}{m_1} \rfloor - 1$ copies of $m_1$ to $X$ and continue in the same manner with the next smallest power of two in $M$.
Question Source : http://cs.stackexchange.com/questions/66136
3200 people like this
Problem Detail:
How many different binary search trees are possible that store the values 1,2,...,n ?
So far I found a recursive formula for the number (by case distinction what's at the root):
$T(n) = 2T(n-1) + \sum_{i=2}^{n-1}T(i-1)T(n-i), n > 1$ and $T(1) = 1$
But I have no idea how to solve this recursion. Our task was only to find the recursion and I believe this to be a correct solution. But I am very interested in a closed formula of it. Can anyone link me to some resources/books or give a general hint on how it can be solved?
###### Answered By : Yuval Filmus
The solution to your recurrence is $$T(n) = \frac{(2n)!}{n!(n+1)!},$$ also known as the Catalan numbers. The quickest way to find this is by computing a few elements of the sequence and using the OEIS to identify the sequence.
Question Source : http://cs.stackexchange.com/questions/66617
3200 people like this
Problem Detail:
I understand that the Backus-Naur Form is utilized to assess and specify type 2 grammar, but is it in itself a form of type 2 grammar?
###### Answered By : André Souza Lemos
Backus-Naur Form is a conventional notation for writing Context-Free Grammars.
Its usefulness comes from the need to distinguish the character encoding standards that are used to write texts in various computer languages from the higher-order alphabets of terminal and non-terminal symbols (corresponding to lexical units and syntactical variables, respectively) used in the definition of grammars.
The very idea of using CFGs to define the syntax of high-level programming languages was remarkably precocius, which is another reason why the work of John Backus and Peter Naur was so appreciated at the time.
Question Source : http://cs.stackexchange.com/questions/66686
3200 people like this
Problem Detail:
A (non-deterministic) finite state automaton for "all strings that contain substring $w$" is very simple. Just make the path for $w$ and add looped transitions for all letters at the first initial state and the last final state.
When asked to make a deterministic automaton in general we might expect that the automaton can be of exponential size (see the warning of Yuval in the case where $w=101$: Write a regex to match string that does NOT contain a certain pattern).
However, Knuth-Morris-Pratt tells us that a pattern match automaton for $w$ is of linear size, and can be constructed in linear time. That means that the deterministic automaton for "contains substring $w$" is of linear size too (failure links can be replaced by transitions, on for each mismatching letter). Luke has used KMP to provide an automaton for the example "not ABCDABD" in his answer Automaton for substring matching.
My question is the following: will the standard determinization of a "substring $w$" automaton always lead to a linear sized automaton? Here I mean the practical version of the construction, where unreachable states are not constructed (so not the full $2^Q$ subset construction). Is the answer direct from the construction, or do we need KMP?
A little observation. In the construction there might be many states that contain the original final state. As they all will accept and also all transitions for ever will lead to accepting states (once we have seen $w$) these states can be grouped into one single state. I do not know whether that is a necessary trick to avoid exponential explosion, see http://cs.stackexchange.com/a/11842/4287
###### Answered By : Yuval Filmus
Consider the following NFA: the states are $q_0,\ldots,q_{|w|}$, there are transitions $q_{i-1} \to q_i$ on $w_i$, there are self-loops on $q_0,q_{|w|}$ (for all alphabet symbols), the initial state is $q_0$, and the unique final state is $q_{|w|}$.
Assuming $w \neq \epsilon$, the determinization generates $2|w|$ states:
• For each location $0 \leq i < |w|$, a state consisting of $q_0$, $q_i$, and $q_j$ for $j < i$ such that $w_1\ldots w_j$ is a suffix of $w_1\ldots w_i$.
• Same as before, with $q_{|w|}$ added.
When $w = \epsilon$, the NFA is already deterministic, and contains only one state.
Question Source : http://cs.stackexchange.com/questions/66535
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Problem Detail:
On page 38 of "Lecture Notes in Computer Science" by Christoph M. Hoffmann, there is an algorithm (ALGORITHM 2).
I have some confusions.
Why it is written that an entry $M_{i,j}, j < i$, cannot be referenced? what is the meaning of "reference" here?
Thanks.
###### Asked By : Mike SQ
One possible explanation is that it is indicating that you should never try to read or use any matrix element $M_{i,j}$ where $i<j$. "Referenced" might be talking about "de-referencing" a pointer, i.e., reading a memory cell, i.e., reading an entry of the matrix. Perhaps the text is indicating that the matrix entries $M_{i,j}$ are only defined for $i \ge j$.
I suggest you read the text with this possible perspective in mind and see if it seems consistent with the surrounding context.
Question Source : http://cs.stackexchange.com/questions/60626
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Problem Detail:
I am using a sensing board able to detect magnetic signals between the board and a display.
I have a set of objects that are represented (each of them) by a unique set of points (magnets) with a particular shape. For example: object #1 is made by three points that form an equilateral triangle with side length 1cm; object #2 is made by three points that form a right triangle with sides 3cm, 4cm, 5cm; object #3 is made by three aligned points with distance 2cm; and so on. I can have a multiplicity of objects with unique patterns.
Now I have a list of points with the coordinates w.r.t. the Cartesian plane, and I need to match them referring to the patterns I got from the objects. I also know that every point must be matched, therefore I can minimize the overlapping errors. In practice, every point in the set can belong to maximum one object, and at the same time also it must belong to an object of the initial set.
Any idea on how to do that in an efficient way?
###### Answered By : Karolis Juodelė
In the general case a problem like this is NP, however in the vast majority of real cases it should be easy.
20 points make 1140 triangles so it shouldn't be hard to pick out the triangles most similar to your basic shapes (unless the shapes can be more complicated). A little ugly backtracking may be needed when the top scoring triangles overlap.
Also, if most magnets move continuously, you can easily map old points to new points and old triangles to new triangles.
What I'm talking about are fairly obvious methods. There may be smarter ways to do this, but you don't necessarily need them.
Question Source : http://cs.stackexchange.com/questions/66144
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Problem Detail:
I've seen the topic of the automorphism group appear in several introductory graph theory books I've looked at. It always feel oddly disjointed and poorly motivated to me.
Is there any practical (or impractical for that matter) applications of knowing the automorphism group of a graph?
###### Answered By : Stella Biderman
Automorphism capture a natural notion of symmetry of graphs. As a result, they can be used to speed up algorithms that would otherwise run slowly by chopping down the search space.
For example, integer programming is usually solved via branch-and-bound. However, if an equation is degenerate this can take far too long to run, because it has to keep checking symmetric parts of the tree. We can use graph automorphisms to compute the orbits of variables in the linear programming problem, and then treat parts with the same orbit as identical. A recent application of such techniques to MILP can be read here.
Question Source : http://cs.stackexchange.com/questions/65391
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Problem Detail:
Since parity game solving is in TFNP ("Total Function Nondeterministic Polynomial") (and the decision version is is NP ∩ coNP), I wonder whether it is contained in PLS ("Polynomial Local Search") or PPA ("Polynomial Parity Argument")? Add PPP ("Polynomial Pigeonhole Principle") if you want, even so this would probably mean that it is already contained in PPAD ("Polynomial Parity Arguments on Directed graphs"), and hence in PPA.
Or is it rather the other way round and parity game solving can be shown to be hard for PLS or PPAD? But that would be surprising, since a recursive algorithm that solves parity games is known (even if it is not efficient in the worst case).
###### Answered By : Rahul Savani
Yes, solving parity games is known to be in PPAD (and thus PPA and PPP too) and PLS, and is thus unlikely to be hard for either (since this would imply containment of one of these classes in the other).
See, e.g.,
Daskalakis, Constantinos, and Christos Papadimitriou. "Continuous local search." Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms. SIAM, 2011.
and combine the membership of Simple Stochastic Games (SSGs) in CLS (which is in PPAD and PLS) with the well-known observation that solving parity games can be reduced to solving SSGs in polynomial time.
The reason that these problems are in PPAD is that they admit "optimality equations", rather like Bellman equations, that characterize solutions as fixed points. The reason these problems are in PLS is that they can be solved with local improvement algorithms like strategy improvement (a two-player generalization of policy iteration for MDPs).
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Problem Detail:
I have 2 search algorithms and I have derived the following tight bound representations:
$$nlog(n)+mlog(n)$$
$$m∗n$$
Now i want to find a function $f(n)$ so that when $m$ is an element of tight bound $f(n)$, both algorithms have equal asymptotic run time.
Now I'm not sure if its just as simple as setting the two equations equal to each other and isolating 'm' or of it is more than tha
###### Answered By : Rick Decker
Your approach, for this problem at least, will work, but there are interesting things happening in the background. Simply setting the two terms equal and solving for $m$ will give you $$m=\frac{n\log n}{n-\log n}$$ However, I doubt that you want to substitute this back into the two original expressions and try to find a $g(n)$ so that each of your original expressions is in $O(g(n))$. Let's try something else.
For no particular reason, let's try letting $m=\log n$. Then your two expressions become \begin{align} n\log n+m\log n&=n\log n+\log^2n \in O(n\log n)\text{, and}\\ n\:m&=n\log n\in O(n\log n) \end{align} Hooray! If $m=f(n)=\log n$ we get a common upper bound with $g(n)=n\log n$.
Let's do the same thing, now with $m=n$. The two expressions are now \begin{align} n\log n+m\log n&=n\log n+n\log n \in O(n\log n)\text{, and}\\ n\:m&=n\; n\in O(n^2) \end{align} Drat. No common upper bound, but by transitivity, we have $n\log n+m\log n\in O(n^2)$ and so again, we find a common upper bound, namely the asymptoticly larger $g(n)=n^2$.
You can do this for all kinds of other $f(n)$, like $f(n)=\log\log n$ and even $f(n)=1$, both of which have common solutions $g(n)=n\log n$. In fact, it seems that we can do this for any $f$ whatsoever.
Question Source : http://cs.stackexchange.com/questions/64012
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Problem Detail:
Are there any special algorithms for maximum independent set of line graphs? Could this special case be in $\mathsf{P}$?
Finding a maximum independent set in $L(G)$ is equivalent to finding a maximum matching in $G$. For more, and a fast polynomial-time algorithm that work for e.g., line graphs, see [1].
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Problem Detail:
I want to add a constraint to a convex program, to guarantee some matrix $A$ to be positive semidefinite. How should I do it?
The library I am working with can cope with linear/ quadratic inequalities only.
By definition, $A$ is positive semidefinite iff $\forall x \in \mathbb C^n : x^T A x \geq 0$, but this is a set of inifinitely many constraints. So, my question is: how can I formulate it using a finitely many set of contraints and using linear/ quadratic inequalities only.
###### Asked By : Dudi Frid
A matrix $A$ is positive semidefinite if and only if there exists a matrix $V$ such that $$A = V^\top V.$$ So, you can use the entries of $V$ as your unknowns, and express each entry of $A$ as a quadratic function of the unknowns. Whenever you want to use $A$, instead rewrite that equation in terms of the entries of $V$.
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Problem Detail:
Ok, this is a follow up question to this, about the Erathostenes Sieve.
If we look at this paper (page 3, footnotes), the author says:
If we start crossing off at $p^2$ rather than $p$, the number of composites we cross off is $\frac{n}{p}−p+1$, but it makes no significant difference to our sum, because it only subtracts an irrelevant $O(n/ \log n)$ factor
The total cost counting crossings is:
$$\sum_{p\leq n}\bigl(\frac{n}{p}-p+1\bigr)=n\sum_{p\leq n}\frac{1}{p}-\sum_{p\leq n}p+\sum_{p\leq n}1$$
The first sum would give us the known $O(n\log\log n)$. But I don't see the second sum to be an "irrelevant $O(n/\log n)$", so how is that?
According to this post in math overflow an upper bound for the sum of the primes up to $n$ is: $$\frac{n^2}{2\log n} + O\left(\frac{n^2}{\log^2 n}\right).$$
###### Answered By : Yuval Filmus
The sum only goes up to $\sqrt{n}$. You can see that in the big display at the bottom of page 3, where only the first $\pi(\sqrt{n})$ primes are considered. Substituting $\sqrt{n}$ for $n$ in your formula, we get $n/\log n + O(n/\log^2 n)$.
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Problem Detail:
Wythoff's game is as follows: there are two players $A$ and $B$ ( $A$ being the first player ) and there are $2$ piles of stones. When his turn a player can remove one or more stones from anyone pile or same number of stones from both the piles. A player unable to make a move loses. Also it is assumed that both the players play optimally.
As mentioned in the link the losing configurations $(n_k,m_k)$ ( $n_k$ stones in one pile and $m_k$ stones in another ) are given by $$n_k = \lfloor( k*\phi) \rfloor$$ $$m_k = \lfloor( k*\phi^2) \rfloor$$ where $k$ is any natural number ( and $n_k \le m_k$ ) and $\phi=\frac{1+\sqrt{5}}{2}$. For example $\{1,2\}$ ( two stones in one pile and one stone in another ) is a losing position.
Modification to Wythoff's game"
In this modified game instead of ${\it2}$ piles there are ${\it 3}$ piles. And when his turn a player can move one or more stones from anyone pile or same number of stones from any $2$ piles or same number of stones from all the $3$ piles.
How do I compute the losing positions for this modified game efficiently ? The inefficient way of course is to find the grundy number of each configuration $\{a,b,c\}$. This method is inefficient because I want to calculate number of losing positions given that number of stones in each pile can be between $1$ and $1000$.
This is a project Euler question ( stone game ). I would appreciate hints only.
There is no closed form for $n=3$ piles. But one can solve the question efficiently using dynamic programming.
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Problem Detail:
Supposed I have a set of simple rules, e.g.
if A is between 10 and 20 and B is between 3 and 5, then use 3
and this rules are used in some kind of (more or less) mathematical formula (for example, A + B = C), would you count this as a form of reasoning or inference of a knowledge based system?
You have a knowledge base (the rules) and you get a result (a decision) by applying this rules in a specific way.
You describe what is called expert system in AI. As such, they are artifacts of articial intelligence research.
As far as the Wikipedia definition goes,
a reasoning system is a software system that generates conclusions from available knowledge using logical techniques such as deduction and induction,
which is vague enough to cover expert systems.
Is there any "intelligence" or independent reasoning in these systems? Certainly not; they are "stupid" in the sense that they just look up values in a table; no learning happens.
You will have to decide which frame of terminolgy reference you want to apply: academic AI or common sense?
Question Source : http://cs.stackexchange.com/questions/59494
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Problem Detail:
I have these 4 symbols with their probabilities:
x P(x) -------- 1 0.3 2 0.3 3 0.2 4 0.2
I built the Huffman tree in this way:
and I obtainded:
x P(x) C(x) ---------------- 1 0.3 0 2 0.3 10 3 0.2 110 4 0.2 111
it's correct? Because according to the solution the results should be:
x P(x) C(x) ---------------- 1 0.3 00 2 0.3 01 3 0.2 10 4 0.2 11
Why? Yet I followed the steps shown here.
###### Answered By : Denis Pankratov
A quick way to check whether your answer has a chance of being correct is to compute the average code length. Your encoding gives the average length of $2.1$, which is greater than using a code of fixed length $2$, so it can't be correct.
If you follow the priority queue algorithm from the source you cite, then you would notice that after merging nodes 3 and 4 you get one supernode of priority 0.4. Now your queue would have three elements of priorities $0.3, 0.3,$ and $0.4$. Thus, you would next merge elements corresponding to priorities $0.3$ and $0.3$ (the algorithm works by merging two nodes with lowest priorities), which happen to be nodes 1 and 2.
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Problem Detail:
I was reading a paper and I came to the following :
"Since independent set is $Ω(n^{1−\epsilon})$-inapproximable unless P=NP (see [19]) for any fixed $\epsilon> 0$, the ..." where [19] is the following article : D. Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In STOC, pages 681–690, 2006.
My question: is it true that independent set is $Ω(n^{1−\epsilon})$-inapproximable unless P=NP? Because I cannot find this result in the cited article. However, I can find the results about max clique and chromatic number. What I am missing?
Yes, this is true.
To see this, notice that cliques and independent sets are complementary. That is, a set of vertices $S$ is independent precisely when $S$ forms a clique in the complement of the graph. Intuitively, this means that if you had an approximation algorithm for finding a maximum clique, you could use it to find a maximum independent set by executing it on the complement graph.
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Problem Detail:
I don't fully understand this build-heap function. Lets assume we have array 3, 4, 5, 13, 16, 32.
It seems like we swap the parent when it is less than the current A[j] but which number does the loop start with? Maybe somebody can go through 2-3 loops and show how the array changes after 1st loop, 2nd loop, 3rd loop. Much appreciated. Oh, also, what would be the runtime?
for i=2 to n j = i while (j>1) and A[parent(j)]<A[j] do swap A(parent(j)] and A[j] j = parent(j)
###### Answered By : Rick Decker
It appears you're building a max-heap, where every element is greater than or equal to its child elements (if any). With that understanding, let's trace the action. First, the parent node of $A[j]$ will be $A[j/2]$ (integer division: discard any remainder) so we'll have $$\begin{array}{r|cccccccc} j & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \dotsc\\ parent(j) & \_ & 1 & 1 & 2 & 2 & 3 & 3 & \dotsc \end{array}$$ To keep things simple, we'll let the initial array be $[3,4,5,13]$:
Insert at $i=2$.
$A[parent(2)]=A[1]=3 < 4=A[2]$ so we swap $A[1]$ and $A[2]$, giving us the array $$[4,3,5,13]$$
Insert at $i=3$.
$A[parent(3)]=A[1]=4 < 5=A[3]$ so we swap $A[1]$ and $A[3]$, giving us the array $$[5,3,4,13]$$
Insert at $i=4$.
$A[parent(4)]=A[2]=3 < 13=A[4]$ so we swap $A[2]$ and $A[3]$, giving us the array $$[5,13,4,3]$$ and now $j=parent(4)=2>1$ so we see if we need another swap.
$A[parent(2)]=A[1]=5 < 13=A[2]$ so we swap $A[1]$ and $A[2]$, giving us the array $$[13,5,4,3]$$ and we're done, the array is now a max-heap.
The runtime of this algorithm is no worse than a multiple of $n\log n$ since none of the elements are further than $\log_2n$ from the root at $i=1$ so you'll need at most $\log n$ swaps for each of the $n$ elements. This, by the way, is not as good as possible: there's different algorithm that builds a heap in no more than a multiple of $n$ time.
Question Source : http://cs.stackexchange.com/questions/54427
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Problem Detail:
I have a string with length N. I would like to know how many segmentations are possible to it. Consider the example abcdc the number of N = 5 All possible segmentations are
['abcdc'] ['abcd', 'c'] ['abc', 'dc'] ['abc', 'd', 'c'] ['ab', 'cdc'] ['ab', 'cd', 'c'] ['ab', 'c', 'dc'] ['ab', 'c', 'd', 'c'] ['a', 'bcdc'] ['a', 'bcd', 'c'] ['a', 'bc', 'dc'] ['a', 'bc', 'd', 'c'] ['a', 'b', 'cdc'] ['a', 'b', 'cd', 'c'] ['a', 'b', 'c', 'dc'] ['a', 'b', 'c', 'd', 'c']
Then what will happen when my N tends to infinity . Any closed form equations?
There are n-1 points where you can break the string. Each is independent of the others. Therefore there are $2^{n-1}$ possibilities to break the string.
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Problem Detail:
Let's say we have a tree like this:
Suppose we are given $N$ distinct elements, $N$ being the number of vertices in the tree (in this case, $N=13$).
In how many ways can we distribute the given $N$ elements in the tree so that it has the max-heap property (any vertex is larger than its children)?
###### Answered By : Yuval Filmus
You can compute this recursively as follows. It will be convenient to denote (binary) trees using the following notation: a tree is either a leaf $\ast$ or is formed from two subtrees $(T_1,T_2)$. For a tree $T$, let $|T|$ denote the number of vertices in $T$, defined inductively by $|\ast|=1$ and $|(T_1,T_2)| = |T_1|+|T_2|+1$. Finally, let $N(T)$ denote the number of ways of allocating the integers $1,\ldots,|T|$ to $T$ so that the max-heap property is satisfied.
The base case is $N(\ast)=1$. Now consider a tree $T=(T_1,T_2)$. Clearly we must put $|T|$ at the root. Every filling out of the rest of the tree can be described as follows:
• Choosing a set $S_1$ of integers as labels for $T_1$. The other elements $S_2$ function as labels for $T_2$.
• Arranging $S_1$ in $T_1$ in a way satisfying the max-heap property.
• Same for $S_2$ and $T_2$.
The number of ways of accomplishing the second bullet is $N(|T_1|)$, and the number of ways of accomplishing the third bullet is $N(|T_2|)$; in both cases, the exact contents of $S_1$ and $S_2$ are not important. The number of choices in the first bullet is $\binom{|T|-1}{|T_1|}$, and we conclude that $$N((T_1,T_2)) = \binom{|T_1|+|T_2|}{|T_1|}N(T_1)N(T_2),$$ with initial case $N(\ast) = 1$.
Using this formula you can easily compute that for your tree the number of possibilities is $304128$ (I believe).
You can also "unroll" the formula: $$N(T) = \prod_{x=(x_1,x_2) \in T} \binom{|x_1|+|x_2|}{|x_1|},$$ where $x$ goes over all internal nodes in $T$. For example, for your tree we get (top to bottom, left to right) $$\binom{12}{5} \binom{4}{3} \binom{6}{5} \binom{2}{1} \binom{4}{1} \binom{2}{1} = 304128.$$
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Problem Detail:
Some time ago, I asked this question but no one quite understood it or was able to answer. I deleted the original question and have since decided to try it again.
As I understand it, all things digital are originally based on ones and zeros - binary code.
However, I have wondered for some time if it would be possible (now or in the future) to use the digits of pi (22/7) in place of the ones and zeros.
So, my question, is it possible? Could it ever be?
No it cannot happen for many reasons.
How would logic look like?
How would you add two numbers?
There is a big problem with telling apart $0$ and $1$ at high frequencies, so adding third option would be harder to manufacture. But encoding it with non-natural basis gets harder.
With non-natural base, all operations that we do are doomed.
If you consider that, try easy example: convert $4$ and $6.5$ into $\pi$ base, add them and write down result.
Since the finite precission kicks in, your basic addition fails. The only operation that would benefit from such base is $\pi + \pi$.
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Problem Detail:
In this paper,
http://www.vision.caltech.edu/malaa/publications/aly08realtime.pdf
the author mentioned "the vertical and horizontal focal lengths" but didn't have any clear definition.
My understanding about "focal length" is the distance between the lens and the image plane.
Does anyone have the idea about these focal lengths?
Thank you.
Note that they assume a pinhole camera model. The term 'focal length' means something different here than it does with a lens camera.
All you ever really need to know about the pinhole camera model can be expressed as: $$x_\mbox{imageplane} := x_\mbox{world}/z_\mbox{world}$$ $$y_\mbox{imageplane} := y_\mbox{world}/z_\mbox{world}$$ This describes how points in the scene project to an image plane 1 unit in front of the camera origin (center of projection & center of world coordinate system), with the optical axis along the z direction.
All the other values usually discussed in connection with pinhole cameras 'just' describe which rectangular part of the image plane is mapped to the rectangle of pixels that is your image. Or, conversely, how pixel coordinates can be transformed back into points on the image plane (and from there, can be transformed to directions in the scene).
Of course this rectangle-on-a-2d-plane can be defined by a 2d point $b$ and two 2d vectors $v_1, v_2$ such that:
$$\begin{bmatrix}x_\mbox{image}\\y_\mbox{image}\end{bmatrix} := \begin{bmatrix}\vec{v_1}&\vec{v_2}\end{bmatrix} \begin{bmatrix}x_\mbox{imageplane}\\y_\mbox{imageplane}\end{bmatrix} + b$$ If we abbreviate this as $(x_\mbox{image},y_\mbox{image}) = f(x_\mbox{imageplane},y_\mbox{imageplane})$ the part of the imageplane that your image depicts is the set $f^{-1}([0,\mbox{width}]\times[0,\mbox{height}])$ which, as I said, is a rectangle1.
The 'other values', for example:
• optical center: $c_x, c_y$ or $c_u, c_v$
• horizontal and vertical 'focal length': $f_x, f_y$ or $f_u, f_v$
• aspect ratio
• field of view angle $\alpha$
• shear (you don't usually want that)
are just used to describe special cases of the above formula.
In your case, $b = (c_u, c_v)$ and $v_1 = (f_u,0), v_2 = (0,f_v)$ such that
$$\begin{bmatrix}x_\mbox{image}\\y_\mbox{image}\end{bmatrix} := \begin{bmatrix}f_u x_\mbox{imageplane} + c_u\\f_v y_\mbox{imageplane} + c_v\end{bmatrix}$$
1 Note that it is not the rectangle with corners $b, b+v_1,b+v_2,b+v_1+v_2$. To find it you'd have to compute the inverse of the function $f$.
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Problem Detail:
So, untyped lambda calculus has the following formal grammar for its terms:
$$e::= x \mid \lambda x . e \mid e_1 e_2$$
Usually this is presented in some ML-esque language as (using de Bruijn indices)
data term = variable Nat | lambda term | apply term term
My question is: apply (variable Nat) term is syntactically valid, but the rator is just a free variable, isn't this an invalid expression? If not, what does it evaluate to?
###### Answered By : Anton Trunov
You can't evaluate (x t), where x is free, since evaluation ($\beta$-reduction) is usually defined in terms of substitution, but here you have neither a bound variable nor a function body for $t$ to be substituted into.
A term that cannot take a further step is called a normal form.
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Problem Detail:
On the 6th slide at https://web.stanford.edu/class/ee364b/lectures/bb_slides.pdf, while defining L2 and U2, why are we taking min for both?
###### Asked By : Deepankar Arya
The slides are correct: remember that we are trying to find a bound on the optimal value of $f$, which is defined as $$\Phi_\mbox{min}({\cal Q}_\mbox{init}) =\min_{x\in{\cal Q}_{\tiny\mbox{init}}}f(x)$$
To find that minimum, we approximate it by upper and lower bounds $\Phi_\mbox{ub}$ and $\Phi_{\mbox{lb}}$. Let's concentrate on $\Phi_{\mbox{ub}}$. Note that if ${\cal Q}={\cal Q}_1\cup{\cal Q}_2$, then both $$\min(\Phi_{\mbox{ub}}(\cal Q_1),\Phi_{\mbox{ub}}(\cal Q_2))\geq\Phi_\mbox{min}(\cal Q)$$ and $$\max(\Phi_{\mbox{ub}}(\cal Q_1),\Phi_{\mbox{ub}}(\cal Q_2))\geq\Phi_\mbox{min}(\cal Q)$$
But we might as well take $\min$, since that is the value that is closer to the global minimum $\Phi_\min(\cal Q)$!
In fact, taking the minimum is going to be necessary if we want the branch and bound method to converge to the minimum as we sub-divide $\cal Q$ into smaller and smaller pieces.
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Problem Detail:
We have Stocks in several discrete positions, let say:
A B C 40 20 80
And Demand, which may be satisfied by one or more Stock positions (if no specified then it would accept any Stock).
Demand for A=20 may be satisfied from A Stock only, and depletes 20 units from it. Demand for AB=20 may be satisfied from any of the A abd B Stocks, and may take partially from both, without any proportion limitations.
So for the Stocks example above:
This Demand may be satisfied (10 strict from A, 20 strict from B, and 30 AB would take from both)
AB B A 30 10 20
And this may not (insufficient B Stocks)
AB B B 30 10 20
This sounds like a more or less common problem which should have a known solution. I would be happy with any reference or idea. Also if the solution is resource heavy a cheaper approximation would be great.
###### Answered By : Tom van der Zanden
You can formulate this as a maximum flow problem, which you can solve using standard techniques.
Turn every stock in to a source node with capacity equal to the amount of stock. Any demand gets turned in to a sink with demand equal to its demand. Add edges between stocks and demands if the stock can fulfill the demand.
If you don't want to have more than one sink or source that is possible too: you create the stock and demand nodes as before, and create an edge between the sink and all stock nodes (with capacity equal to the amount of stock available), similarly create an edge between all demand nodes and the sink (with capacity equal to the demand).
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Problem Detail:
I A many one reducable to B and given A is decidable then is B decidable ?
preparing for an exam and please let me know if this holds
I understood how if B is decidable then A is decidable
and if A is not decidable then B is not decidable
###### Answered By : Rick Decker
If $A$ is reducible to $B$ and $A$ is decidable, then $B$ is not necessarily decidable.
Suppose, for example that $A=\{1\}$ and $B$ is, say, $\{\langle\, B\,\rangle\mid L(M)\text{ is infinite}\}$. It's well-known that $B$ is undecidable and of course $A$ is decidable (any finite language is decidable).
Now let $X$ be a TM for which $L(X)=\Sigma^*$. We could pick $X$ to be a one-state TM where the start state was also an accepting state and on every input symbol there was a transition from the start state to itself, for example. In a similar way, let $Y$ be a particular TM for which $L(Y)=\varnothing$.
We could then produce a many-one map from $A$ to $B$ by: $$f(w)=\begin{cases} \langle\, X\,\rangle & \text{if w=1}\\ \langle\, Y\,\rangle & \text{if w\ne 1} \end{cases}$$ Clearly this is a computable function and $w\in A\Longleftrightarrow f(w)\in B$, establishing a reduction from $A$ to $B$ with $A$ decidable and $B$ undecidable.
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Problem Detail:
I just learned about recurrences and I just can't solve this problem. I have this recurrence relation:
$$T(n) = \begin{cases} k\cdot T(\frac{n}{k}) & n > 0\\ 1 & n = 0\\ \end{cases}$$
where $k$ is a constant number.
I tried drawing a recurrence tree or replacing for lower $n$s but no success. I hope you can help me with an idea!
###### Answered By : Yuval Filmus
Suppose that $n$ is a power of $k$, say $n = k^t$. Then $$T(k^t) = kT(k^{t-1}) = k^2T(k^{t-2}) = \cdots = k^tT(1) = k^t,$$ assuming a base case of $T(1) = 1$. So for powers of $k$, we have $T(n) = n$. You can also prove that by induction: if $T(n/k) = n/k$ then $T(n) = kT(n/k) = k(n/k) = n$.
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Problem Detail:
Just stepping into complexity theory, I am befuddled by this notion of a certificate and can't find any utility of this concept.
From my understanding, a certificate is used when you are trying to ascertain whether a problem is NP...a problem is NP if you can verify a p-time solution exist or not (co-NP). I cannot see why you would need a certificate to perform this solution finding. And how is certificate different from "verification".
For example, if I were to try to prove whether all elements in a set is divisible by some number (say 3). Say I then brute force divide all elements by 3 and sees that indeed all elements in this set is divisible by 3. Now where in this process would a certificate come into play?
Also, how would you go about to physically construct a certificate?
###### Asked By : Beached Whale
Suppose you are the verifier for SUBSET-SUM: "given a set $S$ of integers, is there a subset of $S$ whose elements sum up to exactly zero?" To show the problem is in NP, you must be able to efficiently verify that a solution I give you is correct. As the verifier, you only care about verifying solutions (not "finding solutions"!). You do not care where that solution might come from or who constructs it. Now, when you are given a subset, you can easily sum up the elements and announce whether or not those elements sum up to 0.
The example you mention is a problem that is also in P (clearly, we can solve it in polynomial time). Every problem in P is also in NP. We don't have to care about the certificate, but instead we can simply solve the problem in polynomial time. (Check the definition of NP if this feels confusing, but hopefully it doesn't).
You might also enjoy the question How can I verify a solution to Travelling Salesman Problem in polynomial time?
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Problem Detail:
My confusion is that if the recursive call calls the left nodes, and then adds with the right nodes, how are the nodes that are to to right of the left nodes and vice versa being called?
int size (BinaryNode t) { if (t == null) return 0; else return 1 + size(t.left) + size(t.right); }
###### Answered By : Luke Mathieson
Remember that to get the actual value of size(t.left), you have to evaluate the method size at the node t.left, i.e. assuming t.left has left and right children the algorithm with call size((t.left).left) and size((t.left).right).
Question Source : http://cs.stackexchange.com/questions/39893
3200 people like this
Problem Detail:
recently I've started thinking about caching problems in modern CPUs, where they struggle to adequately fetch program data (not instructions) in time, so that it can be computed further.
So then I came up with the idea, why don't we restrict random access on program data and make it all as a continuous stream (input and all eventual locals), which moves strictly in one direction through some "processing head", where you only have a possibility to read in some x amount of symols, write, let through and append symbols at the head position (this is a data stream, not an instruction stream).
The question is, whether you can effectively run computations in that system:
Is it possible to order all functions and allocate all local data/variables at compile time in such order that they will produce usable stream of data for all further functions, so that you will never need to "loop" the tape to access some particular element? (thus, emulating random access, by waiting for desired element to come by)
I've tried to look up this topic on the internet and found nothing. And maybe it is because this idea is obviously stupid, so no one bothers with it :)
No. Consider an algorithm on a graph (e.g., network flow). It fundamentally requires random access to memory. The restriction you propose would be painful and make many standard algorithms impossible.
Question Source : http://cs.stackexchange.com/questions/25828
3200 people like this
Problem Detail:
While reading Stallings OS Internals and Design, I run into problem. Here is example from the book.
For example, consider a simplified computer in which each instruction occupies one 16-bit word of memory. Assume that the PC is set to the location 300. On succeeding instruction cycles, it will fetch instructions from locations 301, 302, ...
My question is: If the length of instructions is 16bits, and the smallest addressable unit of the memory is 1B, why on succeeding instruction cycles it will fetch instructions from locations 301 and 302, and not multiples of two, 302 and 304?
Is the memory then organised as a sequence of 16bit long memory words?
I too read the book. If Dr. Stallings has quoted this, the memory locations 301 and 302 must be referring to word-wise and not byte-wise locations.
Question Source : http://cs.stackexchange.com/questions/22161
3200 people like this
Problem Detail:
In this table B is foreign key referring to the primary key A with on delete cascade option.What will be the result after deleting (4,3) and how A:1 4 2 9 3 5 B:3 3 4 2 9 4
###### Answered By : Grisha Weintraub
The result will be an empty table.
The steps are :
1. delete (4,3)
2. delete rows with B = 4 --> (2,4), (5,4)
3. delete rows with B = 2, B = 5 --> (9,2)
4. delete rows with B = 9 --> (3,9)
5. delete rows with B = 3 --> (1,3)
Question Source : http://cs.stackexchange.com/questions/14882
3200 people like this
Problem Detail:
I'm still insecure in the section decidability (no proof needed, I want to divine it):
X is decidable and Y is undecidable. Is the intersection of X and Y decidable or undecidable?
X is decidable and Y is a partial quantity of X. Is Y decidable, too or not?
Thanks!
###### Answered By : Rick Decker
For your first question, the answer is that $X\cap Y$ will not necessarily be decidable. Let $Y$ be an undecidable language over an alphabet $\Sigma$ and let $X=\Sigma^*$. Then obviously $X$ will be decidable and $X\cap Y=Y$ will be undecidable. On the other hand, the intersection may be decidable, as we could show by letting $X=\emptyset$.
For your second question, I assume that by "$Y$ is a partial quantity of $X$" you mean that $Y$ is a subset of $X$ (i.e., $Y\subseteq X$). Then we can do a similar construction, letting $X=\Sigma^*$, and $Y$ be undecidable, so again, $Y$ may or may not be decidable.
It's worth mentioning that this latter result frequently trips up students. Very few properties of languages are closed under subset/superset: there's no guarantee that a subset of a regular language will be regular, for example.
Question Source : http://cs.stackexchange.com/questions/27765
3200 people like this
Problem Detail:
Given a system of two Boolean Algebra equalities
a = b. c = d.
one can exhibit a single equation
F(a,b,c,d) = 0.
which is equivalent to the former system. (Symmetric difference is pivotal for constructing such quaternary operation F(a,b,c,d)).
Questions:
• Is there binary operation x ? y (infix notation) such that
a ? c = b ? d.
is equivalent to the original system?
• Is there binary operation x ? y such that
a ? b = c ? d.
is equivalent to the original system?
Edit (June 13): The question is more subtle than I managed to convey. Boolean algebra is a finitely definable variety. I was wondering if we can get a single axiom system by combining identities. Padmanabhan proved in 1968 that every definable class of lattices (such as BA) can be defined by a single identity within the class of lattices. The key observation in his method is equipping the identities
a = b. c = d.
with disjoint sets of variables. Then, simple disjunction
a v c = b v d.
would do.
###### Asked By : Tegiri Nenashi
No. There is no such binary operation.
This is easy to prove with a counting argument. Given a candidate binary operation $?$, count the number $n$ of boolean values $x,y$ such that $x?y$ is true. Out of the 16 possible values for $a,b,c,d$, exactly $m=n^2+(4-n)^2$ of them will satisfy $a?c = b?d$. The possible values for $m$ are $m \in \{16,10,8\}$. However, there are only 4 possible values of $a,b,c,d$ such that $a=b$ and $c=d$, which is not in the set of possible values for $m$.
Or, you could do as David Clarke suggests and write a tiny program to enumerate all possibilities. Frankly, you probably should have done that before asking in any case; that would have helped you ask a more specific question, such as "I know no such binary operation exists; can you give me any insight why not?".
Question Source : http://cs.stackexchange.com/questions/26472
3200 people like this
Problem Detail:
I have a list of millions of pairs of strings, and I want to join all of the pairs that have matching members into lists without duplicates.
Example input:
[["A", "B"], ["A", "D"], ["M", "Q"], ["A", "F"], ["D", "E"], ["Q", "Z"]]
Example output:
[["A", "B", "D", "E", "F"], ["M", Q", "Z"]]
Does anyone know of an efficient algorithm for this? I'm somewhat constrained by memory. Anything that would square the memory from the input would not be an option.
###### Asked By : Cory Gagliardi
You can use a two pass approach:
In the first pass, identify all the different strings appearing in your input. (This can be done in various ways, e.g. hashing, trie, BST)
For the second pass initialize a Disjoint-set data structure with the strings found in the first pass and perform a union operation for each pair in the input.
Question Source : http://cs.stackexchange.com/questions/26400
3200 people like this
Problem Detail:
Input: 2 RB Trees A B, so that both values in a certain range, so that B's range is smaller than A in both sides. Ex. B's range can be [100...200] and A's range is [0...1000]
Output: Unite A and B to one RB Tree AB.
How can I unite these 2 trees in $O(log(n))$ time and with additional space $O(1)$?
I have tried splitting tree A into 3 trees A1,A2,A3, so that: A1 < A2 , B < A3
A2 contains all that values that are equal-bigger than Min(B) and smaller-equal to Max(B). I did it using the RB Tree Split algorithm, and all the splitting sums up to $O(log(n))$ time.
Now I know what to do with A1 and A3 but what should I do with A2 and B? I know I should use the fact that B contains A2's range but how does it help me?
You can't. As shown by Brown and Tarjan on the first page of "A Fast Merging Algorithm", merging two sequences with overlapping ranges has a worst-case bound of $\Theta (m \lg (n/m))$ comparisons.
Question Source : http://cs.stackexchange.com/questions/26265
3200 people like this
Problem Detail:
I'm confused about a solution I saw about the following language not being regular:\begin{equation*} L=\{0^n ~1 ~2^n : n >0\} \end{equation*} The example solution said that $L$ was "the same as":\begin{equation*} L'=\{0^n1^n : n > 0 \} \end{equation*} so it wasn't regular. I understand why $L'$ isn't regular, but I don't understand how $L$ is the same as $L'$. How can you eliminate that middle symbol?
###### Answered By : Yuval Filmus
The language $L'$ is the "same" as $L$ (in fact, it's an asymmetric relation) in two steps. If $L$ were regular, then so would be $L'' = \{0^n 2^n : n > 0 \}$ (take a DFA for $L$ and replace $1$ edges with $\epsilon$ edges). In its turn, $L''$ is the same as $L'$, up to change of symbols.
Question Source : http://cs.stackexchange.com/questions/16122
3200 people like this
Problem Detail:
Let consider well-known problem in distributed computing Broadcast Problem in network with stopping crashes.
The most appropriate solution to the broadcast problem is flooding algorithm, when the source of the message just send it to all available nodes of the network on the first round, if there are crashes on the edges some number of message will fail. Then on the second round, the nodes that received messages from the previous round send it to all available nodes and so on.
Let consider slightly different model, when on every round the source node with message is capable to send a message to only one single node of course stopping failure might occur. It's very useful model, which can represent a sort of controllable or limited spreading.
Unfortunately I don't know the formal name of the model. Do you familiar with result in the model of "limited spreading", I am sure there are must some work in this direction.
Both models you presented are the same with a slight modification. Theoreticians usually don't care about the difference between both except when calculating time complexity (or number of sent messages in radio networks sometimes).
The first model you presented, broadcast communication, the broadcast operation is only allowed. That is, a node $v$ may send at one time slot a message to all its neighbours, denoted $N(v)$.
In the second model, point-to-point communication, for a node to send a message to all its neighbours, it needs to send $|N(v)|$ messages each of which is at a different time slot. Therefore, we emulate the broadcast operation.
In the broadcast model, in order to emulate the point-to-point communication from $v$ to $u$, a node $v$ broadcasts a message that contains the identifier of $u$. If $u$ receives the message and finds that it contains its identifier, then it accept it. Otherwise, it just throw it. | 2021-07-29 00:18:29 | {"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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7360799908638, "perplexity": 534.0195327806609}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153803.69/warc/CC-MAIN-20210728220634-20210729010634-00367.warc.gz"} |
http://mathoverflow.net/questions/116681/lower-bound-for-the-difference-between-the-maximum-eigenvalue-of-a-graph-with-th | # Lower bound for the difference between the maximum eigenvalue of a graph with the one of the one-edge-deleted subgraph
I have proposed very recently a question in the following link concerning the title of the current question: Difference of the maximum eigenvalue of a graph with the one of one-edge-deleted subgraph
I would like to ask my new related question as follows:
Is there a positive function $f(n,m)$ such that for any graph $G$ with $n$ vertices and $m$ edges and any edge $e$ such that $G\setminus e$ is connected the inequality $\lambda(G)-\lambda(G\setminus e)\geq f(n,m)$ holds?
The notations are as in the above link. According to Prof. Chris Godsil's answer to my question posed in the above link one must have $f(n,m)\leq \frac{\pi^2}{(n+1)^2}$.
Edit: thanks to Anthony, I add the assumption $G\setminus e$ is connected.
-
You need to assume that your graph is connected for your question to make sense. Given this, the answer is yes for trivial reasons. For each connected $G$, $G\setminus e$ has strictly smaller dominant eigenvalue, so there is a positive gap. Now since there are finitely many graphs with $n$ vertices and $m$ edges, when you take the minimum of these gaps, you get $f(m,n)>0$. Of course the more interesting question is to give a concrete lower bound. – Anthony Quas Dec 18 '12 at 4:52
@Anthony. Thanks. As you see I add the further assumption. Actually I am asking for the best (greatest lower bound). – Alireza Abdollahi Dec 18 '12 at 5:01
I think you only need to assume that $G$ is connected (not $G\setminus e$). – Anthony Quas Dec 18 '12 at 5:25
Here's a guess for the minimum: Take a large clique and let each vertex of the clique be connected to the left end of a path (like a <a href="en.wikipedia.org/wiki/Lollipop"lollipop</a>!) Now if you remove the edge at the far end of the path, you make a very small difference to the dominant eigenvalue. You'd need to play around with the size of the path and the clique to see what gives the smallest difference. – Anthony Quas Dec 18 '12 at 6:09 | 2015-07-07 22:10:29 | {"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": 0.9093174934387207, "perplexity": 182.1936596661676}, "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/1435375102712.76/warc/CC-MAIN-20150627031822-00038-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/452801/class-article-with-argument-titlepage-changes-symbol-of-thanks-to-arabic-number | # Class article with argument titlepage changes symbol of \thanks to arabic number - how to avoid?
I have the problem that
\documentclass[11pt, a4paper]{article}
\title{A great title \thanks{XYZ}}
\begin{document}
\maketitle
\end{document}
gives a footnote with symbol (*) which I want to have, but when giving
\documentclass[11pt, a4paper, titlepage]{article}
the * is changed to a 1. I want to have the title page with the star, not with the 1. Is this possible?
\documentclass[11pt, a4paper, titlepage]{article} | 2019-11-18 03:20:35 | {"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": 0.7268010377883911, "perplexity": 6764.1590493386975}, "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/1573496669431.13/warc/CC-MAIN-20191118030116-20191118054116-00281.warc.gz"} |
https://techwhiff.com/learn/am-cm-bk-cf-es-fm-md-no-lr-an-ideal-solution-of/403074 | # Am Cm Bk Cf Es Fm Md No Lr An ideal solution of 100. g of...
###### Question:
Am Cm Bk Cf Es Fm Md No Lr An ideal solution of 100. g of water and 2.00 moles of a nonvolatile solute is created at 30.0°C. Given that the vapor pressure of water is 31.8 torr at 30.0°C, what is the vapor pressure of the solution? Your answer should have three significant figures. Provide your answer below: torr 03 PM 2721219 4 Previous
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https://encyclopediaofmath.org/wiki/Epimorphism | # Epimorphism
Jump to: navigation, search
A concept reflecting the algebraic properties of surjective mappings of sets. A morphism $\pi : A \to B$ in a category $\mathfrak{N}$ is called an epimorphism if $\alpha \, \pi = \beta \, \pi$ implies $\alpha = \beta$. In other words, an epimorphism is a morphism that can be cancelled on the right.
Every isomorphism is an epimorphism. The product of two epimorphisms is an epimorphism. Therefore, all epimorphisms of a category $\mathfrak{N}$ form a subcategory of $\mathfrak{N}$ (denoted by $\operatorname{Epi} \mathfrak{N}$).
In the categories of sets, vector spaces, groups, and Abelian groups, the epimorphisms are precisely the surjective mappings, i.e. the linear mappings and the homomorphisms of one set, vector space or group onto another set, vector space or group. However, in the categories of topological spaces or associative rings there are non-surjective epimorphisms (that is, mappings that are not "onto" ).
The concept of an epimorphism is dual to that of a monomorphism.
#### Comments
In the article above, $\alpha$, $\beta$ are assumed to be morphisms $B \to C$ for some $C$. If composition of morphism is written from left to right, as is sometimes done, so that the composite of $\pi : A \to B$ and $\alpha : B \to C$ is written $\pi \, \alpha$, then epimorphisms are of course morphisms that cancel on the left.
The inclusion $\mathbf{Z} \to \mathbf{Q}$ in the category of rings is an example of an epimorphism that is not surjective.
#### References
[a1] B. Mitchell, "Theory of categories" , Acad. Press (1965)
How to Cite This Entry:
Epimorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Epimorphism&oldid=40172
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article | 2022-05-23 02:18:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9675876498222351, "perplexity": 451.7441825304961}, "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-2022-21/segments/1652662552994.41/warc/CC-MAIN-20220523011006-20220523041006-00164.warc.gz"} |
https://sips.synthetix.io/sips/sip-78 | # SIP-78: iSynth limit reset must not trip circuit breaker
Author Justin J Moses Implemented Governance Ethereum TBD TBD 2020-08-19
## Simple Summary
Ensure the repricing of an iSynth does not trigger the decentralized circuit breaker.
## Abstract
When the protocolDAO resets an iSynth, it must also update the pricing for the iSynth in the last exchange price tracker in the Exchanger contract.
## Motivation
SIP-65 introduced a decentralized circuit breaker which is tripped when a price on-chain is detected more than some given factor. When an iSynth is reset with a new entryPoint however, this changes the price quite significantly, and can trip the circuit breaker when there is a previous lastExchangeRate in Exchanger for that synth. This makes the synth unusable unless the factor is bumped up sufficiently to handle the iSynth repricing (or any iSynth repricing if many are performed simultaneously), and since the factor setting is system-wide, a much higher factor makes SIP-65 less useful overall.
## Specification
### Overview
When the protocolDAO resets an iSynth, the lastExchangeRate in Exchanger must be updated to the current iSynth rate with the latest entryPoint, to indicate a refreshed price for the synth.
### Rationale
The protocolDAO already has the power to invoke ExchangeRates.setInversePricing(). It's at this point that the Exchanger must be told to update any previous exchange rate with the one calculated at the time of repricing.
N/A
### Test Cases
• Given iETH is an inverse synth tracking ETH
• And there exists a previous exchange into or out of iETH, ensuring Exchanger.lastExchangeRate(iETH) has an entry
• When the protocolDAO invokes setInversePricing, changing the rate more than the Exchanger.priceDeviationThresholdFactor
• And a user exchanges into or out of iETH
• Then the exchange completes successfully and iETH is not suspended due to a price spike
### Configurable Values (Via SCCP)
N/A
Copyright and related rights waived via CC0. | 2022-10-01 09:25:09 | {"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": 0.44768139719963074, "perplexity": 9754.97683887579}, "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-2022-40/segments/1664030335573.50/warc/CC-MAIN-20221001070422-20221001100422-00026.warc.gz"} |
https://puzzling.stackexchange.com/questions/6235/did-you-forget-to-double-check | # Did you forget to double-check?
Lo! I heard you have gone to war.
Alluding to my utter demise.
Trees are what you used to fear,
blobs of nothing, nothing is best.
Do it no more than once, down these rickety stairs.
Take a notice, ignore the spaces in between.
No, doomed you are not, lest you count the punctuation.
(Count the space here, and ignore the first-born!)
Now, talk, flail, and stop right here.
Now that you've got what I've meant to say, double check.
I would warn you of anything, coming at your door.
Trust me, trust me! And say it once more.
So trust me, trust me! Did you forget to double-check?
Hints:
Do not take the first stanza literally. Very few words in it help you, but oh, how important they are!
The second stanza is really just emphasizing a portion of the first, and does describe a bit about the object.
I think I've said to much $\tiny\text{ But look at the tag. }$
• Wow! This looks weird. – Rand al'Thor Dec 17 '14 at 22:21
• Can somebody do anything with the letters "lteebnntmeessedoutnle" ? (I can't and I'm going to bed now..) If you wonder: I've looked at the title of the riddle and the first 2 paragraphs one. Stop here. – BmyGuest Dec 17 '14 at 22:34
• @BmyGuest How on earth did you get those letters? – Conor O'Brien Dec 18 '14 at 21:23
• All letters which appear twice in any word. Double-checked :-) spontaneous idea, but obviously wrong... – BmyGuest Dec 18 '14 at 22:03
• @BmyGuest Oh, wow! That's really clever! Kudos, kudos! – Conor O'Brien Dec 18 '14 at 22:04
It is
A doorbell
Explanation:
Lo! I heard you have gone to war.
Alluding to my utter demise.
Trees are what you used to fear,
blobs of nothing, nothing is best.
Do it no more than once, down these rickety stairs.
Take a notice, ignore the spaces in between.
No, doomed you are not, lest you count the punctuation.
(Count the space here, and ignore the first-born!)
Now, talk, flail, and stop right here.
Now that you've got what I've meant to say, double check.
I would warn you of anything, coming at your door.
Trust me, trust me! And say it once more.
So trust me, trust me! Did you forget to double-check?
Hopefully no further clarification is needed.
I was originally going to give up on this line of thought, but then I double checked and realized it was backwards. True story.
• Another impressive answer! – Rand al'Thor Dec 19 '14 at 13:54
• @McMagister, you need to remove the "gi" from your username! – pacoverflow Dec 19 '14 at 16:34
• very clever, dear sir :) – zlobi.wan.kenobi Dec 19 '14 at 21:23
• Perfect, nice job! (Truth be told, I got inspired for this whilst falling down a staircase XD) – Conor O'Brien Dec 20 '14 at 2:44
• Also 'doorbell' sounds like 'double' as in double check :-) – Rand al'Thor Dec 20 '14 at 23:26 | 2019-08-22 08:02:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.27966058254241943, "perplexity": 4096.139884198538}, "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/1566027316785.68/warc/CC-MAIN-20190822064205-20190822090205-00101.warc.gz"} |
https://www.nature.com/articles/s41598-022-18672-0?error=cookies_not_supported&code=02c493e8-eff8-486b-b7aa-726dd025c7ed | Introduction
Insect pollinators are currently undergoing population declines linked to a number of factors including land use, agricultural intensification (including agrochemical usage), invasive species, diseases and climate change1,2,3,4,5. In the case of land use change, understanding its impact on resource utilisation by honeybee could provide insights into the long-term viability of pollinator populations. This would also provide an evidence base for developing mitigation strategies that can help address resource deficits for this key group delivering pollination services to crops and wild plants1,4. However, it has proved challenging to assess foraging plant resource utilisation by honeybees at these large scales limiting our understanding of how this could impacts upon population level processes (e.g. across a region the size of Great Britain).
For insect pollinators, in particular bees, the availability of foraging resources at landscape scales has a significant impact on population viability6,7,8,9. There are core calorific and nutritional requirements supported by pollen (to provide principally protein) and nectar (to provide carbohydrates) foraged upon by bees10. Limitations on either being likely to lead to population declines11,12. However, individual plants can vary considerably in nectar availability, as well as protein content and amino acid profiles10,13,14. This variability has the potential to affect key fitness metrics affecting pollinator survival9,14,15. Perhaps the most significant is how diet quality affects susceptibility to economically important parasites and diseases. In honeybees, diverse diets can increase their ability to resist diseases (i.e. immunocompetence)8, whilst the availability of high protein pollens originating from single plant species can improve tolerance to parasitic and viral infections9,15. The negative impact of poor diet on disease susceptibility is unlikely to act in isolation, being directly or indirectly impacted upon by a range of other environmental stressors. Of these, exposure to insecticides may pose a major risk in many agricultural systems where sub-lethal doses of pesticides weaken the immune system of honeybees increasing the likelihood of them becoming susceptible to diseases16. The negative effect of insecticides on bees can also be direct, with exposure resulting in immediate and long term toxicity effects17,18. However, while insecticides may be expected to have negative effects on bees other widely used agrochemicals considered to have low toxicity to honeybees19, such as fungicides and herbicides, may also have negative effects20,21. For example, while the very widely used herbicide glyphosate is considered to have low toxicity to bees from a regulatory perspective, a recent meta-analysis identifies its use with increased mortality of bees21. Similarly, the ubiquitously used azole fungicides (e.g. triazoles) can have unexpected synergistic interactions with other pesticides impacting negatively on bees20,22. Quantifying the extent to which these environmental drivers will impact on honeybees at national scales in real world agricultural systems has important implications for how we will manage these systems in the future.
Whilst diet quality may be of fundamental importance to bee health, land use affected by human activity is known to have negatively affected the availability of foraging resources on which they can feed23. This includes the loss of floristically rich semi-natural habitats from agricultural landscapes, as well as the role of herbicides and crop seed cleaning reducing the prevalence of flowering arable weeds23,24,25,26. This loss of general floral diversity may be compensated to some extent by the prevalence of mass flowering crops, including oilseed rape, that act as a highly abundant monofloral resource for many generalist pollinator species3,27. However, diets dominated by a few species of crop where landscape scale saturation occurs, may potentially lead to a dramatically simplified diet that have negative impacts on health9,15,28. Independent of this, the prevalence of agricultural land use has been linked with a reduction in the diversity of pollen foraged upon by honeybees29, as well as an associated reduction in the protein content of stored hive products vital to support larval development30. A characteristic of many semi-natural and agricultural landscapes are strong seasonal shifts in flower resource availability, leading to boom and bust periods for pollinators31,32,33,34. While pollinators forage dynamically across varying scales to compensate for such temporal resources variability, ultimately landscape diversity and quality places an upper limit on what can be extracted to support population growth31. In the case of honeybees, there is a greater requirement for pollen to support colony growth in the spring, while nectar becomes more important later in the year for maintaining colony size and for storage as honey35. As such, temporal variability in flowering resources may interact with key periods of specific nutritional requirements to have unexpected negative effects.
The quantification of bee diet through the assessment of pollen types returned to nests is a direct approach to understand patterns of resource availability and their impacts on population viability of insect pollinators. Until recently, applying such assessments at large scales has been impractical due to time and expertise limitations associated with microscopy based pollen taxonomy. With the advent of molecular approaches, large-scale quantification of pollen types returned to hives is now viable e.g.36,37,38,39,40. However, to apply this in a systematic way there is a need for model systems. Honeybees (Apis mellifera) are a practical model system, having an existing evidence base for their dependence on pollen diversity and quality15,41, being characterised by foraging ranges as large or larger than other bee species32,42,43, while also being a prominent model system in ecotoxicology44. Also, there exists a network of amateur and professional beekeepers who are willing to provide national scale, spatially and temporally referenced samples of honey from which pollen can be extracted and classified39.
In this study, we utilise 527 honey samples collected across Great Britain as part of the National Honey Monitoring Scheme (https://honey-monitoring.ac.uk/) (Fig. 1). This represents citizen science programs where members of the general public (specifically bee keepers) are vital to the collection of samples that underpin scientific research that would otherwise be impractical or too costly to achieve without their support. This science scheme represents one of many citizen science programs around the world that have engaged with bee keeper communities that often have a vested interest in understanding the ecology and threats posed to the hives and bees they manage45,46,47,48. A subset of the samples provided by these citizen scientist bee keepers were provided with meta-data that included information on prevalence of pests and disease, including Varroa mite infections within the hive. Environmental DNA (eDNA) from pollen suspended within the honey samples was identified using metabarcoding to quantify what the honeybees had foraged upon. Environmental DNA is collected without isolating a specific organism from an environmental sample, i.e. pollen suspended in honey assessed in aggregate rather than by separate species49,50. The environmental context of hive locations was assessed through spatial data on land use, cropping patterns and agrochemical applications. Using this data set we asked the following questions: (1) Does the prevalence of agricultural land use and its associated simplification of the floral community impact on the availability of forage plant species for honeybees and the complexity of their foraging patterns? (2) Does the prevalence of highly attractive sown forage crops, in particular Brassica’s, such as oilseed rape, correlate with lower honeybee diet breath? (3) Does a reduction in resource breath (i.e. the variety of forage plants) and agrochemical use correlate with colony susceptibility to the widespread and economically damaging infestations of the Varroa mite and Deformed Wing Virus. Here we use disease susceptibility as a metric for colony health.
Results
Foraging preferences of the honeybees
The majority of samples were from England (N = 467), with a relatively small number from Wales (N = 31) and Scotland (N = 34). Brassica crops (Brassicaceae), in particular oilseed rape (B. napus), turnip (B. rapa) and cabbage (B. oleraceae), have a close genomic relationship and their separation is unreliable51. Considered as an aggregate, the Brassica group was the dominant forage plants across the 527 samples (85.7%), followed closely by the common hedgerow species aggregate Rubus spp (Rosaceae; 80.2% of samples) as well as Trifolium repens (70.6%) which is ubiquitous in improved and other GB grasslands. The non-native flowering shrub/tree Ligustrum ovalifolium (Oleaceae), which is widely grown in gardens, was the next most dominant forage plant found in 57.3% of honey samples. While these species were also the dominant forage plants in England, this was not true of Scotland and Wales where native species like Filipendula ulmaria (Rosaceae) and Chamaenerion angustifolium (Onagraceae; Scotland only) were more commonly foraged upon than L. ovalifolium. For both Scotland and Wales the aggressively invasive species Impatiens glandulifera (Balsaminaceae) was also within the top 5 most commonly foraged upon plants, found in respectively 58.8 and 67.7% of samples.
Diet breath of the honeybees
The species richness of plants foraged upon by honeybees from unique hives was significantly affected by an interaction between surrounding arable crop cover and the season in which hives were collected (F1,520 = 4.28, p = 0.04). The breadth of plants foraged upon to produce honey during the early season (≤ June) was typically lower than that seen latter on in the year (≤ July) (Fig. 2a and b). There was also far more variability between hives during the late season. However, even with this lower species richness, the breadth of the diet of honeybees during the early season negatively correlated with overall arable cover surrounding hives (Fig. 2a). For late season honey, the strength of this correlation was close to zero (Fig. 2b). The arable cropping rotations within a 2 km radius of hives (F1,520 = 7.17, p = 0.001), defined using a PCA axis of non-insect attractive crops, was negatively correlated with the diet breadth of honeybees. This pattern suggested that crop rotations domination by winter wheat were associated with the lowest diet breadth of foraging plants (Fig. S2a; supplementary information Table S1). There was evidence for a contraction in diet breadth where honeybee diets became increasingly dominated by Brassica crops (including oilseed rape). Here there was a strong negative correlation with the species richness of other forage plants within the diet of the bees and the summed DNA read counts of these crops derived from analysis of the honey samples (F1,520 = 41.4, p < 0.001; Fig. 2c). Finally, there was a negative correlation with flowering habitat and land use (PCA axis 2) (F1,520 = 21.8, p < 0.001; Fig. S2b). This relationship was based on PCA axis scores that suggested that landscapes dominated by improved grasslands were more likely to be characterised by a reduced species richness of foraging plants compared to those dominated by urban land use (supplementary information Table S2). There were no other significant interactions, nor was there evidence of spatial autocorrelation in model residuals (Morran’s I = − 0.053, Exp. = − 0.002, Var. = 0.003, p > 0.05).
Between hive complexity of trophic interactions
Following the derivation of bipartite foraging relationships between hives and flowering plant species, we found that connectance (F1,12 = 40.9, p < 0.001, Figs. 3a and 4), nestedness (F1,12 = 24.3, p < 0.001, Figs. 3b and 4), niche overlap (F1,12 = 15.9, p = 0.002, Figs.3c and 4), and generality (F1,12 = 8.11, p = 0.01, Figs. 3d and 4) all responded to an interaction between arable crop cover and season. For connectance, nestedness and niche overlap this relationship was characterised by strongly positive correlations early in the season, while the later season was, in all cases, characterised by a correlation close to zero (or slightly negative in the case of niche overlap). In contrast, the generality of the hives feeding relationships declined with increasing arable crop cover early in the season, but again was close to zero in the late season.
Disease infection rates
The probability of hives having Varroa infestations at the time of honey harvesting was not found to respond to diet quality in terms of the species richness of plants foraged upon, nor was it affected by the strength of agrochemical exposure (p > 0.05). Agrochemical exposure was defined by the direct effect foliar insecticide (foliar insecticide index), as well as the average application rates within 2 km of hives of the potentially synergistic Triazole fungicides or the herbicide glyphosate (p > 0.05). No interaction terms were found to be significant. The probability of symptomatic expression of the DWV was significantly correlated with an interaction between the foliar insecticide index and season (F1,373 = 3.73, = 0.05; Fig. 5), but not diet breadth. For hives where honey was collected early in the year, there was an increased probability of the symptomatic expression of the virus where the foliar insecticide index of the surrounding landscape increased. This relationship was absent in the latter season. There were no other significant single or interactions effects found to significantly affect the expression of this virus, including those for predicted average Triazole and glyphosate applications. There was no spatial autocorrelation in the model (Morran’s I = 0.049, Exp. = − 0.003, Var. = 0.001, p > 0.05).
Discussion
In this study, we have demonstrated how beekeeper citizen scientists combined with lab based metabarcoding analysis of pollen DNA can provide insights into the factors affecting the viability of honeybees and their associated crop pollination service. While honeybees may potentially compete with wild pollinators52,53,54, they have sufficient fundamental similarities to act as a model system for inferring general impacts of land use intensification on pollinators. Ultimately, they provide access to systematically collected large-scale foraging data at a scale normally outside of the scope of most research programs. We have shown how agricultural land use and management are factors affecting honeybee diets, the resilience of their inter-hive feeding relationships and even impacting susceptibility to disease.
Agricultural land use impacts on forage plant utilisation
In agreement with Alburaki et al.29, the extent of arable agricultural land use surrounding hives had a negative impact on the diet breath of the honeybees in terms of the range of plant species they fed upon during the early part of the year. This effect was slightly more pronounced where cropping rotations were dominated by winter wheat, one of the most widely grown and intensively managed of the arable crops. This negative effect of arable cropping land use was seen in terms of it being not only a predictor of the total number of forage plants, but also as a determinant of complex characteristics of resource overlap and utilisation between competing hives. As the cover of arable cropping increased, hives typically foraged on a greater proportion of the plants utilised by better-connected hives (Nestedness), while the mean effective number of plants foraged upon by individual hives simultaneously decreased (Generality). The similarity of interaction patterns with the types of plants on which different hives foraged also increased with arable cover (niche overlap). These effects are characteristic of a general simplification of resource choices within arable dominated landscapes. This effect has been reported in the US where there is increased patterns of foraging overlap for the most abundant plants within the diet of honeybees55. A likely effect of this is that hives may have been more likely to utilise what limited plant species were present as arable cropping came to dominate landscapes. This impact of arable agriculture on both foraging opportunities with increased arable cover reflects the historic loss from these production systems of flowering plants14,24. It is likely that this has affected not just diet breath by limiting the available number of species to forage upon, but also resource quality in terms of nectar availability and protein content14,30.
While the impact of arable crop cover was pronounced, its effects were only found during the early part of the season, during June or before. This strong seasonal variation in resource utilisation by honeybees has previously been identified31,56. As the season progresses, the availability of plants in flower generally increased in North Western European landscapes and any negative effect of arable land use on measures of hive diet breath or the complexity of foraging relationships may tend to disappear. Although hedgerow plants and some trees are important foraging resources early on in the year, early season arable systems are limited in their availability of flowering plants in general14,33. In contrast, plants flowering in July and August provide some 60% of nectar production available for bees to forage upon in British landscapes14. One driver of this low availability of flowering plants in arable dominated landscapes has been the impact of intensive farm management practices on the persistence of annual weed species14,23,25,26. Such weed species in arable ecosystems are potentially important foraging resources, especially when mass flowering crops are not in flower34. The loss of these often-early flowering species has led to important ‘drought’ periods of floral resource deficiency in these systems. Such early season deficits in resource availability may be a particular threat to honeybees, and potentially other eusocial species, by limiting resource availability as a point in the season where colony growth should be maximised35. Of these other land uses, there was some suggestion that urban environments may be likely to provide a greater diversity of potential plant species to forage upon, something reported in several other studies29,57.
Contraction in diet breath and mass flowering crops
Mass flowering crops, in particular oilseed rape, represents an important foraging resource for many insect pollinators within GB agricultural landscapes3,27. In the case of honeybees, oilseed rape pollen may be particularly rich in essential amino acids compared to other mass flowering crops like field beans, including leucine, valine, and isoleucine13. This nutritional profile of both nectar and amino acids has been seen to drive preferential selection by honeybees13. At least for honeybees, the loss of oilseed rape can reduce survival. Di Pasquale et al.58 suggests that pollen diversity is less important than the loss of such species that individually have a high nutritional value as well as being extremely abundant. We found that as Brassica crops become a more significant part of the diet of honeybees their overall diet range in terms of what other plants they forage upon contracted. It therefore seems likely that in the presence of mass flowering crops, such as oilseed rape, may cause bees to neglect other foraging resources. Such preferential selection by pollinators could reduce diet breath and may have indirect and unexpected impact on bee health. Diets lacking pollens from a range of different plants may have synergistic effects on bee health to such an extent that their loss may destabilize these communities8,15,59. In addition, this may have consequences for populations of wild flowering plants by reducing opportunities for pollination events as bees are drawn to high resource value fields of such mass flowering crops60.
Resource breadth, agrochemicals use and the incidence of pests and disease
There is a significant evidence to suggest polyfloral diets play an important role in supporting bee health8,15,59. However, we found no evidence that diet breadth affected either Varroa mite infestation rates or the symptomatic expression of DWV, the commonest virus reported across the data set. It is likely that there is complexity underlying the role of polyfloral diets, with only a sub-set of plants making a significant contribution to maintaining bee health. This may be through either their improved nutrient profiles or because of direct disease inhibiting effects15,61. However, at least in the latter case, toxic chemical defences of plants may be as likely to have negative consequences for plant-honeybee interactions62. Although hives with broad diets would (through a sampling effect) be more likely to include beneficial species63, it is probable that the relative contribution to the diet of these species is an important determinant of their value61.
We also assessed the impact of agrochemical stressors on symptomatic expression of these two diseases, focusing not only on direct effects associated with foliar insecticides16, but synergistic consequences of exposure to other agrochemical classes21,64,65. While beekeepers do apply acaricides as a means of control rather than complete eradication of Varroa mite infestations (e.g. pyrethroids like flumethrin), the widespread resistance of Varroa to these products has meant a shift towards control using acids (e.g. oxalic or formic acid) and natural oils66. We found no evidence that Varroa infestation rates were affected by exposure risk to any of the considered agrochemicals. However, the symptomatic expression of DWV was positively correlated with the foliar insecticide index during the early part of the year. The identification of this effect in June and before is likely linked to this being the main period of insecticide use on crops, as well as when hives may by more vulnerable to pesticide exposure following the over wintering period. The asymptomatic presence of DWV is thought to be present in many hives, although its symptomatic expression may be considerably increased though synergistic interactions with Varroa infestations16,67. There was no evidence of synergistic interactions between the insecticide use index and the Triazole fungicides or glyphosate herbicide use. The absence of such effects runs counter to a recent meta-analysis of synergisms between agrochemicals65 as well as focused work on these plant protection products in particular21,64. It is possible that our use of proxies to define agrochemical exposure based on application rates surrounding hives68 may lack the sensitivity to identify such synergisms. Indeed, future work focusing on the direct testing of residues from stored hive products may provide a more effective approach to quantifying agrochemical exposure risk69.
Conclusions
Honeybees represent a useful model species to indicate the stressors impacting on insect pollinators more widely, in particular central place foragers such as bumblebees and solitary bees. Indeed, given the typically large foraging ranges of honeybees compared to wild species, it is likely that the observed trends here may well be extrapolated for smaller solitary bees which more typically forage less than 1 km from nests32,70. This study has highlighted that there may be a restriction in the availability of floral resources available to honeybees in the early season, and by extension pollinators in general, in landscapes dominated by arable agriculture. This effect was not found in the later part of the year. The early season encompasses a period during which perennial species, often established though agri-environmental schemes to provide foraging resources for bees, are yet to reach their peak flowering24,33. The impact of limited resource availability within intensively farmed landscapes may also be counteracted by the occurrence of large blocks of early mass flowering crops, such as oilseed rape. While often considered an important resource for bees in agricultural systems3,27, their prevalence in diets may act to further contract diet breadths for species already foraging in a resource-limited landscape. A risk associated with oilseed rape is that while it may be an important source of pollen and nectar in the early season, it is one of the most intensively managed of the GB crops in terms of pesticide use and so may representing a potential exposure risk to bees71. While there was no direct evidence that diet breadth affected bee health, other work has highlighted its importance8,15,59. This study emphasises the need for more careful consideration of management practices to mitigate the impacts of arable crop management on pollinators. In particular, there is a need to identify where temporal mismatch in resource availability occurs and develop new agri-environmental seed mixes that can overcome these deficits. While early flowering annuals may be harder to manage for any famers, needing annual re-establishment, this may be a vital part of this solution for maintaining pollinator populations.
Methods
In 2019 beekeepers across Great Britain provided 527 honey samples (Fig. 1) as part of a citizen science initiative, the National Honey Monitoring Scheme (https://honey-monitoring.ac.uk/). Samples were collected by directly scraping honey from recently filled storage cells on the edges of recently filled combs within hives. These honey samples were then returned to the scheme with associated spatial and temporal meta-data. A smaller subset of these honey samples (N = 377) included additional meta-data on the colony health, such as the presence of Varroa mite infestation (Fig. S1). While Varroa is found in most colonies, the symptomatic expression of viral infections is far more variable making it a useful indicator for understanding disease susceptibility in response to variation in environmental drivers. Although symptomatic expression for a range of viruses was recoded, only Deformed Wing Virus (DWV) was reported with sufficient frequency to be considered in subsequent analyses (52 of 377 samples). Honey samples were split into two temporal batches: early season (Up to June; N = 119) and late season (from July onwards; N = 408). This broadly corresponds with the two main harvests of honey typically produced by apiaries and relates to the seasonal variation in resource availability common to GB agricultural systems24,33. In particular there can be a lack of flowering plants early in the year, although this gap may be filled by oilseed rape production in some agricultural landscapes34.
Forage plant identification by DNA barcoding of pollen in honey
Although honey is derived from nectar, each sample is contaminated with pollen grains derived from the plants that bees foraged upon, as well as trace quantities of other environmental DNA from plant nectar, bacteria and fungi. The pollen grains were extracted allowing subsequent DNA metabarcoding and identification of plant species using an established informatics pipeline described in detail in Oliver et al.39. In summary, pollen was concentrated from honey using a vacuum filtration system (Nalgene) utilising 47 mm mixed cellulose esters membranes (pore size 1.2 µm). Total DNA was extracted using the DNeasy PowerPlant Pro Kit (Qiagen, Hilden, Germany), with an additional proteinase K step alongside homogenisation to aid complete lysis. Resultant DNA was quantified using a Nanodrop One spectrophotometer (Thermo Scientific, Waltham, MA, USA) and ~ 10–20 ng template used in a PCR reaction to amplify the Internal transcribed spacer region 2 (ITS2) of plant nuclear ribosomal DNA. Finally, normalisation and sequencing of amplicons was undertaken through the Illumina MiSeq platform with the MiSeq Reagent Kit v3 (Illumina Inc., San Diego, CA, USA). Raw sequence data were processed using the HONEYPI bioinformatics pipeline as described in Oliver et al.39. Taxonomically comparable amplicon unit sequence variants (ASVs) were phylotyped following species level rarefaction within the phyloseq package within R 3.6.372. From this a data matrix describing honey sample by abundance of DNA reads for each plant species was produced. The summed DNA reads of Brassica crops, a category which included oilseed rape as the main UK mass flowering crop, was derived as a covariate for subsequent analyses.
Landscape structure
Honeybees forage over potentially large distances, although on average feed within 2 km of hives32,42,43. To provide an indication of foraging resource surrounding each hives to this distance, we derived a range of measures of surrounding land use. We used the 2019 UKCEH Land cover map at a 20 m raster pixel resolution combined with the UKCEH Land Cover plus Crops map73 to derive three metrics of land use. The first was arable and horticultural percentage cover (hereafter called arable cover). This represents a fundamental descriptor of land use in the UK. The second was a principal components analysis describing the first two axis of variation determined from the cover of non-insect attractive crops, specifically maize, wheat (winter and spring separately), barley (winter and spring separately), sugar beet, potatoes and other crops (a summed category of crops not defined by the previous). Note this excluded the flowering crops of oilseed rape and field beans. This explains variation beyond simply total arable crop cover, while acknowledging underlying correlations between these cropping patterns that make their use as individual covariates in subsequent models impractical. Thirdly, we derived the first two axis of variation of a principal components analysis based on the land use cover of habitats likely to be important for foraging bees14. These were the cover of mass flowering crops (oilseed rape and field beans), woodlands, flower rich habitats (including heather, heather-grass as well as unimproved calcareous and neutral grassland), improved grassland (receiving inorganic fertiliser) and (sub-) urban land use. Based on an assessment of variable inflation factors in subsequent analyses, the first two axis of variation of both of these PCAs (crop and land use) were characterised by strong multi-colinearity with arable cover VIF > 8.0;74 and so were excluded from subsequent analyses.
Agrochemical exposure risk
The intensity of agrochemical use was defined using the UKCEH Land Cover plus: Pesticides 2012–2017 map68. The mean active ingredient weight per hectare of foliar spray insecticides (alphacypermethrin, deltamethrin, chlorpyrifos, cypermethrin, dimethoate, esfenvalerate, lambdacyhalothrin, pirimicarb, taufluvalinate, zetacypermethrin, fosthiazate, oxamyl, pymetrozine) were determined. This map is based on average applications over a 5-year rotation period and was defined within a 2 km radius of each hive. While ideally pesticide exposure for the 527 honey samples would have been directly assessed by residue analysis this was prohibitively expensive, however a sub-set of 100 samples was assessed for agrochemical residues from 2019. While this represents a limited sub-set of active-ingredients, the probability of detecting these in honey was largely predicted by the application rates around hives based on the UKCEH Land Cover plus: Pesticides 2012–2017 map (Supporting Information Table S7).
UKCEH Land Cover plus: Pesticides 2012–2017 map we derived a Foliar Insecticide Impact index (FII) that represents a composite estimate of the impact of foliar sprayed non-systemic insecticides based on the Environmental Impact Quotient (EIQ)75. This index has previously been used to assess the impacts of insecticides on bees3,17. The FII was defined as:
$$FII = \mathop \sum \limits_{i = 1}^{ai} \left( {Z_{ai} \times P_{ai} } \right) \times \left( {\frac{{M_{ai, R} }}{{A_{R} }}} \right)$$
(1)
where: Following Kovach et al.75 Zai is the toxicity of the active ingredient (ai) to bees where each active ingredient is scored as high (score of 5 where oral LD50 < 1 µg bee−1), medium (score of 3 where 100 µg bee−1 > oral LD50 > 1 µg bee−1) or low (score of 1 where oral LD50 > 100 ug bee−1). This ratio of 5:3:1 is part of the EIQ. Pai is the active ingredient plant surface half-life estimated to be a quarter of the soil deterioration half-life (DT50)76. All values used were derived from the Pesticide Properties Data Base19. Mai, R is the mass of active ingredient applied with the 2 km radius surrounding hives (region R), and AR is the area within that same 2 km radius. Note, while neonicotinoid seed treatments are widely thought to have a negative impact on bee populations they were not in use in arable cropping systems in the UK in 20193.
Although insecticide use represents the immediate risk to honeybees, other agrochemicals of relatively low toxicity may have unexpected interactions that can alter the sensitivity of bees to insecticides65. To account for this we have focused on compounds that have been identified in the literature as being associated with either synergistic negative impacts azole fungicides—e.g.20,77,78 or unexpectedly high real world toxicity to bees not predicted in the regulatory process e.g. glyphosate—21). These two groups of non-insecticide pesticides are ubiquitously used in the UK farming environment, with glyphosate being the dominate herbicide used on the principal mass flowering crop oilseed rape grown in the UK (63.2% respectively by weight of active ingredient), while azole fungicides comprise 82.3% by weight of the fungicides used on this same crop. Both are also widely used on wheat crops, although as non-flowering crops this are not as attractive to bees and so exposure risk is likely lower79. To account for this we also derived the mean application rates of the widely used herbicide glyphosate, as well as the combined application rate of triazole fungicides, both of which have been identified as having interactive effects. Both of these have been linked to potential negative effects on honeybees as part of interactions with insecticides21,64.
The complexity of foraging associations
We derived bipartite food webs based on the foraging associations between individual hives and plants based on the DNA barcoding of pollen from honey. These were derived for randomised sub-sets of hives picked from 14 sub-groups of the overall dataset of 527 hives. These sub-groups were based on hives where honey was collected early (≤ June) or late (≥ July) season within the following band classes of 0–10%, 10–20%, 20–30%, 30–40%, 40–50%, 50–60%, 60–70% and 70–90% arable cover. Again, we focus on arable cover as the major axis of land use variation within GB. The number of hives in each of these 14 sub-groups varied (Supporting Information Table S3). To account for this we undertook 100 random selections of five hives from each of these sub-groups to produce bipartite foraging webs. Using the Bipartite package in R 6.380, we then derived average values of a sub-set of metrics describing bipartite web structure and averaged these across these 100 random picks. These metrics were weighted connectance (realised proportion of possible links between hives and plants), weighted NODF nestedness (the tendency for hives to forage on subsets of plants utilised by better-connected hives, where larger values indicate increased nestedness), niche overlap (mean similarity of interaction patterns for hives with plants) and generality (mean effective number of plants foraged upon per hive) (Supplementary Information Table S5). In all cases, they provide key insights into resource utilisation and specialisation. Metrics were weighted by the number of DNA read counts.
Statistics
We assessed the response of the diet breath (species richness of plants foraged upon by the honeybees determined by DNA barcoding) using generalised least squares regression in response to the covariates: season (categorical early or late), summed arable cover, cropping land use (PCA axis 2), habitat land use (PCA axis 2), Brassica pollen in diet and the foliar insecticide index (FII) (Supporting Information Table S4). All pairwise interactions, as well as triplicate interactions of the latter covariates with season were included. The model was fitted within the nlme package in R 3.6.3 with a spatial correlation structure defined with the ‘corSpatial’ function using X and Y spatial coordinates of hive locations81,82.We accounted for within-group heteroscedasticity with a weighting of 1|Season. A Box-Cox transformation of species richness was used to normalise the response. All models were simplified using deletion of least significant effects and standard model checks to ensure model assumptions were not violated. This included a formal test for spatial autocorrelation of residuals using Morran’s I as well as inspection of a correlelogram.
The probability of hives suffering Varroa infestation and DWV infection (both as separate binomial responses) were assessed in response to: (i) season (early or late); (ii) diet quality, defined by the richness of plants foraged upon (Ln N + 1); (iii) the foliar insecticide index based on a 2 km radius surrounding the hives; (iv) the summed application of Triazole fungicides; (v) summed application of glyphosate (supporting information Table S5). As before, all pairwise interactions were assessed, as well as triplicate interactions with season. This analysis was performed using a general linear model with binomial error and logit link. Model simplification and model checks were as before. As no evidence of spatial auto-correlation was detected, no further correction was required in the model specification to account for this.
Finally, each of the four metrics of bipartite web structure (connectance, NODF nestedness, niche overlap and generality) were tested using a linear model against the covariates of season and arable cover, as well as their pairwise interaction (supporting Information Table S6). Arable cover was defined as the mid-point of the land use categories from which the random bipartite web picks had been drawn (5, 15, 25, 35, 45, 55, 65 and 80%). As each response was an average of multiple randomisations from hives drawn across the country accounting for spatial autocorrelation was not used. In all analyses we weighted the response based on the number of hives in each land use category for the early or late season from which the original randomised hive draws came from. Model simplification and checks followed the same approaches as above. | 2023-03-21 06:00:48 | {"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": 0.47818052768707275, "perplexity": 5761.199026427019}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2023-14/segments/1679296943625.81/warc/CC-MAIN-20230321033306-20230321063306-00678.warc.gz"} |
https://www.wyzant.com/resources/answers/topics/square-root | 69 Answered Questions for the topic square root
Square Root Algebra 1 Prealgebra
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If a natural number is entered into a new toy named Squareroo, the toy will do the following: 1) Find the perfect square closest to the number entered. 2) Take the square root of the perfect square... more
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I get really confused with square rooting and my teacher said the answer I got was wrong. Sqrt (x^10)
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square root stuff is not my friend! please help thank you so much. my teacher said i was wrong.
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Square Root Volume Calculus 3
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Find the volume of the solid bounded by the plane z = 5 and the hyperboloid z = SquareRoot(9+x^2+y^2)
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Find the square root of 49729 by prime factorirzation method chapter of class 8 square and square root solve it.
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How does the square of that approximation compared to 31?
Approximate square root 31 with a decimal to the nearest hundredth. How does the square of that approximation compared to 31?
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options are given below
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Need help with Quadratic Equation that you have to solve with Square Root?
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02/01/16
Solve using the quadratic formula: y = 3x² + 10x + 4
Q. Solve using the quadratic formula: y = 3x² + 10x + 4 (-5 ± sqrt(22)) / 3 (10 ± i sqrt(38)) / 6 (-10 ± 13 sqrt(4)) / 6 (5 ± sqrt(12)) / 3 (-5 ± sqrt(13)) / 3 Thank you!
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Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. | 2020-11-26 04:45:46 | {"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": 0.6216537356376648, "perplexity": 1675.0489637033988}, "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-2020-50/segments/1606141186414.7/warc/CC-MAIN-20201126030729-20201126060729-00353.warc.gz"} |
http://tex.stackexchange.com/questions/linked/2063?page=4&sort=hot | 6k views
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I have started toying around with Martin Scharrer's standalone package because tikz external does not work with todonotes and I feel that the workflow for standalone makes more sense as far as I have ... | 2015-07-28 18:14:13 | {"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": 0.9636896848678589, "perplexity": 3945.916673308245}, "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/1438042982013.25/warc/CC-MAIN-20150728002302-00324-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/89705/modify-biblatex-so-citations-appear-as-author-year-pagenos | Modify Biblatex so citations appear as (author, year: pagenos)
I am really struggling to find a way to modify biblatex authoryear so that it comes out in this modified Harvard style. In case any solution can be helpful to others in the future, this is the style Sage Journals request.
What I want: (author, year: page).Biblatex author year normally produces something like this: (author year, page). By page, I mean the page in the cited document with the referenced material.
Here is a sample bibtex reference and how I would like it to apprear:
@book{Varshney:2003tn,
author = {Varshney, Ashutosh},
title = {{Ethnic Conflict and Civic Life}},
publisher = {Yale University Press},
year = {2003},
series = {Hindus and Muslims in India},
isbn = {9780300100136},
language = {English},
date-modified = {2012-12-14T03:45:28GMT}}
In-text citation:
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua (Varshney, 2003: 83-85).
Bibliography Entry:
Varshney, Ashutosh. (2003) *Ethnic Conflict and Civic Life: Hindus and Muslims in India*. Yale University Press.
Any help is much appreciated!
EDIT: Included an example!
-
Welcome to TeX.sx! Please add a minimal working example (MWE) that illustrates your problem. – Guido Jan 7 '13 at 11:31
Is the pagenos, the page number of the citation (and it would not make sense for a book), or the page number where the material specific to the citation appears. If this is the case, then you can use \cite[pagnos]{citekey}. – Guido Jan 7 '13 at 11:34
Thanks for the replies. I mean the page number of content being referenced (sorry for the confusion). – Michael Davidson Jan 7 '13 at 16:24
I've added an example which will hopefully make it clear. – Michael Davidson Jan 7 '13 at 16:25
Could you explain how this differs from what you currently get with your preferred bib style? – Seamus Jan 7 '13 at 16:32
In general, see Guidelines for customizing biblatex styles. In your specific example, don't capitalize "english" in the .bib file.
\documentclass{article}
\usepackage[english]{babel}
\usepackage[style=authoryear,isbn=false]{biblatex}
\DeclareFieldFormat{postnote}{#1}
\DeclareFieldFormat{multipostnote}{#1}
\usepackage{xpatch}
\xapptobibmacro{date+extrayear}{\nopunct}{}{}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@book{Varshney:2003tn,
author = {Varshney, Ashutosh},
title = {{Ethnic Conflict and Civic Life}},
publisher = {Yale University Press},
year = {2003},
series = {Hindus and Muslims in India},
isbn = {9780300100136},
language = {english},
date-modified = {2012-12-14T03:45:28GMT},
}
\end{filecontents}
\begin{document}
Some text \autocite[83--85]{Varshney:2003tn}.
\printbibliography
\end{document}
Note: I didn't tinker with the formatting of the series field; in your "Bibliographic Entry" snippet, "Hindus and Muslims in India" (italic, colon prepended) resembles a subtitle rather than a series title.
-
Thank you! This is great and I think I am beginning to understand how to edit the biblatex formatting. – Michael Davidson Jan 7 '13 at 18:44 | 2016-02-12 10:00:55 | {"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": 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": 0.7511692643165588, "perplexity": 5054.890987391402}, "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-2016-07/segments/1454701163663.52/warc/CC-MAIN-20160205193923-00015-ip-10-236-182-209.ec2.internal.warc.gz"} |
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# Hyperbolic optics and superlensing in room-temperature KTN from self-induced k-space topological transitions
## Abstract
A hyperbolic medium will transfer super-resolved optical waveforms with no distortion, support negative refraction, superlensing, and harbor nontrivial topological photonic phases. Evidence of hyperbolic effects is found in periodic and resonant systems for weakly diffracting beams, in metasurfaces, and even naturally in layered systems. At present, an actual hyperbolic propagation requires the use of metamaterials, a solution that is accompanied by constraints on wavelength, geometry, and considerable losses. We show how nonlinearity can transform a bulk KTN perovskite into a broadband 3D hyperbolic substance for visible light, manifesting negative refraction and superlensing at room-temperature. The phenomenon is a consequence of giant electro-optic response to the electric field generated by the thermal diffusion of photogenerated charges. Results open new scenarios in the exploration of enhanced light-matter interaction and in the design of broadband photonic devices.
## Introduction
Wave behavior is governed by the topology of the isofrequency surface $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$, the k-space manifold of plane-wave solutions of angular frequency ω that obey the dispersion relation k = k(ω). While standard dielectrics have a closed-surface topology, $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ in hyperbolic media is characterized by a fundamentally different open-surface topology (see k-Space Topology Section in Methods)1,2,3,4,5,6,7,8. Hyperbolic phenomenology can emerge in photonic crystals and waveguide arrays9,10,11, metasurfaces12,13, layered media14,15,16,17, and metamaterials18,19,20,21. A hyperbolic medium placed along an optical circuit then automatically implies one or more k-space topological transitions for the transmitted waves22,23,24,25,26,27,28,29, a photonic equivalent of Bloch-electron Lifshitz transitions30,31,32,33. As discussed in the Self-Induced Topology Section in Methods, the diffusive photorefractive nonlinearity present in near-transition paraelectric ferroelectrics34,35 with a giant electro-optic response36,37,38,39 causes $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ to become $$(1-{\alpha }^{2})({k}_{x}^{2}+{k}_{y}^{2})+{k}_{z}^{2}={k}_{0}^{2}{n}^{2}$$, where the dimensionless parameter α = L/λ is determined by the diffusive length scale $$L=4\pi {n}^{2}{\varepsilon }_{0}\sqrt{g}{\chi }_{PNR}({K}_{B}T/q)$$. Here g is the quadratic electro-optic coefficient, χPNR the low-frequency susceptibility dominated by super-cooled polar-nanoregions (PNRs), and KBT/q is the thermal voltage. This implies a passage from a closed-surface topology for α < 1 (see Fig. 1A, blue surfaces) to an open-surface two-sheet hyperbolic topology for α > 1 (Fig. 1A, yellow surfaces) as α is swept through the α = 1 critical value. The effect of a transition from α = 0 to α > 1 is illustrated in Fig. 1B for the simplified 1+1D case, i.e., when dynamics only depend on one transverse coordinate x and the propagation coordinate z (so that $$(1-{\alpha }^{2}){k}_{x}^{2}+{k}_{z}^{2}={k}_{0}^{2}{n}^{2}$$). As light passes from the elliptical medium (air) to the hyperbolic medium (the ferroelectric with α > 1), the boundary conservation of the transverse component of $${k}_{x}={k}_{x}^{\prime}$$ causes the transmitted wave to undergo negative refraction: the original Poynting vector S is redirected along the normal to the hyperbolic isofrequency surface, $${{{{{{{\bf{S}}}}}}}}^{\prime}$$, opposite to what normally occurs in standard refraction. $${{{{{{{\bf{S}}}}}}}}^{\prime}$$ forms an angle $$\tan \theta \simeq \sqrt{{\alpha }^{2}-1}$$ with the normal to the boundary surface for the part of the hyperbolic branch that can be approximated by an asymptote (dashed line). As illustrated in Fig. 1C, for a localized beam composed of a spectrum of plane waves on the α = 0 → α > 1 boundary (the ‘object’ point), the transition amounts to the splitting of the original beam into two beams, each inheriting only half of the input transverse spatial spectrum (kx > 0 and kx < 0). Insomuch that each half of the spectrum principally generates waves in the α > 1 medium with a $${{{{{{{\bf{k}}}}}}}}^{\prime}$$ that lies on the asymptotic branches, each beam undergoes negligible diffraction and propagates forming and angle ± θ to the normal of the boundary, crossing inside the crystal (the ‘internal focus’). In detail (see Fig. 1C right), the transition implies the topologically non-trivial crossing of the kx > 0 spectrum to the beam with negative x-component Poynting vector, while the negative spectrum beam kx < 0 propagates with a positive x-component. The two crossing beams are topologically protected, that is, they cannot suffer further cascaded splitting in the x direction because they only occupy one of the two arms of the hyperbola corresponding to its half-spectrum. If the beams then suffer an inverse transition α > 1 → α = 0 (exiting the sample), they will once again propagate as the initial localized beam but with a signature non-trivial topological twist that leads to an ensuing crossing point (the ‘image’).
We here demonstrate experimentally self-induced topological transitions from elliptical to hyperbolic k-space manifolds in room-temperature photorefractive KTN.
## Results
To observe the transition, we carried out experiments using the setup illustrated in Supplementary Fig. 1A. A λ = 488 nm y-polarized laser is made to propagate (along the z axis) through a potassium-tantalate-niobate (KTN) crystal whose electro-optic response is enhanced through rapid-cooling (see Material section in Methods). A 100-nm-thick aluminum lithographic mask was deposited onto the polished input xy facet, with an etched circular 8 μm diameter transmission pinhole (see the Scanning-Electron-Microscope (SEM) image in Supplementary Fig. 1B. We observed analogous effects for circular pinholes with diameters of 4, 6, and 10 μm (not reported here). The input localized spot (‘object’) is achieved launching light through the pinhole using a NA = 0.2 lens that forms a ~ 135 μm spot centered on it. The spatial beam intensity distribution at the input and output facets of the crystal are monitored using a CMOS detector array and x50 objective lens (NA = 0.4), while beam power is controlled using neutral density filters at input and measured at output using a silicon power-meter.
The photorefractive effect is caused by the electro-optic response to a photoinduced space-charge electric field. The phenomenon is cumulative in the sample exposure time t, and α = α(t) reaches a steady-state α() = L/λ only after a transient build-up. For $$\alpha (\infty )=\frac{L}{\lambda } \; > \; 1$$, the topological transition that occurs as α passes from α < 1 to α > 1 can be observed as a function of t (see the Controlling α Section in Supplementary Information). Evidence of a self-induced transition is reported in Fig. 2A–D. As α is increased through the transition at α = 1, the propagating beam passes from being a conventional diffracting beam dominated by elliptical closed-surface topology (Fig. 2A), to one undergoing the so-called scale-free optics regime associated to the hybrid flat topology (Fig. 2B, α = 1). In turn, for higher values of α > 1, we find the signature formation of a pair of non-diffracting negative refracting beams, the product of the hyperbolic k-space topology (Fig. 2C, D). In Fig. 2E the full one-axis transition is reported as a function of exposure time t.
In Fig. 3 we report the exploration of the transition as the exposure time t is increased and the input launch power P (as measured after passing through the pinhole) is modified. As expected, an increasing value of α leads to an increase in the angle between the x − directed twin beams. The increase also leads to a characteristic splitting in the second transverse direction y, a phenomenon that becomes ever more evident for higher values of α (third and fourth row of Fig. 3A). For t = 600 s, a rhombus-like non-diffracting distribution emerges (fourth row). While splitting in both the x and y directions is expected, as the wave-system is 2+1D, the fact that the splitting first occurs in the x direction and then in the y indicates an anisotropy in the photorefractive response associated to a thermal gradient on cooldown, as observed in previous experiments34. As discussed in the Anisotropy Section of Methods, the system obeys an anisotropic $$(1-{\alpha }_{x}^{2}){k}_{x}^{2}+(1-{\alpha }_{y}^{2}){k}_{y}^{2}+{k}_{z}^{2}={k}_{0}^{2}{n}^{2}$$ internal dispersion relationship (αy ≠ αx). Hence, the transition will occur first for one direction (the x direction when αx > 1 while αy < 1) and, only after a sufficient nonlinear build-up, also in the orthogonal y direction (when also αy > 1). The result is a characteristic formation of four non-diffracting beams, the x-axis pair separating at an angle such that $$\tan {\theta }_{x}\simeq \sqrt{{\alpha }_{x}^{2}-1}$$, while the y-axis pair separating at a generally smaller angle $$\tan {\theta }_{y}\simeq \sqrt{{\alpha }_{y}^{2}-1}$$. The rhombus-like emission, in turn, signals the occurence of cascaded x − y transitions (see Fig. 3B). Specifically, in distinction to fabricated hyperbolic materials, a self-induced topology acts continuously on the propagating beams, so that a transition can occur also far from the input facet, as determined by the actual intensity at a given position z and the overall exposure at that point t. The effect is naturally strongest at the input of the sample (z 0), since here the intensity of the originally diffracting beam is most intense. However, after each of the four negatively refracting beams have formed, amounting to the vertices of the rhombus-like pattern, each will itself seed further cascaded topological transitions, albeit produced by a lower peak intensity and ensuing smaller effective α and corresponding angle θ (Fig. 3B). Since cascaded transitions in the x direction are protected once a first x − axis twin beam transition has occurred, a second cascaded transition can still occur in the orthogonal y direction, as along this direction each of the two x − directed beams has its full ky spectrum. Similarly, y-directed beams can suffer a cascaded transition and split in the x-direction. In turn, the cascade of the first and second orthogonal topological transitions is, as a whole, topologically protected, since each of the four resulting non-diffracting beams only has a quarter-slice of the original spectrum, a picture that leads to the signature rhombus-like emission. Interestingly, since photoexcitation from donor impurities is negligible for near IR wavelengths in the Cu-doped samples, the self-induced visible topological transition can even be passively inspected using a second IR probe beam (see Supplementary Information).
The full superlensing effect associated to the 2+1D self-induced transition is reported in Fig. 4 (see Superlens section in Methods). Placing a KTN sample in the path of a diffracting beam from an ‘object’ (a focused laser beam, Fig. 4A) entails two separate variations in propagation regimes: one entering the material (Fig. 4B) and propagating in a condition of α > 1 (Fig. 4C), and a second when exiting the material and returning to the conventional α = 0 propagation (Fig. 4D–G). Light, having suffered the double topological transition, now crosses at an external second point, forming a signature ‘image’ (Fig. 4F).
Importantly, superlensing occurs as a consequence of negative refraction in a solely dielectric medium, without the constraints associated to strong absorption. In this, our study sides recent advances in ultra-thin non-metallic metamaterials for light control40. The absence of a metallic component at once guarantees that no wave amplification is required and that no surface plasmons or surface currents are involved so that, combined with the leading linearity in the resulting hyperbolic regime, the standard boundary conditions typical of the Snell law hold. Based on a leading hyperbolic propagation, our scheme involves the passage through the ‘inner focus’ causing the negative refracting waves to have an enlarged (and topologically twisted) output Fourier spectrum, remaining in an overall flat geometry. In these terms, light transmitted through the pin-hole mask, deposited directly on the surface of the sample, will transmit subwavelength features that can be detected, at the output facet, using super-resolution detection schemes. Ideally, a functioning superlens based on a self-induced transition would require a flourescent component in the KTN, as this would convert the high-resolution information directly into propagating waves beyond the sample. Alternatively, the input or output facet of the sample could be sculpted into a curved geometry, as done in a hyperlens41. In these conditions, the spatial resolution limit will be dictated by the validity of the macroscopic diffusion-driven model, that breaks down on spatial scales comparable to the average size of the underlying PNRs. Depending on the specific quenching implemented, this places the limit on the scale of tens to hundreds of nanometers35.
From the applicative perspective, the fact that self-induced hyperbolic dispersion does not involve resonances indicates a method to design broadband flat lenses for visible light. Furthermore, the altered dispersion is spatially resolved, suggesting that it can be used to achieve different transfer functions in parallel across the entire sample.
Conventional nonlinearities, such as the optical Kerr effect, are intensity-dependent, the result being a change in the index of refraction localized in actual space, not in k-space. This can create new topological states, such as self-induced edge states and gap-solitons42,43,44,45. These arise as topological defects and not as changes in the topology of the plane-wave manifold $${{{{{{{{\mathcal{M}}}}}}}}}_{k}$$. The diffusive nonlinearity, in turn, entails the passage from one linear plane-wave regime characterized by the topology of $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ to another, still linear, regime, with a $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{\prime}$$ characterized by a different topology. The nonlinear signature of the diffusive nonlinearity remains in the fact that, in distinction to fabricated hyperbolic metamaterials, the modified $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{\prime}$$ is not the dispersion relation of the KTN crystal, but that of an internal decomposition of the field, a field that is forcibly tied to a localized Gaussian-like waveform. The associated topology of $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{\prime}$$, therefore, represents an optical analog of the local gauge structure of the field, and not a property of freely propagating plane-waves.
## Methods
### k-space topology
Optical waves of wavelength λ in an isotropic dielectric of index of refraction n are bound to a closed-surface $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$, the Ewald sphere, with $${k}_{x}^{2}+{k}_{y}^{2}+{k}_{z}^{2}={k}_{0}^{2}{n}^{2}$$ and k0 = 2π/λ. For a given propagation axis, say the z-axis, only the subset of possible (kx, ky) eigenpairs such that $${k}_{x}^{2}+{k}_{y}^{2}\le {k}_{0}^{2}{n}^{2}$$ leads to propagating waves with a corresponding real $${k}_{z}=\pm \sqrt{{k}_{0}^{2}{n}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}$$, a constraint that limits the minimum transmitted spatial detail to Λ ~ λ/2n. The curvature of the spherical $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ leads to standard diffraction for wavepackets, while the Poynting vector S, normal to $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$, is parallel to the average k. In a metallic-like hyperbolic metamaterial, $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ is a one-sheet hyperboloid, topologically equivalent to $${k}_{x}^{2}+{k}_{y}^{2}-{k}_{z}^{2}={k}_{0}^{2}{n}^{2}$$, and only the high-resolution spectrum (kx, ky) such that $${k}_{x}^{2}+{k}_{y}^{2}\le {k}_{0}^{2}{n}^{2}$$ forms propagating waves along z. In a dielectric-like hyperbolic metamaterial, $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ is topologically equivalent to a two-sheet hyperboloid, such as $$-{k}_{x}^{2}-{k}_{y}^{2}+{k}_{z}^{2}={k}_{0}^{2}{n}^{2}$$, and, as long as the macroscopic effective description holds, all values of (kx, ky) form propagating waves, no diffraction limit exists, and a constraint holds on the resulting allowed direction of k imposed by the minimum value of allowed $${k}_{z}^{2}\le {k}_{0}^{2}{n}^{2}$$. For both dielectric and metallic hyperbolic systems, spectral components that correspond to an asymptotic conical $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ ($${k}_{x}^{2}+{k}_{y}^{2}\gg {k}_{0}^{2}{n}^{2}$$), lead to negligible diffraction and an S, normal to $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$, that is forced along specific topologically protected directions. The connection between hyperbolic dispersion and negative refraction is discussed further in the Supplementary Information.
### Self-induced topology
In unbiased paraelectric photorefractive crystals, the thermal agitation of photogenerated charge carriers leads to a shape-dependent diffusive nonlinearity34,35,46,47,48,49. For a monochormatic Gaussian-like optical field E of wavelength λ, light passes from obeying the standard Helmholtz Equation $$({\partial }_{zz}+{\nabla }_{\perp }^{2}+{k}_{0}^{2}{n}^{2}){{{{{{{\bf{E}}}}}}}}=0$$, to a modified but still linear leading equation $$({\partial }_{zz}+(1-{\alpha }^{2}){\nabla }_{\perp }^{2}+{k}_{0}^{2}{n}^{2}){{{{{{{\bf{E}}}}}}}}=0$$, where α = L/λ and $$L=4\pi {n}^{2}{\varepsilon }_{0}\sqrt{g}{\chi }_{PNR}({K}_{B}T/q)$$ at steady-state (see Methods section in DelRe et al.35). Here g is the effective quadratic electro-optic coefficient, χPNR the effective low-frequency susceptibility of the PNRs, KB the Boltzmann constant, T the temperature, and q the elementary charge. In terms of plane-wave components $${{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}={{{{{{{{\bf{E}}}}}}}}}_{0}({{{{{{{\bf{k}}}}}}}})\exp (i{{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{\bf{r}}}}}}}})$$, $${{{{{{{{\mathcal{M}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}$$ is $${k}_{z}^{2}+(1-{\alpha }^{2})({k}_{x}^{2}+{k}_{y}^{2})-{k}_{0}^{2}{n}^{2}=0$$. For conditions in which α 1, beam dynamics are governed by the so-called scale-free-optics equation $$({\partial }_{zz}+{k}_{0}^{2}{n}^{2}){{{{{{{\bf{E}}}}}}}}=0$$, where the transverse Laplacian is absent. The result, that is equivalent to the flat-band topology in a metal Lifshitz transition, is the propagation of Gaussian beams of arbitrary intensity and width that simply do not diffract, even when they are initially localized into subwavelength spots35. α > 1, in turn, corresponds to a self-induced hyperbolic regime in which the Laplacian along the propagation z axis and the transverse x, y axes has opposite signs. Values of α > 1 require giant values of χPNR > 105 reached by rapidly quenching the sample to TC34,35,36. The role of nonlocal nonlinearity in the topological transition both in direct and k-space is further discussed in the Supplementary Information.
### Material
The crystal was grown using the top seeded solution method. A 4. 9(x) × 12. 3(y) × 2. 2(z)mm sample was cut from the grown boule along the crystallographic [001] axes, and polished to optical grade50. The crystal composition is KTa0.65 Nb0.35 O3, as determined by electron microprobe analysis. The sample was doped, during growth, with traces of Li (less than one substitution of K in 1000) and Cu. Li enhances optical quality close to the phase transition temperature, while Cu supports photorefraction51,52,53. The ratio between Nb and Ta in the solid-solution fixes the value of the Curie Temperature54, that in the sample is TC = 18.4 °C, as determined from the maximum point of the relative dielectric permittivity εr(T) curve. The sample is installed in a specifically tailored crystal-holder that is affixed on top of a temperature controlled thermoelectric element (see Supplementary Fig. 1B), while the temperature control circuit is designed to operate with reduced ringing. To prevent water vapor condensation, the entire optical setup is placed in a box in which a constant flow of dry N2 gas is maintained by overpressure. Prior to each experiment, the crystal is cleansed of residual photorefractive space-charge by illuminating it from the top facet with a laser beam (λ = 532 nm, diameter of ~4.5 mm, 600 mW for 400 s at a temperature of 80 °C). After this exposure, it is left to dwell for 200 s. From the initial temperature T(0) = 80 °C the sample is cooled at a constant cooling rate, ranging between $$\dot{T}=0.3{-}0.{8}^{\circ }$$C/s, to the selected dwell temperature T() = 16 °C. When the temperature reaches T() + 8 °C, the controller slows the cool-down, critically relaxing the sample to T(), where it remains unchanged for the duration of the experiment (exhibiting fluctuations of 0.01 °C). For a fixed T() in proximity of TC, the actual value of L is determined by χPNR, in turn determined by $$\dot{T}$$. For KTN, a model picture suggests that for T > TC the Nb ions emerge from the center of inversion of their respective unit cells forming nanodipoles that hop between the minima of their respective potential field, interact, and create PNRs that fluctuate at random, growing in size on approaching TC. Congruently, cooling to TC, the sample dielectric response exhibits the Vogel-Fulcher-Tammann (VFT) behavior, characteristics of soft condensed matter, behaving as a dipolar-glass-forming-liquid53.
Experiments in samples with slightly different compositions lead to analogous phenomena with modifications in the specific details, such as Curie point and dependence on exposure time for a specific beam intensity (experiments in different samples are not reported here).
### Anisotropy
A first basic anisotropic effect is that the topological transition (as reported in Fig. 2) can be seen for an x-polarized beam, while a y-polarized beam leads only to a weak reduction in diffraction compatible with a low value of associated α < 1. This is in agreement with previous studies for which the χPNR is found to be small for a polarization parallel to the cooldown thermal gradient and an ensuing pyroelectric field (which is, in our case, principally in the y direction) (see Eqs. (29), (30) in the Supplementary Information of DelRe et al.34). The anisotropic dispersion relation recalled in the main article is, in turn, associated to a partial ordering of the PNRs associated to the thermal gradient and the ensuing anisotropic electro-optic response, as discussed in the Supplementary Information.
### Superlens
The superlens experiment was carried out using a linear polarized z propagating Gaussian beam (488 nm, 46 μW power) focused to its minimum 20 μm FWHM waist in a plane 5 mm before the crystal input facet (‘object’ plane, Fig. 4A). In other aspects, the system is the same as that reported in Fig. 2, the sample being of the same composition but now with no deposited mask. The beam size at the input facet is a slightly diffracted Gaussian beam with a 50 μm FWHM (Fig. 4B) that, without a topological transition (T > TC), diffracts to an 80 μm FWHM in a plane 5 mm from the output facet (’image’ plane). Results are reported for t = 100 s, while the sample is rapidly cooled from T(0) = TC + 64 K to T() = TC − 3 K (in this specific case, $$\dot{T}\simeq 0.{5}^{\circ }$$C/s). As expected for the resulting superlens effect, the beam expands as it reaches the output facet (Fig. 4C), after the internal focus point, and refocuses on exiting the crystal (Fig. 4D–F) to then re-expand after the image focus (Fig. 4G). The position of the focus, that in the reported case is 5 mm from the output facet, is dependent on the cooling rate $$\dot{T}$$ and exposure t, and was observed in the range of 2–5 mm from the output facet changing $$\dot{T}$$ in the range 0.3−0.8 °C/s. It is useful to discuss the details in terms of the transverse spatial spectrum of the superlensing in Fig. 4 compared to the experiments in Figs. 2 and 3. Experiments in Figs. 2 and 3 involve a transverse spectrum with Airy rings and a characteristic k ~ 5 × 105 m−1 able to populate, for the values of α inspected, regions of the hyperbolic branches. In turn, in the conditions of Fig. 4, the spectrum is a localized Gaussian (k ~ 5 × 104 m−1), and the propagation is dominated by the central parabolic region of the dispersion hyperbola.
## Data availability
The datasets analyzed in the current study are available from the corresponding author on reasonable request. Source data are provided with this paper.
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## Acknowledgements
Support from the Sapienza-Ricerca di Ateneo 2019 and 2020 projects (C.C. and E.D.R.), the H2020 Fet project PhoQus (C.C. and E.D.R.), the PRIN 2017 PELM (grant number 20177PSCKT, C.C. and E.D.R.) and PRIN 2020 (grant number 2020X4T57A, E.D.R.) projects, the Israel Science Foundation (Grant No. 1960/16, A.J.A.), and the Israel ministry of Science technology and space (Grant No. 3-16816, A.J.A.) is acknowledged.
## Author information
Authors
### Contributions
A.J.A. designed the crystals, that were grown and synthesized by Y.Ga. and G.P.; A.J.A. conceived the propagation experiments, while Y.Ge. and E.D.R. conceived the superlensing experiments. Y.Ge. carried out the principal measurements under the supervision of A.J.A.; E.D.R. elaborated, with the help of Y.Ge., the physical framework and interpretation, discussing ideas with F.D.M., C.C., A.J.A . F.D.M., S.F., and L.F. carried out preliminary experiments. All authors participated in discussions. Y.Ge. and E.D.R. wrote the article, with the help of all authors.
### Corresponding author
Correspondence to Eugenio DelRe.
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Gelkop, Y., Di Mei, F., Frishman, S. et al. Hyperbolic optics and superlensing in room-temperature KTN from self-induced k-space topological transitions. Nat Commun 12, 7241 (2021). https://doi.org/10.1038/s41467-021-27466-3
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• DOI: https://doi.org/10.1038/s41467-021-27466-3 | 2022-05-16 23:06:50 | {"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": 0.6975383758544922, "perplexity": 3305.6048198041253}, "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/1652662512249.16/warc/CC-MAIN-20220516204516-20220516234516-00787.warc.gz"} |
https://byjus.com/question-answer/state-true-or-false-in-bigtriangleup-abc-ad-is-the-median-and-de-is-parallel/ | Question
# State true or false:In $$\bigtriangleup ABC$$, $$AD$$ is the median and $$DE$$ is parallel to $$BA$$, where $$E$$ is a point in $$AC$$ and hence $$BE$$ is parallel to $$BC$$.
A
True
B
False
Solution
## The correct option is B FalseGiven: $$AD$$ is the median of $$\triangle ABC$$.Hence, $$BD = DC$$.Also, $$DE \parallel AB$$ and $$DE$$ is drawn from the mid point of $$BC$$, i.e. $$D$$.Thus, by converse of mid-point theorem, $$DE$$ bisects the third side, which is $$AC$$.Then, $$E$$ is the mid point of $$AC$$.Hence, $$BE$$ is the median of $$\triangle ABC$$.That is, the given statement is false and option $$B$$ is correct.Mathematics
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View More | 2022-01-19 20:02:01 | {"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": 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": 0.8526902794837952, "perplexity": 447.22693738333675}, "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-2022-05/segments/1642320301488.71/warc/CC-MAIN-20220119185232-20220119215232-00180.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-1-section-1-5-linear-inequalities-in-one-variable-1-5-exercises-page-100/40 | ## Intermediate Algebra (12th Edition)
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $10\left(\dfrac{1}{5}x+2 \right) \lt 10\left(\dfrac{1}{5}x+1 \right) ,$ use the Distributive Property and the properties of inequality. For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$ For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Using the Distributive Property and the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} 10\left(\dfrac{1}{5}x\right)+10(2) \lt 10\left(\dfrac{1}{5}x\right)+10(1) \\\\ 2x+20 \lt 2x+10 \\\\ 2x-2x \lt 10-20 \\\\ 0\lt-10 \text{ (FALSE)} .\end{array} Since the solution above ended with a FALSE statement, then there is $\text{ no solution .}$ There is no graph since there is no solution set. | 2018-04-24 18:50:03 | {"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": 0.9628576636314392, "perplexity": 900.2520408017303}, "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-2018-17/segments/1524125947033.92/warc/CC-MAIN-20180424174351-20180424194351-00059.warc.gz"} |
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The four sentences $( \text{labelled} \; 1, 2, 3, 4)$ below, when properly sequenced would yield a coherent paragraph. Decide on the proper sequencing of the order of the sentences and key in the sequence of the four numbers as your answer: A ... why the scale of forest fires in the Amazon basin last year garnered headlines. This is because trees sequester carbon by absorbing carbon dioxide. | 2022-05-20 04:25:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.3518920838832855, "perplexity": 1732.5523583292872}, "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/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00633.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-2-section-2-5-formulas-and-problem-solving-exercise-set-page-150/60 | Algebra: A Combined Approach (4th Edition)
$3.5$ revolutions
$x=\frac{s}{d}$ $x=\frac{87,000}{25,132.74}$ $x=3.461620181$ $$x=3.5$$ | 2018-04-25 09:00:46 | {"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": 0.40661853551864624, "perplexity": 14695.270402669328}, "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-2018-17/segments/1524125947759.42/warc/CC-MAIN-20180425080837-20180425100837-00625.warc.gz"} |
http://mathhelpforum.com/differential-equations/161272-laplace-shifting-problem.html | # Thread: Laplace shifting problem
1. ## Laplace shifting problem
Hi,
I am trying to solve
$y''+4y=0, 0a$
using Laplace transforms.
I end up with
$H(t-a)(exp(-(t-a))-cos(2(t-a))+1/2sin(2(t-a))$
where H(t-a) is the Heaviside/unit step function. Is this correct? If not, where have I gone wrong?
Cheers,
Ben
2. Hmm. That isn't what I get. You're missing the complimentary function, I think.
1. How can you write the original DE in one equation using the Heaviside step function?
2. What do you get when you LT the DE?
3. What are the initial conditions?
4. So what do you get when you solve for the LT of y?
3. Originally Posted by weedingb
Hi,
I am trying to solve
$y''+4y=0, 0a$
using Laplace transforms.
I end up with
$H(t-a)(exp(-(t-a))-cos(2(t-a))+1/2sin(2(t-a))$
where H(t-a) is the Heaviside/unit step function. Is this correct? If not, where have I gone wrong?
Cheers,
Ben
You have proposed a second order linear DE for the solution of which it is necessary to know the values $y(0)$ and $y^{'}(0)$ at the moment not yet specified...
Kind regards
$\chi$ $\sigma$
4. The DE is [in my opinion]...
$y^{''} + 4\ y= \left\{\begin{array}{ll} 0,\,\,0 < x < a{}\\e^{-t} ,\,\, x\ge a\end{array}\right.$ (1)
... and if we suppose $y(0)= y^{'}(0)=0$ in terms of Laplace Transform is written...
$\displaystyle s^{2}\ Y(s) + 4\ Y(s) = \frac{e^{-a\ (s+1)}}{s+1} \implies Y(s)= \frac{e^{-a\ (s+1)}}{(s+1)\ (s^{2}+4)}$ (2)
Now if You consider that is...
$\displaystyle \frac{1}{(s+1)\ (s^{2}+4)} = \frac{1}{5}\ \{\frac{1}{s+1}+ \frac{1-s}{s^{2}+4} \}$ (3)
... You obtain...
$\displaystyle y(t)= \mathcal{L}^{-1} \{Y(s)\} = \frac{e^{-a}}{5}\ \{e^{-(t-a)} + \cos 2(t-a) - \sin 2(t-a)\}\ \mathcal{H} (t-a)$ (4)
Kind regards
$\chi$ $\sigma$ | 2016-08-26 14:47:37 | {"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": 15, "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": 0.8587420582771301, "perplexity": 940.7803496756054}, "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-2016-36/segments/1471982295854.33/warc/CC-MAIN-20160823195815-00157-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://discourse.julialang.org/t/read-only-memory-mapped-files/12098 | I am developing some code in BEDFIles.jl that may be used with very large binary data files. The data are accessed as a read-only memory-mapped Matrix{UInt8} using column-oriented algorithms whenever possible.
As I understand it, there shouldn’t be a problem with having a very large file if I am only accessing a small set of adjacent columns. Suppose that I have 100,000 rows and 10 million columns but I only access the first 10,000 columns. I believe that the columns beyond 10,000 will never need to appear in memory - that they are essentially held as a kind of a promise by the operating system (which would be Linux - I don’t care if Windows does dumb things with memory-mapped files). Is this correct?
If you are opening a memory mapped file, yes that is correct.
Admittedly I am still rather hazy on some of the details, but you can get a partial description here.
It should go without saying that you still have to be careful about actually copying data out of the memory mapped array.
Yes, your expectations are correct. I have mmapped 500GB files on a 16GB machine without any problems, the OS (in my case, Linux) takes care of the memory operations very transparently, paging on demand.
For 10^5\cdot10^4=10^9 UInt8s, that’s 1GB, so chances are the whole section could just fit in memory, making access really fast. | 2022-05-21 02:28:36 | {"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": 0.5176947116851807, "perplexity": 1112.2829441935799}, "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/1652662534773.36/warc/CC-MAIN-20220521014358-20220521044358-00156.warc.gz"} |
http://medicaldevices.asmedigitalcollection.asme.org/article.aspx?articleid=2673583 | 0
Research Papers
# A Precision System for Computed Tomography-Guided Needle Placement in the Thorax and Abdomen—Technical Design and Performance Analysis
[+] Author and Article Information
Maarten M. Arnolli
Enschede 7521 PH, The Netherlands;
Precision Engineering
Science-Based Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands
e-mail: m.m.arnolli@alumnus.utwente.nl
Martijn Buijze, Michel Franken
Enschede 7521 PH, The Netherlands
Ivo A. M. J. Broeders
Robotics and Mechatronics,
MIRA Institute for Biomedical Technology and
Technical Medicine,
University of Twente,
Enschede 7522 NB, The Netherlands
Dannis M. Brouwer
Precision Engineering
Science-Based Engineering,
University of Twente,
Enschede 7522 NB, The Netherlands
1Corresponding author.
Manuscript received May 23, 2017; final manuscript received January 30, 2018; published online April 2, 2018. Assoc. Editor: Carl Nelson.
J. Med. Devices 12(2), 021003 (Apr 02, 2018) (10 pages) Paper No: MED-17-1231; doi: 10.1115/1.4039389 History: Received May 23, 2017; Revised January 30, 2018
## Abstract
A system was developed for computed tomography (CT)-guided needle placement in the thorax and abdomen, providing precise aiming of a needle guide (NG) to reach a user-specified target in a single manual insertion. The objective of this work is to present its technical design and analyze its performance in terms of placement error in air. The individual contributions to the placement error of a fiducial marker based system-to-CT registration system, a two degrees-of-freedom (2DOFs) drive system to aim the NG, and a structural link between NG and CT table were experimentally determined, in addition to the placement error of the overall system. An error contribution of 0.81 ± 0.34 mm was determined for the registration system, <1.2 mm and <3.3 mm for the drive system, and 0.35 mm and 0.43 mm for two load cases of the structural link. The overall unloaded system achieved 1.0 ± 0.25 mm and 2.6 ± 0.7 mm at 100 mm and 250 mm depth, respectively. The overall placement errors in air do not exceed the $≤$5 mm error specified as a clinical user requirement for needle placement in tissue.
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## References
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McCarley, J. , and Soulen, M. , 2010, “ Percutaneous Ablation of Hepatic Tumors,” Semin. Interventional Radiol., 27(3), pp. 255–260.
Winokur, R. , Pua, B. , Sullivan, B. , and Madoff, D. , 2013, “ Percutaneous Lung Biopsy: Technique, Efficacy, and Complications,” Semin. Interventional Radiol., 30(2), pp. 121–127.
Dupuy, D. , and Shulman, M. , 2010, “ Current Status of Thermal Ablation Treatments for Lung Malignancies,” Semin. Interventional Radiol., 27(3), pp. 268–275.
Arnolli, M. , Hanumara, N. , Franken, M. , Brouwer, D. , and Broeders, I. , 2015, “ An Overview of Systems for CT- and MRI-Guided Percutaneous Needle Placement in the Thorax and Abdomen,” Int. J. Med. Rob. Comput. Assisted Surg., 11(4), pp. 458–475.
Barrett, S. , Hanumara, N. , Walsh, C. , Slocum, A. , Gupta, R. , and Shepard, J. , 2005, “A Remote Needle Guidance System for Percutaneous Biopsies,” ASME Paper No. DETC2005-85387.
Stoianovici, D. , Cleary, K. , Patriciu, A. , Mazilu, D. , Stanimir, A. , Craciunoiu, N. , Watson, V. , and Kavoussi, L. , 2003, “ AcuBot: A Robot for Radiological Interventions,” IEEE Trans. Rob. Autom., 19(5), pp. 927–930.
Melzer, A. , Gutmann, B. , Lukoscheck, A. , Mark, M. , Zylka, W. , and Fischer, H. , 2003, “ Experimental Evaluation of an MR Compatible Telerobotic System for CT MR-Guided Interventions,” Suppl. Radiol., 226(409), p. 444.
DEMCON, 2015, “Precise Needle Positioning,” DEMCON, Enschede, The Netherlands, accessed Jan. 30, 2018,
Sprawls, P., Jr ., 1995, Physical Principles of Medical Imaging, 2nd ed., Medical Physics Publishing, Madison, WI.
Arnolli, M. , Franken, M. , and Brouwer, D. , 2013, “ CT Registration: Experimental Determination of Suited Fiducial Marker Material and Registration Errors,” ASME J. Med. Devices, 7(2), p. 020946.
## Figures
Fig. 1
Overview of the system with labeled components
Fig. 2
Illustration of the system's guided method for needle placement. An animation can be found online [9]: (a) the system installed on the patient table, awaiting initial path planning and entry point retrieval, (b) manual placement of the OM such that the RCM coincides with the entry point and push-button controlled locking to the patient table by the LM, (c) CT scanning of patient and OM for OM-to-CT registration using four fiducial markers and for target specification, (d) automated aiming of the NG by the RCM mechanism, (e) manual needle insertion through the NG to specified depth, and (f) CT scanning for placement verification.
Fig. 3
System internals, showing the fiducial markers and the drive system
Fig. 4
Computed tomography scanning of the dummy frame to determine the contribution of the registration system to the placement error: (a) CT scanning of the dummy frame on a foam support and (b) 3D CT image of the dummy frame
Fig. 5
Mean value, standard deviation, minimum and maximum (filled areas) of FREs (Eq. (2)) and TRE (Eq. (8)) as a function of pixel value threshold level used for segmentation, accompanied by the number of successful segmentations out of eight CT scans
Fig. 6
Kinematic diagram of the transmission from worms to segments, converted from the rotational to the translational domain for illustrative purposes. Diagonal cross-hatching indicates connection to the fixed frame of reference of the MB of the OM.
Fig. 7
Experiment setups to determine the error contribution of the drive system: (a) measuring the angle of segment 1 and (b) measuring the angle of segment 2
Fig. 8
Angular errors of the drive system and equivalent placement errors at a target depth of 250 mm: (a) transmission errors for worm 1 to segment 1 and (b) transmission errors for worm 2 to segment 2
Fig. 9
Cross section of the system, labeling the materials forming the structural link
Fig. 10
Experiment setups to determine the compliances of the structural link from NG to table: (a) measurement of deflections dx1, dx2, and dy1 under load FyMB by mass m1 on the MB of the OM, and corresponding deflection diagram and (b) measurement of deflections dy2 and dy3 under load FyS2 by mass m2 on segment 2 (S2), and corresponding deflection diagram
Fig. 11
Determining the placement error of the overall system: (a) the system is installed on the CT table and the OM is locked. The center of a fiducial marker on a PVC rod serves as a target, (b) 3D CT image of the OM and target fiducial marker for OM-to-CT registration and definition of the target coordinates, (c) the target fiducial marker is replaced by a rod with a conical end, to measure the placement error after needle placement, and (d) illustration of ten target positions at 100 and 250 mm depth from the RCM and corresponding placement errors.
## Discussions
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Topic Collections | 2018-07-16 04:27:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.32098543643951416, "perplexity": 11517.630421906431}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589179.32/warc/CC-MAIN-20180716041348-20180716061348-00371.warc.gz"} |
https://tex.stackexchange.com/questions/108418/anki-latex-math-font-size-declaremathsizes | # Anki - LaTeX, Math Font Size, & DeclareMathSizes
I use the Anki flashcard program and render math equations using LaTeX within the program.
The problem is that the font sizes are generally too big, unless the equation is really long, in which case the equation automatically sizes down in order to fit the screen.
This is the default LaTeX header in Anki:
\documentclass[12pt]{article}
\special{papersize=3in,5in}
\usepackage{amssymb,amsmath}
\pagestyle{empty}
\setlength{\parindent}{0in}
\begin{document}
And this is the default footer:
\end{document}
To call LaTeX within Anki when creating a flashcard, you use $and$, which I assume refer to the header and footer respectively. (I'm neither an Anki nor a LaTeX expert.)
Anyway, I've tried inserting the following right before \begin{document}:
\DeclareMathSizes{2}{2}{2}{2}
Yes, I realize this is supposed to make the text really tiny, but more importantly, nothing changes. My equations are still too big (or too small, if the equation is really long).
• welcome to tex.sx. For those of use (most of us I'd guess) without that program could you post the generated latex document including an equation so we can see what markup it is generating and then can suggest a preamble to control the fonts. – David Carlisle Apr 12 '13 at 21:32
• Maybe a simple change to the header is enough. Something like \documentclass[10pt]{article} should change all font sizes. – Alexander Apr 12 '13 at 22:01
• @David, thanks for the welcome. Anki generates an image from the LaTeX input and attaches the image to the flashcard. Not being an Anki expert (nor a big tech expert in general), I don't know how to dig into its source code to see what "markup" it is generating. – MMS Apr 12 '13 at 22:46
• looking at github.com/dae/anki/blob/master/anki/latex.py it deosn't seem to do much with the latex other than append the preamble that you showed, you could change the 200 in the dvipng command on line 11 which will affect the image resolution so (depending on what exactly it means by that it might change the size when it is re-included) – David Carlisle Apr 12 '13 at 23:02
• @Alexander, thanks for the suggestion - it made me discover something interesting, which is that changing 12pt to 8pt did nothing to change the size of the text, even when testing something as simple as $Text$ (i.e., just plain old text). – MMS Apr 12 '13 at 23:02
Tools -> Manage Note Types... -> Options; in the last line of the Header field after \begin{document} just type
\tiny
And your LaTeX will be made smaller. Because previously rendered latex is cached as image files, you may have to delete those files before the change is apparent. On my Mac the files are in ~/Anki/User 1/collection.media/latex-*.png
For me this makes my [$$][/$$] equations the same size as the surrounding Anki font.
I hope this helps!
UPDATE: On Linux, the media directory seems to be located in ~/Documents/Anki/User 1/collection.media/
• On Windows, it's %USERPROFILE%\Documents\Anki\User 1\collection.media\ – Sydius Apr 28 '15 at 7:54
• In my case, I could not enter the \tiny on a new line. It had to immediately follow \begin{document}. – KSHMR Jan 30 '16 at 1:32
• That is, you can't press Enter and expect it to work, but it's okay for too long a line to overlap onto the next line. It seems this isn't designed for newlines. – KSHMR Jan 30 '16 at 1:37
• If you want to render the entire card in LaTeX to make it look more uniform, try changing out the \special{papersize=3in,5in} for \usepackage[paperheight=3in,paperwidth=5in]{geometry} or whatever card size you want, as well. I had to make the geometry smaller for the smaller text to make it look right for me. – Joe Anderson Jun 13 '17 at 13:39
• I settled on using 2in by 3.5in with \scriptsize instead of \tiny. – Joe Anderson Jun 13 '17 at 13:47
I have been struggling with this as well. My solution was to use the LaTeX environment for my entire question and answer. I surround my text with $ and $ and use Latex commands to format the card. This keeps my sizes consistent between text and math, and allows me to use other packages (such as making nice looking tables, compared to the Anki tables!)
For example, my Basic and Reverse cards have this in the preamble:
\documentclass[12pt]{article}
\special{papersize=3in,5in}
\usepackage[T1]{fontenc}
\usepackage[utf8x]{inputenc}
\usepackage{libertine-type1}
\usepackage{biolinum-type1}
\usepackage{libertineMono-type1}
\usepackage[libertine]{newtxmath}
\renewcommand{\familydefault}{\sfdefault}
\usepackage{amssymb,amsmath}
\usepackage{booktabs}
\usepackage{color}
\usepackage{multirow}
\usepackage{rotating}
\usepackage{graphicx}
\usepackage{wasysym}
\pagestyle{empty}
\setlength{\parindent}{0in}
\newcommand{\noun}[1]{\textsc{#1}}
\definecolor{blue}{RGB}{0,130,255}
\begin{document}
This gives me nice tables, math symbols, small caps and default to sans-serif font for graphical display devices.
An example card,
Front: $What is \textbf{finiteness?}$
Back: $A set A is said to be finite \emph{if and only if} \left|A\right|=\mathbb{N} (natural number.) A set that is not finite is said to be infinite.$
Granted, if the displayed card is resized, then the longer card back will also be resized to fit a smaller window. However, I think this can be fixed by changing the papersize in the preamble. I strictly use my laptop and tablet, so I do not have any issues. It may crop up on smaller devices like a phone though. | 2020-06-04 21:28:50 | {"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8670451045036316, "perplexity": 1549.2627135966286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347458095.68/warc/CC-MAIN-20200604192256-20200604222256-00153.warc.gz"} |
https://iris.sissa.it/handle/20.500.11767/4100 | In this thesis we study vector bundles on projective varieties and their moduli spaces. In Chapters 2, 3 and 4 we recall some basic notions as Higgs bundles, decorated bundles and generalized parabolic sheaves and introduce the problem we want to study. In chapter 5, we study Higgs bundles on nodal curves. After moving the problem on the normalization of the curve, starting from a Higgs bundle we obtain a generalized parabolic Higgs bundle. Using decorated bundles we are able to construct a projective moduli space which parametrizes equivalence classes of Higgs bundles on a nodal curve X. This chapter is an extract of a joint work with Andrea Pustetto Later on Chapter 6 is devoted to the study of holomorphic pairs (or twisted Higgs bundles) on elliptic curve. Holomorphic pairs were introduced by Nitsure and they are a natural generalization of the concept of Higgs bundles. In this Chapter we extend a result of E. Franco, O. Garc\'ia-Prada And P.E. Newstead valid for Higgs bundles to holomorphic pairs. Finally the last Chapter describes a joint work with Professor Ugo Bruzzo. We study Higgs bundles over varieties with nef tangent bundle. In particular generalizing a result of Nitsure we prove that if a Higgs bundle $(E,\phi)$ over the variety X with nef tangent remains semisatble when pulled-back to any smooth curve then it discrimiant vanishes.
Some topics on Higgs bundles over projective varieties and their moduli spaces / Lo Giudice, Alessio. - (2013 Sep 27).
### Some topics on Higgs bundles over projective varieties and their moduli spaces
#### Abstract
In this thesis we study vector bundles on projective varieties and their moduli spaces. In Chapters 2, 3 and 4 we recall some basic notions as Higgs bundles, decorated bundles and generalized parabolic sheaves and introduce the problem we want to study. In chapter 5, we study Higgs bundles on nodal curves. After moving the problem on the normalization of the curve, starting from a Higgs bundle we obtain a generalized parabolic Higgs bundle. Using decorated bundles we are able to construct a projective moduli space which parametrizes equivalence classes of Higgs bundles on a nodal curve X. This chapter is an extract of a joint work with Andrea Pustetto Later on Chapter 6 is devoted to the study of holomorphic pairs (or twisted Higgs bundles) on elliptic curve. Holomorphic pairs were introduced by Nitsure and they are a natural generalization of the concept of Higgs bundles. In this Chapter we extend a result of E. Franco, O. Garc\'ia-Prada And P.E. Newstead valid for Higgs bundles to holomorphic pairs. Finally the last Chapter describes a joint work with Professor Ugo Bruzzo. We study Higgs bundles over varieties with nef tangent bundle. In particular generalizing a result of Nitsure we prove that if a Higgs bundle $(E,\phi)$ over the variety X with nef tangent remains semisatble when pulled-back to any smooth curve then it discrimiant vanishes.
##### Scheda breve Scheda completa Scheda completa (DC)
Bruzzo, Ugo
Lo Giudice, Alessio
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/4100 | 2022-09-30 02:39:43 | {"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": 0.7882856130599976, "perplexity": 536.0153507295206}, "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-2022-40/segments/1664030335424.32/warc/CC-MAIN-20220930020521-20220930050521-00524.warc.gz"} |
http://swmath.org/?term=eigenfunctions | • MATSLISE
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• program allowing for calculation of simultaneous eigenfunctions of the energy H^ and spin... | 2017-07-25 12:34:22 | {"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": 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": 0.25603681802749634, "perplexity": 3768.295808532359}, "config": {"markdown_headings": false, "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-2017-30/segments/1500549425193.20/warc/CC-MAIN-20170725122451-20170725142451-00669.warc.gz"} |
https://homework.cpm.org/category/CC/textbook/cca/chapter/11/lesson/11.2.2/problem/11-54 | ### Home > CCA > Chapter 11 > Lesson 11.2.2 > Problem11-54
11-54.
1. For the function f(x) = 7x − 2: Homework Help ✎
1. Find the inverse function.
2. Show that your inverse function is correct. Choose an input number for the function and calculate the output. Use this output as the input for the inverse function. Is your final output the same as your original input?
For additional help, see problems 11-20 and 11-29.
Write down the steps for the function.
Reverse the order.
Write the new steps using algebra and inverse notation.
$\textit{f}^{-1}(\textit{x})= \frac{\textit{x}+2}{7}$ | 2019-08-20 08:00:21 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.5956882834434509, "perplexity": 1638.029919259068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315258.34/warc/CC-MAIN-20190820070415-20190820092415-00017.warc.gz"} |
http://openstudy.com/updates/4fbc3e2ae4b05565343147bc | ## toadytica305 3 years ago List all the real and imaginary zeros of f(x) = x^3+x^2+ 13x - 15... 1 is a zero of the function
1. satellite73
all possible rational roost are number of the form $$\frac{p}{q}$$ where $$p$$ divides the constant ( in your case 15) and $$q$$ divides the leading coefficient, in your case 1
2. satellite73
so the possibilites are $\pm1,\pm3,\pm5,\pm15$
3. satellite73
since you know 1 is a zero, you can factor it as $(x-1)(\text{something})$ and you can find the something by dividing, by synthetic division or by thinking
and to find all real and imaginary zeros
5. satellite73
in this case you get $(x-1) (x^2+2 x+15)$ and the second part you can find the zeros using the quadratic formula they are complex | 2015-07-03 15:41:44 | {"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": 0.7058907747268677, "perplexity": 486.907572704504}, "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/1435375096156.35/warc/CC-MAIN-20150627031816-00270-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://georgia-james.com/solve-using-the-quadratic-formula-w2-160/ | # Solve Using the Quadratic Formula w^2-16=0
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raising to any positive power yields .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
The final answer is the combination of both solutions.
Solve Using the Quadratic Formula w^2-16=0 | 2022-09-28 00:49:27 | {"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": 0.8672318458557129, "perplexity": 2309.5031468160387}, "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-2022-40/segments/1664030335059.31/warc/CC-MAIN-20220927225413-20220928015413-00208.warc.gz"} |
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# If P and Q represent the hundreds and tens digits, ..x=8PQ2.
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23 Jun 2014, 09:49
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If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
(1) P=4
(2) Q=0
OE
[Reveal] Spoiler:
According to Stat. (2), 8P02 will always end with an 02. Remember that For a number to be divisible by 4, the number's last two digits must form a number that is divisible by 4. Since 02 is not a number divisible by 4, the entire number, 8P02 is not divisible by 4. A number that is indivisible by 4 cannot be divisible by 8=4*2, so stat. (2) tells you that the answer to the question stem is a definite "no" - which is sufficient.
If you're not sure, plug in the different digits (0-9) for P and see for yourself how none of the plug-ins (8102, 8202, 8302, etc.) yields a number that is divisible by 8.
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If P and Q represent the hundreds and tens digits, ..x=8PQ2. [#permalink]
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goodyear2013 wrote:
If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
(1) P=4
(2) Q=0
OE
[Reveal] Spoiler:
According to Stat. (2), 8P02 will always end with an 02. Remember that For a number to be divisible by 4, the number's last two digits must form a number that is divisible by 4. Since 02 is not a number divisible by 4, the entire number, 8P02 is not divisible by 4. A number that is indivisible by 4 cannot be divisible by 8=4*2, so stat. (2) tells you that the answer to the question stem is a definite "no" - which is sufficient.
If you're not sure, plug in the different digits (0-9) for P and see for yourself how none of the plug-ins (8102, 8202, 8302, etc.) yields a number that is divisible by 8.
If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
For a number to be divisible by 2 the last digit must be divisible by 2 (so the last digit must be even);
For a number to be divisible by 4 the last two digits must be divisible by 4 (04, 08, 12, 16, ..., 96);
For a number to be divisible by 8 the last three digits must be divisible by 8 (008, 012, 016, ..., );
etc.
(1) P=4. If Q=0, then the answer is NO, because 402 is not divisible by 8, but if Q=3, then the answer is YES, because 432 is divisible by 8. Not sufficient.
(2) Q=0. In this case the last tow digits are 02, so x is not divisible by 4, which means that it's not divisible by 8 either. Sufficient.
Hope it's clear.
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Re: If P and Q represent the hundreds and tens digits, ..x=8PQ2. [#permalink]
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24 Jul 2014, 05:31
Bunuel why we are checking for 4 ... why don't the we check directly for divisibility by 8 .. for last three digits
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Re: If P and Q represent the hundreds and tens digits, ..x=8PQ2. [#permalink]
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24 Jul 2014, 05:46
GmatDestroyer2013 wrote:
Bunuel why we are checking for 4 ... why don't the we check directly for divisibility by 8 .. for last three digits
Which part are you referring to? The second statement?
From (2) we can get that the number cannot be divisible by 4, so no need to check for 8, because it's clear that since it's not divisible by 4 then it also won't be divisible by 8.
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If P and Q represent the hundreds and tens digits, [#permalink]
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07 Aug 2014, 06:03
If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
(1) P=4
(2) Q=0
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Re: If P and Q represent the hundreds and tens digits, [#permalink]
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07 Aug 2014, 06:39
goodyear2013 wrote:
If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
(1) P=4
(2) Q=0
Here's how I solved this question:
I started with statement 2 because it seemed easier to tackle a 0 than a 4.
1) Assuming statement 2, I have to answer: is 8P02 divisible by 8?
2) In doing long division, the first 8 is just a distraction because it will never leave a remainder for the hundreds digit. So I started by just removing it entirely.
3) Now I end up with P02 divisible by 8?
4) I saw that the only multiple of 8 that ends in a "2" is 32. Therefore I would need a "P0" value that leaves a remainder of 3. Since 8 is an even number and so is 0, I know that there is no way for there to be a remainder of 3. Therefore I can say statement 2 is sufficient to answer the question stem: no, X is not divisible by 8. NOTE: This can be tricky, because some people will get to this point and think that because the answer to the question stem is "no" that statement two is not correct. Therein lies the trickery of the data sufficiency.
Here, I can eliminate options A, C, E.
Now I need to test if statement 1 is sufficient.
1) Assuming statement 1, I have to answer: is 84Q2 divisible by 8?
2) Again I can just exclude the 8 to make my calculations simpler.
3) Now I end up with is 4Q2 divisible by 8?
4) Next, I just plugged in a value to test it. Since I know that a value of 0 definitely makes the number NOT divisible by 8 (discovered from my solving of statement 2), I try to look for a way to make the number divisible by 8.
5) As I did my work for statement 2, I had discovered that I would need there to be a remainder of 3 in the 10s digit to make the number divisible by 8. Well, it looks like I can do that by making Q=3. Therefore, I can say that statement 1 is not sufficient.
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If P and Q represent the hundreds and tens digits, [#permalink]
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07 Aug 2014, 21:51
goodyear2013 wrote:
If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
(1) P=4
(2) Q=0
(1) P = 4
X = 8000 + 100P+ 10Q + 2 = 8000+ 400 + 10Q + 2
Since 8400 is divisible by 8, (10Q +2) has to be divisible by 8. Q is from 1 to 9. If you check, there are 32 and 72 that satisfy the requirement. So 8432 is divisible by 8 but 8452 is not divisible by 8.
(2) Q = 0
X= 8000 + 100P + 2
Same reasoning, (100P+2 ) has to be divisible by 8. P is from 1 to 9.
We have, 102, 202,....902. For the numbers to be divisible by 8, they have to be divisible by 4, meaning the last two digits have to be divisible by 4. The last two digits are 02, not divisible by 4. So 8PQ2 is not divisible by 8 --> SUFFICIENT.
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Re: If P and Q represent the hundreds and tens digits, [#permalink]
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08 Aug 2014, 00:41
Criterions of divisibility by 2,4, and 8:
2: the last digit of a number is divisible by 2;
4: the last two digits of a number form the integer divisible by 4
8: the last three digits of a number form the integer divisible by 8
In the problem $$x$$ is divisible by 2, but we don't know about divisibility by 8.
(1) If $$P=4$$, we have number $$x=84Q2$$, which is divisible by 8 if $$4Q2$$ is divisible by 8. The answer depends on $$Q$$: if $$Q$$ is 0, $$x$$ is not divisible by 8, but if $$Q$$ is 3, $$x$$ is divisible by 8. Insufficient
(2) If $$Q=0$$, we have number $$x=8P02$$, which is not divisible by 4, since $$02$$ is not divisible by 4. Hence, $$x$$ is not divisible by 8. Sufficient
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Re: If P and Q represent the hundreds and tens digits, ..x=8PQ2. [#permalink]
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12 Aug 2014, 06:58
goodyear2013 wrote:
If P and Q represent the hundreds and tens digits, respectively, in the four-digit number x=8PQ2, is x divisible by 8?
(1) P=4
(2) Q=0
Merging similar topics.
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Re: If P and Q represent the hundreds and tens digits, ..x=8PQ2. [#permalink]
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Re: If P and Q represent the hundreds and tens digits, ..x=8PQ2. [#permalink] 03 Aug 2016, 06:56
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In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = g –1 ag.This is an equivalence relation whose equivalence classes are called conjugacy classes.. Members of the same ,conjugacy class, cannot be distinguished by using only the group structure, and therefore share many properties.
Center Centralizer Normalizer Subgroup? | Yahoo Answers
25/9/2011, · ,Center, Centralizer, Normalizer Subgroup,? Let (G, *) be a group. For any subset A of G, show that Z(G) is a ,subgroup, of C_G(A) which is a ,subgroup, of N_G(A) which is a ,subgroup, of G. | 2021-10-27 06:37:28 | {"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": 0.40220510959625244, "perplexity": 2491.1015105399947}, "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/1634323588102.27/warc/CC-MAIN-20211027053727-20211027083727-00478.warc.gz"} |
https://email.esm.psu.edu/pipermail/macosx-tex/2020-June/057091.html | # [OS X TeX] hyperref problem
Herbert Schulz herbs at wideopenwest.com
Sat Jun 27 13:06:23 EDT 2020
> On Jun 27, 2020, at 11:48 AM, Nitecki, Zbigniew H. <Zbigniew.Nitecki at tufts.edu> wrote:
>
> It’s several years since I wrote that macro, so I can’t specifically remember why \ignorespaces is there,
> but I suspect that it was because when I first started using that scheme, sometimes the spellchecker in the editor
> would insert a space before or after the colon (remember, the scheme was: a label reads, say \label{sec:intro}
> and the call to that is \refer{thm}{intro} so a space inserted when typing the label would render the reference unknown).
>
Howdy,
But that \ignorespaces appears at the very end of the macro so it doesn't do anything with the part of the reference name before the colon.
PS: I made sure to pass the original problem files onto Dick Koch and he was unable to reproduce the problem. It may be very closely tied to the actual resolution of your screen, etc., although just changing magnification did not get rid of the problem for me.
I consider the problem to be solved too but I'm really curious why removing the \ignorespaces fixes things in TeXShop. I can only guess that the link somehow gets messed up if there is no space (or newline, which is the same thing) between the the link'' (I really don't know how that would be true) and the following text.
Good Luck,
Herb Schulz
(herbs at wideopenwest dot com) | 2023-03-27 09:36:26 | {"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": 0.8730652332305908, "perplexity": 1635.9181807262462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00017.warc.gz"} |
https://www.physicsforums.com/threads/inner-product-space-minimization.683425/ | # Inner product space - minimization.
1. Apr 5, 2013
### binbagsss
The question is : If the vector space C[1,1] of continuous real valued functions on the interval [1,1] is equipped with the inner product defined by (f,g)=$^{1}_{-1}$ $\intf(x)g(x)dx$
Find the linear polynomial g(t) nearest to f(t) = e^t?
So I understand the solution will be given by (u1,e^t).||u1|| + (u2,e^t).||u2||
But I am having trouble understanding what u1 and u2 should be. I understand they must be othorgonal and basis for a subspace S $\in$ C[-1,1].
However Im not too sure what dimension this basis should be of, and not 100% sure what is meant by the vector space C[-1,1].
(The solution uses 1 and t as u1 and u2....)
Many thanks in advance for any assistance.
Last edited: Apr 6, 2013
2. Apr 5, 2013
### haruspex
I think you're making it overcomplicated. Isn't it just asking for the affine function g(t) which minimises the integral for the given f(t)? | 2017-05-25 16:28:58 | {"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": 0.8141109943389893, "perplexity": 832.9456386205936}, "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/1495463608107.28/warc/CC-MAIN-20170525155936-20170525175936-00297.warc.gz"} |
https://www.physicsoverflow.org/18717/why-euler-beta-function-interpreted-scattering-amplitude?show=18718#c18718 | # Why can the Euler beta function be interpreted as a scattering amplitude?
+ 7 like - 0 dislike
3316 views
The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this?
In another word, when the Euler beta function is interpreted as a scattering amplitude, what features does it have that make it able to explain strong force of mesons?
What properties of (or why) the Euler beta function (when interpreted as scattering amplitude) has string behavior?
This post imported from StackExchange Physics at 2014-06-11 07:31 (UCT), posted by SE-user Achmed
Susskind's lecture 6 on string theory discusses the scattering interpretation semi-informally, offset at about 45 minutes into the video, stretched to a length of about 30 minutes.
This post imported from StackExchange Physics at 2015-04-24 12:11 (UTC), posted by SE-user ccorn
Susskind's lecture 6 on string theory discusses the scattering interpretation semi-informally, offset at about 45 minutes into the video, stretched to a length of about 30 minutes.
This post imported from StackExchange Physics at 2014-06-11 07:32 (UCT), posted by SE-user ccorn
I refer you to Schwarz and Witten's textbook 'Superstring Theory,' which provides an excellent analysis of the beta function, and its interpretation as a scattering amplitude.
This post imported from StackExchange Physics at 2015-04-24 12:11 (UTC), posted by SE-user JamalS
I refer you to Schwarz and Witten's textbook 'Superstring Theory,' which provides an excellent analysis of the beta function, and its interpretation as a scattering amplitude.
This post imported from StackExchange Physics at 2014-06-11 07:32 (UCT), posted by SE-user JamalS
Some more details about the calculation of for example the scattering amplitude of four open string tachyons and why it corresponds to the Euler-Beta function are outlined here.
This post imported from StackExchange Physics at 2015-04-24 12:11 (UTC), posted by SE-user Dilaton
@JamalS No love for Green?
This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user 0celo7
@0celo7 Couldn't be bothered typing :)
This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user JamalS
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First I would like to give some more details about what is involved in obtaining scattering amplitudes in string theory and the specific one for the scattering of four open string tachyons.
Generally, for calculating string scattering amplitudes, one starts by drawing the string diagrams (worldsheets) that correspond to the process one is interested in, as Riemann surfaces in the complex plain and brings them by means of conformal transformations into a convenient standard representation. The scattering amplitude of the process is then obtained by integrating over the moduli space that parameterizes all worldsheets with the same initial and outgoing state.
The worldsheet for the scattering of 4 open string tachyons can for example be considered as a disk in the compex plain with four punctures at the locations $P_1$, $P_2$, $P_3$, $P_4$, where the incoming and outgoing particles are inserted. By a conformal transformation, this worldsheet can be mapped to the upper half plain such that the points of insertion appear on the real line
$P_1 \rightarrow x_1 = 0$
$P_2 \rightarrow x_x = \lambda$
$P_3 \rightarrow x_3 = 1$
$P_4 \rightarrow x_4 = \infty$
To define the conformal (or linear fractional) transformation only 3 points are needed, $\lambda$ is left as a free parameter and its range $0 < \lambda < 1$ therefore corresponds to the moduli space of the $2D$ Riemann surfaces with 4 punctures to be integrated over to obtain the scattering amplitude of the process.
The Ansatz for the scattering amplitude depends on the momenta of the 4 incoming and outgoing particles $p_i$ and can be written as
$A(p_1,p_2,p_3,p_4) = g_0^2\int d\mu$
It is proportional to the coupling constant $g_0$ squared because two interaction vertices are involved, and it contains $d\lambda$ to integrate over the moduli space of the relevant worldsheets. Considering momentum conservation and the on-shell condition of the incoming and outgoing particles, and taking into account that the integration measure has to be conformal invariant, the scattering amplitude can be rewritten as ($\alpha'$ is proportional to the string length squared in natural units)
$A(p_1,p_2,p_3,p_4) = g_0^2\int\limits_0^1 d\lambda\lambda^{2\alpha'(p_1,p_2)}(1-\lambda)^{2\alpha'(p_2,p_3)}$
Transforming to the Mandelstem variables $s$, $t$, and $u$, and defining the expressions
$\alpha(s) = \alpha's +1$
$\alpha(t) = \alpha' t +1$
the Veneziano amplitude can finally be written as
$A(p_1,p_2,p_3,p_4) = g_0^2\int\limits_0^1d\lambda \lambda^{-\alpha(s)-1}(1-\lambda)^{-\alpha(t)-1}$
which exactly corresponds to the definition of the Euler beta function as given in the Wikipedia link in the question for example. The symmetry of this amplitude in exchanging s and t (I think it is called channel duality?) is a specific stringy feature.
Somewhat handwavingly, to model the strong interaction by strings mesons are viewed as a pair of quarks sitting at the end of a string, that is represented by a thin tube of color flux lines. The confinement of the strong interaction can then be explained that the fact that when trying to separate the two quarks more and more mesons are produced such that the "binding energy" linearly increases with the separation $E_B = T L$ and $T$ is the string tension.
In addition, it has been observed for hadrons when plotting the angular momentum against the energy (or mass) squared, they appear as points on lines $J = \alpha' E^2$ which can be modelled by a classical rotating string. The discrete mass spectrum (the points on the line) of the hadrons including the offset of the linear relationship can be explained by a quantized open rotating string.
answered Jun 14, 2014 by (6,240 points)
edited Jun 15, 2014 by Dilaton
+ 0 like - 0 dislike
I certainly don't have a complete answer, but what I do know is the following. The Veneziano amplitude is the following: $$\mathcal{A}^{(4)} \propto \lambda\left(B(-\kappa s -1, -\kappa t -1) + B(-\kappa s -1, -\kappa u -1) + B(-\kappa t -1, -\kappa u -1)\right),$$ where $s, t, u$ are the Mandelstem variables and $B(x,y)$ the Euler-Beta function. Note that the above formula is symmetric under exchange of $s,t,u$. I think I read somewhere that physicists expected the strong interaction to be symmetric under such momentum exchanges, but I do not know precisely why. Note that the above formula is what one obtains in string theory for the scattering of open string tachyons, with $\lambda = g_c$ (closed string coupling constant) and $\kappa = \alpha'$.
This post imported from StackExchange Physics at 2014-06-11 07:32 (UCT), posted by SE-user Funzies
answered Jul 5, 2013 by (5 points)
+ 0 like - 0 dislike
I certainly don't have a complete answer, but what I do know is the following. The Veneziano amplitude is the following: $$\mathcal{A}^{(4)} \propto \lambda\left(B(-\kappa s -1, -\kappa t -1) + B(-\kappa s -1, -\kappa u -1) + B(-\kappa t -1, -\kappa u -1)\right),$$ where $s, t, u$ are the Mandelstem variables and $B(x,y)$ the Euler-Beta function. Note that the above formula is symmetric under exchange of $s,t,u$. I think I read somewhere that physicists expected the strong interaction to be symmetric under such momentum exchanges, but I do not know precisely why. Note that the above formula is what one obtains in string theory for the scattering of open string tachyons, with $\lambda = g_c$ (closed string coupling constant) and $\kappa = \alpha'$.
This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user Funzies
answered Jul 5, 2013 by (5 points)
+ 0 like - 0 dislike
A function can be interpretable as a scattering amplitude if that function satisfies the axioms of relativistic S-matrix theory [1]:
1. Lorentz invariance
2. Unitarity (Not realized by the beta function, but may be dropped if the function is interpreted as a Born approximation to the exact amplitude)
3. T, C, P invariance (only for strong nucl. interactions)
4. Analyticity: singularities in invariant energy complex plane correspond to particle poles or thresholds, in a way that doesn't violate causality.
5. Crossing symmetry.
6. Power boundedness
7. Well-behaved particle poles (stable particle masses are positive, and their residues should be negative)
8. Analyticity of the second kind: analytic in the complex angular momentum plane.
Then the function should also match some of the empirical facts gathered by experimentalists. When the beta function was proposed the following list would be conjured:
1. All poles in the complex angular momentum plane move to the right linearly with increasing energy at a universal rate.
2. Diffractive scattering (not realized by the Euler beta function, but Virasoro's amplitude does realize this)
3. Inclusive high energy reaction exhibit scaling (also not realized by Euler beta)
4. Particle spectrum (determined by the poles) in agreement with the quark model (not realized either, but Chan and Paton's method get close)
5. Reproduces data from weak and electromagnetic processes (not realized.)
References: John Schwarz "Dual Resonance Theory" Phys Rep 8, no. 4, (1973) 269-335
This post imported from StackExchange Physics at 2015-04-24 12:12 (UTC), posted by SE-user QuantumDot
answered Aug 12, 2014 by (195 points)
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http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=SOJBB3_2016_v21n3_35 | Research on Methods for Processing Nonstandard Korean Words on Social Network Services
Title & Authors
Research on Methods for Processing Nonstandard Korean Words on Social Network Services
Lee, Jong-Hwa; Le, Hoanh Su; Lee, Hyun-Kyu;
Abstract
Social network services (SNS) that help to build relationship network and share a particular interest or activity freely according to their interests by posting comments, photos, videos,$\small{{\ldots}}$ on online communities such as blogs have adopted and developed widely as a social phenomenon. Several researches have been done to explore the pattern and valuable information in social networks data via text mining such as opinion mining and semantic analysis. For improving the efficiency of text mining, keyword-based approach have been applied but most of researchers argued the limitations of the rules of Korean orthography. This research aims to construct a database of non-standard Korean words which are difficulty in data mining such abbreviations, slangs, strange expressions, emoticons in order to improve the limitations in keyword-based text mining techniques. Based on the study of subjective opinions about specific topics on blogs, this research extracted non-standard words that were found useful in text mining process.
Keywords
Text Mining;Non-standard;Stemming Korean;Unicode;
Language
Korean
Cited by
References
1.
Lee, J. H., "Big Data, Data Mining and Temporary Reproduction," The Journal of Intellectual Property, Vol. 8, No. 4, 2013, pp. 93-125.
2.
Kang, S. J., "Constructing a Large Interlinked Ontology Network for the Web of Data," Journal of Korean Industrial Information Systems Society, Vol. 15, No. 1, 2010, pp. 15-23.
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Park, C. S., Hong, Y. J. and Cho, I. H., "An Analysis on Journalism Characteristics of SNS based on Issued Cases : With Twitter as the Center," Proceedings in 2012 Fall Conference of The Korean Entertainment Industry Association, 2012, pp. 36-40.
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Boyd, D. M. and Ellison, N. B., "Social Network Sites: Definition, History, and Scholarship," Journal of Computer-Mediated Communication, Vol. 13, No. 4, 2007, pp. 210-230.
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Kim, W. S., Lee, J. H., Park, j. W. and Choi, j. H.,"A Technique of the Approval Rating Analysis for Political Party Using Opinion Mining,", Journal of Korean Institute of Information Technology, Vol. 12, No. 10, 2014, pp. 133-141.
6.
Won, J. Y. and Kim, D. G., "Deduction of Social Risk Issues Using Text Mining," Journal of safety and crisis management, Vol. 10, No. 7, 2014, pp. 33-52.
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Lee, J. H. and Lee, H. K., "A Study on Unstructured Text Mining Algorithm through R Programming based on Data Dictionary," Journal of the Korea Society Industrial Information System, Vol. 20, No. 2, 2015, pp. 113-124.
8.
Chang, J. Y., Lee, s. Y. and Han, J. B., "Machine-Learned Classification Technique for Opinion Documents Retrieval in Social Network Services," Proceedings in 2013 Conference of Korean Institute of Information Scientists and Engineers, 2013, pp. 245-247.
9.
Chang, C. Y., Jang, J. H., Kim, S, H., Lee, H. K. and Lee, C. H., "A Study on the Efficient Patent Search Process using Big Data Analysis Tool R," Journal of Korea Safety Management & Science, Vol. 15, No. 4, 2013, pp. 289-294.
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Le, H., and Lee, H. K., "Exploring Relationship Between Social ICT Issues And Academic Research Interests Through Text Mining Analysis," The Journal of Internet Electronic Commerce Research, Vol. 14, No. 5, 2014, pp. 161-180.
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Le, H., Lee, J. H. and Lee, H. K., "Purchase Process Aspect-based Opinion Mining : An Application for Online Shopping Mall," The Journal of Internet Electronic Commerce Research, Vol. 15, No. 2, 2015, pp. 15-28.
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Yun, B. H., "Natural Language Processing based Information Extraction for Newspapers," Journal of Korean Institute of Information Technology, Vol. 6, No. 4, 2008, pp. 188-195.
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Hong, J. P. and Cha, J. W., "Error Correction of Sejong Morphological Annotation Corpora using Part-of-Speech Tagger andFrequency Information," Journal of KISS : Software and Applications, 2013, Vol. 40, No. 7, pp. 417-428.
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Sim, K. S., "Syllable-based POS Tagging without Korean Morphological Analysis," Korean Journal of Cognitive Science, Vol. 22, No. 3, 2011, pp. 327-345.
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An, J. K. and Kim, H. U., "Building a Korean Sentiment Dictionary and Applications of Natural Language Processing," Proceedings in 2014 Summer Conference of Korea Intelligent Information Systems Society, 2014, pp. 177-182.
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Kwon H. R., Na J. H., Yoo J. S. and Cho W. S., "Text-mining Techniques for Metabolic Pathway Reconstruction," Journal of Korean Industrial Information Systems Society, Vol. 12, No. 4, pp. 138-147, 2007.
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URL http://www.unicode.org/ | 2018-05-24 21:57:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.23816582560539246, "perplexity": 6482.937803283263}, "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/1526794866870.92/warc/CC-MAIN-20180524205512-20180524225512-00233.warc.gz"} |
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Manmohan
May 18, 2014
Q4.(iv)places A and B are 100 km. apart on a highway.One car starts from A and another from B at the same time.If the cars travel in the same direction at different speeds,they meet in 5 hours.If they travelled towards each other,they meet in 1 hoyr.What are the speeds of the two cars?
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Aubrey herbert
Sep 27, 2013
Solve x/a + y/b = a+ b x/a2 + y/b2 = 2 Thanks in advance
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Greeny joby
Jul 20, 2013
2) x/a2 + y/b2=2
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Rishabh mehta
Jul 30, 2013
A two digit number 3 times more than 4 times the sum of its digit . If 18 is added to the number the digits are interchanged . Find the number .
Question should be "A two digit number is 3 more than 4 times the sum of its digit . If 18 is added to the number ...
Harsha Bohra
Jun 14, 2014
solve through elimination ax/b-by/a=a+b&ax-by=2ab
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Utpal
Aug 14, 2013
A train travels a distance of 300 km at a uniform speed. If the speed of the train is increased by 5km an hour, the journey would have taken two hours less. Find the original speed of the train.
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Sebastian Sabin
Oct 28, 2013
the pair of equation y=0 and y = - 5 has how many solution
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Sandeep
Jul 24, 2013
Filters | 2021-11-30 09:30:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 40, "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": 0.3152986168861389, "perplexity": 1585.0586795511833}, "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-2021-49/segments/1637964358966.62/warc/CC-MAIN-20211130080511-20211130110511-00233.warc.gz"} |
https://r-mukund.github.io/teaching/fa2021-csci499/ | ## Fall 2021
### Logistics
Classes: Tuesdays and Thursdays, 2–3:50pm · KAP 166 · https://usc.zoom.us/j/94314036860 Scratchpad: Click here Piazza: https://piazza.com/usc/fall2021/csci499 Instructor: Mukund Raghothaman (raghotha@usc.edu) Office hours: Fridays, 1–2:50pm, or by appointment · https://usc.zoom.us/j/93270102560
### Course Description
This course will introduce you to a range of advanced programming paradigms. We will assume an elementary knowledge of programming, such as that covered in CSCI 103 and CSCI 104, and study powerful ways of structuring code with higher-order functions, of providing strong guarantees with static type systems, and ways to liberate the programmer from low-level resource management. By blurring the distinction between programs and data, and widening the gap between a program and its execution, our goal is to blow your mind about what it means to program a machine, and to reinforce your developing sense of computational thinking. The first half of the course can alternatively be seen as an introduction to functional programming with Ocaml, while the second half of the course can be regarded as an introduction to logic programming. [Syllabus], [Registration Calendar]
Note: The syllabus and schedule listed on this webpage are tentative, and may be updated as the course progresses. Please check back regularly!
### Annonucements
• Aug 20: Welcome to CSCI 499! The course website is now live!
### Schedule
#### Unit 1: Functional Programming in Ocaml
Aug 24, 26 Course introduction Overview, motivation and logistics Elementary computations using the REPL and a tour of the programming environment Aug 24: [Recording], [Slides], [L01.ml] Aug 26: [Recording], [Notes] Reading: Richard Bird. Functional Pearl: A Program to Solve Sudoku. JFP 2006. [Link] [Cheatsheet] Aug 31, Sep 2 An introduction to types Structuring data with pairs, tuples, variants and records Pattern matching Recursive definitions and algebraic data types Aug 31: [Recording], [Notes] Sep 2: [Recording], [Notes], [L04-ds.cpp] Reading: [Yaron Minsky's talk on types] Reading: RWO Chapters 5, 6: Records and Variants Sep 7, 9 Abstracting computations with functions Recursive definitions Higher-order functions: Functions as arguments, as return values, and those of the anonymous kind; arrow types Execution model: Evaluation order and scoping rules Sep 7: [Recording]. [Notes], [L05-ds.c], [L05-ss.c], [L05-even.ml] Sep 9: [Recording], [Notes], [L06.java] Reading: RWO Chapter 2: Variables and Types Sep 14, 16 Processing recursive data Processing lists with map and fold Mutable state Sep 14: [Recording], [Notes] Sep 16: [Recording], [Notes], [L08.cpp] Reading: RWO Chapter 3: Lists and Patterns Sep 21, 23 Assorted topics Mutable state, contd. Polymorphism, modules, and programming in the large Sep 21: [Recording], [Notes] Sep 23: [Recording], [Notes] Reading: RWO Chapter 8: Imperative Programming Reading: RWO Chapter 4 and 9: Modules and Functors
#### Unit 2: Implementing a Language Interpreter
Sep 28, 30 Understanding syntax Describing syntax with regular expressions and context-free grammars Lexical analysis with finite state automata Sep 28: [Recording], [Notes] Sep 30: [Recording], [Notes], [L12.mll] Reading: RWO Chapter 18: Parsing Data with Ocaml Oct 5, 7 Syntax, contd. The CYK algorithm for parsing context free grammars Bottom-up parsing using LALR grammars Oct 5: [Recording], [Notes], [L13-expr.ml], [L13-lexer.mll], [L13-parser.mly] Oct 7: [Recording], [Notes], [L14-lexer.mll], [L13-parser.mly] Oct 12 Midterm released, Disambiguating grammars, understanding the lexical analyzer [Recording], [Notes], [L15-expr.ml], [L15-lexer.mll], [L15-parser.mly] Oct 14 Fall recess, no class Oct 19, 21 Syntax, contd. 2: Parsing algorithms for regular expressions and context-free grammars Oct 19: [Recording], [Notes], [L16-regex.ml] Oct 21: [Recording], [Notes] Oct 26, 28 An elementary understanding of types and the runtime Oct 26: [Recording], [Notes] Oct 28: [Recording], [Notes]
#### Unit 3: Programming with Relations
Nov 2, 4 Spreadsheets The computational model: Cells, values, formulas and dependence graphs Pivot tables, array formulas and lookups Turing-completeness of spreadsheets as a programming medium Nov 2: [Recording], [Notes], [L20.c] Nov 4: [Recording], [Notes], [L21.ods] Nov 9, 11 The relational data model Relational algebra: SPJ queries and set operations Non-recursive queries with SQL Nov 9: [Recording], [Notes], [Chinook_Sqlite.sqlite] Nov 11: [Recording], [Notes] Nov 16, 18 An introduction to recursive query languages Querying graph data with Cypher Rule-based queries using Datalog Bottom-up query evaluation Nov 16: [Recording], [Notes] Nov 18: [Recording], [Notes] Nov 23, 30 More on recursive query languages Logic programming with Prolog Top-down query evaluation by backtracking Nov 23: [Recording], [Notes], [cousins.dl], [parent.facts], [path.dl] Nov 30: [Recording], [Notes], [scc.dl], [sgen.dl], [paradox.dl], [nopath.dl] Nov 25 Thanksgiving break, no class Dec 2 Conclusion and review What was this course all about? Reflections on the future of programming Dec 9 Final exam (2–4pm)
The first half of the course will follow the Real World Ocaml textbook. This is the only required textbook for this course. We will assign additional supplementary readings as appropriate.
1. Yaron Minsky, Anil Madhavapeddy, and Jason Hickey. Real World Ocaml. 2nd edition. O'Reilly, 2020.
We will be using drafts of the second edition of the book which is currently in preparation and freely available at here.
2. Harold Abelson, Gerald Jay Sussman, and Julie Sussman. Structure and Interpretation of Computer Programs. 2nd edition. The MIT Press, 1996.
The book is freely available here.
3. Leon Stirling and Ehud Shapiro. The Art of Prolog. 2nd edition. The MIT Press, 1994.
The book is freely available here.
### Development Environment
2. Datalog: Obtain Souffle from here, and follow the build instructions described here.
3. Prolog: Install SWI Prolog either from your operating system package manager (sudo dnf install pl, brew install swi-prolog, sudo apt install swi-prolog, or similar), or directly from its website.
1. Homework assignments: 4×15% = 60%
2. Midterm exam: 20%
3. Final exam: 20%
### On Collaboration
You are welcome to discuss homework assignments with a partner. However, each of you will turn in your submissions separately. You will each be responsible for independently writing and physically typing the solutions in your submission. Please identify your discussion partner, if any, in your submission.
### Policies
From Dornsife's academic conduct policy: Plagiarism—presenting someone else's ideas as your own, either verbatim or recast in your own words—is a serious academic offense with serious consequences. Please familiarize yourself with the discussion of plagiarism in SCampus in Part B, Section 11, "Behavior Violating University Standards." Other forms of academic dishonesty are equally unacceptable. See additional information in SCampus and university policies on scientific misconduct, here.
Discrimination, sexual assault, intimate partner violence, stalking, and harassment are prohibited by the university. You are encouraged to report all incidents to the Office of Equity and Diversity / Title IX Office here, and / or to the Department of Public Safety. This is important for the health and safety of the whole USC community. Faculty and staff must report any information regarding an incident to the Title IX Coordinator who will provide outreach and information to the affected party. The sexual assault resource center webpage fully describes reporting options. Relationship and Sexual Violence Services webpage provides 24/7 confidential support.
Note on collaborative work: For collaborative projects, students are expected to have equal distribution of work. If there is any perceived imbalance in the collaborative project, the student should bring this to the attention of the instructor or the teaching assistant.
Assistance with writing and disabilities: Several USC's schools provide support for students who need help with scholarly writing. Check with your advisor or program staff to find out more. Students whose primary language is not English should check with the American Language Institute which sponsors courses and workshops specifically for international graduate students. The Office of Disability Services and Programs provides certification for students with disabilities and helps arrange the relevant accommodations. If an officially declared emergency makes travel to campus infeasible, USC Emergency Information will provide safety and other updates, including ways in which instruction will be continued by means of Blackboard, teleconferencing, and other technology.
Last updated: Thu Dec 2 12:58:54 PM PST 2021 | 2022-05-20 08:29:15 | {"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": 0.1794065535068512, "perplexity": 13364.078020077248}, "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-2022-21/segments/1652662531762.30/warc/CC-MAIN-20220520061824-20220520091824-00349.warc.gz"} |
https://dsp.stackexchange.com/questions/2977/projection-of-2d-signal/2979 | # Projection of 2D signal
Paper describes that a 2D signal(say video) can be projected to 1D for time series analysis by using a transformation function fx=Ax+constant and fy=Ay+constant where Ax , Ay represents the (x,y) coordinates of a single trajectory A. Then,these fx vs t can be plotted and various analysis can be performed. Going by their justification,is it logical to think that a coordinate position of an object can only be described by a single coordinate when doing time series analysis and signal compression?
Also,they mention about taking cartesian product.Why is that?
• You can look for cartesian product directly on Wikipedia, they have an entry for that. – heltonbiker Jul 26 '12 at 20:27
• Thank you for the reply. I went through that earlier but do not understand what is the need of doing that?Do we have to do cartesian product since we are projecting 2D into 1D? – Priya M Jul 26 '12 at 20:36
• From what I understood, the paper tries to do a lot of complex things, and many of them are not related to your specific question. Actually, it seems you are still trying to get the motivation of the authors instead of needing help to answer a specific question. – heltonbiker Jul 26 '12 at 21:00
• Well, the authors claim that(through lemma),Fig2 and in page5 that 2D can be reduced/converted to 1D. So I wanted to confirm is it really this process if we want to perform time series analysis on multidimensional signals say video we convert from n dimensions to 1D dimension or is there any other way as I do not know.Also, I fail to understand why cartesian product is required in this case. – Priya M Jul 26 '12 at 21:06 | 2019-11-12 05:27:33 | {"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": 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": 0.6403245329856873, "perplexity": 474.76791244264444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664752.70/warc/CC-MAIN-20191112051214-20191112075214-00389.warc.gz"} |
http://blog.udanax.org/2009/12/serialize-printing-of-helloworld-using.html | ## December 9, 2009
### Serialize Printing of HelloWorld using BSP of Hama
Apache Hama Team made the BSP package, which is a computational model based on the concept of supersteps on the top of Hadoop for perform matrix/graph computations with better performance. It provide more flexible programming model than Map/Reduce, and more simple APIs than MPI. The vertical system structure of the BSP is as below:
Sequential composition of "supersteps". - Local computation - Process Communication - Barrier Synchronization
* More detailed will be announced very soon.
To helping understand the Synchronization and Superstep, I made a "Serialize Printing of Hello World" example.
The below codes create 10 BSPPeerThreads. Each thread will have a shuffled ID number from 0 to 9.
BSPPeerThread thread;
int[] randomSequence = new int[] { 2, 3, 4, 5, 0, 1, 6, 7, 8, 9 };
for (int i = 0; i < NUM_PEER; i++) {
conf.set("bsp.peers.num", String.valueOf(NUM_PEER));
conf.set(BSPConstants.PEER_HOST, "localhost");
conf.set(BSPConstants.PEER_PORT, String
.valueOf(30000 + randomSequence[i]));
System.out.println(randomSequence[i] + ", " + thread.getName());
}
for (int i = 0; i < NUM_PEER; i++) {
list.get(i).join();
}
Try to run below codes, then you'll see the output in order of thread ID. (It'll take 10 steps)
for (int i = 0; i < NUM_PEER; i++) {
if (myId == i) {
System.out.println("Hello BSP from " + i + " of " + NUM_PEER + ": "
+ getName());
}
try {
peer.sync();
} catch (IOException e) {
e.printStackTrace();
} catch (KeeperException e) {
e.printStackTrace();
} catch (InterruptedException e) {
e.printStackTrace();
}
} | 2018-02-21 13:20:25 | {"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": 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": 0.17264896631240845, "perplexity": 7410.898026755928}, "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/1518891813622.87/warc/CC-MAIN-20180221123439-20180221143439-00582.warc.gz"} |
https://www.kix1055.com/k3m8l7/4g5u3.php?page=supply-definition-economics-66d10e | This relates closely to the demand for a good or service at a specific price; all else being equal, the supply provided by producers will rise if the price rises because all firms look to maximize profits. The supply equation is the explicit mathematical expression of the functional relationship. g If there is an increase in demand for beef, then the supply of beef will rise. Perloff, Microeconomics Theory & Applications with Calculus (Pearson 2008) at 19. r The coefficient of A-Level Model Essays £8.00 . For example in the case of time, supply is not transferred to one agent from another, but one agent may offer some other resource in exchange for the first spending time doing something. Flashcards. If supply is low and demand is high, the price will also be high. PES > 1), then producers can increase output without a rise in cost or a time delay; If supply is inelastic (i.e. j Some economic models in the field of behavioural economics assume that self-interested individuals behave altruistically because they get some benefit, or utility, from doing so. [19] If the linear supply curve intersects the quantity axis PES will equal zero at the point of intersection and will increase as one moves up the curve;[18] however, all points on the curve will have a coefficient of elasticity less than 1. y k The opposite of supply-side is demand-driven Keynesian theory. Personalized courses, with or without credits. The concept of supply in economics is complex with many mathematical formulas, practical applications and contributing factors. This can vary based on which type of money supply one is discussing. Supply can relate to the amount available at a specific price or the amount available across a range of prices if displayed on a graph. The definition and meaning of joint supply refers to a product that can end up being at least two other types of goods. {\displaystyle Q=40P-2P_{rg}} – Producer Surplus: this is the difference between how much a supplier sold something for and how cheaply he or she would have gone (minimum selling price). P A-Level revision guide £7.95 . Supply chain finance is often made possible through a technology-based platform, and is affecting industries such as the automobile and retail sectors. Definition of Market Supply: The market supply is the total quantity of a good or service that all producers are willing to supply at the prevailing set of relative prices during a defined period of time.It is understood that "Supply" means Market Supply, unless it … "Marshallian Cross Diagrams and Their Uses Before Alfred Marshall: The Origins of Supply and Demand Geometry," Page 3. {\displaystyle Q_{\text{s}}=325+P-30P_{\text{rg}}} = where Related terms and concepts to supply in today’s context include supply chain finance and money supply. y Supply functions, then, may be classified according to the source from which they come: consumers or firms. Definition, Example with Infographic. However, these factors are held constant (according to the law of supply) to alleviate the effect of the law of supply especially with relation with quantity supplied and the supply … P f McGraw-Hill 2008. 4.Taxes and Subsidies: Taxes make supply decrease and subsidies make supply increase. One of the most important factors that affects supply is the good’s price. Supply Shifters- T.O.N.E.R.S. . PLAY. I Png, Managerial Economics (Blackwell 1999). This model will be used to examine some of the interactions among supply, demand and price. Test. Definition: Law of supply states that other factors remaining constant, price and quantity supplied of a good are directly related to each other.In other words, when the price paid by buyers for a good rises, then suppliers increase the supply of that good in the market. amy_edwards57. Flashcards. The neutrality of money is an economic theory stating that changes in the aggregate money supply only affect nominal variables. Determinants of supply in economics are the factors that influence producer supply cause the supply curve to shift. Both supply and demand curves are best used for studying the economics of the short run. p Supply is the value that market participants such as firms and individuals are willing to provide at a price level. Supply's counterpart is demand; it measures how many co… [1] Some of the more important factors affecting supply are the good's own price, the prices of related goods, production costs, technology, the production function, and expectations of sellers. to The law of supply and demand is a theory that describes how supply of a good and the demand for it interact. STUDY. The price a consumer will pay for a good determines how much of the good’s supply is sold. Supply is the source of economic activity. If you produce beef you will get leather as a side effect. Supply can be used to measure demand. These factors that influence the supply are called the determinants of supply. The portion of the SRMC below the shutdown point is not part of the supply curve because the firm is not producing any output. + × ( Key Concepts: Terms in this set (41) (Supply) definition of supply. Switch to. Supply and Demand in a Single-Product Market (Exercise Prepared for the In economics, the supply of a particular good or service is simply the quantity of the item that is produced and offered for sale. {\displaystyle P_{\text{rg}}} The supply of a product is influenced by various determinants, such as price, cost of production, government policies, and technology. Joint supply occurs when two goods are supplied together. 2. The principle that suppliers will normally offer more for sale at higher prices and less at lower prices. While supply can refer to anything in demand that is sold in a competitive marketplace, supply is most used to refer to goods, services, or labor. Defined. Learn. [18] The coefficient of elasticity decreases as one moves "up" the curve. + Melvin & Boyes, Microeconomics 5th ed. The supply of a product is influenced by various determinants, such as price, cost of production, government policies, and technology. For example, a cow in a farm can be used for meat, milk, cheese, yogurt, and leather. ( AS-Level Revision guide £4.00. (Sharpe 2009) at 83. Supply chain finance aims to effectively link all tenets of a transaction, including the buyer, seller, financing institution—and by proxy the supplier—to lower overall financing costs and speed up the process of business. 1 Spell. It is calculated for discrete changes as If the opposite is true, they are a consumer of j. GCSE Revision Guide £7.49. ; Supply curve, in economics, graphic representation of the relationship between product price and quantity of product that a seller is willing and able to supply. (Prentice-Hall 2001) at 336. https://en.wikipedia.org/w/index.php?title=Supply_(economics)&oldid=975365964, Articles with unsourced statements from October 2009, Articles to be expanded from November 2018, Creative Commons Attribution-ShareAlike License, This page was last edited on 28 August 2020, at 03:27. The law of supply and demand, one of the most basic economic laws, ties into almost all economic principles in some way. The law of supply and demand is a fundamental and foundational principle of economics. Q 1 Supply and production are very similar terms and are often used interchangeably. Supplyis the producer's willingness and ability to supply a given good at various price points, holding all else constant. The supply model assumes that price and quantity supplied are directly related. When the price of a product is high, the supply is high. + p Generally, if supply is high and demand low, the corresponding price will also be low. Supply is the willingness and ability of producers to create goods and services to take them to market. Law of supply. The law of supply dictates that all other things remaining equal, an increase in the price of the good in question results in an increase in quantity supplied. s In economics, the amount of a good that sellers are willing to provide in the market, Marginal costs and short-run supply curve, Aggregate supply and demand in macroeconomics, Melvin & Boyes, Microeconomics 5th ed. Learn. Melvin & Boyes, Microeconomics 5th ed. Supply: is the total amount of goods and services that producers are willing and able to purchase at a given price in a given time period.. Supply is the amount of goods available, and demand is how badly people want a good or service. By using Investopedia, you accept our, Investopedia requires writers to use primary sources to support their work. quantity supplied. Marshall gave laws of economics definition. Spell. [15], The market supply curve can be translated into an equation. = ; What is the formula for calculating price elasticity of supply? (Houghton Mifflin 2002) at 60. credibility is due to the managers at work. The formula for price elasticity of supply is: Percentage change in quantity supplied divided by the percentage change in price When Pes > 1, then supply is price elastic When Pes < … M2 by contrast includes all of M1 but also includes short-term deposits and certain types of market funds. {\displaystyle P={\tfrac {Q}{40}}+{\tfrac {P_{rg}}{20}}} quantity supplied. A supply schedule is a table which shows how much one or more firms will be willing to supply at particular prices under the existing circumstances. Definition: The total stock of money circulating in an economy is the money supply. k more Law of Supply and Demand Definition (Houghton Mifflin 2002). Supply in economics and finance is often, if not always, associated with demand. President Reagan used supply-side economics to combat stagflation. [20] Perfect competition is the only market structure for which a supply function can be derived. Ayers & Collins, Microeconomics (Pearson 2003) [20] There is simply not a one-to-one relationship between price and quantity supplied. Melvin & Boyes, Microeconomics 5th ed. Wheat production also delivers straw, which farmers, racetracks horse owners and other animal owners purchase for their stables, and biofuel (bioethanol). The advent of the industrial revolution and the ensuing British economic powerhouse, which included heavy production, technological innovation and an enormous amount of labor, has been a well-discussed cause. Get the detailed answer: What is the definition of supply in economics? = Supply is the value that market participants such as firms and individuals are willing to provide at a price level. ( ∑ The LDMR states that as production increases eventually a point (the point of diminishing marginal returns) will be reached after which additional units of output resulting from fixed increments of the labor input will be successively smaller. In a perfectly competitive market the price is given by the marketplace from the point of view of the supplier; a manager of a competitive firm can state what quantity of goods will be supplied for any price by simply referring to the firm's marginal cost curve. = In economics, we have two forces: the producer, who makes things, and the consumer, who buys them. Your dashboard and recommendations. In this way, consumers are able to influence prices through their demand. Supply The law of supply. Quantity demanded is used in economics to describe the total amount of a good or service that consumers demand over a given period of time. ¯ Booster Classes. S market supply curve a graph of the quantity supplied of a good by all suppliers at different prices 3 factors every business owner must consider labor and output, production costs, and setting output As consumers buy up the supply of a product without decreasing their demand, the price increases. Supply refers to the quantity of a good that the producer plans to sell in the market. rg Jodi Beggs. = [10] A shift in the supply curve, referred to as a change in supply, occurs only if a non-price determinant of supply changes. For example, the percentage change the amount of the good supplied caused by a one percent increase in the price of a related good is an input elasticity of supply if the related good is an input in the production process. Aggregate supply is used to show the amount of goods that can be produced at different price levels in a given time period – usually one year. 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supply definition economics
Old Man Logan Ant-man, Types Of Work Measurement, Green Tomato Chutney Maharashtrian Style, Can I Stream The Weather Channel Live, Infrastructure As Code, Milwaukee Circular Saw With Laser, How To Use Hair Color Remover, Kérastase Initialiste Serum, | 2021-05-06 15:31:48 | {"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": 0.41779279708862305, "perplexity": 2304.5333001314757}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988758.74/warc/CC-MAIN-20210506144716-20210506174716-00127.warc.gz"} |
http://docs.mathjax.org/en/latest/input/tex/extensions/colorV2.html | # colorV2¶
The colorV2 extension defines the \color macro to be the non-standard macro that is the default in MathJax version 2, namely, it takes two arguments, one the name of the color (or an HTML color of the form #RGB or #RRGGBB), and the second the math to be colored. This is in contrast to the standard LaTeX \color command, which is a switch that changes the color of everything that follows it.
This extension is not loaded automatically when the autoload extension is used. To load the color extension explicitly, add '[tex]/color' to the load array of the loader block of your MathJax configuration, and add 'color' to the packages array of the tex block.
window.MathJax = {
loader: {load: ['[tex]/colorV2']},
tex: {packages: {'[+]': ['color']}}
};
or, use \require{colorV2} in a TeX expression to load it dynamically from within the math on the page, if the require extension is loaded.
Alternatively, you can configure the autoload package to load colorV2 when \color is used rather than the (LaTeX-compatible) color extension:
window.MathJax = {
tex: {
autoload: {
color: [], // don't autoload the color extension
colorV2: ['color'] // autoload colorV2 on the first use of \color
}
}
}; | 2020-02-21 22:16:55 | {"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": 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": 0.9092890024185181, "perplexity": 5214.257191561131}, "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-2020-10/segments/1581875145538.32/warc/CC-MAIN-20200221203000-20200221233000-00020.warc.gz"} |
https://quant.stackexchange.com/questions/51311/how-do-we-determine-the-correct-measure | # How do we determine the “correct measure”?
Frequently I come across the statement that the "correct measure" for a product is this-or-that measure. For example,
1. Eurodollar Futures or Stock returns - Risk neutral measure
2. Libor forward rate - T-forward measure
3. Libor in arrears - T-1-forward measure etc.
4. Swaption - annuity measure
The explanation for this is that the payoff is a martingale under this measure. But I do not understand the logic here - are we making an assumption about the martingale property, based on some reasonable justification? How does one go about finding the right measure for a product?
• is it still unclear? Is there anything you do not understand? – Daneel Olivaw Feb 27 '20 at 18:05
• @Bravo: if you like Daneel's (or mine) answer below, could you pls click on the "tick mark" next to one of the answers so that this question can be marked as "answered"? – Jan Stuller Jun 9 '20 at 16:12
Recall that any traded asset divided by a numéraire is a martingale under the measure associated to that numéraire. For the 3 interest rates you mention, the natural measure (namely the one that makes those processes martingales) is deduced from the structure of the rate.
Always keep in mind that the value at $$t_0$$ of a cash flow $$C$$ paid at $$T$$ is equal to the conditional expectation of the discounted cash flow under the risk-neutral measure $$\mathcal{Q}$$: $$C(t_0)=E_{t_0}^\mathcal{Q}\left(\frac{B_{t_0}}{B_T}C(T)\right)$$ where $$B_t$$ is the money market account: $$B_t=e^{rt}$$, namely the numéraire under the risk-neutral measure. You thus notice that: $$D(t_0,T)=\frac{B_{t_0}}{B_T}$$ where $$D(t_0,T)$$ is the discount factor from $$t_0$$ to $$T$$. The theory of numéraire change developed by Geman, El Karoui and Rochet in [1] establishes the following equivalency between measures: $$E_{t_0}^\mathcal{Q}\left(\frac{B_{t_0}}{B_T}C(T)\right) =E_{t_0}^\mathcal{N}\left(\xi(t_0,T)\frac{B_{t_0}}{B_T}C(T)\right) =E_{t_0}^\mathcal{N}\left(\frac{N_{t_0}}{N_T}C(T)\right)$$ where $$N_t$$ is another asset which might be used as a numéraire, $$\mathcal{N}$$ its associated measure, and $$\xi(t_0,T)$$ the associated Radon-Nikodym derivative to change from one measure to the other: $$\xi(t_0,T)=\frac{B_TN_{t_0}}{B_{t_0}N_T}$$
Forward LIBOR rate: you probably know that the forward LIBOR rate is equal to: $$L(t,T,T+\delta)=\frac{1}{\delta}\left(\frac{P(t,T)}{P(t,T+\delta)}-1\right)$$ where $$P(t,T)$$ is a zeron-coupon bond with maturity $$T$$. Now, such a product is a traded asset with a strictly positive price and no dividends, therefore it can be used as a numéraire. Hence under the $$T+\delta$$ measure the LIBOR rate is a martingale: \begin{align} E_{t_0}^{T+\delta}\left(L(t,T,T+\delta)\right) &=E_{t_0}^{T+\delta}\left(\frac{1}{\delta}\left(\frac{P(t,T)}{P(t,T+\delta)}-1\right)\right) \\ &=\frac{1}{\delta}\left(E_{t_0}^{T+\delta}\left(\frac{P(t,T)}{P(t,T+\delta)}\right)-1\right) \\ &=\frac{1}{\delta}\left(\frac{P(t_0,T)}{P(t_0,T+\delta)}-1\right) \\[7pt] &=L(t_0,T,T+\delta) \end{align}
LIBOR-in-arrears: in this case you are paid at $$T$$ the LIBOR for the period $$[T,T+\delta]$$. There is no measure under which the LIBOR-in-arrears is a pure martingale. The value of this flow is: \begin{align} E_{t_0}^\mathcal{Q}\left(\frac{B_{t_0}}{B_T}L(T,T,T+\delta)\right) &=P(t_0,T)E_{t_0}^T\left(L(T,T,T+\delta)\right) \\ &=P(t_0,T+\delta)E_{t_0}^{T+\delta}\left(\frac{L(T,T,T+\delta)}{P(T,T+\delta)}\right) \\[4pt] &=P(t_0,T+\delta)\left(L(t_0,T,T+\delta)+\delta E_{t_0}^{T+\delta}\left( L(T,T,T+\delta)^2\right)\right) \end{align} where the term $$\delta E_{t_0}^{T+\delta}(L(T,T,T+\delta)^2)$$ is a convexity adjustment.
Swap rate: the value of the swap rate $$S(t_0)$$ at time $$t_0$$ is derived by setting equal the values of the two legs of a swap starting at $$t_0$$, namely: $$\sum_{i=1}^n\delta_i^SS(t_0)P(t_0,t_i)=\sum_{j=1}^m\delta_j^LL(t_0,t_{i-1},t_i)P(t_0,t_i)$$ Rearranging: $$S(t_0)=\frac{\sum_{j=1}^m\delta_j^LL(t_0,t_{i-1},t_i)P(t_0,t_i)}{A^S(t_0,t_n)}$$ where the swap annuity $$A(t_0,t_n)$$ is defined as: $$A^S(t_0,t_n)=\sum_{i=1}^n\delta_i^SP(t_0,t_i)$$ The annuity is a portfolio of zero-coupon bonds (traded assets), thus it can be used as a numéraire. You therefore see that under the measure $$\mathcal{A}$$ associated to the annuity, the swap rate will be a martingale by a similar argument to the forward LIBOR rate: \begin{align} E_{t_0}^\mathcal{A}\left(S(t)\right) &=E_{t_0}^\mathcal{A}\left(\frac{\sum_{j=1}^m\delta_j^LL(t,t_{i-1},t_i)P(t,t_i)}{A^S(t,t_n)}\right) \\ &=\frac{\sum_{j=1}^m\delta_j^LL(t_0,t_{i-1},t_i)P(t_0,t_i)}{A^S(t_0,t_n)} \\[7pt] &=S(t_0) \end{align}
The forward LIBOR and swap rate cases clearly show that the proper martingale measure depends on the structure of the product being considered.
Note also that products like swaptions are quoted on a Bachelier/Black implied-volatility basis, that is the swaption is quoted with the implied volatility that corresponds to its market price. This implied volatility is obtained by inverting the Bachelier/Black formulas through numerical methods. Now, these formulas assume the underlying market factor (i.e. the swap rate) is a martingale under the pricing measure, thus the term “natural measure”: it is the measure implied by the market’s quotation convention.
References
[1] Geman, H., El Karoui, N., Rochet, J.C. (1995). "Changes of Numéraire, Changes of Probability Measures and Pricing of Options", on Journal of Applied Probability, Vol. 32, pg 443-458.
• The def of martingale is: $E(M_t\mid F_{t_0})=M_{t_0}$. How can we say: $E_{t_0}^{T+\delta}\frac{1}{P(t,T+\delta)}=\frac{1}{P(t_0,T+\delta)}$, as the function is in the denominator (won't Jensen's inequality apply?) – Bravo Feb 25 '20 at 11:21
• The theory of change of numéraire demonstrates that the asset divided by the numéraire is a martingale. In this case, the numéraire is the zero-coupon bond with maturity $T+\delta$, i.e. $P(\cdot,T+\delta)$. – Daneel Olivaw Feb 25 '20 at 11:26
• Note that the equation in your comment is not correct: an asset with a payoff of $1$ at some time $t’$ corresponds to a zero-coupon bond with maturity $t’$, thus its price at $t\leq t’$ is given by $P(t,t’)$ with $P(t’,t’)=1$. – Daneel Olivaw Feb 25 '20 at 11:29
• So if your asset is $X(t)$ and your numéraire is $N(t)$, it is the process $X(t)/N(t)$ which is a martingale under the measure $\mathcal{N}$ induced by the numéraire $N(t)$. – Daneel Olivaw Feb 25 '20 at 11:31
• I don’t understand why you mention Jensen’s inequality. – Daneel Olivaw Feb 25 '20 at 11:33
Your question: "How does one go about finding the right measure for a product?"
Answer: One should choose any measure that will make it easy and convenient to compute the pricing at hand.
We are free to use whichever measure we would like. For example, it is possible to derive the process for the Forward Libor $$L(t,T_1,T_2)$$ under a different measure than the one associated with $$P(t,T_2)$$ as Numeraire. However, such process would be a lot more complicated. So if we were to price options on Forward Libor under a different measure, we would make things unnecessarily more complex for ourselves.
Specific example:
$$1 + \delta L(t,T_1,T_2) = \frac{P(t,T_1)}{P(t,T_2)}$$
Therefore:
$$L(t,T_1,T_2) = \frac{1}{\delta} \left( \frac{P(t,T_1)-P(t,T_2}{P(t,T_2)}\right)$$
Re-arrange:
$$L(t,T_1,T_2)P(t,T_2) = \frac{1}{\delta} \left( P(t,T_1)-P(t,T_2) \right)$$
We know the right-hand side is a linear combination of traded assets (i.e. zero coupon bonds with different maturities) so we know that these have to be a Martingale under a numeraire of our choice. Chose $$P(t,T_1)$$ as numeraire:
$$\mathbb{E}^{P_{T(1)}} \left[ \frac{1}{\delta} \frac{P(t,T_1)-P(t,T_2)}{P(t,T_1)} \right] = martingale = \mathbb{E}^{P_{T(1)}} \left[ \frac{L(t,T_1,T_2)P(t,T_2)}{P(t,T_1)} \right]$$
Notice that on the RHS, we have the Libor process $$L(t,T_1,T_2)$$ multiplied by the bond $$P(t,T_2)$$ and divided by the bond $$P(t,T_1)$$ and this whole expression has to be a martingale for no-arbitrage pricing: so the above is not very helpful in the sense that we now have to worry about coming up with mathematical processes for $$L(t,T_1,T_2)$$, $$P(t,T_2)$$ and $$P(t,T_1)$$ such that their fraction is a martingale.
But, what if, instead of using $$P(t,T_1)$$ as Numeraire, we decide to use $$P(t,T_2)$$ as Numeraire?
$$\mathbb{E}^{P_{T(2)}} \left[ \frac{1}{\delta} \frac{P(t,T_1)-P(t,T_2)}{P(t,T_2)} \right] = martingale = \\ = \mathbb{E}^{P_{T(2)}} \left[ \frac{L(t,T_1,T_2)P(t,T_2)}{P(t,T_2)} \right] = \mathbb{E}^{P_{T(2)}} \left[ L(t,T_1,T_2)\right]$$
We can now directly deduce the process for $$L(t,T_1,T_2)$$ (using $$P(t,T_2)$$ as Numeraire) as:
$$L(t,T_1,T_2)=L(t_0,T_1,T_2)exp\left( -0.5 \sigma^2t + \sigma W(t) \right)$$
Because we know that under the $$P(t,T_2)$$ Numeraire, $$L(t,T_1,T_2)$$ alone must be a martingale.
We could have chosen $$P(t,T_1)$$ as Numeraire, but we'd have made things a lot more difficult for ourselves (just to stress the point again: because we'd have to think out the process for $$L(t,T_1,T_2)$$ such that $$\frac{L(t,T_1,T_2)P(t,T_2)}{P(t,T_1)}$$ is a martingale, rather than just $$L(t,T_1,T_2)$$ being a martingale).
Conclusion: the change of measure technique is all about convenience and computability. It is a mathematical technique that allows one to simplify the pricing task at hand.
• If anyone has any other good examples of why a certain change of measure is particularly helpful, in addition to the examples already posted, please do share: I feel this is a relevant topic. For example, any specific examples of when the measure associated with the Stock-price Numeraire is particularly beneficial? – Jan Stuller Jun 8 '20 at 18:00
• Strictly speaking unrelated to derivative pricing, but here is a nice example: Probability in different measures. – Daneel Olivaw Jun 11 '20 at 8:25
Via a combination of the Cameron-Martin-Girsanov Theorom and the Martingale Representation Theorom you can find the equivalent martingale measure. | 2021-03-05 09:32:34 | {"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": 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": 59, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9418780207633972, "perplexity": 639.1742115829269}, "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-2021-10/segments/1614178370752.61/warc/CC-MAIN-20210305091526-20210305121526-00261.warc.gz"} |
https://gateoverflow.in/1725/gate1998-11 | 1.8k views
Suppose $A = \{a, b, c, d\}$ and $\Pi_1$ is the following partition of A
$\Pi_1 = \left\{\left\{a, b, c\right\}\left\{d\right\}\right\}$
1. List the ordered pairs of the equivalence relations induced by $\Pi_1$.
2. Draw the graph of the above equivalence relation.
3. Let $\Pi_2 = \left\{\left\{a\right\}, \left\{b\right\}, \left\{c\right\}, \left\{d\right\}\right\}$
$\Pi_3 = \left\{\left\{a, b, c, d\right\}\right\}$
and $\Pi_4 = \left\{\left\{a, b\right\}, \left\{c,d\right\}\right\}$
Draw a Poset diagram of the poset, $\left\langle\left\{\Pi_1, \Pi_2, \Pi_3, \Pi_4\right\}, \text{ refines } \right\rangle$.
recategorized | 1.8k views
+6
.
................................
Interchange
$\Pi_2\ <-> \Pi_4\\ \Pi_1\ <-> \Pi_3$
0
Tuhin Dutta $\prod$3 refines both $\prod$4 and $\prod$1
am i right?
0
a. For Calculating Ordered Pairs Just multiply the partition with itself.
$\{a,b,c\} * \{a,b,c\} = \{ (a,a),(a,b),(a,c),(b.a),(b,b),(b,c),(c,a),(c,b),(c,c)\}$
$\{d\}*\{d\} = \{(d,d)\}$
$\therefore$ The ordered pairs of the equivalence relations are $= \{ (a,a),(a,b), (a,c), (b,a), (b,b), (b,c), (c,a), (c,b), (c,c),(d,d) \}$
b. For each ordered pair $(a,b)$ in the equivalence relation just make a directed edge from a towards $b$.
c. Suppose we have two partitions of a set $S:P_1={A_1,A_2,A_3...}\ and\ P_2={B_1,B_2,B_3...}$
We say that $P_1$ is a refinement of $P_2$ if every $A_i$ is a subset of some $B_j$ . Or We can say that $P_1$ refines $P_2$
by Boss (19k points)
edited by
0
@Satbir Not the literal meaning. What is the concept behind listing the ordered pairs as you did?
My main doubt here is why are we not multiplying {a,b,c} with {d}.
+2
0
Got it. In a partition, all the elements are related to each other and in two different partitions, no two elements are related to each other.
+1
@Satbir In the last statement where you have written If P1 is a refinement of P2 then it means P1 refines P2.
Is this correct or do you mean P2 refines P1?
0
first statement
0
partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all the points of P and possibly some other points as well; the partition Q is said to be “finer” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order.
So in last statement it should be p2 refines p1
0
We say P1 refines P2 or P1 is finer than P2 if every partition of P1 is subset of some partition of P2.
0
suppose P = {A,B} and Q = {{A},{B}}
then Q refines P or we can write Q is a refinement of P.
0
@Satbir Yes, You can write so...because every partition of Q is subset of partitions of P. And now you have corrected the statement in your answer.
0
YES ...i got little confused with harshpeswani's comment.
(a) The ordered pairs of the equivalence relations induced $= \{ (a,a),(a,b), (a,c), (b,a), (b,b), (b,c), (c,a), (c,b), (c,c),(d,d) \}$
PS: equivalence relations = each partition in power set - $\phi$
by Active (2.7k points)
+20
we can find the same by { {a,b,c} ${\times }$ {a,b,c} , {d} ${\times }$ {d} }
= { (a,a), (a,b), (a,c), (b,a), (b,b), (b,c), (c,a), (c,b) ,(c,c), (d,d) }
0
what about (b) and (c) ?
0
c.) Make the graph according to the given connected vertices below.
PI1----->PI3
PI4----->PI3
PI2----->PI4
PI2----->PI1
+1
Equivalence relation is reflexive, symmetric and transitive.
But that does not means that all symmetric and transitive pairs should be there, right? We can just have
reflexive pairs also, right? Like this: $\{(a,a),(b,b),(c,c),(d,d)\}$
Will it make difference if we notice that the question does not at all make use of word "equivalence class", but use word "equivalence relation", what you say? I guess it makes difference if we think of the following three points:
1. "equivalence relation" has no such restriction that each elements in the set should be related to each other, which is the restriction put by "equivalence class".
2. Set of all "equivalence classes" form partition, but the converse is not true, that is
3. Partition is made of sets which may or may not be "equivalence classes". The only criteria is that they should be disjoint and their union should be original set.
I guess after considering these facts, we are allowed to have only reflexive pairs, or even some symmetric and transitive pairs, if not all, right?
+6
Hasse diagram
You can interchange $\prod_1$ and $\prod_4$ as they are unrelated
0
@Ayush I am not understanding the a) part of the question please explain it
+3
@Prince -You must be knowing the two way theorem
Every Equivalence relation induces a partition on a set if and only if Every partition of the set induces an equivalence relation on the set.
Now to find the ordered pairs from the partition, simply take cross product of the partitions among themselves.
+2
A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finerthan ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that α ≤ ρ.
https://en.wikipedia.org/wiki/Partition_of_a_set#Refinement_of_partitions
0
@Ayush Upadhyaya you mean to say cross product of equivalence classes? coz equivalence relation divide set into disjoint classes.
0
Yes, and those classes form the biggest equivalence relation possible on the set.
0
,how we will draw part (b)
0
@codingo1234-Do you know what refines relation means?
0
@,earlier I was not knowing what "refines relation" means ,but I saw 's comment and came to know about when a partition is refinement of other
"A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that α ≤ ρ." ,
but I was confused regarding drawing graph of equivalence relation of part(a) and I was thinking about hasse digaram(god knows why everything mixes up in mind),I think it will be directed graph,am i correct?
0
@Ayush Upadhyaya Can you please provide a resource to read two-way theorem
by Active (2.6k points)
1
2 | 2019-10-14 13:57:00 | {"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": 0.6848692893981934, "perplexity": 686.0135373333794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986653247.25/warc/CC-MAIN-20191014124230-20191014151730-00238.warc.gz"} |
https://www.oreilly.com/library/view/numerical-linear-algebra/9780123944351/B9780123944351000193_2.xhtml | ## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.
No credit card required
## 19.6 The Divide-And-Conquer Method
This presentation is a summary of the divide-and-conquer method. For more in-depth coverage of this algorithm, see Refs. [1, pp. 216-228], [19, pp. 359-363], and [26, pp. 229-232].
The recursive algorithm divides a symmetric tridiagonal matrix into submatrices and then applies the same algorithm to the submatrices. We will illustrate the method of splitting the problem into smaller submatrix problems using a 5 ×5 matrix
$T=\left[\begin{array}{ccccc}{a}_{1}& {b}_{1}& 0& 0& 0\\ {b}_{1}& {a}_{2}& {b}_{2}& 0& 0\\ 0& {b}_{2}& {a}_{3}& {b}_{3}& 0\\ 0& 0& {b}_{3}& {a}_{4}& {b}_{4}\\ 0& 0& 0& {b}_{4}& {a}_{5}\end{array}\right].$
Write A as a sum of two matrices as follows:
$T=\left[\begin{array}{ccccc}{a}_{1}& {b}_{1}& 0& 0& 0\\ {b}_{1}& {a}_{2}-{b}_{2}& 0& 0& 0\\ 0& 0& {a}_{3}-{b}_{2}& {b}_{3}& 0\\ 0& 0& {b}_{3}& {a}_{4}& {b}_{4}\\ 0& 0& 0& {b}_{4}& {a}_{5}\end{array}\right]+\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& {b}_{2}& {b}_{2}& 0& 0\\ 0& {b}_{2}& {b}_{2}& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right].$
If we let
${T}_{1}=$
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No credit card required | 2018-12-18 14:06:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.19655293226242065, "perplexity": 2887.815476208376}, "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-2018-51/segments/1544376829399.59/warc/CC-MAIN-20181218123521-20181218145521-00613.warc.gz"} |
https://collegephysicsanswers.com/openstax-solutions/jet-airplane-750-m-wingspan-flying-280-ms-what-emf-induced-between-wing-tips-if | Question
(a) A jet airplane with a 75.0 m wingspan is flying at 280 m/s. What emf is induced between wing tips if the vertical component of the Earth’s field is $3.00 \times 10^{-5} \textrm{ T}$? (b) Is an emf of this magnitude likely to have any consequences? Explain.
1. $0.63 \textrm{ V}$
2. This is an insignificant voltage. It will have no effect on the electronics of the plane.
Solution Video | 2019-02-22 07:11:55 | {"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": 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": 0.3758520185947418, "perplexity": 805.0616754239354}, "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/1550247513661.77/warc/CC-MAIN-20190222054002-20190222080002-00025.warc.gz"} |
https://planilhafluxodecaixa.com/au70a/article.php?page=if-a-and-b-are-2x2-matrices-then-ab%3Dba-0f1fee | # if a and b are 2x2 matrices then ab=ba
If A and B are two matrices such that then (A) 2AB (B) 2BA (C) A+B (D) AB 1:08 188.3k LIKES. \end{pmatrix}=\begin{pmatrix} AB = BA for any two square matrices A and B of the same size. Find the a b c and d Q-15 If a=[ -2 4 5] and b=[1 3 -6] verify that (ab)'=b'a'? If so, prove it. 3 & 1 &0 In linear transformation terms, if two matrices $AB$ and $BA$ are equal, it means that the compound linear transformation that first applies the linear transformation $B$ and then applies the linear transformation $A$ is equivalent to the one where the linear ⦠The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B. so then A^2=A and the same applies for B; B ⦠1&1 Consider the system of simultaneous differential... Find all values of k, if any, that satisfy the... Types of Matrices: Definition & Differences, Singular Matrix: Definition, Properties & Example, Cayley-Hamilton Theorem Definition, Equation & Example, Eigenvalues & Eigenvectors: Definition, Equation & Examples, How to Solve Linear Systems Using Gauss-Jordan Elimination, How to Find the Distance between Two Planes, Complement of a Set in Math: Definition & Examples, Finding the Equation of a Plane from Three Points, Horizontal Communication: Definition, Advantages, Disadvantages & Examples, Addressing Modes: Definition, Types & Examples, What is an Algorithm in Programming? If A and B are (2x2) matrices, then AB = BA. Suppose that #A,B# are non null matrices and #AB = BA# and #A# is symmetric but #B# is not. 1&1 AB ≠ BA For every matrix A, it is true that (A^T)^T = A. 2. Then, taking traces of both sides yields. 1 &1 \\ Thus B must be a 2x2 matrix. 0 &0 \\ [a-b. If A and B are square matrices of the same order, then tr(AB) = tr(A)tr(B) AB is symmetric → AB = BA. (In fact, any 2x2 matrices A and b with the property that AB and BA aren't the same, will work.) If A and B are 2x2 matrices, then AB=BA. For every matrix A, it is true that (A^T)^T = A. Then I choose A and B to be square matrices, then A*B = AB exists. Therefore, AB = BA. If multiplying A^2, then it's asking you to multiply the identity matrix by itself, giving you the identity matrix. True. For a given matrix A, we find all matrices B such that A and B commute, that is, AB=BA. False. Sciences, Culinary Arts and Personal Determine whether (BA)2 must be O as well. #B^TA^T-BA=0->(B^T-B)A=0->B^T=B# which is an absurd. A(B+C) = AB + AC ≠ (B+C)A = BA + CA I hope this helps! then. \end{bmatrix} A = 3 X 3 matrix. let A be the 2x2 matrix with first row 1,0 and second row 0,0, and let B be the 2x2 matrix with first row 0,1 and second row 0,0. \end{bmatrix} \\\\ \rule{20mm}{.5pt}& \rule{20mm}{.5pt} & \rule{20mm}{.5pt} Answer to: AB = BA for any two square matrices A and B of the same size. The array of quantities or expressions set out by rows and columns; treated as a single element and manipulated according to rules. 4 & -3 & 4\\ If A and B are matrices of same order, then (AB'- BA') is a (A) skew symmetric matrix (B) null matrix (C) symmetric matrix (D) unit matrix. If A and B are 2x2 matrices with columns a1,a2 and b1,b2 respectively, then ab = [a1b1 a2b2] false each column of AB is a linear combo of the columns of B … The set of 2x2 matrices that contain exactly two 1's and two 0's is a linearly independent in M22. Try matrices B that have lots of zero entries. Therefore, AB is symmetric. 4 &-3 & -1\\ If multiplying A^2, then it's asking you to multiply the identity matrix by itself, giving you the identity matrix. Next you want to multiply A times B, and B times A, which should give you 18 different equations. Unlike general multiplication, matrix multiplication is not commutative. For every matrix A, it is true that (A^T)^T = A. 1 &1 \\ -1 & -1 & 1\\ True B. (i) Begin your proof by letting. Each matrix represents a transformation also matrix can bethink as the composition of their corresponding transformation. If A and B are 2x2 matrices, then AB = BA. Neither A nor B can be the identity matrix. {/eq} and {eq}BA = \begin{bmatrix} 2:32 3.0k LIKES. Prove that your matrices work. 1 &1 \\ tr(AB - BA) = tr(I) ==> tr(AB) - tr(BA) = 2, since I is 2 x 2 ==> 0 = 2, since tr(AB) = tr(BA). If A and B are 2x2 matrices, then AB=BA. {/eq}, then. If A and B are 2x2 matrices, then AB = BA. \rule{20mm}{.5pt}& \rule{20mm}{.5pt} & \rule{20mm}{.5pt} {eq}AB = BA 2x2 matrices are most commonly employed in describing basic geometric transformations in a 2-dimensional vector space. (ii) The ij th entry of the product AB ⦠2a+c]=[-1 5]. Multiplying A x B and B x A will give different results. I have an extra credit problem for linear algebra that I need help with: There are the 2x2 matrices A and B (A,B e M(2x2)) such that A+B=AB Show that AB=BA From a different problem, I have that (AB)^T=B^T(A^T) is true, so A^T(B^T )= (BA)^T = (AB)^T = B^T(A^T) Is this essentially the same question, or is there something that I'm missing with an identity matrix ⦠\end{pmatrix}. \end{pmatrix}\begin{pmatrix} False. As we know the composition of matrices may not commute so the product of two matrices need not commute also. 0&0 Matrices are widely used in geometry, physics and computer graphics applications. If A is an invertible matrix, then A transpose is also invertible and the inverse of the transpose equals the transpose of … The resulting product matrix will have the same number of rows as matrix A and the same number of columns as B. First of all, note that if $AB = BA$, then $A$ and $B$ are both square matrices, otherwise $AB$ and $BA$ have different sizes, and thus wouldn't be equal. 2) Hence then for the matrix product to exist then it has to live up to the row column rule. Then if A is non singlar and I replace B with A^-1 and since we know that AB = I, then A is invertible. 1 &1 \\ The technique involves creating a 2×2 matrix with opposing characteristics on each end of the spectrum. Click hereto get an answer to your question ️ If AB = A and BA = B then B^2 is equal to The answer is only A+B because when multiplying the identity matrix with any other matrix, the same numbers in the matrix that isn't the identity matrix will be unchanged and the answer. If A=\begin{bmatrix} 5&-6\\ -6& 3 \end{bmatrix},... 1. \end{pmatrix}=\begin{pmatrix} If A and B are (2x2) matrices, then AB = BA. The statement is in general not true. Matrix calculations can be understood as a set of tools that involves the study of methods and procedures used for collecting, classifying, and analyzing data. The set of 2x2 matrices that contain exactly two 1's and two 0's is a linearly independent in M22. Expert Answer . 0 &0 \\ Note. In the matrix multiplication AB, the number of columns in matrix A must be equal to the number of rows in matrix B. n matrices. Matrix multiplication is associative. 1 ? X = 4 \left( \begin{array} {... a) Does the set S span \mathbb{R}^{3}? Unlike general multiplication, matrix multiplication is not commutative. If A and B are 2x2 matrices with columns a1,a2 and b1,b2 respectively, then ab = [a1b1 a2b2] false each column of AB is a linear combo of the columns of B using weights from the corresponding columns of A To ask Unlimited Maths doubts download Doubtnut from - https://goo.gl/9WZjCW Suppose A and B are two nonsingular matrices such that AB=BA^2 and `B⦠[2a-b. There are many pairs of matrices which satisfy $AB=BA$, where neither of $A,B$ is a scalar matrix. 2x2 Matrix Multiplication Calculator is an online tool programmed to perform multiplication operation between the two matrices A and B. \end{pmatrix},B\begin{pmatrix} This last line is clearly a contradiction; hence, no such matrices exist. 0 &0 \\ If A is an invertible matrix, then A transpose is also invertible and the inverse of the transpose equals the transpose of the inverse. To solve this problem, we use Gauss-Jordan elimination to ⦠I hope this helps! False. Show that , if A and B are square matrices such that AB=BA, then . False. Let us take {eq}A=\begin{pmatrix} First we have to specify the unknowns. For a particular example you could e.g. 3) For A to be invertible then A has to be non-singular. AB = (AB)^t; since AB is symmetric = B^tA^t; by how the transpose "distributes". The multiplicative identity matrix is so important it is usually called the identity matrix, and is usually denoted by a double lined 1, or an I, no matter what size the identity matrix is. {/eq}, So both A,B are squire matrix but {eq}AB\ne BA. 77.4k VIEWS. \rule{20mm}{.5pt} & \rule{20mm}{.5pt} & \rule{20mm}{.5pt}\\ In many applications it is necessary to calculate 2x2 matrix multiplication where this online 2x2 matrix multiplication calculator can help you to effortlessly make your calculations easy for the respective inputs. It doesn't matter how 3 or more matrices are grouped when being multiplied, as long as the order isn't changed The 2×2 Matrix is a visual tool that consultants use to help them make decisions. 1&1 False. In (a) there are lots of examples. - Definition, Examples & Analysis, GED Math: Quantitative, Arithmetic & Algebraic Problem Solving, High School Geometry: Homeschool Curriculum, NY Regents Exam - Geometry: Tutoring Solution, McDougal Littell Geometry: Online Textbook Help, McDougal Littell Algebra 2: Online Textbook Help, Prentice Hall Geometry: Online Textbook Help, WEST Middle Grades Mathematics (203): Practice & Study Guide, TExMaT Master Mathematics Teacher 8-12 (089): Practice & Study Guide, SAT Subject Test Mathematics Level 1: Tutoring Solution, Biological and Biomedical {/eq} and {eq}B The multiplicative identity matrix for a 2x2 matrix is: The following will show how to multiply two 2x2 matrices: 1. The team then sorts their ideas and insights according to where they fall in the matrix. It is called either E or I Consider the following $2\times 2$ matrices. We give a counter example. \end{bmatrix} 1 &3 & 2\\ The only difference is that the order of the multiplication must be maintained True. True. False. Hint: AB = BA must hold for all B. The multiplicative identity matrix is a matrix that you can multiply by another matrix and the resultant matrix will equal the original matrix. 3c+2]=[0 13]. abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix ⦠Find the value of x. if A and B is a symmeyric, proof that AB-BA is a skew symmetric {/eq}. A = [a ij] and B = [b ij] be two diagonal n? There are matrices ⦠For every matrix A, it is true that (A^T)^T = A. False. Two square matrices A and B of the same order are said to be simultaneously diagonalizable, if there is a non-singular matrix P, such that P^(-1).A.P = D and P^(-1).B.P = D', where both the matrices D and D' are diagonal matrices⦠AB = BA.. Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. Services, Working Scholars® Bringing Tuition-Free College to the Community. If B is a 3X3 matrix then we will have a matrix containing a,b,c,d,e,f,g,h,i where these letters are the unknowns representitive of the coefficients in the B matrix. True. 0&0 Suppose to the contrary that AB - BA = I for some 2 x 2 matrices A and B. #AB = (AB)^T = B^TA^T = B A#. Matrix multiplication is NOT commutative in general IA = AI = A Click hereðto get an answer to your question ï¸ If A and B are symmetric matrices of same order, prove that AB - BA is a symmetric matrix. 0&0 2.0k VIEWS. All rights reserved. Then I choose A and B to be square matrices, then A*B = AB exists. A(BC) = (AB)C So #B# must be also symmetric. 2) Hence then for the matrix product to exist then it has to live up to the row column rule. 4. If any matrix A is added to the zero matrix of the same size, the result is clearly ⦠For the product AB, i) I already started by specifying that A = [aij] and B = [bij] are two n x n matrices ii) and I wrote that the ijth entry of the product AB is cij = ∑(from k=1 to n of) aik bkj Now the third part (and the part I'm having trouble with) says to evaluate cij for the two cases i ≠ j and i = j. No, AB and BA cannot be just any two matri- ces. A. In (a) there are lots of examples. \end{bmatrix} 2x2 Matrix Multiplication Calculator is an online tool programmed to perform multiplication operation between the two matrices A and B. There are specific restrictions on the dimensions of matrices that can be multiplied. Some people call such a thing a âdomainâ, but not everyone uses the same terminology. False. 2.0k SHARES. \rule{20mm}{.5pt} & \rule{20mm}{.5pt} & \rule{20mm}{.5pt}\\ By continuing with ncalculators.com, you acknowledge & agree to our, 4x4, 3x3 & 2x2 Matrix Determinant Calculator, 4x4 Matrix Addition & Subtraction Calculator, 2x2 Matrix Addition & Subtraction Calculator. 0 &0 \\ False. If not, give a counter example. If {eq}A = \begin{bmatrix} let A be the 2x2 matrix with first row 1,0 and second row 0,0, and let B be the 2x2 matrix with first row 0,1 and second row 0,0. {eq}AB = \begin{bmatrix} If it's a Square Matrix, an identity element exists for matrix multiplication. IA = AI = A \rule{20mm}{.5pt} &\rule{20mm}{.5pt} & \rule{20mm}{.5pt}\\ -4 &-3 & 2 Our experts can answer your tough homework and study questions. Multiplying A x B and B x A will give different results. All other trademarks and copyrights are the property of their respective owners. = AB; by assumption. Find all possible 2 × 2 matrices A that for any 2 × 2 matrix B, AB = BA. If A and B are square matrices of the same order, then tr(AB) = tr(A)tr(B) This last line is clearly a contradiction; hence, no such matrices exist. Check Answ {/eq} and {eq}B = \begin{bmatrix} row 1 [1 1 1] row 2 [1 2 3] row 3 [1 4 5] Find a 3 X 3 matrix B, not the identity matrix or the zero matrix such that AB = BA. 3. Solve the following system of equations using the... A) A = \begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 0 ... For A = \begin{pmatrix} -2 & 0 \\ 4 & 1 \\ 7 & 3... solve for the values of u'1 and u'2 . False. All matrices which commute with all 2 × 2 matrices (3 answers) Closed 3 years ago. 77.4k SHARES. If AB+BA is defined, then A and B are square matrices of the same size. 1&1 Then, taking traces of both sides yields. Thus, if A and B are both n x n symmetric matrices then AB is symmetric ↔ AB = BA. \end{pmatrix}. If #A# is symmetric #AB=BA iff B# is symmetric. 3) For A to be invertible then A has to be non-singular. \rule{20mm}{.5pt} &\rule{20mm}{.5pt} & \rule{20mm}{.5pt}\\ Matrix multiplication is associative, analogous to simple algebraic multiplication. Suppose to the contrary that AB - BA = I for some 2 x 2 matrices A and B. {/eq}. Given A = [ 1 1 \\ 2 1 ], B = [ ? Dear Teachers, Students and Parents, We are presenting here a New Concept of Education, Easy way of self-Study. Then if A is non singlar and I replace B with A^-1 and since we know that AB = I, then … For a particular example you could e.g. Previous question Next question Get more help from Chegg. \[A=\begin{bmatrix} 0 & 1\\ Click hereðto get an answer to your question ï¸ If AB = A and BA = B then B^2 is equal to 0&0 BA=\begin{pmatrix} Solution. {/eq}, Then {eq}AB=\begin{pmatrix} The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B. {/eq} for any two square matrices {eq}A tr(AB - BA) = tr(I) ==> tr(AB) - tr(BA) = 2, since I is 2 x 2 ==> 0 = 2, since tr(AB) = tr(BA). The multiplicative identity matrix obeys the following equation: True or False: If A, B are 2 by 2 Matrices such that (AB)2 = O, then (BA)2 = O Let A and B be 2 × 2 matrices such that (AB)2 = O, where O is the 2 × 2 zero matrix. Prove that if A and B are diagonal matrices (of the same size), then AB = BA. Hope this helps! © copyright 2003-2020 Study.com. = BA; since A and B are symmetric. (In fact, any 2x2 matrices A and b with the property that AB and BA aren't the same, will work.) \end{pmatrix}\begin{pmatrix} but #A = A^T# so. Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. Prove that if A and B are diagonal matrices (of the same size), then. The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B. 2x2 matrices are most commonly employed in ⦠\end{pmatrix} In any ring, $AB=AC$ and $A\ne 0$ implies $B=C$ precisely when that ring is a (not necessarily commutative) integral domain. Write the matrix representation for the given... Let A = \begin{bmatrix} 2 & 4\\ 4 & 9\\ -1 & -1... Find \frac{dX}{dt}. = BA; since A and B are symmetric. {/eq} of the same size. If AB+BA is defined, then A and B are square matrices of the same size. Get 1:1 help now from expert Precalculus tutors Solve it with our pre-calculus problem solver and calculator Favorite Answer For AB to make sense, B has to be 2 x n matrix for some n. For BA to make sense, B has to be an m x 2 matrix. They must have the same determinant, where for 2 × 2 matrices the determinant is deï¬ned by det a b c d = ad â bc. Find two 2x2 matrices A and B so that AB=BA. The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of B. The answer is only A+B because when multiplying the identity matrix with any other matrix, the same numbers in the matrix that isn't the identity matrix will be unchanged and the answer.
Olá,
Podemos Ajudar? | 2021-04-13 10:44:45 | {"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": 2, "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": 0.9163517951965332, "perplexity": 578.0338483165655}, "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/1618038072180.33/warc/CC-MAIN-20210413092418-20210413122418-00222.warc.gz"} |
https://simpleelastix.readthedocs.io/RigidRegistration.html | Rigid Registration¶
A rigid transform can register objects that are related by rotation and translation. For example, if you are registering images of a patient’s bones, you can often assume that a rigid transform is sufficient to align these structures. In fact, it is often advantageus to chose a simple transform if problems that allows it, as this constrains the solution space and ensures no spurious non-rigid local minima affect your results. Think of it as a way of embedding expert knowledge in the registration procedure.
Tip
Rigid registration is one of the simplest of methods in the catagory of linear transformation models and is often used as initialization for affine- and non-rigid transforms.
The rigid transform is selected using (Transform "EulerTransform"). Consider the images in Figure 8.
Figure 8. The original image fixedImage.nii (left) and translated and rotated image movingImage.nii (right).
The image on right has been rotated 10 degrees and translated 13 pixels in the x-direction and 17 pixels in the y-direction. Using the EulerTransform we may correct for this misalignment.
import SimpleITK as sitk
elastixImageFilter = sitk.ElastixImageFilter() | 2019-06-25 22:38:24 | {"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": 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": 0.5695297718048096, "perplexity": 1111.0577251780167}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999948.3/warc/CC-MAIN-20190625213113-20190625235113-00144.warc.gz"} |
https://www.physicsforums.com/threads/laplaces-equation-in-cylindrical-coordinates.215807/ | # Laplace's Equation in Cylindrical Coordinates
1. Feb 16, 2008
### G01
1. The problem statement, all variables and given/known data
A long copper pipe, with it's axis on the z axis, is cut in half and the two halves are insulated. One half is held at 0V, the other at 9V. Find the potential everywhere in space.
2. Relevant equations
$$\nabla^2V=0$$
3. The attempt at a solution
Alright. This is a laplace's equation problem in two dimensions, since the potential should be independent of z because the pipe is infinite. Using separation of variables( $$V=R\Phi$$) the solution to the two dimensional Laplace's EQ in cylindrical coordinates is:
$$R(r)=ar^k+br^-k$$
$$\Phi(\phi)=A\sin(k\phi) + B\cos(k\phi)$$
w/ $$V=R(r)\Phi(\phi)$$
I am getting confused when I try to apply boundary conditions to this problem. Here is what I think the boundary conditions should be:
1) V should go to zero as r goes to infinity. (I don't think this is right, since the pipe is infinite, but I am not sure.)
2)V=0 for phi between 0 and pi, r=radius of pipe
3)V=9 for phi between 0 and 2pi, r=radius of pipe.
I can't figure out how to actually use these to eliminate any of the constants. Any help and hints would be greatly appreciated.
2. Feb 16, 2008
### Rainbow Child
First of all you have to break the problem into two regions $V_{in}$ inside the pipe and $V_{out}$ outside the pipe. In order for $V$ to be finite to each region you can eleminate $\alpha$ or $\beta$ in eah region.
The potential $V$ must be periodical for $\phi$ thus you know $k$.
Lastly you have to use the superposition principle and Fourier analysis for the remaining constants.
3. Feb 16, 2008
### G01
OK. It seems the part screwing me up is finding k. The potential is periodic in Phi, but it seems more like a step function. I guess I'm saying that I don't understand how can represent that potential as a sine or cosine term, no matter what k happens to be.
Could you elaborate on that part of the problem?
4. Feb 16, 2008
### Rainbow Child
Since the potential must be periodic in $\phi$ you must have $k\in \mathbb{N}$. Thus the solution is
$$V_{in}(r,\phi)=\frac{A_0}{2}+\sum_{n=1}^\infty r^n\left(A_n\,\cos(n\,\phi)+B_n\,\sin(n\,\phi)\right)$$
$$u(R,\phi)=\left\{\begin{matrix} 0 & 0<\phi<\pi\cr 9 & \pi<\phi<2\,\pi \end{matrix}$$
The constants $A_n,\,B_n$ can be found by Fourier analysis. Can you do that? | 2017-10-21 01:51:38 | {"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": 0.7376284599304199, "perplexity": 373.085582483691}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824537.24/warc/CC-MAIN-20171021005202-20171021025202-00169.warc.gz"} |
https://homework.cpm.org/category/CC/textbook/cc1/chapter/3/lesson/3.1.1/problem/3-19 | ### Home > CC1 > Chapter 3 > Lesson 3.1.1 > Problem3-19
3-19.
Jing Ya takes the bus across town to school each morning. Last week, he timed his trips and found that the time varies day to day. The times (in minutes) are listed below.
$15, 10, 11, 13, 11$
1. If you had to use one number to tell someone how long it took Jing Ya to get to school, what would you say?
Try arranging the times in order and choosing one which is in the middle or appears most often.
$11$ or $12$ minutes. Sometimes it takes Jing Ya longer to get to school, and sometimes shorter.
However, a typical trip to school will take $11$ or $12$ minutes.
2. Jing Ya does not want to be late. If he needs to get to school by 8:00 a.m. each day, what is the latest time he should get on the bus (assuming one is waiting for him at any given time)? Explain how you got your answer.
• It may help to look at the longest amount of time it took Jing Ya to get to school. | 2020-12-06 02:01:24 | {"extraction_info": {"found_math": true, "script_math_tex": 5, "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": 0.550895094871521, "perplexity": 1206.512020760464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141753148.92/warc/CC-MAIN-20201206002041-20201206032041-00047.warc.gz"} |
https://sncosmo.readthedocs.io/en/latest/registry.html | # Registry¶
## What is it?¶
The registry (sncosmo.registry) is responsible for translating string identifiers to objects, for user convenience. For example, it is used in sncosmo.get_bandpass and sncosmo.get_source to return a Bandpass or sncosmo.Model object based on the name of the bandpass or model:
>>> sncosmo.get_bandpass('sdssi')
<Bandpass 'sdssi' at 0x28e7c90>
It is also used in methods like bandflux to give it the ability to accept either a Bandpass object or the name of a bandpass:
>>> model = sncosmo.Model(source='hsiao')
>>> model.bandflux('sdssg', 0.) # works, thanks to registry.
Under the covers, the bandflux method retrieves the Bandpass corresponding to 'sdssg' by calling the sncosmo.get_bandpass function.
The registry is actually quite simple: it basically amounts to a dictionary and a few functions for accessing the dictionary. Most of the time, a user doesn’t need to know anything about the registry. However, it is useful if you want to add your own “built-ins” or change the name of existing ones.
## Using the registry to achieve custom “built-ins”¶
There are a small set of “built-in” models, bandpasses, and magnitude systems. But what if you want additional ones?
Create a file mydefs.py that registers all your custom definitions:
# contents of mydefs.py
import numpy as np
import sncosmo
wave = np.array([4000., 4200., 4400., 4600., 4800., 5000.])
trans = np.array([0., 1., 1., 1., 1., 0.])
band = sncosmo.Bandpass(wave, trans, name='tophatg')
sncosmo.registry.register(band)
Make sure mydefs.py is somewhere in your \$PYTHONPATH or the directory you are running your main script from. Now in your script import your definitions at the beginning:
>>> import sncosmo
>>> import mydefs
>>> # ... proceed as normal
>>> # you can now use 'tophatg' as a built-in
## Changing the name of built-ins¶
To change the name of the 'sdssg' band to 'SDSS_G':
# contents of mydefs.py
import sncosmo
band = sncosmo.get_bandpass('sdssg')
band.name = 'SDSS_G'
sncosmo.register(band)
## Large built-ins¶
What if your built-ins are really big or you have a lot of them? You might only want to load them as they are needed, rather than having to load everything into memory when you do import mydefs. You can use the sncosmo.registry.register_loader function. Suppose we have a bandpass that requires a huge data file (In reality it is unlikely that loading bandpasses would take a noticeable amount of time, but it might for models or spectra.):
# contents of mydefs.py
import sncosmo
# ...
# read data from filename, create a Bandpass object, "band"
# ...
return band
filename = 'path/to/datafile/for/huge_tophatg'
Now when you import mydefs the registry will know how to load the Bandpass named 'huge_tophatg' when it is needed. When loaded, it will be saved in memory so that subsequent operations don’t need to load it again. | 2020-05-26 16:11:22 | {"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": 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": 0.2885999083518982, "perplexity": 3333.0705135063995}, "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/1590347391277.13/warc/CC-MAIN-20200526160400-20200526190400-00305.warc.gz"} |
https://blog.tharinduhasthika.com/is-this-a-prime-number | # .css-df1pn7{display:block;width:16rem;}
Photo by Towfiqu barbhuiya on Unsplash
# Is this a Prime Number?
## Simple Approaches to Check for Prime
Tharindu Hasthika
·May 3, 2018·
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Prime Numbers are an interesting set of numbers, there is no general expression to find a particular prime number. But when we are given a number to check for its primality, we can do it very easily (although we would need to do some serious computation for larger numbers). In this article I'll show you some methods of checking if a number is prime.
There are many ways of checking if a number is prime, below are some of them.
### A Naive Approach
This is the most simple and straight forward way of checking primality.
def is_prime(n):
# prime numbers >= 2
if n <= 1:
return False
# loop i from 2 to n - 1
for i in range(2, n):
# if n is divisible by i and the remainder is 0
# then it is surely not prime
if n % i == 0:
return False
return True
The Running Time: $$O(n)$$
So This in't that bad for small integers but if the number is a very large number, like ten or hundred million it would slow down.
So What can we do to reduce our running time?
### A Slightlty Smart Approach
If you can remember your maths classes, you could have noticed that for every factor there is another factor that when multiplied together gives the number in question. More formally,
if $$a$$ is a factor $$n$$
then there is some $$b$$ such that $$a*b=n$$
So let's see some examples and figure out what can be done to make the algorithm faster.
For example lets take 20
• $$1 * 20 = 20$$
• $$2 * 10 = 20$$
• $$4 * 5 = 20$$
• $$5 * 4 = 20$$
• $$10 * 2 = 20$$
• $$20 * 1 = 20$$
As you can see we are finding the other factor which makes up the number in a separate step, which is unnecessary. So we only need to look for the first numbers that are less than or equal to $$\sqrt{n}$$.
So with that intuition we could write a new function which is much more faster for larger numbers. While coding you should remember that finding the sqrt of a number is more costly that multiplication so we could say that $$i*i \leq n$$.
def is_prime(n):
# prime numbers >= 2
if n <= 1:
return False
# loop i from 2 to sqrt(n)
i = 2
while i * i <= n:
# if n is divisible by i and the remainder is 0
# then it is surely not prime
if n % i == 0:
return False
i += 1
return True
### Some Optimizations to the Function
There are some other optimizations that can be applied to the function to reduce the running time even further.
Check if $$n$$ is divisible by $$2$$, and if $$n \ne 2$$ then it is not prime.
If a number is not divisible by $$2$$ then we can be sure that it will not be divisible by any other even number. So we can cut down the iteration count in half!
So with those Optimizations the final function is as follows.
def is_prime(n):
# prime numbers >= 2
if n <= 1:
return False
if n == 2:
return True
# reject all even numbers exept for 2
if n % 2 == 0:
return False
# loop i from 3 to sqrt(n)
i = 3
while i * i <= n:
# if n is divisible by i and the remainder is 0
# then it is surely not prime
if n % i == 0:
return False
# loop over odd numbers only
i += 2 | 2022-12-05 14:23:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5982903242111206, "perplexity": 488.83809058308555}, "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/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00035.warc.gz"} |
http://jmre.ijournals.cn/cn/ch/reader/view_abstract.aspx?file_no=20210601&flag=1 | The Intersection Problem for Kite-GDDs of Type $2^{u}$
DOI：10.3770/j.issn:2095-2651.2021.06.001
作者 单位 安永红 呼伦贝尔学院继续教育学院, 内蒙古 呼伦贝尔 021008 张桂芝 呼伦贝尔学院科学技术处, 内蒙古 呼伦贝尔 021008
Kite-可分组设计的相交数问题是确定所有可能的元素对$(T,s)$, 使得存在一对具有相同组型 $T$ 的Kite-可分组设计 $(X,{\cal H},{\cal B}_1)$ 和$(X,{\cal H},{\cal B}_2)$ 满足$|{\cal B}_1\cap {\cal B}_2|=s$. 本文研究组型为 $2^u$ 的Kite-可分组设计的相交数问题, 设 $J(u)=\{s:\exists$ 组型为 $2^u$ 的Kite-可分组设计相交于$s$ 个区组\}, $I(u)=\{0,1,\ldots,b_{u}-2,b_{u}\}$,其中 $b_u=u(u-1)/2$ 是组型为$2^u$ 的Kite-可分组设计的区组个数. 我们将给出对任意整数 $u\ge 4$ 都有$J(u)=I(u)$ 且 $J(3)= \{0,3\}$.
The intersection problem for kite-GDDs is the determination of all pairs $(T,s)$ such that there exists a pair of kite-GDDs $(X,{\cal H},{\cal B}_1)$ and $(X,{\cal H},{\cal B}_2)$ of the same type $T$ satisfying $|{\cal B}_1\cap {\cal B}_2|=s$. In this paper the intersection problem for a pair of kite-GDDs of type $2^u$ is investigated. Let $J(u)=\{s:$ $\exists$ a pair of kite-GDDs of type $2^u$ intersecting in $s$ blocks$\}$; $I(u)=\{0,1,\ldots,b_{u}-2,b_{u}\}$, where $b_u=u(u-1)/2$ is the number of blocks of a kite-GDD of type $2^u$. We show that for any positive integer $u\geq 4$, $J(u)=I(u)$ and $J(3)= \{0,3\}$. | 2021-12-02 19:15:27 | {"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": 0.9585712552070618, "perplexity": 2488.166892457565}, "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/1637964362287.26/warc/CC-MAIN-20211202175510-20211202205510-00520.warc.gz"} |
http://mathoverflow.net/questions/2144/a-single-paper-everyone-should-read?answertab=active | # A single paper everyone should read? [closed]
Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.
Do you have such an example?
Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.
And as usual let's do one per post and vote freely, vote a lot.
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## closed as off topic by Kevin H. Lin, Andres Caicedo, Felipe Voloch, Hailong Dao, Bill JohnsonJul 9 '11 at 15:01
Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.
Why are so many answers big-picture papers and philosophical tracts? I'm sure many of them are good papers, but is this really what the question was about? Am I right in suspecting that posters only read the title of the question and not the question itself? – Thierry Zell Sep 4 '10 at 0:23
Perhaps it's time to close this question. – S. Carnahan Oct 22 '10 at 17:40
Agreed, as Thierry and Tobias say, there are too many recommendations for punditry. – Robin Chapman Nov 17 '10 at 11:48
PDE as a Unified Subject by Sergiu Klainerman.
An essay on partial differential equations written by a leading expert in the field, I strongly recommend to anyone who aspires to know more on the subject as well as to those who are not interested strictly in PDE's, but would like to get a grasp of interactions between Mathematics and Physics. There are also many interesting references.
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Perhaps not really a paper, but i think a "must-read" is A Mathematician's Lament by Paul Lockhart.
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It might not be a bad idea to read Scott Aaronson's thoughts on it afterwards, though: scottaaronson.com/blog/?p=410 – Qiaochu Yuan Oct 25 '09 at 16:10
I wish all math teachers read that link. – Ilya Nikokoshev Oct 25 '09 at 18:40
I strongly disagree with the "must-read" label. Lockhart's article is mostly composed of half-truths and empty dramatic language. – S. Carnahan Nov 7 '09 at 5:30
IMHO, one can say "I prefer geometric proofs/arguments to algebraic ones" in shorter space than 25 pages. – Ketil Tveiten Nov 17 '10 at 11:26
Absolutely, the only way to reintroduce honesty into the prevailing math curriculum is to abolish the requirement that everyone must study math. – Michael Hardy Nov 17 '10 at 13:53
One paper that I want to share with any of my colleagues, although it is not in my field, is Doyle and Conway, Division by Three, math/0605779v1.
To emphasize why this paper is so great, let me quote the entirety of the conclusion (saving you the trouble of reading the rest of the paper):
### What’s wrong with the axiom of choice?
Part of our aversion to using the axiom of choice stems from our view that it is probably not ‘true’. A theorem of Cohen shows that the axiom of choice is independent of the other axioms of ZF, which means that neither it nor its negation can be proved from the other axioms, providing that these axioms are consistent. Thus as far as the rest of the standard axioms are concerned, there is no way to decide whether the axiom of choice is true or false. This leads us to think that we had better reject the axiom of choice on account of Murphy’s Law that ‘if anything can go wrong, it will’. This is really no more than a personal hunch about the world of sets. We simply don’t believe that there is a function that assigns to each non-empty set of real numbers one of its elements. While you can describe a selection function that will work for finite sets, closed sets, open sets, analytic sets, and so on, Cohen’s result implies that there is no hope of describing a definite choice function that will work for ‘all’ non-empty sets of real numbers, at least as long as you remain within the world of standard Zermelo-Fraenkel set theory. And if you can’t describe such a function, or even prove that it exists without using some relative of the axiom of choice, what makes you so sure there is such a thing?
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat doubtful whether large natural numbers (like 805000, or even 2200) exist in any very real sense, and we’re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic—and hence also of set theory—are inconsistent. (See [E. Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.]) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today.
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Or the first phrase: "In this paper we show that it is possible to divide by three." – Ilya Nikokoshev Nov 8 '09 at 13:01
"An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" by Jonathan Shewchuk at UC Berkeley
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If you are a geometer I would say it is worth to read the paper of Gromov, called "Spaces and Questions", this paper is not about one single result, it rather gives a point of view on geometry, which seems very inspiring, at least to me, here is the refference: http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf
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Well, the main purpose of the question was to mention "papers everyone should read", and not just geometers; I started Gromov's paper not being a geometer myself and didn't find it very inspiring... – Jose Brox Nov 19 '09 at 17:50
If you ever - as in my case - quoted a textbook to your students claiming that pointwise convergence of Fourier series for piecewise continuous functions is difficult and subtle, you'll feel stupid after reading Paul Chernoff's two-page paper "Pointwise Convergence of Fourier Series."
I can't find a free online copy of it, but you should be able to read it here with university access: JSTOR (Actually, you can see the first page for free, which already proves the main result.)
(Or get it from the library: The American Mathematical Monthly, Vol. 87, No. 5 (May, 1980), pp. 399-400.)
EDIT: There is a free copy here: http://math.berkeley.edu/~strain/118.S10/chernoff.pointwise.convergence.of.fourier.series.pdf
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Pointwise convergence for continuous functions is indeed very difficult and subtle. Chernoff's paper is about the far easier differentiable case. – Richard Borcherds Sep 3 '10 at 16:31
I am surprised to see that so many people suggest meta-mathematical articles, which try to explain how one should do good mathematics in one or the other form. Personally, I usually find it a waste of time to read these, and there a few statements to which I agree so wholeheartedly as the one of Borel:
"I feel that what mathematics needs least are pundits who issue prescriptions or guidelines for presumably less enlightened mortals."
The mere idea that you can learn how to do mathematics (or in fact anything useful) from reading a HowTo seems extremely weird to me. I would rather read any classical math article, and there are plenty of them. The subject does not really matter, you can learn good mathematical thinking from each of them, and in my opinion much easier than from any of the above guideline articles. Just to be constructive, take for example (in alphabetical order)
• Atiyah&Bott, The Yang-Mills equations over Riemann surfaces.
• Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts.
• Furstenberg, A Poisson formula for semi-simple Lie groups.
• Gromov,Groups of polynomial growth and expanding maps.
• Tate, Fourier analysis in number fields and Hecke's zeta-functions.
I am not suggesting that any mathematician should read all of them, but any one of them will do. In fact, the actual content of these papers does not matter so much. It is rather, that they give an insight how a new idea is born. So, if you want to give birth to new ideas yourself, look at them, not at some guideline.
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"Rigor and Proof in Mathematics: A Historical Perspective" by Israel Kleiner. Mathematics Magazine December 1991, 64:291-314.
This paper gives a very nice overview of how the understanding of rigor in mathematics has evolved from the early ages to the 20th century.
http://www.jstor.org/sici?sici=0025-570X%28199112%2964%3A5%3C291%3ARAPIMA%3E2.0.CO%3B2-Z
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Not technically a paper but a lecture (in pdf form) full of pretty pictures and cool ideas:
We all know what it means for a set to have 6 elements, but what sort of thing has -1 elements, or 5/2? Believe it or not, these questions have nice answers. The Euler characteristic of a space is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive real values. These concepts shed new light on basic mathematics. For example, the space of finite sets turns out to have homotopy cardinality e, and this explains the key properties of the exponential function. Euler characteristic and homotopy cardinality share many properties, but it's hard to tell if they are the same, because there are very few spaces for which both are well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series---and the two then agree! The challenge of unifying them remains open.
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I highly recommend this lucid, little book (with the length of a paper):
Mathematics: A very short introduction, by Fields Medalist Timothy Gowers
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## The Unreasonable Effectiveness of Mathematics in the Natural Sciences
by Eugene Wigner
Although Wigner is physicist, I consider this article about mathematical physics very important both for physicists and mathematicians. It's a wonderful feeling to realize to what extent our world can be mathematical.
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. E.Wigner
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This is not only a historically important paper,it focuses on an aspect of mathematics that Western culture has hesitated to return to after the wholesale rejection of it during the Bourbaki era. – Andrew L Oct 22 '10 at 19:04
Even if bashing Bourbaki seems to have become hip again (as it was actually during the Bourbaki era as well), and thus the myth of the "wholesale rejection" of applications of mathematics "during the Bourbaki era" has become generally accepted in certain circles, it still remains a myth, which like all myths contains a germ of truth surrounded by a lot of prejudices, misunderstandings and plainly wrong statements. – Tobias Hartnick Nov 17 '10 at 13:44
Two notes on notation by Knuth. This paper discusses "Iverson" notation, which is of use to almost all mathematicians, and good notation for Stirling numbers.
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On the theorem of Pythagoras by by E.W. Dijkstra. (Did you know that in every plane triangle sgn$(\alpha + \beta - \gamma)$ = sgn$(a^2 + b^2 - c^2)$, a "theorem, say, 4 times as rich [as the original]"?)
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I don't think this is any news to most mathematicians. This is even in some good German schoolbooks from the 1960's-70's. Often when there is a statement like "if $a=b$ then $c=d$" one could check whether $a\leq b$ implies $c\leq d$, and lots of geometric inequalities have been created this way from identities. – darij grinberg Nov 17 '10 at 14:08
That's easy just off the top of my head,Illya: Nets And Filters In Topology by the late Robert G. Bartle;appearing in the 1955 Volume 62 of American Mathematical Monthly. I remember having a friend in the Stanford mathematics honor society who'd published papers by age 20,but had never heard of either nets or filters. I recommended it to him right on the spot.
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Birds and Frogs by Freeman Dyson, which explains nicely that the world of mathematics is both , broad and deep.
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Toen's course on stacks. I don't know if this counts as a paper, but courses 2,3, and 4 introduce a really interesting approach to geometry using the functor of points approach that I've not seen before.
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I had recommended to me from several prominent faculty the paper:
The Yang-Mills Equations over Riemann Surfaces Author(s): M. F. Atiyah and R. Bott Source: Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 Published by: The Royal Society Stable URL: http://www.jstor.org/stable/37156
One professor called it "the basis for truly 21st century mathematics." It is also reportedly accessible by beginning graduate students with some exposure to differential geometry and suitable for independent study or as a reading course. It is a 93 page paper and develops a lot of fundamental constructions and ideas from scratch. Here is Martin Guest's review on MathSciNet.
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For about 5 years I carried my copy with me everywhere I went, in an increasingly decrepit 3-ring binder weighed down by page after page of my own notes and explanations. One day, at a conference, a dispute arose over whether the main result of the paper held with integral coefficients or required one to work over the rationals. In the flash of an eye, four or five of us pulled out our copies and opened to the relevant page. Luckily, I was right: integral coefficients. The first time I left home without the paper, it felt like a rite of passage. Or at least that's the way I remember it. – Dan Ramras Sep 4 '10 at 4:49
Imre Lakatos "Proofs and Refutations". Great book about origin of mathematical reasoning and rise of formal theories.
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Paul Halmos How to Write Mathematics
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Advice to a Young Mathematician in the Princeton Companion to Mathematics
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"On Computable Numbers, with an Application to the Entscheidungsproblem", Alan Turing, 1936. A great mind and a great paper.
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Two additional papers in combinatorics (That I managed to find on line) each having a beautiful and simple result.
On the Shannon Capacity of a Graph by Laszlo Lovasz
The Upper Bound Conjecture and Cohen Macaulay Rings by Richard Stanley
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2N Noncollinear Points Determine at Least 2N Directions, by Peter Ungar. This is a beautiful short paper that proves the result in the title.
A general remark: If you have to choose a single paper (or a single paper of a mathematician selected in other answers), I would recommend more strongy to choose original papers of important basic results rather than large survey papers or "meta" paper about mathematics. (This is also closer to the original intention of the question.)
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One paper that I've read a few times and always loved was Who Can Name the Bigger Number? (also available in Spanish and French, for those who prefer to read in those). It discusses how our concept of "big numbers" has evolved over time, and talks about Turing machines and the "busy beaver" numbers, which represent a non-computable function.
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Proofs from the Book! (Ok it's a book rather than a paper, but just pick any chapter.) Every line is amazing.
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Missed Opportunities, Freeman Dyson
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Really up to the point. After reading it, I also think any mathematician (or physicist) should do the same. Thanks! – Jose Brox Nov 19 '09 at 19:19
"On the Electrodynamics of Moving Bodies", Albert Einstein
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I love Minkowski's rebuttal to that paper. – Ryan Budney Nov 19 '09 at 2:27
Stallings's How Not To Prove the Poincare Conjecture is the funniest paper I've ever read.
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"On the Number of Primes Less Than a Given Magnitude", B. Riemann.
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In recent years Manin has put out several philosophical writings on mathematics, physics, and other related topics:
Mathematical knowledge: internal, social and cultural aspects
The notion of dimension in geometry and algebra
Georg Cantor and his heritage
Von Zahlen und Figuren
There's also a book, Mathematics as Metaphor, that collects even more of Manin's philosophical material.
These are all very nice reads and I would recommend them to almost anyone, mathematician/physicist or not.
- | 2015-03-02 04:04:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5999948978424072, "perplexity": 810.7042500821299}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462700.28/warc/CC-MAIN-20150226074102-00005-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://folia.unifr.ch/unifr/documents/301481 | # Effects of high-order correlations on personalized recommendations for bipartite networks
• Liu, Jian-Guo Research Center of Complex Systems Science, University of Shanghai for Science and Technology, China - Department of Physics, University of Fribourg, Switzerland - Department of Modern Physics, University of Science and Technology of China, Hefei, China
• Zhou, Tao Research Center of Complex Systems Science, University of Shanghai for Science and Technology, China - Department of Physics, University of Fribourg, Switzerland - Department of Modern Physics, University of Science and Technology of China, Hefei, China
• Che, Hong-An Research Center of Complex Systems Science, University of Shanghai for Science and Technology, China
31.10.2009
Published in:
• Physica A. - 2010, vol. 389, no. 4, p. 881-886
##### Collaborative filtering
English In this paper, we introduce a modified collaborative filtering (MCF) algorithm, which has remarkably higher accuracy than the standard collaborative filtering. In the MCF, instead of the cosine similarity index, the user–user correlations are obtained by a diffusion process. Furthermore, by considering the second-order correlations, we design an effective algorithm that depresses the influence of mainstream preferences. Simulation results show that the algorithmic accuracy, measured by the average ranking score, is further improved by 20.45% and 33.25% in the optimal cases of MovieLens and Netflix data. More importantly, the optimal value $\lambda\textsubscript{opt}$ depends approximately monotonously on the sparsity of the training set. Given a real system, we could estimate the optimal parameter according to the data sparsity, which makes this algorithm easy to be applied. In addition, two significant criteria of algorithmic performance, diversity and popularity, are also taken into account. Numerical results show that as the sparsity increases, the algorithm considering the second-order correlation can outperform the MCF simultaneously in all three criteria.
Faculty
Faculté des sciences
Department
Physique
Language
• English
Classification
Physics | 2022-12-07 19:03:11 | {"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": 0.43594890832901, "perplexity": 798.3493854556821}, "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/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00392.warc.gz"} |
http://tex.stackexchange.com/questions/35909/pgfplot-accuracy-of-tick-labels | # pgfplot: accuracy of tick labels
the y-axis of my diagram uses small numbers like 0,005. I don't want to use a multiplier like 10^-3 which appears by default. If I remove it with
scaled x ticks = false, x tick label style={/pgf/number format/fixed},
the tick labels are not accurate enough and show ony 0 or 0,1:
Code used:
\begin{tikzpicture}
\begin{axis}[%
scaled x ticks = false,
x tick label style={/pgf/number format/fixed},
scaled y ticks = false,
scale only axis,
width=6cm,
height=5cm,
xmin=0, xmax=30000,
ymin=-0.001, ymax=0.006,
ymajorgrids,
axis on top]
How can i get this right?
-
By indenting code with four spaces you get a code block, as I did in my edit. When pasting in code, you can select the entire block and hit Ctrl + K (or click the {} button above the text field) to get everything indented. Also, it's always best to provide a complete, compilable, yet minimal example, a minimal working example (MWE). – Torbjørn T. Nov 23 '11 at 15:23
yticklabel style={/pgf/number format/fixed,
/pgf/number format/precision=3}
as an option to the axis.
Left, with /pgf/number format/precision=3, right without. Complete code:
\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[%
scaled x ticks = false,
x tick label style={/pgf/number format/fixed},
scaled y ticks = false,
yticklabel style={/pgf/number format/fixed,
/pgf/number format/precision=3},
scale only axis,
width=6cm,
height=5cm,
ymajorgrids,
axis on top]
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[%
scaled x ticks = false,
x tick label style={/pgf/number format/fixed},
scaled y ticks = false,
yticklabel style={/pgf/number format/fixed},
scale only axis,
width=6cm,
height=5cm,
ymajorgrids,
axis on top] | 2015-03-01 19:49:19 | {"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": 0.5265507698059082, "perplexity": 10222.996817627425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462548.53/warc/CC-MAIN-20150226074102-00170-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://math.libretexts.org/TextMaps/Calculus_Textmaps/Map%3A_Calculus_(OpenStax)/Calculus_I/Chapter_2%3A_Limits/2.1%3A_A_Preview_of_Calculus |
# 2.1: A Preview of Calculus
As we embark on our study of calculus, we shall see how its development arose from common solutions to practical problems in areas such as engineering physics—like the space travel problem posed in the chapter opener. Two key problems led to the initial formulation of calculus: (1) the tangent problem, or how to determine the slope of a line tangent to a curve at a point; and (2) the area problem, or how to determine the area under a curve.
## Feature
As we embark on our study of calculus, we shall see how its development arose from common solutions to practical problems in areas such as engineering physics—like the space travel problem posed in the chapter opener. Two key problems led to the initial formulation of calculus: (1) the tangent problem, or how to determine the slope of a line tangent to a curve at a point; and (2) the area problem, or how to determine the area under a curve.
## The Tangent Problem and Differential Calculus
Rate of change is one of the most critical concepts in calculus. We begin our investigation of rates of change by looking at the graphs of the three lines $$f(x)=−2x−3,g(x)=\frac{1}{2}x+1$$, and $$h(x)=2$$, shown in Figure.
The rate of change of a linear function is constant in each of these three graphs, with the constant determined by the slope.
As we move from left to right along the graph of $$f(x)=−2x−3$$, we see that the graph decreases at a constant rate. For every 1 unit we move to the right along the x-axis, the y-coordinate decreases by 2 units. This rate of change is determined by the slope (−2) of the line. Similarly, the slope of 1/2 in the function $$g(x)$$ tells us that for every change in x of 1 unit there is a corresponding change in y of 1/2 unit. The function $$h(x)=2$$ has a slope of zero, indicating that the values of the function remain constant. We see that the slope of each linear function indicates the rate of change of the function.
Compare the graphs of these three functions with the graph of $$k(x)=x^2$$ (Figure). The graph of $$k(x)=x^2$$ starts from the left by decreasing rapidly, then begins to decrease more slowly and level off, and then finally begins to increase—slowly at first, followed by an increasing rate of increase as it moves toward the right. Unlike a linear function, no single number represents the rate of change for this function. We quite naturally ask: How do we measure the rate of change of a nonlinear function?
The function $$k(x)=x^2$$ does not have a constant rate of change.
We can approximate the rate of change of a function $$f(x)$$ at a point $$(a,f(a))$$ on its graph by taking another point $$(x,f(x))$$ on the graph of $$f(x)$$, drawing a line through the two points, and calculating the slope of the resulting line. Such a line is called a secant line. Figure shows a secant line to a function $$f(x)$$ at a point $$(a,f(a))$$.
The slope of a secant line through a point $$(a,f(a))$$ estimates the rate of change of the function at the point $$(a,f(a))$$.
We formally define a secant line as follows:
### Definition:
The secant to the function $$f(x)$$ through the points $$(a,f(a)$$ and $$(x,f(x))$$ is the line passing through these points. Its slope is given by
$$m_{sec}=\frac{f(x)−f(a)}{x−a}.$$
The accuracy of approximating the rate of change of the function with a secant line depends on how close $$x$$ is to a. As we see in Figure, if $$x$$ is closer to a, the slope of the secant line is a better measure of the rate of change of $$f(x)$$ at a.
As x gets closer to a, the slope of the secant line becomes a better approximation to the rate of change of the function $$f(x)$$ at a.
The secant lines themselves approach a line that is called the tangent to the function $$f(x)$$ at a (Figure). The slope of the tangent line to the graph at a measures the rate of change of the function at a. This value also represents the derivative of the function $$f(x)$$ at a, or the rate of change of the function at a. This derivative is denoted by $$f′(a)$$. Differential calculus is the field of calculus concerned with the study of derivatives and their applications.
For an interactive demonstration of the slope of a secant line that you can manipulate yourself, visit this applet (Note: this site requires a Java browser plugin):
Solving the Tangent Problem: As x approaches a, the secant lines approach the tangent line.
Example illustrates how to find slopes of secant lines. These slopes estimate the slope of the tangent line or, equivalently, the rate of change of the function at the point at which the slopes are calculated.
#### Example Exercise:
1) Finding Slopes of Secant Lines
Estimate the slope of the tangent line (rate of change) to $$f(x)=x^2$$ at $$x=1$$ by finding slopes of secant lines through $$(1,1)$$ and each of the following points on the graph of $$f(x)=x^2$$.
a. $$(2,4)$$
b. $$(\frac{3}{2},\frac{9}{4})$$
Solution:
Use the formula for the slope of a secant line from the definition.
a. $$m_{sec}=\frac{4−1}{2−1}=3$$
b. $$m_{sec}=\frac{\frac{9}{4}−1}{\frac{3}{2}−1}=\frac{5}{2}=2.5$$
The point in part b. is closer to the point $$(1,1)$$, so the slope of 2.5 is closer to the slope of the tangent line. A good estimate for the slope of the tangent would be in the range of 2 to 2.5 (Figure).
The secant lines to $$f(x)=x^2$$ at $$(1,1)$$ through (a) $$(2,4)$$ and (b) $$(\frac{3}{2},\frac{9}{4})$$ provide successively closer approximations to the tangent line to $$f(x)=x^2$$ at $$(1,1)$$.
2) Estimate the slope of the tangent line (rate of change) to $$f(x)=x^2$$ at $$x=1$$ by finding slopes of secant lines through $$(1,1)$$ and the point $$(\frac{5}{4},\frac{25}{16})$$ on the graph of $$f(x)=x^2$$.
Solution: 2.25
We continue our investigation by exploring a related question. Keeping in mind that velocity may be thought of as the rate of change of position, suppose that we have a function, $$s(t)$$, that gives the position of an object along a coordinate axis at any given time t. Can we use these same ideas to create a reasonable definition of the instantaneous velocity at a given time $$t=a?$$ We start by approximating the instantaneous velocity with an average velocity. First, recall that the speed of an object traveling at a constant rate is the ratio of the distance traveled to the length of time it has traveled. We define the average velocity of an object over a time period to be the change in its position divided by the length of the time period.
### Definition:
Let $$s(t)\ be the position of an object moving along a coordinate axis at time t. The average velocity of the object over a time interval \([a,t]$$ where $$a<t$$ (or $$[t,a]$$ if $$t<a)$$ is
$$v_{ave}=\frac{s(t)−s(a)}{t−a}$$.
As t is chosen closer to a, the average velocity becomes closer to the instantaneous velocity. Note that finding the average velocity of a position function over a time interval is essentially the same as finding the slope of a secant line to a function. Furthermore, to find the slope of a tangent line at a point a, we let the x-values approach a in the slope of the secant line. Similarly, to find the instantaneous velocity at time a, we let the t-values approach a in the average velocity. This process of letting x or t approach a in an expression is called taking a limit. Thus, we may define the instantaneous velocity as follows.
### Definition:
For a position function $$s(t)$$, the instantaneous velocity at a time $$t=a$$ is the value that the average velocities approach on intervals of the form $$[a,t]$$ and $$[t,a]$$ as the values of t become closer to a, provided such a value exists.
Example illustrates this concept of limits and average velocity.
#### Example Exercise:
1) Finding Average Velocity
A rock is dropped from a height of 64 ft. It is determined that its height (in feet) above ground t seconds later (for $$0≤t≤2$$) is given by $$s(t)=−16t^2+64$$. Find the average velocity of the rock over each of the given time intervals. Use this information to guess the instantaneous velocity of the rock at time $$t=0.5$$.
a. [$$0.49,0.5$$]
b. [$$0.5,0.51$$]
Solution:
Substitute the data into the formula for the definition of average velocity.
a. $$v_{ave}=\frac{s(0.49)−s(0.5)}{0.49−0.5}=−15.84$$
b. $$v_{ave}=\frac{s(0.51)−s(0.5)}{0.51−0.5}=−16.016$$
The instantaneous velocity is somewhere between −15.84 and −16.16 ft/sec. A good guess might be −16 ft/sec.
2) An object moves along a coordinate axis so that its position at time t is given by $$s(t)=t^3$$. Estimate its instantaneous velocity at time $$t=2$$ by computing its average velocity over the time interval [$$2,2.001$$].
Hint: Use $$v_{ave}=\frac{s(2.001)−s(2)}{2.001−2}$$.
Solution:12.006001
## The Area Problem and Integral Calculus:
We now turn our attention to a classic question from calculus. Many quantities in physics—for example, quantities of work—may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the x-axis over an interval (Figure)?
The Area Problem: How do we find the area of the shaded region?
As in the answer to our previous questions on velocity, we first try to approximate the solution. We approximate the area by dividing up the interval $$[a,b]$$ into smaller intervals in the shape of rectangles. The approximation of the area comes from adding up the areas of these rectangles (Figure).
The area of the region under the curve is approximated by summing the areas of thin rectangles.
As the widths of the rectangles become smaller (approach zero), the sums of the areas of the rectangles approach the area between the graph of $$f(x)$$ and the x-axis over the interval $$[a,b]$$. Once again, we find ourselves taking a limit. Limits of this type serve as a basis for the definition of the definite integral. Integral calculus is the study of integrals and their applications.
#### Example Exercise:
1) Estimation Using Rectangles
Estimate the area between the x-axis and the graph of $$f(x)=x^2+1$$ over the interval $$[0,3]$$ by using the three rectangles shown in Figure.
The area of the region under the curve of $$f(x)=x^2+1$$ can be estimated using rectangles.
Solution: The areas of the three rectangles are 1 unit2, 2 unit2, and 5 unit2. Using these rectangles, our area estimate is 8 unit2.
2) Estimate the area between the x-axis and the graph of $$f(x)=x^2+1$$ over the interval $$[0,3]$$ by using the three rectangles shown here:
Solution: 16 $$unit^2$$
## Other Aspects of Calculus:
So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in Figure. These are only a few of the types of questions that can be asked and answered using multivariable calculus. Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit.
We can use multivariable calculus to find the volume between a surface defined by a function of two variables and a plane.
## Key Concepts:
• Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit.
• Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
• Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.
## Key Equations:
• Slope of a Secant Line
$$m_{sec}=\frac{f(x)−f(a)}{x−a}$$
• Average Velocity over Interval [a,t]
$$v_{ave}=\frac{s(t)−s(a)}{t−a}$$
For the following exercises, points $$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1.$$
-------------------------------------------------------------------------------------------------
## Exercise:
1) [T] Complete the following table with the appropriate values: y-coordinate of Q, the point $$Q(x,y)$$, and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
$$x$$ $$y$$ $$Q(x,y)$$ $$m_{sex}$$ 1.1 a. e. i. 1.01 b. f. j. 1.001 c. g. k. 1.0001 d. h. l.
Solution: a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000
2) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at $$x=1$$.
3) Use the value in the preceding exercise to find the equation of the tangent line at point P. Graph $$f(x)$$ and the tangent line.
Solution: $$y=2x$$
For the following exercises, points $$P(1,1)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^3$$.
1) [T] Complete the following table with the appropriate values: y-coordinate of Q, the point $$Q(x,y)$$, and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
$$x$$ $$y$$ $$Q(x,y)$$ $$m_{sec}$$ 1.1 a. e. i. 1.01 b. f. j. 1.001 c. g. k. 1.0001 d. h. l.2
2) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at $$x=1$$.
Solution: 3
3) Use the value in the preceding exercise to find the equation of the tangent line at point P. Graph $$f(x)$$ and the tangent line.
For the following exercises, points $$P(4,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=\sqrt{x}$$.
1) [T] Complete the following table with the appropriate values: y-coordinate of Q, the point $$Q(x,y)$$, and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
$$x$$ $$y$$ $$Q(x,y)$$ $$m_{sec}$$ 4.1 a. e. i. 4.01 b. f. j. 4.001 c. g. k. 4.0001 d. h. l.
Solution: a. 2.0248457; b. 2.0024984; c. 2.0002500; d. 2.0000250; e. (4.1000000,2.0248457); f. (4.0100000,2.0024984); g. (4.0010000,2.0002500); h. (4.00010000,2.0000250); i. 0.24845673; j. 0.24984395; k. 0.24998438; l. 0.24999844
2) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at $$x=4$$.
3) Use the value in the preceding exercise to find the equation of the tangent line at point P.
Solution: $$y=\frac{x}{4}+1$$
For the following exercises, points $$P(1.5,0)$$ and $$Q(ϕ,y)$$ are on the graph of the function \9f(ϕ)=cos(πϕ)\).
1) [T] Complete the following table with the appropriate values: y-coordinate of Q, the point $$Q(x,y)$$, and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
$$x$$ $$y$$ $$Q(x,y)$$ $$m_{sec}$$ 1.4 a. e. i. 1.49 b. f. j. 1.499 c. g. k. 1.4999 d. h. l.
2) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to f at $$x=4$$.
Solution: $$π$$
3) Use the value in the preceding exercise to find the equation of the tangent line at point P.
For the following exercises, points $$P(−1,−1)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=\frac{1}{x}$$.
[T] Complete the following table with the appropriate values: y-coordinate of Q, the point $$Q(x,y)$$, and the slope of the secant line passing through points P and Q. Round your answer to eight significant digits.
$$x$$ $$y$$ $$Q(x,y)$$ $$m_{sec}$$ -1.05 a. e. i. -1.01 b. f. j. -1.005 c. g. k. -1.001 d. h. l.
Solution: a. −0.95238095; b. −0.99009901; c. −0.99502488; d. −0.99900100; e. (−1;.0500000,−0;.95238095); f. (−1;.0100000,−0;.9909901); g. (−1;.0050000,−0;.99502488); h. (1.0010000,−0;.99900100); i. −0.95238095; j. −0.99009901; k. −0.99502488; l. −0.99900100
2) Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to f at $$x=−1$$.
3) Use the value in the preceding exercise to find the equation of the tangent line at point P.
Solution: $$y=−x−2$$
For the following exercises, the position function of a ball dropped from the top of a 200-meter tall building is given by $$s(t)=200−4.9t^2$$, where position s is measured in meters and time t is measured in seconds. Round your answer to eight significant digits.
1) [T] Compute the average velocity of the ball over the given time intervals.
a. [4.99,5]
b. [5,5.01]
c. [4.999,5]
d. [5,5.001]
2) Use the preceding exercise to guess the instantaneous velocity of the ball at $$t=5$$ sec.
Solution: −49 m/sec (velocity of the ball is 49 m/sec downward)
For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15 m/sec. Its height in meters at time t seconds is $$h(t)=15t−4.9t^2$$.
1) [T] Compute the average velocity of the stone over the given time intervals.
a. [1,1.05]
b. [1,1.01]
c. [1,1.005]
d. [1,1.001]
2) Use the preceding exercise to guess the instantaneous velocity of the stone at $$t=1$$ sec.
Solution: 5.2m/sec
For the following exercises, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $$h(t)=600+78.4t−4.9t^2$$, where t is measured in seconds.
1) [T] Compute the average velocity of the rocket over the given time intervals.
a. [9,9.01]
b. [8.99,9]
c. [9,9.001]
d. [8.999,9]
2) Use the preceding exercise to guess the instantaneous velocity of the rocket at $$t=9$$ sec.
Solution: -9.8m/sec
For the following exercises, consider an athlete running a 40-m dash. The position of the athlete is given by $$d(t)=\frac{t^3}{6}+4t$$, where d is the position in meters and t is the time elapsed, measured in seconds.
1) [T] Compute the average velocity of the runner over the given time intervals.
a. [1.95,2.05]
b. [1.995,2.005]
c. [1.9995,2.0005]
d. [2,2.00001]
2) Use the preceding exercise to guess the instantaneous velocity of the runner at $$t=2$$ sec.
Solution:6 m/sec
For the following exercises, consider the function$$f(x)=|x|$$.
1) Sketch the graph of f over the interval [$$−1,2$$] and shade the region above the x-axis.
2) Use the preceding exercise to find the exact value of the area between the x-axis and the graph of f over the interval [$$−1,2$$] using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer.
Solution: Under, 1 $$unit^2$$; over: 4 $$unit^2$$. The exact area of the two triangles is $$\frac{1}{2}(1)(1)+\frac{1}{2}(2)(2)=2.5 units^2$$.
For the following exercises, consider the function $$f(x)=\sqrt{1−x^2}$$. (Hint: This is the upper half of a circle of radius 1 positioned at ($$0,0$$).)
1) Sketch the graph of f over the interval [$$−1,1$$].
2) Use the preceding exercise to find the exact area between the x-axis and the graph of f over the interval [$$−1,1$$] using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer.
Solution: Under, 0.96 $$unit^2$$; over, 1.92 $$unit^2$$. The exact area of the semicircle with radius 1 is $$\frac{π(1)^2}{2}=\frac{π}{2} unit^2$$
For the following exercises, consider the function $$f(x)=−x^2+1$$.
1) Sketch the graph of f over the interval [$$−1,1$$].
2) Approximate the area of the region between the x-axis and the graph of f over the interval [$$−1,1$$].
Solution: Approximately 1.3333333 $$unit^2$$
## Glossary
average velocity
the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval [$$t,a$$] (if $$t<a$$ or [$$a,t$$] if $$t>a$$), with a position given by $$s(t)$$, that is $$v_{ave}=\frac{s(t)−s(a)}{t−a}$$
differential calculus
the field of calculus concerned with the study of derivatives and their applications
instantaneous velocity
The instantaneous velocity of an object with a position function that is given by $$s(t)$$ is the value that the average velocities on intervals of the form [$$t,a$$] and [$$a,t$$] approach as the values of t move closer to $$a$$, provided such a value exists
integral calculus
the study of integrals and their applications
limit
the process of letting x or t approach a in an expression; the limit of a function $$f(x)$$ as x approaches a is the value that $$f(x)$$ approaches as x approaches a
multivariable calculus
the study of the calculus of functions of two or more variables
secant
A secant line to a function $$f(x)$$ at a is a line through the point ($$a,f(a)$$) and another point on the function; the slope of the secant line is given by $$m_{sec}=\frac{f(x)−f(a)}{x−a}$$
tangent
A tangent line to the graph of a function at a point ($$a,f(a)$$) is the line that secant lines through ($$a,f(a)$$) approach as they are taken through points on the function with x-values that approach a; the slope of the tangent line to a graph at a measures the rate of change of the function at a | 2017-01-16 19:20:53 | {"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": 0.7962924838066101, "perplexity": 219.3547839435814}, "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-2017-04/segments/1484560279248.16/warc/CC-MAIN-20170116095119-00248-ip-10-171-10-70.ec2.internal.warc.gz"} |
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Title: Combinatorial principles in second-order theories of bounded arithmetic Author(s): De Castro, Rodrigo Doctoral Committee Chair(s): Jockusch, Carl G., Jr. Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: An attempt is made to study the mathematical strength of the weak second order theories of Bounded Arithmetic U$\sbsp{2}{i}$ and V$\sbsp{2}{i}$, i $\geq$ 0, introduced by S. Buss. It is first shown that U$\sbsp{2}{1}$ can $\Sigma\sbsp{1}{1,b}$-define the functions in the second class of Grzegorczyk, $\varepsilon\sp2$, or, equivalently, the class of $\Sigma\sbsp{1}{1,b}$-definable functions in U$\sbsp{2}{1}$ is closed under bounded recursion.It is shown next that U$\sbsp{2}{1}$ proves the $\Delta\sbsp{1}{1,b}$-pigeonhole principle. Two general combinatorial principles, the $\Delta\sbsp{1}{1,b}$-partition principle and the $\Delta\sbsp{1}{1,b}$-equipartition principle, are obtained from it thereby demonstrating that the $\Delta\sbsp{1}{1,b}$-PHP embodies a strong notion of cardinality. The introduction of these principles is motivated with some examples, notably by showing that Euler's theorem is provable in U$\sbsp{2}{1}$. The provability of Euler's theorem in weak first order fragments of Peano Arithmetic is an open problem.A theory of polynomials is developed in U$\sbsp{2}{1}$ and it is proved that U$\sbsp{2}{1}$ + B $\vdash$ "existence of primitive roots" where formula B asserts that a nontrivial polynomial of degree n can have at most n solutions modulo p if p is a prime. By an essential use of the $\Delta\sbsp{1}{1,b}$-PHP and the $\Delta\sbsp{1}{1,b}$-partition principle it is shown that V$\sbsp{2}{1}\/\vdash$ B, hence V$\sbsp{2}{1}\/\vdash$ "existence of primitive roots". Issue Date: 1992 Type: Text Language: English URI: http://hdl.handle.net/2142/20320 Rights Information: Copyright 1992 De Castro, Rodrigo Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9236438 OCLC Identifier: (UMI)AAI9236438
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