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https://math.stackexchange.com/questions/2560990/do-non-convex-platonic-solids-exist | # Do non-convex platonic solids exist?
Consider a solid with the following properties -
1. It is composed of congruent, regular polygons.
2. At each vertex, the same number of edges and faces meet.
This is the same as the requirement for the Platonic solids, but the solid need not be convex. Of course, the five Platonic solids will satisfy these conditions, but are there any others?
EDIT: consider the topological proof given in the Wikipedia article on Platonic solids - https://en.wikipedia.org/wiki/Platonic_solid
If we require the Euler characteristic to be 1 instead of 2 (as in the proof) we get -
$$\frac{1}{p} + \frac{1}{q} = \frac{1}{2} + \frac{1}{2E}$$
This still keeps open the possibility (using the same argument as for the Platonic solids given in that proof) of five such solids with Euler characteristic 1 (so they won't be convex). Question is, do these solids exist?
• If you drop the regularity requirement, you get Kepler-Poinsot polyhedra. – Noah Schweber Dec 11 '17 at 2:49
• @NoahSchweber - these are interesting, but I don't think they satisfy the requirement that each vertex should have the same number of faces and edges intersecting at it. – Rohit Pandey Dec 11 '17 at 2:53
• They do, actually… the faces or vertex arrangements are regular star polyhedra. – Parcly Taxel Dec 11 '17 at 2:55
• @ParclyTaxel can you provide an example of such a polyhedron? – Rohit Pandey Dec 11 '17 at 3:01
• The Kepler-Poinsot polyhedra do satisfy the requirements because the non-convex "vertices" are not considered vertices of the polyhedra, they merely happen to be points where three faces intersect. For example, the great dodecahedron is considered to have only 12 vertices and 12 pentagonal faces. However, since the faces pass through each other, I suppose you cannot really call it a "solid". It does not have a clearly defined interior, it is merely an arrangement of 12 pentagons with nice regularity properties. – Rahul Dec 11 '17 at 3:16 | 2020-01-19 00:04:12 | {"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": 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": 0.7639356851577759, "perplexity": 485.4379795240115}, "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-2020-05/segments/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00062.warc.gz"} |
https://solvedlib.com/n/which-of-the-following-best-shows-how-the-results-ono-way,6382721 | # Which of the following; best shows how the results ono-Way Anova would be correctly reported in & scientific journal?NOTE: In
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https://projecteuclid.org/euclid.pm/1498701621 | Publicacions Matemàtiques
Sur la Séparation des Caractères par les Frobenius
Abstract
In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field $K$, by the Frobenius of a prime ideal ${\mathfrak p}$ of $\mathcal{O}_K$. We first recall an upper bound for the norm ${\mathrm N}({\mathfrak p})$ of the smallest such prime ${\mathfrak p}$, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number $p$ for which $P$ $(\operatorname{mod} p)$ has a certain type of factorization in ${\mathbb F}_p[X]$, where $P\in {\mathbb Z}[X]$ is a monic, irreducible polynomial of square-free discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group $A_n$ over a fixed number field $K$. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two irreducible unramified characters of the alterning group $A_n$, for $n=5,7,13$.
Article information
Source
Publ. Mat., Volume 61, Number 2 (2017), 475-515.
Dates
Revised: 10 October 2016
First available in Project Euclid: 29 June 2017
https://projecteuclid.org/euclid.pm/1498701621
Digital Object Identifier
doi:10.5565/PUBLMAT6121706
Mathematical Reviews number (MathSciNet)
MR3677869
Zentralblatt MATH identifier
06781949
Citation
Euvrard, Charlotte; Maire, Christian. Sur la Séparation des Caractères par les Frobenius. Publ. Mat. 61 (2017), no. 2, 475--515. doi:10.5565/PUBLMAT6121706. https://projecteuclid.org/euclid.pm/1498701621 | 2019-10-13 22:21:18 | {"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": 0.7501677870750427, "perplexity": 470.12195864079524}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986648343.8/warc/CC-MAIN-20191013221144-20191014004144-00275.warc.gz"} |
https://stats.stackexchange.com/questions/486504/beta-distribution-with-support-from-0-to-2 | # Beta distribution with support from 0 to 2
Is there a way to make the Beta distribution have support from 0 to 2, instead of 0 to 1?
• $$C'(x,\alpha, \beta) = C(x/2, \alpha, \beta)$$
• $$P'(x,\alpha,\beta) = P(x/2,\alpha,\beta)/2$$
where $$C$$ and $$P$$ are the original Beta CDF and PDF and $$C'$$ and $$P'$$ are the new ones (and $$\alpha$$ and $$\beta$$ are the two shape parameters). | 2022-06-30 17:48:17 | {"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8546598553657532, "perplexity": 172.9248804004502}, "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-2022-27/segments/1656103850139.45/warc/CC-MAIN-20220630153307-20220630183307-00637.warc.gz"} |
http://mathhelpforum.com/algebra/97052-solved-could-someone-tell-me-if-my-answers-correct-please.html | # Math Help - [SOLVED] Could someone tell me if my answers are correct please?
1. ## [SOLVED] Could someone tell me if my answers are correct please?
Find the slope and y-intercept of the following.
y=3x-5
slope=3, y-intercept=-5
----------
Write an equation for the line with slope 1/3 and y-intercept 4.
y=1/3x+4
Write an equation for the line with slope 3 going through point (1,-2).
y=3x+(-2)
I also need help getting the formula to solve a question:
(I have already arrived at an answer, but I used logic and not a formula to get to it)
One eighth of the square root of 7 less than a number is 2 what is the number?
263
I got to that answer by saying that if 2 was one eighth the square root of the number, minus seven, then 16=the square root and since 16^2=256 then the number must be 256+7=263
Thank you again for your help!
Drakmord
2. Originally Posted by drakmord
Write an equation for the line with slope 3 going through point (1,-2).
y=3x+(-2)
You can use the formula $y-y_1=m(x-x_2)$
So you have:
$y-(-2)=3(x-1)$
$y+2=3x-3$
$y=3x-5$
3. Originally Posted by drakmord
I also need help getting the formula to solve a question:
(I have already arrived at an answer, but I used logic and not a formula to get to it)
One eighth of the square root of 7 less than a number is 2 what is the number?
$\frac{1}{8}\times\sqrt{x-7}=2$
$\sqrt{x-7}=16$
$x-7=256$
$x=263$ | 2015-09-02 17:34:11 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8827617764472961, "perplexity": 504.4223501996987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645281115.59/warc/CC-MAIN-20150827031441-00132-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/523469/georgi-decomposition-of-representations-into-subgroups | # Georgi - decomposition of representations into subgroups
I have long been unable to follow section 12.3 of Georgi - Lie algebras in particle physics. This section deals with how irreps of $$SU(3)$$ decompose as irreps of subgroups $$H \subset SU(3)$$ and is later generalised to $$SU(N)$$ in section 13.5. Although I understand the concept and have seen other treatments that I follow, I would like some help understanding Georgi's treatment.
First Georgi says (as far as I can see without justification) that the fundamental, $$\mathbf{3}$$, of $$SU(3)$$ decomposes as an $$SU(2)\times U(1)$$ doublet with hypercharge $$1/3$$ and a singlet of hypercharge $$-2/3$$. Is it clear why? How does this generalise for the $$\mathbf{N}$$ decomposing in arbitrary subgroups $$H \subset SU(N)$$?
Following this Georgi considers an arbitrary Young Tableau (i.e. irrep) of $$SU(3)$$ with $$n$$ boxes; I believe with arbitrary symmetry of the indices. He assumes that $$j$$ indices of the tensor transform as $$SU(2)$$ doublets and (n-j) as singlets -- but why is this the only possibility? Is it because any irrep of $$SU(2)$$ can be formed from tensor products of the doublet?
We go on to represent the $$n-j$$ singlets by a Young Tableau of $$n-j$$ boxes in a row: does Georgi mean $$SU(3)$$ Tableaux or $$SU(2)$$ Tableaux? From figure 12.6 it seems the are $$SU(3)$$ Tableaux but then why must they be rows?
For the actual algorithm Georgi says, without proof,
To determine whether a given $$SU(2)$$ rep, $$\alpha$$, appears in the decomposition we take the tensor product of $$\alpha$$ with the $$n-j$$ boxes.
I need some help with this. Firstly does this mean writing the $$SU(2)$$ rep as a Young Tableau and taking the $$SU(3)$$ tensor product with $$n-j$$ boxes in a row? And what value of $$j$$ do we choose?
Now in the examples (12.6 onwards) I don't understand the notation. In (12.6) we are looking for how the $$6$$ (two boxes in a row) decomposes. What is the notation below? Is it $$\left( SU(2) \textrm{ irrep } \, \, \, SU(3) \textrm{ irrep } \right)$$ where the $$SU(3)$$ irrep is row of some number $$n-j$$ of boxes for different $$j$$? In that case how was $$n$$ chosen?
OP asks many questions, so we will be somewhat sketchy.
1. Actually the Lie group $$G~:=~SU(2) \times U(1)$$ is not a subgroup of $$SU(3)$$ but $$G/\mathbb{Z}_2$$ is. This is because the group homomorphism $$G~\ni~ (g,\alpha)~~\stackrel{\Phi}{\mapsto}~~ \begin{pmatrix} \alpha g & \mathbb{0}_{2\times 1} \cr \mathbb{0}_{1\times 2} & \alpha^{-2}\end{pmatrix}_{3\times 3}~\in~SU(3)$$ has kernel $${\rm Ker}(\Phi)~=~\{\pm (\mathbb{1}_{2\times 2},1)\}~\cong~\mathbb{Z}_2.$$
2. Here we will argue at the level of Lie algebras $$su(2) \oplus u(1)\subseteq su(3).$$ In detail, we identify $$su(3)~\cong~ {\rm span}_{\mathbb{R}}(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6,\lambda_7,\lambda_8)$$ with Hermitian traceless $$3\times 3$$ matrices; the isospin $$su(2)~\cong~ {\rm span}_{\mathbb{R}}(\lambda_1,\lambda_2,\lambda_3)$$ with Hermitian traceless $$2\times 2$$ block matrices in rows/columns 1,2; while the hypercharge $$u(1)~\cong~ {\rm span}_{\mathbb{R}}(\lambda_8)$$ is generated by the diagonal traceless matrix $${\rm diag}(1,1,-2)$$ times an real number. (Here $$\lambda_a$$ denotes Gell-mann matrices). In other words, up to normalization of the hyperchange, the fundamental representation decomposes as $${\bf 3}~\to~{\bf 2}_{1/3}\oplus {\bf 1}_{-2/3}.$$
3. Concerning OP's questions on arbitrary $$su(3)$$ representations, for intuition purposes, the tensor picture is perhaps helpful: Each tensor index (corresponding to a box) takes values $$1,2,3$$. The index values $$1,2$$ correspond to an $$su(2)$$ dublet, while the index value $$3$$ corresponds to an $$su(2)$$ singlet. This leads to a distributive property, where a box (i.e. a $$su(3)$$ triplet) can turn into an $$su(2)$$ dublet or an $$su(2)$$ singlet. This is what Georgi tries to indicate in eqs. (12.16-19). As OP already anticipates, the $$j$$th $$su(2)$$ irrep can be realized as a symmetric tensor product $$({\bf 2j+1})\cong {\bf 2}^{\odot 2j}$$ of the $$su(2)$$ dublet $${\bf 2}$$ alone.
4. Concerning the generalization to $$SU(n) \times SU(m) \times U(1)$$ in Georgi's section 13.5, see e.g. my Phys.SE answer here.
• given the OP asks about the more general case, it would be nice to expand a bit on the cosetting by $\mathbb{Z}_2$ given the literature (see for instance Hagen, C. R., and A. J. Macfarlane. "Reduction of representations of SUm+ n with respect to the subgroup SUm⊗ SUn." Journal of Mathematical Physics 6.9 (1965): 1366-1371. ) tends to omit this subtle point. – ZeroTheHero Jan 7 at 16:49
• I updated the answer. – Qmechanic Jan 8 at 16:00
• Very helpful - what about the quoted part from georgi... The algorithm is given without proof so even though now I understand it I'd like to know how the combinatorics works out correctly – lux Jan 11 at 23:16 | 2020-05-30 05:08:22 | {"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": 63, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8950660824775696, "perplexity": 482.3314925662227}, "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/1590347407289.35/warc/CC-MAIN-20200530040743-20200530070743-00402.warc.gz"} |
https://economics.stackexchange.com/tags/preferences/new | # Tag Info
To follow up on the answer of @VARulle let me give you some conditions for which the indifference curve is path connected. The argument can also be found in the book Mathematical Methods and Models for Economists by Angel de la Fuente. Preferences are monotone if $x > y$ implies $x \succ y$ and that preferences are continuous if $x_n \succeq y_n$, $x_n \... 7 Given your last comment above it seems that what you are really asking is whether the indifference sets of a continuous preference relation on$\mathbb R^n_+$are path-connected. The answer is No. Let$n=1$and let the preference relation be represented by$u(x)=(1-x)^2$. Then the indifference set e.g. for$u=1$is$\{0\}\cup\{2\}\$, which is not path-... | 2021-07-25 07:27:09 | {"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": 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": 0.9494661092758179, "perplexity": 440.2795304575034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046151638.93/warc/CC-MAIN-20210725045638-20210725075638-00036.warc.gz"} |
https://blender.stackexchange.com/questions/31350/re-do-view-of-a-photo-via-dem-data-and-blender | # Re-do view of a photo via DEM data and Blender
I'm trying to re-render -
I hopped to generate some jewelery out of the rendered view. So I found some DEM-data (DEM=digital elevation model) of the region and added a camera and a plane to mirror the mountains there.
Here's the blend file of the scene I created so far
I just can't seem to recreate the view from the photo, or even something close to it. I first tried with just the plane with a displacement modifier on the UV-data and tried to move the camera in to the region, but this didn't came to fruition. After some other things I tried I can kind of recreate the view with the one side of the UV-map (of the plane) being scaled to zero - trying to mirror the view from the camera. (you also see this in the blender file, this is where the shot was approximately made)
I was wondering if someone did something similar to this and might help me.
The most straight forward thing I came up with was that I massively scaled the plane up and tried to physically move the camera in to the region, but this just didn't work. The blender cycles camera just skewed the view too much, the angles and everything seemed off...
edit: To be clear: my problem doesn't lie wit generating the mesh with displacement modifiers, subsurfing, etc. But more with how to then make the camera see more or less the same as in the photo
• Make sure the aspect ratio and lense simulation is the same as was present in your camera.
– VRM
May 23 '15 at 16:11
• duplicate?: blender.stackexchange.com/questions/27451/…
– user1853
May 23 '15 at 17:02
• Do you mean something like using the photo as an semi-transparent reference in your camera view? Jul 24 '15 at 8:48
• If you mean view from camera looking somehow like this, then you can accomplish that by using Lens Distortion node (of course, camera itself is tilted slightly) Aug 24 '15 at 22:07
Start with a plane. Subdivide it like 100 times. In Top ortho view unwrap it using project from view bounds.
to the plane add a displace modifier that uses your hieght map as texture. Make sure the texture coordinates are set to UV.
For a smoother surface add a subsurf modifier.
From there just set the camera where you want.
• My problem isn't really with these steps - I managed to do so too - that was what I meant with "first tried with just the plane with a displacement modifier on the UV-data and tried to move the camera in to the region" My problem is with the next step - how to get the camera to render the same view as on the image. - But thanks for trying to answer anyways. May 25 '15 at 7:52
If you're trying to do (render a scene that perfectly mimics the real-life photo) what I think you're trying to do, then there's a lot more involved than what we'd reasonably be able to write here. But there are a few steps that can get you closer.
• A good place to start is Blender's 2D Camera Tracking documentation. You'll want to look at the metadata for the photograph, or otherwise learn a bit about your camera. There are a lot of parameters that go into making a photo look the way it does.
• It looks like your camera used an 'HDR' setting to get the image's colors and contrast to look so stellar. You may have some difficulty making this happen in blender, but it can be done.
• As for the orientation and location of your camera, you're right to start by blowing the mesh up considerably (it will help make the light and shadows more realistic). Moreover, the world is really really big. It may take a lot of scaling before your geometry even starts to approach the correct scale. Whenever I'm having difficulty with scale (it's easy to get 'lost' in Blender-units) I add a roughly human-shaped object (normally just a tapered cylinder with an icosphere head) and put it in my scene. Then I scale objects until I could see them being the right height. (It's hard here because mountains are big... But it'll help you when you're orienting the camera. It'll probably go a lot closer to the mesh — that is, 'lower' to earth — than you're imagining.)
• Before you even begin with textures and colors, though, you need to get the geometry right. I'd suggest making this image your camera's background reference image and using this to establish how to position your camera to get the right image.
• Note that your heightmap may not be scaled properly in the z-direction: Once you get the camera in the right position, I often find that my mountains look a bit mole-hilly. sz to scale in the z-direction only; with a camera at 'eye level' this may give you a better-looking middle-distance.
• Use clues in the photo to orient your camera. Clearly the camera is pointed at a certain set of mountains from a certain direction, so start by putting your camera in the right geographical location. Point it in the right direction. Now look at your horizon in the photograph — it's definitely angled! Orient your camera so that the xy plane is at the same angle, and your image will start to come together.
My parting suggestion is to play around with the focal length: I find a much lower angle will make the environment appear larger (with the requirement that you move the camera 'back' to maintain the contents of the frame). Play around with that slider and see the results; I think that may be why you're not getting the right 'feel', even though the geometry is all there. | 2021-12-08 00:30:29 | {"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": 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": 0.27664628624916077, "perplexity": 884.7678140200005}, "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-2021-49/segments/1637964363420.81/warc/CC-MAIN-20211207232140-20211208022140-00308.warc.gz"} |
https://www.cs.sfu.ca/~ggbaker/zju/math/complexity.html | Complexity
• The whole point of the big-O/Ω/Θ stuff was to be able to say something useful about algorithms.
• So, let's return to some algorithms and see if we learned anything.
• Consider this simple procedure that sums a list (of numbers, we assume):
• First: is the algorithm correct? Does it solve the problem specified?
• Second: is it fast?
• To evaluate the running time of an algorithm, we will simply ask how many “steps” it takes.
• In this case, we can count the number of times it runs the += line.
• For a list with $$n$$ elements, it takes $$n$$ steps.
• Or is counting the += line the right thing to do?
• When implementing the for loop, each iteration requires an add (for the loop index) and a comparison (to check the exit condition). We should count those.
• Also, the variable initialization and return steps.
• So, $$3n+2$$ steps.
• But, not all of those steps are the same.
• How long does an x86 ADD instruction take compared to a CMP or RET instruction?
• Will the compiler keep both i and total in registers, or will one/both be in RAM? (A factor of ~10 difference.)
• How do those instructions interact in the pipeline? Which can be sent through parallel pipelines in the processor and executed concurrently?
• The answer to those is simple: I don't know and you don't either.
• That's part of the reason we're asking about algorithms, not programs.
• But both $$n$$ and $$3n+2$$ are perfectly reasonable proposals for the answer.
• Deciding between them requires more knowledge about the actual implementation details than we have.
• Good thing we have the function growth notation.
• Remember: this is easy for $$n=5$$ elements. A good or bad algorithm will both be fast then.
• We want to know how the algorithm behaves for large $$n$$.
• Finally our answer: the sum procedure has running time $$\Theta(n)$$.
• We'll say that this algorithm has time complexity $$\Theta(n)$$, or “runs in linear time”.
• Both $$n$$ and $$3n+2$$ are $$\Theta(n)$$, and so is any other “exact” formula we could come up with.
• The easy answer (count the += line) was just as correct as the very careful one.
• The big-Θ notation hides all of the details we can't figure out anyway.
• Another example: print out the sum of each two numbers in a list.
• That is, given the list [1,2,3,4,5], we want to find 1+2, 1+3, 1+4, 1+5, 2+3, 2+4,….
• Pseudocode:
• For a list with $$n$$ elements, the for j loop iterates $$n-1$$ times when it is called with i==0, then $$n-2$$ times, then $$n-3$$ times,…
• So, the total number of times the print step runs is \begin{align*} (n-1)+(n-2)+\cdots+2+1 &= \sum_{k=1}^{n-1} k\\ &= \frac{n(n-1)}{2}\\ &= \frac{n^2}{2}-\frac{n}{2}\,. \end{align*}
• If we had counted the initialization of the for loops, counter incrementing, etc, we might have come up with something more like $$\frac{3}{2}n^2 + \frac{1}{2}n + 1$$.
• Either way, the answer we give is that it takes $$\Theta(n^2)$$ steps.
• Or, the algorithm “has time complexity $$\Theta(n^2)$$” or “has $$\Theta(n^2)$$ running time” or “has quadratic running time”.
• The lesson: when counting running time, you can be a bit sloppy.
• We only need to worry about the inner-most loop(s), not the number of steps in there, or work in the outer levels.
• … because they are going to disappear anyway as constant factors and lower-order terms when they go into a big-O/Ω/Θ anyway.
Average and Worst Case
• Consider a linear search: we want to find an element in a list and return its (first) position, or -1 if it's not there.
• How many steps there?
• The answer is: it depends.
• If the thing we're looking for is in the first position, it takes $$\Theta(1)$$ steps.
• If it's at the end, or not there, it takes $$\Theta(n)$$ steps.
• The easiest thing to count is usually the worst case: what is the maximum steps required for any input of size $$n$$?
• The worst case is that we go all the way to the end of the list, but don't find it and return -1.
• The only line that makes sense to count here is the if line. It's in the inner-most loop, and is executed for every iteration.
• We could also count the number of comparisons made: the == and the implicit comparison in the for loop.
• That is either $$n$$ or $$2n+1$$ steps, so $$\Theta(n)$$ complexity.
• The other useful option is the average case: what is the average steps required over all inputs of size $$n$$?
• Much harder to calculate, since you need to consider every possible input to the algorithm.
• Even if we assume the element is found, the possible number of comparisons are:
• On average, the number of comparisons is: $\frac{2+4+\cdots+2n}{n} = n+1\,.$
• Again, we have $$\Theta(n)$$ complexity.
• … but it's a good thing we checked. Some algorithms are different.
• We have said that these running times are important when it comes to running times of algorithm.
• But we are throwing away a lot of information when we look only at big-O/Ω/Θ.
• The lower-order terms must mean something.
• The leading constants definitely do.
• Assuming one million operations per second, this is the approximate running time of an algorithm given running time, with an input of size $$n$$:
• Maybe that gives a little idea why we'll only worry about complexity
• … at least at first.
• A summary:
• If you can get $$O(\log n)$$ life is good: hand it in and go home.
• $$O(n\log n)$$ is pretty good: hard to complain about it.
• $$O(n^k)$$ could be bad, depending on $$k$$: you won't be solving huge problems. These are polynomial complexity algorithms for $$k\ge 1$$.
• $$\Omega(k^n)$$ is a disaster: almost as bad as no algorithm at all if you have double-digit input sizes. These are exponential complexity algorithms for $$k\gt 1$$.
• A problem that has a polynomial-time algorithm is called tractable.
• No polynomial time algorithm: intractable.
• There is a big category of problems that nobody has a polynomial-time algorithm for, but also can't prove that none exists: the NP-complete problems.
• Some examples: Boolean satisfiability, travelling salesman, Hamiltonian path, many scheduling problems, Sudoku (size $$n$$).
• If you have an algorithm with a higher complexity than necessary, no amount of clever programming will make up for it.
• No combination of these will make a $$O(n^2)$$ algorithm faster than an $$O(n\log n)$$: faster language, better optimizer, hand-optimization of code, faster processor.
• Important point: the complexity notations only say things about large $$n$$.
• If you always have small inputs, you might not care.
• Algorithms with higher complexity class might be faster in practice, if you always have small inputs.
• e.g. Insertion sort has running time $$\Theta(n^2)$$ but is generally faster than $$\Theta(n\log n)$$ sorting algorithms for lists of around 10 or fewer elements.
• The most important info that the complexity notations throw away is the leading constant.
• There is a difference between $$n^2$$ instructions and $$100n^2$$ instructions to solve a problem.
• Once you have the right big-O, then it's time to worry about the constants.
• That's what clever programming can do.
• When we're talking about algorithms (and not programming), the constants don't usually matter much.
• It's rare to have an algorithm with a big leading constant.
• So it's not really possible to decide between the algorithms.
• Usually it's a choice between $$4n\log n$$ or $$5n\log n$$: you probably have to implement, compile, and profile to decide for sure.
• Example: sorting algorithms. There are several algorithms to sort a list/array.
• Insertion/Selection/Bubble Sorts: $$\Theta(n^2)$$.
• Merge/Heap Sorts: $$\Theta(n\log n)$$.
• Quicksort: $$\Theta(n\log n)$$ average case but (very rarely) $$\Theta(n^2)$$ worst case.
• But quicksort is usually faster in practice.
• … except when it isn't.
• Several recent languages/libraries have implemented a heavily-optimized mergesort (e.g. Python, Perl, Java ≥JDK1.3, Haskell, some STL implementations) instead of Quicksort (C, Ruby, some other STL implementations).
Space Complexity
• We have only been talking about running time/speed so far.
• It also makes good sense to talk about the complexity of other things.
• Most notably, memory use by an algorithm.
• An algorithm that uses $$\Theta(n^{3})$$ space is bad. Maybe as bad as $$\Theta(n^{3})$$ time.
• An algorithm that uses $$O(1)$$ extra space (in addition to space needed to store the input) is called in-place.
• e.g. selection sort is in-place, but mergesort ($$\Theta(n)$$ extra space) and Quicksort ($$\Theta(\log n)$$ extra space, average case) aren't. | 2020-08-11 12:05:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7991116642951965, "perplexity": 1116.3977124226399}, "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-2020-34/segments/1596439738777.54/warc/CC-MAIN-20200811115957-20200811145957-00596.warc.gz"} |
https://codereview.stackexchange.com/questions/219722/leetcode-maximum-depth-of-n-ary-tree/220468#220468 | # LeetCode: maximum-depth-of-n-ary-tree
https://leetcode.com/problems/maximum-depth-of-n-ary-tree/
Given a n-ary tree, find its maximum depth.
The maximum depth is the number of nodes along the longest path from the root node down to the farthest leaf node.
For example, given a 3-ary tree:
We should return its max depth, which is 3.
Note:
The depth of the tree is at most 1000. The total number of nodes is at most 5000.
Here is my code, and also 1 test as an example Please comment about space and time complexity, I also made a recursive answer but I tend to think about BFS in an iterative way while using queue, and DFS in recursive way, although you can use a stack for it as well. so I a real interview I would probably go with BFS with a queue.
using System;
using Microsoft.VisualStudio.TestTools.UnitTesting;
using System.Collections.Generic;
namespace TreeQuestions
{
// Definition for a Node.
public class Node
{
public int val;
public IList<Node> children;
public Node()
{
}
public Node(int _val, IList<Node> _children)
{
val = _val;
children = _children;
}
}
/// <summary>
/// https://leetcode.com/problems/maximum-depth-of-n-ary-tree/
/// </summary>
[TestClass]
public class MaximumDepthOfN_aryTree
{
public int MaxDepth(Node root)
{
if (root == null)
{
return 0;
}
int maxDepth = 0;
Queue<Node> queue = new Queue<Node>();
queue.Enqueue(root);
while (queue.Count > 0)
{
maxDepth++;
var queueSize = queue.Count;
for (int i = 0; i < queueSize; i++)
{
var current = queue.Dequeue();
if (current.children != null)
{
foreach (var child in current.children)
{
queue.Enqueue(child);
}
}
}
}
return maxDepth;
}
[TestMethod]
public void MaximumDepthOfN_aryTreeTest()
{
List<Node> node3 = new List<Node>();
List<Node> node1 = new List<Node>();
Node root = new Node(1, node1);
int result = MaxDepth(root);
Assert.AreEqual(3, result);
}
}
}
1. MaxDepth() could/should be static. The containing class too.
2. The LeetCode question does not require breadth-first traversal.
Depth first would not require an(y) additional data structure (no queue, but would implicitly use the stack) which might be optimised away by tail-recursion (if C# is smart enough).
The following is therefore not criticism but a suggested alternative.
- I would not expect (much of) a performance difference
- Space-advantage, if significant (due to tail-recursion optimisation, compiler-and/or run-time environment dependent) would have to be measured
- So the main consideration in (not) choosing depth-first traversal would be a (probably) religious argument about brevity/readability of code
public static int MaxDepth2(IEnumerator<Node> nodeEnum = null, int depth = 0)
{
// No current node
if (nodeEnum == null || !nodeEnum.MoveNext()) return depth;
// The greater of current and maximum of siblings
using (var nodeEnumInner = nodeEnum.Current?.children?.GetEnumerator())
return Math.Max(MaxDepth2(nodeEnumInner, depth + 1), MaxDepth2(nodeEnum, depth));
}
:
:
using (var nodeEnum = new List<Node> { root }.GetEnumerator())
depth2 = MaximumDepthOfN_aryTree.MaxDepth2(nodeEnum);
• This code is rather confusing, and isn't an up-to-scratch implementation of an IEnumerator<T> consumer. What advantage does this have over using a foreach loop or LINQ's Max? This code also introduces different null checking behaviour (nodeEnum.Current?.) which may obscure bugs. Without CTO (which is in no way guaranteed in C#, and can't be performed without an understanding that Max is associative), this will have significantly worse memory characteristics than a simple DFS. May 6 '19 at 10:05
• @VisualMelon: On matters religious: Criticism accepted... 1. It is confusing 2. Null-checking syntax is inconsistent. I should have spent more time on writing "inspection-worthy code". I don't get your "up-to-scratch consumption" chirp... and would be keen to see it improved (by all means with LINQ or foreach, but preserving the point I was trying to make about the - arguably - more intuitive depth-first approach please). On matters empirical: I'd need to see something more substantial to support an opinion about "memory characteristics". May 6 '19 at 10:24
• It's fine to provide an alternative suggestion, but it needs to be clear why you think it would be better option in some way (and ideally identify any deficiencies, which can be hard to see in questions like this because there are no real requirements). You ought to be disposing the IEnumerator<T> (e.g. in a finally block); I'm afraid I can't find a sensible reference this minute. I suggest we continue this discussion in chat so that we don't flood the comment section. May 6 '19 at 10:31
• @VisualMelon: Keeping it off chat for the moment because the observation about disposing the enumerator is vital. Thanks. I've updated my code. As for motivating the suggestion, I think I did that. May 6 '19 at 10:52
• Fair enough. I wrote some comments concerning the memory usage in chat if you wanted to see those. May 6 '19 at 11:22
It looks optimized to me, but you can add a check for maxDepth >= 1000 and break the while loop if true.
A recursive approach that takes into account the threshold of max depth 1000. The threshold of 5000 nodes is ambigious, because what behavior do you expect when there are more nodes?
using System;
using System.Linq;
using System.Text;
using System.Collections.Generic;
using System.Globalization;
using System.Text.RegularExpressions;
public class Program
{
public static void Main()
{
var node5 = new Node();
var node6 = new Node();
var node3 = new Node(0, new List<Node> { node5, node6 });
var node2 = new Node();
var node4 = new Node();
var node1 = new Node(0, new List<Node> { node3, node2, node4 });
Console.WriteLine("Max Depth = " + MaximumDepthOfN_aryTree.MaxDepth(node1, 1000, 1));
}
public class MaximumDepthOfN_aryTree
{
public static int MaxDepth(Node root, int maxDepthThreshold, int depth)
{
if (root.children == null || !root.children.Any()) {
return depth;
}
if (depth == maxDepthThreshold) {
return depth;
}
return root.children.Max(x => MaxDepth(x, maxDepthThreshold, depth++));
}
}
public class Node
{
public int val;
public IList<Node> children;
public Node() {
}
public Node(int _val, IList<Node> _children) {
val = _val;
children = _children;
}
}
} | 2021-09-25 16:02:46 | {"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": 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": 0.27425962686538696, "perplexity": 7779.278505219537}, "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/1631780057687.51/warc/CC-MAIN-20210925142524-20210925172524-00339.warc.gz"} |
https://www.physicsforums.com/threads/balancing-magnetic-forces-on-rods.849953/ | # Balancing magnetic forces on rods
Tags:
1. Dec 29, 2015
### Will Freedic
1. The problem statement, all variables and given/known data
If two parallel rods are placed 1mm apart, with one directly above the other, each carrying a current I in opposite directions, of length L what is the mass of the upper rod as to balance the force
2. Relevant equations
F=ILxB
3. The attempt at a solution
I have little idea of how to go about attempting this solution, thankyou!
2. Dec 29, 2015
### Mister T
You need to assume the lower wire is held in a fixed position. It exerts a magnetic force $IL\times B$ on the upper wire.
3. Dec 29, 2015
### Will Freedic
Thankyou, but how do you work the magnetic field strength?
4. Dec 29, 2015
### cnh1995
Biot-Savart's law or Ampere's law will be useful here.
5. Dec 29, 2015
### Mister T
I suggest you look in a textbook. There should be a section on force between long parallel current-carrying wires. | 2018-03-20 12:58:02 | {"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": 0.41672483086586, "perplexity": 1257.3675383008215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647406.46/warc/CC-MAIN-20180320111412-20180320131412-00097.warc.gz"} |
https://camath.fudan.edu.cn/camb/ch/reader/view_abstract.aspx?file_no=41B104&flag=1 | Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type Citation： Yu-Feng YAO.Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type[J].Chinese Annals of Mathematics B,2020,41(1):49~60 Page view： 107 Net amount： 121 Authors： Yu-Feng YAO; Foundation： This work was supported by the National Natural Science Foundation of China (Nos.11771279, 11671138) and the Natural Science Foundation of Shanghai (No.16ZR1415000). Abstract： Let $F$ be an algebraically closed field of prime characteristic, and $W(m,n,\textbf{1})$ be the simple restricted Lie superalgebra of Witt type over $F$, which is the Lie superalgebra of superderivations of the superalgebra $\fraka(m;\textbf{1})\otimes \wedge(n)$, where $\fraka(m;\textbf{1})$ is the truncated polynomial algebra with $m$ indeterminants and $\wedge(n)$ is the Grassmann algebra with $n$ indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of $W(m,n,\textbf{1})$ with atypical weights of type I. Keywords： Restricted Lie superalgebra, Witt type Lie superalgebra, Restricted& representation, Typical (atypical) weight, Character formula Classification： 17B10, 17B50 Download PDF Full-Text | 2020-04-08 18:18:17 | {"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": 0.8073558807373047, "perplexity": 618.2403218700905}, "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/1585371821680.80/warc/CC-MAIN-20200408170717-20200408201217-00192.warc.gz"} |
https://www.iacr.org/cryptodb/data/paper.php?pubkey=23353 | ## CryptoDB
### Paper: Every Vote Counts: Ensuring Integrity in Large-Scale DRE-based Electronic Voting
Authors: Feng Hao Matthew Nicolas Kreeger URL: http://eprint.iacr.org/2010/452 Search ePrint Search Google The Direct Recording Electronic (DRE) system commonly uses touch-screen technology to directly record votes. It can provide several benefits in large-scale electronic voting, including usability, accessibility and efficiency. Unfortunately, a lack of tallying integrity in many existing products has largely discredited the entire approach along with its merits. To address this problem, we propose a cryptographic protocol called DRE-i, where i stands for integrity. We take a broad interpretation of the DRE: which includes not only touch-screen machines, as deployed at polling stations, but also remote voting systems conducted over the Internet or mobile phones. In all cases, the system records electronic votes directly, although the implementations are different. Our DRE-i protocol provides a drop-in solution to add integrity assurance to any DRE voting system without altering the voter's intuitive voting experience. It preserves election tallying integrity even if the DRE machine is completely corrupted, although in that case, vote secrecy will be compromised. The protocol requires a medium (e.g., an attached printer, email, or SMS) that the DRE machine can write the commitment data to. In addition, it requires a public bulletin board that everyone can read. Whilst past electronic voting protocols generally assume trusted computing or rely on trustees (i.e., tallying authorities), our proposal depends on neither. The protocol is self-tallying -- that is, anyone can tally the votes, without involving tallying authorities at all.
##### BibTeX
@misc{eprint-2010-23353,
title={Every Vote Counts: Ensuring Integrity in Large-Scale DRE-based Electronic Voting},
booktitle={IACR Eprint archive},
keywords={Electronic Voting, DRE, internet voting},
url={http://eprint.iacr.org/2010/452},
note={ haofeng66@gmail.com 14845 received 20 Aug 2010, last revised 24 Aug 2010},
author={Feng Hao and Matthew Nicolas Kreeger},
year=2010
} | 2019-10-15 08:04:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4681013822555542, "perplexity": 12413.975709394825}, "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/1570986657586.16/warc/CC-MAIN-20191015055525-20191015083025-00270.warc.gz"} |
https://socratic.org/questions/what-is-the-distance-between-2-2-6-and-5-1-1 | What is the distance between (–2, 2, 6) and (–5, –1, 1) ?
Mar 7, 2018
See a solution process below:
Explanation:
The formula for calculating the distance between two points is:
$d = \sqrt{{\left(\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}\right)}^{2} + {\left(\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}\right)}^{2} + {\left(\textcolor{red}{{z}_{2}} - \textcolor{b l u e}{{z}_{1}}\right)}^{2}}$
Substituting the values from the points in the problem gives:
$d = \sqrt{{\left(\textcolor{red}{- 5} - \textcolor{b l u e}{- 2}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{1} - \textcolor{b l u e}{6}\right)}^{2}}$
$d = \sqrt{{\left(\textcolor{red}{- 5} + \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{- 1} - \textcolor{b l u e}{2}\right)}^{2} + {\left(\textcolor{red}{1} - \textcolor{b l u e}{6}\right)}^{2}}$
$d = \sqrt{{\left(- 3\right)}^{2} + {\left(- 3\right)}^{2} + {\left(- 5\right)}^{2}}$
$d = \sqrt{9 + 9 + 25}$
$d = \sqrt{43}$ | 2019-07-21 19:14:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.9187169075012207, "perplexity": 569.369755907872}, "config": {"markdown_headings": false, "markdown_code": false, "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-2019-30/segments/1563195527196.68/warc/CC-MAIN-20190721185027-20190721211027-00358.warc.gz"} |
http://www.hongliangjie.com/2012/04/11/new-map-estimation-of-latent-dirichlet-allocation/ | # New MAP Estimation of latent Dirichlet allocation
A recent published paper entitled “On Estimation and Selection for Topic Models” by Matthew A. Taddy on AISTATS 2012 is interesting.
Firstly, this paper tries to introduce a new way to do inference for latent Dirichlet allocation (LDA). Traditionally, variational inference and Gibbs sampling are widely used to perform Bayesian inference for LDA. For a simpler version of LDA, Probabilistic Latent Semantic Analysis (PLSA), a Maximum Likelihood Estimation (MLE) via EM algorithm is usually used where no prior distributions are used at all. Both Bayesian inference and MLE can be treated as two extremes to learning a LDA or PLSA as MLE may be prone to over-fitting but Bayesian inference is a little bit overly complicated. The parameterization of the model is slightly different from normal settings. Let $$\mathbf{x}_{i}$$ be a vector of counts in $$V$$ categories (terms) for document $$i$$. Basically, each element in this vector is the number of times term $$v$$ appearing in document $$i$$. We also have the total term count $$m_{i} = \sum_{j}^{V} x_{i,j}$$. Thus, the $$K$$-topic model has the following generative story:
\begin{aligned} \mathbf{x}_{i} \sim \mbox{Multinomial}(\boldsymbol{\theta}_{i,1} \boldsymbol{\phi}_{1} + \cdots + \boldsymbol{\theta}_{i,K} \boldsymbol{\phi}_{K}, m_{i}) \end{aligned}
where $$\boldsymbol{\theta}_{i}$$ is a per-document distribution over topics and $$\boldsymbol{\phi}_{k}$$ is a distribution over words. Of course, this setting is indeed exactly same as previously published settings. However, we can easily view the term counts as a function of total counts explicitly. Again, Dirichlet distributions are put on $$\boldsymbol{\theta}$$ and $$\boldsymbol{\phi}$$. The paper discusses a joint posterior maximization by using EM algorithm. The author treats the term count vector $$\mathbf{x}_{i}$$ as:
\begin{aligned} \mathbf{x}_{i} \sim \mbox{Multinomial}(\boldsymbol{\phi}_{1}, t_{i,1}) + \cdots + \mbox{Multinomial}(\boldsymbol{\phi}_{K}, t_{i,K}) \end{aligned}
where $$\mathbf{t}_{i} \sim \mbox{Multinomial}(\boldsymbol{\theta}_{i}, m_{i})$$ and these $$\mathbf{t}_{i}$$ are treated as missing-data for EM algorithm. Thus, missing-data is no longer single topic assignments for each term but aggregated topic counts for the whole documents. The detailed derivations of the corresponding EM algorithm is in the paper.
The second point of the paper is to propose some new method to perform model selection, which is always difficult for topic models in some sense. The whole idea is to find $$K$$ that can maximize $$P(\mathbf{X} \, | \, K)$$, which is usually approximated through MCMC. In this paper, the author details Laplace’s method to approximate this probability. Again, details are in the paper. The last point made by the paper is to use multinomial dispersion to evaluate whether the model fits the data, which is standard in statistics but never used in topic modeling literature.
The experiments are somewhat interesting. The author demonstrated that MAP is almost always comparable to variational EM (VEM) and sometimes slightly better than Gibbs sampling. Sometimes, MAP is even better than VEM.
Note, this is NOT the first paper to propose a MAP learning for topic models. Here’s one at least:
J. T. Chien, J. T. Chien, M. S. Wu, and M. S. Wu. Adaptive Bayesian Latent Semantic Analysis. Audio, Speech, and Language Processing, IEEE Transactions on [seealso Speech and Audio Processing, IEEE Transactions on], 16(1):198–207, 2008. | 2017-10-20 05:20:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.67255699634552, "perplexity": 907.8858138035122}, "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/1508187823731.36/warc/CC-MAIN-20171020044747-20171020064747-00018.warc.gz"} |
http://mathhelpforum.com/algebra/199082-sat-multiples-problem.html | # Math Help - SAT multiples problem
1. ## SAT multiples problem
If $S$ is the set of positive integers that are multiples of $7$, and if $T$ is the set of positive integers that are multiples of $13$, how many integers are in the intersection of $S$ and $T$?
I said there is 1 integer in the intersection, since only 91 is a common multiple of 7 and 13. However, according to the answer,
This is the set of all positive integers that are multiples of $7*13 = 91$. There are an infinite number of positive integers that are multiples of $91$, so there are more than thirteen integers in the intersection of $S$ and $T$.
91's multiples include 186, 278, etc, so I'm guessing the infinity is referring to that. However, I thought the problem only wanted an integer that was only multiple of 7 and 13?
2. ## Re: SAT multiples problem
Originally Posted by m58
If $S$ is the set of positive integers that are multiples of $7$, and if $T$ is the set of positive integers that are multiples of $13$, how many integers are in the intersection of $S$ and $T$?
I said there is 1 integer in the intersection, since only 91 is a common multiple of 7 and 13. However, according to the answer,
91's multiples include 186, 278, etc, so I'm guessing the infinity is referring to that. However, I thought the problem only wanted an integer that was only multiple of 7 and 13?
You have dropped and "s"! The problem said "multiples" of 7 and 13, not just the product of 7 and 13. The set of "positive multiples of 7" are 7, 14, 21, 28, etc. and the set of "positive multiples of 13" are 13, 26, 39, etc.
3. ## Re: SAT multiples problem
Originally Posted by HallsofIvy
You have dropped and "s"! The problem said "multiples" of 7 and 13, not just the product of 7 and 13. The set of "positive multiples of 7" are 7, 14, 21, 28, etc. and the set of "positive multiples of 13" are 13, 26, 39, etc.
So the multiples of 7 and 13 that are x≥91 are included in the intersection, meaning there are infinitely many of them. | 2015-02-28 12:49:56 | {"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": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.853576123714447, "perplexity": 200.18131984690447}, "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/1424936461944.75/warc/CC-MAIN-20150226074101-00205-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://nanoscale.blogspot.com/2016/07/ask-me-something.html | ## Saturday, July 30, 2016
I realized that I haven't had an open "ask me" post in almost two years. Is there something in particular you'd like me to write about? As we head into another academic year, are there matters of interest to (grad or undergrad) students?
Grumpy said...
How do you feel about the traditional lecture format for sophomore level physics? Do you have any tips for delivering the best learning environment with the least amount of prep?
Anonymous said...
Why don't my professors care about my personal and professional development? Investing in the human capital of young, promising researchers is literally their job. I regret graduate school. Keeping this anonymous because I don't want this coming back to me.
Anonymous said...
Ok, here's a real question: What's the best way to explain solid state physics phenomena like band gaps and excitations and conduction *without* relying on the single-particle fiction? I feel like we have to lie to students all the time (or at least speak very imprecisely) since we lack the language and understanding of multi-electron wavefunctions. Thoughts?
Anonymous said...
In a graduate course I learned about Green Functions as elementary building blocks for many particle physics in condensed matter. The lecture started with linear response theory (Kubo) and developed the Feynman concept for diagrams in order to address interacting Hamiltonians.
I clearly lack the understanding how a formalism like Kubo's can be used as basis for approximations given that Kubo itself is a drastic simplification (first order).
Moreover, how can we measure (!) single particle Green's functions and what information do they really provide? We spent tons of time to develop a formalism for them but nobody motivated the practical and experimental relevance in further detail. :/
Jake Steele said...
What is your opinion on 2D materials and their applications? I've noticed several fantastic claims by both media and researchers.
Anonymous said...
1. Can you shed light on out-of-plane plasmonic resonances which have not been studied much ?
2. What future do you see for the plasmonic based research in the coming years ?
Douglas Natelson said...
I'll try to take these from the top, though it may take me a bit to get through them.
Grumpy, it depends a bit what you mean by "sophomore level". Are you referring to waves + optics, and some lead in to quantum? Or Lagrangians/Hamiltonians? The more technically detailed and matehmatical the material (and the farther removed from classroom demos), the more challenging it is to move toward nontraditional formats, it seems to me. In terms of "least amount of prep", I assume you mean from the faculty side, and further that there is an unwritten "yet still having the course be well run and pedagogically valuable". Some kind of flipped lecture scheme, with recorded video snippets watched with a rubric by students pre-class, and then class-as-discussion-and-examples would probably work long-term, though the initial investment of time and effort to set this up and make it good would be large. An example of the online component that worked well is here, by my colleague Jason Hafner.
Anon@6:01, I'm sorry to hear that things aren't going well. I can only speak for myself when I say that I definitely care about the professional development of my students, and I believe that this is true of the large majority of my colleagues. Altruism is part of this (it is our job, as you say), and even self-serving professors should care - successful students = good for the lab, and while not as obvious as things like publications and citation counts, the post-PhD careers of students is one way that some professors "keep score". That being said, professors are almost never trained in how to manage students, and I now have a much greater appreciation for how the mentoring relationship can go off the rails. Take advantage of support structure available to you - other members of your committee or department, an ombudsperson. Feel free to contact me by email if that would be helpful.
Anon@6:03, the thing is, very often the single-particle language isn't really (much of) a fiction. I mean, would you call it a lie to talk about the 6s electrons in gold, when technically speaking the true description of the electrons in an atom of Au requires a totally antisymmetric wavefunction involving 79 electrons, including relativistic corrections? I know what you mean, though. I ran into this when writing my book. People talk about "the wavefunction of an electron in a quantum dot" like it's a particle-in-a-box, for example, when what they really mean is the "envelope" function that is a good approximation to constructing the true electronic state out of Bloch functions that take into account the atoms in the lattice, and that true state is really closer to some totally antisymmetric linear combination of Slater determinants, etc. etc. I think the best we can do is make sure that students know "single-particle state" is an approximation that is good a lot of the time, and to remind them periodically that it's an approximation as well as linguistic shorthand for something more complicated. I'm not sure this helps, but it's the best I can do. Others may have more to say on this.
Grumpy said...
Hi Doug,
Thank you very much for the thorough answer!! I just watched the edx videos by your colleague. Wow they are great. I am really in over my head (first big lecture course as an assistant professor). Yes this is the waves/SR/baby quantum version.
Would you mind clarifying what sort of "rubric" the students should use when watching the videos?
I am a big fan of the flipping deal, but to be honest I never took a class that way. I rarely got much out of lectures as a student; I mostly learned by doing problem sets at cafés.
Paul Anzel said...
What steps, if any, would you want to take to reform undergraduate physics education?
Anonymous said...
What is the meaning of life?
Anonymous said...
Read Erwin Schrodinger's masterpiece " What is life ? "
type in google , to get pdf file.
Douglas Natelson said...
Anon@11:36pm, I might not be the best person to answer this, and if a theorist reading this has a better answer, I invite their contribution. I think you already have said part of the answer - when trying to figure out response to some perturbation, a conventional approach is to start with linear response basically as an assumption (though the validity of the linear response treatment can be established with some rigor within its formalism, I believe). Then you seek some way of looking at additional terms in the perturbative expansion beyond first order. Feynman diagrams are one formalism for this. If you're lucky, you can find ways to sum some kinds of terms to all orders (basically trying to leverage 1/(1-x) expands to 1 + x + x^2 +.... for example). As for ways of measuring the single-particle Green's fn, that's tricky, as you've surmised. Photoemission is the canonical way to do something like this (you can see how many true single electrons are extracted to infinity as a function of energy and momentum transfer), though this gives something more like the spectral function of the retarded Greens fn - see here. Personally I think of the Greens fn in much the way I think of the wavefunction itself. Hard to measure the whole thing directly, but part of a formalism that lets you calculate other measurable quantities (correlation fns measured through scattering experiments, for example).
Jake, yes, there is an enormous amount of hype floating around regarding 2d materials. They really do have some remarkable properties, and thanks to advances in growth techniques, patterning methods, and the development of new characterization tools and theoretical methods, people are making much more progress with these systems than they did ~ 40 years ago, when transition metal dichalcogenides also had a mini-boom in popularity. It's important to remember that Si and III-V semiconductors are ubiquitous because of many decades of effort by thousands of people. Introducing a new materials system and having it be rapidly translated into technologies, especially in electronics or optoelectronics, is usually a slow process, so statements about immediate technological impact have to be taken with a big grain of salt. Also, the semiconductor industry in particular is incredibly rigid (for a variety of economic reasons) about adopting new materials. Still, the whole cycle of materials growth -> characterization -> theoretical modeling -> new growth is potentially much faster now, so there is reason for measured optimism.
Anonymous said...
I'm not an undergrad or grad student anymore, but I'm interested in condensed matter physics.
One important question regarding future technology is about batteries:
What are the current obstacles in building more efficient and higher capacity batteries, and is there a theoretical limit to how much energy a unit volume of battery may be able to hold? Is there a breakthrough around the corner?
I'm only interested in somewhat conventional and safe battery technologies, not exotic ones that rely on nuclear fission or fusion or other nonsense.
Anonymous said...
I know you've addressed the academic job market process before, but do you have any advice to postdocs of when to start going on the job market? I'm just drafting my first postdoc manuscript, which I'm hoping to have submitted right before this season of applications begin.
I guess a lot of this is stemming from after starting my postdoc, I really do not see the benefit of it. It's too short of time and extremely stressful to be in such temporary employment. The only benefit I see is to the PI's getting fast labor out of someone. The longer I'm in this postdoc, the more my confidence to do, and stay in, science declines.
Douglas Natelson said...
Anon@2:56am, I'm not really familiar with out-of-plane plasmon modes - I'd have to read up on them. Do you have a particular reference in mind? Regarding plasmons in general, there are many directions being pursued - energy harvesting (photovoltaics); plasmon-assisted photodetection; part of the palette of tools for metamaterials and metasurfaces; photothermal heat generation; flavors of quantum optics using plasmons; hot electron photochemistry and photocatalysis; the now-usual surface-enhanced spectroscopies. It's still a bit murky which of these will pan out into technologies, but a deep understanding the dynamics and the roles of disorder and symmetry breaking are important basic issues.
Grumpy, I'll pass along your kind words. By "rubric", I mean something like what UIUC does in their flipped freshman physics classes. Students watch brief video segments in a specific order and have to answer brief questions in between before getting to see the next segment. The students are expected to have watched everything and answered the questions prior to class. Then the instructor can see what the answers look like and can tailor conversation appropriately. (Full disclosure: I have not done this flipped video approach myself, so you shouldn't take what I say as anything like the gospel. I, too, got the most out of working on problem sets, though I did appreciate very well-prepared lectures like the ones I had in my undergrad quantum sequence.)
Paul, that's a big question. I have my personal tastes, but everyone has their own opinions. Answers depend strongly on the target audience: would-be doctoral candidates, or people interested more broadly in science and technical fields. I think a looming issue is making sure that undergrads get an appreciation that physics is a living field and that there's more to it than string theory stuff on the one hand and ancient received wisdom on the other.
Emil Prodan said...
It will be interesting a post on the recent experiments which sent the field of topological insulators into a crisis. These are the observation of perfectly conducting edge states in 2D QSH insulators even in the presence of strong magnetic fields as high as 12T [PRL 114, 096802 (2015) and NATURE COMMUNICATIONS 6, 7252 (2015)], and the discovery that magnetic impurities can localize the entire surface spectrum of 3D TRS topological insulators (as opposed to opening a harmless tiny gap) [Nature Communications 7, 10559 (2016)]. These are both in stark contrast with the theoretical predictions.
Matt said...
If you were a beginning grad student now, what field would you go into and why?
If you happened to have a spare million to invest in a nanoscale-related field, where would you put it?
Thanks for the great blog!
Douglas Natelson said...
Anon@9:55, those are good questions. Battery materials (anodes and cathodes) should ideally be able to take in or lose large amounts of lithium (assuming we continue down that road and don't switch to sodium or something) rapidly, with minimal structural damage or other irreversibility, thousands of times. The electrolyte that transports the lithium ions has to be electrochemically stable while all this is going on (perhaps not counting the formation of the Mysterious Magic Goop - that is, the solid-electrolyte interphase - that appears on the two electrodes after a few cycles). The separator has to be boring, inert, and exceedingly stable in this environment. All of these constituents need to be reliably and inexpensively manufacturable. It's a pain. As far as I know, the biggest obstacle to greatly improved Li-ion battery capacities is the lack of a really good high capacity cathode material, though others more in the field may disagree.
As for the ultimate limits of energy density available in batteries, a rough estimate would be on the order of what you have in gasoline. That's a very dense packing of covalent bonds that get undone and rearranged when combustion takes place. It's hard to imagine that you could do better than that via electrochemical lithiation/delithiation, given the need to have some kind of structurally meaningful electrodes.
I don't know of any impending breakthrough, though materials discovery could result in an improved cathode at any time. More to the point, there are far more people with far more resources and economic motivation working on this problem than ever before. That has to improve the odds....
Anon@11:13, your questions are deserving of a longer response than I can really give right now. When you say two rounds, do you mean in subsequent years? The short version: You should go on the market when you are most likely to be competitive. A postdoc appointment is a chance to learn something new beyond your PhD and demonstrate that you can shift gears a bit. Producing nice result(s) as a postdoc helps support that your doctoral work's awesomeness is a reflection at least in large part of you, rather than the adviser. Competitiveness for jobs depends on the area. In many engineering disciplines faculty are hired with no postdoc experience. In physics, it would be very unusual for someone to be strongly competitive on the job market without that additional post-PhD demonstration of productivity these days.
David Brown said...
Does the discovery described in the following publication have economically important implications?
"Lifshitz transitions and zero point lattice fluctuations in sulfur hydride showing near room temperature superconductivity" by Bianconi & Jarlborg, 2015
Gautam Menon said...
Is there a case to be made for the importance of quantum mechanical effects in biological systems?
Anonymous said...
A damaging misconceptions ..... that research dominates and diminishes teaching.
starts the write up way back in 1996 in the following pdf and then concludes the para ' teaching makes for better research"
http://roaldhoffmann.com/sites/all/files/research_strategy_teach.pdf
The general feeling by many are teaching and research are split wide open for evaluating the academics capability even for tenure.
The questions are
1) Is it due to heavy emphasis on metrics ?
Many students have reported that many academics have a Jack Nicholson ( As good as it gets) type compulsive obsessive disorder ( or is it an order) { COD or COO ?) with metrics.
2) Do you feel that attention span of undergrads has come down, due to technology overwhelming and academics find it difficult to reign the undergrads attention for sustained period of teaching.
3) The large class size ( which many universities have adopted) is it responsible for decline in academics teaching. online can be done for a large class , but the number of queries which come by email from a large class for each different question takes a lot of time to answer . In pre technology days , a good academic would one day in the morning, answer the volley of questions and sign off. email queries and answers discourages cohort learning because it is one is to one.
4) Some universities , only some univ give importance to good teaching for tenure? Why not all?
Roald Hoffman first para in the pdf has come true.
Douglas Natelson said...
Emil, nice to hear from you! Yes, I should probably write a longer post at some point about the issues in these systems. In the InAs/GaSb case, there is quite some controversy about the nature of edge state transport (e.g., can you get "trivial" edge state transport in these systems due to band bending that can look a lot like interesting edge state transport? What is the best way to determine unambiguously that you are in the topologically nontrivial regime? What limits ballistic transport in that regime?). I am actually working on a proposal involving that system, so I'm a little leery of saying too much in this format b/c I don't want to oversimplify or misstate anything.
Matt, great question. Assuming we're talking physics, I probably would still go into condensed matter (I really like the scale of the work, the physics involved, and the puzzle solving aspects of the experiments), though I've always had a love of astronomy/observational astrophysics. I still remember going to one of my undergrad advisers fall of senior year for advice about grad school, and this highly successful and recognized condensed matter guy spent twenty minutes telling me how we were entering the golden age of astrophysics :-)
As for where I'd invest a spare $1M in nano, apart from my own research program (hey, that should net something like$50K/yr, right?), I'd look at heteroepitaxial growth of 2d materials. If you were wondering about investing in nano-related companies or businesses, the safest investment would be a "tool maker" - there's an ever-expanding market for electron microscopes, and for increasingly high end rf generators and analyzers. [obviously no one should take investment advice here seriously!]
Douglas Natelson said...
David, regarding H3S superconductivity: I haven't read that paper in detail. Superconductivity at high temperatures in H3S under pressure is very neat - challenging experiments and an exciting result that shows that there really is no reason in principle why we can't have room temperture superconductivity. As far as I know, it seems to be consistent with established theoretical understanding, meaning BCS and related physics (as in this paper). Direct economic impact is unlikely, since the pressure conditions needed to do this in H3S require specialized apparatus (diamond anvil cells). Still, between this and other work on nonequilibrium superconductivity-like response, we seem to be getting closer....
Gautam, I believe (without digging up references) that there is strong evidence of quantum effects in particular processes (photosynthesis; the way light detection works in the eye). I am extremely skeptical that there are (electronically) quantum coherent effects involved in consciousness as some people try to argue. Obviously quantum mechanics is directly responsible for chemistry and all the noncovalent interactions in biological systems, too, but presumably that's not what you mean....
Anon@8:50, there is no question that the incentive structure at research universities strongly rewards success in research and at best only moderately considers pedagogy and classroom teaching. Part of this may have to do with it being easier to quantify research (\$ brought in, papers published, citation counts, etc.), while serious assessment of teaching is difficult. I don't think there has been a big change in attention span of students, but there are certainly more distractions now (ubiquitous internet devices) and a shift such that mere facts are available virtually instantly. To get student buy-in for teaching, we have to convey that we're teaching how to think critically and analytical skills, not just transferring facts, which they can look up on their phones. Regarding class size, there is near-universal recognition that smaller classes are better, and many places are pushing toward smaller sections and fewer giant lectures. As for online instruction, I think we are all still finding out what approach works best under which circumstances. There is no substitute for student-instructor interaction, and you can't run an effective online course at a serious level (I don't mean some fun thing, but something with real assignments that is the equivalent of a for-credit class) without an appropriate amount of personnel support to have those interactions.
Emil Prodan said...
Hi Doug, is very simple to decide that. QSH is a distinct phase, one should identify the phase boundary as one usually does when studying a (bulk!) phase transition. One should take note from IQHE where this has been pushed experimentally to the limits (so much that the experimental scaling analysis looks like textbook theory). Chern insulators are now at par with IQHE, i.e. the critical point between trivial and topological phases has been identified beyond any doubt. This was never achieved for QSH class, hence QSH cannot be declared a distinct condensed matter phase. It is amazing how the experimentalists missed this point and never looked into it seriously.
David Schroeder said...
Have you ever heard of Podkletnov's work with high temperature superconductors? In those experiments Podkletnov (back between 1996 and 2001) claimed to detect, at first small (.05 g) acceleration signals from spinning superconductors. Later (2001) he claimed totally fantastic (and unbelievable) reports of 1000 g impulses, persisting for 1/10,000th of a second, when his YBCO superconductor was subjected to 2 million volt discharges. It would be great if you could do a write-up on this.
I was totally fascinated by these claims, while keeping Carl Sagan's cautionary mantra: "Extraordinary claims require extraordinary evidence" in mind. Indeed, I was so intrigued that I ran some simple experiments with commercial one inch YBCO superconductors. On a couple of occasions I detected .0001 g signals. But I'm convinced that these resulted from an electromagnetic pulse as 1.2 megawatts (600 volts, 2000 amps) coursed through the superconductor, affecting my accelerometer (I've completely isolated the acoustic 'pop' from expanding cryofluid with electronic timing circuits). Nonetheless I'm going to try additional experiments with improved shielding of the ADXL203 accelerometer.
Luis Gregório Dias said...
Hello Doug. Here's my question: how do you manage your time?
For instance: do you set aside a particular time to write/read e-mails? Do you use "to-do-list" apps such as Wunderlist, Google Keep, etc.? Any tips for keeping a high productivity overall while juggling with all the demands of the academic life (doing good science, writing/reviewing papers, advising students, writing blog posts, teaching, etc.)? ;)
Douglas Natelson said...
David, you've already hit on the key aspects of Podkletnov and his more recent kin, those that are looking at "EM Drive" and "Mach Effect" devices: extraordinary claims require extraordinary evidence, and sensitive force measurements in the presence of large oscillating or pulsed currents are tricky.
Luis, the short answer is "not as well as I should". I keep lists - I got into that habit in grad school. I use google calendar. I do try to stick with some routines, and once I had kids I tried to figure out what kind of work I can do well at home vs. what I need to do at the office/lab. It's hard.
sam north said...
Hello Douglas
Could you give an opinion on this report.
http://lenr-canr.org/acrobat/MosierBossinvestigat.pdf
Thanks
Sam
Douglas Natelson said...
Sam, I flipped through it. The report does not really conform to what one would expect for a scientific paper, and the author takes it basically as a given that there are definitely low energy nuclear reactions happening, going all the way back to Pons and Fleischman in '89. My views on this stuff have not changed since I wrote this or this, the passion of LENR supporters notwithstanding. The TL/DR version: It would be incredible and awesome if it were true, but I have yet to see experimental work that actually meets the standards of reproducibility one would expect for a real effect.
sam north said...
Thanks for reply Douglas.If you do not know.
Bob Greenyer is working with Aarhus University.
http://www.e-catworld.com/2016/09/03/bob-greenyer-reports-on-mfmp-activity-at-aarhus-university/
Andrea Rossi Robert E Godes and others
keep plugin away at LENR.
I will post on your blog if anything important
happens.
Regards
Sam
sam north said...
http://www.e-catworld.com/2016/09/19/report-stable-excess-heat-100-per-cent-reproducible-in-lenr-experiment-at-tohoku-university-japan/
sam north said...
Hi Douglas
Do you follow DR Randell Mills
and his Sun Cell technology.
https://youtu.be/AhIoDxjaibQ
Have a good New Year
Sam
Douglas Natelson said...
Sam, Randall Mills has been pushing hydrinos since before I was in grad school. I actually read through his manifesto/long paper on this back in 1994. It's just not correct, nor are hydrinos backed up by any experimental evidence taken in serious labs. | 2017-01-21 19:40:26 | {"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": 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": 0.4472770094871521, "perplexity": 1464.8281180266192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281202.94/warc/CC-MAIN-20170116095121-00177-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.wiglafjournal.com/corporate/2014/03/abenomics-why-what-and-now-what/ | # Abenomics: Why, What, and Now What?
March 2014 1 Comment
Economic policies advocated by Japan prime-minister Shinzo Abe, since his return to the helm after the 2012 general elections, is referred to as Abenomics. This article aims to provide a first-level understanding of the principles of Abenomics and its effects on the global business environment.
### Background
The Japanese economy was in a crisis situation for over 2 decades. As per commentators and analysts, the crisis initiated with the burst of Japanese asset bubble in 1991-92. What followed is commonly referred to as Ushinawareta Jūnen or the Lost Decade – a period of sluggish or no economic growth. In fact the “lost decade” was far longer than a decade. Some writers go to the extent of calling it “lost two decades”. The severity of the Lost Decade is, however, debatable. Nobel Laureate Economist Paul Krugman maintains that the story of Japan’s decline is an overstatement. His interpretation Japan’s slow economic growth was merely an effect of demographics (in this case Japan’s aging population leading to a dip in prospective workforce).
Japan was hit severely during the global economic recession in the 2008-09. In 2008, Japan suffered a real-GDP loss of 0.7% (global: 3.1% hike), while in 2009 the loss was a staggering 5.2% (global: 0.7% loss). Exports from Japan also dipped by 27% ($746.5B to$545.3B) from 2008 to 2009.
Japanese exporters faced tremendous challenge from Chinese and Korean competitors who were fast invading key Western markets.
### Enter Abe
Shinzo Abe was first elected as prime minister in the 2006. His first stint was a short one, ending abruptly with his resignation in 2007. In September 2012 he was re-elected as the president of the Liberal Democratic Party. In December 2012 LDP aced the Lower House election, winning 294 seats out of 480.
Abe announced his return with the memorable words, “With the strength of my entire cabinet, I will implement bold monetary policy, flexible fiscal policy, and a growth strategy that encourages private investment, and with these three policy pillars, achieve results.”
Thus began Abenomics.
### Challenges Abe faced
Abe’s key economic challenges at the time of his re-election were beating deflation, correcting a strong Yen, and achieving economic growth. Before proceeding on to the core of Abenomics, let’s first evaluate these challenges individually.
1. Beating Deflation In economics, deflation is a decrease in the general price level of goods and services. In Japan the deflation situation in the decade preceding the end of 2010 is captured well in the diagram below.
Deflation leads to problems both on the demand side as well as the supply side. Demand falls because customers defer purchases, anticipating a price fall in the future. Moreover, deflation increases the real burden of debt on debtors. On the supply side, deflation causes revenue drop for firms (due to price cuts), which leads to wage reduction for employees and increased unemployment.
2. Correcting strong yen The yen strengthened over time, from an exchange rate of 1USD = 120 JPY in 2007 to 1USD = 77JPY in 2012. Due to this strengthening, Japanese exporters faced a tough situation as Japanese goods got more expensive in the global market. The price challenge from Chinese and South Korean competitors further worsened this situation. The strengthening of the yen also led to lowering of foreign direct investment (FDI) in Japan.
3. Economic growth As already mentioned, since the beginning of the lost decade GDP growth in Japan has struggled: by 2013 the Nominal GDP in Japan was almost at the same level as that in 1991.
As a result of these challenges, Japanese firms faced tremendous difficulty to survive in business. Due to deflation, domestic-focused firms lost revenue. To make things worse, the exports were also not going well due to the strengthening of the yen. This combination led to job-cuts and unemployment. This, in turn, further depressed consumption levels and led the economy to a vicious cycle of depression.
Abenomics responds to the situation with a three-pronged approach (popularly called “the three arrows”)
• Monetary policies
• Fiscal policies
• Structural policies
### The three arrows
To begin with we need to take note of the monetary policies. As a first step, BOJ (Bank of Japan) announced an inflation target of 2% (Abe had spoken about curbing their independence if they did not act). The method adopted for achieving the same was quantitative easing. In simple words, BOJ would increase (double) Japan’s monetary base and also buy more (twice as many) Japanese government bonds. As a result it is expected that the money that the banks get by selling off bonds would be used to loan money to borrowers. With this easing of difficulty in getting loans, consumers and businesses would both have more money at disposal and thus spending (consumption) would increase, leading to inflation. With inflation the businesses would make more revenue and hence it would lead to decreased joblessness and increased salary.
BOJ’s quantitative easing would solve two key challenges:
1. Beating deflation
2. Weakening the JPY (or rather correcting the strengthening yen)
As a part of the second arrow (fiscal policies), the Japanese government announced a hefty 10.3 trillion yen short-term stimulus package (cleared by cabinet in January 2013) for infrastructure projects. The intent behind this was to increase revenue for businesses and to allow for job creation. In October 2013 Abe also announced an increase in consumption tax from 5% to 8% to boost government earnings.
The third arrow (structural policies) remains the missing piece in the plan. This requires not just overhaul of regulations in key sectors but also a plan to overcome the growth-hampering effect of the troubled Japan demographics (aging population and decline in workforce). While the first two arrows are relatively easy to implement, structural policies are not. Especially because it leads to contradictions in message. Let’s consider a hypothetical example: If structural reforms demand deregulation in an industry and the government implements it then it will end up contradicting itself if it starts recommending pay hikes/employment in that industry (which again is a critical requirement for the success of monetary policies).
### The Abe effect
The effects of the monetary policies were visible quickly after they were implemented. The yen weakened by over 20%, and in a couple of quarters the exchange rate had shifted to 1USD = >100JPY. The yen weakening provided relief to a whole bunch of Japanese companies who earned a significant portion of revenues through exports. A case in point would be Renesas Electronics. Renesas was almost headed toward bankruptcy when Abe took over. Renesas first began making operating profit and in the recently concluded fiscal quarter (Q3 2014 for Renesas) have managed to make 23 billion yen net profit. Those who follow the global semiconductor industry will attribute Renesas’s turnaround to a whole host of factors such as organizational changes, opex reduction, et cetera. One point however needs to be noted:
In Q3 2013 Renesas’s revenue was 191B yen and in Q3 2014 their revenue was 215.6B yen. In percentage terms, this translates to 13% growth YoY. If we evaluate the same numbers in USD instead of JPY, however, we note that there has actually been a revenue decline of 10%.
Renesas was not the only company to report advantage due to Abenomics. The Bank of Japan Tankan survey of business sentiment showed business sentiment rising to a six-year high. Riding the positive recovery story of these companies, Nikkei stock exchange made an annual gain of 57% in 2013—its biggest gain in 42 years.
Japan’s core consumer prices (excluding non-volatile food prices) recorded the biggest annual rise in five years (0.8% increase between August 2012 and August 2013).
However, on the flip side, Abe’s vision that companies would pass on the additional profits to employees in form of salary hikes (such that the salaried Japanese individuals don’t feel the pinch of inflation) has so far fallen short of expectations. A recent survey by Reuters revealed that only 11% of Japanese companies plan to increase employees’ total compensation (base pay plus bonus). 66% of companies expressed intent to increase only the bonus component. Though prices have gone up by 1.3% year-over-year, the increase in salaries is 0.8%. The reluctance among firms to increase salaries could possibly be dangerous for Abenomics, as it would restrict consumption increase by the masses (only 2.4% increase in private consumption).
Further, while companies are investing, the current rate of 1.3% rise in capital investment falls short of typical expectations from a recovering economy.
It is impossible for government to control the spending patterns of firms without contradicting itself given its structural reforms, the third arrow of Abenomics, are aimed at relaxing fiscal controls, loosening the labor market, and making it easy for a business to start and operate in Japan. Akira Amari, the minister of state for Economic Revitalization, had remarked (earlier in 2013), “I want to create an environment where it’s embarrassing for a company not to raise wages.” In fact, the actual government response to the situation was linking corporate tax structure to salary increases. The response no doubt is fair, but is contradictory to the initial promises.
### Global business impact and forward outlook
The global impact can be judged by the mixed global reactions thus far. Abenomics has found support from countries who intend to increase FDI in Japan. On the other hand, countries whose key export areas are same as those of Japan have expressed concerns about the potential damage to their export businesses.
Hyun Oh-Seok, finance minister of South Korea, commented “We need some kind of coordinated efforts to prevent these kinds of unintended side effects from [Japan’s new] monetary policy. Whether it is intended or not, the result [of the depreciation of the Japanese yen] is quite quick.” South Korea’s top 10 exports compete directly with Japan. Another country facing similar concerns is Germany.
Companies that export to Japan (and have Japanese competitors) have reported feeling the pinch of Abenomics. This is because these companies in quite a few cases could not increase prices in yen to their customers. Instead they had to lower the prices in their own accounting currency.
On the other hand, the has been a sharp increase in the prices of commodities imported by Japan. In fact, the increase in money spent for importing has beaten the incremental money earned through increased exports. This has resulted in record trade deficits for Japan. This, in turn, is making the world question Japan’s long-term economic recovery prognosis. Credit Suisse AG has cut their Japan growth forecasts (Feb 2014) for the year 2014 from 2.2% to 1.6%.
### References
• Nilohit
brilliant stuff anirban. Very well compiled!
Anirban is a core-team member at Lifkart (an Early stage Indian Construction Start-up). Prior to the current gig he worked for about 5 years as a pricing manager at Cypress Semiconductor. He holds a BE in Electrical Engineering from National Institute of Technology , India and an MBA in Marketing from Symbiosis Centre for Management and Human Resource Development (SCMHRD), Pune, India. | 2017-03-24 06:12:36 | {"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": 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": 0.36109212040901184, "perplexity": 8341.732478451058}, "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-2017-13/segments/1490218187717.19/warc/CC-MAIN-20170322212947-00635-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://web.lemoyne.edu/~craigda/Math/MathJaxLocal/Sandbox/play.html | MathJax Playground
Some MathJax Math
MathJax lets you employ $$\TeX$$ and $$\LaTeX$$ in web pages to render mathematics. Yay!
Inline $$\TeX$$ mathematics uses the delimiters $$x^2/y^2 = \mathrm{aarghh!}$$.
Display mathematics uses the standard $$\LaTeX$$ delimiters. $\frac{\int f(x) dx}{\int f(x)\cdot g(x) dx}$ or $$\sum_{n=0}^{\infty} \frac{1}{n!}x^n.$$ or $$\int_{-\infty}^{+\infty} \hspace{-1em} dx f(x)$$
Do I need $$\TeX$$ delimiters when loading environments? Does not appear so. \begin{eqnarray*} f(z) = \frac{1}{2\pi i}\int_{\mathcal{C}} \frac{f(z')}{z-z'} dz. \end{eqnarray*}
Here was the clue how to set up left alignment of equations. (And, in passing, a pointer to the MathJax-users Google group.)
Sample $$\TeX$$ macros: $$\RR$$. $$\caltxt{TESTIFYZ}$$ vs. $$\mathcal{TESTIFYZ}$$ $$a \newrel b$$ $$a \newrelz b$$ $$\bold{test}$$
For more details, visit the MathJax documentation pages. $$\LaTeX$$ command support is detailed here.
Go back home. | 2018-11-18 06:10:35 | {"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": 2, "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": 0.23841679096221924, "perplexity": 6171.799556764285}, "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-2018-47/segments/1542039743968.63/warc/CC-MAIN-20181118052443-20181118074443-00100.warc.gz"} |
http://finalfantasy.wikia.com/wiki/Topaz_(item) | # Topaz (item)
20,935 pages on
this wiki
Yellow gemstone in the shape of a Hume heart. Fills its holder with bloodlust.
Topaz (トパーズ, Topāzu?) is a recurring item in the series.
## AppearancesEdit
### Final Fantasy IXEdit
IX Topaz is an item and an add-on obtained by synthesizing it on disc 4 at Daguerreo, Black Mage Village, or in Memoria at Hades for 100 gil, an Ore, and an Eye Drops. It can be found in the Cold Lagoon Chocograph (x19) or by using a Dead Pepper at the foaming water below Quan's Dwelling (x15), or dropped from Grimlock (Blue Head), Wraith (Red Flame), Veteran, or Vepal (Red). It teaches the abilities Ifrit and Fira, and when used as an item, restores HP equal to the number of Topazes in the player's inventory x92.
The more Topazes in the player's inventory, the more powerful Ifrit's Flames of Hell becomes, according to the following formula:
$Flames of Hell attack power = 42 + No. of Topazes$
### Final Fantasy XIEdit
This article or section is a stub about an item in Final Fantasy XI. You can help the Final Fantasy Wiki by expanding it.
### Final Fantasy XII: Revenant WingsEdit
This article or section is a stub about an item in Final Fantasy XII: Revenant Wings. You can help the Final Fantasy Wiki by expanding it.
### Final Fantasy XIVEdit
XIV Topaz is a rare type of gem used by master Goldsmiths to craft very high level jewelry for tank classes.
### Final Fantasy Crystal Chronicles: Echoes of TimeEdit
FFCCEoT Topaz is a jewel that grants MDEF Grow 3 when equipped to a piece of equipment.
This article or section is a stub about an item in Final Fantasy Crystal Chronicles: Echoes of Time. You can help the Final Fantasy Wiki by expanding it.
### Dissidia Final FantasyEdit
Handle carefully. Shocks can cause cracks.
—Description
Topaz is a trade accessory that provides Luck +3 when equipped. It can be obtained from the Falcon Course in the Duel Colosseum or from the shop by trading 1,800 gil and Elixir x3.
This article or section is a stub about an item in Dissidia Final Fantasy. You can help the Final Fantasy Wiki by expanding it.
### Final Fantasy: The 4 Heroes of LightEdit
A gem dropped by monsters, infused with very, very great power.
—Description
This article or section is a stub about an item in Final Fantasy: The 4 Heroes of Light. You can help the Final Fantasy Wiki by expanding it.
## EtymologyEdit
Topaz is a silicate mineral of aluminium and fluorine. It is the birthstone for November. | 2017-02-20 20:11:40 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "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": 0.3021124303340912, "perplexity": 13065.912788398906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170609.0/warc/CC-MAIN-20170219104610-00298-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/rate-of-climb-calculations-w-4-engine-jet-powered-aircraft.323617/ | # Rate of Climb Calculations (w/ 4 engine jet powered aircraft)
1. Jul 6, 2009
### waealu
I have this problem and it seems like it has a simple solution, however I can't seem to figure it out.
Using the following values and the previous equation calculate the rate of climb for a
typical airplane at an altitude of 8000m.
4 Engines each with 150kN of thrust.
Aircraft mass is 330 tonnes.
Drag is 500kN
Aircraft Mach number is 0.6.
I know I need to use something like dh/dt=((T-D)V)/W , but I seem to have a problem. I can figure out the airspeed, and get something like dh/dt=24,648/W , but I'm not sure where to go from there. I am not sure of the measurement units or if I need to convert anything.
Thank you
2. Jul 8, 2009
### Ja4Coltrane
Wellll... your question wasn't given too clearly. For us to help you, you should probably give more details. Regardless, I'm going to try to guess what you mean.
I'm assuming that the plane in facing at a slightly different angle from the one it is actually moving in.
So let $$\alpha$$ be the angle from the horizontal to the plane nose, and let $$\theta$$ be the angle from the horizontal to the direction of the planes velocity.
Let $$F_T$$ be the thrust force (total--sum of forces from engines) and let $$F_D$$ be the drag force. Then, you have to solve the following equations:
$$F_T \cos{\alpha} - F_D \cos{\theta} = 0$$
$$F_T \sin{\alpha} - F_D \sin{\theta} - Mg = 0$$
3. Jul 9, 2009
### waealu
Thanks for the help, but I looked over the problem yesterday and figured it out. It was easier that I originally thought. However, I forgot to return to the forum to remove the question. Thank you for the response, though. | 2017-08-22 22:48:40 | {"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": 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": 0.6689887046813965, "perplexity": 429.9944632511165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886116921.7/warc/CC-MAIN-20170822221214-20170823001214-00198.warc.gz"} |
https://math.stackexchange.com/questions/3926029/bundles-of-relative-scalars?noredirect=1 | # Bundles of relative scalars
Let $$M$$ be a smooth manifold with an atlas $$\{(U_i,\psi_i)\}$$. If $$x\in U_i\cap U_j$$, let $$a_{ji}(x)$$ denote the Jacobian matrix at $$\psi_i(x)$$ of the smooth map $$\psi_j^{-1}\circ \psi_i:\psi_i(U_i\cap U_j)\to \psi_j(U_i\cap U_j)$$. Then $$a_{ji}$$ is a smooth map $$U_i\cap U_j\to GL(n,\Bbb R)$$ and satisfies the cocycle condition $$a_{ki}=a_{kj}a_{ji}$$.
Now define $$g_{ji}=a_{ji}^m$$, where $$m$$ is a fixed positive integer, and let $$E\to M$$ be a smooth vector bundle that has the transition functions $$\{g_{ji}\}$$. Then $$E$$ is called the the bundle of relative scalars over $$M$$ of weight $$m$$, according to Steenrod's The Topology of Fibre Bundles, section 6.
I see that if $$m=1$$, then $$E\to M$$ is the tangent bundle of $$M$$. But for $$m>1$$, what is $$E$$? I mean, I haven't seen these kind of definitions in textbooks about smooth manifolds, for example, Lee's Introduction to Smooth Manifolds. But maybe these bundles have another familiar names nowadays, I hope.
I think you're misinterpreting Steenrod's definition. (Admittedly, his exposition is hard to follow on this point.) What he calls the bundle of relative scalars of weight $$\boldsymbol w$$ is the rank-$$1$$ vector bundle with transition functions $$\det(a_{ij})^w$$, where $$(a_{ij})$$ are the transition functions for the tangent bundle as you defined them above.
Modern mathematicians usually call this the bundle of densities of weight $$\boldsymbol w$$. See my answer to this question for a detailed explanation. | 2021-10-21 03:29:32 | {"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": 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": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.990779459476471, "perplexity": 101.0893705521264}, "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/1634323585380.70/warc/CC-MAIN-20211021005314-20211021035314-00171.warc.gz"} |
https://cs.stackexchange.com/questions/148108/nc-with-nearest-neighbor-gates | # NC with nearest neighbor gates
Consider a circuit belonging to the class $$\text{NC}^i$$, as defined here.
From my understanding, the circuit consists of AND, OR ar NOT gates, each of bounded fan in --- without loss of generality, let the fan in be $$2$$ --- with $$\log^{i} n$$ depth and polynomial size.
Now, let's say we define a different circuit class, $$\text{NearestNeighbourNC}^i$$, where we have AND, OR, and NOT gates, again of fan in $$2$$, with the additional restriction that each gates can only act on nearest neighbor lines. Let's say we also have nearest neighbor SWAP gates.
Is it true that compiling a general $$\text{NC}^i$$ circuit with a circuit of this new class will blow up the depth and the depth would now be $$\text{poly}(n)$$?
$$\text{NC}^i$$ has no guarantee on the locality of the inputs and whether they are nearest neighbour, so I think it should be true. Is there any clever way of compiling $$\text{NC}^i$$ circuits that preserve locality without blowing up the depth?
For each circuit in the class $$\text{NearestNeighbourNC}^i$$, consider starting with a $$\sqrt{n} \times \sqrt{n}$$ 2D grid. We will build a circuit on this grid, in layers
Each vertex denotes an input node, storing some value.
When an AND or OR gate acts on two vertices, it transforms them into one vertex storing the output value. A SWAP gate swaps the value of two vertices. A NOT gate complements the value of a vertex.
The restriction is that the gates are only between adjacent vertices.
Now, consider constructing the circuit in levels.
At each level, we apply AND, OR, or SWAP gates to some pairs of adjacent vertices, NOT gates to some vertices, and do nothing to other vertices.
After each level, we are left with some graph.
In the final level, one of the vertices is the vertex storing the final output value.
• I just mean that the inputs to each gate should come from lines that are spatially local. As in, if we have three lines, spatially separated, and an AND gate, the gate can only act on input lines 1 and 2 or 2 and 3 -- we cannot apply the gate directly to 1 and 3. Jan 3 at 7:29
• Is it clearer now? Jan 3 at 10:00
• That helps a lot, thanks. SWAP gates are not part of NC circuits, and they don't have a single output, so they don't fit in this formalism. By "adjacent" do you mean "horizontally adjacent", i.e., also must both be in both layers? It's not clear what it means for a gate to be "between" vertices; do you mean to act on two vertices?
– D.W.
Jan 3 at 21:04
• In a given level, can gate 1 at level $\ell$ take inputs from outputs 100,101 at level $\ell-1$ and gate 2 take inputs from outputs 1,2 at level $\ell-1$? Or do you also intend that the graph be planar?
– D.W.
Jan 3 at 21:07 | 2022-01-18 20:49:28 | {"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": 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": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7318491339683533, "perplexity": 580.1202079695478}, "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/1642320300997.67/warc/CC-MAIN-20220118182855-20220118212855-00085.warc.gz"} |
https://deepai.org/publication/subexponential-algorithms-for-variants-of-homomorphism-problem-in-string-graphs | # Subexponential algorithms for variants of homomorphism problem in string graphs
We consider the complexity of finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy for the problem: if H has no two vertices sharing two common neighbors, then the problem can be solved in time 2^O(n^2/3 n), otherwise there is no algorithm working in time 2^o(n), even in intersection graphs of segments, unless the ETH fails. This generalizes several known results concerning the complexity of computatational problems in geometric intersection graphs. Then we consider two variants of graph homomorphism problem, called locally injective homomorphism and locally bijective homomorphism, where we require the homomorphism to be injective or bijective on the neighborhood of each vertex. We show that for each target graph H, both problems can always be solved in time 2^O(√(n) n) in string graphs. For the locally surjecive homomorphism, defined in an analogous way, the situation seems more complicated. We show the dichotomy theorem for simple connected graphs H with maximum degree 2. If H is isomorphic to P_3 or C_4, then the existence of a locally surjective homomorphism from a string graph with n vertices to H can be decided in time 2^O(n^2/3^3/2 n), otherwise the problem cannot be solved in time 2^o(n), unless the ETH fails. As a byproduct, we obtain several results concerning the complexity of variants of homomorphism problem in P_t-free graphs. In particular, we obtain the dichotomy theorem for weighted homomorphism, analogous to the one for string graphs.
## Authors
• 11 publications
• 29 publications
• ### Almost all string graphs are intersection graphs of plane convex sets
A string graph is the intersection graph of a family of continuous arcs...
03/18/2018 ∙ by János Pach, et al. ∙ 0
• ### Complexity of the list homomorphism problem in hereditary graph classes
A homomorphism from a graph G to a graph H is an edge-preserving mapping...
10/07/2020 ∙ by Karolina Okrasa, et al. ∙ 0
• ### Optimality Program in Segment and String Graphs
Planar graphs are known to allow subexponential algorithms running in ti...
12/24/2017 ∙ by Édouard Bonnet, et al. ∙ 0
• ### Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs
For graphs G and H, a homomorphism from G to H is an edge-preserving map...
06/19/2019 ∙ by Karolina Okrasa, et al. ∙ 0
• ### Computing Maximum Independent Set on Outerstring Graphs and Their Relatives
A graph G with n vertices is called an outerstring graph if it has an in...
03/17/2019 ∙ by Prosenjit Bose, et al. ∙ 0
• ### On grounded L-graphs and their relatives
We consider the graph class Grounded-L corresponding to graphs that admi...
08/13/2018 ∙ by Vít Jelínek, et al. ∙ 0 | 2021-09-21 03:24:19 | {"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": 0.9190101027488708, "perplexity": 1285.117403846522}, "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/1631780057131.88/warc/CC-MAIN-20210921011047-20210921041047-00540.warc.gz"} |
https://worldbuilding.stackexchange.com/questions/64437/the-government-builds-an-ark-before-an-apocalypse-how-does-it-get-people-to-co/64443 | # The government builds an Ark before an apocalypse — how does it get people to come?
It’s an alternate 2023 (oh, 2012 is just so cliché), and Kanye West, in all his megalomania has nudged a toxic comet of death into a collision course with Earth. US Government hacks conclude that the comet will render the surface of the planet uninhabitable, and so draw up plans to build a massive national shelter somewhere in the Rockies (make it underground or perhaps an enormous transparent dome, but the catastrophe is such that it has to be a contained habitat). They expect the survivors to live there for decades. Space is extremely limited, so every non-essential inhabitant needs to be very well justified.
Such an operation clearly has a staggering list of essential, indispensable, and downright mission-critical personnel to make it work, else the entire colony fails. There’s the guy that wrote the stupendously complex software that keeps the dome from exploding, the brilliant scientist who is the only one that knows how to grow the wonder crops that feed the colony, the original architect who would be really nice to have around if we ever had questions about that thingamajig that holds the structure up, and that hilarious comedian who we would all be really depressed if we didn’t have him and his heart-lifting performances to cheer us all up.
All great, we have the A-team to preserve the continuity of the United States, yes?
Except no one wants to come. A surprisingly large number of personnel would rather spend their last moments with family members and loved ones, rather than save themselves and live out a regimented, grueling life in a dank, musty habitat underneath Mount Yuck.
The Government is now stuck with a dilemma of an Ark that is woefully understaffed, as no one wants to leave their loved ones behind! How does it come up with incentives to get people to come? Perhaps it lets people bring a certain number of friends and family members, but how do we make this into a workable policy? We wish to maximize the amount of psychological support colony members can expect to recieve on the “other side”, while minimizing dead weight on the colony.
Some edge example cases to consider (humans are weird, after all):
### Mel
Mel wrote the software that keeps the piranhas out of the colony’s water supply. The algorithms are so complicated (piranhas are surprisingly difficult to detect) that Mel is basically the only person who understands it. Mel and his wife belong to some religious sect, and they have twenty-three biological children.
### Dianna
Dianna is a world-class civil engineer who designs the air-filtration systems for the colony. There are only a handful of people in the world who can do what Dianna does. But Dianna is gay, and she lives in Alabama, where the county clerks are really reluctant to grant her and her partner Taylor’s marriage license, even though they’ve been together for 9 years and 34 days. (Taylor insists it was 35 days since she first asked Dianna out at 11:58 PM.) No one doubts they’re a couple, but the bureaucrats in charge of the colony say they’re not “official” without the license.
### Shifty Steve and Lil’ Marco
Shifty Steve is a one-in-a-million genetic candidate for the colony, without him the gene pool just wouldn’t be the same. Shifty Steve has no immediate family, but he makes a deal with his…er…friend, Lil’ Marco, where Steve gets all the hookers and cocaine he wants in the year preceding the comet’s arrival, as long as Steve figures out a loophole to get Lil’ Marco a spot in the colony with him.
### Rachael Straugnum–Alina and her boyfriend who she loves very dearly (he’s bae-goals)
Rachael is an agricultural scientist who pioneered miraculous advances in crop science that might just make our colony viable for feeding thousands of people post-apocalypse. Rachael is your typical urban hipster, for her and her bae, marriage is just so you know, ughhh. That’s not to say Rachael doesn’t love her partner; much to the contrary, if something were to ever happen to bae, she would probably just sit in her habitation eating ice-cream and watching depressive post-apocalyptic Netflix, crops be dammed. Rachael and her bae suppose that, since the world is ending, they might as well go to the courthouse and get that socially constructed piece of paper, but whatever.
### Lieutenant Brown
Lieutenant Brown is in charge of keeping the colony running smoothly, and his leadership skills are indispensable to the colony’s social and functional cohesion. Lieutenant Brown has a old Army buddy, who he’s been through hell and back with, and the two are practically brothers. The Lieutenant is a soldier, and he says he can deal with the loss of his best buddy, but we still worry it might adversely affect his performance as a post-apocalyptic leader if he knows the government intentionally left his best friend behind.
### Secretary Thompson
Secretary Thompson is the guy in charge of the entire operation, the big kahuna. Secretary Thompson really wants to bring his favorite sugar-baby with him (in fairness, it would be quite a waste to leave dat ass behind) even though she is of zero value to the colony.
### Mr. and Mrs. Norman
Mrs. Norman is a famous architect who designed the dome structure, and it would be really nice if we had her around to help if something ever went wrong with the dome. She has no love for her husband, Mr. Norman, and the two not-so-secretly despise one another, but if given an opportunity, Mrs. Norman will bring Mr. Norman with her for the sake of appearances, even though she knows she will probably be a lot happier letting him die in the apocalypse.
### @LUClDITYxo
LUClDITYxo (real name: Alyssa Parkland) is a world-famous beloved Twitter comedian, and no one is better at brightening up our day than her. Economists estimate that GDP would be approximately 0.14% lower were it not for the morale boost she gives millions of American workers each day. LUClDITYxo’s favorite mutual is a girl artlesszaddy_ (real name: Meghan Stearns), and the two are practically sisters even though they met on the internet. LUClDITYxo is the famous one (though artlesszaddy_ has a respectable 12.8k following), but most of her creativity actually comes from her wild online conversations with artlesszaddy_.
Keep in mind that we need a policy for who goes in the bunker, not just case-by-case judgments for the example characters given in the question. We need to select among thousands of candidates, each one as idiosyncratic as the few examples given. If case-by-case judgment is really the best way to go, we need to know how to set up the vetting process to minimize cries of “bias!” and whatnot. If there is a reviewing panel, who gets to sit on it, and what rubric are they given? Do members of the panel themselves automatically get to go?
We need a legitimate (it does not need to be fair, but must be at the least perceived as legitimate) system to sort out all the people like them, keeping in mind that there is limited space available. While no policy is perfect, we need one that will result in the most optimal outcome; for example, if we say “take your spouse at the time of the apocalypse”, do we let Rachael and Dianna get married right before the comet hits, with the cost of letting Lil’ Marco tag along (he “marries” Shifty Steve)? How do we account for relationships that might be just as essential for the psychological well-being of the survivors, but might not count as traditional “kin”?
Possible answers might take the form of:
• Incentives or “motivation” to get people like Mel or Steve to come without bringing excessive tag-alongs. Perhaps midnight kidnappings are in order?
• Mitigating the bus factor so that the colony doesn’t depend on so many essential personnel in the first place
• Smartly designed criteria for who is recruited for the colony so that people who aren’t wanted but would still be covered under a naïve policy won’t get taken (like Mr. Norman), and vice versa (for people such as Dianna and Rachael)
• Additional recruitment efforts to find replacements for “expensive” personnel like Mel
• A list of Yes/No’s for the 8 specific test cases. They are only meant to be representatives for the variety of situations that might arise. Remember that this is a colony with thousands of niche jobs that need to be filled.
• Handwaving away the technology needed to run the facility. Yes, the flesh eating piranhas are a problem and we assume that it is not possible to apply a hardware solution (read: a grate) to it. Having Mel train other people to maintain the software is fair game. Having the government spend a few million and hire new programmers to develop a more maintainable and sensible framework is fair game. Saying the software is not needed in the first place isn’t.
Side note: LUClDITYxo is included as a test case to test for bias. A good policy accounts for the fact that the people making the judgments are aware of their limited perspectives. For example, if a vetting panel is proposed as an answer, what happens if it is filled with so-called “old white men” who don’t see the value in bringing LUClDITYxo? How might this affect the perception of the project pre-apocalypse?
• How did the original Noah get kangaroos and penguins to come? – JDługosz Dec 13 '16 at 8:53
• I think the long list of people doesn’t add to the question. – JDługosz Dec 13 '16 at 8:55
• There is a close reason explained as “too story based”. That kind of detail is not worldbuilding, but plot and character, and off topic. – JDługosz Dec 13 '16 at 9:01
• Maybe you should edit your question to make it more clear that your focusing on a general policy who's to join the bunker and who not. You're asking too many questions yourself in the next to last paragraph, so the focus on the policy gets lost. – Alexander von Wernherr Dec 13 '16 at 9:33
• I did not see the original, but in its current state the list of characters does not seem to add anything at all to this question. – GrinningX Dec 13 '16 at 11:47
The first problem is what all of these people already know. Presumably, they know that the apocalypse is coming. Given that a dome/habitat has been made, they also have some idea of what kind of apocalypse it will be. That's either really good or really bad. If it's the kind of apocalypse that starts at breakfast and is over by brunch, it's all good. If it's the kind that starts at Christmas and ends at no-one-even-knows-what-that-is-anymore-because-it-was-so-amazingly-long-ago, that's bad. See, the trouble with these people is that they all love someone. It's really easy to tell them all that they can bring one person, but they will never get past what it will do to the ones they leave behind. With that in mind, the goal should be to minimize the discomfort associated with leaving one's loved ones to die horrible deaths.
First off, control information. Whatever these people don't absolutely need to know, don't tell them. Sugar-coat it. "Yes, the comet will hit in a blaze of fantastic glory and everything outside of this cool dome thing will be instantly vaporized." That's the line you want. Make it okay for the relatives that get left behind. No pain, no suffering, they won't even know it.
Secondly, everyone wants to spend time with their loved ones, so allow them their freedom until, say, three days before the big bad is about to go down. Make sure the dome is up and running well before then, but don't require the specials to be there until basically the last minute. Until then, send them home. Send them home with cash. Lots and lots of cash. Basically, give these people the absolute best last days with their families. Make it idyllic, and impress upon them the fact that, if they truly love these people, they shouldn't tell them what's coming. Just hug them, kiss them, tell them you love them, and then promise you'll be back in a few days. In short, give everyone the goodbye they wanted.
Of course, every special gets to bring exactly one other person with them. Really, these specials can't be the only people in the dome or there's really no reason to have the dome, so if each of them gets a plus one, not really a big problem. Except Mrs. Norman. For whatever reason, Mr. Norman absolutely is not allowed because he has some something or other that would endanger the colony. Mrs. Norman has to come, otherwise it would look bad, but she can't bring her husband. Also, these secondary specials do not get to know that the apocalypse is coming or you'll have the same problem with them. These secondaries only get to know that they are going somewhere with their lovers/best friends for a few days.
Also, just a thought here, but make sure your Twitter stars can actually talk to people. I mean, I don't know who these people are, but there is that not-baseless stereotype about amazing online personalities that simply cannot deal with actual in-the-flesh-people. If the point of bringing them into the dome is to entertain the inhabitants, make sure they will actually be able to once the internet goes down.
I know this creates a secondary population of seemingly useless people, but really, we don't fully understand what makes a person's genes worth keeping. It's possible that hidden in the genetic folds of "dat ass" is coding for incredibly IQ, or massive strength, or some other unexpressed trait. Every person holds the possibility of a beneficial mutation or genetic twitch, and just as you shouldn't breed two natural-bob-tail Australian shepherds, you shouldn't breed only the humans humans decided should be bred. We simply don't know enough, so even if you wanted to optimize the gene pool, you should include a few wild cards, and you might as well kill two birds with one stone. If nothing else, they can be taught useful skills. If absolutely nothing else, they'll breath carbon dioxide for your plants.
For general vetting purposes, these same rules apply. Figure out exactly what you need, find the best people for the job, tell them as little as possible while conveying what they need to know, allow them to bring ONE person if they so choose, and give them the best good-bye period you can.
As for what you'll want, picture a scaled down United States. You'll need a governing body, enforced by a police structure of some kind. You'll need food workers from growing to processing to cooking. You'll need sewers and weavers and builders and herdsmen and blacksmiths. The important consideration is how big the dome is. However many people it can support at maximum, figure on half that population, then half of that population is your specials. The secondaries will all be apprenticed to an appropriate occupation. If you end up with people who refuse to cooperate, kick them out into the post-apocalyptic wasteland. Moreover, let them know that if they get the boot, their loved one does as well.
• Larry niven messed with the idea that include the spouses just in case there is a gene for luck, becasue those people obviously have it. – John Dec 13 '16 at 18:52
• Yeah, but the luck that policy bred screwed over everyone nearby, so it didn't work out that well for Niven's characters :P – SPavel Dec 13 '16 at 20:00
Distribute N unique fist-sized rocks randomly across the country. Announce their locations and declare that anyone showing up at the gate carrying one gets in along with four other people of their choice. And then let them in when people show up.
The combat and cunning required to make it to the gate will give you a wide cross-section of humanity, include people who are driven to survive, healthy enough to travel, and fairly selected.
• that would turn America into something remeniscent of World War Z's Jerusalem Wall scene. And the cunning and strong wouldn't work because anyone with a gun can shoot down the smartest and strongest person from 1 KM away. – arthurz12345 Apr 15 '17 at 1:19
• Part of being smart is having longer range guns. :-) – SRM Apr 15 '17 at 13:50
The vast majority of normal people would prefer survival to death, so getting the plebs in line is as simple as saying you are only taking X people. You will get 100*X people to show up, demanding they be saved (make sure you have a military ready to handle these people). You can be extremely picky, and write a policy for the general public (doesn't apply to your VIPs) for who gets in and not. And if anyone complains, the Kanye-Teroid will take care of them.
## Mel
As a software engineer, I can tell you that you don't need Mel. You need a good senior engineer who is unattached and not a religious nut, a solid test environment, and another good senior engineer who is unattached and not a religious nut....heck, give them a couple environments to test things in, you don't want to cheap out. Also, there's probably a simpler solution to your problem than "crazy code only one person ever can understand." Relying on "crazy code only one person can ever understand" is a failure point. I'd take my chance with the Kanye-teroid. Even with redundant engineers, I've saved you 23 people (and they were kids at that). You can even give each of the engineers a harem of 6 mates and still come out ahead by 11 people.
## Dianna
If for some reason being gay is against your code, find one of the other "few" people who can do what Dianna does and get them to the Ark. Otherwise, there's no issue. Let both Dianna and her new missus in the Ark. As mentioned above, super complicated air filters only a few people in the world understand is a failure point, and there's probably a better way.
## Shifty Steve and Lil’ Marco
Promise Lil' Marco a spot, and instead take him to a bunker and shoot him. Or, better yet, kidnap Steve and spend the next year harvesting his spunk and shoot him too.
## Rachael Straugnum–Alina and her boyfriend who she loves very dearly (he’s bae-goals)
The good thing about crops is that they are really easy to reproduce. Also, as said above, miracle crop only one person understands and can grow ever is a failure point.
## Lieutenant Brown
This guy can't be the only person capable of running the colony. If he is truly the only person that can do it (see failure points above), then hire a biker gang to kill his army buddy, have a big trial and ask Brown what he wants done to the bikers.
## Secretary Thompson
If he's in charge, what does it matter?
## Mr. and Mrs. Norman
What do you know, that biker gang killed two people instead of just that old army guy. Also, what is with this world that there are so many areas of science critical to this colony that only one person understands? I would think that there's only so much crazy that can go into a building.
## @LUClDITYxo
Hell is living with only one comedian. Besides the GDP will drop a lot after the Kanye-teroid hits. Find some people with useful skills who are also funny.
Your colony, as described, has a lot of points of failure, and a lot of "hit by a bus" factor. If one of these people dies before they get into the colony, or a freak accident in the colony happens, then you are in a bad shape. Each of these points of failure should have 2-3 people capable of doing the job, and the ability to train more. What happens if Mel has an aneurysm? Shifty Steve gets AIDS from all the hookers and drugs? The twitter person isn't funny in more than 145 characters?
• the bus factor is the essence of the question, as you push the limits with experimental tech, there are fewer and fewer people capable of doing what you need (we’re talking about maintaining a habitat meant to withstand a global apocalypse). The question is how do we mitigate the bus factor to make the colony succeed despite its reliance on advanced experimental technology that few people understand, at least at the onset of the scenario – taylor swift Dec 13 '16 at 19:09
• If you cannot find more than one person, or the ability to teach more than one person, this specialized knowledge, then you cannot. – Marshall Tigerus Dec 13 '16 at 19:53
• Your habitat explodes (or implodes) because no one understands how it actually works. Highly coupled systems are inherently fragile in nature, and only insane people would consider this to be an acceptable situation or solution. I suspect there would be many smaller shelters with less closely coupled systems waiting for the apocalyptic failure to come and scavenge any usable things left. – Thucydides Dec 14 '16 at 6:42
The only way to make the policy workable is to take the loved ones as well. If you're going to build a long term self contained colony you also have to think about the psychological wellbeing of the personnel, that means relationships, it means friendships, it means not taking people who are going to really get other people's backs up.
You can't ask people to leave their lovers and children behind, you have to take them as well. For each person you want to take, you need to take the entire immediate family. If this hasn't been considered then whoever ran the project is an idiot and deserves to fail epically.
You can leave the pseudo celebs behind, especially when you're talking about leaving all the ordinary people anyway. Much more important to take Randall Munroe.
Apart from the useless pseudo celebrities, you're unlikely to get much dead weight. People in the intellectual upper 10% don't tend to hang around with people from the bottom 10%. Whoever they ask to bring is probably going to be useful in the long term, if not the short term. And even then, any colony requires rubbish collection and cleaners, hairdressers and telephone sanitisers. There's work to be done that doesn't require an IQ of over 130.
Let's consider team selection, since that's the game we're playing here. In my spare time I do team sports (I know that's terribly bad form for a geek but that's how I roll) and as often as not, I'm the team captain so I have to choose my team.
1) The best player. He knows he's the best player, he likes to make sure everyone else knows he's the best player, you've all met him, he has the looks and the ego to match, but he's not the best player because his ego gets in the way of him being a team player. So he's out and still doesn't understand why.
2) The guy who turns up to everything and never seems to get any better. He might be in or out, depending how much other people like him, in my case he's a real charmer and usually goes. I took him to the last tournament and he played an absolute stormer, it happens.
3) The really good player with a +1 who will only play if she can as well, but she's not as good. This is where I'm leading to with this example because it's the one that matches your situation. This player isn't one person but two, sometimes two who work really well together even if the individual skills aren't up to the spec of no 1, the average and team benefit is greater than taking any two others.
You're not considering the skills of the individual but the combined skills and combined resource drain. The guy with a wife and a dozen kids, is he really worth 14 places when compared to a guy with 2 kids who's nearly as good? Probably not, if the second guy's wife has a similar or complementary skillset, definitely not. Every single position and person has to be considered like this. Not as a single person, but a combined skillset and cost of the total group. Is Steve worth Marco? That's up to how much you want Steve, but at one in a million, I suggest taking one of the other 300 candidates in the country.
The answer in most cases for special requirements versus someone else who's nearly as good but lower cost, probably not worth it.
Ultimately what this comes down to is that, as with any team selection, there are people who, on paper, you'd like to take, but you'll have to leave behind, and people who wouldn't be your first choice, but ultimately you will be taking them. People aren't numbers, you can't treat them that way, and the tighter your resources the more you'll be spending on reviewing each person as an individual, not under a general rule.
• this doesn’t really answer the question, which is how we formulate a policy for deciding who counts as a “family member”, to maximize psychological support and minimize dead weight on the colony – taylor swift Dec 13 '16 at 9:00
• That comment might have been a better question — general and short. In fact, it might have been addressed here already, I have some recollection. – JDługosz Dec 13 '16 at 9:03
• @taylorswift, that's not your question, it would be a better question, but you've asked for policy on getting people to come, not selecting who should come as well. – Separatrix Dec 13 '16 at 9:24
• @Separatrix i assumed they would go hand-in-hand, as the premise was that the main reason people wouldn’t want to come was because they would have to leave people behind. But as a limiting factor, there are severe space and resource constraints, as stated in the original question, so you can’t just say “take everyone” – taylor swift Dec 13 '16 at 9:27
• @taylorswift, it sounds like you want three ark ships, let's call them A,B and C. In ship A you put all the elites, in ship C you put the engineers and other people who actually do stuff and in ship B you put the useless people. We all know how that story ends. – Separatrix Dec 13 '16 at 9:31
First:
Leave the celebs and criminals at home, so no LUClDITYxo, no Shifty Steve and Lil' Narco. They don't add up anything usefull
Second:
IMH, you can leave Mel behind, because he'd come with 24 mouths to feed and why are there piranhas in a colony water supply that cannot be kept out by a simple fence?
Bring Dianna, don't care if she's gay, here wife and all other civil engineers who are as good as her, we'll definitely need them in the new home.
Bring Rachel and her boyfriend. Agricultural scientist may come handy if you want to plant crops.
Bring Lt. Brown and a dozen of other young soldiers. Even if we may not have to fight in the new colony, they're valuable workers.
Bring Secretary Thompson and his sugar babe. Otherwise the whole ark will be blocked by his ego.
About policy: Every mentioned person may bring their kids, brothers and parents as long as they're not older than 50. This way the colony as a good balance of workforce and experience.
Young engineers may have great ideas, but they all are worthless if they don't know how to hold a shovel, so you also need experienced workers.
Third:
Bring experts from every scientific field, amount depending on the size of the ark.
Bring cooks, but no 5* guys, but ordinary cooks who know how to make a good stew.
And at least one barkeeper. For reasons.
• the problem with this answer is that it doesn’t give a policy, only case-by-case judgments for the example characters given in the question. We need to select among thousands of candidates, each as idiosyncratic as the examples given – taylor swift Dec 13 '16 at 9:17
I think let the Ai did it. Count every important person that would be needed including some who should do the hardwork, but also lets make the Ai cross analyze their social media, and criminal records (if we want to restart the earth, make sure we just bring the good seeds). Last, make sure the system can work with no human interference, and no one can change it, and don't let the programmer knows what the Ai real use.
So lets not get everyone with the drama of who should or shouldn't be in the ark. Lets make it cold but fair, that some of us will be picked by force of the machine and put there to prolong humanity whether we liked it or not.
I still don't understand why you need a policy more detailed than immediate family with a review board for borderline cases.
But if you need to automate it you just need to give a numeric value to each person's value to the colony then get the average of your group, then you compare your numbers to see who gets in.
example: Mel is a 6 (skill only partially useful in colony) with a his wife having no beneficial skills (and maybe a negative for the religious thing) being a 1, each kid is a 1-2 depending on age. that makes mes group. 6+1+24(1.5)=43 43/26=1.65
Dianna is a 9 (skill useful in colony and after colony, exemplar in the field) and let's say her sig other is a 3-4 becasue she is lets say a school teacher. useful but not exemplary. so Dianna's family gets a score of 9+3.5=12.5 12.5/2=6.25
So mel's family is a 1.65 vs Dianna's 6.25 clearly you take Dianna and leave Mel. in fact Mel is not getting in even with a personal perfect score.
You need to work up a ranking system but that's not too hard.
Starting points
2. potentially useful/ self sufficient
3. useful skill set
Modifiers
-1 poor health/genetics
-1 negative criminal history/ negative social effects
-1 History of violence
+1 exemplar in the field
+1 positive social effect: musician, comedian, ect
+1 instrumental to colony construction or other special consideration
+2 Critical skill necessary for rebuilding
+2 Critical skill necessary in colony function
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#### Colby
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##### Delta-Gamma-Theta Approximation
« on: March 23, 2010, 08:50:52 PM »
## Delta-Gamma-Theta Approximation
The definition of the Taylor Series is:
$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+\ldots$
Let's do some one-to-one substitutions to make the Taylor Series fit our subject, option pricing. Let $x=S_t$ and $E=S_t-S_0$ then if the "output" is the Option Price ($f(x)=C(s_t)$), the 1st derivative with respect to stock price will be Delta and the 2nd derivative with respect to stock price will be Gamma. The infinite amount of terms following the 3rd term would, in most cases, be relatively small. Therefore, in this case, those terms can be replaced by one simple error term.
Rewriting the Taylor Series equation gives us:
$C(S_t)=S_0+\Delta E + \frac{1}{2}\Gamma E^2 + \text{error term}$
Example
Using the same parameters from the 2nd example, estimate the change in call's value if the stock price increases to 210.
Recall that $S_0=200, r=0.05,\sigma=0.2$ and $\Delta=0.8554$ , so the approximate value of the call is:
$C(S_t)=C(S_0)+\Delta E + \frac{1}{2}\Gamma E^2$
$C(S_t)=27.95+0.8554(10)+0.5\Gamma(100)=36.504+50\Gamma$
The value of the call is highly dependent upon Gamma, the 2nd derivative of option price with respect to stock price. Concavity and convexity will only forecast if the stock will level off or change even more in value, so those aspects of Calculus are important for forecasting stock prices. What can we expect Gamma to be in this example?
There really isn't enough information given to calculate or predict the Gamma. All that we know is that Gamma will be the same whether it is a call or a put. Assuming Gamma to be zero makes the option follow a more linear pattern which is not a good estimation of the option itself, so an arbitrarily small value for Gamma will suffice.
Let's assume that: $\Gamma=0.02$
$36.504+50\Gamma=37.504$
The call value is expected to increase by 9.554, which is less than 10, the change in stock price. This follows the laws of arbitrage and the increase in the call can be expected with the increase in the stock price.
The Error Term in Hedging
The approximation used in the last example is actually Delta-Gamma approximation. To apply Delta-Gamma-Theta approximations to option values, the parameter of time must be introduced to the equation. Specifically, the Greek of Theta must be used. The additional error term, albeit small, will most often reduce the approximate option value because Theta is usually negative. Why is Theta usually negative? Theta measures the increase in option price with respect to the decrease in time to maturity, and options increase in value as maturity time increases due to extra room for volatility.
Less maturity time => Lower variance => Decreased expected value
The error term is measured in days, and time in the Delta-Gamma-Theta approximation is measured in years, so in order to keep all time variables equal, it must be converted like so:
$C(S_t)=S_0+\Delta E+\frac{1}{2}\Gamma E^2 + \frac{t\theta}{365}$
The number of days in a year is a matter of convention. The banker may use 360 days whereas the actuary would use 365.25 days. The investor may use actual number of days, 365 or 366, depending whether the current year is a leap year or not.
### References
W. McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006
### Contributors
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« Last Edit: November 05, 2010, 06:35:16 PM by Colby »
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May 22, 2017, 10:12:35 AM
• irbwilleo: anyone have interest in joining a college football league?
May 22, 2017, 02:55:10 PM
• kidd5jersey: How does the CFB league work?
May 22, 2017, 03:27:53 PM
May 22, 2017, 04:03:47 PM
• kidd5jersey: Irb , I would possibly be interested if I could be Jacksonville State, my alma mater
May 22, 2017, 05:28:27 PM
• irbwilleo: haha, you certainly could.... I think you may be at a disadvantage, but we could make that work
May 22, 2017, 05:43:38 PM
• irbwilleo:
May 22, 2017, 05:43:58 PM
• corbydoc: Can I get Alabama?
May 23, 2017, 01:11:34 AM
• kidd5jersey: Just got in from Cubs game a little bit ago. I'll take a look at it Irb
May 23, 2017, 02:46:44 AM
• irbwilleo: Corbydoc - You certianly can
May 23, 2017, 10:35:33 AM
• irbwilleo: Kidd5jersey - Sounds good man, let me know
May 23, 2017, 10:35:49 AM
• kidd5jersey: irb PM
May 23, 2017, 03:22:49 PM
• pandoval172: how do i disable email notifications or change my email address?
May 23, 2017, 04:14:39 PM
• kidd5jersey: Click your 'profile'. From there, click the 'modify account' tab on the left
May 23, 2017, 04:34:01 PM
• RyanJames5: I'm writing for a prospect website if anyone is interested in checking out my first article. [link]
May 26, 2017, 10:25:19 AM
• indiansnation: Nice job ryan
May 26, 2017, 03:36:02 PM
• indiansnation: looking to/move vets in busch,wcb2, and abl
May 26, 2017, 11:00:53 PM
• fantasyboi: hey extensions for TT NFL is AAV right? not total?
Yesterday at 11:24:45 PM
• Mets Donations Accepted: Indians I'm still waiting to hear back from you
Yesterday at 11:49:58 PM
• fantasyboi: mets replied
Yesterday at 11:56:20 PM
• Billy: @fantasyboi TT extensions are their highest ranking at their position over the past 2 years. It is taken by their total fantasy points scored for the entire season (so someone who missed a week or 2 due to injury might be lower than his average per week shows)
Today at 12:52:51 AM
• fantasyboi: i know that. im asking if the contract value is AAV or total contract? im pretty sure its AAV
Today at 12:58:59 AM
• Billy: The contract value you get from extensions is the yearly contract. So if they rank #1 at QB they get a $15m per year extension. Then because the salary is greater than$9.9m, they get a max of 5 year deal
Today at 02:00:08 AM
• BHows: Tarheels-PM
Today at 11:40:10 AM
• chrisetc21: Is Fantraxed being a pain in the ass for anyone else?
Today at 05:26:13 PM | 2017-05-29 20:41:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 12, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5513701438903809, "perplexity": 8278.349983388569}, "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/1495463612553.95/warc/CC-MAIN-20170529203855-20170529223855-00275.warc.gz"} |
http://openstudy.com/updates/559eba34e4b05bcb08a12494 | ## anonymous one year ago Which inequality will have a solid boundary line and a shaded region above its graph? x − y ≥ 3 2x − 3y ≤ 3 3y − x < 2 2x + y < 7
1. pooja195
$\huge~\rm~< = dotted ~~~line$ $\huge~\rm~\le=Solid~~line$
2. pooja195
Which options can we elminate ?
3. anonymous
the bottom two
4. mathstudent55
$$y \lt mx + b$$ means shade below dashed line $$y \gt mx + b$$ means shade above dashed line $$y \le mx + b$$ means shade below solid line $$y \ge mx + b$$ means shade above solid line Solve each inequality for y, and compare with 4 statements above. Remember that in solving an inequality, if you multiply or divide both sides by a negative number, the inequality sign changes direction.
Find more explanations on OpenStudy | 2016-10-28 04:42:26 | {"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": 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": 0.6704439520835876, "perplexity": 1556.9919069123239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721555.54/warc/CC-MAIN-20161020183841-00521-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2592062/seeking-for-proof-of-v-f-oplus-f-perp-without-the-use-of-orthogonal-basis | # Seeking for proof of $V=F\oplus F^\perp$ without the use of orthogonal basis
Let $V$ a finite dimensional linear space and $F$ a subspace of $V$. I want to prove that the direct sum of $F$ and its orthogonal complement $F^\perp$ is the whole space $V$: $$F\oplus F^\perp=V$$ I know that the proof is pretty easy if we use an orthogonal basis, but how we prove it without using the orthogonal basis? The problem is how to prove that $F+F^\perp=V$?
• Is the claim true for infinite-dimensional $V$? How would you distinguish finite from infinte-dimensional without referencing dimension and hence bases? – Hagen von Eitzen Jan 4 '18 at 18:53
• Here's a try, assuming we're working over the reals: given $v$ in $V$, let $w$ be the element of $F$ closest to $v$, and show that $v-w$ is in $F^{\perp}$. – Gerry Myerson Jan 4 '18 at 18:58
• Do you get to know about dual spaces? Even without an inner product, for finite-dimensions, we can get the right dimension-counting via dimension theorem applied to the image of the algebraic transpose of some linear map $T$ whose kernel is $F$. – Christopher A. Wong Jan 4 '18 at 19:03
Let $(r_1,\ldots,r_p)$ a basis of $F$ and denote $A=(r_1,...,r_p)^T\in \mathcal M_{p,n}(\Bbb R)$ the matrix where $r_i$ is its $i$-th row and $n=\dim V$. We have $$x\in F^\perp \iff \langle r_i,x\rangle =0 \forall i\iff Ax=0\iff x\in\ker A$$ So we get $F^\perp=\ker A$, hence due to rank-nullity theorem $$\dim F^\perp= \dim \ker A=n-rk(A)=n-rk(A^T)=n-p$$ Finally, with the fact $F\cap F^\perp =\{0\}$ we can conclude the desired results. | 2019-12-14 13:34:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9357615113258362, "perplexity": 136.27334702644245}, "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-2019-51/segments/1575541157498.50/warc/CC-MAIN-20191214122253-20191214150253-00550.warc.gz"} |
https://cognitate.co.uk/maths-mixed-questions-7-9/ | Mixed questions grades 7-9
Calculators are allowed for all questions unless indicated by the question. There are 5 questions for this topic.
Question 1:
Calculate the equation of the line perpendicular to the line $2y +3x = 8$ and passes through the point $(6, 1)$ (3 marks)
Rearranging the equation of line in the form $y = mx + c$:
$y =-1.5x + 4$
The new gradient is $\frac{2}{3}$ (1 mark)
Using the given coordinates:
$1 = \frac{2}{3}(6) + c$
$1 = 4 + c$
$c = -3$ (1 mark)
$y = \frac{2}{3}x -3$ (1 mark)
Question 2: (Non-calculator)
Write the following expression as a single fraction:
$\frac{x+1}{3} + \frac{2x+2}{6}$ (3 marks)
Multiply the first term by 2 in the numerator and the denominator:
$\frac{x+1}{3} = \frac{2x+2}{6}$ (1 mark)
Multiply the second term by 3 in the numerator and the denominator:
$\frac{x-6}{2}= \frac{3x-18}{6}$ (1 mark)
Now they both have a common denominator we can add them together:
$\frac{2x+2}{6} + \frac{3x-18}{6} = \frac{5x-16}{6}$ (1 mark)
Question 3:
$f(x) = 3x +2, g(x) = 1-3x$
A. Calculate $g(3)$(1 mark)
B. Calculate $gf(3)$ (3 marks)
$g(3) = 1-3(3) = -8$ (1 mark)
$f(3) = 3(3) +2 = 11$ (1 mark)
$g(11) = 1-3(11) = -32$ (2 marks)
Question 4:
Josh has 500ml of paint to paint a model house. The real house is 40 times larger than the model.
Calculate the amount of paint needed for the real house in litres. (2 marks)
Paint needed is $500 \times 40^2 = 800000ml = 800$ litres (2 marks)
40 is squared because the relationship between the surface area of the real house and the model house is not linear, it is squared. If it was the volume, it would be cubed.
Question 5:
A colony of bacteria grows by 9% every hour. The mass at the start point is $4.5g$
A. Determine the formula for mass ($M$) in $g$, at time $t$ hours since the start point. (2 marks)
B. Estimate the mass of the bacteria after 4 hours and 30 minutes to 2dp. (2 marks)
C. Why can you not calculate the exact value for part B? (1 mark)
The mass increases by 9% every hour.
The original mass is $4.5g$, as the colony is growing exponentially. The rate of increase is $1.09^t$ (1 mark)
Therefore $M = 4.5 \times 1.09^t$ (1 mark)
$M = 4.5 \times 1.09^t , t = 4.5$ (1 mark)
Therefore $M = 4.5 \times 1.09^{4.5} = 6.63g$ (1 mark)
The colony grows 9% every hour. This doesn’t imply the colony grows 4.5% every half hour. If it did, then that colony would grow 4.5% for the second half hour, which results in 9.2% growth an hour, not 9%. We don’t know the rate of growth for half an hour, only the growth of an hour. (1 mark)
Accept any reasonable answer showing understanding of exponential growth. | 2020-10-29 13:38:15 | {"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": 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": 0.7923848032951355, "perplexity": 1103.9365963457221}, "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-2020-45/segments/1603107904287.88/warc/CC-MAIN-20201029124628-20201029154628-00305.warc.gz"} |
https://www.groundai.com/project/synthesis-of-linear-quantum-stochastic-systems-via-quantum-feedback-networks/ | A Proof of Theorem 5
# Synthesis of linear quantum stochastic systems via quantum feedback networks1
## Abstract
Recent theoretical and experimental investigations of coherent feedback control, the feedback control of a quantum system with another quantum system, has raised the important problem of how to synthesize a class of quantum systems, called the class of linear quantum stochastic systems, from basic quantum optical components and devices in a systematic way. The synthesis theory sought in this case can be naturally viewed as a quantum analogue of linear electrical network synthesis theory and as such has potential for applications beyond the realization of coherent feedback controllers. In earlier work, Nurdin, James and Doherty have established that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simpler one degree of freedom quantum harmonic oscillators, together with a direct interaction Hamiltonian which is bilinear in the canonical operators of the oscillators. However, from an experimental perspective and based on current methods and technologies, direct interaction Hamiltonians are challenging to implement for systems with more than just a few degrees of freedom. In order to facilitate more tractable physical realizations of these systems, this paper develops a new synthesis algorithm for linear quantum stochastic systems that relies solely on field-mediated interactions, including in implementation of the direct interaction Hamiltonian. Explicit synthesis examples are provided to illustrate the realization of two degrees of freedom linear quantum stochastic systems using the new algorithm.
## 1 Introduction
Linear quantum stochastic systems (see, e.g., [1, 2, 3, 4]) arise as an important class of models in quantum optics [5] and in phenomenological models of quantum RLC circuits [6]. They are used, for instance, to model optical cavities driven by coherent laser sources and are of interest for applications in quantum information science. In particular, they have the potential as a platform for realization of entanglement networks [7, 8, 9] and to function as sub-sytems of a continuous variable quantum information system (e.g., the scheme of [10]). Recently, there has also been interest in the possibilities of control of linear quantum stochastic systems with a controller which is a quantum system of the same type [11, 2, 3, 12] (thus involving no measurements of quantum signals), often referred to as “coherent-feedback control”, and an experimental realization of a coherent-feedback control system for broadband disturbance attenuation has been successfully demonstrated by Mabuchi [12]. Linear quantum stochastic systems are a particularly attractive class of quantum systems to study for coherent control because of their simple structure and complete parametrization by a number of matrix parameters.
A natural and important question that arose out of the studies on coherent control is how an arbitrary linear quantum stochastic system can be built or synthesized in a systematic way, in the quantum optical domain, from a bin of quantum optical components like beam splitters, phase shifters, optical cavities, squeezers, etc; this can be viewed as being analogous to the question in electrical network synthesis theory of how to synthesize linear analog circuits from basic electrical components like resistors, capacitors, inductors, op-amps, etc. The synthesis problem is not only of interest for quantum control, but is a timely subject in its own right given the current intense research efforts in quantum information science (see, e.g., [13]). These developments present significant opportunities for investigations of a network synthesis theory (e.g., [14] for linear electrical networks) in the quantum domain as a significant direction for future development of circuit and systems theory. In particular, such a quantum synthesis theory may be especially relevant for the theoretical foundations, development and design of future linear photonic integrated circuits.
As a first step in addressing the quantum synthesis question, Nurdin, James and Doherty [4] have shown that any linear quantum stochastic system can, in principle, be synthesized by a cascade of simpler one degree of freedom quantum harmonic oscillators together with a direct interaction Hamiltonian between the canonical operators of these oscillators. Then they also showed how these one degree of freedom harmonic oscillators and direct bilinear interaction Hamiltonian can be synthesized from various quantum optical components. However, from an experimental point of view, direct bilinear interaction Hamiltonians between independent harmonic oscillators are challenging to implement experimentally with current technology for systems that have more than just a few degrees of freedom (possibly requiring some complex spatial arrangement and orientation of the oscillators) and therefore it becomes important to investigate approximate methods for implementing this kind of interaction. Here we propose such a method by exploiting the recent theory of quantum feedback networks [15]. In our scheme, the direct interaction Hamiltonians are approximately realized by appropriate field interconnections among oscillators. The approximation is based on the assumption that the time delays required in establishing field interconnections are vanishingly small, which is typically the case in quantum optical systems where quantum fields propagate at the speed of light.
The paper is organized as follows. Section 2 provides a brief overview of linear quantum stochastic systems and the concatenation and series product of such sytems. Section 3 recalls the notion of a model matrix and the concatenation of model matrices, while Section 4 recalls the notion of edges, their elimination, and reduced Markov models. Section 5 reviews a prior synthesis result from [4]. Section 6 presents the main results of this paper and an example to illustrate their application to the realization of a two degrees of freedom linear quantum stochastic system. In Section 7 the special class of passive linear quantum stochastic systems is introduced and it is shown that such systems can be synthesized by using only passive sub-systems and components. Another synthesis example for a passive system is also presented. Finally, Section 8 offers the conclusions of this paper. In order to focus on the results, all proofs are collected together in the Appendix.
## 2 Linear quantum stochastic systems
This section serves to recall some notions and results on linear quantum stochastic systems that are pertinent for the present paper. A relatively detailed overview of linear quantum stochastic systems can be found in [4] and further discussions in [1, 2, 16, 3], thus they will not be repeated here.
Throughout the paper we shall use the following notations: , will denote the adjoint of a linear operator as well as the conjugate of a complex number, if is a matrix of linear operators or complex numbers then , and is defined as , where denotes matrix transposition. We also define, and and denote the identity matrix by whenever its size can be inferred from context and use to denote an identity matrix. Similarly, denotes a matrix with zero entries whose dimensions can be determined from context, while denotes a matrix with specified dimension with zero entries. Other useful notations that we shall employ is (with square matrices) to denote a block diagonal matrix with on its diagonal block, and ( a square matrix) denotes a block diagonal matrix with the matrix appearing on its diagonal blocks times.
Let be the canonical position and momentum operators of a many degrees of freedom quantum harmonic oscillator satisfying the canonical commutation relations (CCR) . The integer will be referred to as the degrees of freedom of the oscillator. Letting then these commutation relations can be written compactly as:
xxT−(xxT)T=2iΘ,
with and . Here a linear quantum stochastic system is a quantum system defined by three parameters: (i) A quadratic Hamiltonian with , (ii) a coupling operator where is an complex matrix and (iii) a unitary scattering matrix . We also assume that the system oscillator is in an initial state with density operator . For shorthand, we write or . The time evolution, in the interaction picture, of () is given by the quantum stochastic differential equation (QSDE):
dX(t) =AX(t)dt+B[dA(t)dA(t)#];X(0)=x. dY(t) =CX(t)dt+DdA(t), (1)
with
A =2Θ(R+I{K†K});B=2iΘ[−K†SKTS#]; C =K;D=S,
where is a vector of input vacuum bosonic noise fields and is a vector of output fields that results from the interaction of with the harmonic oscillator. Note that the dynamics of and are linear. We refer to as the system matrices of . For the case when , we shall often refer to the linear quantum stochastic system as a one degree of freedom (open quantum harmonic) oscillator.
Elements of and may be partitioned into blocks. For example, may be partitioned as and as , where and are vectors of bosonic input and output field operators of length and , respectively, such that . We refer to and as the multiplicity of and , respectively. It is important to keep in mind that the sum of the multiplicities of all input and output partitions sum up to , the total number of all input and output fields. With this partitioning, a linear quantum stochastic system may be viewed as a quantum device having input ports and output ports as illustrated in Figure 1. The multiplicity of a port is then defined as the multiplicity of the input or output fields coming into or going out of that port.
Let us also recall the notion of the concatenation () and series products () between open quantum systems developed in [17]. The concatenation product between and defines another quantum linear system given by:
G1⊞G2=(diag(S1,S2),[LT1LT2]T,H1+H2),
while if and have the same number of input (and output) fields the series product defines another linear quantum system given by:
G2◃G1=(S2S1,L2+S2L1,H1+H2+I{L†2S2L1}).
Note that in the definition of the concatenation and series product, it is not required that the elements of and and of and are commuting. That is, and may possibly be sub-components of the same system. Also both operations are associative, but neither operations are commutative. By the associativity, the operations and are unambiguously defined.
The concatenation product corresponds to collecting together the parameters of and to form one larger concatenated system, and the series product is a mathematical abstraction of the physical operation of cascading onto , that is, passing the output fields of as the input fields to . The cascaded system is then another linear quantum stochastic system with parameters given by the series product formula.
## 3 The model matrix and concatenation of model matrices
The system can also be represented using a so-called model matrix [15]. This representation will be particularly useful for the goals of the present paper. For the model matrix representation is given by (the partitioned matrix):
M(G)=[−iH−12L†L−L†SLS], (2)
with the understanding that if is partitioned as with and and is partitioned accordingly as with and , then the model matrix above can be expressed with respect to this partition as:
M(G)=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−iH−12∑noutj=1L†jLj−∑noutj=1L†jSj1…−∑noutj=1L†jSjninL1S11…S1nin⋮⋮⋱⋮LnoutSnout1…Snoutnin⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (3)
Also, with respect to a particular partitioning of , it is convenient to attach a unique label to each row and column of the partition. For example, for the partitioning (3) we may give the labels for the first, second, …, -th row of , respectively, and for the first, second, …, -th column of , respectively. Moreover, with respect to this labelling (and analogously for any other labelling scheme chosen), elements of the blocks are denoted as:
Ms0r0(G) =−iH−12nout∑j=1L†jLj;Ms0rk(G)=−nout∑j=1L†jSjk,k>0; Msjr0(G) =Lj,j>0;Msjrk=Sjk,j,k>0.
Since is another representation of a physical system described by , and can be identified directly from the entries of , we will often omit the and for brevity write a model matrix simply as and denote its entries by , with ranging over row labels and ranging over the column labels. Thus, we also refer to the triple in (2) as parameters of the model matrix .
Several model matrices can be concatenated to form a larger model matrix. Such a concatenation corresponds to collecting together the model parameters of the individual matrices in a larger model matrix and is again denoted by the symbol . If and then the concatenation is defined as:
M(G1)⊞M(G2) =⎡⎢ ⎢⎣−iH1−iH2−12L†1L1−12L†2L2−L†1S1−L†2S2L1S10L20S2⎤⎥ ⎥⎦ =M(G1⊞G2).
## 4 Edges, Elimination of Edges, and Reduced Markov Models
Following [15], a particular row partition labelled with in a model matrix can be associated with an output port (having multiplicity ) while a particular column partition with can be associated with an input port (having multiplicity ). In a system which is the concatenation of several sub-systems, it is possible to connect an output port from one sub-system to an input port of another sub-system (possibly the same sub-system to which the output port belongs) to form what is called an internal edge denoted by . For this connection to be possible, the ports and must have the same multiplicity. Such an edge then represents a channel from port to port . All ports which are connected to other ports to form an internal edge or channel are referred to as internal ports and fields coming into or leaving such ports are called internal fields. All other input and output ports that are not connected in this way are viewed as having semi-infinite edges (since they do not terminate at some input or output port, as appropriate) and are referred to as external ports and the associated semi-infinite edges are referred to as external edges. Fields coming into or leaving external ports are called external fields. From a point of view in line with circuit theory, one may think of a linear quantum stochastic system as being a “node” on a network and quantum fields as quantum “wires” that can connect different nodes.
In any internal edge , there is a finite delay present due to the time which is required for the signal from port to travel to port . As a consequence of these finite time delays, concatenated systems with internal edges cannot be represented by a Markov model such as presented in Section 2 (see [4] and the references therein for an overview of Markov models). However, as shown in [15], the non-Markov model converges to a reduced Markov model in the limit that the time delay on all internal edges go to zero. That is, for negligibly small time delays, the reduced Markov model acts as an approximation of the non-Markov model. In particular, such a reduced model serves as a powerful approximation of quantum optical networks in which signals travel at the speed of light and the time delay can be considered to be practically zero if the internal input and output ports are not extremely far apart. We recall the following results:
###### Theorem 1
[15, Theorem 12 and Lemma 16] Let , , be the time delay for an internal edge and assume that is invertible. Then in the limit that , with the edge connected reduces to a simplified model matrix with input ports labelled and output ports labelled (i.e., the connected ports and are removed from the labelling and the associated row and column removed from ). The block entries of are given by:
(Mred)αβ=Mαβ+Mαrj(1−Skj)−1Mskβ,
with and . is the model matrix of a linear quantum stochastic system with parameters:
(Sred)pq =Spq+Spj(I−Skj)−1Skq (Lred)p =Lp+Spj(I−Skj)−1Lk Hred =H+nout∑p=1I{L†pSpj(I−Skj)−1Lp},
for all and .
Several internal edges may be eliminated one at a time in a sequence leading to a corresponding sequence of reduced model matrices. The following result shows that such a procedure is unambiguous:
###### Theorem 2
[15, Lemma 17] The reduced model matrix obtained after eliminating all the internal edges in a set of internal edges one at a time is independent of the sequence in which these edges are eliminated.
Suppose that can be partitioned as:
⎡⎢ ⎢⎣−iH−1/2L†iLi−1/2L†eLe−L†iSii−L†iSei−L†eSie−L†iSiiLiSiiSieLeSeiSee⎤⎥ ⎥⎦, (4)
where the subscript refers to “internal” and to “external”. That is, parameters with subscript or pertain to internal ports, those with subscript or pertain to external ports, while and pertain to scattering of internal fields to external fields and vice-versa, respectively. Interconnection among internal input and output ports can then be conveniently encoded by a so-called adjacency matrix. Let denote the total multiplicity of internal input and output ports and let us view a port with multiplicity as distinct ports of multiplicity . Suppose that these multiplicity 1 ports are numbered consecutively starting from 1, then an adjacency matrix is an square matrix whose entries are either or with ( only if the -th output port and the -th input port form a channel or internal edge. Note that at most only a single element in any row or column of can take the value 1. Internal edges can be simultaneously eliminated as follows:
###### Theorem 3
[15, Section 5] Suppose that has a partitioning based on internal and external components as in (4) and that connections between internal ports have been encoded in an adjacency matrix . If exists, then the reduced model matrix after simultaneous elimination of all internal edges has the parameters:
Sred =See+Sei(η−Sii)−1Sie Lred =Le+Sei(η−Sii)−1Li Hred =H+∑j=i,eI{L†jSji(η−Sii)−1Li}.
## 5 Prior work
Let be a linear quantum stochastic system with and (). Let so that and satisfies the commutation relations of Section 2. Then the following result holds:
###### Theorem 4
[4] Let . Then
G=(0,0,Hd)⊞(Gn◃…◃G2◃G1),
where and .
The theorem says that can in principle be constructed as the cascade connection of the one degree of freedom oscillators , together with a direct bilinear interaction Hamiltonian which is the sum of bilinear interaction Hamiltonians between each pair of oscillator and . This construction is depicted in Figure 2. It was then shown how each can be constructed from certain basic quantum optical components and how can be implemented between any pair of oscillators. However, a drawback of this approach, based on what is feasible with current technology, is the challenging nature of implementing . This may possibly involve complex positioning and orientation of the oscillators and thus practically challenging for systems with more than just a few degrees of freedom. Although advances in experimental methods and emergence of new technologies may eventually alleviate this difficulty, it is naturally of immediate interest to also explore alternative methods of implementing this interaction Hamiltonian, at least approximately. In the next section, we propose such an alternative synthesis by exploiting the theory of quantum feedback networks that has been elaborated upon in preceding sections of the paper.
## 6 Main synthesis results
For , let for , and , with , and , , and with . Here is as defined in the previous section. Let for , and note that with , , and as already defined.
Consider now the model matrix for the concatenated system , see Figure 3. With respect to the natural partitioning of induced by the ’s (via their concatenation), we label the first rows of as , the next rows as , the rows after that and so on until the last rows are labelled . Similarly, we label the first columns of as , the next columns as , the columns after that and so on until the last columns . On occasions, we will need to write a bracket around one of both of the subscripts of or (e.g., as in or ).
###### Theorem 5
Let the output port be connected to the input port to form an internal edge/channel for all . Assuming that is invertible , then the reduced model matrix obtained by allowing the delays in all internal edges go to zero has parameters given by:
Sred = diag(S11,S22,…,Snn) Lred = (LT11,LT22,…,LTnn)T Hred = n∑k=1Hk+n−1∑j=1n∑k=j+1I{[L†jkL†kj][I−Sjk−SkjI]−1[LjkLkj]}.
The proof of the theorem is given in Appendix A. We will also exploit the following lemma:
###### Lemma 6
For any real matrix and unitary complex numbers and satisfying , there exist complex matrices and such that . In fact, a pair and satisfying this is given by with an arbitrary non-zero real number and , where . Or, alternatively, and
.
See Appendix B for a proof of the lemma. As a consequence of the above theorem and lemma, we have the following result:
###### Corollary 7
Let whenever , and for all . Also, let , and with satisfying , and ( and given), where , and the pair () be given by:
Kjk = [κjkiκjk] Kkj = 2i[10][−K†jkΔ∗jkKTjkΔjk]−1(Rjk−I{KTjK#k}), (5)
or
Kkj = [κjkiκjk] Kjk = 2i[10][K†kjΔjk−KTkjΔ∗jk]−1(Rjk−I{KTjK#k})T, (6)
where , and is an arbitrary non-zero real constant for all . Then the reduced Markov model has the decomposition with and for . Moreover, the network formed by forming the series product of within the concatenated system and defined by is a linear quantum stochastic system with parameters given by:
Snet = Im; Lred = Kx,K=[K1K2…Kn]; Hred = 12xTRx,R=[Rjk]j,k=1,…,n.
In other words, realizes a linear quantum stochastic system with the above parameters.
###### Remark 8
Note that the series connection can be viewed as forming and futher eliminating the internal edges in or and, hence, is in essence a special case of a reduced Markov model [15].
The proof of the corollary is given in Appendix C. The corollary shows that an arbitrary linear quantum stochastic system can be realized by a quantum network constructed according to Theorem 5 and the corollary, with an appropriate choice of the parameters , and (). From here, any system can then be easily obtained as . That is, as a cascade of a static network that realizes the unitary scattering matrix and the system (see [17, 4]).
###### Example 9
Consider the realization problem of a two degrees of freedom linear quantum stochastic system () with
K =[K1K2]=[3/2i/21i](Kj∈C1×2,j=1,2); R =[R11R12RT12R22]=⎡⎢ ⎢ ⎢⎣20.5110.53−1−11−1101−101⎤⎥ ⎥ ⎥⎦(Rjk∈R2×2,j,k=1,2).
Let and . Define and , with certain parameters still to be determined. Choose and , so that as required in Corollary 7. Then set , , and . Compute and set . Then by Corollary 7 we set and compute , and . Thus, we have determined all the parameters of and . Labelling the ports of and according to the convention adopted in this section, can be implemented by concatenating and and eliminating the internal edges and to form and then eliminating the edge (cf. Remark 8) to obtain as an approximate realization of . This realization is illustrated in Figure 4. and can then be physical realized in the quantum optics domain following the constructions proposed in [4]. Using the schematic symbols of [4], a quantum optical circuit that is a physical realization of is depicted in Figure 5.
## 7 Synthesis of passive systems
Let us now consider a special class of linear quantum stochastic systems that we shall refer to as passive linear quantum stochastic systems, for a reason that is explained below, and show that any such system can be built from passive components. This class of systems has also been considered in, e.g., [18, 19].
For let be the annihilation operators for mode and define . Then satisfies the CCR
Unknown environment 'array% =diag(In,−In).
Moreover, note that:
[aa#]=[ΣΣ< | 2019-06-25 07:23:48 | {"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": 0.8478984236717224, "perplexity": 432.6601898936887}, "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-2019-26/segments/1560627999814.77/warc/CC-MAIN-20190625072148-20190625094148-00167.warc.gz"} |
http://jocelynchi.com/gettingtothebottom/software.html | # Getting to the Bottom
A Series for Optimization Methods for Statistics and Machine Learning
## Software
This is the companion R package for the Getting to the Bottom series of articles for learning optimization methods for statistics and machine learning written for Statisticsviews.com by Jocelyn Chi and Eric Chi.
### Requirements
You’ll need R to use this package. If you don’t already have R, you can download it at the following links for Mac OS X or Windows. We also highly recommend the RStudio IDE.
### Installation
install.packages("gettingtothebottom")
### Updating the Package
Since we’ll be adding new functions for each article, you’ll want to make sure that you’re running the latest version of the package when replicating the examples in the articles.
If you’re using RStudio, you can check for updates by selecting the Tools menu, selecting Check for Package Updates, and then checking the packages you’d like to update.
If you’re not using RStudio, you can also update via the update.packages() command. This command will go through each of the packages you have installed and prompt you for whether or not you want to update it. You can press the esc key to exit this function at any time.
### Documentation
After loading the package with library(gettingtothebottom), you can browse the documentation for function usage and examples via the help pages with help(gettingtothebottom).
Additional detail on the functions and datasets included in this package can also be found in the accompanying Getting to the Bottom article for each optimization algorithm. | 2019-04-22 02:14:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.23687581717967987, "perplexity": 1152.056254773306}, "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-2019-18/segments/1555578533774.1/warc/CC-MAIN-20190422015736-20190422041736-00289.warc.gz"} |
https://www.earthdoc.org/content/papers/10.3997/2214-4609.201405971 | 1887
### Abstract
This paper addresses two basic today's issues, namely the release of CO2 in the atmosphere from fossil fuels combustion and the depletion of fossil fuels resources. The objective here is double: mitigate the CO2 emissions and increase the resources of fossil fuels.
/content/papers/10.3997/2214-4609.201405971
2001-06-11
2022-06-30 | 2022-06-30 08:24:58 | {"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": 0.8530434370040894, "perplexity": 2165.1283154557345}, "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-2022-27/segments/1656103669266.42/warc/CC-MAIN-20220630062154-20220630092154-00394.warc.gz"} |
https://academy.vertabelo.com/course/python-data-structures/tuples/list-of-tuples/lists-of-tuples | On January 21th at 14:15 UTC , progression through exercises will be unavailable for 10 minutes due to a planned maintenance break.
Deals Of The Week - hours only!Up to 80% off on all courses and bundles.-Close
Recap
Tuples as return types in functions
Tuples as records
List of tuples
10. Lists of tuples
Summary
## Instruction
Good job! We mentioned that tuples are often used to represent real-world objects. Because we typically have multiple objects in any IT system, we need a way of keeping them together. That's why we often create lists of tuples. Take a look:
cars_to_sell = [
('Volvo', 18000, 137000),
('BMW', 23000, 80000),
('Opel', 12000, 75000),
('Ford', 14500, 100000)
]
In the list above, each element is a tuple that represents a single car to sell.
## Exercise
Create a list named new_hires that will contain three tuples, each representing a single new employee. Each tuple should have three elements: the employee's name, position, and monthly salary. Include the following people:
1. Mark Adams, an SQL Analyst who earns $4,000 2. Leslie Burton, an HR Specialist who earns$2,300
3. Dorothy Castillo, a UX Designer who earns \$3,100
### Stuck? Here's a hint!
The first tuple should look like this:
('Mark Adams', 'SQL Analyst', 4000) | 2021-01-25 19:05:18 | {"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": 0.18822267651557922, "perplexity": 6331.342990558018}, "config": {"markdown_headings": true, "markdown_code": false, "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-2021-04/segments/1610703644033.96/warc/CC-MAIN-20210125185643-20210125215643-00719.warc.gz"} |
https://socratic.org/questions/what-is-the-slope-of-the-line-whose-equation-is-y-2x-7 | # What is the slope of the line whose equation is y=2x+7?
Feb 13, 2017
The slope is $\textcolor{red}{m = 2}$
#### Explanation:
Because this equation is already in the slope-intercept form we can obtain the slope directly from the equation.
The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$
Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.
Or, for this equation:
$y = \textcolor{red}{2} x + \textcolor{b l u e}{7}$
Therefore the slope is $\textcolor{red}{m = 2}$ | 2021-11-28 18:56:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.7034677267074585, "perplexity": 364.53787316960154}, "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/1637964358570.48/warc/CC-MAIN-20211128164634-20211128194634-00213.warc.gz"} |
http://www.math-only-math.com/sum-of-angles-of-a-quadrilateral.html | # Sum of Angles of a Quadrilateral
Sum of angles of a quadrilateral are discussed here. We have now two triangles in the below figure.
We know, the sum of the angles of a triangle = 180°
Since there are two triangles,
therefore, the sum of two triangles is 180° + 180° = 360°
Note: The sum of the four angles is 360°.
For Example:
1. In a quadrilateral ABCD, ∠A = 100°, ∠B = 105° and ∠C = 70°, find ∠D.
Solution:
Here the sum of the four angles
or, ∠A + ∠B + ∠C + ∠D = 360°
We know, ∠A = 100°, ∠B = 105° and ∠C = 70°
or, 100° + 105° + 70° + ∠D = 360°
or, 275° + ∠D = 360°
∠D = 360° - 275°
Therefore, ∠D = 85°
2. Find the measure of the missing angles in a parallelogram, if ∠A = 70°.
Solution:
We know the opposite angles of a parallelogram are equal.
So, ∠C will also measure 70°
Sum of angles = 360°
∠A + ∠B + ∠C + ∠D = 360°
or, 70° + ∠B + 70° + ∠D = 360° (We know, ∠A = ∠C )
or, ∠B + ∠D + 140° = 360°
or, ∠B + ∠D = 360° - 140°
or, ∠B + ∠D = 220°
But ∠B = ∠D (Because opposites angles of a parallelogram are equal)
∠B = ∠D
= 220° ÷ 2
= 110°
Therefore, ∠B = 110°, ∠C = 70° and ∠ D = 110°
Parallel Lines.
Perpendicular Lines.
Construction of Perpendicular Lines by using a Protractor.
Sum of Angles of a Quadrilateral. | 2017-09-25 13:25:37 | {"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": 0.8299437165260315, "perplexity": 7435.267573697119}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818691830.33/warc/CC-MAIN-20170925130433-20170925150433-00505.warc.gz"} |
https://www.physicsforums.com/threads/acceptible-answer.250647/ | 1. Aug 18, 2008
### franky2727
starting with question of find the general sollution of the differential equation
X2y'=y2+3xy+X2 would an acceptable answer be y=-x(ln|x|+c+1) i would show all my working but my camera isnt working so i'll save you must the trouble and just skip to a part that i know is correct where dx/x=(v+1)-2dv
where v=y/x. thanks in advance i think its right just want a second oppinion before plowing it into an exam
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Aug 18, 2008
### Defennder
I don't understand your notation. Is 'X' different from 'x'?
3. Aug 19, 2008
### franky2727
no just a bit daft with the caps lock button
4. Aug 19, 2008
### HallsofIvy
Staff Emeritus
Well, if y= y=-x(ln|x|+c+1), then y'= -(ln|x|+ c+1)- 1. Putting those into the differential equation, x2y'=y2+3xy+x2 would give you x2(-ln|x|+ c+ 1)- 1)= (ln|x|+ c+ 1)2+ 3x(-ln|x|+ c+ 1)+ x2. Since the right side is clearly going to involve "(ln|x|)2", I don't see how those are going to be equal.
I suspect you have integrated incorrectly. Yes, this is a homogenous equation and the substitution v= y/x gives
$$\frac{dv}{(v+1)^2}= \frac{dx}{x}$$
Integrating both sides of that gives
$$-\frac{1}{v+1}= ln|x|+ C$$
so
$$v+1= -\frac{1}{ln|x|+ C}$$
$$\frac{y}{x}= -1-\frac{1}{ln|x|+ C}$$
$$y= -x-\frac{x}{ln|x|+ C}$$
5. Aug 19, 2008
### Defennder
I worked it out and did not get the same answer as you did. ln(x)+c is supposed to be the denominator of some fraction in y.
6. Aug 19, 2008
### franky2727
got it now, was just checking more to see if it would be acceptible to leave logs and things in the answer but made an error in the question | 2017-11-17 18:38:48 | {"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": 0.6983039379119873, "perplexity": 2475.999133036633}, "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-2017-47/segments/1510934803848.60/warc/CC-MAIN-20171117170336-20171117190336-00246.warc.gz"} |
https://stackoverflow.com/questions/29478838/whats-the-best-way-to-calculate-a-numerical-derivative-in-matlab | What's the best way to calculate a numerical derivative in MATLAB?
(Note: This is intended to be a community Wiki.)
Suppose I have a set of points xi = {x0,x1,x2,...xn} and corresponding function values fi = f(xi) = {f0,f1,f2,...,fn}, where f(x) is, in general, an unknown function. (In some situations, we might know f(x) ahead of time, but we want to do this generally, since we often don't know f(x) in advance.) What's a good way to approximate the derivative of f(x) at each point xi? That is, how can I estimate values of dfi == d/dx fi == df(xi)/dx at each of the points xi?
Unfortunately, MATLAB doesn't have a very good general-purpose, numerical differentiation routine. Part of the reason for this is probably because choosing a good routine can be difficult!
So what kinds of methods are there? What routines exist? How can we choose a good routine for a particular problem?
There are several considerations when choosing how to differentiate in MATLAB:
1. Do you have a symbolic function or a set of points?
2. Is your grid evenly or unevenly spaced?
3. Is your domain periodic? Can you assume periodic boundary conditions?
4. What level of accuracy are you looking for? Do you need to compute the derivatives within a given tolerance?
5. Does it matter to you that your derivative is evaluated on the same points as your function is defined?
6. Do you need to calculate multiple orders of derivatives?
What's the best way to proceed?
• Good work putting this together! However I suspect that this topic might be too broad for a SO Q&A, as the best way will highly depend on the situation. – knedlsepp Apr 6 '15 at 21:09
These are just some quick-and-dirty suggestions. Hopefully somebody will find them helpful!
1. Do you have a symbolic function or a set of points?
• If you have a symbolic function, you may be able to calculate the derivative analytically. (Chances are, you would have done this if it were that easy, and you would not be here looking for alternatives.)
• If you have a symbolic function and cannot calculate the derivative analytically, you can always evaluate the function on a set of points, and use some other method listed on this page to evaluate the derivative.
• In most cases, you have a set of points (xi,fi), and will have to use one of the following methods....
2. Is your grid evenly or unevenly spaced?
• If your grid is evenly spaced, you probably will want to use a finite difference scheme (see either of the Wikipedia articles here or here), unless you are using periodic boundary conditions (see below). Here is a decent introduction to finite difference methods in the context of solving ordinary differential equations on a grid (see especially slides 9-14). These methods are generally computationally efficient, simple to implement, and the error of the method can be simply estimated as the truncation error of the Taylor expansions used to derive it.
• If your grid is unevenly spaced, you can still use a finite difference scheme, but the expressions are more difficult and the accuracy varies very strongly with how uniform your grid is. If your grid is very non-uniform, you will probably need to use large stencil sizes (more neighboring points) to calculate the derivative at a given point. People often construct an interpolating polynomial (often the Lagrange polynomial) and differentiate that polynomial to compute the derivative. See for instance, this StackExchange question. It is often difficult to estimate the error using these methods (although some have attempted to do so: here and here). Fornberg's method is often very useful in these cases....
• Care must be taken at the boundaries of your domain because the stencil often involves points that are outside the domain. Some people introduce "ghost points" or combine boundary conditions with derivatives of different orders to eliminate these "ghost points" and simplify the stencil. Another approach is to use right- or left-sided finite difference methods.
• Here's an excellent "cheat sheet" of finite difference methods, including centered, right- and left-sided schemes of low orders. I keep a printout of this near my workstation because I find it so useful.
3. Is your domain periodic? Can you assume periodic boundary conditions?
• If your domain is periodic, you can compute derivatives to a very high order accuracy using Fourier spectral methods. This technique sacrifices performance somewhat to gain high accuracy. In fact, if you are using N points, your estimate of the derivative is approximately N^th order accurate. For more information, see (for example) this WikiBook.
• Fourier methods often use the Fast Fourier Transform (FFT) algorithm to achieve roughly O(N log(N)) performance, rather than the O(N^2) algorithm that a naively-implemented discrete Fourier transform (DFT) might employ.
• If your function and domain are not periodic, you should not use the Fourier spectral method. If you attempt to use it with a function that is not periodic, you will get large errors and undesirable "ringing" phenomena.
• Computing derivatives of any order requires 1) a transform from grid-space to spectral space (O(N log(N))), 2) multiplication of the Fourier coefficients by their spectral wavenumbers (O(N)), and 2) an inverse transform from spectral space to grid space (again O(N log(N))).
• Care must be taken when multiplying the Fourier coefficients by their spectral wavenumbers. Every implementation of the FFT algorithm seems to have its own ordering of the spectral modes and normalization parameters. See, for instance, the answer to this question on the Math StackExchange, for notes about doing this in MATLAB.
4. What level of accuracy are you looking for? Do you need to compute the derivatives within a given tolerance?
• For many purposes, a 1st or 2nd order finite difference scheme may be sufficient. For higher precision, you can use higher order Taylor expansions, dropping higher-order terms.
• If you need to compute the derivatives within a given tolerance, you may want to look around for a high-order scheme that has the error you need.
• Often, the best way to reduce error is reducing the grid spacing in a finite difference scheme, but this is not always possible.
• Be aware that higher-order finite difference schemes almost always require larger stencil sizes (more neighboring points). This can cause issues at the boundaries. (See the discussion above about ghost points.)
5. Does it matter to you that your derivative is evaluated on the same points as your function is defined?
• MATLAB provides the diff function to compute differences between adjacent array elements. This can be used to calculate approximate derivatives via a first-order forward-differencing (or forward finite difference) scheme, but the estimates are low-order estimates. As described in MATLAB's documentation of diff (link), if you input an array of length N, it will return an array of length N-1. When you estimate derivatives using this method on N points, you will only have estimates of the derivative at N-1 points. (Note that this can be used on uneven grids, if they are sorted in ascending order.)
• In most cases, we want the derivative evaluated at all points, which means we want to use something besides the diff method.
6. Do you need to calculate multiple orders of derivatives?
• One can set up a system of equations in which the grid point function values and the 1st and 2nd order derivatives at these points all depend on each other. This can be found by combining Taylor expansions at neighboring points as usual, but keeping the derivative terms rather than cancelling them out, and linking them together with those of neighboring points. These equations can be solved via linear algebra to give not just the first derivative, but the second as well (or higher orders, if set up properly). I believe these are called combined finite difference schemes, and they are often used in conjunction with compact finite difference schemes, which will be discussed next.
• Compact finite difference schemes (link). In these schemes, one sets up a design matrix and calculates the derivatives at all points simultaneously via a matrix solve. They are called "compact" because they are usually designed to require fewer stencil points than ordinary finite difference schemes of comparable accuracy. Because they involve a matrix equation that links all points together, certain compact finite difference schemes are said to have "spectral-like resolution" (e.g. Lele's 1992 paper--excellent!), meaning that they mimic spectral schemes by depending on all nodal values and, because of this, they maintain accuracy at all length scales. In contrast, typical finite difference methods are only locally accurate (the derivative at point #13, for example, ordinarily doesn't depend on the function value at point #200).
• A current area of research is how best to solve for multiple derivatives in a compact stencil. The results of such research, combined, compact finite difference methods, are powerful and widely applicable, though many researchers tend to tune them for particular needs (performance, accuracy, stability, or a particular field of research such as fluid dynamics).
• As described above, one can use the diff function (link to documentation) to compute rough derivatives between adjacent array elements.
• MATLAB's gradient routine (link to documentation) is a great option for many purposes. It implements a second-order, central difference scheme. It has the advantages of computing derivatives in multiple dimensions and supporting arbitrary grid spacing. (Thanks to @thewaywewalk for pointing out this glaring omission!)
• I used Fornberg's method (see above) to develop a small routine (nderiv_fornberg) to calculate finite differences in one dimension for arbitrary grid spacings. I find it easy to use. It uses sided stencils of 6 points at the boundaries and a centered, 5-point stencil in the interior. It is available at the MATLAB File Exchange here.
Conclusion
The field of numerical differentiation is very diverse. For each method listed above, there are many variants with their own set of advantages and disadvantages. This post is hardly a complete treatment of numerical differentiation.
Every application is different. Hopefully this post gives the interested reader an organized list of considerations and resources for choosing a method that suits their own needs.
This community wiki could be improved with code snippets and examples particular to MATLAB.
• Very comprehensive. +1. – rayryeng - Reinstate Monica Apr 6 '15 at 20:11
• The gradient function is missing, in my opinion one of the best choices. – thewaywewalk Apr 6 '15 at 20:26
• I think there should be a mention of automatic differentiation (en.wikipedia.org/wiki/Automatic_differentiation) since it is often more accurate and faster than other methods. There's no method for doing it supplied with matlab, but plenty of solutions available online for both forward and reverse mode automatic differentiation. – yhenon Apr 6 '15 at 21:06
• Hi, I'd be happy too, give me a day or so to write something, it's a great technique but not necessarily easy for people to grasp. – yhenon Apr 6 '15 at 21:15
• Yes. I wouldn't consider the reconstruction of the function from data points as part of numerical differentiation. Thus for me it's odd that you list FDM as a method of doing numerical derivatives, as those are methods that solve differential equations and must make use of (very specific) numerical derivates. (Maybe you really wanted to link to this and that's where my confusion stems from...) – knedlsepp Apr 6 '15 at 22:10
I believe there is more in to these particular questions. So I have elaborated on the subject further as follows:
(4) Q: What level of accuracy are you looking for? Do you need to compute the derivatives within a given tolerance?
A: The accuracy of numerical differentiation is subjective to the application of interest. Usually the way it works is, if you are using the ND in forward problem to approximate the derivatives to estimate features from signal of interest, then you should be aware of noise perturbations. Usually such artifacts contain high frequency components and by the definition of the differentiator, the noise effect will be amplified in the magnitude order of $i\omega^n$. So, increasing the accuracy of differentiator (increasing the polynomial accuracy) will no help at all. In this case you should be able to cancelt the effect of noise for differentiation. This can be done in casecade order: first smooth the signal, and then differentiate. But a better way of doing this is to use "Lowpass Differentiator". A good example of MATLAB library can be found here.
However, if this is not the case and you're using ND in inverse problems, such as solvign PDEs, then the global accuracy of differentiator is very important. Depending on what kind of bounady condition (BC) suits your problem, the design will be adapted accordingly. The rule of thump is to increase the numerical accuracy known is the fullband differentiator. You need to design a derivative matrix that takes care of suitable BC. You can find comprehensive solutions to such designs using the above link.
(5) Does it matter to you that your derivative is evaluated on the same points as your function is defined? A: Yes absolutely. The evaluation of the ND on the same grid points is called "centralized" and off the points "staggered" schemes. Note that using odd order of derivatives, centralized ND will deviate the accuracy of frequency response of the differentiator. Therefore, if you're using such design in inverse problems, this will perturb your approximation. Also, the opposite applies to the case of even order of differentiation utilized by staggered schemes. You can find comprehensive explanation on this subject using the link above.
(6) Do you need to calculate multiple orders of derivatives? This totally depends on your application at hand. You can refer to the same link I have provided and take care of multiple derivative designs. | 2019-11-19 02:35:46 | {"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": 0.687207818031311, "perplexity": 474.6350711152295}, "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-2019-47/segments/1573496669967.80/warc/CC-MAIN-20191119015704-20191119043704-00456.warc.gz"} |
https://mathematica.stackexchange.com/questions/200889/compute-unstable-integral-with-high-precision/200896 | # Compute unstable integral with high precision
I want to calculate $$\int_{\mathbb{R}^n} e^{-t_1^4-t_2^4- ...-t_n^4-(1-t_1-t_2-...-t_n)^4} \text{d}t_1 \text{d}t_2 ... \text{d}t_n$$ with the highest possible n.
I tried with
Module[{n = 3, intVars, poly},
intVars = Table[{Subscript[t, i], -∞, ∞}, {i, 1, n - 1}];
poly = Sum[Subscript[t, i]^4, {i, 1, n - 1}] + (1 - Sum[Subscript[t, i], {i, 1, n - 1}])^4;
Print[E^-poly];
NIntegrate[E^-poly, ##, Method -> {Automatic, "SymbolicProcessing" -> 0}] & @@ intVars
]
which seems to work well only for n<6. Is there any trick I can use?
• Play with NIntegrate[E^-poly, ##, Method -> {"QuasiMonteCarlo", "SymbolicProcessing" -> 0}, AccuracyGoal -> 3, PrecisionGoal -> 3, WorkingPrecision -> 30] & @@ intVars]. It works for $n=19$. – user64494 Jun 23 at 17:35
• You cannot calculate a high-dimensional integral with high precision. Read the tutorial: "For low-dimensional integrals, the default setting for PrecisionGoal is related to WorkingPrecision. For high-dimensional integrals, it is typically taken to be a fixed value, usually 2. " – Alex Trounev Jun 23 at 17:38
• @Alex Trounev: You are right. My idea is unsufficiently considered. – user64494 Jun 23 at 17:42
One idea I like to use for these kinds of integrals is to add an auxiliary variable and a Dirac delta function, convert the Dirac delta function to it's integral formulation, and then do a bunch of simple 1D integrals. For your case, this would proceed as follows, starting from:
$$\underset{t\in \mathbb{R}^n}{\int }e^{-t_1^4-t_2^4-\ldots \ -t_n^4-\left(1-t_1-t_2-\ldots -t_n\right){}^4}$$
Introduce the auxiliary variable $$s$$ and add a Dirac delta function:
$$\underset{t\in \mathbb{R}^n}{\int }\underset{s\in \ \mathbb{R}}{\int }e^{-t_1^4-t_2^4-\ldots -t_n^4-s^4} \delta(1-t_1-t_2-\ldots -t_n-s)$$
Next, introduce the integral formulation of the Dirac delta function:
$$\frac{1}{2 \pi }\underset{t\in \ \mathbb{R}^n}{\int }\underset{s\in \mathbb{R}}{\int }\underset{u\in \ \mathbb{R}}{\int }e^{-t_1^4-t_2^4-\ldots -t_n^4-s^4} e^{i u-i t_1 u-i t_2 \ u-\ldots -i t_n u-i s u}$$
Finally, we can do all of the $$t$$ and $$s$$ integrals to obtain:
$$\frac{1}{2 \pi } \int_{-\infty \ }^{\infty } e^{i u} g(u)^{n+1} \, du$$
where:
$$g(u)=\int_{-\infty }^{\infty } e^{-i t u-t^4} \, dt$$
Now, let's have Mathematica do these integrals:
g[u_] = Sqrt[2 Pi] FourierTransform[Exp[-t^4], t, u]
2 Gamma[5/4] HypergeometricPFQ[{}, {1/2, 3/4}, u^4/256] - 1/4 u^2 Gamma[3/4] HypergeometricPFQ[{}, {5/4, 3/2}, u^4/256]
So, the desired integral has become:
int[n_, opts:OptionsPattern[NIntegrate]] := (1/(2 Pi)) NIntegrate[
Cos[u] g[u]^(n+1),
{u, -Infinity, Infinity},
opts,
]
Now, g[u] is real:
Refine[g[u] ∈ Reals, u ∈ Reals]
True
so I use Cos[u] instead of Exp[I u] since the integral is real. I also customize the integration method. Let's check:
int[2, WorkingPrecision->20]
int[2, WorkingPrecision->40]
int[2, WorkingPrecision->60]
1.4733172914977911077
1.473317291497785926905017339845596712841
1.47331729149778592690501733984559670949096610342311667206502
The result seems correct, and improves with higher working precision. Now, for higher orders:
int[4, WorkingPrecision -> 20]
int[5, WorkingPrecision -> 20]
int[6, WorkingPrecision -> 20]
int[40, WorkingPrecision -> 20]
4.4732305211180348293
7.7543594355221995796
13.461688085347942892
4.0351905913672630176*10^9
• You change the order of the integration in the above. This is not a simple matter for improper integrals. Next, in the integral presentation of DiracDelta the principal value is taken. – user64494 Jun 23 at 19:54
• @user64494 What makes you say g[u] is complex valued? I don't think it is. As for order of integration, I haven't been rigorous in proving that it is possible to do so, but the results seem correct, and that is sufficient for me. – Carl Woll Jun 23 at 19:58
• You are right concerning g[u]. However, other remarks of me remain open. The statement "the results seem correct" is not based. – user64494 Jun 23 at 20:02
• You are right: I don't execute g[u_]. It works well. Sorry for the trouble. – user64494 Jun 23 at 20:37
• @CarlWoll your solution is very elegant and it seems to do the job excellently. Thank you very much for the effort invested! Just for curiosity, where is used the method of inserting delta and then Fourier transform? To decouple integration variables? Do you have an intuitive interpretation for this? – m137 Jun 23 at 21:28
We can increase n, reducing the accuracy of calculations, for example
f[m_, p_] :=
Block[{n = m, $$MinPrecision = p,$$MaxPrecision = p},
intVars =
Table[{Subscript[t, i], -\[Infinity], \[Infinity]}, {i, 1, n}];
poly = Sum[
Subscript[t, i]^4, {i, 1,
n}] + (1 - Sum[Subscript[t, i], {i, 1, n}])^4;
NIntegrate[E^-poly, ##, PrecisionGoal -> p/2,
AccuracyGoal -> p/2] & @@ intVars]
f[4, 6]
(* 4.47148*)
f[5, 4]
(* 7.75615*)
f[6, 2]
(* 15.8289*) | 2019-11-20 22:52:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7504702210426331, "perplexity": 3326.1038491262143}, "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/1573496670635.48/warc/CC-MAIN-20191120213017-20191121001017-00401.warc.gz"} |
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## The evolution of cooperation in an iterated survival game Wakeley, John | CIRM H
Post-edited
Research talks;Mathematics in Science and Technology;Probability and Statistics
A new type of a simple iterated game with natural biological motivation is introduced. Two individuals are chosen at random from a population. They must survive a certain number of steps. They start together, but if one of them dies the other one tries to survive on its own. The only payoff is to survive the game. We only allow two strategies: cooperators help the other individual, while defectors do not. There is no strategic complexity. There are no conditional strategies. Depending on the number of steps we recover various forms of stringent and relaxed cooperative dilemmas. We derive conditions for the evolution of cooperation.
Specifically, we describe an iterated game between two players, in which the payoff is to survive a number of steps. Expected payoffs are probabilities of survival. A key feature of the game is that individuals have to survive on their own if their partner dies. We consider individuals with simple, unconditional strategies. When both players are present, each step is a symmetric two-player game. As the number of iterations tends to infinity, all probabilities of survival decrease to zero. We obtain general, analytical results for n-step payoffs and use these to describe how the game changes as n increases. In order to predict changes in the frequency of a cooperative strategy over time, we embed the survival game in three different models of a large, well-mixed population. Two of these models are deterministic and one is stochastic. Offspring receive their parent’s type without modification and fitnesses are determined by the game. Increasing the number of iterations changes the prospects for cooperation. All models become neutral in the limit $(n \rightarrow \infty)$. Further, if pairs of cooperative individuals survive together with high probability, specifically higher than for any other pair and for either type when it is alone, then cooperation becomes favored if the number of iterations is large enough. This holds regardless of the structure of pairwise interactions in a single step. Even if the single-step interaction is a Prisoner’s Dilemma, the cooperative type becomes favored. Enhanced survival is crucial in these iterated evolutionary games: if players in pairs start the game with a fitness deficit relative to lone individuals, the prospects for cooperation can become even worse than in the case of a single-step game.
A new type of a simple iterated game with natural biological motivation is introduced. Two individuals are chosen at random from a population. They must survive a certain number of steps. They start together, but if one of them dies the other one tries to survive on its own. The only payoff is to survive the game. We only allow two strategies: cooperators help the other individual, while defectors do not. There is no strategic complexity. There ...
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## Evolutionary branching: trade-offs and magic traits Kisdi, Eva | CIRM H
Multi angle
Research talks;Mathematics in Science and Technology
Adaptive dynamics has shaped our understanding of evolution by demonstrating that, via the process of evolutionary branching, ecological interactions can promote diversification. The classical approach to study the adaptive dynamics of a system is to specify the ecological model including all trade-off functions and other functional relationships, and make predictions depending on the parameters of these functions. However, the choice of trade-offs and other functions is often the least well justified element of the model, and examples show that minor variations in these functions can lead to qualitative changes in the model predictions. In the first part of this talk, I shall revisit evolutionary branching and other evolutionary phenomena predicted by adaptive dynamics using an inverse approach: I investigate under which conditions a trade-off function exists that yields a given evolutionary outcome.
Evolutionary branching can amount to the birth of new species, but only if reproductive isolation evolves between the emerging branches. Recent studies show that mating is often assortative with respect to the very trait that is under ecological selection. Such "magic traits" can ensure reproductive isolation, yet they are by far not free tickets to speciation. In the second half of my talk, I discuss the consequences of sexual selection emerging from assortative mating, and show how a perfect female should search for mates.
Adaptive dynamics has shaped our understanding of evolution by demonstrating that, via the process of evolutionary branching, ecological interactions can promote diversification. The classical approach to study the adaptive dynamics of a system is to specify the ecological model including all trade-off functions and other functional relationships, and make predictions depending on the parameters of these functions. However, the choice of ...
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## Economie théorique Montbrial, Thierry de | Presses Universitaires De France 1971
Ouvrage
- xv; 206 p.
Systèmes-décisions. Section 3: mathématiques et statistiques pour la décision
Localisation : Ouvrage RdC (MONT)
microéconomie # macroéconomie # prise de décision # calcul économique # économie politique # mathématiques économiques
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## Non-cooperative equilibria of Fermi systems with long range interactions Bru, J.-B. ; de Siqueira Pedra, W. | American Mathematical Society 2012
Ouvrage
- xi; 155 p.
ISBN 978-0-8218-8976-3
Memoirs of the american mathematical society , 1052
Localisation : Collection 1er étage
équilibre thermodynamique # règle de phase # physique mathématique # intéraction à long terme # théorème de Choquet # approximation de Bogoliubov # système de spin quantique
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## Security and game theory:algorithms, deployed systems, lessons learned Tambe, Milind | Cambridge University Press 2012
Ouvrage
- xiv; 319 p.
ISBN 978-1-107-09642-4
Localisation : Ouvrage RdC (TAMB)
sécurité informatique # théorie des jeux # modèle mathématique # sécurité internationale
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## Advances in dynamic game theory :numerical methods, algorithms, and applications to ecology and economics Jorgensen, Steffen ; Quincampoix, Marc ; Vincent, Thomas L. | Springer 2007
Ouvrage
- 717 p.
ISBN 978-0-8176-4399-7
Annals of the international society of dynamic games , 0009
théorie des jeux dynamiques # jeux coopératifs et non coopératifs # jeux stochastiques # jeux d'évolution # jeux différentiels # jeux de poursuite # application de la théorie des jeux à l'économie # modèles théoriques # méthodes numériques
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## Models of adaptive behaviour :an approach based on state Houston, Alasdair I. ; McNamara, John M. | Cambridge University Press 1999
Ouvrage
- 378 p.
ISBN 978-0-521-65539-2
Localisation : Ouvrage RdC (HOUS)
adaptation # évolution # modèle mathématique # maximisation # jeu dynamique # optimisation dynamique # énergie # sélection naturelle
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## An introduction to game-theroretic modelling Mesterton-Gibbons, Michael | American Mathematical Society 2000
Ouvrage
- 368 p.
ISBN 978-0-8218-1929-6
Student mathematical library , 0011
Localisation : Collection 1er étage
théorie des jeux # modèle théorique de jeux # théorie de la décision # jeu non-coopératif # jeu coopératif # dilemme du prisonnier # jeu de stratégie
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## The game's afoot ! game theory in myth and paradox Mehlmann, Alexander ; Kramer David | American Mathematical Society 2000
Ouvrage
ISBN 978-0-8218-2121-3
Student mathematical library , 0005
Localisation : Collection 1er étage
calcul stratégique # dilemme du prisonier # jeu à deux personnes # loisir # mathématique récréative # modèle théorique de jeu # paradoxe # théorie des jeux
#### Filtrer
##### Codes MSC
Ressources Electroniques (Depuis le CIRM)
Books & Print journals
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Z | 2019-10-16 09:06:55 | {"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": 0.35438618063926697, "perplexity": 7152.439129801347}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986666467.20/warc/CC-MAIN-20191016063833-20191016091333-00511.warc.gz"} |
http://stator.imag.fr/w/index.php?title=ICMS_2014_session_on_automated_proofs_by_induction&diff=next&oldid=152&printable=yes | # Difference between revisions of "ICMS 2014 session on automated proofs by induction"
## Aims and scope
In many contexts one proves properties of the form ${\displaystyle \forall n\in \mathbb {N} ~P(n)}$. For instance, formal verification of software proves safety properties of the form “for any number of program steps, the output satisfies the specification”.
In easy cases, ${\displaystyle P(n)}$ can be proved by induction: ${\displaystyle P(0)}$ holds and ${\displaystyle P(n)\Rightarrow P(n+1)}$ holds for all ${\displaystyle n}$. In most cases, however, one must invent some property ${\displaystyle Q}$ that is “inductive” (${\displaystyle Q(0)}$ holds and ${\displaystyle Q(n)\Rightarrow Q(n+1)}$ holds for all ${\displaystyle n}$) and such that ${\displaystyle Q(n)\Rightarrow P(n)}$ holds. In software verification, such a property is called an “inductive invariant”.
Over the years, a variety of approaches have been proposed to automatically or semi-automatically obtain inductive properties (model-checking, abstract interpretation, predicate abstraction), with building blocks such as Craig interpolation, Kleene and policy iterations, quantifier elimination etc.
The session aims at bringing together designers of software verification tools and automated theorem provers, and users of these tools.
## Publications
• A short abstract will appear on the permanent conference web page (see below) as soon as accepted.
• An extended abstract will appear on the permanent conference web page (see below) as soon as accepted. It will also appear on the proceedings that will be distributed during the meeting. | 2022-08-09 22:21:28 | {"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": 0.7635860443115234, "perplexity": 1082.9739338464938}, "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/1659882571090.80/warc/CC-MAIN-20220809215803-20220810005803-00068.warc.gz"} |
https://blog.jle.im/ | # in Code
## Justin Le
Welcome! My name is Justin Le. I’m an Applied Math/Computational Science PhD with a background with physics, currently working at SimSpace doing Haskell for cyber-security platforms. This is just my weblog covering my various adventures in programming and explorations in the vast worlds of computation, physics, engineering, mathematics, and knowledge. Expect a healthy curiosity and an only slightly unhealthy obsession for finding new ways to marvel, wonder, and create. Join me if you wish!
Check out some of my most popular articles: First-Class Statements, “IO Monad” Considered Harmful, and Purely Functional Typed Models!
## Recent Entries
• ### Breaking a Degenerate Hyper-Dimensional Game of Life
tldr: Demonstrated with interactive visualizations and simulations — over the course of a month, we were able to discover successive new mathematical properties of a “degenerate” hyper-dimensional game of life" to take a “7 dimensions may just barely be possible on a commercial PC, could we ever reach 10 dimensions?” to “10 dimensions is easy enough to be run on any modern browser (jump to spoilers here), and 60 dimensions can be reached with a compiled language”.
This is a story about breaking a degenerate hyper-dimensional game of life via interactive exploratory visualizations and math!
T’was the night before Thursday, December 17, 2020, the release of “Conway Cubes”. It was Day 17 of Advent of Code 2020, a series of fun little themed coding puzzles building up to Christmas; I always enjoyed these puzzles because they are so self-contained and tidy that they are often open-ended in the interesting ways you can solve them or expand on them (which I’ve written many blog posts on).
On the surface, Day 17 seemed to be a straightforward extension of Conway’s Game Of Life (“GoL”). GoL is a simulation played out on a 2D grid, where cells are “on” and “off”, and at each step of the simulation the states spread and propagate in interesting ways based on the state of their neighbors (a 2D cellular automaton). The twist of the Advent of Code puzzle is it asks what would happen if we played out the rules of GoL in 3D instead, and then 4D.
I submitted my solution on my assigned puzzle input with a naive implementation (placing 66 and 66 on the leaderboards for that day), concluding the “competitive” part. Of course, the real fun always starts after. When discussing with some friends (on the subreddit and freenode’s ##adventofcode channel), we started talking about the trade-offs of different implementations and realized that the extra dimensionality was no joke: as you upped the number of dimensions, the number of points you have to consider grow exponentially, and so does the number of neighbors at each point to check. 4D can be solved naively, but anything higher is going to be strained. My naive solution on 6D took three minutes, and 7D in a reasonable amount of time (requiring as much as 612,220,032 points with 2,186 neighbors each) seemed impossible on commercial consumer hardware because of the sheer number of points in 7D space. But I thought…what if a breakthrough in optimization was possible? I set an (arbitrary) personal goal of reaching 10D (3,570,467,226,624 points with 59,048 neighbors each), not knowing if it would ever be possible.
And soon…a breakthrough did come! Someone brought up that if we look at the 3d version, we see there’s actually a mirror symmetry! Because everything starts off on the xy plane, with z=0, the resulting progression must be symmetrical on both sides (positive and negative z).
• ### Advent of Code 2020: Haskell Solution Reflections for all 25 Days
Merry Christmas and Happy New Years, to all!
Once again, every year I like to participate in Eric Wastl’s Advent of Code! It’s a series of 25 Christmas-themed puzzles that release every day at midnight — there’s a cute story motivating each one, usually revolving around saving Christmas. Every night my friends and I (including the good people of freenode’s ##advent-of-code channel) talk about the puzzle and creative ways to solve it (and also see how my bingo card is doing). The subreddit community is also pretty great as well! And an even nicer thing is that the puzzles are open-ended enough that there are often many ways of approaching them…including some approaches that can leverage math concepts in surprising ways, like group theory, galilean transformations and linear algebra, and more group theory. Many of the puzzles are often simple data transformations that Haskell is especially good at!
Speaking of Haskell, I usually do a write-up for every day I can get around to about unique insights that solving in Haskell can provide to each different puzzle. I did them in 2017, 2018, and 2019, but I never finished every day. But 2020 being what it is, I was able to finish! :D
You can find all of them here, but here are links to each individual one. Hopefully you can find them helpful. And if you haven’t yet, why not try Advent of Code yourself? :) And drop by the freenode ##advent-of-code channel, we’d love to say hi and chat, or help out! Thanks all for reading, and also thanks to Eric for a great event this year, as always!
• ### Roll your own Holly Jolly streaming combinators with Free
Hi! Welcome, if you’re joining us from the great Advent of Haskell 2020 event! Feel free to grab a hot chocolate and sit back by the fireplace. I’m honored to be able to be a part of the event this year; it’s a great initiative and harkens back to the age-old Haskell tradition of bite-sized Functional Programming “advent calendars”. I remember when I was first learning Haskell, Ollie Charles’ 24 Days of Hackage series was one of my favorite series that helped me really get into the exciting world of Haskell and the all the doors that functional programming can open.
All of the posts this year have been great — they range from insightful reflections on the nature of Haskell and programming in Haskell, or also on specific language features. This post is going to be one of the “project-based” ones, where we walk through and introduce a solidly intermediate Haskell technique as it applies to building a useful general toolset. I’m going to be exploring the “functor combinator style” where you identify the interface you want, associate it with a common Haskell typeclass, pick your primitives, and automatically get the ability to imbue your primitives with the structure you need. I’ve talked about this previously with:
and I wanted to share a recent application I have been able to use apply it with where just thinking about the primitives gave me almost all the functionality I needed for a type: composable streaming combinators. This specific application is also very applicable to integrate into any composable effects system, since it’s essentially a monadic interface.
In a way, this post could also be seen as capturing the spirit of the holidays by reminiscing about the days of yore — looking back at one of the more exciting times in modern Haskell’s development, where competing composable streaming libraries were at the forefront of practical innovation. The dust has settled on that a bit, but it every time I think about composable streaming combinators, I do get a bit nostalgic :)
This post is written for an intermediate Haskell audience, and will assume you have a familiarity with monads and monadic interfaces, and also a little bit of experience with monad transformers. Note — there are many ways to arrive at the same result, but this post is more of a demonstration of a certain style and approach that has benefited my greatly in the past.
• ### Shuffling things up: Applying Group Theory in Advent of Code
So it’s November, and Advent of Code season is in the air! It’s time for everyone’s favorite Santa-based light hearted learn-to-program-or-a-new-language holiday season programming challenge series. Every year a bunch of us gather around the fireplace, roast chestnuts, and brainstorm all of the interesting ways we can solve these cute themed puzzles every day. These puzzles are designed to accessible enough for most new programmers, but deep enough to provide entertainment for experienced ones. I’ve written many blog posts on some of the interesting insight some of the puzzles have yielded, and I also post my reflections on as many puzzles I can while solving them in Haskell. And if you’re solving things in Haskell, I also published an open-sourced rate-limited API library so you can fetch and submit answers from the comfort of your command line.
To kick off the season, I’ve decided to write about one of my favorite puzzles from Advent of Code 2019 – Day 22: Slam Shuffle. To me, it stands out because it’s a perfect example of how Haskell’s approach to mathematical abstraction nudges you into the direction of an efficient solution — in a way that other languages would obscure or make less obvious.
So, let’s dive in! In the end, hopefully this post can get you excited for this wonderful season, and maybe also shed some insight into what it means when we say that Haskell can help you leverage math to find good solutions to your real problems.
Of course, this post has spoilers for Advent of Code 2019 Day 22, if you are planning on trying to figure it out from yourself. If you haven’t tried it, I recommend you give it a shot and come back after! :D
• ### Enhancing Functor Structures Step-By-Step (Part 2)
Welcome to Part 2 of the “Enhancing Functor Structures” series! Here we are taking a base structure describing a data type schema and enhancing it step-by-step with new functory capabilities: first, covariant capabilities (to generate parsers), then contravariant capabilities (to generate serializers)…who knows what might be in store next?
Please do check out Part 1 if you haven’t already, since this post pretty much jumps straight into things!
• ### Enhancing Functor Structures Step-By-Step (Part 1)
A style of Haskell programming that I’ve been pretty excited about with over the past two years or so is something that I can maybe call a “functor structure” design pattern. In this post we’re going to be exploring the idea of enhancing normal data types with different types of functor structures step-by-step, by starting with a simple useful structure and enhancing it piece by piece in order to reap incremental benefits. This process reflects a lot of the way I personally work through these things — I normally don’t get the whole powerful structure all the way; instead I incrementally add things as I see how things fit together.
We’re going build the tools to describe a data type schema, which can represent algebraic data types — sums and products. We’ll start off just building things we can use to describe the schema (by printing out documentation), and by the end of the journey we’ll also be able to use our schema to generate parsers and serializers through json.
This interest in functor structures culminated in my Functor Combinatorpedia post last year and the functor-combinators library. But personally I had never really explored the less commonly used lowercase-f functor abstractions in Hask — contravariant functors and invariant functors until recently.
This series is designed for an intermediate Haskeller with familiarity in things like product/sum types, using Applicative/Alternative, and monadic parser combinators, and is written in sync with functor-combinators-0.3.6.0.
• ### Introducing the mutable library
mutable: documentation / reference / github
(Note: This post has been heavily revised to reflect mutable-0.2.0.0, as of July 2020. For reference, the original post is available on github.)
I’m excited to announce the release of the mutable library!
The library offers what I call beautiful mutable values1 — automatic, composable piecewise-mutable references for your data types. Sort of like an automatically generated MVector, but for all your ADTs.
My high-level goal was a composable and overhead-free solution for dealing with mutable values in Haskell in a type-safe and clean way. After all, why do imperative languages have to have all the fun? In Haskell, we can have the best of both worlds: efficient and clean mutable algorithms and type safety.
The official documentation and homepage is here, so it’s a good read if you want to be introduced to how to use the library and where it is most effective. But I’m going to use this blog post to talk about why I wrote the library, some of the neat things you can do with it, and the techniques that went into writing it.
1. Okay so I don’t actually think the library is beautiful, I just like the way that “beautiful mutable values” sounds when you say it out loud.↩︎
• ### Adjunctions in the wild: foldl
I recently made a few connections that linked some different concepts in Haskell that I hadn’t realized before. They deal with one of my favorite “practical” libraries in Haskell, and also one of the more “profound” category theory-inspired abstractions in Haskell. In the process, it made the library a bit more useful to me, and also made the concept a bit more concrete and understandable to me.
This post mainly goes through my thought process in finding this out — it’s very much a “how I think through this” sort of thing — in the end, the goal is to show how much this example made me further appreciate the conceptual idea of adjunctions and how they can pop up in interesting places in practical libraries. Unlike most of my other posts, it’s not about necessarily about how practically useful an abstraction is, but rather what insight it gives us to understanding its instances.
The audience of this post is Haskellers with an understanding/appreciation of abstractions like Applicative, but be aware that the final section is separately considered as a fun aside for those familiar with some of Haskell’s more esoteric types. The code samples used here (along with exercise solutions) are available on github. | 2022-12-01 13:43:33 | {"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": 0.3431835472583771, "perplexity": 1445.306795684367}, "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-2022-49/segments/1669446710813.48/warc/CC-MAIN-20221201121601-20221201151601-00500.warc.gz"} |
https://www.doubtnut.com/question-answer-physics/if-the-maximum-intensity-of-radiation-for-a-black-is-found-at-265-mu-m-what-is-the-temperature-of-th-363446471 | # If the maximum intensity of radiation for a black is found at 2.65 mu m, What is the temperature of the radiating body ? (Weins constant =2.9 xx 10^(-3) mK)
Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.
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Wavelength corresponding to maximum intensity , lambda_(max)=2.65 mu m =2.65xx10^(-6)m. <br> Weins constant, b=2.90xx10^(-3) mk. <br> From Weins Law , T=(b)/(lambda_(m))=(2.90xx10^(-3))/(2.65xx10^(-6)=1094K. | 2022-01-21 20:51:53 | {"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": 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": 0.23854424059391022, "perplexity": 13441.51347369383}, "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/1642320303709.2/warc/CC-MAIN-20220121192415-20220121222415-00618.warc.gz"} |
https://www.zbmath.org/serials/?q=se%3A112 | ## Inverse Problems
Short Title: Inverse Probl. Publisher: IOP Publishing, Bristol ISSN: 0266-5611 Online: http://iopscience.iop.org/0266-5611 Comments: Indexed cover-to-cover
Documents Indexed: 4,518 Publications (since 1985) References Indexed: 917 Publications with 35,522 References.
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### Authors
73 Yamamoto, Masahiro 42 Klibanov, Michael V. 40 Ikehata, Masaru 36 Bertero, Mario 35 Somersalo, Erkki 33 Ramlau, Ronny 31 Borcea, Liliana 31 Burger, Martin 31 Colton, David Lem 31 Scherzer, Otmar 30 Cakoni, Fioralba 30 Kaltenbacher, Barbara 29 Cheney, Margaret 28 Bal, Guillaume 28 Hofmann, Bernd 27 Haddar, Houssem 27 Kaipio, Jari P. 27 Nakamura, Gen 27 Papanicolaou, George C. 26 Aktosun, Tuncay 26 Kirsch, Andreas 26 Kress, Rainer 26 Ramm, Alexander G. 24 Arridge, Simon R. 24 Louis, Alfred Karl 24 Monk, Peter B. 24 Siltanen, Samuli 23 Li, Peijun 23 Uhlmann, Gunther Alberto 23 Zou, Jun 22 Jin, Bangti 22 Piana, Michele 21 Hu, Guanghui 21 Rundell, William 20 Calvetti, Daniela 20 Hohage, Thorsten 20 Isakov, Victor M. 20 Lesselier, Dominique 20 Seo, Jin Keun 19 Fokas, Athanassios S. 19 Jin, Qinian 19 Lassas, Matti J. 19 Liu, Hongyu 19 Novikov, Roman G. 19 Schuster, Thomas 18 Boccacci, Patrizia 18 de Hoop, Maarten V. 18 Garnier, Josselin 18 Haltmeier, Markus 18 Liu, Xiaodong 18 Zhang, Bo 17 Bao, Gang 17 Boiti, Marco 17 Cheng, Jin 17 Haber, Eldad 17 Hyvönen, Nuutti 17 Kunyansky, Leonid A. 17 Pempinelli, Flora 17 Pike, E. R. 17 Potthast, Roland W. E. 16 El Badia, Abdellatif 16 Lionheart, William Robert Breckon 15 Defrise, Michel 15 McLaughlin, Joyce Rogers 15 Sabatier, Pierre C. 15 Tsogka, Chrysoula 14 Đinh Nho Hào 14 Dorn, Oliver 14 Grünbaum, Francisco Alberto 14 Han, Bo 14 Kang, Hyeonbae 14 Leitão, Antonio 14 Maass, Peter 14 Schönlieb, Carola-Bibiane 14 Shen, Chao-Liang 14 Sun, Jiguang 14 Symes, William W. 14 Yagle, Andrew E. 13 Akduman, Ibrahim 13 Engl, Heinz W. 13 Harrach, Bastian 13 Imanuvilov, Oleg Yu. 13 Kolehmainen, Ville 13 Lechleiter, Armin 13 Pereverzev, Sergei V. 13 Schotland, John C. 13 Stefanov, Plamen D. 13 Stuart, Andrew M. 13 van den Berg, Peter M. 12 Abubakar, Aria 12 Bellassoued, Mourad 12 Borden, Brett 12 Hasanoǧlu, Alemdar 12 Herman, Gabor T. 12 Hettlich, Frank 12 Mathé, Peter 12 Palamodov, Victor Pavlovitch 12 Rieder, Andreas 12 Sini, Mourad 12 Tautenhahn, Ulrich ...and 4,625 more Authors
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2,435 Partial differential equations (35-XX) 1,808 Numerical analysis (65-XX) 846 Optics, electromagnetic theory (78-XX) 577 Biology and other natural sciences (92-XX) 364 Information and communication theory, circuits (94-XX) 354 Operator theory (47-XX) 327 Integral transforms, operational calculus (44-XX) 295 Statistics (62-XX) 266 Geophysics (86-XX) 265 Fluid mechanics (76-XX) 264 Ordinary differential equations (34-XX) 260 Mechanics of deformable solids (74-XX) 231 Integral equations (45-XX) 199 Dynamical systems and ergodic theory (37-XX) 186 Calculus of variations and optimal control; optimization (49-XX) 157 Operations research, mathematical programming (90-XX) 145 Computer science (68-XX) 138 Quantum theory (81-XX) 91 Probability theory and stochastic processes (60-XX) 82 Harmonic analysis on Euclidean spaces (42-XX) 80 Systems theory; control (93-XX) 69 Global analysis, analysis on manifolds (58-XX) 54 Linear and multilinear algebra; matrix theory (15-XX) 48 Differential geometry (53-XX) 48 Statistical mechanics, structure of matter (82-XX) 43 Classical thermodynamics, heat transfer (80-XX) 40 Astronomy and astrophysics (85-XX) 37 Potential theory (31-XX) 33 Mechanics of particles and systems (70-XX) 30 Functional analysis (46-XX) 28 Functions of a complex variable (30-XX) 24 Difference and functional equations (39-XX) 22 Special functions (33-XX) 21 General and overarching topics; collections (00-XX) 19 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 14 Approximations and expansions (41-XX) 12 Real functions (26-XX) 11 Relativity and gravitational theory (83-XX) 10 Abstract harmonic analysis (43-XX) 8 Combinatorics (05-XX) 6 Algebraic topology (55-XX) 4 History and biography (01-XX) 4 Topological groups, Lie groups (22-XX) 4 Sequences, series, summability (40-XX) 4 Convex and discrete geometry (52-XX) 3 Measure and integration (28-XX) 3 General topology (54-XX) 2 Nonassociative rings and algebras (17-XX) 2 Geometry (51-XX) 1 Number theory (11-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Manifolds and cell complexes (57-XX)
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Inverse problems in elasticity. Zbl 1070.35118
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Structural identification of an unknown source term in a heat equation. Zbl 0917.35156
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Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Zbl 0983.35151
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Thermoacoustic tomography with variable sound speed. Zbl 1177.35256
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Analysis versus synthesis in signal priors. Zbl 1138.93055
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Krasnoselski–Mann iteration for hierarchical fixed-point problems. Zbl 1128.47060
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2000
Level-set function approach to an inverse interface problem. Zbl 0986.35130
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2001
Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Zbl 1044.78513
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A simple method using Morozov’s discrepancy principle for solving inverse scattering problems. Zbl 0902.35123
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1999
Ensemble Kalman methods for inverse problems. Zbl 1311.65064
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Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates. Zbl 1205.65194
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Recent progress in the boundary control method. Zbl 1126.35089
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Optical tomography: forward and inverse problems. Zbl 1188.35197
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Linear spectral problems, nonlinear equations and the $${\bar \partial}$$- method. Zbl 0685.35080
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Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations. Zbl 0805.65133
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Nonparametric statistical inverse problems. Zbl 1137.62323
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Generalized KM theorems and their applications. Zbl 1117.65081
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A scaled gradient projection method for constrained image deblurring. Zbl 1155.94011
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Inverse spectral problems for Sturm-Liouville operators on graphs. Zbl 1089.34009
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Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. Zbl 0860.34007
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Nonlinear evolution equations, rescalings, model PDEs and their integrability. I. Zbl 0645.35087
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An inverse source problem in potential analysis. Zbl 0963.35194
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Cloaking via change of variables in electric impedance tomography. Zbl 1153.35406
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Inverse scattering problems with multi-frequencies. Zbl 1332.78019
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Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Zbl 0805.35157
Potthast, Roland
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Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit. Zbl 1086.35095
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MAP estimators and their consistency in Bayesian nonparametric inverse problems. Zbl 1281.62089
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On the method of Lavrentiev regularization for nonlinear ill-posed problems. Zbl 1005.65058
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Analytical and computational methods for transmission eigenvalues. Zbl 1192.78024
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On a global estimate in a linear inverse hyperbolic problem. Zbl 0862.35141
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Fréchet derivatives in inverse obstacle scattering. Zbl 0821.35147
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Uniqueness in inverse obstacle scattering. Zbl 0787.35119
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The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media. Zbl 1027.35158
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Prolongation algebras and Hamiltonian operators for peakon equations. Zbl 1020.35096
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Recovery of coefficients for a weighted $$p$$-Laplacian perturbed by a linear second order term. Zbl 1460.35386
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On Calderón’s inverse inclusion problem with smooth shapes by a single partial boundary measurement. Zbl 1467.35355
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Recovering a quasilinear conductivity from boundary measurements. Zbl 1457.65172
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Unique continuation from a generalized impedance edge-corner for Maxwell’s system and applications to inverse problems. Zbl 1456.35192
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Some inverse problems for wave equations with fractional derivative attenuation. Zbl 1459.35396
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Monotonicity principle in tomography of nonlinear conducting materials. Zbl 1462.78012
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Laboratory application of sampling approaches to inverse scattering. Zbl 1466.62451
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Adaptive regularisation for ensemble Kalman inversion. Zbl 1456.65095
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Oversmoothing Tikhonov regularization in Banach spaces. Zbl 1483.65086
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Phaseless inverse scattering with background information. Zbl 1473.35140
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Stability estimates for the relativistic Schrödinger equation from partial boundary data. Zbl 1483.35344
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A direct method for solving inverse Sturm-Liouville problems. Zbl 07305940
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Unique determination for an inverse problem from the vortex dynamics. Zbl 1462.34041
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Non-stationary multi-layered Gaussian priors for Bayesian inversion. Zbl 1457.60097
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Ultrasound modulated bioluminescence tomography with a single optical measurement. Zbl 1458.35420
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Deep synthesis network for regularizing inverse problems. Zbl 07310609
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Tikhonov regularization for polynomial approximation problems in Gauss quadrature points. Zbl 1461.65010
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The maximum entropy on the mean method for image deblurring. Zbl 07310613
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Inverse initial boundary value problem for a non-linear hyperbolic partial differential equation. Zbl 1457.58016
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Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review. Zbl 1461.62129
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Sparsity-based nonlinear reconstruction of optical parameters in two-photon photoacoustic computed tomography. Zbl 1460.78011
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Numerical solution of an inverse random source problem for the time fractional diffusion equation via PhaseLift. Zbl 1475.35432
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A range-relaxed criteria for choosing the Lagrange multipliers in the iterated Tikhonov Kaczmarz method for solving systems of linear ill-posed equations. Zbl 07323241
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Optimal-order convergence of Nesterov acceleration for linear ill-posed problems. Zbl 07357681
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3D Compton scattering imaging with multiple scattering: analysis by FIO and contour reconstruction. Zbl 1466.35372
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Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems. Zbl 07362502
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A projective two-point gradient Kaczmarz iteration for nonlinear ill-posed problems. Zbl 07362507
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NETT: solving inverse problems with deep neural networks. Zbl 1456.65038
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Stable determination of polygonal inclusions in Calderón’s problem by a single partial boundary measurement. Zbl 1445.35334
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Convergence analysis of (statistical) inverse problems under conditional stability estimates. Zbl 07153871
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Uniqueness in inverse cavity scattering problems with phaseless near-field data. Zbl 1460.35246
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Half-inverse Sturm-Liouville problem on a time scale. Zbl 1433.93084
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Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems. Zbl 07220290
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Stability for inverse source problems by Carleman estimates. Zbl 1459.35393
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Unique reconstruction of the potential for the interior transmission eigenvalue problem for spherically stratified media. Zbl 1436.35328
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The Calderón problem for the fractional magnetic operator. Zbl 1443.35194
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Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs. Zbl 1443.62213
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Numerical schemes to reconstruct three-dimensional time-dependent point sources of acoustic waves. Zbl 1471.65122
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Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem. Zbl 1445.35330
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$$\ell_1-\alpha\ell_2$$ minimization methods for signal and image reconstruction with impulsive noise removal. Zbl 1468.94021
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Preconditioning inverse problems for hyperbolic equations with applications to photoacoustic tomography. Zbl 07153865
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Persistent homology detects curvature. Zbl 07161824
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Deep unfolding of a proximal interior point method for image restoration. Zbl 1452.94003
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Modulus-based iterative methods for constrained $$\ell_p$$-$$\ell_q$$ minimization. Zbl 1451.65072
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Monotonicity in inverse obstacle scattering on unbounded domains. Zbl 1458.35469
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Geometry of the phase retrieval problem. Zbl 1451.49037
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An inverse problem for the fractional Schrödinger equation in a magnetic field. Zbl 1473.35500
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The quadratic Wasserstein metric for inverse data matching. Zbl 1465.49029
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Image reconstruction through metamorphosis. Zbl 1447.94005
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The bound-state soliton solutions of the complex modified KdV equation. Zbl 1447.35290
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Bayesian approach to inverse time-harmonic acoustic scattering with phaseless far-field data. Zbl 1443.76205
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Convexification for an inverse parabolic problem. Zbl 1445.35333
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Hölder-logarithmic stability in Fourier synthesis. Zbl 1453.42008
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135 Yamamoto, Masahiro 81 Uhlmann, Gunther Alberto 78 Fu, Chuli 77 Klibanov, Michael V. 76 Nguyen Huy Tuan 74 Ammari, Habib M. 74 Lesnic, Daniel 71 Yang, Chuanfu 62 Yurko, Vjacheslav Anatoljevich 60 Scherzer, Otmar 59 Liu, Hongyu 58 Dehghan Takht Fooladi, Mehdi 58 Hofmann, Bernd 58 Ma, Wen-Xiu 56 Lassas, Matti J. 55 Yao, Jen-Chih 55 Yao, Yonghong 53 Liu, Jijun 52 Ramm, Alexander G. 52 Xiong, Xiangtuan 52 Yin, Zhaoyang 51 Wei, Ting 50 Kaltenbacher, Barbara 49 Ceng, Lu-Chuan 47 Hu, Guanghui 46 Li, Peijun 46 Zou, Jun 45 Liou, Yeongcheng 45 Shehu, Yekini 44 Nakamura, Gen 43 Banks, Harvey Thomas 43 de Hoop, Maarten V. 43 Han, Bo 43 Jin, Bangti 42 Fan, Engui 41 Bellassoued, Mourad 41 Geng, Xianguo 40 Cakoni, Fioralba 38 Garnier, Josselin 38 Haddar, Houssem 38 Hasanoǧlu, Alemdar 37 Bao, Gang 37 Bondarenko, Natalia P. 37 Fokas, Athanassios S. 37 Kian, Yavar 37 Kress, Rainer 37 Salo, Mikko 37 Stuart, Andrew M. 36 Bal, Guillaume 36 Marin, Liviu 36 Pereverzev, Sergei V. 36 Reichel, Lothar 36 Sakhnovich, Alexander L. 35 Lu, Shuai 35 Yang, Fan 34 Chang, Shih-Sen 34 Kumam, Poom 34 Liu, Chein-Shan 34 Oksanen, Lauri 34 Ramlau, Ronny 33 Burger, Martin 33 Kang, Hyeonbae 33 Mathé, Peter 33 Vessella, Sergio 32 Cheng, Jin 32 George, Santhosh 32 Haltmeier, Markus 32 Hon, Yiu-Chung 32 Van der Mee, Cornelis Victor Maria 32 Wen, Chingfeng 32 Yan, Guozheng 31 Choulli, Mourad 31 Huang, Ting-Zhu 31 Kunisch, Karl 31 Wei, Guangsheng 30 Hyvönen, Nuutti 30 Li, Jingzhi 30 Ma, Fuming 30 Novikov, Roman G. 30 Novotny, Antonio André 30 Park, Won-Kwang 30 Sini, Mourad 30 Stefanov, Plamen D. 29 Censor, Yair 29 Cholamjiak, Prasit 29 Korotyaev, Evgeny L. 29 Mewomo, Oluwatosin Temitope 29 Xu, Hong-Kun 28 Belishev, Mikhail Igorevitch 28 Harrach, Bastian 28 Jin, Qinian 28 Trong, Dang Duc 28 Wang, Jenn-Nan 27 Aktosun, Tuncay 27 Borcea, Liliana 27 Buterin, Sergey Alexandrovich 27 Cheng, Xiaoliang 27 Cho, Yeol Je 27 Colton, David Lem 27 Đinh Nho Hào ...and 14,700 more Authors
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### Cited in 769 Journals
732 Inverse Problems 483 Inverse Problems in Science and Engineering 467 Journal of Mathematical Physics 413 Journal of Inverse and Ill-Posed Problems 410 Journal of Computational and Applied Mathematics 402 Applied Mathematics and Computation 356 Journal of Mathematical Analysis and Applications 351 Applicable Analysis 324 Inverse Problems and Imaging 303 Journal of Computational Physics 232 Computers & Mathematics with Applications 211 SIAM Journal on Imaging Sciences 199 Journal of Differential Equations 178 Numerical Algorithms 168 Applied Mathematics Letters 158 SIAM Journal on Scientific Computing 157 Mathematical Methods in the Applied Sciences 146 Computer Methods in Applied Mechanics and Engineering 144 Applied and Computational Harmonic Analysis 138 Journal of Nonlinear Mathematical Physics 135 Theoretical and Mathematical Physics 130 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 128 Journal of Scientific Computing 126 Numerical Functional Analysis and Optimization 125 Journal of Inequalities and Applications 124 SIAM Journal on Mathematical Analysis 123 Applied Numerical Mathematics 123 Linear Algebra and its Applications 120 SIAM Journal on Applied Mathematics 118 Abstract and Applied Analysis 114 Fixed Point Theory and Applications 110 Wave Motion 109 Mathematical Problems in Engineering 105 Journal of Mathematical Imaging and Vision 104 Applied Mathematical Modelling 102 Optimization 99 Journal of Optimization Theory and Applications 97 Communications in Mathematical Physics 97 Engineering Analysis with Boundary Elements 89 Numerische Mathematik 88 Communications in Partial Differential Equations 85 International Journal of Computer Mathematics 80 Computational Optimization and Applications 73 Journal of Functional Analysis 71 Physica D 68 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 67 Computational and Applied Mathematics 67 Boundary Value Problems 64 Advances in Computational Mathematics 64 SIAM/ASA Journal on Uncertainty Quantification 63 Journal of Integral Equations and Applications 62 Chaos, Solitons and Fractals 62 Communications in Nonlinear Science and Numerical Simulation 62 Computational Geosciences 61 Mathematics of Computation 60 Mathematics and Computers in Simulation 58 SIAM Journal on Numerical Analysis 58 Journal of Mathematical Sciences (New York) 56 Journal de Mathématiques Pures et Appliquées. Neuvième Série 56 Nonlinear Analysis. Real World Applications 56 Comptes Rendus. Mathématique. Académie des Sciences, Paris 55 Journal of Applied Mathematics 54 Physics Letters. A 54 The Journal of Fourier Analysis and Applications 53 Studies in Applied Mathematics 53 Analysis and Mathematical Physics 49 Journal of Geometry and Physics 48 Proceedings of the American Mathematical Society 48 Results in Mathematics 48 Transactions of the American Mathematical Society 48 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 47 The Annals of Statistics 46 Acta Applicandae Mathematicae 46 Advances in Difference Equations 45 Letters in Mathematical Physics 45 BIT 45 Mathematical and Computer Modelling 45 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 44 Journal of Nonlinear Science and Applications 43 Archive for Rational Mechanics and Analysis 42 Advances in Mathematical Physics 41 Applied Mathematics and Optimization 41 SIAM Journal on Optimization 40 Integral Equations and Operator Theory 40 Acta Mathematicae Applicatae Sinica. English Series 39 Discrete and Continuous Dynamical Systems 39 Differential Equations 39 Electronic Journal of Statistics 37 Mathematical Notes 37 Advances in Mathematics 37 Foundations of Computational Mathematics 36 International Journal for Numerical Methods in Engineering 36 Quarterly of Applied Mathematics 34 The Journal of Geometric Analysis 33 Computational Mechanics 33 GEM - International Journal on Geomathematics 32 Reports on Mathematical Physics 32 SIAM Journal on Control and Optimization 32 Mathematical Programming. Series A. Series B 32 Journal of Fixed Point Theory and Applications ...and 669 more Journals
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### Cited in 61 Fields
6,830 Partial differential equations (35-XX) 5,911 Numerical analysis (65-XX) 2,020 Operator theory (47-XX) 1,343 Calculus of variations and optimal control; optimization (49-XX) 1,287 Dynamical systems and ergodic theory (37-XX) 1,266 Optics, electromagnetic theory (78-XX) 1,174 Information and communication theory, circuits (94-XX) 1,140 Operations research, mathematical programming (90-XX) 1,123 Ordinary differential equations (34-XX) 981 Statistics (62-XX) 967 Biology and other natural sciences (92-XX) 953 Mechanics of deformable solids (74-XX) 817 Fluid mechanics (76-XX) 803 Computer science (68-XX) 538 Integral equations (45-XX) 514 Quantum theory (81-XX) 479 Integral transforms, operational calculus (44-XX) 449 Systems theory; control (93-XX) 436 Linear and multilinear algebra; matrix theory (15-XX) 420 Geophysics (86-XX) 417 Harmonic analysis on Euclidean spaces (42-XX) 397 Probability theory and stochastic processes (60-XX) 280 Classical thermodynamics, heat transfer (80-XX) 278 Global analysis, analysis on manifolds (58-XX) 225 Differential geometry (53-XX) 211 Approximations and expansions (41-XX) 210 Functional analysis (46-XX) 195 Statistical mechanics, structure of matter (82-XX) 182 Special functions (33-XX) 145 Real functions (26-XX) 144 Difference and functional equations (39-XX) 141 Mechanics of particles and systems (70-XX) 140 Potential theory (31-XX) 139 Functions of a complex variable (30-XX) 129 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 69 Combinatorics (05-XX) 66 Nonassociative rings and algebras (17-XX) 65 Algebraic geometry (14-XX) 50 General topology (54-XX) 47 Convex and discrete geometry (52-XX) 45 Algebraic topology (55-XX) 45 Astronomy and astrophysics (85-XX) 43 Relativity and gravitational theory (83-XX) 42 Abstract harmonic analysis (43-XX) 31 General and overarching topics; collections (00-XX) 28 Measure and integration (28-XX) 24 Topological groups, Lie groups (22-XX) 21 Several complex variables and analytic spaces (32-XX) 20 Number theory (11-XX) 19 Associative rings and algebras (16-XX) 19 Manifolds and cell complexes (57-XX) 15 History and biography (01-XX) 11 Geometry (51-XX) 9 Commutative algebra (13-XX) 8 Mathematics education (97-XX) 6 Group theory and generalizations (20-XX) 6 Sequences, series, summability (40-XX) 5 Field theory and polynomials (12-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Category theory; homological algebra (18-XX) 1 Mathematical logic and foundations (03-XX) | 2022-05-24 14:59:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5548378229141235, "perplexity": 7474.511367235816}, "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/1652662573053.67/warc/CC-MAIN-20220524142617-20220524172617-00618.warc.gz"} |
https://eprints.soton.ac.uk/416266/ | The University of Southampton
University of Southampton Institutional Repository
# The enigmatic spin evolution of PSR J0537-6910: r-modes, gravitational waves and the case for continued timing
Andersson, N., Antonopoulou, D., Espinoza, C.M., Haskell, B. and Ho, W.C.G. (2018) The enigmatic spin evolution of PSR J0537-6910: r-modes, gravitational waves and the case for continued timing. The Astrophysical Journal, 864 (2), 1-8, [137-[8pp]].
Record type: Article
## Abstract
We discuss the unique spin evolution of the young X-ray pulsar PSR J0537-6910, a system in which the regular spin down is interrupted by glitches every few months. Drawing on the complete timing data from the Rossi X-ray Timing Explorer (RXTE, from 1999-2011), we argue that a trend in the inter-glitch behaviour points to an effective braking index close to $n=7$, much larger than expected. This value is interesting because it would accord with the neutron star spinning down due to gravitational waves from an unstable r-mode. We discuss to what extent this, admittedly speculative, scenario may be consistent and if the associated gravitational-wave signal would be within reach of ground based detectors. Our estimates suggest that one may, indeed, be able to use future observations to test the idea. Further precision timing would help enhance the achievable sensitivity and we advocate a joint observing campaign between the Neutron Star Interior Composition ExploreR (NICER) and the LIGO-Virgo network.
Text
rmode0537_apj - Author's Original
Text
Andersson 2018 ApJ 864 137 - Version of Record
Accepted/In Press date: 28 July 2018
e-pub ahead of print date: 10 September 2018
## Identifiers
Local EPrints ID: 416266
URI: http://eprints.soton.ac.uk/id/eprint/416266
ISSN: 1538-4357
PURE UUID: 0d773d27-1695-4139-bb58-83d9e9da5ce3
ORCID for W.C.G. Ho: orcid.org/0000-0002-6089-6836
## Catalogue record
Date deposited: 11 Dec 2017 17:30
## Contributors
Author: D. Antonopoulou
Author: C.M. Espinoza
Author: W.C.G. Ho | 2020-01-18 12:10:03 | {"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": 0.27486610412597656, "perplexity": 8587.553127822894}, "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/1579250592565.2/warc/CC-MAIN-20200118110141-20200118134141-00527.warc.gz"} |
http://principia-m.com/ts/0004/ | Think Stitch
# Dynamic Creation of Processes 2
This is the second article for the Dynamic Creation of Processes in SyncStitch and CSP.
## Continuation of the Parent Process
When child processes are created by the parallel composition, the parent process gets into the wait state for the child processes, and terminates after the terminations of all child processes. However, it is often the case that we want the parent process to continue after the terminations of the child processes in order to do another jobs. We can achieve this using the parallel compositions and the sequential compositions.
The process P described below creates two child processes when it receives an event c. At the same time, P specifies the continuation process after the terminations of the child processes with the sequential composition seq. When all of the child processes terminate, the seq shifts to the continuation process.
(define-process P
(! c
(seq
(par '()
(! a1 SKIP)
(! a2 SKIP))
(! b SKIP))))
The computation tree of P is as follows. The state labelled with A is the state in which two child processes have terminated. From the state A, the par terminates, the process tree is folded, and the seq shifts to the continuation (! b SKIP), which is the state labelled with B.
In the case that you want to pass data from the state A to the state B, you can do it with a memo process.
## Parent-Child Processes Revisited
Consider the process which creates a child process Q and waits for the termination like a shell in UNIX-like operating systems again. This time we executes the process Q twice.
(define-process P
(! c
(par (list b)
Q ; the first instance
(! b ; waits for the termination
(! c
(par (list b)
Q ; the second instance
(! b SKIP))))))) ; waits for the termination and terminates.
(define-process Q
(! a (! b SKIP)))
The computation tree is as follows:
In the selected state, the second instance of the process Q is about to generate an event b, the termination notification. However it cannot occur. You can see this in that the state is colored with green, which means there is no transition from the state, even though the parent process is ready to accept the event b. Can you spot the problem?
Since the both two par nodes on the process tree have the event b in the synchronization lists, the occurrence of the event b needs participation of the first instance of the process Q which had already terminated. In other words, the occurrence of the event b is the simultaneous synchronization of three processes. In general, more than two processes can synchronize on an event in CSP. Unfortunately, this characteristics of CSP caused the problem.
## Solution 1
One of the solutions to this problem is to use different events for different instances of the process Q. Suppose, for example, we have events b1 and b2. We also need to divide the definitions of the instances of the process Q since the notification events differ.
(define-process P
(! c
(par (list b1)
Q1
(! b1
(! c
(par (list b2)
Q2
(! b2 SKIP)))))))
(define-process Q1
(! a (! b1 SKIP)))
(define-process Q2
(! a (! b2 SKIP)))
No deadlock is found this time.
The second instance of the process Q can successfully terminate.
## Solution 2
If two child processes run concurrently, it is necessary to use different event for each child process to distinguish which child is terminated. However, if child processes are executed sequentially, it seems to be a waste to use two events. Since the processes are executed sequentially, the problem is solved using the sequential compositions.
We can combine the execution of the two instances of the process Q by seq. By doing this, we can share a event b in the two executions.
(define-process P
(seq
(! c
(par (list b)
Q
(! b SKIP)))
(! c
(par (list b)
Q
(! b SKIP)))))
(define-process Q
(! a (! b SKIP)))
3 August, 2013 | 2017-05-23 01:06:29 | {"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": 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": 0.215628981590271, "perplexity": 1730.9243381201584}, "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/1495463607245.69/warc/CC-MAIN-20170523005639-20170523025639-00510.warc.gz"} |
https://b-ok.org/book/651360/16a68b | Main The Fundamental Principles of Quantum Mechanics
# The Fundamental Principles of Quantum Mechanics
First Edition, fifth Impression
Year: 1937
Edition: 1ST
Publisher: McGraw-Hill
Language: english
Pages: 611 / 630
Series: International Series in Physics
File: DJVU, 17.42 MB
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1
### Солитоны и нелинейные волновые уравнения
Year: 1988
Language: russian
File: PDF, 24.59 MB
2
### Теория вихрей
Year: 1936
Language: russian
File: PDF, 13.61 MB
INTERN A TION AL SERIES IN PHYSICS LEE A. DuBRIDGE, CONSULTING EDITOR t. 9055 THE FUNDAMENTAL PRINCIPLES OF QUANTUM MECHANICS ""1," . , ',> 1 ...,,' ''''.1'._, 4 "' '.'J"'" .,.. . ... . . ..,,' ,...t. . . ,:. . : The quality of the materials used in the manufacture of this book is governed by continued postwar shortages.
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Bacher and Goudsmit-ATOMIC ENERGY STATES Bitter-INTRODUCTION TO FERROMAGNETISM Clark-ApPLIED X-RAYS Condon and M orse-QU
TUM MECHANICS Curtis-ELECTRICAL MEASUREMENTS DafJev--CRYSTAL STRUCTURE AND ITS APPLICATIONS Edwards-ANALYTIC AND VECTOR MECHANICS Eldridge-THE PHYSICAL BASIS OF THINGS Hardy and Perrin-THE PRINCIPLES OF OPTICS Harnwell-PRINCIPLES OF ELECTRICITY AND ELECTRO- MAGNETISM Harnwell and Livifl,good-ExPERIMENTAL ATOMIC PHYSICS Houston-PRINCIPLES OF MATHEMATICAL PHYSICS Hughes and DuBridge-PHoTOELECTRIC PHENOMENA Hund-HIGH-FREQUENCY MEASUREMENTS Hund-PHENOMENA IN HIGH-FREQUENCY SYSTEMS Kemble-THE FUNDAMENTAL PRINCIPLES OF QUANTUM MECHANICS Kennard-KINETIC THEORY OF GASES' Koller-THE PHYSICS OF ELECTRON TUBES Morse-VrnRATI9N .t\ND SOUND- Muskat-THE FLOW'OF HOMOGENEOUS FLmbi:J'
HROUGH POROUS MEDIA /.
l'Jl Pauling and Goudsmit- THE STRUC
VREj
,LINE SPECTRA Richtmyer and Kennard-INTRODUCTION TO MODERN PHYSICS Ruark and Urey-AToMs, MOLECULES AND QUANTA Seitz-THE MODERN THEORY OF SOLIDS Slater-INTRODUCTION TO CHEMICAL PHYSICS Slater- MICROWAVE TRANSMISSION Slater and Frank-INTRODUCTION TO THEORETICAL PHYSICS Smythe-STATIC AND DYNAMIC ELECTRICITY Stratton-ELECTROMAGNETIC THEORY White-INTRODUCTION TO ATOMIC SPECTRA Williams-MAGNETIC PHENOMENA
Dr. F. K. Richtmyer was consulting editor of the BerieB from ita ipception in 1929 until hiB death in 1939.
THJ1: FUN D AMENT ALPRIN CIPLES OF QUANTUM MECHANICS
With Elementary Applications
BY EDWIN C. KEMBLE
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Profe8sor of Physics, Harvard University
FIRST EDITION FOU}{TIf 1.\1 PRI<
SRrON
McGRA"T-HILL 'BOOK COMPANY. INC.
NEW YOHK AND LONDON 1937
COPYRIGHT, 1937, BY THE MCGRAW-HILL BOOK COMPANY, INC.
PRINTED IN THE UNITED STATES OF AMERICA
All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers.
DEDICATED TO
DAYTON CLARENCE MILL
R
PREFACE
This volume was originally intended to be an expansion of a summary of the elements of quanturn mechanics
ritten sorne years ago for the Reviews of Modern Physics by the author in collaboration with Professor E. L. Hill. The point of view is es
entially the sarnp as in the summary, but as the present work has grown in my hands it has lost most of its resemblance to the initial pattern. The method of approach was dictated by the desire to meet the needs of graduate students of physics. For this reason the argulnent is induc- tive in form and applications of the theory have been interwoven with the development of the basic mathematical structure. In order to minimize the necessity for frequent consultation of Hlathematical refer- ence books, a good deal of background mathematical material is included in Chap. IV. In reading other treatises on quantum theory I have frequently been distressed by the tendency to gloss over the numerous mathelnatical uncertainties and pitfalls which abound in the subject. From the stand- point of the beginner there is much to be said for this practice of minimiz- ing the
_e(ects of the theory in order to exhibit its main outlines in a compact and attractive form. Nevertheless it has seemed to me that a book which deliberately called attention to the "rpak spots in the argu- ment would be of considerable value to teachers and to students of the more mature type. The work of the mathematician von Neulnann provides a masterly antidote to the lack of rigor r haracteristic of the average physicist, but by common consent this "lork is too difficult for any but the most mathernatical students of this subject. Hence I have been led to try my hand at bridging the gap bet,veen the exacting tech- nique of von Neumann and the usual less rigorous formulations of the theory. In carrying this project through I have restricted the discussion to such elementary Inathpmatical methods as are the common property of physicists today. The reader must judge my success in avoiding the Scylla of sloppy thinking and the Charybdis of tedious complexity. Fine print, starred sections, and appendices indicate portions of the material which may well be omitted or briefly scanned on first reading. A feature of the present volunle on the physical and philosophical side is its consistent emphasis on the operational point of view and on the fundamental importance of Gibbsian assemblages of independent sys- tems in the physical interpretation of the mathenlatical formalism. vn
viii
PREFACE
A considerable collection of references indicates the author's indebted- ness to the ideas of others, but the list is by no means exhaustive. I have borrowed freely from other books and am particularly indebted to those of von Neumann, Dirac, and of Born and Jordan. \t is a pleasure to thank my colleagues and former colleagues, Dr. Eugene Feenberg, Dr. W. H. Furry, Professor J. C. Slater, and Professor J. H. Van Vleck for invaluable suggestions and generous assistance. I am particularly indebted to Professor Van Vleck for reading the entire manuscript and for his constant encouragement. Dr. Montgomery H. Johnson is responsible for much of the work on the continuous spectrum in Sec. 31, while Dr_ Bela I.Jengyel and Dr. Charles H.Fay have at various tin1es spent long hours in checking equations and other technica] assistance. 'Tlhe author is very grateful to the librarian of the Harvard Physics Laboratory, lVII's. Miner T. Patton, for her cheerfulness and accuracy in the repeated typing of successive editions of the manuscript. rro the Milton v"und of Harvard University I am indebted for a generous grant for technical help in preparing the manuscript for the printer.
EDWIN C. KEMBLE.
PEACHAM, V ERMON'r, August, 1937.
CONTENTS
PREFACE.
PAGE . . Vll
NOTATION
. xvii
REFERENCE ABBREVIATIONS
. . . . . . . . . . XVlll
CHAPTER I INTRODUCTION TO DUAI.JISTIC THEORY OF MATTER; I)EVELOPMENT OF SCHROD- INGER'S WAVE EQUATION. . . . . . . . 1 1. Historical Introduction. . . . . . . . . . . . . . . . . . . . 1 2. The Dualistic Theory of Radiation. . . . . . . . . . . . . . . 4 3. An Analogy bet\veen Geolnetrical Optics and Classical Mechanics. 7 4. Wave Packets and Group Velocity. . . . . . . . . 10 5. The Schrodinger Wave Equation for a Single Particl{' . 14 5a. The Time-free Equation . . . . . . . . . . 14 5b. The Second (General) Schrodinger Equation. . 15 *6. The Application of the Restricted Relativity Principle. 19 7. The Wave Equation for a System of Many Particles. . 21 7a. Formulation of Equation . . . . . . . . . . . 21 7b. Relation of Schrodinger Equation to the Classical Hamiltonian Function . . . . . . . . . . . . . . . . . . . . . . . . . 23 *7c
/The Schrodinger Equation and the Hamilton-Jacobi Equation . . 24 *7d. The Wave Equation for a System of Charged Particles in a Classical External Electromagnetic Field . . . . . . . . . . . . . . . 26 8. The Physical Interpretation of the Wave Function and the NOTmalization Condition. . . . . . . . . . . . . . . . . 2
8a. Probability and Quadratic Integrability. . . . . 29 8b. Normalization and Mass Current Density. 31 8c. A System Consisting of Two Independent Parts. 33
CHAPTER II
WAVE PACKETS AND 'rHE RELA'rION BETWEEN CLASSICAL MECHANICS AND W AVIJ MECHANICS. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9. Wave Packets and Group Veloeity in a One-dilncnsional Homogeneous Medium . . . . . . . . . . . . . . . 35 9a. The Fourier Integral Theorf'ffi. . . . 35 9b. Derivation of Group-velocity Formula 37 10. Wave Packets in Three Dimensions . . . . 41 *11. Wave Surfaces and the Hamilton-Jacobi Equation of the Classical Dyna- mICS . . . . . . . . . . . . . . . . . . . . . . . . . . 43 *12. Wave Packets and the Motion of Particles in a Force Field; Ferlnat's Principle. . . . . . . . . <. . . . . . ., ..... 46 13. Direct Rigorous Proof of Newton's Second La'w of Motion for Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 * An asterisk before the nunlber of a section or subsection indicates that the section or suhsection so marked may be omitted or skimmprl to advan.tage on first reading. L_
x
CONTENTS
14.
The Statistical Interpretation of the Wave Theory of Matter. 14a. Review qf Assurn ptions . . . . . . . . . . . . . . . 14b. Necessity of Introducing Assemblages . . . . . . . . 14c. Multiplicity of Energy and Momentum Values for a Definite State. The Wave Function and Measurements of Linear Momentum. . . . . 15a. Operational Definition of Momentum for Free Particles. . . . . 15b. Computation of Momentum Probabilities from a Wave Function. *15c. Momentum of Center of Gravity. . . . . . . . . . . . . . . *15d. The Measurement of the Momenta of Particles Moving in a Force Field. . . . . . . . . . . . . . . . . . . . . . . . . . . *15e. Individual Momenta of Particles in a System . . . . . . . . . . 15f. Summary: The Determination of the Wave F.unction of an Assem- blage . . . . . . . . . . . . The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . .
PAGE 51 51 53 56 58 58 60 63
15.\
66 68
16.
69 72
CHAPTER III ONE-DIMENSIONAL ENERGY-LEVEL PROBLEMS . . . . . . . . . . . . . . . . 78 17. Boundary and Continuity Conditions; Eigenvalues and Eigenfunc- tions. . . . . . . . /. . . . . . . . . . .. ........ 78 18. The One-Dimensional Anharmonic Oscillator . .. ........ 81 19. The Qualitative Behavior of the Integral Curves: Existence of Class A Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 82 19a. Behavior of Integral Curves in Regions of Positive and Negative Kinetic Energy . . . . . . . . . . . . . . . . 82 19b. The Discrete Eigenfunctions. . . . . . . . . . . . 84 19c. The Continuous Spectrum of Class B Eigenfunctions. 85 19d. The Paradox of the Nodes . . . . . 86 20. The Planck Ideal Linear Oscillator. . . . . 87 20a. The Sommerfeld Polynomial Method. 87 20b. Deterlnina tion of the Eigenvalues . . 89 20c. The Eigenfunctions and Their Properties . 89 21. An Approximation Method vVhich Correlates the Eigenvalues of Wave Mechanics "with the Energy Levels of the Bohr Theory. 90 21a. The B. W. K. Approximations for t/;(x). . . . 90 21b. Application to Eigenvalue Problem. . . . . . 93 *21c. Zwaan's Method and the Stokes Phenomenon. 95 *21d. Analysis of the Stokes Phenomenon . . . . . 97 *21e. Derivation of the Connection Formulas. . . . 100 *21f. Derivation of the Sommerfeld Phase-integral Quantum Condition. 103 *21g. Higher Approximations. . . . . . . . . . . . . . . . . .. 107 *21h. Modification of Method for Radial Motion in Two-particle Problern 107 21i. Asymptotic Agreenlent of Wave Theory and Classical Theory Regarding Position of Particle. . . . . . . . . . . . . . . . 108 21j. The Transmission of Progressive Matter Waves through a Potential Hill . . . . . . . . . . .. ............. 109
CHAPTER IV
THE MATHEMATICAL THEORY OF COMPLETE SYSTEMS OF ORrl'HOGONAL FUNC- TIONS . . . . . . . . . . . . . . . . . . . . .
22. Scalar Products and Systems of Orthogonal Functions. . 22a. Expansion in a Series of Functions. . . . . . . . 22b. Comparison of Properties of Vectors and Functions
113 113 113 . 114
CONTE1VTS
23.
22c. Scalar Products of Vectors . . . . . . . . . . . . . . . . . 22d. Scalar Products of Quadratically Integrable Functions of m Variables 22e. Spaces of Infinitely Many Dimensions . . . . . . . . . . 221. Proof of Orthogonality of Eigenfunctions of the One-dimensional Anharmonic Oscillator Problern . . . . . . . . . . . . . . Self-adjoint Operators and Equations. The Sturn}-Liouville Problen1. 23a. Self-adjoint Differential Operators in One Dimension. 23b. Orthogonality ,vith Respect to a Density Function p. 23c. The Sturm-Liouville Problem . . . . . . . . . . . 23d. Singular-point Boundary Conditions . . . . . . . . *23e. Existence of Discrete Eigenvalues for Sturm-Liouville Problenls with Singular End Points. . . . . . . . . . . . . . . . . . Reduction of Eigenvalue Problems Based on Self-adjoint Differential Equations to Variational Form . . . . . . . . . . . . . . . . . . Completeness of System of Discrete Eigenfunctions of a Sturm-Liouville Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . *25a. The Eigenvalues as Absolute Minima. . . . . . . . . . . . . . *25b. The Expansion of Arbitrary Functions in Terms of Eigenfunctions.
24.
25.
Xl
PAGE 115 116 119
120 121 121 123 124 125
128
130
132 132 135
CHAPTER V THE DISCRETE ENERGY SPECTRUM OF THE TvVO-PAH/fICLE CENTRAL-FIELD PROBLEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 26. The Behavior of Solutions of an Ordinary Second-order Differential Equa- tion near a Singular Point . . . . . . . . . . 140 27. The Legendre Polynomials . . . . . . . . . . . . . . . . . . 143 27a. General Properties of the Legendre Equation . . . . . . . 143 27b
/'Explicit Determination of Eigenvalues and Eigenfunctions. 144. 28. The/Energy Levels of the Two-particle Problem. 146 / 28a. The Wave Equation . . . . . . . . . . . 146 28b. Separation of the Variables . . . . . . . . 146 28c. The Azimuthal Factor of the Wave Function 147 28d. Deterrnination of 8(0) and Its Eigenvalues . 148 28e. Completeness of System of Eigenfunctions . 149 28f. Physical Interpretation of Quanturn Numbers land m 150 28g. Behavior of Radial Wave Functions at Boundary Points 152 28h. The Dumbbell Model of the Diatornic Moleeule. 155 29. The Hydrogenic Atom. . . . . . . . . . 157 29a. Application of the B. W. K. Method. 157 29b. Application of Polynomial Method. 158 29c. Generalized Laguerre Polynon1ials 160 29d. The Most General Eigenfunction. . 161
CHAPTER VI THE CONTINUOUS SPECTRUM AND THE BASIC PROPERTIES OF SOLUTIONS OF THE MANY-PARTICLE PROBLEM . . . . . . . . . . . . . . . . . . . . . . . 162 30. The Continuous Spectrum in One-dimensional Problems . . . . . . . . 162 30a. The Nature and Use of the Eigenfunctions of the Continuous Rpec- trum . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 30b. The Weyl Theory . . . . . . . . . . . . . . . . . . . . . . 163 *3a.r.'
Formal Treatment of Continuous Spectrum as the Limit of a Dis- crete Spectrum . . . . . . . . . . . . . . . 165 *30d. The Spacing of Energy Levels in Problems {3 and a. . . . . . . . 168
.
XlI CONTENTS
PAGE *30e. The Eigendifferentials . . . . . . . . . . . . . . . . . . . . 169 *30f. Passage 'from the Completeness Theorem for Problem {j to That for Problem a. . . . . . . . . . . . . . . . . . . . . 171 30g. The Fourier Integral Formulas a Special Case. . . . . 173 30h. Normal Packet Functions in One Dimension. . . . . . 174 \ 30i. Normal Packet Functions for the Two-particle Problem 174 *30j. Normalization of the Class B Radial Eigenfunctions for the Hydro- genic Atom . . . . . . . . . . . . . . . . . . . . . . . . 176 31. Weak Quantization. Theory of Radioactive Emission of Alpha Particles 178 31a. Weak Quantization in General. . . . . . 178 31b. A Model for Alpha Particle Disintegration . . . 179 31c. Resonant Energy Intervals . . . . . . . . . . 181 31d. Energy Distribution in Weakly Quantized States 186 31e. The Disintegration Process . . . . . . . . . . 187 *31f. Complex Eigenvalues. . . . . . . . . . . . . 192 32. The Existence and Properties of Solutions of the Many-particle Schrod- inger Eigenvalue-eigenfunction Problem . . . . . . . . . . . . . . 195 32a. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 195 32b. New Boundary C6nditions for Physically Admissible Wave Func- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . 197 *32c. Approximating Arbitrary Quadratically Integrable Functions by Means of Class D Functions. . . . . . . . . .. .... 201 32d. Hermitian Character of the Hamiltonian Opera tor.. .... 202 32e. Reduction of the Eigenvalue-eigenfunction Problem for Discrete Spectra to Variational Form. . . . . . . . . .. .... 206 32f. A Lower Bound for the Energy Integral. . . . . .. .... 207 *32g. Behavior of Solutions of the Differential Equation at Singular Domains . . . . . . . . . . . . . . . . . . . . . 208 *32h. The Auger Effect . . . . . . . . . . . . . . . . . . 213 32i. The Discrete Eigenfunctions of the Differential Equation as Mini- mizing Functions. . . . . . , . . . . . . . . . . . . 214 32j. The Continuous Spectrum and the Completeness of the System of Eigenfunctions. . . . .. . 215 32k. Degeneracy. . . . . . . . . . . . . . . . . . . . . . . . . 217
CHAPTER VII DYNAMICAL VARIABLES AND OPERATORS . . . . . . . . . . . . . . . . . . 219 33. The Mean Values of the Cartesian Coordinates and Conjugate Linear Momenta. . . . . . . . . . . . . . . . . . . . 219 33a. The Statistical Mean Values of the Coordinates . 219 33b. The Linear Momentum Operator. . . 220 " 34. The Angular-momentum Operators . . . . . . . . . 224 34a. Definition of Opera tors. . . . . . . . . . . . 224 34b. Hermitian Character of Angular-momentum Operators. 225 34c. The Expansion Theorem . . . . . . . . . . 226 34d. Angnlar Momentum of a System of Particles . . . . . 227 34e. Mean Values . . . . . . . . . . . . . . . . . . . . . 229 34f. The Vector Angular Momentum and Its Square; the Symmetric Top 230 35. The Energy Operators . . . . . . . . . . . . . . . . . . 234 35a. Calculation of Probabilities and Mean Values of Energy 234 *35b. Transformation of Hamiltonian Operator. . . 237 36. Dynamical Variables in General. . . . . . . . 240
COJ.V TEN TS
XUl
PAGE 36a. Remarks on the Value of the General Theory . . . . . . . . . . 240 36b. Possibility of Defining Physical Quantities by Operators . . . . . 242 36c. The Transformation of Probability Amplitudes and Dynamical Variables . . . . .'. . . . . . . . . . 245 36d. Type 1 Operators as Dynamical Variables. . . . . . . . .. 248 36e. Calculation of Probabilities . . . . . . . . . . . . . . .. 256 36f. Type 2 Operators as Dynamical Variables; the Method of von Neumann. . . . . '. . . . . . . . . . . 259 36g. The Method of Dirac and Jordan . . . . . . . . . . . " 265 *36h. Multiplication Operators in Many Dimensions. . . . . . .. 268 36i. Transformation of Probability Amplitudes from One Arbitrary Coor- dina te Scheme to Another. . . . . . . . . . 270 36j. Dynamical Variables with Complex Eigenvalues. . . . . . . . . 275
CHAPTER VIII COMMUTATION RULES AND RELATED MATTERR . . . . . . . . . . . . . . . 278 37. Simultaneous Eigenfunctions and the Commutation of Dynamical Vari- ables. . . . . . . . . . . . . . 278 37a. Operator Algebra . . . . . . 278 37b. Functions of a Single Operator. 279 37 c. Commutative ()perators . . . 281 37d. Functions of a Normal Set of Commuting Dynamical Variables. 286 38. The Conservation Laws . . . . . . . . . . . . . . . . . . .. 288 38a. Conservation of Energy. . . . . . . . . . . . . . . . .. . 288 38b. Variation of Energy When the Hamiltonian Depends on the Time. 289 38e. Conservation of an Arbitrary Dynamical Variable . . . . . . . . 290 38d..' Commutation Properties of the Hamiltonian and the Angular Momentum. . . . .. ................ 291 39. Conjugate Dynan1ical Variables and Quantum-mechanical Equations of Motion. . . . . . . . . . . . . . . . . . . . 293 39a. Conjugate l)ynamical Variables . . . . . . . . . . . 293 39b. Functions of Non-commuting Linear Operators . . . . 300 3ge. An Operator Form of Hamilton's Equations of Motion. 301 40. Symmetry Properties of the Wave Equation 303 40a. Symmetry Properties in General. 303 40b. The Reflection Operators . . 305 40c. The Rota tion Operator. . . . . 306 40d. The Permutation Operators. . . 308 40e. Degeneracy and the Integrals of the Schrodingcr Equation. 310 40f. The Normal Degeneracy of the Energy Levels of Free Atomic Systems. . . . . . . . . . . . . . . . . . 313
CHAPTER IX
THE MEASUREMENT OF DYNAMICAL VARIABLES. . . . . . . . 41. General Theory of Measurement. . . . . . . . . . 41a. Fundamental Characteristics of Measurements. . 4)Jr. Pure States and Mixtures. . . . . . . . . . . 41c. Postulates Regarding Retrospective and Predictive Measurements. 41d. The Reduction of the Wave Packet .. .... 41e. Classical Orbits and Wave Packets. . 42. More About Measurements. . . . . . . . .
. 318 . . 318 318 320 322 326 331 334
XIV 42a. 42b. 42c. \ 42d. \ 42e.
CONT1!}NTS
PAGE Conjugate Variablps and Measurements. . . . . . . . . . .. 334 Impossibility of Measurcrnents Which Imply Distinction between Particles of Same Species. . . 335 A Classification of Observations . . . . . . . . . 341 Measurements as Correlations. . . . . . . . . . 342 The Observing Mechanism Not Entirely Classical 343
CHAPTER X MATRIX THEORY . . . . . . 348 43. Matrix Algebra . . . . 348 44. Matrices and Operators 352 44a. The Derivation of Matrices from Operators. 352 44b. Canonical Matrix Transformations. . . . . 355 44c. Matrix Form of the Eigenvalue-eigenfunction Problem. 359 *44d. Matrices with Continuous Elernents . . . . . 363 45. The Matrix Theory of Heisenberg, Born, and Jordan . . . . 366 45a. Fundamental Postulates . . '. . . . . . . . . . . . 366 45b
' Correlation of the Heisenberg and Schrodinger 1
heories . 368 45c. Solution of Matfix Equations of Motion for an Ideal Linear Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . 370 45d. Reduction of the Fundamental Problem of Matrix J\,fechanics to a Principal-axis Transformation . . . . . . . . . . . . . . . 372 46. The Bohr Correspondence Principle and Its Relationship to Matrix Theory 374 46a. The Bohr Postulates . . . . . . . . . . . . . . . . . 374 46b. The Bohr Correspondence Principle and the Heisenberg Matrix Theory. . . . . . . . . . . . . . . . . . . . 375
CHAPTER XI THEORY OF PERTURBATIONS WHICH Do NOT INVOLVE THE TIME 380 47. The Perturbation Theory for Nondegenerate Problems. 380 47 a. First-order Perturbations . . . . . . 380 47b. Second-order Perturbations . . . . . . . . 384 47 c. An Exam pIe: The Dia tonlic Molecule. . . . 386 48. The Perturbation Theory for Degenerate Problems 388 48a. First-order Energy Perturbations. . . . . . 388 48b. Second-order Energy Perturbations. . . . . 391 *48c. Van Vleck's Method for Second-order Perturbations . 394 48d. Simplification of Perturbation Calculations by Means of Integrals of the Perturbed IIarniltonian . . . . . . . . . . . . . . . . . 396 49. The Energy Levels of an Hydrogenic Atonl in a Uniform Magnetic Field (Spin Neglected) . . . . . . . . . . . . 398 49a. Derivation of Hamiltonian Operator . . . . . . . . . . . . . . 398 49b. Legitimacy of the Perturbation Method. . . . . . . . . . . . . 399 49c. First-order Energy Correction; Relation to Magnetic Moment and Larmor Precession . . . . . . . . . . . . . . . . . . .. 400 49d. The Second-order Energy Correction. . . . . . . . . . . ., 402 50. The Energy Levels of an Hydrogenic Atom in a Unifornl Electric Field. 403 51. The Variational Method . . . . . . . . . . . . . . . . . 408 51a. Reduction of the Variational Problenl to Algebraic Form. 408 51b. The Ritz Method . . . . . . . . . . . . . . . . . 410 *51c. IIigher Roots of the Secular Equation . . . . . . . . . 415 51d. General Observations Regarding the Use of the Variational Method 416
CONTENTS
xv
PAGE 51e. Modifications of the Method; Construction of Eigenfunctions from Non-orthogonal System. . . . 418 52. The Problem of the Hydrogen Molecule . . 419 52a. The Fixed-nuclei Problem. . . . . . 419 52b. The Heitler and London Calculation . 420 52e. The Method of James and Coolidge . 425
CHAPTER XII QUANTUM STATIS'l'ICAL MECHANICS AND THE EINSTEIN TRANSITION PROBABILITIES 427 53. Quantum Statistical Mechanics. . . . . . . . . . . . . . . . . . . 427 53a. The General Theory of Perturbations Which May Involve the Time 427 53b. The Adiabatic Theorenl. . . . . . . . . . . . . . . . .. 43] 53e. The Fundamental Problerns of Quantum Statistical Mechanics . . 432 53d. The Conventional Characterization of a Chaotic Assemblage . . . 434 53e. Transition Probabilities and Statistical Equilibrium for Chaotic Assemblages. . . . . . . . . . . . . . . . . . . . . . . . 439 53!. The Gibbs Canonical Assenlblage for Systenls of the Most General Type. . . . . . . . . . . . . . . . . . . . . . . . . . . 446 54. The Absorption and Enlission of Radiation: Perturbation of an Atomic System by a Classical Radiation Field . . . . . . . . . . . . . . . 448 54a. The Einstein Derivation of the Planck Radiation Fonnula . . . . 448 54b. Elementary Approaches to the Quantum Theory of the Einstein Transition Probabilities. . . . . . . . . . . . . . . . 450 54e. The Perturbing Hamiltonian for a Classical Radiation Field 452 54d. The Born Transition Probability. . . 454 54e. The Einstein Transition Probabilities. . . . . . . . 458 54/. ,Spectroscopic Stability. . . . . . . . . . . . . . 462 *54)1.' Magnetic Dipole and Electric Quadrupole Radiation . 462 55. Some Elementary Selection Rules for Electric Dipole Radiation. 469 55a. The Harmonic Oscillator . . . . . . . . . . . . . . . 469 5.5b. Selection Rules for the Two-particle Problenl . . . . . . . 470 55e. Fine Structure and Polarization of Spectrunl Lines in Simple Zeenlan Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 471
CHAPTER XIII INTRODUCTION TO THE PROBLEM OF ATOMIC STRUCTURE: I
LECTRON SPIN. 474 56. The Atomic Problem as a T\vo-particle Problenl. . . . . . . . 474 56a. The Empirical Basis for the Idealized Bohr Atom Model . 474 56b. Derivation of the Ritz Fornlula . . . . . . . . . . . . 478 57. The Bohr Assignnlent of Electronic QuantuIll Numbers. . . . . .. 481 57 a. The Quantum Numbers of the Valence Electrons in the Spectra of the Alkalies and Alkaline Earths. . .'. . . . . . . . . . . . 481 57b. Perturbation Theory and the Significance of an Assignnlent of Quan- tum N urn bers to Inner Electrons. . . . . 484 58. The Electron-spin Hypothesis. . . . . . . . . . . . 491 58a. The Empirical Fine Structure of Spectrunl Lines. 491 58b. The Combination of Angular Momenta. . . . . 495 58e. The Lande Magnetic Core 1"'heory . . . . . . . 498 58d. Solution of the Fine-structure Problem by tht' Electron-spin I-Iypo- thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 500 59. The Fine Structure of the Spectra of A tornic Systems with a Single Valence Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
XVI
CONTENTS
60. 61.
The ApproxiIr}.ate Relativistic Theory of the Hydrogen Atom. . . . . The Pauli Wave-mechanical Forrnulation of the Theory of Electron Spin 61a. Nature of the Configuration Space and Wave Functions . . 6J.b. Preliminary Discussion of Spin Operators and Spin Matrices 61c. Application of the Pauli Theory to the Alkali Doublets
PAGE 507 510 510 512 519
\ \
CHAPTER XIV
THE THEORY OF THE STRUCTURE OF MANy-ELECTRON ATOMS. 62. General Formulation of the Problem. 62a. The Configuration Space. . . . . . . . . . . 62b. The Hamiltonian Operator . . . . . . . . . . 62c. The Perturbation Form of the General Atomic Problem 63. Problem B: The Spin-orbit Energy Neglected. . . . . . . . 63a. Integrals of the Motion. . . . . . . . . . . . . . . 63b. Antisymmetric Functions and the Empirical Pauli Exclusion Rule 63c. Closed Shells . . . . . . . . . . . . . . 63d. Terms Originating in a Given Configuration. 64. Selection Rules for Electric Dipole Radiation . . . i 64a. The Laporte Rul
. . . . . . . . . . . . 64b. Selection Rules for the Central-field Problem 64c. Selection Rules for Problems Band C 65. The Helium Atom and Exchange Energy. 65a. Two-electron Atoms . . . . . . . . 65b. The Exchange Phenomenon. . . . . 66. Diagonal Sums and the Problem B Energy Levels.
523 523 523 524 526 528 528 533 536 . 537 . 540 540 541 543 547 547 553 . 555
ApPENDICES . . . . . . . . . . . . . . . . . . . . 557 A. The Calculus of Variations and the Principle of Least Action . . . 557 B. Derivation of Equation (15.7). . . . . . . . . . . . 564 C. Theorems Regardin-g the Linear Oscillator Problem. . . . . . 568 D. Mathematical Notes on the B.W.K. Method . . . . . . . . . 572 E. The Reduction of Certain Boundary-value Problenls Based on Self-adjoint Differential Equations to Variational Form. . . . . . . . F. The Legendre Polynomials and Associated Legendre Functions G. The Generalized Laguerre Orthogonal Functions. . . . . H. Two Theoreuls Relating to the Continuous Spectruln . . 1. Concerning the Expansion of Hf in Spherical Harmonics . J. The Jacobi Polynomials. K. Schlapp's Method
579 583 . 585 588 592 594 . 596
NAME INDEX. . .
SUBJECT INDEX. . .
. ........ . . .
. 599 603
NOTATIO
The nun1ber of different physical and mathematical quant
ties to be represented by separate symbols in this book is embarrassingly large in comparison with the avail[1ble letters of the Roman and Greek alphabets. For this reason the establishment of a one-to-one correspondence of symbols and meanings has proved impracticable. The author has endeavored to keep the notation consistent within each chapter and, with a fevv exceptions vvhich should not be confusing, has used only one symbol for each well-defined and recurrent meaning. The following notes may be of use to the reader who attempts to dip into the middle of the book. An asterisk * used as a superscript denotes the complex conjugate of the number or function in question. Ordinarily the symbol '11 denotes a time-dependent wave function, while -f indicates the time-free space factor of a monochromatic or single- energy '11. At times t/; is also used for the instantaneous form of a general wave function. Vectors are indicated by superior arrows. Three-dimensional vector and scalar products are indicated by the
conventipnal X and ., e.g., A X B and A . B. . The scalar products of many-dimensional cOlnplex vectors and of functions are denoted by heavy parentheses, e.g.,
(A, B) =
kAkBk *, (1jI (x), (f(x» = f 1jI(f*dx. In Chap. IV the norrn of a function f, viz., (f,f), is indicated by Nf, "hilA the magnitude, or square root, of the norm is indicated by Ilfll.
a' denotes a lnixed process of sumn1ation and integration over all eigenvalue points in a'-space. Cf. p. 246. Matrices are denoted by boldface type or by a typical element enclosed in double vertieal rules. Thus, H = IIHCm,n)]/. The first of the two indi('es of the typical elernent of an ordinary two- dimensional matrix indicates the ro'u), while the second denotes the column. The Dirac notation for an eigenfunction of a in x'-space, viz., (x'ja'), is introduced in See. 36h, while the l)irac notation for rnatrix elements, J.g., (/3/11,,1/3'), appears in Sec. 44d. r-rhe Dirac
ynlboliHIn is elnployed only at points where it is particularly convenient. XVll
D. P.
E. Q.
M. G. Q.
M. M. P.
P. Q. 11[.
,
Q. M.
T. A. S.
REFERENCE ABBREVIATIONS
Dijferen.tialgleichungen der Physik, Riemann- Weber, Braun- schweig, edition of 1927. Elementare Quantenmechanik, M. Born and P. Jordan, Berlin, 1930.
M athematische Grundlagen der Quanten'mechanik, J. v. N eu- mann, Berlin, 1932. Methoden der Mathernatischen Physik I, R. Courant and D. Hilbert, Berlin, 2d ed., 1931. / The Principles of Quantum Mechanics, P. A. M. Dirac, Oxford, 1st ed., 1930; 2d ed., 1935. Quantum Mechanics, E. U. Condon and P. M. Morse, New York, 1929. The Theory of Atomic Spectra, }1
. U. Condon and G. H. Short- If\Y, Oxford, 1935.
XVIU
FUNDAMENTAL PRINCIPI
ES OF QUAN1'UM MECHANICS WITH ELEMENTARY APPLI(
\ TIONS
CHAPTER I INTRODUCTION TO THE DUALISTIC THEORY OF MATTER; DEVELOPMENT OF SCHRODINGER'S WAVE EQUATION 1. HISTORICAL INTRODUCTION The first step toward the forrnulation of the quantum theory waR jlade in 1900 by Max PlaIlck in the course of a theoretical investigation of the laws of thermal radiation. 1 His probleln was to explain the distribution of energy in the continuous spectrurn of a heated black body as a function of its temperature. The experimental fact is that the intensity per unit frequency interval rises from zero at very low fre- quencies to a maximum value whose position and ll1agnitude depend upon the t
Jrlperature-then falls again, approaching zero at very high t' frequencieS. The drop in intensity in the high-frequency region is in / violent conflict with a theoretical result previously obtained bY' the elder Lord Rayleigh 2 on the basis of the equipartition theorem derived from the classical statistical lllechanics and of thp
Tave theory of light. Planck attributed the discrepancy to the breakdown of the equipartition theorem when applied to high-frequency oscillations and made the . brilliant suggestion that, if the vibrating matter particles which emit radiation have Inotions restricted to certain diserete energy values, or energy levels, there would be a departure from the laws of the classical statistical mechanics of the sort required by the experimental facts. On the basis of this hypothesis Planck was able to derive a formula for the intensity of the . radiation in terms of temperature and frequency which fits the empirical data within the limits of experimental uncertainty. The acceptance of Planck's suggestion meant a complete revolution in physics since it was incoll1patible with both the Newtonian mechanics and the electromagnetic theory of light. As corollaries, one nlay infer at once that the n1echanics of the collisiolls between atoms are completely J M. PLANCK, Verh. d. deut. physik. GeseU.. 2. 237 (1900); Ann. d. Physik 4, 553 (1901). 2 RAYLJ4
:r(J:H1 Phil. Mag. 49, 539 (1900)
1
2 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I
nonclassical and that the light is emitted and absorbed in discrete "parcels" or "quanta." A natural inference is that radiant energy is co
puscular in character. Planck himself was unwilling to entertain so 'radical an hypothesis and spent much time and energy in an ultimately fru
tless attempt to save the wave theory of light by a modification of hie original energy-level assumption. In fact so firln was the hold of the wave theory on the minds of all physicists that it was not until 1905 that Einstein,l then a young man of 26, seriously revived the corpuscular conception of the nature of light. The application of Planck's hypothesis to the problem of the structure of matter was still further delayed. Although Einstein showed in 1907 that it contained the key to the problem of the low-temperature specific heats of solids, it was not until 1913 that Bohr 2 united the Rutherford nuclear conception of the atom with the energy-level hypothesis to formulate his famous theory of the structure and spectrum of hydrogen. The initial success of /the young Dane 3 was followed by a period of rather feverish, but fruitJul, activity for both experimental and theoretical physicists. New discoveries followed one another in bewildering suc- cession, and in a short time scientific understanding of the nature of matter was immeasurably deepened. Prior to 1924, however, the theoretical developments were largely of an essentially provisional character. At that time the battle between the advocates of the wave theory of light and the' proponents of the corpuscular theory had led to an unsatisfactory stalemate. Those who favored the corpuscular theory had made it abundantly clear that radiation has many of the properties
. 1
. EINSTEIN, Ann. d. Physik (4) 17, 132 (1905); 20, 199 (1906). Einstein's first paper on the theory of relativity was sent to the publishers less than 4 months after his first paper on the corpuscular theory of light! 2 N. BOHR, Phil. Mag. 26, 476, 857 (1913). 3 Bohr was twenty-eight years old when he published his first paper on the theory of the hydrogen spectrum. In fact, the quantum theory has been from first to last a development by young men. Einstein, as already mentioned, wrote his initial paper on the corpuscular theory of light at the age of twenty-six. Heisenberg was twenty- four years old when he lai-d the foundation of the matrix mechanics. Dirac and Jordan wrote their first important papers at the ages of twenty-four and twenty- three, respectively. W. Pauli,. Jr., was already a figure of importance in theoretical physics when at twenty-five he formulated the exclusion principle which bears his - name. Uhlenbeck and Goudsmit were twenty-five and twenty-three years of age, respectively, when they invented the spinning electron. L. de Broglie published his first paper on the wave theory of
he electron at thirty-one, while Schrodinger's most important papers on wave mechanics were written at the relatively advanced age of thirty-nine. Of course the contributions of older men, especially Sommerfeld and I Born, have been exceedin
Jy valuable, but one cannot but be impressed with the importance to science of a system of education which eaables young men to finish their preliminary training and start their career of productivity while the extraordinary mental energy of youth is still in fun vigor.
SEC. 1]
HISTORICAL INTRODUCTION
3
to be expected from their model. But no satisfactory way of accounting for- the characteristic wave phenolnena of interference and diffraction on the basis of a pure particle theory had been found. There was abundant evidence of the reality of the energy levels postulated by Planck and Bohr-yet it had also become clear that Bohr's makeshift combination of classical mechanics and "quantum conditions" was inadequate for the vvorking out of an exact theory of atomic structure. Moreover, aperiodic phenomena and the problelll of the interaction of atoms in the formatioIl of molecules and solids were practically untouched. The temporary retardation in the progress of theoretical physics brought about by the limitations inherent in the Bohr theory was finally broken by the introduction of new fundamental hypotheses by Louis de Broglie 1 and Werner Heisenberg. 2 To de Broglie we owe the sugges- tion that matter may share the dualistic characteristics of radiation by combining the properties of waves with those of corpuscles. To Heisenberg we owe a scheme for the exact description of atomic dynamical systems by means of a new kinenlatics based on Bohr's" correspondence principle." De Broglie's hypothesis in the hands of Schrodinger 3 received the definitive form now generally accepted, and Heisenberg's method was converted into a powerful matrix calculus by Born and Jordan. 4 Both suggestions, despite their extreme dissinlilarity, proved to be of great value. Fused into a single theory which we call the "quantum l!lechanics," they correct the deficiencies of the Bohr theory as a tool/for investigating the structure of matter, relate the newly discovered diffraction of electron beams to the problem of locating atomic energy levels, and go a long way toward removing the dilemma regarding the nature of radiant energy. In the fornl of quantum theory now most generally accepted, the dualistic nature of radiation is treated as a fact to be described rather than explained or exorcised. In accordance with de Broglie's hypothesis, a similar dualistic nature is ascribed to matter, and thus a unification in the treatment of matter and radiation is attained. The fundamental similarities between the assumed characteristics of matter and radiation form one of the most striking features of present physical theory. Differ- ences remain, to be sure-and we can by no stretch of the imagination
dentify these two modes of existence-but the analogy is far reaching enough to permit the use of observations regarding the characteristics of radiation as guides in the construction of a theory of matter. The de Broglie-Schroding
r wave mechanics is the result of a conscious 1 L. DE BROGLIE, Nature 112, 540 (1923); Thesis, Paris, 1924; Ann. de Physique (10)..3, 22 ,(1925). 2 W. HEISENBERG, Zeits. j. Physik 33, 879 (1925). 3 E. SCHRODINGER, Ann. d. Physik 79, 361, 489 (1926). 4 M. BORN and P. JORDAN, Zeits. f. Physik 34, 858 (1925).
4 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I \
attempt to follow such guides and affords a relatively easy method of approach to the general theory.
- 2. THE DUALISTIC rHEORY OF RADIATION the importance of optical analogy in the development of quantuln mecH
nics lies primarily in the fact that the dualistic nature of light is much more obvious than the dualistic nature of matter. In the region of long waves the wavelike characteristics of radiation are strongly predominant, while in the X-ray region ,the corpuscular characteristics are more obvious. 'As the transition from one part of the spectrum to the other is continuous, the dualism is inescapable. On the other hand, serious technical difficulties stand in the way of a direct experimental study of long wave length--matter waves-so that it is small wonder that the wavelike aspect of the nature of matter was discovered at a very late date. Let us therefore begiJ)' our study of the quantum mechanics with a. preliminary examination (of the properties of radiation. As previously stated, the result of the conflict between the wave theory of radiation and . the corpuscular theory up to 1924 was a draw. The electromagnetic theory of Maxwell gave a simple and accurate account of interference, diffraction, and dispersion,' besides making proper /co'nnectioIl with quasi-static elec
romagnetic phenomena in the limiting region of very long waves. The corpuscular t
eory gave a simple an,d accurate account of the fundamental laws of the photoelectric effect and the Compton effect. It could be regarded as a logical corollary of the fundamental law of spectroscopy,
E' - E" = hv ,
and it seemed necessary in order to account for the abrupt changes in momentum experienced by emit.ting and absorbing atoms and molecules in a radiation field.! Neither point of view gave a satisfactory descrip- tion of the whole field of optics. In the case of the Doppler effect, 2 the predictions of the two theories were identical and in agreement with experiment. In some ways the two theories supplemented each other. For example, in the case of the in verse photoelectric effect (i.e., the production of the continuous X-ray spectrum) the corpuscular theory was needed to account for the sharply defined high-frequency limit to the spectrum, but
he help of th
\vave theory was needed to account for the polarization of the radiation. Either point of view gave a qualita- tive explanation of the variation in hardness with direction of emission, which is, in fact, a kind of Doppler effect.
1 A. EINSTEIN, Physik. Zeits. 18, 121 (1917). 2 E. SCHRODI
GER, Physik. Zeits. 23, 301 (1922).
SEC. 2]
5
To obtain a satisfactory theory of light, one 11lUst formulate a descrip- tion of its behavior which ernbodies the characteristics of both of these conflicting points of view. As a first step toward the formulation of such a description, we observe that a similar controversy arose long ago in connection with matter. In bulk it has properties which are con- veniently described from the continuum point of view. In particular, it may be the vehich
of sound waves which act like ,vaves in a continuous medium. If matter be molecular in structure, however, we must expect this fact to be most evident in the properties of low-density gases. Experiments made on such gases do favor the molecular hypothesis and are regarded aSAcrucial since high-density matter must in any case act in some ways like a continuum. Similarly, the corpuscular properties of radiation, if they exist, must be most evident if the corpuscles are of great energy and fe,v in number as in the 'case of low-intensity beams of hard X-rays. Precisely here the evidence for atomicity through the C. T. R. Wilson ray-track experiment and the Duane-Geiger point counter is most positive and definite. On the other hand, to get a test of one of the predictions of the wave theory one must have a record of the absorption of a quantity of light containing on the basis of the particle theory a very large number of photons. Thus one may say that inter- ference experiments show that statistically light has the properties of waves \vithout in any way directly disproving its granular structure. In other wQrds, the experiments which were initially regarded as evidence / against jJfe corpuscular theory are not to the point but actually show nlerely that the properties of the corpuscles are different from those to be expected by analogy with classical mechanics. We then lay down as an initial postulate the hypothesis of the ato- micity of radiation. As a second postulate, we assume in accordance \vith experiment that the wave theory gives a correct description of the aVE/rage intensity distributioll in ordinary interference and diffraction experiments. Thus, in predictipg or describing the results of optica] experiments, we make use of both concepts with the related mathe- lnatical machinery.! For the detailed correlation of the energy E and Inomentum p with the frequency 1/ and wave length X we make use of the fundamental formulas due to Einstein:
E = hv , h p = -. A
(2.1 ) (2.2)
lOne may, if one likes, assume that light consists of both waves and corpuscles, or he may say that it consists of corpuscles guided by a "ghost" electromagnetic wave field [cf. W. F. G. Swann, Science 61,433 (1925)]. The writer would prefer to regard both \vaves and corpuscles ultinlately as mental aids in the description and prediction of empirica.1 results, leaving all questions rep;arding their objective reality to the? philosophers.
6 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I
Here h denotes PI
nck's constant as usual. These equations mean that radiation which is monochromatic from the point of view of the wave theolry (i.e.,' gives a single sharp spectrum line vvhen analyzed by a spect
rometer) consists of photons of energy E = hv and momentum p = h/x. We further suppose that in the case of a plane progressive wave fhe direction of the momentum of the associated particles is that of the forward normal to the vvave front. It is immediately evident that these assumptions create a theory of radiation which up.ites the partial successes of the Maxwell theory with those of the corpuscular point of view. They do not answer all questions regarding the interaction of light and matter, but that is not to be expected without a fully dev
loped theory of matter. N either do they make any atteillpt to answer the question: "Why does light act in some respects like an assemblage of corpuscles and in other respects like a spreading-wave phenomenon?" Instead, they describe the dualistic behavior of light. As it is now generally recognized that description rather than explanation is the true function of physical theory, this procedure is entirely correct. Furthermore, the assumptions carry with theln the tacit or explicit admission that deterministic models 1 are of little use in dealing with the radiation problerp. In fact, the problem of the nature of radiation as seen by physicists prior to 1925 loses its point as soon as such deterministic models are cast aside. Statistics and indeterminism enter the theory of light when one assumes that the distribution of discrete corpuscles in space is to be calculated by means of continuous wave functions. This can only mean that the intensity of light of frequency v in any small volume G as computed from the wave functions is a measure of the probable number of photons of energy hv in G. If the energy in G were measured n times, the individual measurements would necessarily show flu'ctuations or departures from the lllean-although, according to the theory, the mean itself would approach the computed value as n becomes very large. 2 " Corresponding to this theoretical i
determinism, there is an experi- mental indeterminism eyidenced; for example, in the haphazard devel- opment of the grains on a photographic plate. Tpe iCleal test for determinism or indeterminislll would be, of course, to perform repeatedly the same experiment with the" initial conditions" exactly controlled z and then see whether or not the results are identical. As this is in practice
1 We here use the term "model" in "a very broad sense to describe either the classical concept of a particle or of a wave ,vith all the tacit assumptions formerly bound up with these concepts. 2 The necessity for indeterminism is also evident if we consider the problem of reflection. Waves can ahvays divide themselves in a definite way at th
interface of two media, but each individual photon must either be reflected as a whole or pass acros
the boundary.
SEC. 3] GEOMETRICAL OPTICS AND CLASSICAL MECHANICS 7
impossible, we can only examine the results under conditions as similar as possible to see whether the deviations from the mean of the results are commensurate with the uncertainty regarding the initial conditions. In the case of photons, the best we can do is to throw a beam of plane- parallel light on a small aperture and allow the emergent beam to fall upon a photographic plate. There is then an uncertainty regarding the point at which any individual developable photographic grain will appear rneasured by the effective diameter of the illuminated portion of the plate. This uncertainty can be decreased to a certain limiting value by decreasing the area of the aperture. But,.if the opening is too small, diffraction causes the illuminated area to increase once more and thus presents a complete barrier to an indefinite reduction of the experi- mental uncertainty. Thus an exact control of the" future" is impossible in such an optical experiment. Whether this is due to the fact that the initial conditions for the various photons cannot be exactly repeated, or to the fact that the initial conditions do not exactly control the future, is a futile qqestion since incapable of experimental investigation. From a practical standpoint, the field of optics is indeterministic and must remain so, unless some new mode of experiment is discovered which permits a more exact control over the behavior of photons than any we now hav(. If the union of particle-theory and
Tave-statistics outlined above is fundamentally correct, no such mode of experiment can exist. 3. AN ANALOGY BETWEEN GEOMETRICAL OPTICS AND CLASSICAL / MECHANICS In order to formulate a theory of matter paralleling the dualistic theory of radiation sketched in the preceding section, we must invent a suitable differential equation for the wave function. This equation may be assumed to have a form similar to that of the wave equation of optics but must be so designed as to harmonize with the Newtonian mechanics in the limiting case when diffraction effects are negligible. If it is possible to set up such an equation, there should be a certain silnilarity between the Newtonian mechanics and diffractionless, or geometrical, optics. The desired similarity does exist and is evident at once if one compares the principle of least time in optics (F"ermat's principle) with the principle of least action in mechanics. 1 The optical principle states that the path of a ray of light (wave front normal)
from a. point A to a point B is always such as to give the in tegral
f BdS , / AW 1 The principle of least action originated in an attempt by Maupertuis to obtain for the_corpuscular theory of light a theorem analogous to Fermat's principle (cj. E. T. Whittaker, Analytical Dynamic.
, 2d ed., p. 248, Calnbrid
e, 1917).
8 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I'
an extreme value (usually a minimum) with respect to the integrals over all other
onceivable paths for rays of the same color, or frequency. In this formula W denotes the local phase velocity of light and is a function bf the frequency and the space coordinates, say x, y, z. As the frequency \ is treated as a constant in varying the integral, and as the local wave length A is equal to wi v, we may substitute X for w in the statement of 'the principle. Using the notation of the calculus of variations we have
o fB ds = 0 JA XCv, X, y, z) . This means that the true path of the ray is characterized by the fact that for it the first variation oi the path length measured in wave lengths is zero.
(v unvaried)
(3.1 )
The usual elementary derivation of Fermat's principle 1 is valid only for light rays moving inhomogeneous media bounded by plane surfaces of ! discontinuity and is based on the assumptions of geometrical optics, viz., that light rays travel in straight lines in such media except at thf\ boundaries where they are regularly reflected or refracted in accordance with the sine law
sin () I sin () 2
WI
W2
The extension of the law to inhomogeneous media may be obtained by a limiting process in which the 'inhomogeneous medium is approximated by a heterogeneous aggregation of homogeneous volume elements, each differing slightly in index of refraction from its neighbors. Since the spreading of light waves is fully determined by the "wave equation" t'i2 _ 1 a 2 <{) v cp - -, w 2 (x,y ,z, v) at 2
(3.2)
we may conclude that Fermat's principle is deducible under suitable restrictions fronl this equation. The mathematical verification of this theorem wil1 be given at a later stage of our argument (Sec. 12, p. 46). Let us now cornpare Fern1.at's principle in the form of Eq. (3.1) with the principle of least action. In the case of a single particle of total energy E, kinetic energy T, and mass fJ., moving through a force field with potential energy V(x,y,z), the latter principle requires that the value of the action integral
s = f: 2Tdt = f: V2p.(E - V)ds
(3.3)
1 Cf. P. DRUDE, Theory of Optics, Chap. I, Art.. 2, English translation by Mann and Millikan, New York, 1917.
SEC. 3] GEOMETRICAL OPTICS AND CLASSICA_L MECHANICS 9
for the natural, or mechanical, path between the two points A and B shall be an extremal as compared with its values for adjacent paths an..d the same value of E. For such a part icle the integrand in the second form of the integral, viz., y2JJ-(E - V), is the absolute value of the instantaneous momentum which the particle would assume at x, y, z if it had the energy E. Denoting this quantity by p(E,x,y,z), we may state the principle in the form (cf. Appendix A)
of: p(E,x,y,z)ds = o.
(E unvaried)
(3.4)
It will be convenient to call p(E,x,y,z) the classical local momentum to distinguish it froln the true quantuill n1echanical momentum to be defined in Sees. 14 and 15 and from the local lllomentum of Y oung. 1 Evidently Eqs. (3,1) and (3'4) present formally identical mathematical problems. In fact, the paths of particles in the Newtonian dynamics may be identified with the "rays" of a probleln in geometrical optics in which the wave length is adjusted to make the integrand of the integral of Eq. (3'1) proportional to the integrand of Eq. (3'4). To be precise, the condition imposed on the wave problem is that
c Cv
/ A = W = p(E,x,y,z) = 'V 2J.t[E - V(x,y,z»),
(3'5)
where C
ay be any function of v, and v in turn must be a function of E. COI)Iparing Eqs. (3'5) and (2'2), we observe that the required relation between the classical local momentum of the particle and the correspond- ing local wave length is identical, except for a possible constant factor, with the relation between momel1-tum and wave length in the dualistic theory of light. The analogy between the variational principles of geometrical optics and particle dynamics "vas seized upon by Sir William Hamilton in the early part of the nineteenth century and used as a guide in the develop- ment of dynamical and optical theory. The Hamilton-Jacobi partial differential equation is the fruit of this development, but Hamilton himself regarded the analogy as an analogy only. It remained for de Broglie and Schrodipger to show that it may profitably be used as a stepping stone for the development of a true wave mechanics similar in form to physical rather than geometrical optics. If Hamilton had ,vorked a little later, he might even have discovered Schrodinger's wave equation, for he would almost certainly have been led to it, if he had sought a wave equation for his waves. But in his time, the wave theory of light was just beginning to be investigated, and it was not yet the fashion to describe waves as solutions of a partial 1 L. A. YOUNG, Phys. Rev. 38, 1612 (1931), 39, 455 (1932). I
10
INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I \
differential equation. Hamilton contented himself, as was the custom in optics, with invcestigating the positions of the wave fronts of his waves. He found the wave length of mechanical waves as it depends on position, the'index of-refraction as a function of the potential energy, and so on- the \
atter being simply the carrying over to mechanics of Newton's idea of the optical index of refraction as a function of potential. But he did' not try to set up a wave equation. And neither, ,at first, did de Broglie" led so much later to precisely similar conceptions. The final step of writing out such an equation for matter waves and applying it both to large-s
cale mechanics and to the mechanics of atomic systems was left to Schrodinger. 4. WAVE PACKETS AND GROUP VELOCITY The parallelism between the principle of least action and Fermat's principle does not complete the reduction of the laws of Newton's mechanics to a form similar to that of the laws of geon1etrical optics. These principles deal solely with the paths of particles and rays, saying nothing about the time required to traverse the path. In order to extend the discussion to take in the time, we must define the relation between the motion of a large-scale body and the corresponding matter waves in the limiting case where a sharply defi
ed orbit exists. Following optical analogy we postulate that the intensity of the matter waves associated with any particle at any space-time point x,y,z,t measures the probability that the particle is in the neighborhood of x,y,z at the time t. Then if the particle is to have a fairly definite orbit, it must be associated with a localized ,vave distur
ance which moves with it over a definite path. Such disturbances are familiar to students of physical optics or of other types of wave motion. They are usually called ( ( wave packets" because they can be analyzed into superpositions of infinite plane monochromatic waves involving a narrow range of wave lengths and directions of wave normal, and because this analysis is of fundamental importance in under- standing their behavior as time goes on (ef. Chap. II, Secs. 9 to 12). Optical wave packets can be formed from monochromatic optical wave trains if a diaphragm is used to cut them off laterally and a shutter to cut them off longitudinally. In a non-dispersive medium where the speed is independent of the wave length it is not necessary to start with a monochromatic train. Thus Fizeau in measuring the velocity of light used wave packets formed by white light passing through apertures in the rim of a revolving toothed wheel. To use this method in a medium with appreciable dispersion it would be essential to start with approxi- mately monochromatic light, since the different colors have different speeds. The essential requirements to be imposed upon a wave disturbance in order that it shall constitute a wave packet are that (a) it shall occupy a
SEC. 4]
WAVE PACKETS AND GROUP VELOCITY
11
1 Cf. A. SCHUSTER and J. W. NICHOLSON, Theory of Optics, 3d ed., p. 326, London, 1928; T. H. HAVELOCK, The Propagation of Disturbances in dispersive Media, Cam. bridge, 1914. Havelock ascribes the first discussion of group velocity to Hamilton. Proc. Roy___ Irish Acad. 1, 267,341 (1839).
12 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I
too far if ;it is not to conflict with the second requirement for tt wave packet. By a suitable cOlnpromise, then, we can devise wave _disturbances appropriate to a dispersive medium which satisfy all three requirements I for a wave packet. Such a disturbance will necessarily have a fairly \ well defined wave length and direction of motion, although the Fourier \:1 n alysis would resolve it into a superposition of a continuous spectrum 01 infinite plane wave systems whose wave lengths and ,vave normals are spread out over a very narrow range grouped about the wave length and wave normal appropriate to the interior of the packet. We shall return to this Fourier analysis in the next chapter.
"'or the present it will suffice to observe that the lack of sharpness in the definitions of the position, wave length, and direction of motion of such a packet is to be correlated with a corresponding lack of sharpness in the position, momen- tum, and direction of motion of the associated photon or matter corpuscle. The classical velocity and orbit of the corpuscle must then be identified with the velocity and/ orbit of the centroid of the wave packet. Thus classical mechanics is leventually to be regarded as a limiting case of the mechanics of matter wave packets. It follows that if the analogy between the principle of least time and the principle of least action really means anything it must be possible to show that the orbit and orbital velocity of a large-scale "particle" 1 in the Newtonian mechanics are identical with the orbit and orbital velocity of a wave packet in a suitably defined wave problem. To make use of the analogy of the preceding article we ought to show (a) that wave packets travel along the rays of geometric optics, and (b) that the speed of the packet is the saine as that of the corresponding mechanical particle. To avoid interrupting the main argument here we shall assume prop- osition (a) without proof 2 for the present and will base our discussion of proposition (b) on the familiar formula for the group velocity of a finite train of waves in a homogeneous dispersive medium. 3 Denoting the average, or "interior" wave length of the group or packet by "A and the phase velocity by w, the usual expression for the group velocity V g is - - ( dW ) V g = w("A) - "A - _. d"A A=A
1 The motion of the center of n1ass of a system of particles follows the same laws as the motion of a single particle both classically and in the quantum mechanics (ef. Sec. 15, p. 64). 2 The identity of the paths of wave packets or narrow beams of light with the rays defined by the normals to the corresponding extended wave systems is commonly assumed without proof in textbooks on physical optics. The assumption is validated in Sec. 12, Chap. II. 3 Cf. footnote 1, p. 11.
SEC. 4]
WAVE PACKETS AND GROUP VELOCITY
13
Using the relation between wave length, frequency, and phase velocity we readily convert this expression into the more compact form,
:0 = [ :v (O l=
J
(4-1)
where the differentiation is carried out with v,x,y,z acting as independent variables. Proofs of the group-velocity formula for lnatter waves are given in Sees. 9, 10, and 12. If the energy of the particle and the wave length are related as in Eq_ (3'5), and if the group velocity is identified with the speed of the particle v, Eqs. (3-5) and (4-1) give
= :v(
}
As p is a function of E,x,y,z and as the spatial coordinates are independent of v
1
_E- dC + :1 ap dE . C 2 dv CaEdv
v
But
ap _ J.l _ 1_ aE - p - v'
hence
dE _ C = pv dC _ dv C dv The product pv depends upon x, y, and z, but the left-hand member of the above equation is independent of these variables_ Hence dC / dv must vanish_ C is a constant, and Eq. (4'2) reduces tor'
(4'2)
dE dv = C = pX.
(4'3)
The linear relationship between energy and frequency thus derived suggests the hypothesis that the Einstein energy-frequency relation (2'1) holds for matter as well as for radiation. If we assume the validity of Eq_ (2-1) for matter, Eq. (4'3) requires.that Eq_ (2'2) shall also apply to matter_ The relation thus obtained between the classical local momentum of a particle and the length of the associated matter waves was first suggested by de Broglie. 2 It has been confirmed experimentally by the 1 The proof here given of th: linear relationship between E and" is closely related to that given by F. D. Murnaghan and K. F. Herzfeld, Proc. Nat. A cad. Sci. 13, 330 (1927) . 2 Loc. cit., footnote 1, page 3.
14 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I
\ electron diffraction experiments of Davisson and Germer,t Thomson,2 and' Rupp. 3 When Eq. (2'2) is applied to bodies of macroscopic dimen- sions, the wave lengths obtained are exceedingly small. In the case pf a golf.ball weighing 47 grams 'and trav'eling with a speed as low as a millimeter in 10 sec., the wave length is 1.4 X 10- 26 em.! This means t
at diffraction effects are hopelessly beyond the reach of experiment in ,the case of large-scale bodies. On the other hand, the computed wave "- length becomes appreciable if Eq. (2'2) is applied to atomic or molecular problems. An oxygen molecule, for example, with a speed corresponding to the mean thermal energy of 300 o K., has a wave length of approximately 1.5 X 10- 8 ern., while an electron with a "10-volt" kinetic ellergy has a wave length of 5.3 X 10- 8 em. As these dimensions are of the order of magnitude of atomic diameters and X-ray wave lengths, it is clear that diffraction effects must play a prominent part in atomic dynamics_
It is important to note that by introducing a vector 0' having the magnitude l/X and the direction of the wave normal, we can throw Eq. (2'2) into the vector form
p = hcr.
(4-4)
We shall call 0' the vector wave number. Its components O'x, cry, O'z denote the number of waves per centimeter crossed by lines parallel to the x, y, and z axes respectively_
6. THE SCHRODINGER WAVE EQUATION FOR A SINGLE PARTICLE' 5a. The Time-free Equation.- In view of the above results we may assume that a differential equation of the type of Eq. (3'2) is valid for matter waves, the phase velocity w being determined in accordance with Eqs_ (2'1) and (2'2). Then, E E W = A/I = P = V2jL[E _ V(x,y,z)] ' (5'1)
or
J"V2 + V v + V w = = - -0 J"V 2}J.v
(5 0 2)
(The relativistic formulation of the theory given in the next article yields a different expression for w.) Denoting the wave function for mhtter waves by '11, we may then write the wave equation in the form V' 2 'lt = 2J"[E - V (x,y -,
)l a 2 'lt . ( 5-3 ) E2 at 2 . 1 C. DAVISSON and L. H. GERMER, Phys. Rev. 30, 705 (1927); Proc. Nat. Acad. Sci. 14, 317 (1928). 2 G. P. THOMSON, Proc. Roy. Soc. Al17, 600 (1928); Al19, 651 (1928). 3 E. Rupp, Ann. d. Physik 86, 981 (1928).
SEC. 5]
THE SCHRODINGER WAVE EQUATION
IH
As in the optical case, the differential equation iR applicable to mono- chroma tic, or "mono-energetic' , wave functions only. This restriction means that all solutions of ]jjq. (5.3) which have physical significance are also to be solutions of the differential equation for a harmonic function of t:
1 a 2'1t --- 'It at 2 Combining Eqs. (5.3) and (5.4), we obtain
- 47r 2 p2 =
47r 2 E 2 h 2
(5.4)
87r 2 jJ- V 2 '1t + ¥ [E - V(x,y,z)]'lt = o.
(5.5)
This is the first form of Schrodinger's wave equation for a single particle. Since' the variable parameter E enters into the equation explicitly, Eq. (5.5) really includes a whole family of differential equations for each type of potential energy function V. We shall at times refer to (5.5) as the time-free wave equation to distinguish it from the equation (5.10) of Sec.' 5b. Equation (5.5) has the optical analogue
47r 2 47r 2 n 2 V' 2 <p = - A2 <P = -- Ao2 cp,
(5.6)
where n denotes the index of refraction and Ao is th
wave length in a vacuum. ,rhus one may say that the essential feature
f the Schrodinger equation/1:s that it makes the index of refraction for \:matter waves at each point of space proportional to the mom entum which the particle would have at that point, or to V2jJ-(E - V). In the case of a particle moving under the influence of the earth's gravitational field, for example, the index of refraction will increase as the point under consideration approaches the earth's surface. From the standpoint of the wave Inechanics the parabolic path of the orbit is to be attributed to the bending of the matter waves on account of the resulting inhomogeneity of space with respect to their propagation. It is precisely analogous to the bending of the sun's rays as they pass obliquely through the earth's inhomogeneous atmosphere. 6b. The Second (General) Schrodinger Equation.-Equation (5.5) is adequate as it stands for the investigation of the energy levels (fre- quency values) in a one-particle atomic problem, but in the case of the \vave packets of Sec. 4, and in many other cases, non-monochromatic wave functions must be used. Hence we need a more general differentia) equation which does not contain the parameter E or its equivalent v. Such a differential equation becomes a necessity when we have to do with problems in which the potential pnergy depends upon the time
16 INTRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I explicitly, or in which for any reason the energy of the system is not, conserved. A most important example is the perturbation of an atom by an external light wave which is the basis of the theory of dispersion. Here the assumption that we have to do with a single monochromatic w
ve function, or a fixed linear combination of such functions, breaks do\vn entirely. 'The obvious procedure for deriving a differential equation applicable to a wave packet, or to the sum of several monochromatic waves, is to eliminate E from Eq. (5'5) by differentiation. To do this we first write Eq. (5.5) in the form [ V 2 _ 871"2,u V J w = _ 871"2,uE 'lr h 2 h 2 ' Applying the operator [ \72 - 87/";:: V J to each side of the equation and reducing the right-hand member with the aid of Eq. (5'4), we obtain [ 2/_ 871"2,u V J 2 _ _1671"2,u2 a 2 ,¥ \7 h 2 '¥ - h 2 at 2 (5'7) This equation is linear and is valid for any monochromatic solution of Eq. (5.5), independent of the value of E. Hence it is satisfied by the sum of any finite or uniformly c011vergent infinite series/of monochromatic solutions of Eqs. (5'3i and (5'5). We may therefore assume that every physically admissible wave function is a solution of Eq. (5'7). The converse proposition is not plausible, however, since the above equation is of the fourth order and admits of solutions which are not linear com- binations of solutions of the family of equations (5'5). Moreover Eq. (5'7) is difficult to generalize for use in connection with noncon- servative systems. 1 Fortunately there is a simpler equation than (5'7) which is adequate for our needs. Let us assume that the values of the wave function '¥ are complex numbers, or pairs of real numbers representable as complex numbers. Let us further assume that every adlnissible wave function is a linear combination of monochromatic functions of the special type
27riEt -- '¥ = \f;(x,y,z)e h 05'8) where i = (-1)
2 and \f;(x,y,z) is in general complex. 2 This represents
1 Cj. E. SCHRODINGER, Ann. d. Physik (4) 81, 109 (1926). 2 If ..p is expressed in the form ..p = A(x,y,z)ei
(X,lI,z), where A and cp are real, we can resolve '¥ into real and imaginary parts by the fornnlla
[ 27r Et ] . . [ 27r Et ] '¥ = A. cos h - cp -
A. SIn h - cp ·
SEC. 5]
THE SCHRODINGER WAVE EQUATION
11
a standing wave system if -.j; is real and a progressive wave system if -.j; has an appropriate complex form. In the case of a monochromatic wave of the type of Eq. (5.8) the exponential time factor in Eq. (5'5) may be canceled out, yielding the equation
r 2 1/t +
[E - V(x,y,z)]1/t = 0.)
(5.9)
The factor -.j;(x,y,z) is sometimes called the amplitude or "space factor" of the wave function. When we have to do with monochromatic wave functions, a knowledge of -.j; is equivalent to a knowledge of the complete function '11, for the two functions have the same absolute value and satisfy the same differential equation. The determination of the -.j; functions for any problem involves the evaluation of the corresponding energies, so that the complete functions can be set up if desired. Hence we shall speak of -.j; as a "time-free monochromatic wave function," or if no ambiguity is involved, we shall apply to it the simpler term" wave function. " We can now eliminate E from Eq. (5.5) by means of the relation h a'¥ E'¥ = - 2n-i at '
thus obtaining the alternative wave equation
I V 2 \f1 - 81r 2 ,u V\fI + 41r,ui o\fl = oj h 2 h at .
(5'10)
This latter equation is much easier to handle than (5.7) as it is of the first order in t and the second order in the space coordinates. We shall call it Schrodinger's second equation for a single particle. Any linear combination of solutions of the family of Eqs. (5.5) having the form (5.8) is a solution of (5'10). Not all solutions of Eqs. (5'9) and (5,10) are of direct physical interest and one of the important problems to be dealt with in Chap. III is that of defining a suitable class of solutions useful for physical purposes which we shall describe as physically admissible. We assume provisionally that all such physically admissible solutions of Eq. (5 0 10) can be expressed as linear combinations of solutions of Eq. (5'5) which are of the form specified by (5'8).1 The justification of the restriction of Eq. (5 0 8) is that complex waves of the type which it defines are easy to handle mathematically and adequate to the needs of our problem. Our problem is, it will be remem-
1 Here an integral over a continuum of solutions and the limit of an infinite series of discrete .solutions are included within the scope of the phrase "linear conlbination of solutions."
18 INTRfJDUCTION TO DUALISTIC TIIEORY OF MA'l'TER [CHAP. I bered, nlerely that of fornlulating a mathematical deRcription of a type of waves which_ will describe the facts of classical mechanics in limitiJ?g cases where diffraction effects are negligible. There is nothing in the situation which requires '¥ to .be either a scalar or any particular kind of a vector. The wave function for sound waves (density or pressure) is \a scalar, whereas the waves of the Maxwell electromagnetic theory
consist of two three-dimensional real vectors G and X which may be united into a single three-dimensional complex vector. In neither case can one formulate a single second-order partial differential equation like (5.10) which summarizes the properties of either monochromatic or non-monochronlatic waves in a dispersive medium. Hence the complex waves here introduced are mathematically simpler than either sound or light waves in a dispersive medium. Of course our freedom to use these complex waves is dependent on the fact that while the intensity of the waves, measured by 1'lt1 2 , has direct physical meaning; (ef. p. 6), '¥ itself does not. It follows that the complex conjugate of any wave function 'It would serve equally well to describe the same physical situation. We shall indicate the conjugate of any complex' number by an asterisk. Thus
21/ i Et '¥* = t/;*(x,y,z)e h . Evidently '¥* satisfies the differential equation V 2 '1t* _ 87r 2 ,u V 'It * _ 47r,ui a'lt* = 0 h 2 h at .
(5.11)
(5.12)
I
Th8 choice of 'It rather than 'It* as the wave function is purely a matter of convention since the same physical results would be obtained by reversing the choice. 1 We shall make use of both functions in the development of the theory. As regards the properties of the wave equation (5'10), ,ve may observe here that it is formally similar to the equation for the diffusion of heat since it is of the first order in t. Owing to this circumstance the complete wave function in any particular case, if analytic in t, is determined by the special form of the wave equation and by the instan- taneous form of 'It at SOllIe initial instant, say t = 0. 2 On the other hand,
/_---1 The opposite convention has been adopted by many writers and was used by /Prof. E. L. Hill and the author in their articles "General Principles of Quantum Mechanics," Rev. Mod. Phys. 1, 157 (1929),2, 1 (1930). The present choice is sanc- tioned by convenience and more
eneral use. 2 To prove the above proposition we must show that if any two solutions of Eq. (5.10), say 'JI 1 and 'l1 2 , have identical values at t = 0, their difference is identically zero. Let 'It denote the difference function '1! 1 - 'l1 2 . Then 'l' is a solution of
SEC. 6] APPLICATION OF RESTRICTED RELATIVITY PRINCIPLE 19
the imaginary coefficient of a 'It / at in Eq. (S.IO) gives its solutions an undamped wave form quite different from the solutions of the differential equation for thermal diffusion.
*6. THE APPLICATION OF THE RESTRICTED RELATIVITY PRINCIPLE Historically the formulation of the Schrodinger wave equations was antedated by de Broglie's application of the restricted relativity principle to the problem of the correlation of waves and free particles. 1 His argument will be reviewed here, since it leads to a very different expres- sion for the phase velocity from that given in Eq. (S.2). Symmetry demands that a stationary particle be associated with a stationary rather than a progressive wave system. Hence we nlay postulate the form
'It = !(xo,Yo,zo)e-21rivot
(6.1)
for the wave function of a particle referred to a Rystem of coordinates Xo,Yo,Zo with respect to which it is at rest. To get the corresponding wave form for a free particle moving in the direction of the z axis with a speed v, de Broglie applies the Lorentz transformation
z - vt Zo =
1
t _ vz c 2
to -
. 2' V c 2
Xo,Yo = x,y,
1
, v 2 c 2
which yields
'It = ! ( x, Y, z - vt ) J v' ::;:/C2 (t-
) ] vI - V 2 /C 2 This expression may be made to describe an infinite plane "\vave system or a localized disturbance according to the hypothesis regarding the space factor !(xo,Yo,zo). In either case the frequency defined by the phase factor is
(6.2)
Po VI - V 2 /C 2
(5.12) which vanishes identically in x, y, z at t = o. Thus
( a'lt ) = 0 at t=o .
Differentiating both sides of (5.10) with respect to t, inverting the order of differentia- tion and setting t = 0, we see that (a 2 'lt / at 2 )o must also vanish. In the same .way we can prove that all the tiIne derivatives of 'It vanish at t = o. rrhus all terms in the expansion of 'It in po\vers of t yanish for all values of x, y, z and hence '¥ must be identically equal to zero if it is analytic in t (cf. Sec. 32b). 1 LQC. cit., footnote 1, p. 3.
20 IN,TRODUCTION TO DUALISTIC THEORY OF MATTER [CHAP. I
\ If the zero level of energy is fixed in accordance with the usual relativity . expreSSIon _
fJ-oC 2 Er = fJ-C 2 = , vI - V 2 /C 2 Er and v transform in the same way so that the fundamental relation (2.1) IS unaffected by the change of coordinate systems. This fact was the first great contribution of de Broglie. The wave length for a free particle required by Eq. (6.2) is
(6.3)
_ c 2 vi - V 2 /C 2 _ c 2 h _ h X - - - - - -, v Vo Er v fJ-V
in agreement with Eq.--{2.2). The phase velocity on the other hand is W = AV = (:V)(JL
2) =
2 . (6.4) / The discrepancy between this e
pression for the phase velocity and the value v/2 derived for a free particle from the nonrelativistic point of view [cf. Eq. (5.2)] suggests that absolute phase velocity is without physical significance. As a matter of fact there is no way in which it can be measured experimentally. If the particle is not free, but moves under the influence of a force field with potential energy V(x,y,z) we can still take into account the relativistic variation of mass with speed. The principle of least action in the form of Eq. (3.4) is still valid (cf. Appendix A), although the expres- sion for the momentum in terms of the potential and total energies has the more complicated form
1 p(Er,x,y,z) = cV(E r - V)2 - fJ-02C 4 .
(6.5)
A comparison of this principle with Fermat's principle shows that the mechanical orbits and the rays of the wave problem agree if C IX = p, where C is a constant as in
q. (3.5). As before we find that ap _ fJ- _ 1 aE r - P - V .
(6.6)
It follows from Eq. (4.1) that the group velocity of the waves is equal to the speed of the particle if we identify C with aEr/dV or h. The monochromatic wave equation corresponding to the above expression for p is
47r 2 47r 2 V 2 'lt = - h 2 p2i' = - h 2 c 2 [(E r - V)2 - fJ-02C 4 )i'. (6.7)
SEC. 7] WAVE EQUA_TION FOR SYSTEM OF MANY PARTICLES 21
Introducing the ordinary energy E = Er - MoC 2 we readily reduce (6'7) to the form 1
V' 2 'lt + 87r 2 j.to (E _ V)it = - 47r 2 (E - V)2it. (6'8) c 2 h 2 j.toC 2
1:'he right-hand member here appears a correction term to (5'5) which is usually small. We shall use this equation for the relativistic treatment of hydrogenic atoms in Chap. XIII. It is to be observed, however, that the assumption that the forces are derivable from a potential energy function is not valid in relativistic dynamics except in certain special cases and then only for a single frame of reference. 2 Hence a thorough- going relativistic wave equation cannot be formulated on the basis of the action function given by (6'5). Moreover, the fact that the energy E enters nonlinearly into (6'8) makes a difficulty which ultimately spoils the possibility of basing a satisfactory quantum mechanics on (6'8). For this reason we shall not take the space here to discuss the extension of the wave equation (6,7) to the non-monochromatie case.
7. THE WAVE EQUATION FOR A SYSTEM OF MANY PARTICLES 7a. Formulation of Equation.-Equations (5'5) and (5'10) can be generalized without difficulty to include the case of a system of n particles moving under the influence of conservative forces. The generalization, however, involves one important break with optical analogy. In optics a wave fUllction sp.read out in ordinary three-dimensional space can describ,e( the statistical behavior of any nun1ber of coexistent photons. The r{hotons apparently exert no influence upon each other so that the forIn of an interference pattern is independent of the intensity of the light used. vVe infer that the same wave function may be used to describe the behavior of one photon or ten thousand. In the case of Inatter corpuscles, however, ,ve must consider the II forces" which they exert on each other. III a .system of n particles the nlotion of each dapends on the coordinates of all,3 since the potential energy V is a function of the coordinates of all. Thus the wave function it which is to describe the behavior of a system of n particles must depend on the 3n coordinates of the system .. and the time. We n1ay say, if we like, that it is spread out over a 3n-dimensional 'l coordinate" or II configuration" space in which each point represents a possible configuration of the system as a whole. But the use of such geometrical language is not essential and means
1 Cf. O. KLEIN, Zeitsf. Physik 37,895 (1926); W. GORDON, Zeits.f. Physik 40,117 (1926). 2 Cf. W. PAULI, JR., "Relativitatstheorie," Encyklopaedie der M athematischen Wissenschaften, XIX, p. 678. 3 Classically, and hence also in the quantum mechanics.
22 INTRODf]CTIO}l TO DfJALISTIC TIIEORY OP MA TTF)R [CHAP. I
rnerely that,
ince '¥ depends on 3n independent variables, it could he laid out aR a "point" function only in a space having the corresponding nunlber of dimen..sions. In practice the wave function for any particular case is always derived and applied by purely analytical Inethods inde- pendent of the concept of configuration space. \The generalization of Eqs. (5.5) and (5.10) for a system of n particles \ can' be derived "\I\Tith the aid of the appropriate form of the principle of least action and a suitable extension of Fermat's principle to 3n-dimen- sional vvaves. 1 Inspection of the three-dimensional equations suffices, ho\vever, to suggest the correct generalization. Let Eq. (5.5) be written first in the f orIn
( 1 a2 1 a 2 1 a 2 ) 87r2
ax 2 +
ay2 +
az 2 'IF + Jt2.[E - V(x,y,z)]'lF = o. (7'1) It will be observed that each coordinate enters the equation through the potential energy function V and also through the corresponding 1 a 2 1 a 2 1 a 2 operat.or - ---;;, -- -;--, or -+ -, as the case may be. We assume that the }.t ax.. }.t ay2 J.l az 2 classical potential energy function for the problem in hand is known and use it also in the ,vave equation. The obvious procedure for generalizing Eq. (7.1) iR then to add a corresponding differential operator for each added coordinate. Let the ordinary three-dimensional Cartesian coordinates of the particles be labeled as a single series Xl, X2, · . . , X3n, and let the corresponding masses be }.Ll, }.L2, . . . , jJ,3n, where, of course, the three masses of anyone particle are the same. Then the expanded equations, analogous to (5.5) and (5.10), are
(
1 a 2 ) 871"2
- - iF -l- -[E - V(XI . . . X3n)]'lF - O.
}.tk aXk 2 h 2 , k=l
(7'2)
(
3n
2 _) 'TI _ 8 ".,.2TT (Xl . . . X ) 4 ".,.11 a ,T,.
II Y 3n '¥ +
---=- = 0 }.tk ax/.;2 h 2 h at . k=l
(7.3)
As noted ,above, these equations may be justified in the lilniting case where diffraction effects arc negligible by showing that they are in har- mony with the principle of least action. An alternative procedure is to use Eq. (7.3) to derive Halnilton's canonical classical equations of motion for sharply defined wave packets (ef. Chap. VIII, Sec. 39c). The final, general justification of Eqs. (7 '2) and (7 '3) must obviously come through the agreement between results derived from then1 and the facts of experiment in the domain of atomic and molecular physics.
1 Cf. E. C. KEMBLE, Rev. 111 od. F)hys. 1, 16() (1929).
SEC. 7]
WAVE EQUA TIOJ-l FOR SY8TE2v[ OF .LMANY PAlt-TlCLE 1 S
23
Implicit in the above exteuRiou of the wave pquationH iH the assump- tion that the total energy of the Systpl11 of particles and the frequency of the associated wave system are related by the rule (2'1), E = hv. By means of this rule we can pass frOIn Eq. (7 '2) to Eq. (7 '3) or from Eq. (7'3) to Eq. (7,2). ThuR, starting from Eq. (7'3), let us seek a monochroI11atic solution of the form
\]f = f(Xl, X2, . . . X3n)e- 2 n-ipt. Substitution of this expreRsion yields (
1 iJ2 ) 87r2
Mk aXk 2 '¥ + h 2 (hv - V)\]f - o. k=l With the aid of (2'1) this reduces to Eq. (7,2). 7b. Relation of Schrodinger Equation to the Classical Hamiltonian Function.-Equation (7 '2) is formally related to the claRsical I-Iamilton- ian function for the systenl under consideration and can be dedueed frOln it by the following rule of thulnb. First set up the energy equation using the classical Han1iltonian function in Cartesian coordinates: 1 3n
H(p,q) .
_ 2 Pk2 + V(X1 . . . X3n) = E.
Mk k=l 1 The condition for the existence of a classical Hanliltonian function is that the / equations oj 'motion of the systenl under consideration are reducible to the Lagrangian form /
(7'4)
aL d ( dL ) _ 0 k = 1 2 . .. f aqk - dt dqk - , '" ref. Eqs. (A9), Appendix A] where L is a suitably defined function of the generalized coordinates qI, q2, . . . , qj, their velocities, and the tirHe. If the kinetie energy '1' is a homogeneous quadratic function of the velocities as in i he Newtonian nonrela- tivistic dynalnics, and if the forces acting on the' systern arc derivable frolll a potential function V(ql, . . . , qj, t) \vhich does not depend on the vplocities, such a reduction of the equations of lnotion is possible ,vi th L = T - TT. In nlore general cases, e.g., whc'n the relativistic variation of rnass is taken into account, or ,,'"hen the particles are acted on by forces of rnag;netic origin dppending directly on the velocities, it is necesKary to invent an appropriate Lagrangian funetion L. 'Vhen the equations have been thro\vn into Lagrangian fonn, the rnornenta conju- gate to the coordinates ql, . . . ,qj are defined hy the equations aL P k = ----;--, k = 1 2 . .. f aq k '" and the IIamiltonia.ll function H(p,q,t) is derived fro'Dl the function "2:kPkQk - L(q,q,t) by elinlinating; thE' velocities \vith the' aid of the eorrE'sponding monlenta. In the Ne\vtonian theory of a conservative systerll \vhpre L = T - V, the lnornen""
24 INTRODUCTiON TO DliALISTIC THEORY OF MATTER rCHAP. J
Second conv
t each side of Eq. (7.4) into an operator l by the substitution h a P k
-. -, k = 1 2 . .. 3n - 27rt ax k '" v
operation of multiplying by V, E
operation of multiplyjng by E. Fi
ally, let each member of the operator equation act on 'It(Xl, . . .', X3n, t). The resulting differential equation is equivalent to (7.2) and is frequently expressed in the operator form
H( a:k ' Xk)W = Ew, (7.5) where H(a/aXk, Xk), or H(a7ax, x), stands for the operator obtained from the classical Hamiltonian by the above substitution. In case no ambi- guity is involved, this operator, or any modification of it which plays the same role in the the
ry, is designated by the simple symbol H. To set up Eq. (7.3) one proceeds as above, except that one replaces E h a by the operator - 27ri at . *7c. The Schrodinger Equation and the Hamilton-Jacobi Equation.- These substitutions form an obvious parallel to those which are made in setting up the Hamilton-Jacobi partial differential equation of classical dynamics. 2 The latter equation has two forms similar to (7.2) and (7.3) respectively. In case the Hamiltonian function H does not depend explicitly on the time, it remains constant during any natural motion of the system and is identified with the energy of the system. The first form of the Hamilton-Jacobi equation is obtained frQm the energy equation H(p,q) = E by means of the substitution
as Pk
-, aqk
k=12 ... f " ,
tum Pk reduces to aT / aqk, and the Hamiltonian function is equal to the total energy E. In the case of a system of n particles with Cartesian coordinates, Pk becomes P,JcXk and the Hamiltonian function reduces to the simple form (7.4). Whatever the special forms of Land H may be, the f second-order Lagrangian equations given above are equivalent to the 2f first-order canonical equations of Hamilton
dqk _ aH. dPk _ aH (jj - apk ' dt -- - aqk . 1 An operator is a rule for the transformation of one function into another. The transformed function, or "transforn)," can have the same arguments as the origina. function, or a different set. 2 Cf. any standard text in advanced analytical dynami:::
SEC. 7] WAVE EQUATION FOR SYSTEM OF MANY PARTICLES 25
Any system of generalized coordinates ql, q2, . . . , qr is permissible. A solution of the resulting inhomogeneous first-order partial differential equation
H ( as , . .. as , q 1 . .. qr) = E aq 1 ' aqr' , is sometimes called an " action function," although it is not to be identified with the action integral of the principle of least action (cf. footnote 1, p. 44). The second form of the Hamilton-Jacobi equation is applicable even when H involves t explicitly and the energy is not conserved. It is obtained from H(p,q,t) by replacing Pk by aAjaqk for all values of k and equating the resulting expressio
to -aAjat. ThuH
(7 . 6a )
H (:
' . · . ,
, ql, · · · , qIJ t) = - aa
· The parallelism between this classical equation and the wave equation
(7 '6b)
H ( 2 h . a a , . . . , 2 h -. a a , Xl, . · . , X r ) '¥ = - 2 h . a a '¥ t 7r
Xl 7r
xr 7r
(7 '3a)
is striking. We have not followed Schrodinger 1 in deriving the general wave equation (7 '3a) directly fronl (7 '6b) but the reader will find a discussion of the intimate relation between these equations in Chap. II, /' Sees. 11, 1
:2 The apparently formal parallelisnl between the general / wave equation and the second form of the Hamilton-Jacobi equation actually gives a basis for proving the asymptotic agreement between the classical mechanics and quantunl mechanics in cases which lie outside the range of the principle of least action, e.g., in the ease of nonconserva- tive systems where the forces depend on the velocities or are explicit functions of the time. Furthermore it resolves the apparent ambiguity in the sign of the pxponent in Eq. (5'8). If this sign is negative, we must 1 b . . E y h a . . (7 3) (7 3 ) h UMe t 1e su stItutIon -7 - 27ri at In settIng up . or . a , w ereas a positive sign would correspond to the substitution E -7 2 h . i. If the 7r
at Hamiltonian operator H(
z :i ' q) is real, the two choices are equivalent as indicated by Eq. (5'12). However, in the more general case of a complex Halniltonian operator, such as the one to be introduced in Eq. (7'S), they are not equivalent. In order to maintain the parallelism between (7 '3a) and (7 '6b) in such cases it is necessary to use the sub-
1 Cf. footnote 3, p. 3. 2 Cf. alBo G. D. BIRKHOFF, Proc. ..Vat. Acrtd Sri. 19, 339 (1933).
26 INTRODUCTION TO DUALI81 1 IC TIIEORY OF MATTER [CHAP. I
\ t a t t ' E h a I ' 1 h a I ' s 1 u Ion
- 27ri at a ong WIt 1 Pk
27ri aqk or e se reverse sIgns throughout, 1 FroIn one point of view the sign of the operator substituted for E is
omewhat puzzling on first examination, It is well known that the fun
tion S of Eq. (7'6a) is the generating function of a contact trans- for
tion which replaces the variable
ql, . . . , qj by a new set one of which is the tiTne, or djffers from it by a constant. The momentum conjugate to t in this llPW set of coordinates is E. Since we Rubstitute + 2 h . a a for the nlOJnentunl conjugate to Xk, one might argue that to be 7r
x k '" consistent we ought to substitute +2
2 :t for E. The argument is fallacious, however, for the-classical Hamiltonian from which vve form the Hamiltonian operator in F
q. (7 '3a) has not been subjected to the above-Inentioned transformation. E is not conjugate to t when the independent variables are! ql, ' . , ,qf and we cannot make substitutions corresponding\ to tvvo different sets of independent variables in the same equation, It is possible,. however, to get a Ratisfactory classical analogue for the substitutions u
ed in setting up Eq. (7'3a) without .using the Hamil- ton-Jacobi equation, 1"0 do so one nlakes use of a classical scheme for treating the time on the saIne fornlal basis as the spatial coordinates, 2 In this schenIe a paralneter T is introduced as the independent variable and the nUlnber of independent coordinates for an f-dinlensional system is stepped up to f + 1 by adding the time to the ordinary coordinates ql, . . . , qf' 2f + 2 Hamiltonian equations arc then set up ,vith - E playing the role of n10In(
ntunl conj ugate to the coordinate t. *7d. The Wave Equation for a System of Charged Particles in a Clas- sical External Electromagnetic Field.---For tIRe in connection ,vith the study of the Zeernall and Stark effects and for a discussion of the absorp-
1 It is much eftsier to t ransfonn the IIarniltonian operator and the first-order IIamilton-.Jaeobi equation fnn}) onp ('oordinnte ::,ystenl to another than to make the corresponding transfornlation of the
('cond-order ""ave equation (7.3a). Hence there is an obviou8 ternptation to 8E't up thE' wave equation in a generalized coordinate system p,q by transforrning; the ela8si
al Harniltonian and subsequently rnaking the substitution
h a 'r. J! --* -- -. I' , 2 .,' 7r
d q k
h a E -
-- -. 27ri at
This is not permissible in genpral, ho\vever, and it is necessary to set up the wave equation in Cartesian eoordinates first and apply the transformation afterward, or to make use of some ITlethod definitely proved to he equivalent (ef. VII, Sec. 35b).
See the article on IIanlilton-Jacobi theory by L. Nordheirn and E. Fues, in t}eiger and Scheel's Handbnch der Physik Band V, Rap. 3, Ziff. 4, Berlin, 1927.
SEC. 7] WAVE EQUA.TION FOR SYSTEM OF MANY [)ARTICLES 27
tion of radiation we shall need a wave equation applicable to a f3ystem of charged particles in a classical electromagnetie field. Although the derivation of this equation is son1e\vhat technical for an introductory chapter, we insert it here to avoid repetition in l
ter chapters. As we have introduced the subject of wave 111ec!tanics by a considera- tion of the corpuscular theory of light, i.e., with a preliminary study of the quantum properties of the electrolnagnetic field, the reader vvill be inclined to raise his eyebrows at the attenlpt to cornbine a quantun1 theory of the atom \vith a clas
ical picture of an interacting electro- magnetic field. Our excu
e for the eonstruction of snch a hybrid theory lies partly in the extreme diffieulty of forrnulating a
atisfactory thorough- going quantum theory of the interaction bet-vvcen matter and radiation and partly in the observation that in t,llC lin1iting case of very long ,vave lengths-static or quasi-static fields-the corpuscular properties of the electromagnetic field recede into the background while the classical properties dominate. Hence 'V8 can reasonably hope that such a classical treatment of the field will be in asyn1ptotic agreement \vith cxperilnellt as the wave lengths under consideration beeome very large. As a Inatter of fact the absorption formulas "rhieh the theory yields have proved satisfactory over a very wide range of the Rpectruln. Following the method sketchpd above, ,ve shall begin by constructing the appropriate classical Halniltonian function in Cartcsian coordinates, converting it/into an operator aR before and using the operator to forn1 an equatiqn' of the forIll (7 '3a). If the external fi{\ld varies \vith the tirne, the Hamiltonian will involve t explicitly and ,ve ha\Te to do ,vith a ca
e in which the energy is not conserved. Consider a systen1 of n charged particles D10ving in an externaJ classical electromagnetic field ,vith the scalar potential <p(x, y, z, t) and
the vector potential a(x, y, Z, t). I.Jet CIx(f), CIy(i), a/i) denote the cOIl1ponents of the vector potential at the point :tj, Yh Zj, ,vh('re the jth particle is located. I..Ict J.lJ. and Cj denote respectively the nlaHS llnd the algebraic value of the charge of that particle. The classical Hamiltonian function then takes the form 1
n H(p,x,t) =
(2
) [ (Px(j) -
('tx(})Y + (P/ iJ -
j ('ty<i)Y j=l
n + (pz(j) - :' ('tz(})Y J + V (Xl, . . . ,z,.) +
Pj<p(j). (7 '7) j=l Here V denotes as usual the internal potential energy of the systeIll -vvhich, 1 Cf. VAN VLECK, The Theory of E'lectric and 4
lagnetic SHs('eptibilities, Chap. I, p. 7, | 2020-02-29 10:49:36 | {"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": 0.8617064356803894, "perplexity": 808.5900274106995}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875148850.96/warc/CC-MAIN-20200229083813-20200229113813-00420.warc.gz"} |
https://hongtaoh.com/en/2020/05/25/remote-origin-already-exists-error-shooting/ | # How to Solve the Problem of "Remote Origin Already Exists"
Hongtao Hao / 2020-05-25
Here is my situation: I want to use another GitHub account. I need to transfer a repo from my old account to the new one. Maybe there is an easy way to transfer, but it is beyong my ability now (I guess it’s complicated so I didn’t even bother searching for an answer).
So I git clone the repo from my old account and tried to push it to my new account. Then, a error occured:
fatal: remote origin already exists.
There are two ways to solve this problem:
1. Change the URL
Here is where I found the answer. Many thanks to kahowell .
Simply use:
git remote set-url origin https://github.com/new.url.here
However, if you just want to remove the remote (and add a new one later), you can use the following line of code to remove remote:
git remote remove origin
1. Creating a new folder
Just create a new folder, and then drag all the files from the old folder into the new folder. The original git remote will be automatically erased.
Done. | 2021-01-17 05:17:10 | {"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": 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": 0.2059078961610794, "perplexity": 1641.062030372193}, "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-2021-04/segments/1610703509973.34/warc/CC-MAIN-20210117051021-20210117081021-00096.warc.gz"} |
https://puzzling.stackexchange.com/questions/109591/champion-with-a-smirk | # Champion, with a smirk
The answer to this puzzle is a 16-letter sentence. Good luck!
Text version of the letter grids:
ERETILUI UEUSTIRE BIRACACL
ITURERRQ RNQTRANI RLERECBI
SQETQAIP QSIEUSEQ ECRLAECL
ALTULUEI ARIRTSIA ABLIBCIB
NEIQEATL UTQNEANU RBEKKERC
CAQLALRC SSIRNSUT NKTARUIT
NEUTIUQA QNTKPLRI IRCNSKCL
ARURATLI ASULQMAU ICAKRCAE
RHRPEDPR YOLPEHNH AIMRIRTP
SEPSDSHH LNGGHOLY EORIONGR
REPRSDDS XGLHLNOX GTENMEOI
PEHERHSH EYXWXAGP TNMENOTE
FIATKTOA UHNLGLWW MORIMOEN
PSREPSHD WGXYOYGO OEGRGMNG
RDRSREHR EOHWLOLX IMNTOTRM
DEDSHSRE UNGEWNOI UTRITNTU
• Do you happen to have a text version of the letter grids? – Deusovi Apr 20 at 13:48
• No, I don't. But I can fix that. Will not be online for a while though. So will have to do it later then. – Prim3numbah Apr 20 at 13:56
• I can't for the life of me figure out a way to find a starting position. There's one game especially suited for this kind of grid and those correct squares, but in that game, the previous position cannot be uniquely determined from the current position on the grid, so I guess I'll just have to wait and see if someone can figure this out. – Bass Apr 23 at 19:27
• @Bass Your wish is my command ;-) In all honesty, your remark here about a 'game' suddenly helped me make sense of the 'odd ones out' I had previously found (and got totally stuck thinking about), so thank you for providing the spark! – Stiv Apr 23 at 21:10
The first thing to do is work out what shape each diagram represents. Across each row, these are in turn:
EQUILATERAL TRIANGLE, UNIT SQUARE, BLACK CIRCLE,
RED SPHERE, YELLOW HEXAGON, GREEN TETROMINO.
Next:
Highlight the cells in each grid which contain letters that do not appear in the name of the corresponding shape:
What do we now have? The highlighted cells on these 8x8 grids of squares:
Are the squares on a chessboard that represent the furthest a chess piece can (theoretically) move from its starting position! We can see this more clearly by highlighting the start positions for the pieces in question (thereby fulfilling the enigmatic Step 1 of the puzzle instructions):
Now, we have identified 16 letters of interest - the same number as the number of "+/-" commands in Step 2 of the puzzle instructions. What we must do next, therefore, is:
Augment the letters in our 16 yellow squares by the number of alphabet positions indicated in the command list, like so:
U-18, T-18, Q-12, C+0, A+10, R-5, D-3, R+2, S-14, R+1, E+4, H+10, R-9, E+18, U-12, U-7
...which then spells out the message we seek (punctuation added)...
CHECKMATE, SIR - I WIN! - a remark fit for a gloating 'champion, with a smirk', as per the title!
• Well done! I was thinking of adding a suitable hint, guess I don't need to now :) – Prim3numbah Apr 24 at 5:56 | 2021-07-27 09:01:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5188747048377991, "perplexity": 2896.4009162635834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153223.30/warc/CC-MAIN-20210727072531-20210727102531-00355.warc.gz"} |
https://docs.readthedocs.io/en/stable/config-file/v2.html | # Configuration File V2¶
Read the Docs supports configuring your documentation builds with a YAML file. The configuration file must be in the root directory of your project and be named .readthedocs.yaml.
All options are applied to the version containing this file. Below is an example YAML file which shows the most common configuration options:
# .readthedocs.yaml
# Read the Docs configuration file
# Required
version: 2
# Set the version of Python and other tools you might need
build:
os: ubuntu-20.04
tools:
python: "3.9"
# You can also specify other tool versions:
# nodejs: "16"
# rust: "1.55"
# golang: "1.17"
# Build documentation in the docs/ directory with Sphinx
sphinx:
configuration: docs/conf.py
# formats:
# - pdf
# Optionally declare the Python requirements required to build your docs
python:
install:
- requirements: docs/requirements.txt
## Supported settings¶
Read the Docs validates every configuration file. Any configuration option that isn’t supported will make the build fail. This is to avoid typos and provide feedback on invalid configurations.
Warning
When using a v2 configuration file, the local settings from the web interface are ignored.
### version¶
Required
true
Example:
version: 2
Warning
If you don’t provide the version, v1 will be used.
### formats¶
Formats of the documentation to be built.
Type
list
Options
htmlzip, pdf, epub
Default
[]
Example:
version: 2
# Default
formats: []
version: 2
# Build PDF & ePub
formats:
- epub
- pdf
Note
You can use the all keyword to indicate all formats.
version: 2
# Build all formats
formats: all
Warning
pdf, epub, and htmlzip output is not supported when using MkDocs.
### python¶
Configuration of the Python environment to be used.
version: 2
python:
install:
- requirements: docs/requirements.txt
- method: pip
path: .
extra_requirements:
- docs
- method: setuptools
path: another/package
system_packages: true
#### python.version¶
Warning
This option is now deprecated and replaced by . See for the description of this option.
#### python.install¶
List of installation methods of packages and requirements. You can have several of the following methods.
Type
list
Default
[]
##### Requirements file¶
Install packages from a requirements file.
The path to the requirements file, relative to the root of the project.
Key
requirements
Type
path
Required
true
Example:
version: 2
python:
version: "3.7"
install:
- requirements: docs/requirements.txt
- requirements: requirements.txt
Warning
If you are using a environment to manage the build, this setting will not have any effect. Instead add the extra requirements to the environment file of Conda.
##### Packages¶
Install the project using python setup.py install or pip install.
The path to the package, relative to the root of the project.
Key
path
Type
path
Required
true
The installation method.
Key
method
Options
pip, setuptools
Default
pip
Extra requirements section to install in addition to the package dependencies.
Warning
You need to install your project with pip to use extra_requirements.
Key
extra_requirements
Type
list
Default
[]
Example:
version: 2
python:
version: "3.7"
install:
- method: pip
path: .
extra_requirements:
- docs
- method: setuptools
path: package
With the previous settings, Read the Docs will execute the next commands:
span.prompt1:before {
content: "\$ ";
}
pip install .[docs]
python package/setup.py install
#### python.system_packages¶
Type
bool
Default
false
Warning
If you are using a environment to manage the build, this setting will not have any effect, since the virtual environment creation is managed by Conda.
### conda¶
Configuration for Conda support.
version: 2
conda:
environment: environment.yml
#### conda.environment¶
The path to the Conda environment file, relative to the root of the project.
Type
path
Required
true
### build (beta specification)¶
Warning
This functionality is in beta. If you find any inconsistencies or have feedback, please open an issue.
Configuration for the documentation build process. This allows you to specify the base Read the Docs image used to build the documentation, and control the versions of several tools: Python, Node.js, Rust, and Go.
version: 2
build:
os: ubuntu-20.04
tools:
python: "3.9"
nodejs: "16"
rust: "1.55"
golang: "1.17"
#### build.os¶
The Docker image used for building the docs. Image names refer to the operating system Read the Docs uses to build them.
Note
Arbitrary Docker images are not supported.
Type
string
Options
ubuntu-20.04
Required
true
#### build.tools¶
Version specifiers for each tool. It must contain at least one tool.
Type
dict
Options
python, nodejs, rust, golang
Required
true
#### build.tools.python¶
Python version to use. You can use several interpretes and versions, from CPython, PyPy, Miniconda, and Mamba.
Note
If you use Miniconda3 or Mambaforge, you can select the Python version using the environment.yml file. See our Conda Support guide for more information.
Type
string
Options
• 2.7
• 3 (last stable CPython version)
• 3.6
• 3.7
• 3.8
• 3.9
• 3.10
• pypy3.7
• miniconda3-4.7
• mambaforge-4.10
#### build.tools.nodejs¶
Node.js version to use.
Type
string
Options
14, 16
#### build.tools.rust¶
Rust version to use.
Type
string
Options
1.55
#### build.tools.golang¶
Go version to use.
Type
string
Options
1.17
#### build.apt_packages¶
List of APT packages to install. Our build servers run Ubuntu 18.04, with the default set of package repositories installed. We don’t currently support PPA’s or other custom repositories.
Type
list
Default
[]
version: 2
build:
apt_packages:
- libclang
- cmake
Note
When possible avoid installing Python packages using apt (python3-numpy for example), .
### sphinx¶
Configuration for Sphinx documentation (this is the default documentation type).
version: 2
sphinx:
builder: html
configuration: conf.py
fail_on_warning: true
#### sphinx.builder¶
The builder type for the Sphinx documentation.
Type
string
Options
html, dirhtml, singlehtml
Default
html
Note
The htmldir builder option was renamed to dirhtml to use the same name as sphinx. Configurations using the old name will continue working.
#### sphinx.configuration¶
The path to the conf.py file, relative to the root of the project.
Type
path
Default
null
If the value is null, Read the Docs will try to find a conf.py file in your project.
#### sphinx.fail_on_warning¶
Turn warnings into errors ( and options). This means the build fails if there is a warning and exits with exit status 1.
Type
bool
Default
false
### mkdocs¶
Configuration for Mkdocs documentation.
version: 2
mkdocs:
configuration: mkdocs.yml
fail_on_warning: false
#### mkdocs.configuration¶
The path to the mkdocs.yml file, relative to the root of the project.
Type
path
Default
null
If the value is null, Read the Docs will try to find a mkdocs.yml file in your project.
#### mkdocs.fail_on_warning¶
Turn warnings into errors. This means that the build stops at the first warning and exits with exit status 1.
Type
bool
Default
false
### submodules¶
VCS submodules configuration.
Note
Only Git is supported at the moment.
Warning
You can’t use include and exclude settings for submodules at the same time.
version: 2
submodules:
include:
- one
- two
recursive: true
#### submodules.include¶
List of submodules to be included.
Type
list
Default
[]
Note
You can use the all keyword to include all submodules.
version: 2
submodules:
include: all
#### submodules.exclude¶
List of submodules to be excluded.
Type
list
Default
[]
Note
You can use the all keyword to exclude all submodules. This is the same as include: [].
version: 2
submodules:
exclude: all
#### submodules.recursive¶
Do a recursive clone of the submodules.
Type
bool
Default
false
Note
This is ignored if there aren’t submodules to clone.
## Schema¶
You can see the complete schema here.
## Legacy build specification¶
The legacy build specification used a different set of Docker images, and only allowed you to specify the Python version. It remains supported for backwards compatibility reasons. Check out the above for an alternative method that is more flexible.
version: 2
build:
image: latest
apt_packages:
- libclang
- cmake
python:
version: "3.7"
The legacy build specification also supports the apt_packages key described above.
Warning
When using the new speficiation, the build.os and python.version options cannot be used. Doing so will error the build.
### build (legacy)¶
#### build.image (legacy)¶
The Docker image used for building the docs.
Type
string
Options
stable, latest
Default
latest
Each image support different Python versions and has different packages installed, as defined here:
• stable: 2, 2.7, 3, 3.5, 3.6, 3.7, pypy3.5
• latest: 2, 2.7, 3, 3.5, 3.6, 3.7, 3.8, pypy3.5
#### python.version (legacy)¶
The Python version (this depends on ).
Type
string
Default
3
Note
Make sure to use quotes (") to make it a string. We previously supported using numbers here, but that approach is deprecated.
Warning
If you are using a environment to manage the build, this setting will not have any effect, as the Python version is managed by Conda.
## Migrating from v1¶
### Changes¶
• The version setting is required. See .
• The default value of the setting has changed to [] and it doesn’t include the values from the web interface.
• The top setting requirements_file was moved to python.install and we don’t try to find a requirements file if the option isn’t present. See .
• The setting conda.file was renamed to conda.environment. See .
• The build.image setting has been replaced by build.os. See . Alternatively, you can use the legacy build.image that now has only two options: latest (default) and stable.
• The settings python.setup_py_install and python.pip_install were replaced by python.install. And now it accepts a path to the package. See .
• The setting python.use_system_site_packages was renamed to python.system_packages. See .
• The build will fail if there are invalid keys (strict mode).
Warning
Some values from the web interface are no longer respected, please see if you have settings there.
## Migrating from the web interface¶
This should be pretty straightforward, just go to the Admin > Advanced settings, and find their respective setting in .
Not all settings in the web interface are per version, but are per project. These settings aren’t supported via the configuration file.
• Name
• Repository URL
• Repository type
• Language
• Programming language
• Project homepage
• Tags
• Single version
• Default branch
• Default version
• Show versions warning
• Privacy level
• Analytics code | 2021-10-25 08:10:20 | {"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": 0.24012133479118347, "perplexity": 12254.926004613}, "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/1634323587655.10/warc/CC-MAIN-20211025061300-20211025091300-00521.warc.gz"} |
https://keet.wordpress.com/page/2/ | # Gastrophysics and follies
Yes, turns out there is a science of eating, which is called gastrophysics, and a popular science introduction to the emerging field was published in an accessible book this year by Charles Spence (Professor [!] Charles Spence, as the front cover says), called, unsurprisingly, Gastrophysics—the new science of eating. The ‘follies’ I added to the blog post title refers to the non-science parts of the book, which is a polite term that makes it a nice alliteration in the pronunciation of the post’s title. The first part of this post is about the interesting content of the book; the second part about certain downsides.
The good and interesting chapters
Given that some people don’t even believe there’s a science to food (there is, a lot!), it is perhaps even a step beyond to contemplate there can be such thing as a science for the act of eating and drinking itself. Turns out—quite convincingly in the first couple of chapters of the book—that there’s more to eating than meets the eye. Or taste bud. Or touch. Or nose. Or ear. Yes, the ear is involved too: e.g., there’s a crispy or crunchy sound when eating, say, crisps or corn flakes, and it is perceived as an indicator of the freshness of the crisps/cornflakes. When it doesn’t crunch as well, the ratings are lower, for there’s the impression of staleness or limpness to it. The nose plays two parts: smelling the aroma before eating (olfactory) and when swallowing as volatile compounds are released in your throat that reach your nose from the back when breathing out (i.e., retronasal).
The first five chapters of the books are the best, covering taste, smell, sight, sound, and touch. They present easily readable interesting information that is based on published scientific experiments. Like that drinking with a straw ruins the smell-component of the liquid (and so does drinking from a bottle) cf drinking from a glass that sets the aromas free to combine the smell with the taste for a better overall evaluation of the drink. Or take the odd (?) thing that frozen strawberry dessert tastes sweeter from a white bowl than a black one, as is eating it from a round plate cf. from an angular plate. Turns out there’s some neuroscience to shapes (and labels) that may explain the latter. If you think touch and cutlery don’t matter: it’s been investigated, and it does. The heavy cutlery makes the food taste better. It’s surface matters, too. The mouth feel isn’t the same when eating with a plain spoon vs. from a spoon that was first dipped in lemon juice and then in sugar or ground coffee (let it dry first).
There is indeed, as the intro says, some fun fact on each of these pages. It is easy to see that these insights also can be interesting to play with for one’s dinner as well as being useful to the food industry, and to food science, be it to figure out the chemistry behind it or how to change the product, the production process, or even just the packaging. Some companies did so already. Like when you open a bag of (relatively cheap-ish) ground coffee: the smell is great, but that’s only because some extra aroma was added in the sealed air when it was packaged. Re-open the container (assuming you’ve transferred it into one), and the same coffee smell does not greet you anymore. The beat of the background music apparently also affects the speed of masticating. Of course, the basics of this sort of stuff were already known decades ago. For instance, the smell of fresh bread in the supermarket is most likely aroma in the airco, not the actual baking all the time when the shop is open (shown to increase buying bread, if not more), and the beat of the music in the supermarket affects your walking speed.
On those downsides of the book
After these chapters, it gradually goes downhill with the book’s contents (not necessarily the topics). There are still a few interesting science-y things to be learned from the research into airline food. For instance, that the overall ‘experience’ is different because of lower humidity (among other things) so your nose dries out and thus detects less aroma. They throw more sauce and more aromatic components into the food being served up in the air. However, the rest descends into a bunch of anecdotes and blabla about fancy restaurants, with the sources not being solid scientific outlets anymore, but mostly shoddy newspaper articles. Yes, I’m one of those who checks the footnotes (annoyingly as endnotes, but one can’t blame the author for that sort of publisher’s mistake). Worse, it gives the impression of being research-based, because it was so in the preceding chapters. Don’t be fooled by the notes in, especially, chapters 9-12. To give an example, there’s a cool-sounding section on “do robot cooks make good chefs?” in the ‘digital dining’ chapter. One expects an answer; but no, forget that. There’s some hyperbole with the author’s unfounded opinion and, to top it off, a derogatory remark about his wife probably getting excited about a 50K GBP kitchen gadget. Another example out of very many of this type: some opinion by some journalist who ate some day, in casu at über-fancy way-too-expensive-for-the-general-reader Pairet’s Ultraviolet (note 25 on p207). Daily Telegraph, New York Times, Independent, BBC, Condiment junkie, Daily Mail Online, more Daily Mail, BBC, FT Weekend Magazine, Wired, Newsweek etc. etc. Come on! Seriously?! It is supposed to be a popsci book, so then please don’t waste my time with useless anecdotes and gut-feeling opinions without (easily digestible) scientific explanations. Or they should have split the book in two: I) popsci and II) skippable waffle that any science editor ought not to have permitted to pass the popsci book writing and publication process. Professor Spence is encouraged to reflect a little on having gone down on a slippery slope a bit too much.
In closing
Although I couldn’t bear to finish reading the ‘experiential meal’ chapter, I did read the rest, and the final chapter. As any good meal that has to have a good start and finish, the final chapter is fine, including the closing [almost] with the Italian Futurists of the 1930s (or: weird dishes aren’t so novel after all). As to the suggestions for creating your own futurist dinner party, I can’t withhold here the final part of the list:
In conclusion: the book is worth reading, especially the first part. Cooking up a few experiments of my own sounds like a nice pastime.
Conjuring up or enhancing a new subdiscipline, say, gastromatics, computational gastronomy, or digital gastronomy could be fun. The first term is a bit too close to gastromatic (the first search hits are about personnel management software in catering), though, and the second one has been appropriated by the data mining and Big Data crowd already. Digital gastronomy has been coined as well and seems more inclusive on the technology side than the other two. If it all sounds far-fetched, here’s a small sampling: there are already computer cooking contests (at the case-based reasoning conferences) for coming up with the best recipe given certain constraints, a computational analysis of culinary evolution, data mining in food science and food pairing in Arab cuisine, robot cocktail makers are for sale (e.g., makr shakr and barbotics) and there’s also been research on robot baristas (e.g., the FusionBot and lots more), and more, much more, results over at least the past 10 years.
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# The isiZulu spellchecker seems to contribute to ‘intellectualisation’ of isiZulu
Perhaps putting ‘intellectualisation’ in sneer quotes isn’t nice, but I still find it an odd term to refer to a process of (in short, from [1]) coming up with new vocabulary for scientific speech, expression, objective thinking, and logical judgments in a natural language. In the country I grew up, terms in our language were, and still are, invented more because of a push against cultural imperialism and for home language promotion rather than some explicit process to intellectualise the language in the sense of “let’s invent some terms because we need to talk about science in our own language” or “the language needs to grow up” sort of discourses. For instance, having introduced the beautiful word geheugensanering (NL) that captures the concept of ‘garbage collection’ (in computing) way better than the English joke-term for it, elektronische Datenverarbeitung (DE) for ‘ICT’, técnicas de barrido (ES) for ‘sweep line’ algorithms, and mot-dièse (FR) for [twitter] ‘hashtag’, to name but a few inventions.
Be that as it may, here in South Africa, it goes under the banner of intellectualisation, with particular reference to the indigenous languages [2]; e.g., having introduced umakhalekhukhwini ‘cell/mobile phone’ (decomposed: ‘the thing that rings in your pocket’) and ukudlulisa ikheli for ‘pass by reference’ in programming (longer list of isiZulu-English computing and ICT terms), which is occurring for multiple subject domains [3]. Now I ended up as co-author of a paper that has ‘intellectualisation’ in its title [4]: Evaluation of the effects of a spellchecker on the intellectualization of isiZulu that appeared just this week in the Alternation journal.
The main general question we sought to answer was whether human language technologies, and in particular the isiZulu spellchecker launched last year, contribute to the language’s intellectualisation. More specifically, we aimed to answer the following three questions:
1. Is the spellchecker meeting end-user needs and expectations?
2. Is the spellchecker enabling the intellectualisation of the language?
3. Is the lexicon growing upon using the spellchecker?
The answers in a nutshell are: 1) yes, the spellchecker does meet end-user needs and expectations (but there are suggestions further improving its functionality), 2) users perceive that the spellchecker enables the intellectualisation of the language, and 3) non-dictionary words were added, i.e., the lexicon is indeed growing.
The answer to the last question provides some interesting data for linguists to bite their teeth in. For instance, a user had added to the spellchecker’s dictionary LikaSekelaShansela, which is an inflected form of isekelashansela ‘Vice Chancellor’ (that is recognised as correct by the spellchecker). Also some inconsistencies—from a rule-of-thumb viewpoint—in word formation were observed; e.g., usosayensi ‘scientist’ vs. unompilo ‘nurse’. If one were to follow consistently the word formation process for various types of experts in isiZulu, such as usosayensi ‘scientist’, usolwazi ‘professor’, and usomahlaya ‘comedian’, then one reasonably could expect ‘nurse’ to be *usompilo rather than unompilo. Why it isn’t, we don’t know. Regardless, the “add to dictionary” option of the spellchecker proved to be a nice extra feature for a data-driven approach to investigate intellectualisation of a language.
Version 1 of the isiZulu spellchecker that was used in the evaluation was ok and reasonably could not have interfered negatively with any possible intellectualisation (average SUS score of 75 and median 82.5, so ‘good’). It was ok in the sense that a majority of respondents thought that the entire tool was helpful, no features should be removed, it enhances their work, and so on (see paper for details). For the software developers among you who have spare time: they’d like, mainly, to have it as a Chrome and MS Word plugin, predictive text/autocomplete, and have it working on the mobile phone. The spellchecker has improved in the meantime thanks to two honours students, and I will write another blog post about that next.
As a final reflection: it turned out there isn’t a way to measure the level of intellectualisation in a ‘hard sciences’ way, so we concluded the other answers based on data that came from the somewhat fluffy approach of a survey and in-depth interviews (a ‘mixed-methods’ approach, to give it a name). It would be nice to have a way to measure it, though, so one would be able to say which languages are more or less intellectualised, what level of intellectualisation is needed to have a language as language of instruction and science at tertiary level of education and for dissemination of scientific knowledge, and to what extent some policy x, tool y, or activity z contributes to the intellectualization of a language.
References
[1] Havránek, B. 1932. The functions of literary language and its cultivation. In Havránek, B and Weingart, M. (Eds.). A Prague School Reader on Esthetics, Literary Structure and Style. Prague: Melantrich: 32-84.
[2] Finlayson, R, Madiba, M. The intellectualization of the indigenous languages of South Africa: Challenges and prospects. Current Issues in Language Planning, 2002, 3(1): 40-61.
[3]Khumalo, L. Intellectualization through terminology development. Lexikos, 2017, 27: 252-264.
[4] Keet, C.M., Khumalo, L. Evaluation of the effects of a spellchecker on the intellectualization of isiZulu. Alternation, 2017, 24(2): 75-97.
# Orchestrating 28 logical theories of mereo(topo)logy
Parts and wholes, again. This time it’s about the logic-aspects of theories of parthood (cf. aligning different hierarchies of (part-whole) relations and make them compatible with foundational ontologies). I intended to write this post before the Ninth Conference on Knowledge Capture (K-CAP 2017), where the paper describing the new material would be presented by my co-author, Oliver Kutz. Now, afterwards, I can add that “Orchestrating a Network of Mereo(topo) logical Theories” [1] even won the Best Paper Award. The novelties, in broad strokes, are that we figured out and structured some hitherto messy and confusing state of affairs, showed that one can do more than generally assumed especially with a new logics orchestration framework, and we proposed first steps toward conflict resolution to sort out expressivity and logic limitations trade-offs. Constructing a tweet-size “tl;dr” version of the contents is not easy, and as I have as much space here on my blog as I like, it ended up to be three paragraphs here: scene-setting, solution, and a few examples to illustrate some of it.
Problems
As ontologists know, parthood is used widely in ontologies across most subject domains, such as biomedicine, geographic information systems, architecture, and so on. Ontology (the philosophers) offer a parthood relation that has a bunch of computationally unpleasant properties that are structured in a plethora of mereologicial and meretopological theories such that it has become hard to see the forest for the trees. This is then complicated in practice because there are multiple logics of varying expressivity (support more or less language features), with the result that only certain fragments of the mereo(topo)logical theories can be represented. However, it’s mostly not clear what can be used when, during the ontology authoring stage one may want to have all those features so as to check correctness, and it’s not easy to predict what will happen when one aligns ontologies with different fragments of mereo(topo)logy.
Solution
We solved these problems by specifying a structured network of theories formulated in multiple logics that are glued together by the various linking constructs of the Distributed Ontology, Model, and Specification Language (DOL). The ‘structured network of theories’-part concerns all the maximal expressible fragments of the KGEMT mereotopological theory and five of its most well-recognised sub-theories (like GEM and MT) in the seven Description Logics-based OWL species, first-order logic, and higher order logic. The ‘glued together’-part refers to relating the resultant 28 theories within DOL (in Ontohub), which is a non-trivial (understatement, unfortunately) metalanguage that has the constructors for the glue, such as enabling one to declare to merge two theories/modules represented in different logics, extending a theory (ontology) with axioms that go beyond that language without messing up the original (expressivity-restricted) ontology, and more. Further, because the annoying thing of merging two ontologies/modules can be that the merged ontology may be in a different language than the two original ones, which is very hard to predict, we have a cute proof-of-concept tool so that it assists with steps toward resolution of language feature conflicts by pinpointing profile violations.
Examples
The paper describes nine mechanisms with DOL and the mereotopological theories. Here I’ll start with a simple one: we have Minimal Topology (MT) partially represented in OWL 2 EL/QL in “theory8” where the connection relation (C) is just reflexive (among other axioms; see table in the paper for details). Now what if we add connection’s symmetry, which results in “theory4”? First, we do this by not harming theory8, in DOL syntax (see also the ESSLI’16 tutorial):
logic OWL2.QL
ontology theory4 =
theory8
then
ObjectProperty: C Characteristics: Symmetric %(t7)
What is the logic of theory4? Still in OWL, and if so, which species? The Owl classifier shows the result:
Another case is that OWL does not let one define an object property; at best, one can add domain and range axioms and the occasional ‘characteristic’ (like aforementioned symmetry), for allowing arbitrary full definitions pushes it out of the decidable fragment. One can add them, though, in a system that can handle first order logic, such as the Heterogeneous toolset (Hets); for instance, where in OWL one can add only “overlap” as a primitive relation (vocabulary element without definition), we can take such a theory and declare that definition:
logic CASL.FOL
ontology theory20 =
theory6_plus_antisym_and_WS
then %wdef
. forall x,y:Thing . O(x,y) <=> exists z:Thing (P(z,x) /\ P(z,y)) %(t21)
. forall x,y:Thing . EQ(x,y) <=> P(x,y) /\ P(y,x) %(t22)
As last example, let me illustrate the notion of the conflict resolution. Consider theory19—ground mereology, partially—that is within OWL 2 EL expressivity and theory18—also ground mereology, partially—that is within OWL 2 DL expressivity. So, they can’t be the same; the difference is that theory18 has parthood reflexive and transitive and proper parthood asymmetric and irreflexive, whereas theory19 has both parthood and proper parthood transitive. What happens if one aligns the ontologies that contain these theories, say, O1 (with theory18) and O2 (with theory19)? The Owl classifier provides easy pinpointing and tells you the profile: OWL 2 full (or: first order logic, or: beyond OWL 2 DL—top row) and why (bottom section):
Now, what can one do? The conflict resolution cannot be fully automated, because it depends on what the modeller wants or needs, but there’s enough data generated already and there are known trade-offs so that it is possible to describe the consequences:
• Choose the O1 axioms (with irreflexivity and asymmetry on proper part of), which will make the ontology interoperable with other ontologies in OWL 2 DL, FOL or HOL.
• Choose O2’s axioms (with transitivity on part of and proper part of), which will facilitate linking to ontologies in OWL 2 RL, 2 EL, 2 DL, FOL, and HOL.
• Choose to keep both sets will result in an OWL 2 Full ontology that is undecidable, and it is then compatible only with FOL and HOL ontologies.
As serious final note: there’s still fun to be had on the logic side of things with countermodels and sub-networks and such, and with refining the conflict resolution to assist ontology engineers better. (or: TBC)
As less serious final note: the working title of early drafts of the paper was “DOLifying mereo(topo)logy”, but at some point we chickened out and let go of that frivolity.
References
[1] Keet, C.M., Kutz, O. Orchestrating a Network of Mereo(topo)logical Theories. Ninth International Conference on Knowledge Capture (K-CAP’17), Austin, Texas, USA, December 4-6, 2017. ACM Proceedings.
# Logic, diagrams, or natural language for representing temporal constraints in conceptual modeling languages?
Spoiler alert of the answer: it depends. In this post, I’ll trace it back to how we got to that conclusion and refine it to what it depends on.
There are several conceptual modelling languages with extensions for temporal constraints that then will be used in a database to ensure data integrity with respect to the business rules. For instance, there may be a rule for some information system that states that “all managers in a company must have been employees of that company already” or “all postgraduate students must also be a teaching assistant for some time during their studies”. The question then becomes how to get the modellers to model this sort of information in the best way. The first step in that direction is figuring out the best way to represent temporal constraints. We already know that icons aren’t that unambiguous and easy [1], which leaves the natural language rendering devised recently [2], or one of the logic-based notations, such as the temporal Description Logic DLRUS [3]. So, the questions to investigate thus became, more precisely:
• Which representation is preferred for representing temporal information: formal semantics, Description Logics (DL), a coding-style notation, diagrams, or template-based (pseudo-)natural language sentences?
• What would be easier to understand by modellers: a succinct logic-based notation, a graphical notation, or a ‘coding style’ notation?
To answer these questions, my collaborator, Sonia Berman (also at UCT) and I conducted a survey to find out modeller preference(s) and understanding of these representation modes. The outcome of the experiment is about to be presented at the 36th International Conference on Conceptual Modeling (ER’17) that will be held next week in Valencia, Spain, and is described in more detail in the paper “Determining the preferred representation of temporal constraints in conceptual models” [4].
The survey consisted mainly of questions asking them about which representation they preferred, a few questions on trying to model it, and basic questions, like whether they had English as first language (see the questionnaire for details). Below is one of the questions to illustrate it.
One of the questions of the survey
Its option (a) is the semantics notation of the DLRUS Description Logic, its option (b) the short-hand notation in DLRUS, option (c) a coding-style notation we made up, and option (e) is the natural language rendering that came out of prior work [2]. Option (d) was devised for this experiment: it shows the constraint in the Temporal information Representation in Entity-Relationship Diagrams (TREND) language. TREND is an updated and extended version of ERVT [5], taking into account earlier published extensions for temporal relationships, temporal attributes, and quantitative constraints (e.g., ‘employee receives a bonus after two years’), a new extension for the distinction between optional and mandatory temporal constraints, and the notation preferences emanating from [1].
Here are some of the main quantitative results:
The top-rated representation modes and `dislike’ ratings.
These are aggregates, though, and they hide some variations in responses. For instance, representing ‘simple’ temporal constraints in the DL notation was still ok (though noting that diagrams were most preferred), but the more complex the constraints got, the more the preference for the natural language rendering. For instance, take “Person married-to Person may be followed by Person divorced-from Person, ending Person married-to Person.” is deemed easier to understand than $\langle o , o' \rangle \in marriedTo^{\mathcal{I}(t)} \rightarrow \exists t'>t. \langle o , o' \rangle \in divorcedFrom^{\mathcal{I}(t')} \land \langle o , o' \rangle \not\in marriedTo^{\mathcal{I}(t')}$ or $\diamond^+\mbox{{\sc RDev}}_{{\sf marriedTo,divorcedFrom}}$. Yet, the temporal relationship ${\sf marriedTo \sqsubseteq \diamond^* \neg marriedTo}$ was deemed easier to understand than “The objects participating in a fact in Person married to Person do not relate through married-to at some time”. Details of the experiment and more data and analysis are described in the paper [4]. In sum, the evaluation showed the following:
1. a clear preference for graphical or verbalised temporal constraints over the other three representations;
2. ‘simple’ temporal constraints were preferred graphically and complex temporal constraints preferred in natural language; and
3. their English specification of temporal constraints was inadequate.
Overall, this indicates that what is needed is some modeling tool that has a multi-modal interface for temporal conceptual model development, with the ability to switch between graphical and verbalised temporal constraints in particular.
If I hadn’t had teaching obligations (which now got cancelled due to student protests anyway) and no NRF funding cut in the incentive funding (rated researchers got to hear from one day to the next that it’ll be only 10% of what it used to be), I’d have presented the paper myself at ER’17. Instead, my co-author is on her way to all the fun. If you have any questions, suggestions, or comments, you can ask her at the conference, or drop me a line via email or in the comments below. If you’re interested in TREND: we’re working on a full paper with all the details and have conducted further modeling experiments with it, which we hope to finalise writing up by the end of the year (provided student protests won’t escalate and derail research plans any further).
References
[1] T. Shunmugam. Adoption of a visual model for temporal database representation. M. IT thesis, Department of Computer Science, University of Cape Town, South Africa, 2016.
[2] Keet, C.M. Natural language template selection for temporal constraints. CREOL: Contextual Representation of Events and Objects in Language, Joint Ontology Workshops 2017, 21-23 September 2017, Bolzano, Italy. CEUR-WS Vol. (in print).
[3] A. Artale, E. Franconi, F. Wolter, and M. Zakharyaschev. A temporal description logic for reasoning about conceptual schemas and queries. In S. Flesca, S. Greco, N. Leone, and G. Ianni, editors, Proceedings of the 8th Joint European Conference on Logics in Artificial Intelligence (JELIA-02), volume 2424 of LNAI, pages 98-110. Springer Verlag, 2002.
[4] Keet, C.M., Berman, S. Determining the preferred representation of temporal constraints in conceptual models. 36th International Conference on Conceptual Modeling (ER’17). Mayr, H.C., Guizzardi, G., Ma, H. Pastor. O. (Eds.). Springer LNCS vol. 10650, 437-450. 6-9 Nov 2017, Valencia, Spain.
[5] A. Artale, C. Parent, and S. Spaccapietra. Evolving objects in temporal information systems. Annals of Mathematics and Artificial Intelligence, 50(1-2):5-38, 2007.
# Part-whole relations and foundational ontologies
Part-whole relations seem like a never-ending story—and it still doesn’t bore me. In this case, the ingredients were the taxonomy of part-whole relations [1] and a couple of foundational ontologies and the aim was to link the former to the latter. But what started off with the intention to write just a short workshop note, for seemingly clear and just in need of actually doing it, turned out to be not so straightforward after all. The selected foundational ontologies were not as compatible as assumed, and creating the corresponding orchestration of OWL files was a ‘non-trivial exercise’.
What were (some of) the issues? On the one hand, there are multiple part-whole relations, which are typically named differently when they have a specific domain or range. For instance, to relate a process to a sub-process (e.g., eating involves chewing), to relate a region to a region it contains, relating portions of stuff, and so on. Those relations are fairly well established in the literature. What they do demand for, however, is clarity as to what those categories really are. For instance, with the process example, is that to be understood as Process as meant in the DOLCE ontology, or, say, Process in BFO? What if a foundational ontology does not have a category needed for a commonly used part-whole relation?
The first step to answer such questions was to assess several foundational ontologies on 1) which of the part-whole relations they have now, and which categories are present that are needed for the domain and range declarations for those common part-whole relations. I assessed that for DOLCE, BFO, GFO, SUMO, GIST, and YAMATO. This foundational ontology comparison is summarised in tables 1 and 2 in the paper that emanated from the assessment [2], entitled “A note on the compatibility of part-whole relations with foundational ontologies” that I recently presented at FOUST-II: 2nd Workshop on Foundational Ontology, Joint Ontology Workshops 2017 in Bolzano, Italy. In short: none fits perfectly for various reasons, but there are more and less suitable ontologies for a possible alignment. DOLCE and SUMO were evaluated to have the best approximations. It appeared at the workshops presentation’s Q&A session, where two of the DOLCE developers were present, that the missing Collective was an oversight, or: the ontology is incomplete and it was not an explicit design choice to exclude it. This, then, would make DOLCE the best/easiest fit.
I’ll save you the trials and tribulations creating the orchestrated OWL files. The part-whole relations, their inverses, and their proper parthood versions were manually linked to modules of DOLCE and SUMO, and automatically linked to BFO and GFO. That was an addition of 49 relations (OWL object properties) and 121 logical axioms, which were then extended further with another 11 mereotopological relations and its 16 logical axioms. These files are accessible online directly here and also listed with brief descriptions.
While there is something usable now and, by design at least, these files are reusable as well, what it also highlighted is that there are still some outstanding questions, as there already were for the top-level categories of previously aligned foundational ontologies [3]. For instance, some categories seem the same, but they’re in ‘incompatible’ parts of the taxonomy (located in disjoint branches), so then either not the same after all, or this happened unintentionally. Only GIST has been updated recently, and it may be useful if the others foundational ontologies were to be as well, so as to obtain clarity on these issues. The full interaction of part-whole relations with classical mereology is not quite clear either: there are various extensions and deviations, such as specifically for portions [4,5], but one for processes may be interesting as well. Not that such prospective theories would be usable as-is in OWL ontology development, but there are more expressive languages that start having tooling support where it could be an interesting avenue for future work. I’ll write more about the latter in an upcoming post (covering the K-CAP 2017 paper that was recently accepted).
On a last note: the Joint Ontology Workshops (JOWO 2017) was a great event. Some 100 ontologists from all over the world attended. There were good presentations, lively conversations, and it was great to meet up again with researchers I had not seen for years, finally meet people I knew only via email, and make new connections. It will not be an easy task to surpass this event next year at FOIS 2018 in Cape Town.
References
[1] Keet, C.M., Artale, A. Representing and Reasoning over a Taxonomy of Part-Whole Relations. Applied Ontology, 2008, 3(1-2):91-110.
[2] Keet, C.M. A note on the compatibility of part-whole relations with foundational ontologies. FOUST-II: 2nd Workshop on Foundational Ontology, Joint Ontology Workshops 2017, 21-23 September 2017, Bolzano, Italy. CEUR-WS Vol. (in print)
[3] Khan, Z.C., Keet, C.M. Foundational ontology mediation in ROMULUS. Knowledge Discovery, Knowledge Engineering and Knowledge Management: IC3K 2013 Selected Papers. A. Fred et al. (Eds.). Springer CCIS vol. 454, pp. 132-152, 2015. preprint
[4] Donnelly, M., Bittner, T. Summation relations and portions of stuff. Philosophical Studies, 2009, 143, 167-185.
[5] Keet, C.M. Relating some stuff to other stuff. 20th International Conference on Knowledge Engineering and Knowledge Management (EKAW’16). Blomqvist, E., Ciancarini, P., Poggi, F., Vitali, F. (Eds.). Springer LNAI vol. 10024, 368-383. 19-23 November 2016, Bologna, Italy.
# Figuring out the verbalisation of temporal constraints in ontologies and conceptual models
Temporal conceptual models, ontologies, and their logics are nothing new, but that sort of information and knowledge representation still doesn’t gain a lot of traction (cf. say, formal methods for verification). This is in no small part because modelling temporal information is not easy. Several conceptual modelling languages do have various temporal extensions, but most modellers don’t even use all of the default language features yet [1]. How could one at least reduce the barrier to adoption of temporal logics and modelling languages? The two principle approaches are visualisation with a diagrammatic language and rendering it in a (pseudo-)natural language. One of my postgraduate students looked at the former, trying to figure out what would be the best icons and such, which showed there was still a steep learning curve [2]. Before examining whether that could be optimised, I wondered whether the natural language option might be promising. The problem was, that no-one had yet tried to determine what the natural language counterpart of the temporal constraints were supposed to be, let alone whether they be ‘adequate’ or the ‘best’ way of rendering the temporal constraints in tolerable natural language sentences. I wanted to know that badly enough that I tried to find out.
Given that using templates is a tried-and-tested relatively successful approach for atemporal conceptual models and ontologies (e.g., for ORM, the ACE system), it makes sense to do something similar, but then for some temporal extension. As temporal conceptual modelling language I used one that has a Description Logics foundation (DLRUS [3,4]) for that easily links to ontologies as well, added a few known temporal constraints (like for relationships/DL roles, mandatory) and removing others (some didn’t seem all that interesting), which resulted in 34 constraints, still. For each one, I tried to devise more and less reasonable templates, resulting in 101 templates overall. Those templates were evaluated on semantics and preference by three temporal logic experts and five ‘mixed experts’ (experts in natural language generation, logic, or modelling). This resulted in a final set of preferred templates to verbalise the temporal constraints. The remainder of this post first describes a bit about the templates and then the results of which I think they are most interesting.
Templates
The basic idea of a template—in the context of the verbalisation of conceptual models and ontologies—is to have some natural language for the constraint where then the vocabulary gets slotted in at runtime. Take, for instance, simple named class subsumption in an ontology, $C \sqsubseteq D$, for which one could define a template “Each [C] is a(n) [D]”, so that with some axiom $Manager \sqsubseteq Employee$, it would generate the sentence “Each Manager is an Employee”. One also could have devised the template “All [C] are [D]” and then it would have generated “All Managers are Employees”. The choice between the two templates in this case is just taste, for in both cases, the semantics is the same. More complex axioms are not always that straightforward. For instance, for the axiom type $C \sqsubseteq \exists R.D$, would “Each [C] [R] some [D]” be good enough, or would perhaps “Each [C] must [R] at least one [D]” be better? E.g., “Each Professor teaches some Course” vs “Each Professor must teach at least one Course”.
The same can be done for the temporal constraints. To get there, I did a bit of a linguistic detour that informed the template design (described in the paper [5]). Let us take as first example for templates temporal class that has a semantics of $o \in C^{\mathcal{I}(t)} \rightarrow \exists t' \neq t. o \notin C^{\mathcal{I}(t')}$; for instance, UndergraduateStudent (assuming they graduate and end up as alumni or as drop outs, and weren’t undergrads from birth):
1. If an object is an instance of entity type [C], then there is some time where it is not a(n) [C].
2. [C] is an entity type whose objects are, for some time in their existence, not instances of [C].
3. [C] is an entity type of which each object is not a(n) [C] for some time during its existence.
4. All instances of entity type [C] are not a(n) [C] for some time.
5. Each [C] is not a(n) [C] for some time.
6. Each [C] is for some time not a(n) [C].
Which one(s) do you think captures the semantics, and which one(s) do you prefer?
A more elaborate constraint for relationships is ‘dynamic extension for relationships, past, mandatory], which is formalised as $\langle o , o' \rangle \in \mbox{{\sc RDexM}-}_{R_1,R_2}^{\mathcal{I}(t)} \rightarrow (\langle o , o' \rangle \in{\tt R_1}^{\mathcal{I}(t)} \rightarrow \exists t' where $\langle o , o' \rangle \in \mbox{{\sc RDex}}_{R_1,R_2}^{\mathcal{I}(t)} \rightarrow ( \langle o , o' \rangle \in{\tt R_1}^{\mathcal{I}(t)} \rightarrow \exists t'>t. \langle o , o' \rangle \in {\tt R_2}^{\mathcal{I}(t')})$.; e.g., every passenger who boards a flight must have checked in for that flight. Two options could be:
1. Each ..C_1.. ..R_1.. ..C_2.. was preceded by ..C_1.. ..R_2.. ..C_2.. some time earlier.
2. Each ..C_1.. ..R_1.. ..C_2.. must be preceded by ..C_1.. ..R_2.. ..C_2.. .
I’m not saying they are all correct; they were some of the options given, which the participants could choose from and comment on. The full list of constraints and template options are available in the supplementary material, which also contains a file where you can fill in your own answers, see what the (anonymised) participants said, and it has the final list of ‘best’ constraints.
Results
The main aggregate quantitative results are shown in the following table.
Many observations can be made from the data (see the paper for details). Some of the salient aspects are that there was low inter-annotator agreement among the experts, despite that they know each other (temporal logics is a small community) and that the ‘mixed group’ deemed many sentences correct that the experts deemed wrong in the sense of not properly capturing the semantics of the constraint. Put differently, it looks like the mixed experts, as a group, did not fully grasp some subtle distinction in the temporal constraints.
With respect to the templates, the preferred ones don’t follow the structure of the logic, but are, in a way, a separate rendering, or: there’s no neat 1:1 mapping between axiom type and template structure. That said, that doesn’t mean that they always chose the shortest template: the experts definitely did not, while the mixed experts leaned a bit toward preferring templates with fewer words even though they were surely not always the semantically correct option.
It may not look good that the experts preferred different templates, but in a follow-up interview with one of the experts, the expert noted that it was not really a problem “for there is the logic that does have the precise meaning anyway” and thus “resolves any confusion that may arise from using slightly different terminology”. The temporal logic expert does have a point from the expert’s view, fair enough, but that pretty much defeats my aim with the experiment. Asking more non-experts may not be a good strategy either, for they are, on average, too lenient.
So, for now, we do have a set of, relatively, ‘best’ templates to verbalise temporal constraints in temporal conceptual models and ontologies. The next step is to compare that with the diagrammatic representation. This we did [6], and I’ll describe those results informally in a next post.
I’ll present more details at the upcoming CREOL: Contextual Representation of Events and Objects in Language Workshop that is part of the Joint Ontology Workshops 2017, which will be held next week (21-23 September) in Bolzano, Italy. As the KRDB group at FUB in Bolzano has a few temporal logic experts, I’m looking forward to the discussions! Also, I’d be happy if you would be willing to fill in the spreadsheet with your preferences (before looking at the answers given by the participants!), and send them to me.
References
[1] Keet, C.M., Fillottrani, P.R. An analysis and characterisation of publicly available conceptual models. 34th International Conference on Conceptual Modeling (ER’15). Johannesson, P., Lee, M.L. Liddle, S.W., Opdahl, A.L., Pastor López, O. (Eds.). Springer LNCS vol 9381, 585-593. 19-22 Oct, Stockholm, Sweden.
[2] T. Shunmugam. Adoption of a visual model for temporal database representation. M. IT thesis, Department of Computer Science, University of Cape Town, South Africa, 2016.
[3] A. Artale, E. Franconi, F. Wolter, and M. Zakharyaschev. A temporal description logic for reasoning about conceptual schemas and queries. In S. Flesca, S. Greco, N. Leone, and G. Ianni, editors, Proceedings of the 8th Joint European Conference on Logics in Artificial Intelligence (JELIA-02), volume 2424 of LNAI, pages 98-110. Springer Verlag, 2002.
[4] A. Artale, C. Parent, and S. Spaccapietra. Evolving objects in temporal information systems. Annals of Mathematics and Artificial Intelligence, 50(1-2):5-38, 2007.
[5] Keet, C.M. Natural language template selection for temporal constraints. CREOL: Contextual Representation of Events and Objects in Language, Joint Ontology Workshops 2017, 21-23 September 2017, Bolzano, Italy. CEUR-WS Vol. (in print).
[6] Keet, C.M., Berman, S. Determining the preferred representation of temporal constraints in conceptual models. 36th International Conference on Conceptual Modeling (ER’17). Springer LNCS. 6-9 Nov 2017, Valencia, Spain. (in print)
# Round 2 of the search engine, browser, and language bias mini-experiment
Exactly a year ago I did a mini-experiment to see whether search engine bias exist in South Africa as well. It did. The notable case was that Google in English on Safari on the Mac (GES) showed results for ‘politically interesting searches’ that had less information and was leaning to the right-side of the political spectrum in a way that raised cause for concern, as compared to Google in isiZulu in Firefox (GiF) and Bing in English in Firefox (BEF). I repeated the experiment in the exact same way, with some of the same queries and a few more new ones that take into account current affairs; the only difference being using my Internet connection at home rather than at work. The same problem still exists, sometimes quite dramatically. As recommendation, then: don’t use Google in English on Safari on the Mac unless you want to be in an “anti-government Democratic Alliance as centre-of-the-world” bubble.
To back it all up, I took screenshots again, with the order fltr GiF, GES, BEF, so you can check for yourself what users with different configurations see on the first page of the search results. The set of clearly different/biased results are listed first.
• EFF”, which in South Africa is a left populist opposition party, and internationally the abbreviation of the electronic frontier foundation:
“EFF” search
GiF lists it as political party; GES in relation to the DA first and then as political party; BEF as political party and electronic frontier foundation.
• jacob zuma”, the current president of the country: GiF first has a google ad to oust zuma, then general info and news; GES with a google ad to oust zuma, comment by JZ’s son
“jacob zuma” search
blaming the whites (probably fuelling racial divisiveness), then general info and news; BEF has general info and news.
• ANC”, currently the largest political party nationally and in power: GiF has first a link to ANC site, one
“ANC” search
news, and for the rest contact info; GES has first ‘bad press’ for the ANC as top stories, then twitter, then the ANC website; BEF lists first the ANC site, then news and info.
• Manana”, who is the Higher Education deputy minister who faces allegations of mistreatment by female
“Manana” search
staff members in his department: GiF with news about the accusations; GES has negative news about the ANC women’s league and DA actions; BEF shows info about Manana and mixed it up with the Spanish mañana.
• The autocomplete function when typing “ANC” was somewhat surprising: GiF also associates it with ‘eff news’, and ‘zuma’;
exploring the autocomplete on “ANC”
GES doesn’t have ‘eff news’ to suggest, so autocomplete also seems to be determined by the client-side configuration; BEF has all sorts of things.
• white monopoly capital” (long story): GiF shows general info and news; GES also shows general info
“white monopoly capital” search
and news, but with that inciting blaming the whites news item; BEF shows general info and news as well, but differently ordered from Google’s result.
• DA”, which in South Africa refers to the abbreviation of the Democratic Alliance opposition party
“DA” search
(capitalist, for the rich): GiF lists the DA website and some news; GES shows news on DA action and opinion, then the DA website; BEF lists the DA site, some general info and disambiguation.
• motion of no confidence”, which was held last week against Jacob Zuma
“motion of no confidence” search
(the motion failed, but not by a large margin): GiF has again that Google ad for the organization to oust Zuma, then info and mostly news (with 1 international news site [Al Jazeera]); GES has info then SA opinion pieces rather than news; BEF has news and info.
• FeesMustFall”, which was one of the tags of the student protests in 2015
“FeesMustFall” search
and 2016 (for free higher education): GiF has general info and news; GES shows first two ads to join the campaign, then general info and news; BEF has info and news. So, this seems flipped cf. last year.
Then the set of searches of which the results are roughly the same. I had expected this for “Law on cookies in South Africa” and “Socialism”, for they were about the same last year as well. I wasn’t sure about “women’s month” (this month, August), given its history; there are slight differences, but not much. The interesting one, perhaps, was that “state capture gupta” also showed similar results across the three configurations, all of them showing results to pages that treat it as fact and at least some detailed background reading on it.
“Law on cookies in South Africa” search
“Socialism” search
“women’s month” search
“state capture gupta” search
Finally, last year the mini-experiment was motivated by lecture preparations for the “Social Issues and Professional Practice” block of CSC1016S that I’m scheduled to teach in the upcoming semester (if there won’t be protests, that is). As compared to last year, now I can also add a note on the Algorithmic Transparency and Accountability statement from the ACM, in addition to the ‘filter bubble’ and ‘search engine manipulation’ items. Maybe I should cook up an exercise for the students so we can get data rather still being in the realm of anecdotes with my 20 searches and three configurations. If you did the same with a different configuration, please let me know. | 2018-04-22 17:58:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 9, "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": 0.46194395422935486, "perplexity": 3234.271667489391}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945637.51/warc/CC-MAIN-20180422174026-20180422194026-00357.warc.gz"} |
https://www.business-science.io/code-tools/2020/09/09/five-minute-time-series-modeling-data.html | # Time Series in 5-Minutes, Part 6: Modeling Time Series Data
Written by Matt Dancho on September 9, 2020
Have 5-minutes? Then let’s learn time series. In this short articles series, I highlight how you can get up to speed quickly on important aspects of time series analysis.
In this article we walk through modeling time series data using the modeltime package. In part 1-5 of the series we learned how to use timetk to visualize, wrangle, and feature engineer time series data, and in this article you’ll see how simple it is is to prepare the data for modeling using the timetk package.
## Time Series in 5-Mintues Articles in this Series
I just released timetk 2.0.0 (read the release announcement). A ton of new functionality has been added. We’ll discuss some of the key pieces in this article series:
# Have 5-Minutes? Then let’s learn Time Series Modeling
Forecasting with tidymodels made easy! This short tutorial shows how you can use:
• Modeltime models like arima_reg(), arima_boost(), exp_smoothing(), prophet_reg(), prophet_boost(), and more
• Parsnip models like linear_reg(), mars(), svm_rbf(), rand_forest(), boost_tree() and more
To perform classical time series analysis and machine learning in one framework! See “Model List” for the full list of modeltime models.
## The Modeltime Workflow
Here’s the general process and where the functions fit.
Just follow the modeltime workflow, which is detailed in 6 convenient steps:
1. Collect data and split into training and test sets
2. Create & Fit Multiple Models
3. Add fitted models to a Model Table
4. Calibrate the models to a testing set.
5. Perform Testing Set Forecast & Accuracy Evaluation
6. Refit the models to Full Dataset & Forecast Forward
Let’s go through a guided tour to kick the tires on modeltime.
Load libraries to complete this short tutorial.
## Time Series Forecasting Example
Load libraries to complete this short tutorial.
### Step 1 - Collect data and split into training and test sets.
We can visualize the dataset.
Let’s split the data into training and test sets using initial_time_split()
### Step 2 - Create & Fit Multiple Models
We can easily create dozens of forecasting models by combining modeltime and parsnip. We can also use the workflows interface for adding preprocessing! Your forecasting possibilities are endless. Let’s get a few basic models developed:
• ARIMA
• Exponential Smoothing
• Linear Regression
• MARS (Multivariate Adaptive Regression Splines)
Important note: Handling Date Features
Modeltime models (e.g. arima_reg()) are created with a date or date time feature in the model. You will see that most models include a formula like fit(value ~ date, data).
Parsnip models (e.g. linear_reg()) typically should not have date features, but may contain derivatives of dates (e.g. month, year, etc). You will often see formulas like fit(value ~ as.numeric(date) + month(date), data).
#### Model 1: Auto ARIMA (Modeltime)
First, we create a basic univariate ARIMA model using “Auto Arima” using arima_reg()
#### Model 2: Boosted Auto ARIMA (Modeltime)
Next, we create a boosted ARIMA using arima_boost(). Boosting uses XGBoost to model the ARIMA errors. Note that model formula contains both a date feature and derivatives of date
• ARIMA uses the date
• XGBoost uses the derivatives of date as regressors
Normally I’d use a preprocessing workflow for the month features using a function like step_timeseries_signature() from timetk to help reduce the complexity of the parsnip formula interface.
#### Model 3: Exponential Smoothing (Modeltime)
Next, create an Error-Trend-Season (ETS) model using an Exponential Smoothing State Space model. This is accomplished with exp_smoothing().
#### Model 4: Prophet (Modeltime)
We’ll create a prophet model using prophet_reg().
#### Model 5: Linear Regression (Parsnip)
We can model time series linear regression (TSLM) using the linear_reg() algorithm from parsnip. The following derivatives of date are used:
• Trend: Modeled using as.numeric(date)
• Seasonal: Modeled using month(date)
#### Model 6: MARS (Workflow)
We can model a Multivariate Adaptive Regression Spline model using mars(). I’ve modified the process to use a workflow to standardize the preprocessing of the features that are provided to the machine learning model (mars).
OK, with these 6 models, we’ll show how easy it is to forecast.
### Step 3 - Add fitted models to a Model Table.
The next step is to add each of the models to a Modeltime Table using modeltime_table(). This step does some basic checking to make sure each of the models are fitted and that organizes into a scalable structure called a “Modeltime Table” that is used as part of our forecasting workflow.
We have 6 models to add. A couple of notes before moving on:
• Note that some of the models have tunable parameters.
• It’s expected that tuning and parameter selection is performed prior to incorporating into a Modeltime Table.
• If you try to add an unfitted model, the modeltime_table() will complain (throw an informative error) saying you need to fit() the model.
### Step 4 - Calibrate the model to a testing set.
Calibrating adds a new column, .calibration_data, with the test predictions and residuals inside. A few notes on Calibration:
• Calibration is how confidence intervals and accuracy metrics are determined
• Calibration Data is simply forecasting predictions and residuals that are calculated from out-of-sample data.
• After calibrating, the calibration data follows the data through the forecasting workflow.
### Step 5 - Testing Set Forecast & Accuracy Evaluation
There are 2 critical parts to an evaluation.
• Visualizing the Forecast vs Test Data Set
• Evaluating the Test (Out of Sample) Accuracy
#### 5A - Visualizing the Forecast Test
Visualizing the Test Error is easy to do using the interactive plotly visualization (just toggle the visibility of the models using the Legend).
From visualizing the test set forecast:
• Models 1&2: ARIMA & ARIMA Boost are performing well. Both models have “auto” components because we used Auto ARIMA. The XGBoost component has parameters that were specified. We can possibly get better accuracy by tuning, but because the ARIMA component is working well on this data, additional improvement may be low.
• Model 3: ETS(M,A,A) is performing the best. The 80% confidence interval is the most narrow of the bunch, indicating the hold out set is modeled well.
• Model 4: PROPHET is comparable to the ARIMA models, but has a slightly wider test error confidence interval.
• Model 5: LM is over-shooting the local trend. This is because the trend component is a simple linear line, which doesn’t account for the change points.
• Model 6: EARTH is overfitting the local trend. This is because we did not tune the number of change points, so the algorithm is auto-calculating the change points.
#### 5B - Accuracy Metrics
We can use modeltime_accuracy() to collect common accuracy metrics. The default reports the following metrics using yardstick functions:
• MAE - Mean absolute error, mae()
• MAPE - Mean absolute percentage error, mape()
• MASE - Mean absolute scaled error, mase()
• SMAPE - Symmetric mean absolute percentage error, smape()
• RMSE - Root mean squared error, rmse()
• RSQ - R-squared, rsq()
These of course can be customized following the rules for creating new yardstick metrics, but the defaults are very useful. Refer to default_forecast_accuracy_metrics() to learn more.
To make table-creation a bit easier, I’ve included table_modeltime_accuracy() for outputing results in either interactive (reactable) or static (gt) tables.
From the accuracy metrics:
• Model 3: ETS is clearly the winner here with MAE of 77
• Model 6: MARS is over-fitting the local trend. This comes out in the R-Squared of 0.55.
### Step 6 - Refit to Full Dataset & Forecast Forward
The final step is to refit the models to the full dataset using modeltime_refit() and forecast them forward.
## Refitting - What happened?
The models have all changed! (Yes - this is the point of refitting)
• The LM model looks much better now because the linear trend line has now been fit to new data that follows the longer term trend.
• The EARTH model has a trend that is more representative of the near-term trend.
• The PROPHET model has a trend that is very similar to the EARTH model (this is because both modeling algorithms use changepoints to model trend, and prophet’s auto algorithm seems to be doing a better job at adapting).
• The ETS model has changed from (M,A,A) to (A,A,A).
• The ARIMA model have been updated and better capture the upswing.
This is the (potential) benefit of refitting.
More often than not refitting is a good idea. Refitting:
• Retrieves your model and preprocessing steps
• Refits the model to the new data
• Recalculates any automations. This includes:
• Recalculating the long-term trend for Linear Model
• Recalculating the changepoints for the Earth Model
• Recalculating the ARIMA and ETS parameters
• Preserves any parameter selections. This includes:
• XGBoost Parameters in the Boosted ARIMA min_n = 2, learn_rate = 0.015.
• Any other defaults that are not automatic calculations are used.
# Have questions on using Modeltime for time series?
Make a comment in the chat below. 👇
And, if you plan on using modeltime for your business, it’s a no-brainer - Join the Time Series Course. | 2020-11-28 04:55:28 | {"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": 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": 0.47018197178840637, "perplexity": 4352.170662970279}, "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/1606141195069.35/warc/CC-MAIN-20201128040731-20201128070731-00231.warc.gz"} |
https://homework.cpm.org/category/MN/textbook/cc3mn/chapter/10/lesson/10.4.2/problem/10-136 | ### Home > CC3MN > Chapter 10 > Lesson 10.4.2 > Problem10-136
10-136.
Thirty coins, all dimes and nickels, are worth $\2.60$. How many nickels are there?
Write two equations. One for the total number of coins and one for what the amount of coins are worth.
$d=$ dimes, $n=$ nickels
$22$ dimes and $8$ nickels. | 2021-09-20 08:27:45 | {"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": 0.43583738803863525, "perplexity": 3825.9934694477383}, "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/1631780057033.33/warc/CC-MAIN-20210920070754-20210920100754-00483.warc.gz"} |
https://www.sparrho.com/item/scaling-laws-for-weakly-interacting-cosmic-superstring-and-p-brane-networks/98186c/ | # Scaling laws for weakly interacting cosmic (super)string and p-brane networks
Research paper by P. P. Avelino, L. Sousa
Indexed on: 28 Feb '12Published on: 28 Feb '12Published in: arXiv - Astrophysics - Cosmology and Nongalactic Astrophysics
#### Abstract
In this paper we find new scaling laws for the evolution of $p$-brane networks in $N+1$-dimensional Friedmann-Robertson-Walker universes in the weakly-interacting limit, giving particular emphasis to the case of cosmic superstrings ($p=1$) living in a universe with three spatial dimensions (N=3). In particular, we show that, during the radiation era, the root-mean-square velocity is ${\bar v} =1/{\sqrt 2}$ and the characteristic length of non-interacting cosmic string networks scales as $L \propto a^{3/2}$ ($a$ is the scale factor), thus leading to string domination even when gravitational backreaction is taken into account. We demonstrate, however, that a small non-vanishing constant loop chopping efficiency parameter $\tilde c$ leads to a linear scaling solution with constant $L H \ll 1$ ($H$ is the Hubble parameter) and ${\bar v} \sim 1/{\sqrt 2}$ in the radiation era, which may allow for a cosmologically relevant cosmic string role even in the case of light strings. We also determine the impact that the radiation-matter transition has on the dynamics of weakly interacting cosmic superstring networks. | 2021-06-22 11:11:09 | {"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": 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": 0.696661651134491, "perplexity": 1421.204834898436}, "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-2021-25/segments/1623488517048.78/warc/CC-MAIN-20210622093910-20210622123910-00293.warc.gz"} |
https://math.stackexchange.com/questions/3272678/integral-by-the-residue-theorem | Integral by the residue theorem
Check that the value of $$I(\alpha)=28\int_{-\infty}^\infty \frac{1-x^2}{2+3x^2+2\alpha^2x^4}\text{d}x$$ is $$I(\alpha)=14\pi\left(1-\frac{1}{\alpha}\right)\sqrt{\frac{2}{4\alpha+3}}$$ for some $$\alpha>0$$.
I've tried to apply the Residue Theorem to this integral with the contour $$C_R\cup C$$, where $$C_R$$ is the semicircle of radius $$R$$ centered at the origin (with initial point $$(R,0)$$ and final point $$(-R,0)$$) and $$C$$ is the segment $$(-R,R)$$ (with initial point $$(-R,0)$$ and final point $$(R,0)$$).
$$\textbf{Lemma}$$. Suppose $$f(z)$$ is defined in the upper half-plane. If there is an $$a > 1$$ and $$M > 0$$ such that $$|f(z)| <\frac{M}{|z|^a}$$ for large $$|z|$$. Then, $$\lim_{R\to \infty} \int_{C_R} f(z)\text{d}z=0$$
The roots of the polynomial $$p(x)=2+3x^2+2\alpha^2x^4$$ are $$x_1=-\frac{1}{2}\sqrt{\frac{-\sqrt{9-16\alpha^2}-3}{\alpha^2}}, \qquad x_2=\frac{1}{2}\sqrt{\frac{-\sqrt{9-16\alpha^2}-3}{\alpha^2}}$$ $$x_3=-\frac{1}{2}\sqrt{\frac{\sqrt{9-16\alpha^2}-3}{\alpha^2}}, \qquad x_4=\frac{1}{2}\sqrt{\frac{\sqrt{9-16\alpha^2}-3}{\alpha^2}}$$ By the previous lemma and the residue theorem, we have that $$I(\alpha)=\int_{-\infty}^\infty \frac{28(1-x^2)}{2+3x^2+2\alpha^2x^4}\text{d}x=\lim_{R\to \infty} \int_C \frac{28(1-x^2)}{2+3x^2+2\alpha^2x^4}\text{d}x=2\pi i \sum_{i=1}^n\text{Res}(f,x_i)$$ However, I don't know how to discuss what roots are in the upper half-plane due to the parameter $$\alpha$$.
Can anyone continue from here or give some hints to follow?
• The choice depends on the branch of $\sqrt{}$ you take, but exactly one of $x_1,x_2$ and exactly one of $x_3,x_4$ is in the upper half plane, assuming $\alpha\neq\frac34$. – user10354138 Jun 24 '19 at 13:33
You are on the right track, and only needs a bit of extra input to get to the answer. Assume for a moment that $$\alpha > 3/4$$. Then the zeros of $$2+3t+3\alpha^2 t^2 = 0$$ are
$$t = \frac{-3 \pm i\sqrt{16\alpha^2-9}}{4a^2} = \left( \frac{\sqrt{4\alpha-3} \pm i\sqrt{4\alpha+3}}{\sqrt{8}\alpha} \right)^2.$$
Now, among the square roots of these values, the only ones with positive imaginary parts are
$$x_{(1)} = \frac{\sqrt{4\alpha-3} + i\sqrt{4\alpha+3}}{\sqrt{8}\alpha} \qquad \text{and} \qquad x_{(2)} = \frac{-\sqrt{4\alpha-3} + i\sqrt{4\alpha+3}}{\sqrt{8}\alpha}.$$
Then, after some painful algebra,
$$I(\alpha) = 2\pi i \sum_{k=1}^{2} \operatorname{Res}(f, x_{(k)}) = 14\pi \left(1 - \frac{1}{\alpha}\right) \sqrt{\frac{2}{4\alpha+3}}.$$
At this point, this equality is established only for $$\alpha > 3/4$$. But since both sides are analytic near $$(0, \infty)$$, it extends to all of $$(0, \alpha)$$ by the principle of analytic continuation.
• Amazing! But what "both sides" are you refering the last part? – user326159 Jun 24 '19 at 16:29
• @user326159, Oh, by 'both sides' I was referring $I(\alpha)$ and $14\pi(1-\alpha^{-1})\sqrt{2/(4\alpha+3)}$ as functions of $\alpha$. – Sangchul Lee Jun 24 '19 at 17:02
• Great! I understand now. Thank you very much – user326159 Jun 24 '19 at 17:06 | 2021-01-27 03:04:03 | {"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": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9443744421005249, "perplexity": 87.60482934924998}, "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-2021-04/segments/1610704820894.84/warc/CC-MAIN-20210127024104-20210127054104-00142.warc.gz"} |
https://ai.stackexchange.com/questions/10051/additive-attention-in-convolutional-networks | # Additive Attention in Convolutional Networks
Attention has been used widely in recurrent networks to weight feature representations learned by the model. This is not a trivial task since recurrent networks have a hidden state that captures sequence information. The hidden state can be fed into a small MLP that produces a context vector summarizing the salient features of the hidden state.
In the context of NLP, convolutional networks are not as straightforward. They have the notion of channels that are different feature representations of the input, but are channels the equivalent to hidden states? Particularly, this raises two questions for me:
• Why use attention in convolutional networks at all? Convolutions have shown to be adept feature detectors––for example, it is known that higher layers learn small features such as edges while lower layers learn more abstract representations. Would attention be used to sort through and weight these features?
• In practice, how would attention be applied to convolutional networks? The output of these networks is usually (batch, channels, input_size) (at least in PyTorch), so how would the attention operations in recurrent networks be applied to the output of convolutional networks?
References
Convolutional Sequence to Sequence Learning, Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, Yann N. Dauphin, 2017 | 2020-04-07 14:44:12 | {"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": 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": 0.7265631556510925, "perplexity": 1884.7350364156293}, "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/1585371799447.70/warc/CC-MAIN-20200407121105-20200407151605-00210.warc.gz"} |
http://mathrefresher.blogspot.com/2006/03/parallelograms.html | ## Thursday, March 23, 2006
### Parallelograms
In today's blog, I review some basic proofs from Euclid's Elements relating to Parallelograms. These extend the results on parallel lines and are needed for the proofs on similar triangles which I use in my discussion about sin and cosin.
The diagrams are taken from David Joyce's web site on Euclid's Elements which I highly recommend.
Definition: Parallelogram
A parallelogram is any four-sided shape where opposite sides are parallel to each other.
Lemma 1: In parallelograms, opposite sides and opposite angles are congruent.
Proof:
(1) Let ABCD be a parallelogram.
(2) AD is parallel to BC [Definition of Parallelogram]
(3) Alternate angles are congruent [see Lemma 2 here] gives us:
∠ DAC ≅ ∠ BCA
∠ DCA ≅ ∠ BAC
(4) Since line AC is congruent to itself, we can use the ASA lemma (see here) to conclude that triangle DAC ≅ triangle BCA
(5) But then corresponding sides are congruent which gives us (see here for definition of Congruent Triangles):
AB ≅ DC
(6) And opposite angles are congruent since:
∠ ADC ≅ ∠ CBA [see here for definition of Congruent Triangles]
We can assume that ∠ DAB ≅ ∠ DCB since we could apply the same arguments #1 thru #6 to the diagonal DB as well.
QED
Lemma 2: Parallelograms on the same base and in the same parallel are equal to each other.
Proof:
(1) Let ABCD and EBCF be parallograms that share the same base BC and are colinear on AF.
AD ≅ BC [Since they are opposite sides of ABCD from Lemma 1 above]
EF ≅ BC [Since they are opposite sides of EBCF from Lemma 1 above]
(3) AE ≅ DF since:
DF = EF + DE
AD ≅ EF (from the previous step)
(4) Now we can use Postulate 1 to conclude triangle ABE ≅ triangle DCF since:
AB ≅ DC [Since they are opposite sides of ABCD, from Lemma 1 above]
AE ≅ DF [Step #3]
∠ EAB ≅ ∠ FDC [since AB is parallel to DC and since Corresponding angles are congruent for parallel lines -- see here]
(5) We note that the trapezoid ABGD has the same area as EGCF since:
Both are formed from subtracting the area of DGE.
(6) But this implies that that the parallelogram ABCD is congruent to EBCF since they both formed by adding GBC to each trapezoid above.
QED
Lemma 3: Triangles with equal bases in the same parallels are equal to each other.
If ABC, DEF are triangles with BC ≅ EF; if AD is parallel to BF; and if C,E lie on BF; then, ABC ≅ DEF.
Proof:
(1) Let G be a point colinear with AD such that BG is parallel to AC.
(2) Let H be a point colinear with AD such that FH is parallel to DE.
(3) Then GACB and DHFE are parallelograms [Definition of parallelograms]
(4) Then the area of GACB is equal to the area of DHFE [See Lemma 2 above]
(5) The area of triangle ABC is half the area of GACB; and the area of triangle DEF is half the area of DHFE since:
(a) From Lemma 1 above, we know that each triangle such as DEF is congruent to its other half (in the case of DEF its other half is FHD)
(b) But if both triangles are congruent, then each triangle is (1/2) the total area, that is, the area of each triangle is half the area of each parallelogram.
(6) Since GACB ≅ DHFE (#4), we have (in terms of areas):
ABC = (1/2)GACB
DEF = (1/2)GACB
So we can see that ABC ≅ DEF.
QED
Corollary 3.1: If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.
Proof:
(1) Let ABCD be a parallelogram
(2) Triangle ABC ≅ triangle EBC from Lemma 3 above.
(3) And triangle ABC ≅ triangle CDA by Side-Angle-Side (see here) since:
(a) AD ≅ BC (By Lemma 1 above)
(b) DC ≅ AB (By Lemma 1 above)
(c) ∠ ADC ≅ ABC (By Lemma 1 above)
(4) Since triangle ABC ≅ triangle ECB ≅ triangle CDA is follows that parallelogram ABCD is double the area of triangle ECB.
QED
References
#### 1 comment :
karol , mer && art said...
thank you so much ! i am a geometry student and this helped so much {: | 2017-05-29 04:06:21 | {"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": 0.9156473875045776, "perplexity": 2803.9188877583892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612013.52/warc/CC-MAIN-20170529034049-20170529054049-00011.warc.gz"} |
https://soffer801.wordpress.com/2011/09/21/spans/ | # Spans
Sometimes mathematicians are lazy. We want to be rigorous and accurate, but sometimes we really just want to say “Come on! You know what I mean.” So we have a way to do that. Last time I gave an example of an “exotic” subspace of $\mathbb C$ (thought of as a vector space over $\mathbb R$. It was:
$\{a\cdot\pi + a\cdot i\mid a \in \mathbb R\}$
But, come on. That’s a lot to write. Can’t I just say “you know, the subspace that has the vector $\pi+i$? Strictly speaking, no. There is more than one such subspace, but we can do pretty much that. I can say the subspace spanned by $\pi +i$. You can think of this in two equivalent ways:
1. Think of it as the smallest subspace containing $\pi + i$.
2. Think of it as taking $\pi + i$ and then whatever else you need to make it a vector space.
Another example, the subspace of $\mathbb R^3$ spanned by $(2.5,3,0)$. What else must we have. We need exactly the points $a\cdot (2.5,3,0)$ for every $a$. That is,
$\mbox{span}((2.5,3,0))=\{(2.5a,3a,0)\mid a\in\mathbb R\}$.
We use the notation $\mbox{span}(\cdot)$.
But it gets more complicated when you have more points. What if I want to say “you know what I mean” about the points $p=(1,0,0)$ and $q=(0,0,1)$ in $\mathbb R^3$? That is, what is $\mbox{span}(\{p,q\})$?
Well, we need everything that looks like $(a,0,0)$ and everything that looks like $(0,0,b)$. But we need even more. We need every possible combination of these two. That is, to be a subspace, we need to be able to add any two vectors and get another one in the subspace. So it turns out, hopefully not surprisingly, that the subspace we want is
$\mbox{span}(\{p,q\})=\{(a,0,b)\mid a,b\in\mathbb R\}$.
You can imagine these sorts of things can get very complicated, and so having this notation makes talking about subspaces simpler.
One more thing to whet your appetite for what is to come. In $\mathbb R^3$, $\mbox{span}(\{(0,1,-1),(0,-\pi,\pi)\})=\mbox{span}((0,1,-1))$. Make sense? | 2018-01-22 23:35:18 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 22, "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": 0.8558225631713867, "perplexity": 182.97068142137002}, "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-2018-05/segments/1516084891546.92/warc/CC-MAIN-20180122232843-20180123012843-00400.warc.gz"} |
http://math.stackexchange.com/questions/276841/deciding-whether-a-formula-is-provable-with-a-fixed-number-of-universal-generali | # Deciding whether a formula is provable with a fixed number of universal generalizations
Let $y$ be Godel number of some formula which is derivable in some first-order logic. $F(y,n)$ is true if and only if the number of usage $Gen$(universal generalization) inference rule in any derivation of that formula is less or equal to $n$.
Please help to show that $F(y,n)$ is recursive function. A hint or link to such proofs will be highly appreciated.
-
What if the formula with Gödel number $y$ can't be derived? What if it can but (as will in fact always be the case) there are different proofs, with different numbers of generalizations? – Chris Eagle Jan 13 at 11:13
@ChrisEagle see updated version – Ashot Jan 13 at 16:54
@Ashot That still won't work: for any $n$, any theorem has a derivation with more than $n$ uses of $Gen$. Also $F$ is a relation, so trivially isn't a recursive function. And it can't be a recursive relation either as it isn't even defined for when $y$ numbers a non-theorem. – Peter Smith Jan 13 at 17:01
My guess is that the solution involves knowing enough proof theory to manage the proofs - perhaps they need to be in cut free form. But, as a start, I would suggest taking the specific set of inference rules you are working with and proving a special case: the set of $\phi$ that are provable with no uses of generalization is decidable. – Carl Mummert Jan 14 at 12:35
@Peter Smith: I read the question as: the relation $F(\phi,n)$ holds if there exists a proof of $\phi$ using $n$ or fewer applications of generalization. Prove that $F$ is decidable. If $\psi$ is not provable then $F(\psi,n)$ is simply false for all $n$. – Carl Mummert Jan 14 at 12:37 | 2013-05-20 22:06:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8531526923179626, "perplexity": 284.5401544836144}, "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/1368699273641/warc/CC-MAIN-20130516101433-00038-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://minecraft-forum.net/tag/decay/ | This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More]
Work against the clock as the world is slowly eaten away by void decay. The corruption will eat everything in it?s path leaving nothing but void in its wake so work fast if you want to stay ahead of it. Once you?ve progressed far enough you...[Read More]
This small mod makes minecraft leaves decay much faster. It should also be compatible with most modded leaves. If you don’t like the default speed you can also change the decay speed & fuzz (How “random” the decay will be) in the config...[Read More] | 2017-09-24 21:17:26 | {"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": 0.9020094871520996, "perplexity": 2002.3051606219312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690211.67/warc/CC-MAIN-20170924205308-20170924225308-00634.warc.gz"} |
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3200/2/a/bk/ | Properties
Label 3200.2.a.bk Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$
Related objects
Newspace parameters
Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.a (trivial)
Newform invariants
Self dual: yes Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$
$q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + (\beta - 1) q^{7} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (b - 1) * q^7 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} + (\beta - 1) q^{7} + (2 \beta + 3) q^{9} - 2 q^{11} + 2 \beta q^{13} + 2 \beta q^{17} + 2 \beta q^{19} + 4 q^{21} + ( - \beta - 7) q^{23} + (2 \beta + 10) q^{27} + 2 q^{29} + (2 \beta - 2) q^{31} + ( - 2 \beta - 2) q^{33} + ( - 4 \beta - 2) q^{37} + (2 \beta + 10) q^{39} + ( - 2 \beta + 8) q^{41} + ( - 3 \beta + 1) q^{43} + (\beta - 5) q^{47} + ( - 2 \beta - 1) q^{49} + (2 \beta + 10) q^{51} + ( - 2 \beta - 4) q^{53} + (2 \beta + 10) q^{57} + (2 \beta - 4) q^{59} + 6 q^{61} + (\beta + 7) q^{63} + (3 \beta - 1) q^{67} + ( - 8 \beta - 12) q^{69} + ( - 2 \beta - 2) q^{71} + 2 \beta q^{73} + ( - 2 \beta + 2) q^{77} + (4 \beta - 4) q^{79} + (6 \beta + 11) q^{81} + ( - 3 \beta - 3) q^{83} + (2 \beta + 2) q^{87} + (4 \beta - 6) q^{89} + ( - 2 \beta + 10) q^{91} + 8 q^{93} + ( - 2 \beta + 12) q^{97} + ( - 4 \beta - 6) q^{99}+O(q^{100})$$ q + (b + 1) * q^3 + (b - 1) * q^7 + (2*b + 3) * q^9 - 2 * q^11 + 2*b * q^13 + 2*b * q^17 + 2*b * q^19 + 4 * q^21 + (-b - 7) * q^23 + (2*b + 10) * q^27 + 2 * q^29 + (2*b - 2) * q^31 + (-2*b - 2) * q^33 + (-4*b - 2) * q^37 + (2*b + 10) * q^39 + (-2*b + 8) * q^41 + (-3*b + 1) * q^43 + (b - 5) * q^47 + (-2*b - 1) * q^49 + (2*b + 10) * q^51 + (-2*b - 4) * q^53 + (2*b + 10) * q^57 + (2*b - 4) * q^59 + 6 * q^61 + (b + 7) * q^63 + (3*b - 1) * q^67 + (-8*b - 12) * q^69 + (-2*b - 2) * q^71 + 2*b * q^73 + (-2*b + 2) * q^77 + (4*b - 4) * q^79 + (6*b + 11) * q^81 + (-3*b - 3) * q^83 + (2*b + 2) * q^87 + (4*b - 6) * q^89 + (-2*b + 10) * q^91 + 8 * q^93 + (-2*b + 12) * q^97 + (-4*b - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9 $$2 q + 2 q^{3} - 2 q^{7} + 6 q^{9} - 4 q^{11} + 8 q^{21} - 14 q^{23} + 20 q^{27} + 4 q^{29} - 4 q^{31} - 4 q^{33} - 4 q^{37} + 20 q^{39} + 16 q^{41} + 2 q^{43} - 10 q^{47} - 2 q^{49} + 20 q^{51} - 8 q^{53} + 20 q^{57} - 8 q^{59} + 12 q^{61} + 14 q^{63} - 2 q^{67} - 24 q^{69} - 4 q^{71} + 4 q^{77} - 8 q^{79} + 22 q^{81} - 6 q^{83} + 4 q^{87} - 12 q^{89} + 20 q^{91} + 16 q^{93} + 24 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^7 + 6 * q^9 - 4 * q^11 + 8 * q^21 - 14 * q^23 + 20 * q^27 + 4 * q^29 - 4 * q^31 - 4 * q^33 - 4 * q^37 + 20 * q^39 + 16 * q^41 + 2 * q^43 - 10 * q^47 - 2 * q^49 + 20 * q^51 - 8 * q^53 + 20 * q^57 - 8 * q^59 + 12 * q^61 + 14 * q^63 - 2 * q^67 - 24 * q^69 - 4 * q^71 + 4 * q^77 - 8 * q^79 + 22 * q^81 - 6 * q^83 + 4 * q^87 - 12 * q^89 + 20 * q^91 + 16 * q^93 + 24 * q^97 - 12 * q^99
Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
−0.618034 1.61803
0 −1.23607 0 0 0 −3.23607 0 −1.47214 0
1.2 0 3.23607 0 0 0 1.23607 0 7.47214 0
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
Atkin-Lehner signs
$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.a.bk 2
4.b odd 2 1 3200.2.a.bf 2
5.b even 2 1 640.2.a.j yes 2
5.c odd 4 2 3200.2.c.v 4
8.b even 2 1 3200.2.a.be 2
8.d odd 2 1 3200.2.a.bl 2
15.d odd 2 1 5760.2.a.cd 2
20.d odd 2 1 640.2.a.l yes 2
20.e even 4 2 3200.2.c.x 4
40.e odd 2 1 640.2.a.i 2
40.f even 2 1 640.2.a.k yes 2
40.i odd 4 2 3200.2.c.w 4
40.k even 4 2 3200.2.c.u 4
60.h even 2 1 5760.2.a.bw 2
80.k odd 4 2 1280.2.d.m 4
80.q even 4 2 1280.2.d.k 4
120.i odd 2 1 5760.2.a.ci 2
120.m even 2 1 5760.2.a.ch 2
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.a.i 2 40.e odd 2 1
640.2.a.j yes 2 5.b even 2 1
640.2.a.k yes 2 40.f even 2 1
640.2.a.l yes 2 20.d odd 2 1
1280.2.d.k 4 80.q even 4 2
1280.2.d.m 4 80.k odd 4 2
3200.2.a.be 2 8.b even 2 1
3200.2.a.bf 2 4.b odd 2 1
3200.2.a.bk 2 1.a even 1 1 trivial
3200.2.a.bl 2 8.d odd 2 1
3200.2.c.u 4 40.k even 4 2
3200.2.c.v 4 5.c odd 4 2
3200.2.c.w 4 40.i odd 4 2
3200.2.c.x 4 20.e even 4 2
5760.2.a.bw 2 60.h even 2 1
5760.2.a.cd 2 15.d odd 2 1
5760.2.a.ch 2 120.m even 2 1
5760.2.a.ci 2 120.i odd 2 1
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3200))$$:
$$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{7}^{2} + 2T_{7} - 4$$ T7^2 + 2*T7 - 4 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} - 20$$ T13^2 - 20
Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 20$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} + 14T + 44$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 4T - 16$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - 16T + 44$$
$43$ $$T^{2} - 2T - 44$$
$47$ $$T^{2} + 10T + 20$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 8T - 4$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} + 2T - 44$$
$71$ $$T^{2} + 4T - 16$$
$73$ $$T^{2} - 20$$
$79$ $$T^{2} + 8T - 64$$
$83$ $$T^{2} + 6T - 36$$
$89$ $$T^{2} + 12T - 44$$
$97$ $$T^{2} - 24T + 124$$ | 2022-06-25 11:42:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.992027759552002, "perplexity": 4144.495064100607}, "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-2022-27/segments/1656103034930.3/warc/CC-MAIN-20220625095705-20220625125705-00416.warc.gz"} |
https://tex.stackexchange.com/questions/352623/unicode-char-u200b | # Unicode char {U+200B}
When I try to cite more than two authors, I got this:
! Package inputenc Error: Unicode char (U+200B)
(inputenc) not set up for use with LaTeX.
\documentclass[a4paper,oneside,BCOR=10mm,12pt,titlepage]{scrreprt}
\usepackage[polutonikogreek,ngerman,english]{babel}
\usepackage[maxlevel=4,autostyle,german=guillemets]{csquotes}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[style=mla]{biblatex}
\DeclareMultiCiteCommand{\parencites}[\mkbibparens]{\parencite}{\multicitedelim}
\begin{document}
\parencites[cf.][]{key1}{key2}{key3}
\end{document}
• unrelated to the error you are missing a { before key3} – David Carlisle Feb 7 '17 at 23:49
## 2 Answers
U+200B is zero width space which you have here:
elim}
from the end of this line
\DeclareMultiCiteCommand{\parencites}[\mkbibparens]{\parencite}{\multicitedelim}
U+0065 LATIN SMALL LETTER E e
U+006c LATIN SMALL LETTER L l
U+0069 LATIN SMALL LETTER I i
U+006d LATIN SMALL LETTER M m
U+200c ZERO WIDTH NON-JOINER ‌
U+200b ZERO WIDTH SPACE ​
U+007d RIGHT CURLY BRACKET } } \rbrace
delete that line and re-type without the invisible control characters between the m and }
• Nice! Thank you so much! Now it works perfectly! – user124801 Feb 7 '17 at 23:53
• So this worked perfectly. Thank you. I deleted a whole line and retyped it and it worked. How did David generate the output in the bottom box with the character codes? I tried blindly finding the ZWS blindly but didn't manage until I deleted the whole line. – Cyrille Dec 19 '17 at 22:18
• @Cyrille I cut the text where I suspected the character into w3c.github.io/xml-entities/unicode-names.html and hit "convert" (a javacscript page that I wrote some time ago) – David Carlisle Dec 19 '17 at 22:32
• Thanks @DavidCarlisle that will be a helpful resource in the future. I have gone ahead and added the link in your answer. – Cyrille Dec 21 '17 at 9:40
• I just had same problem and fix was same. I don't understand how the invisible character gets into a raw text document that I'm editing with Emacs. Caused by cut-and-paste from some other format? I found out later I could view the TeX file in Texworks and it did show the U+200b literally on screen, so I could see it was there. It did not show in Emacs, but I could move cursor left and right and you can tell there is something invisible because column number changes as cursor seems to stay on same spot. – pauljohn32 May 26 '18 at 18:26
This is one of the top hits for U+200B and LaTeX, so I’ll post solutions here.
Take the following example:
\tracinglostchars=2
\documentclass{article}
\pagestyle{empty}
\begin{document}fl fl fl\end{document}
In LuaLaTeX, it compiles to:
The first fl has no ligature because I inserted U+200B, a zero-width space. The second has no ligature because I inserted U+200C, a zero-width non-joiner. These might have been in the original source you copied from intentionally: a zero-width space could mean a potential line break, such as after a slash, and a zero-width non-joiner disables a ligature. For example, the fi in Elfin or the fl in Halfling are (according to pedants like me) not supposed to be ligated, since they belong to different pieces of a compound word. Almost no one bothers to actually do that, but it is much more common in some other languages.
If you try to compile it in PDFLaTeX, you will get the error message that brought you here:
! Package inputenc Error: Unicode character (U+200B)
(inputenc) not set up for use with LaTeX.
There are several ways to fix it.
### Clean Your Source by Hand
This is what most people on this site recommend. Your editor might have a way to display special characters so you can delete them all. But really, isn’t this a job for a computer?
### Clean your Source in your Editor
This is harder with an invisible, zero-width character, but you might be able to copy zero-width space from a character map, open up your search-and-replace dialogue box and paste the character into the search field. You can then replace it with something like ZWS or {\hskip 0pt}.
### Clean Your Source with Perl
The following one-line Perl script will create a new source file with all zero-width spaces removed:
perl -CSD -pe "s/\N{U+200B}//gu" < U200B.tex > noU200B.tex
If it’s easier to remember, you could also write this as
perl -CSD -pe "s/\N{ZERO WIDTH SPACE}//gu" < U200B.tex > noU200B.tex
The -CSD option selects UTF-8 unconditionally, even if you don’t have UTF-8 as your default locale. The -pe option runs the given Perl script on the input file and prints to the output file. The s command does substitution, the \N{...} is a regular expression matching zero-width space, the empty field between // means replace with nothing, and gu means replace all instances globally in the unicode string. Then, the < and > operators select the input and output files.
Either of these produce a file that compiles to:
It’s also possible to automatically remove all characters outside a given subset. The script
perl -CSD -pe "s/[^\p{Word}\p{Punct}\p{Symbol}\p{Mark}\p{PerlSpace}]//gu"
allows only the following: Unicode “word” characters, punctuation, symbols, accents and a few kinds of spaces. It erases most invisible characters. An even more restrictive version would be
perl -CSD -pe "s/[^\p{ASCII}]//gu"
This cleans out all characters but for the ASCII originally allowed in TeX (including “ instead of double backtick).
And yes, we could replace zero-width space by something instead of nothing. The script
perl -CSD -pe "s/\N{ZERO WIDTH SPACE}/{\\\\hskip 0pt}/gu; s/\N{ZERO WIDTH NON-JOINER}/{}/gu"
given the above MWE as input, produces the following output:
\tracinglostchars=2
\documentclass{article}
\pagestyle{empty}
\begin{document}f{\hskip 0pt}l f{}l fl\end{document}
### Teach LaTeX to Understand Zero-Width Space
If the problem is that U+200B is “not set up for use with LaTeX,” but it’s equivalent to a TeX command—\hskip 0pt or \hspace{0pt} are zero-width spaces that prevent ligatures and enable a potential line break—we can set the character up to use that command.
\tracinglostchars=2
\documentclass{article}
\usepackage{iftex}
\pagestyle{empty}
\ifTUTeX
\usepackage{fontspec}
\else
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} % The default since 2018
\DeclareUnicodeCharacter{200B}{{\hskip 0pt}}
\fi
\begin{document}fl fl fl\end{document}
Although the \DeclareUnicodeCharacter command is in inputenc, the LaTeX kernel has loaded it by default since 2018. So, we could have skipped declaring it. | 2020-09-27 04:27:32 | {"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": 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": 0.8370505571365356, "perplexity": 4129.20035667226}, "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-2020-40/segments/1600400250241.72/warc/CC-MAIN-20200927023329-20200927053329-00636.warc.gz"} |
https://cbyte.net/loan-calculator | # Loan Calculator
## Loan & Interest Calculator
Loan Calculator
It is possible to cope with bank loans with unlimited needs such as going out for a holiday, housing, having a car, having a car, establishing a business. With the help of the bank you can get what you need with your credit. Differences in credits affect loan calculation tools. The requirement loan is a type of credit that banks provide when they provide needs such as marriage, purchasing goods, vacation etc.. The most attractive feature is that you get collective currency and is a monthly payment. You can calculate the loan based on the interest rate. You can get detailed information about the sample payment schedule, installment amount, interest you will pay and other amounts according to the credit you have calculated. However, there are a number of points to note when applying for any individual or institutional credits.
Unpaid loan debts have a bad effect on your credit score. Especially when you enter the black list, you can be deprived of many services from credit cards to shopping credits for about 5 years. Therefore, it will be useful calculate loan before you get the loan.
How can you calculate loan
Suppose you get a $24 loan with 12 months maturity at 1% interest per month. Your monthly payments are calculated as$ 8.88. One month after you receive the credit, you will be owed $1 of 24$ interest.
The 'Interest' value to be used in the form is the interest rate, not the percentage interest. For example, if the monthly interest rate is 1%, it should be 0,01 in form and 0,0079 in 0,79%. | 2019-04-18 11:17:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42716366052627563, "perplexity": 1542.887386248145}, "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-2019-18/segments/1555578517558.8/warc/CC-MAIN-20190418101243-20190418123243-00154.warc.gz"} |
https://math.stackexchange.com/questions/1858246/vector-bundles-and-de-rham-cohomology | # Vector bundles and de Rham cohomology
So $M$ is a compact manifold and I am asked to either prove the following statement or give a counterexample:
if $\pi: E \rightarrow M$ is a vector bundle, then $H^2(E) \simeq H^2(M)$.
I know the definition of a vector bundle, and know how the de Rham cohomology is defined, but that's all I have. After some research, I found that it might have to do with 'homotopy equivalence' but I don't understand it. Can someone explain this without going too deep in technical stuff? Thanks in advance!
• Think about shrinking $\mathbf R^n$ down to a point. Now do this simultaneously for all the fibres of $\pi$: this shrinks $E$ down to $M$. (Formally we say $\pi$ is a "homotopy equivalence".) "Homotopy invariance of cohomology" means that this operation does not change the cohomology. – Nefertiti Jul 13 '16 at 14:50
• Indeed a vector bundle is homotopy equivalent to the base space, and this would show that all cohomology groups are isomorphic, not just $H^2$. Without using that... I'm not sure where to start, but given that you're looking only at $H^2$, maybe there is some more context? – Callus - Reinstate Monica Jul 13 '16 at 14:57
• It's for an assignment on symplectic geometry. The previous question was to show that if $M$ admits a symplectic structure, that $H^2(M)$ is non-empty, which I showed by using the fact that on a symplectic manifold, there is a canonical volume form. This holds for all $H^{2n}(M)$, so in particular for $n=1$. But for this question, I can't assume that $M$ admits a symplectic structure. – user353840 Jul 13 '16 at 15:02
This is an elaboration of Nefertiti's comment. One of the most basic (and useful) properties of de Rham cohomology (and, indeed, any cohomology theory) is the homotopy axiom: two homotopic maps induce the same map in cohomology (see, for instance, Bott & Tu, $\S$4). Intuitively: small perturbations don't affect cohomology! Here are some relevant definitions:
Let $X,Y$ be spaces, $I$ the unit interval. By a "map" I mean a continuous map. A homotopy between two maps $f,g:X \to Y$ is a map $F:X \times I \to Y$ such that $F(x,0) = f(x)$ and $F(x,1) = g(x)$. If we think of $I$ as a time scale, then $F$ is a continuous transition in time between $f$ and $g$. We say that $f$ is homotopic to $g$ and write $f \simeq g$.
A homotopy equivalence $f:X \to Y$ is a map with a homotopy inverse $g:Y \to X$: this means that $g \circ f$ is homotopic to the identity on $X$ and $f \circ g$ is homotopic to the identity on $Y$.
Proposition: The de Rham cohomology satisfies the homotopy axiom: $f \simeq g$ implies that the induced maps $f^* = g^*$ coincide.
Proof in Bott & Tu, Corollary 4.1.2.
Corollary: Two homotopy equivalent spaces have isomorphic cohomology.
Proof. Let $f:X \to Y$ be a homotopy equivalence with homotopy inverse $g:Y \to X$. Then by functoriality of de Rham cohomology, $f^* \circ g^* = (g \circ f)^* = 1_X^* = 1_{H^*(X)}$ and $g^* \circ f^* = (f \circ g)^* = 1_Y^* = 1_{H^*(Y)}$. $\square$
A deformation retract of a space $X$ onto a subspace $A$ is a homotopy $F:X \times I \to X$ between the identity on $X$ and a retraction $r:X \to X$ onto $A$. This means that $F(x,0) = x$, $F(x,1) \in A$, and $F(a,1) = a$ for all $x \in X$, $a \in A$. A deformation retract provides a homotopy equivalence between $X$ and $A$, where the retraction with codomain restricted to $A$ is homotopy inverse to the inclusion of $A$ into $X$.
With this in hand, all that we need to show is that the total space of a vector bundle $\pi:E \to M$ deformation retracts onto the $0$-section of the base space, which is homeomorphic to the base space itself. This is done concisely by Andrew D. Hwang here: Total space of vector bundle deformation retracts onto 0-section of base space so I see no need to repeat the argument.
• Thanks for this! I was able to figure it out. – user353840 Jul 16 '16 at 21:07 | 2020-01-20 03:57:37 | {"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": 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": 0.9355751276016235, "perplexity": 106.61621715250101}, "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/1579250597230.18/warc/CC-MAIN-20200120023523-20200120051523-00375.warc.gz"} |
http://quant.stackexchange.com/questions/14339/simulate-non-stationary-time-series-with-cointegration | # Simulate non-stationary time series with cointegration
how can I simulate/generate two non-stationary time series (with unit root) so that they can be also cointegrated (using R or Matlab).
-
– user508 Aug 10 '14 at 19:18
Two cointegrated series contain a single unit root. Each series can be formulated as the sum of a common unit root plus a stationary component. Most textbooks covering cointegration will cover such formulations - see Hamilton's (1994) discussion of Phillips' "triangular representation" of a cointegrated vector, for example.
Simulating is likely to be easy (depending on other features you may need, of course). For example, simulate one non-stationary series, and construct a second series as the sum of the first and a stationary variable. Viola! Cointegrated.
- | 2015-07-31 21:31:34 | {"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": 0.9219300150871277, "perplexity": 2231.555672469912}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988312.76/warc/CC-MAIN-20150728002308-00203-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://libros.duhnnae.com/2017/jul6/150067339471-The-yields-of-light-meson-resonances-in-neutrinonuclear-interactions-at-E-nu-10-GeV-High-Energy-Physics-Experiment.php | # The yields of light meson resonances in neutrinonuclear interactions at <E nu> = 10 GeV - High Energy Physics - Experiment
The yields of light meson resonances in neutrinonuclear interactions at <E nu> = 10 GeV - High Energy Physics - Experiment - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.
Abstract: The total yields of the all well-established light mesonic resonances up tothe $\phi$1020 meson are estimated in neutrinonuclear interactions at < E nu> = 10 GeV, using the data obtained with SKAT bubble chamber. For someresonances, the yields in the forward and backward hemispheres in the hadronicc.m.s. are also extracted. From the comparison of the obtained and availablehigher-energy data, an indication is obtained that the resonance yields risealmost linearly as a function of the mean mass < W > of the neutrinoproducedhadronic system. The fractions of pions originating from the light resonancedecays are inferred.
Autor: N.M.Agababyan, V.V.Ammosov, M.Atayan, N.Grigoryan, H.Gulkanyan, A.A.Ivanilov, Zh.Karamyan, V.A.Korotkov
Fuente: https://arxiv.org/ | 2017-09-25 02:49:07 | {"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": 0.8254052996635437, "perplexity": 13337.92164838067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690307.45/warc/CC-MAIN-20170925021633-20170925041633-00603.warc.gz"} |
https://doc.sagemath.org/html/en/reference/categories/sage/categories/semisimple_algebras.html | # Semisimple Algebras¶
class sage.categories.semisimple_algebras.SemisimpleAlgebras(base, name=None)
The category of semisimple algebras over a given base ring.
EXAMPLES:
sage: from sage.categories.semisimple_algebras import SemisimpleAlgebras
sage: C = SemisimpleAlgebras(QQ); C
Category of semisimple algebras over Rational Field
This category is best constructed as:
sage: D = Algebras(QQ).Semisimple(); D
Category of semisimple algebras over Rational Field
sage: D is C
True
sage: C.super_categories()
[Category of algebras over Rational Field]
Typically, finite group algebras are semisimple:
sage: DihedralGroup(5).algebra(QQ) in SemisimpleAlgebras
True
Unless the characteristic of the field divides the order of the group:
sage: DihedralGroup(5).algebra(IntegerModRing(5)) in SemisimpleAlgebras
False
sage: DihedralGroup(5).algebra(IntegerModRing(7)) in SemisimpleAlgebras
True
class FiniteDimensional(base_category)
WithBasis
class ParentMethods
radical_basis(**keywords)
Return a basis of the Jacobson radical of this algebra.
• keywords – for compatibility; ignored.
OUTPUT: the empty list since this algebra is semisimple.
EXAMPLES:
sage: A = SymmetricGroup(4).algebra(QQ)
super_categories()
sage: Algebras(QQ).Semisimple().super_categories() | 2019-06-18 15:03:29 | {"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": 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": 0.31953421235084534, "perplexity": 10698.639532484827}, "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-2019-26/segments/1560627998755.95/warc/CC-MAIN-20190618143417-20190618165417-00477.warc.gz"} |
https://www.numerade.com/questions/finding-a-particular-solution-in-exercises-37-44-find-the-particular-solution-of-the-differential--7/ | Meet students taking the same courses as you are!Join a Numerade study group on Discord
# Finding a Particular Solution In Exercises $37-44,$ find the particular solution of the differential equation that satisfies the initial condition(s).$f^{\prime \prime}(x)=x^{-3 / 2}, f^{\prime}(4)=2, f(0)=0$
## $-4 \sqrt{x}+3 x$
Integrals
Integration
### Discussion
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##### Heather Z.
Oregon State University
##### Kristen K.
University of Michigan - Ann Arbor
##### Michael J.
Idaho State University
Lectures
Join Bootcamp
### Video Transcript
for real aren't getting double crime. Thanks equals negative X to the neighboring. Three have and Rome four equals team and of zero hero. They justify the admission condition. For the first thing you do is take the logo of a double crime. And then from eggs He was the anti roll, huh, ext? Negative. Three halves. Yeah, Add one. My, I have X, a negative one. Bye bye, Bill. Maybe the U So negative too. It's the 1/2 the 1/2 makes it a square root. The negative puts it in the denominator. So every time we'll see. Now we know that when prime when exits for that crime is to something. Two. I have negative too. Oh, this here water. Let's see. And that gives me two over. Negative one to be one negative one plus c four c b three. So they are going to take The integral eggs will now eat will be in control. Um oh, negative to x with maybe 1/2 hour. Yes. Give me negative too. Add one. That's gonna be a positive 1/2 and multiplied by the reciprocal and plus three. I really have plus three x to the zeros. So I'm gonna have see And then we have to find out what See? So we know that every Becks zero when x zero, Why is the room so I have negative were X to the 1/2 plus three x Last thing equals my f of X. So when X equals zero, everything's going to kids a lot and see with them after equals zero. So my function my ex, it's just people negative core that's a plus.
University of Houston
#### Topics
Integrals
Integration
##### Heather Z.
Oregon State University
##### Kristen K.
University of Michigan - Ann Arbor
##### Michael J.
Idaho State University
Lectures
Join Bootcamp | 2021-03-08 09:08:44 | {"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": 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": 0.53668612241745, "perplexity": 3543.774372032827}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178383355.93/warc/CC-MAIN-20210308082315-20210308112315-00291.warc.gz"} |
https://www.tutorialspoint.com/php-string-cast-vs-strval-function-which-one-should-i-use | # PHP string cast vs strval function, which one should I use?
PHPServer Side ProgrammingProgramming
A value can be converted into a string with the help of (string) cast or the strval() function.
The strval() function is a function call whereas (string) cast is an internal type casting method.
Unless there is some specific dataset or use case, both of these can be used interchangeably.
This is because PHP uses automatic type conversion, due to which a variable's type is determined based on the context in which it is used.
The strval($var) function returns the string value of$var whereas the (string)$var explicitly converts the "type" of$var during the process of evaluation.
The \$var can be any scalar type or an object that implements the __toString method.
The strval() can't be used on arrays or on objects that do not implement this __toString method.
In general, (string)cast is relatively quicker.
Published on 06-Apr-2020 08:55:30 | 2021-12-04 11:50:32 | {"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": 0.23644199967384338, "perplexity": 2727.3774112664455}, "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/1637964362969.51/warc/CC-MAIN-20211204094103-20211204124103-00615.warc.gz"} |
https://blogs.mathworks.com/steve/2016/03/21/matlab-image-display-autoscaling-values-with-imshow/?from=jp | # MATLAB image display – autoscaling values with imshow
Last week I talked about displaying gray-scale and binary images. In that post, I showed how to control the grayscale range. For example, the call imshow(I,[0.4 0.6]) displays the matrix I as a gray-scale image so that the value 0.4 gets displayed as black, and the value 0.6 gets displayed as white.
Brett, a MathWorks application engineer and frequent File Exchange contributor, correctly pointed out that I neglected to discuss a common and useful syntax: imshow(I,[]). This syntax automatically determines the grayscale range of the display based on the minimum and maximum values of I. It is equivalent to imshow(I,[min(I(:)) max(I(:))]).
Here's an example. I have a matrix of terrain elevation values, in meters, near Mt. Monadnock in New Hampshire, USA. The peak of Mt. Monadnock is about 960 meters.
load mt_monadnock
The matrix is stored in the MAT-file using the variable name Zc.
whos Zc
Name Size Bytes Class Attributes
Zc 3042x3042 37015056 single
min(Zc(:))
ans =
141.5285
max(Zc(:))
ans =
963.1366
To display these elevation values as a gray-scale image with autoscaling, just call imshow(Zc,[]):
imshow(Zc,[])
Warning: Image is too big to fit on screen; displaying at 33%
And, while we're here, I'll point out that you can install any colormap you like once a gray-scale image is displayed. Want some color?
colormap(parula)
Published with MATLAB® R2016a
|
### コメント
コメントを残すには、ここ をクリックして MathWorks アカウントにサインインするか新しい MathWorks アカウントを作成します。 | 2023-02-05 23:15:26 | {"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": 0.39162328839302063, "perplexity": 5404.297617243287}, "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/1674764500294.64/warc/CC-MAIN-20230205224620-20230206014620-00175.warc.gz"} |
https://gamedev.stackexchange.com/questions/105004/is-there-a-formula-to-know-how-many-enemies-i-can-place-before-jumping-character | # is there a formula to know how many enemies i can place before jumping character?
My game is very similar to the hidden game in chromes web browser. I have a character that needs to jump over incoming enemies. Is there any way i can know when to send an incoming enemy? Some of the factors to take in to account i guess are the speed of enemies, height of jump, speed of jump, and dimensions of bitmaps (that represent character, and enemies).
Thanks
• I'm guessing you're talking about an infinite runner? – jzx Aug 3 '15 at 8:42
minReactionTime = 0.5; //Give the player half a second to react | 2021-07-29 14:25:44 | {"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": 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": 0.42649421095848083, "perplexity": 1103.3521720075596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153860.57/warc/CC-MAIN-20210729140649-20210729170649-00485.warc.gz"} |
https://www.mersenneforum.org/showthread.php?s=351efcf0b3a2e231cad4f6c2715eb8c8&t=24158 | mersenneforum.org Sale!! 1/2 Price Fermat Test
Register FAQ Search Today's Posts Mark Forums Read
2019-03-09, 21:59 #1 a1call "Rashid Naimi" Oct 2015 Out of my Body 181810 Posts Sale!! 1/2 Price Fermat Test Well it doesn't really cost 1/2 the core time but the exponent is 1/2 as large: Mod(b,q)^(q-1) == 1 is equivalent to : Mod(b,q)^((q-1)/2) == 1 || Mod(b,q)^((q-1)/2) == q-1 for all positive odd q.
2019-03-10, 01:46 #2
paulunderwood
Sep 2002
Database er0rr
7·467 Posts
Quote:
Originally Posted by a1call Well it doesn't really cost 1/2 the core time but the exponent is 1/2 as large: Mod(b,q)^(q-1) == 1 is equivalent to : Mod(b,q)^((q-1)/2) == 1 || Mod(b,q)^((q-1)/2) == q-1 for all positive odd q.
Euler knew a better definition some 300 years ago
2019-03-10, 01:59 #3 a1call "Rashid Naimi" Oct 2015 Out of my Body 2·32·101 Posts Yes, that guy was quite something. I think the only reason he is not considered the most intelligent person ever, is because most people do not understand what he says. Not so With the Einstein's and the Newton's. Incidentally thank you for the link but I can't figure anything out of that. ETA: OK, so I had to dig in all the way to "Legendre symbol" to figure it out. Thanks again for the reference. At the level of i=2, Mod(b,q)^((q-1)/(2^i)) the test returns true for all primes and some pseudoprimes. For values of i>2 the test returns true for a subset of primes given 2^i | q-1. the test's performance can be improved as: (Mod(b,q)^/((q-1)/(2^i)))^(2^j) for j
2019-03-10, 02:16 #4
paulunderwood
Sep 2002
Database er0rr
7·467 Posts
Quote:
Originally Posted by a1call Yes, that guy was quite something. I think the only reason he is not considered the most intelligent person ever, is because most people do not understand what he says. Not so With the Einstein's and the Newton's. Incidentally thank you for the link but I can't figure anything out of that.
Pari/GP has the function kronecker() which covers jacobi. So you would write:
Code:
Mod(b,q)^((q-1)/2) == kronecker(b,q)
Note also that if the number is congruent to 1 mod 4 and the root is 1, you can take another square root and the answer should be +1 or -1. If it is 1 and the number is congruent to 1 mod 8 then you can take yet another square root and so on -- this "strong test" is the basis of the Miller-Rabin (M-R) test.
2019-03-10, 02:35 #5 a1call "Rashid Naimi" Oct 2015 Out of my Body 111000110102 Posts Thank you very much Paul for your great insights. Hopefully a Pari-GP code will make my gibberish more clear: Code: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\CQL-100-A - Low cost PRP test by Rashid Naimi q=2 theFlag=0 i=26 theCheckDepth = 13; while (!theFlag,{ q=nextprime(q+1); d=2^i*q+1; \\d=2^38*q+1; m= (Mod (2,d)^q); theRealFlag =0; for(j=1,theCheckDepth , if( lift(m^j)==1 || lift(m^j)==d-1, theRealFlag =1; print("**** Found a (Probable) Prime at Depth: ",j); print(theRealFlag ," > ",d); next(1); ); ); if ( theRealFlag , print (q); print ("**** 2^",i,"*",q,"+1"); print (d); print (#digits (d)," dd"); print ( isprime (d)); \\next (19); ); }) ## Code: **** Found a (Probable) Prime at Depth: 4 1 > 282649554904416257 **** Found a (Probable) Prime at Depth: 8 1 > 282649554904416257 **** Found a (Probable) Prime at Depth: 12 1 > 282649554904416257 4211806579 **** 2^26*4211806579+1 282649554904416257 18 dd 1 ETA Don't know of any case where "lift(m^j)==1 " is applicable. Last fiddled with by a1call on 2019-03-10 at 02:40
2019-03-10, 03:39 #6 danaj "Dana Jacobsen" Feb 2011 Bangkok, TH 2·11·41 Posts For the case of base 2, you can go even one step further. It is even more discriminating than the Fermat and Euler tests, though less than a full strong pseudoprime (i.e. Miller-Rabin) test. To my knowledge this was first noted by Colin Plumb in 1995. It is measurably faster than the other two tests over a variety of input, though nothing crazy (it reduces the exponent by either 2 or 4 and because of base 2 we don't need to calculate the Jacobi / Kronecker symbol). See, e.g. bnlib 1.1 and the discussion in this Mersennseforum thread.
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Sun Jul 5 08:30:20 UTC 2020 up 102 days, 6:03, 1 user, load averages: 1.03, 1.19, 1.17 | 2020-07-05 08:30:20 | {"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": 0.37857452034950256, "perplexity": 4747.625985181276}, "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/1593655887046.62/warc/CC-MAIN-20200705055259-20200705085259-00399.warc.gz"} |
https://www.albert.io/ie/ap-computer-science-a/employee-cell-phone-variable | Free Version
Easy
# Employee Cell Phone Variable
APCSA-E1ZAGE
You are asked to create a class to define an Employee. One piece of information that needs to be tracked for each employee is his/her primary cell phone number (including the area code).
Of the choices given below, which would be the BEST option for declaring a variable to store this information?
A
private String cellPhone;
B
private int cellPhone;
C
private double cellPhone;
D
private String[] cellPhone;
E
private int[] cellPhone; | 2017-01-17 07:02:22 | {"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": 0.1731109321117401, "perplexity": 2961.3162592482727}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279489.14/warc/CC-MAIN-20170116095119-00015-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/383939/previous-section-name | # previous section name
Is there a way to get the title [short] and the number of the previous and the next sections?
\sectiontitle % where is it defined I wonder
\thesection
So, I can calculate, for example, the counter of the previous section:
\the\numexpr\thesection-1\relax
What I'm missing is something like
\prevsectiontitle, \nextsectiontitle ?
• This deliberately depends on the document class and requires hacking into the section commands – user31729 Jul 30 '17 at 6:28
• You should not rely on the output of \thesection giving the pure number. Rather use \the\numexpr\value{section}-1\relax for example – user31729 Jul 30 '17 at 8:38
A version with saving automatic generated labels containing the information on section titles to the .aux file and extracting the previous, the current and the next section title.
The counter sectioncntr is never reset automatically, it continues to count the number of sections in the document and at each \refstepcounter of this counter a \zlabel is set, which stores the sectiontitle property which has been defined before with \zref@newprop.
The short names \Xtitle where X stands for previous, current or next do not insert links to the section header, whereas the long names \Xsectiontitle insert links. See the example with the page footer for an application.
\documentclass{article}
\usepackage[user,hyperref]{zref}
\usepackage{fancyhdr}
\usepackage{blindtext}
\fancypagestyle{plain}{%
\fancyhf{}
\fancyfoot[L]{\previoussectiontitle}
\fancyfoot[C]{\currenttitle}
\fancyfoot[R]{\nextsectiontitle}
}
\usepackage{xpatch}
\makeatletter
\zref@newprop{sectiontitle}[unknown]{\@currentsectiontitle}
\newcounter{sectioncntr}
\newcommand{\previoustitle}{%
\zref@ifrefundefined{sectiontitle:\the\numexpr\c@section-1}{}{%
\zref@extract{sectiontitle:\the\numexpr\c@section-1}{sectiontitle}%
}%
}
\newcommand{\currenttitle}{%
\zref@ifrefundefined{sectiontitle:\number\value{section}}{}{%
\zref@extract{sectiontitle:\number\value{section}}{sectiontitle}%
}%
}
\newcommand{\nexttitle}{%
\zref@ifrefundefined{sectiontitle:\the\numexpr\c@section+1}{}{%
\zref@extract{sectiontitle:\the\numexpr\c@section+1}{sectiontitle}%
}%
}
\newcommand{\previoussectiontitle}{%
\zref@ifrefundefined{sectiontitle:\the\numexpr\c@section-1}{}{%
}%
}
\newcommand{\currentsectiontitle}{%
\zref@ifrefundefined{sectiontitle:\number\value{section}}{}{%
}%
}
\newcommand{\nextsectiontitle}{%
\zref@ifrefundefined{sectiontitle:\the\numexpr\c@section+1}{}{%
}%
}
\xpatchcmd{\@sect}{%
\@tempskipa #5\relax
}{%
\def\@currentsectiontitle{#7}
\ifnum0=\pdfstrcmp{#1}{section}
\refstepcounter{sectioncntr}\zlabel{sectiontitle:\the\c@section}%
\fi
\@tempskipa #5\relax
}{%
\typeout{Patch Success}
}{%
\typeout{Patch Failure}
}
\makeatother
\usepackage{hyperref}
\pagestyle{plain}
\begin{document}
\section{Foo}
\subsection{Foo subsection}
\blindtext[20]
\section{Foobar}
\blindtext[20]
\section{Next Foo}
\blindtext[20]
\end{document}
• I assumed an article - like class here. book will be a little bit more complicated in order to prevent sections from previous or next chapters to be shown as well – user31729 Jul 30 '17 at 7:00
• It is designed for \section, not for \section* however – user31729 Jul 30 '17 at 7:12
• Thank you, Christian. I'm using a book-like class (with a single chapter per document though). Your \xpatchcmd{\@sect} fails. Also, eventually I'm redefining \section for auto-labeling as follows \LetLtxMacro{\oldsection}{\section} \RenewDocumentCommand \section{sm} {\IfBooleanTF{#1} {\oldsection*{#2}\label{sec:#2}} {\oldsection{#2}\label{sec:#2}} } – user612313 Jul 31 '17 at 8:41
• I'm using extbook – user612313 Jul 31 '17 at 8:53
• @user612313: Yes, that's why it is important to provide a complete example and not only wishes. Patching must fail with xparse macros! You should update your question with the minimal working document – user31729 Jul 31 '17 at 9:17 | 2019-10-17 06:33:05 | {"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": 0.8354731798171997, "perplexity": 2761.4626723552783}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986672723.50/warc/CC-MAIN-20191017045957-20191017073457-00261.warc.gz"} |
https://syn.tools/syn/SDDs/0006-commodore-configuration-catalog-compilation.html | # SDD 0006 - Commodore Configuration Catalog Compilation
Author Simon Gerber Owner Simon Gerber Reviewers (SIG) Date 2019-10-18 Status implemented
Summary The catalog compilation engine collects configuration information from different global and customer-specific Git repositories, collates this data into an inventory for Kapitan and compiles the set of manifests (the "catalog") for a SYN-managed cluster.
## Motivation
We want to be able to define defaults for the SYN platform configuration in a hierarchical fashion and compile a catalog for a specific cluster from a hierarchical set of input config repositories.
### Terminology
• Engine: the catalog compilation engine described in this document
• Inventory: the Kapitan inventory used by the engine
• Component: an Engine component consisting of Kapitan component templates, a Kapitan component class, and engine postprocessing filters
• Catalog: the set of rendered templates described by the inventory
### Goals
• The engine is responsible for collating parts of a Kapitan inventory from different Git repositories
• The engine gets some information to help with this task directly from the Lieutenant API
• The engine is responsible for fetching all the Kapitan components that make up the SYN platform from Git repositories. The list of components is generated from the inventory
• The engine is responsible for ensuring that the inventory is a valid Kapitan inventory
• The engine runs Kapitan to produce a set of YAML files that can be applied to a cluster (the "catalog")
• The engine commits the compiled catalog to the cluster catalog repository on a branch
• The engine pushes the branch to the upstream catalog repository and creates a merge request
• It must be possible to override versions for SYN platform Kapitan components for a cluster
• We want to be able to define default versions for SYN platform Kapitan components
### Non-Goals
• The exact way of how and where the engine runs is outside the scope of this SDD
• Currently, supporting configuration sources which aren’t a Git repository is outside the scope of the compilation engine
## Design Proposal
The high level idea is to build a tool around Kapitan which can collect fragments of a Kapitan inventory from different Git repos and organize them in such a way that Kapitan can compile a catalog for the cluster.
### Background
Kapitan is a tool to render templates in different templating languages (currently Jsonnet, and Jinja2 are fully supported, Kadet and Helm are in alpha status. Kadet is a python-based, Jinja-inspired templating language for YAML). Kapitan works with components and an inventory. Kapitan components are collections of templates which should be rendered. The Kapitan inventory is similar to Puppet’s hieradata and consists of "classes" and "targets." A Kapitan target describes a unit which will be compiled by Kapitan (for example all the manifests required to deploy an application on a cluster). Each Kapitan component also requires a "component class" which describes how the component should be rendered. This includes listing which inputs to render, and where to save the outputs, the input format (on of the supported templating languages), and the output format (for example YAML or JSON). Additionally, Kapitan itself can download dependencies of a component, for example the helm chart if using helm as the templating language.
### Overall architecture for the compilation engine
For the catalog compilation engine, we define all of the platform configuration in a single Kapitan target named "cluster," which is constructed from data provided by the Lieutenant API. The target pulls in a generic inventory class for global configuration (named `global.common`), cloud-specific configuration (for example class `global.cloud.aws`) and cluster distribution specific configuration (for example `global.distribution.rancher`). Additionally the target also pulls in a class for the cluster itself (for example class `thecustomer.thecluster`). Finally, the compilation engine traverses the inventory to gather all the components which are referenced, downloads them, and ensures that a class `components.thecomponentname` and `defaults.thecomponentname` (to define component defaults) is available in the Kapitan inventory. See Component Repositories below for a more detailed description of how we manage Kapitan components.
During compilation, we enable Kapitan’s dependency fetching to allow components to use the dependency mechanism to download their internal dependencies. We configure Kapitan’s search path to include the `dependencies` directory. This allows components to omit the leading `dependencies/` in their compilation descriptions and enables the framework to provide extra Jsonnet (or other templating) libraries that are available to all components. The compilation engine downloads a list of libraries (defined in the inventory) and makes them available under `dependencies/lib` which in turn is visible under `lib` from the perspective of components.
The engine also provides a postprocessing step which runs after Kapitan. This postprocessing step can modify the rendered output, for example to add an explicit namespace into a rendered Helm chart, as helm doesn’t support adding namespace information when using `helm template` . The postprocessing step works similar to Kapitan’s compile step, and is heavily inspired by it. Postprocessing filters can be defined in component repositories alongside the implementation of the Kapitan component. This proposal defines Jsonnet as the only templating language which is available for postprocessing filters. Postprocessing filters have access to the Kapitan inventory in the same way as the Kapitan component templates. in addition to inventory access, postprocessing filters are provided with a method to load the contents of a YAML file into Jsonnet. A postprocessing filter should have the same structure as a Kapitan Jsonnet template. The engine renders the Jsonnet, and collects all the keys of the resulting JSON document. The engine then writes the content of each key into a file with the same name as the key. The component’s `filters.yml` can define an output directory per filter which is used as the base directory for the file names returned by the filter.
Kapitan’s secrets management is leveraged to ensure no secrets appear in any Git repository (input or output) and the in-cluster GitOps engine is configured to run `kapitan refs --reveal` on the cluster catalog before applying the manifests to the cluster. The Kapitan secrets management is configured to retrieve the secrets from Vault. The Vault instance is configurable in the inventory.
### Inventory Repositories
The inventory is collected from various configuration Git repositories, which are either global or in the scope of a customer. A customer may have one or more configuration repositories for their different SYN platforms. The inventory repositories aren’t versioned, as they always reflect the desired state. It must be possible to pull global and customer-specific repositories from different Git hosting platforms.
Inventory repositories are cloned directly into the Kapitan `inventory` directory to make their contents available as inventory classes in Kapitan.
Component versions are tracked in the inventory parameters in key `component_versions` and default to `master`. An example:
``````parameters:
component_versions:
argocd:
version: v1.0.0``````
Component versions can be any git tree-ish.
### Component Repositories
• Components are versioned as part of the inventory repositories, which comes from global defaults and the cluster / customer specific repository
• Each component is managed in an individual Git repository, comparable to a Puppet module.
• Components bundle the Kapitan component templates (for example Jsonnet) with the component class that will be referenced in the Kapitan inventory and optional postprocessing filters which are applied on the compiled Kapitan catalog.
• The engine downloads the component repository into `dependencies` and symlinks the component class into `inventory/classes/components` to make the class available in the Kapitan inventory structure.
• A component repository must have the file `class/<component-name>.yml` which contains the Kapitan component class which defines how the component is compiled
• A component repository must define default parameters for the templates in the file `class/defaults.yml` in order to allow components to be included at any level in the configuration hierarchy without overriding component parameters which have been configured in the hierarchy earlier than the component was included. The `defaults.yml` file of a component is symlinked to `inventory/classes/defaults/<compoent-name>.yml` and included in the Kapitan target before `global.common`.
• If a component has postprocessing filters, it must have the file `postprocess/filters.yml` defining which postprocessing filters the compilation engine should apply.
• A component repository must be named the same as the component.
#### Repository Discovery
A default base URL is configurable to define the base path where component repositories will be searched. If a component is hosted on a different Git server or namespace, it’s full URL can be configured in the global config file. If no specific URL is provided for a component, the default base URL and the component name are concatenated to form the Git repository URL:
``ssh://git@git.example.com/components/ + <component-name> + .git``
The inventory can override the URL of a component by configuring the `url` parameter under `component_versions`:
``````parameters:
component_versions:
argocd:
version: feature/my-pr
url: ssh://git@git.example.com/my-user/my-fork.git``````
This allows for example using forks of components for some part of the hierarchy.
### Secrets management
Commodore automates the more tedious parts of Kapitan’s secret management, allowing users to simply specify Kapitan secret references (denoted by `?{…}`) in the configuration parameters. Commodore currently only supports Vault KV as back-end for storing secrets. Commodore eliminates the need for users to manually create Kapitan secret files (using `kapitan refs --write …`), by scanning the configuration parameters (everything defined under `parameters:` in Kapitan classes) for secret references and generating secret files in `catalog/refs` before running `kapitan compile`. This ensures that the secret files and the catalog are always in sync. All secret references MUST be made in the configuration parameters, otherwise Commodore can’t discover them. Secret references can use reclass references to define dynamic defaults, for example `?{vaultkv:${cluster:tenant}/${cluster:name}/thesecret/thekey}`.
#### Secret file generation
Commodore generates the secret files and their contents according to specific rules. A Kapitan secret reference, for example `?{vaultkv:path/to/secret/thekey}`, always refers to a key named thekey in a secret named `path/to/secret` in Vault’s KV back-end. The address of the Vault instance and the name of the back-end are configurable:
``````parameters:
secret_management:
# Name of the back-end (called mount in Vault)
vault_mount: kv``````
For the secret reference mentioned above, Commodore generates a Kapitan secret file in `catalog/refs/path/to/secret/thekey` with `path/to/secret:thekey` as the reference to the Vault secret.
Kapitan’s `vaultkv` secret engine is configured in the class `global.common` under the dict `secret_management`. The configuration defaults to vault.syn.vshn.net and a back-end with name `kv`. This can be overridden at any level of the inventory. The GitOps engine on the cluster uses the Vault Kubernetes authentication (in a sidecar) to lease and renew a token which can be used by Kapitan.
### Implementation Details/Notes/Constraints
Currently the engine is implemented in Python. The implementation provides a local mode where it reuses the existing (probably downloaded by the engine) directory structure to run only local operations. This allows local development of new components without having to re-download all the information on every run of the engine.
The engine uses the Python Jsonnet library to render the postprocessing filters and provides native callbacks to Jsonnet for accessing the Kapitan inventory and for loading YAML files in Jsonnet.
Component versions are extracted from the Kapitan inventory using Kapitan’s `inventory_reclass` method.
#### Directory structure
The current implementation of the engine produces and manages the following directories:
• `compiled` contains the Kapitan output
• `catalog` is a checkout of the catalog repository which is updated from the contents of `compiled/cluster`
• `dependencies` contains all the dependencies downloaded by the engine (component repositories, template libraries, …)
• `dependencies/libs/<libname>` contains downloaded template library repositories
• `dependencies/<component-name>` is the checked out component repository for component "component-name"
• `dependencies/lib` contains symlinks to the template libraries
• `inventory` contains the collated Kapitan inventory. Configuration repositories are directly cloned into inventory
• `inventory/targets/cluster.yml` contains the Kapitan target definition (generated by Commodore)
• `inventory/classes/global` contains globally managed configuration classes
• `inventory/classes/<customer-name>` contains customer-specific configuration classes
#### Constraints
• Currently, a Commodore component can define only one Kapitan class
### Risks and Mitigations
• RISK: The current architecture is built around generating an inventory and classes to use with Kapitan. Thus we’ve a hard dependency on Kapitan in the current design iteration.
• MITIGATIONS:
• Kapitan is written in Python, and currently under active development. If worst comes to worst, we potentially could keep a Kapitan fork alive.
• It would be entirely possible to replace Kapitan with another tool (either 3rd party or developed in the context of SYN) which does the inventory resolution and template rendering.
## Drawbacks
• Templates for the output of a Kapitan component (K8s YAMLs for SYN) are mainly written in Jsonnet. Kapitan is developing an alternative called Kadet which is based on YAML. Having the component templates in YAML would potentially make more sense as we’re more familiar with writing YAML than JSON.
## Alternatives
• It would be possible to put much more of the inventory value lookup magic behind an API (Lieutenant or otherwise), f.e. with hiera. With hiera, this approach would use the same Hieradata structure as we’ve in VMSFv1, with all its advantages and drawbacks.
• Early experimental versions of the engine tried different approaches in regard to how the inventory and components are organized:
• The first experimental version of the engine collected four Git repositories which define classes (one each for "global," "cluster," "customer," and "components") and put them into predefined spots in the Kapitan inventory. This implementation used the SYNventory API to determine which source repository to clone for each of the four categories. Each of the source repositories would define non-specific classes (for example "cluster.distribution"), which allowed for writing non-specific Kapitan targets. This was deemed "too magic," as it has too much configuration magic stored in the SYNventory API.
• The second experimental version of the engine collected a number of Git repositories which define classes that are specific for a particular configuration (for example "cluster.rancher") and kept the "components" repository which contained Kapitan classes for each component. The generation of the Kapitan target was outsourced to the SYNventory API. The API then would provide the engine with a customized target which contains the exact classes that are required for a customer’s configuration. This version briefly also contained customer-specific configuration in the target itself.
• The current design of the engine closely resembles the second experimental version, as the target specification is retrieved from the Lieutenant API in addition to some information about the cluster’s project and stage. The big change from the second experimental version is that the different Kapitan components are now all managed in self-contained repositories, the contents of which are strategically inserted in the right places in the Kapitan directory structure. This makes the overall workflow simpler, as updating a component happens within a single repository. | 2022-10-06 01:19:27 | {"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": 0.20416590571403503, "perplexity": 4723.892016849435}, "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/1664030337680.35/warc/CC-MAIN-20221005234659-20221006024659-00246.warc.gz"} |
http://tm.durusau.net/?cat=323 | ## Archive for the ‘Gremlin’ Category
### TinkerPop 2.3.0 has been unleashed
Thursday, March 21st, 2013
TinkerPop 2.3.0 has been unleashed by Marko A. Rodriguez.
Release notes for:
Blueprints
Pipes
Gremlin
Frames
Rexster
Enjoy!
### Distributed Graph Computing with Gremlin
Thursday, March 7th, 2013
Distributed Graph Computing with Gremlin by Marko A. Rodriguez.
From the post:
The script-step in Faunus’ Gremlin allows for the arbitrary execution of a Gremlin script against all vertices in the Faunus graph. This simple idea has interesting ramifications for Gremlin-based distributed graph computing. For instance, it is possible evaluate a Gremlin script on every vertex in the source graph (e.g. Titan) in parallel while maintaining data/process locality. This section will discuss the following two use cases.
• Global graph mutations: parallel update vertices/edges in a Titan cluster given some arbitrary computation.
• Global graph algorithms: propagate information to arbitrary depths in a Titan cluster in order to compute some algorithm in a parallel fashion.
Another must read post from Marko A. Rodriguez!
Also a reminder that I need to pull out my Oxford Classical Dictionary to add some material to the mythology graph.
### Titan-Android
Wednesday, December 26th, 2012
Titan-Android by David Wu.
From the webpage:
Titan-Android is a port/fork of Titan for the Android platform. It is meant to be a light-weight implementation of a graph database on mobile devices. The port removes HBase and Cassandra support as their usage make little sense on a mobile device (convince me otherwise!). Gremlin is only supported via the Java interface as I have not been able to port groovy successfully. Nevertheless, Titan-Android supports local storage backend via BerkeleyDB and supports the Tinkerpop stack natively.
Just in case there was an Android under the tree!
I first saw this in a tweet by Marko A. Rodriguez.
### Big Graph Data on Hortonworks Data Platform
Thursday, December 13th, 2012
Big Graph Data on Hortonworks Data Platform by Marko Rodriguez.
The Hortonworks Data Platform (HDP) conveniently integrates numerous Big Data tools in the Hadoop ecosystem. As such, it provides cluster-oriented storage, processing, monitoring, and data integration services. HDP simplifies the deployment and management of a production Hadoop-based system.
In Hadoop, data is represented as key/value pairs. In HBase, data is represented as a collection of wide rows. These atomic structures makes global data processing (via MapReduce) and row-specific reading/writing (via HBase) simple. However, writing queries is nontrivial if the data has a complex, interconnected structure that needs to be analyzed (see Hadoop joins and HBase joins). Without an appropriate abstraction layer, processing highly structured data is cumbersome. Indeed, choosing the right data representation and associated tools opens up otherwise unimaginable possibilities. One such data representation that naturally captures complex relationships is a graph (or network). This post presents Aurelius‘ Big Graph Data technology suite in concert with Hortonworks Data Platform. Moreover, for a real-world grounding, a GitHub clone is described in this context to help the reader understand how to use these technologies for building scalable, distributed, graph-based systems.
If you like graphs at all or have been looking at graph solutions, you are going to like this post.
### Conditional Traversals With Gremlin
Sunday, July 8th, 2012
Conditional Traversals With Gremlin by Max Lincoln.
An eligibility test that depends upon the ability to traverse to a particular node in the graph.
Reminded me of my musings on transient properties/edges.
Is not choosing an edge is the same thing as the edge not being present? For all cases?
Max mentions that NoSQL Distilled says this use case isn’t the typical one for graphs.
My suggestion is to experiment and rely on your own requirements and experiences.
Authors have to paint with a very broad brush or their books would all look like the Oxford English Dictionary (OED). Fascinating but not for the faint of heart.
BTW, when looking up the reference for the Oxford English Dictionary, the wikipedia reference mentioned that:
The Dutch dictionary Woordenboek der Nederlandsche Taal, which has similar aims to the OED, is the largest and it took twice as long to complete.
I don’t read Dutch but the dictionary is reported to be available for free at: http://gtb.inl.nl/
If you read Dutch, please confirm/deny the report. I would like to send a little note along the the OED crowd about access as a public service. (Like they would care what I think. Still, doesn’t hurt to comment every now and again.)
### Short Intro to Graph Databases, Manipulating and Traversing With Gremlin
Monday, May 28th, 2012
Short Intro to Graph Databases, Manipulating and Traversing With Gremlin
Alex Popescu at myNoSQL captures a slide deck by Pierre De Wilde, “A Walk in Graph Databases.”
Has extensive examples using Gremlin after a short graph theory introduction.
Amusing graphic of everything looking like a table if all you have is a relational database.
Truth is that everything looks like a graph from a certain point of view.
Design question: What graph qualities, if any, are appropriate for your data and goals?
Always possible that graph representation or properties are inappropriate for your project.
### TinkerPop2 Release
Thursday, May 24th, 2012
A message from Marko Rodriguez announced the release of TinkerPop2 with notes on the major features of each:
Blueprints: https://github.com/tinkerpop/blueprints/wiki/Release-Notes
- Massive changes to blueprints-core API
Pipes: https://github.com/tinkerpop/pipes/wiki/Release-Notes
- TreePipe added for exposing the spanning tree of a traversal
Gremlin: https://github.com/tinkerpop/gremlin/wiki/Release-Notes
- Automatic path and query optimizations
Frames: https://github.com/tinkerpop/frames/wiki/Release-Notes
- FramedGraph is simply a wrapper graph in the Blueprints sense
Rexster: https://github.com/tinkerpop/rexster/wiki/Release-Notes
- Synchronicity with the Blueprints API
BTW, Marko says:
As you may know, there are big changes to the API: package renaming, new core API method names, etc. While this may be shocking, it is all worth it. In 2 weeks, there is going to be a release of something very big for which TinkerPop2 will be a central piece of the puzzle. Stay tuned and get ready for a summer of insane, crazy graph madness.
So, something to look forward to!
### Visualizing a set of Hiveplots with Neo4j
Wednesday, March 28th, 2012
Visualizing a set of Hiveplots with Neo4j by Max De Marzi.
Max writes:
If you want to learn more about Hive Plots, take a look at his website and this presentation (it is quite large at 20 MB). I cannot do it justice in this short blog post, and in all honestly haven’t had the time to study it properly.
Today I just want to give you a little taste of Hiveplots. I am going to visualize the github graphs of nine languages you might not have heard of: Boo, Dylan, Factor, Gosu, Mirah, Nemerle, Nu, Parrot, Self. I’m not going to show you how to create the graph this time, because this is real data we are using. You can take a look at it on the data folder in github.
The graph is basically: (Language)–(Repository)–(User). There are two relationships between Repository and User, wrote and forked.
Hive plots are an effort by Martin Krzywinski to enable viewers of a graph visualization to distinguish between two or more graphs and to recognize key features of those graphs. His website is: http://www.hiveplot.com/.
### Exploring Wikipedia with Gremlin Graph Traversals
Saturday, March 10th, 2012
Exploring Wikipedia with Gremlin Graph Traversals by Marko Rodriguez.
From the post:
There are numerous ways in which Wikipedia can be represented as a graph. The articles and the href hyperlinks between them is one way. This type of graph is known a single-relational graph because all the edges have the same meaning — a hyperlink. A more complex rendering could represent the people discussed in the articles as “people-vertices” who know other “people-vertices” and that live in particular “city-vertices” and work for various “company-vertices” — so forth and so on until what emerges is a multi-relational concept graph. For the purpose of this post, a middle ground representation is used. The vertices are Wikipedia articles and Wikipedia categories. The edges are hyperlinks between articles as well as taxonomical relations amongst the categories.
If you aren’t interested in graph representations of data before reading this post, it is likely you will be afterwards.
Take a few minutes to read it and then let me know what you think.
### A Well-Woven Study of Graphs, Brains, and Gremlins
Friday, February 24th, 2012
A Well-Woven Study of Graphs, Brains, and Gremlins by Marko Rodriguez.
From the post:
What do graphs and brains have in common? First, they both share a relatively similar structure: Vertices/neurons are connected to each other by edges/axons. Second, they both share a similar process: traversers/action potentials propagate to effect some computation that is a function of the topology of the structure. If there exists a mapping between two domains, then it is possible to apply the processes of one domain (the brain) to the structure of the other (the graph). The purpose of this post is to explore the application of neural algorithms to graph systems.
Entertaining and informative post by Marko Rodriguez comparing graphs, brains and the graph query language Gremlin.
I agree with Marko on the potential of graphs but am less certain than I read him to be on how well we understand the brain. Both the brain and graphs have many dark areas yet to be explored. As we shine new light on one place, more unknown places are just beyond the reach of our light.
### Gremlin vs Cypher Initial Thoughts @Neo4j
Wednesday, February 22nd, 2012
Gremlin vs Cypher Initial Thoughts @Neo4j
Romiko Derbynew writes:
The Neo4jClient now supports Cypher as a query language with Neo4j. However I noticed the following:
• Simple graph traversals are much more efficient when using Gremlin
• Queries in Gremlin are 30-50% faster for simple traversals
• Cypher is ideal for complex traversals where back tracking is required
• Cypher is our choice of query language for reporting
• Gremlin is our choice of query language for simple traversals where projections are not required
• Cypher has intrinsic table projection model, where Gremlins table projection model relies on AS steps which can be cumbersome when backtracking e.g. Back(), As() and _CopySplit, where cypher is just comma separated matches
• Cypher is much better suited for outer joins than Gremlin, to achieve similar results in gremlin requires parallel querying with CopySplit.
• Gremlin is ideal when you need to retrieve very simple data structures
• Table projection in gremlin can be very powerful, however outer joins can be very verbose
So in a nutshell, we like to use Cypher when we need tabular data back from Neo4j and is especially useful in outer joins.
Excellent comparison of Gremlin vs. Cypher. Both have their advantages.
### Max Flow with Gremlin and Transactions
Wednesday, February 22nd, 2012
Max Flow with Gremlin and Transactions
Max De Marzi writes:
The maximum flow problem was formulated by T.E. Harris as follows:
Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, a nd a maximal flow from one given city to the other.
Back in the mid 1950s the US Military had an interest in finding out how much capacity the Soviet railway network had to move cargo from the Western Soviet Union to Eastern Europe. This lead to the Maximum Flow problem and the Ford–Fulkerson algorithm to solve it.
If you’ve been reading the Neo4j Gremlin Plugin documentation, you’ll remember it has a section on Flow algorithms with Gremlin. Let’s add a couple of things and bring this example to life.
If that sounds like an out-dated Cold War problem, consider Max’s conclusion:
The max flow and related problems manifest in many ways. Water or sewage through underground pipes, passengers on a subway system, data through a network (the internet is just a series of tubes!), roads and highway planning, airline routes, even determining which sports teams have been eliminated from the playoffs.
What else can be modeled as max flow or related problems? Drug/weapons smuggling? Oil/gas/electricity transport? Others?
### Bio4j release 0.7 is out !
Wednesday, January 11th, 2012
Bio4j release 0.7 is out !
A quick list of the new features:
• Expasy Enzyme database integration
• Node type indexing
• Amazon web services Availability in all Regions
• New CloudFormation templates
• Bio4j REST server
• Explore you database with the Data browser
• Run queries with Cypher
• Querying Bio4j with Gremlin
Wait! Did I say Cypher and Gremlin!?
Looks like this graph querying stuff is spreading.
Even if you are not working in bioinformatics, Bio4j is worth more than a quick look.
### Graph Algorithms
Thursday, January 5th, 2012
Graph Algorithms
I ran across this Wikipedia book while working on one of the data structures posts for today.
I think you may find it useful but some cautions:
First, being a collection of Wikipedia articles, it doesn’t have a consistent editorial voice. That is more than being fussy, the depth and usefulness of explanations will vary from article to article.
Second, you will find topics that are “stubs,” and hence not very useful.
Third, I think with the advent of Neo4j, Grelim, Cypher and other graph databases/software, future entries should have in addition to text, exercises that users can perform with common software to reinforce their understanding of entries.
### A TinkerPop Story
Wednesday, December 14th, 2011
A TinkerPop Story
From the post:
In a time long, long right now and a place far, far within, there exists a little green gremlin named…well, Gremlin. Gremlin lives in a place known as TinkerPop. For those who think of a “place” as some terrestrial surface coating a sphere that is circling one of the many massive fiery nuclear reactors in the known universe, TinkerPop is that, yet at the same time, a wholly different type of place indeed.
In a day of obscure (are there any other kind?) errors and annoyances, this is an absolute delight!
Highly recommended!
### TinkerPop 2011 Winter release!
Friday, December 9th, 2011
TinkerPop 2011 Winter release!
Which includes:
New homepage design: http://tinkerpop.com
Gremlin 1.4 (Ain’t No Thing But a Chicken Wing):
https://github.com/tinkerpop/gremlin/wiki/Release-Notes
You didn’t really want to spend all weekend holiday shopping and hanging out with relatives did you?
### Multilingual Graph Traversals
Thursday, December 8th, 2011
OK the real title is: JVM Language Implementations. I like mine better.
From the webpage:
Gremlin is a style of graph traversing that can be hosted in any number of languages. The benefit of this is that users can make use of the programming language they are most comfortable with and still be able to evaluate Gremlin-style traversals. This model is different than, lets say, using SQL in Java where the query is evaluated by passing a string representation of the query to the SQL engine. On the contrary, with native Gremlin support for other JVM languages, there is no string passing. Instead, simple method chaining in Gremlin’s fluent style. However, the drawback of this model is that for each JVM language, there are syntactic variations that must be accounted for.
The examples below demonstrate the same traversal in Groovy, Scala, and Java, respectively.
Seeing is believing.
### Pilot
Sunday, October 23rd, 2011
Pilot
From the readme file:
Pilot is a graph database operator that allows you to perform common application-level operations on graph databases without delving into the details of their implementation or requiring knowledge of the component technologies.
Pilot aims to support graph databases conforming to the property graph model. Pilot employs technologies from the Tinkerpop stack — specifically Blueprints and Gremlin — for general access and manipulation of the underlying graph database, but also uses native graph database APIs to further optimize performance for certain operations. In addition, Pilot also handles multithreading and transaction management, while keeping all of these abstracted away from the calling application. As such, Pilot is ideally suited for use in concurrent web applications.
• Supported graph database providers:
• OrientDB
• Neo4j
• Tinkergraph (the Blueprints in-memory reference implementation)
• (others may be added in future if there is demand)
• Some of the functionality currently supported by Pilot include:
• Get edges between given vertices
• Get neighbors of a given vertex
• Retrieving vertices corresponding to some properties (see Property Graph Model)
• Transaction management
• Large commit optimization
• Application profiling
Graph databases aren’t a new idea. I don’t have the reference at hand but once ran across a relational database that was implemented as a hypergraph. It may be that computing power has finally gotten to the point that graph databases, or at least their capabilities, will be the common expectation.
### An Introduction to Tinkerpop
Saturday, October 8th, 2011
An Introduction to Tinkerpop by Takahiro Inoue.
Excellent introduction to the Tinkerpop stack.
### Furnace — A Property Graph Algorithms Package
Sunday, September 25th, 2011
Furnace — A Property Graph Algorithms Package
Marko Rodriguez posted the following note to the Grelim-users mailing list today:
Hello,
For many months, the TinkerPop community has been trying to realize the best way to go about providing a graph analysis package to the TinkerPop stack ( http://bit.ly/qCMlcP ). With the increased flexibility and power of Pipes and the partitioning of Gremlin into multiple JVM languages, we feel that the stack is organized correctly now to support Furnace — A Property Graph Algorithms Package.
http://furnace.tinkerpop.com
( https://github.com/tinkerpop/furnace/wiki if the domain hasn’t propagated to your DNS yet )
The project is currently just stubbed, but overtime you can expect the ability to evaluate standard (and non-standard) graph analysis algorithms over Blueprints-enabled graphs in a way that respects explicit and implicit associations in the graph. In short, it will implement the ideas articulated in:
This will be possible due to Pipes and the ability to represent abstract relationships using Pipes, Gremlin_groovy (and the upcoming Gremlin_scala). Moreover, while more thought is needed, there will be a way to talk at the Frames-levels (http://frames.tinkerpop.com) and thus, calculate graph algorithms according to one’s domain model. Ultimately, in time, as Furnace develops, we will see a Rexster-Kibble that supports the evaluation of algorithms via Rexster.
While the project is still developing, please feel free to contribute ideas and/or participate in the development process. To conclude, we hope people are excited about the promises that Furnace will bring by raising the processing abstraction level above the imperative representations of Pipes/Gremlin.
Thank you,
Marko.
http://markorodriguez.com
You have been waiting for the opportunity to contribute to the Tinkerpop stack, particularly on graph analysis, so here is your chance! Seriously, you need to forward this to every graph person, graph project and graduate student taking graph theory.
We can use simple graphs and hope (pray?) the world is a simple place. Or use more complex graphs to model the world. Do you feel lucky? Do you?
### A Graph-Based Movie Recommender Engine
Thursday, September 22nd, 2011
A Graph-Based Movie Recommender Engine by Marko A. Rodriguez.
From the post:
A recommender engine helps a user find novel and interesting items within a pool of resources. There are numerous types of recommendation algorithms and a graph can serve as a general-purpose substrate for evaluating such algorithms. This post will demonstrate how to build a graph-based movie recommender engine using the publicly available MovieLens dataset, the graph database Neo4j, and the graph traversal language Gremlin. Feel free to follow along in the Gremlin console as the post will go step-by-step from data acquisition, to parsing, and ultimately, to traversing.
As important as graph engines, algorithms and research are at present, and as important as they will become, I think the Neo4j community itself is worthy of direct study. There are stellar contributors to the technology and the community, but is that what makes it such an up and coming community? Or perhaps how they contributed? It would take a raft (is that the term for a group of sociologists?) of sociologists and perhaps there are existing studies of online communities that might have some clues. I mention that because there are other groups I would like to see duplicate the success of the Neo4j community.
Marko takes you from data import to a useful (albeit limited) application in less than 2500 words. (measured to the end of the conclusion, excluding further reading)
And leaves you with suggestions for further exploring.
That is a blog post that promotes a paradigm. (And for anyone who takes offense at that observation, it applies to my efforts as well. There are other ways to promote a paradigm but you have to admit, this is a fairly compelling one.)
Put Marko’s post on your read with evening coffee list.
### Tinkerpop Stack Releases
Sunday, September 18th, 2011
Marko Rodriguez announced a new round of Tinkerpop Stack Releases today:
The TinkerPop stack went through another round of releases this morning.
• Blueprints 1.0 (Blueprints): = https://github.com/tinkerpop/blueprints/wiki/Release-Notes
• Pipes 0.8 (Cleaner): = https://github.com/tinkerpop/pipes/wiki/Release-Notes
• Frames 0.5 (Beams): = https://github.com/tinkerpop/frames/wiki/Release-Notes
• Gremlin 1.3 (On the Case): = https://github.com/tinkerpop/gremlin/wiki/Release-Notes
• Rexster 0.6 (Dalmatian): = https://github.com/tinkerpop/rexster/wiki/Release-Notes
• Rexster-Kibbles 0.6 = http://rexster-kibbles.tinkerpop.com
For those using Gremlin, Pipes, and Rexster, be sure to look through the release notes as APIs have changed slightly. Here are the main points of this release:
• Blueprints now has transaction buffers and Neo4jBatchGraph for bulk loading a Neo4j graph.
• Pipes makes use of FluentPipeline and PipeFunction which yields great expressivity and further opens up the framework to other JVM languages.
• Gremlin is ~2.5x faster in many situations and has relegated most of its functionality to Pipes and native Java.
• Rexster supports Neo4j High Availability and more updates to its REST API.
### Spring Data Graph 1.1.0 with Neo4j support released
Sunday, August 21st, 2011
Spring Data Graph 1.1.0 with Neo4j support released
From the wiki:
We are pleased to announce that the second release (1.1.0.RELEASE) of the Spring Data Graph project with Neo4j support is now available!
After the first public release of Spring Data Graph in April 2011 we mainly focused on user feedback.
With the improved documentation around the tooling and an upgraded AspectJ version we addressed many of the AspectJ issues that where reported by users. With the latest STS and Eclipse and hopefully with Idea11 it is possible to develop Spring Data Graph applications without the red wiggles. To further ease the development we also provided sample build scripts for ant/ivy and a plugin for gradle.
Of course we kept pace with development of Neo4j, currently using the latest stable release of Neo4j (1.4.1).
During the last months of Neo4j development the improved querying (Cypher, Gremlin) support was one of the important aspects.
So we strove to support it on all levels. Now, it is possible to execute Cypher queries from Spring Data Graph Repositories, from the Neo4j-Template but also as part of dynamic field annotations and via the introduced entity methods. The same goes for Gremlin scripts. What’s possible with this new expressive power? Let’s take a look. …
OK, better? Worse? About the same? Projects can’t improve without your feedback. Issues discussed only around water coolers can’t be addressed. Yes?
There’s some famous so-and-so’s Law about non-reported comments but I can’t find the reference. You?
### On the Nature of Pipes
Sunday, August 7th, 2011
On the Nature of Pipes by Marko Rodriguez.
From the post:
Pipes is a data flow framework developed by TinkerPop. The graph traversal language Gremlin is a Groovy-based domain-specific language for processing Blueprints-enabled graph databases with Pipes. Since the release of Pipes 0.7 on August 1, 2011, much of the functionality in Gremlin has been generalized and made available through Pipes. This has opened up the door for other JVM languages (e.g. JRuby, Jython, Clojure, etc.) to serve as host languages for graph traversal DSLs. In order to promote this direction, this post will explain Pipes from the vantage point of Gremlin.
You may not be a graph database enthusiast after reading Marko’s post but you will increase your understanding of them.
That you are not then a graph database enthusiast will be your own fault.
### TinkerPop – New Releases
Monday, August 1st, 2011
Good news from Marko Rodriguez:
TinkerPop just released a new round of stable releases.
Blueprints 0.9 (Mavin) – https://github.com/tinkerpop/blueprints/wiki/Release-Notes
Pipes 0.7 (PVC) – https://github.com/tinkerpop/pipes/wiki/Release-Notes
Frames 0.4 (Studs) – https://github.com/tinkerpop/frames/wiki/Release-Notes
Gremlin 1.2 (New Sheriff in Town) – https://github.com/tinkerpop/gremlin/wiki/Release-Notes
Rexster 0.5 (Dog Star) – https://github.com/tinkerpop/rexster/wiki/Release-Notes
Here is the main points with each release:
• Blueprints:
• Vertex API changed so now you have Vertex.getInEdges(String… labels) and Vertex.getOutEdges(String… labels)
• Heavy development on GraphSail which turns any IndexableGraph into Sail RDF store
• Pipes:
• Introduced PipeClosure pattern which allows for closure-based pipes in native Java
• Migrated all “Gremlin-specific pipes” (closure-based) to Pipes
• Opening up the stage for data flow traversal languages for any JVM language
• Frames:
• Added helper interfaces VertexFrame and EdgeFrame
• Gremlin:
• Support the easy definition of new steps with
Gremlin.defineStep()
• Mass migration of all “Gremlin-specific pipes” to Pipes
• Support for processing closures in aggregate, groupCount, and paths
• Rexster:
• Added RexPro (the future foundation for the Rexster’s multi-protocol infrastructure).
• Added rexster-console.sh (RexsterConsole) to allow remote “mysql>”-style interactions via any JSR 223-based JVM language
• JSON serialization inherited from Blueprints (consistent throughout TinkerPop stack)
### The Pathology of Graph Databases
Saturday, July 23rd, 2011
The Pathology of Graph Databases by Marko A. Rodriguez.
If you want to learn Gremlin as a graph traversal language you would be hard pressed to find a better starting place.
### bulbflow
Saturday, July 16th, 2011
bulbflow: a Python framework for the graph era
From the Overview:
Bulbs is an open-source Python persistence framework for graph databases and the first piece of a larger Web-development toolkit that will be released in the upcoming weeks.
It’s like an ORM for graphs, but instead of SQL, you use the graph-traveral language Gremlin to query the database.
You can use it to connect to any Blueprints-enabled
database, including TinkeGraph, Neo4j, OrientDB, Dex, and OpenRDF (and there is an InfiniteGraph implementation in development).
This means your code is portable because you can to plug into different graph database backends without worrying about vendor lock in.
Bulbs was developed in the process of building Whybase, a startup that will open for preview this fall. Whybase needed a persistence layer to model its complex relationships, and Bulbs is an open-source version of that framework.
You can use Bulbs from within any Python Web-development framework, including Flask, Pyramid, and Django.
Will be watching for future developments!
### OrientDB 1.0rc3 – Graph(Ed)
Friday, July 15th, 2011
OrientDB 1.0rc3 – Graph(Ed)
From the webpage:
This is a special edition of OrientDB with these TinkerPop technologies in bundle:
• Blueprints provides a collection of interfaces and implementations to common, complex data structures. In short, Blueprints provides a one stop shop for implemented interfaces to help developers create software without being tied to particular underlying data management systems.
• Gremlin is a Turing-complete, graph-based programming language designed for key/value-pair multi-relational graphs. Gremlin makes use of an XPath-like syntax to support complex graph traversals. This language has application in the areas of graph query, analysis, and manipulation.
• Pipes is a graph-based data flow framework for Java 1.6+. A process graph is composed of a set of process vertices connected to one another by a set of communication edges. Pipes supports the splitting, merging, and transformation of data from input to output.
The graph community just keeps getting stronger.
### Graphs, Brains, and Gremlin
Thursday, July 14th, 2011
Graphs, Brains, and Grelim
From the post:
What do graphs and brains have in common? First, they both share a relatively similar structure: Vertices/neurons are connected to each other by edges/axons. Second, they both share a similar process: traversers/action potentials propagate to effect some computation that is a function of the topology of the structure. If there exists a mapping between two domains, then its possible to apply the processes of one domain (the brain) to the structure of the other (the graph). The purpose of this post is to explore the application of neural algorithms to graph systems.
As only Marko could answer the question: “What do graphs and brains have in common?”
Highly recommended.
I am particularly interested in the the use of spreading activation for subject recognition. How do we capture such a recognition and/or communicate it to others?
### Cloning Tinkerpop Repositories
Friday, July 1st, 2011
Instructions on creating a local copy of the Gremlin wiki (posted to the gremlin-users@googlegroups.com mailing list by Pierre De Wilde).
The instructions (with minor formatting changes) from his post:
For those who want a local copy of Gremlin wiki:
cd gremlin
git clone https://github.com/tinkerpop/gremlin.wiki.git
doc/wiki
cd doc/wiki
gollum
Open your browser at http://localhost:4567 and ta-da…
Moreover, the wiki is searchable and (unlike the github version) it’s printer-fiendly.
Gollum is a simple wiki system built on top of Git that powers GitHub Wikis.
https://github.com/github/gollum
To install Gollum, use RubyGems (http://rubygems.org/):
[sudo] gem install gollumcd cd
Of course, the same procedure may be applied for other Tinkerpop repositories (blueprints, pipes, frames, rexster, rexster-kibbles).
Unfortunately, gollum cannot access multiple repositories at once, so you will need to launch several versions with a different port (gollum -port xxxx)
Thanks Pierre! | 2013-05-20 18:21:06 | {"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": 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": 0.20199733972549438, "perplexity": 4183.726363947315}, "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-2013-20/segments/1368699186520/warc/CC-MAIN-20130516101306-00047-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Concept_Development_Studies_in_Chemistry_(Hutchinson)/12%3A__The_Kinetic_Molecular_Theory | # 12: The Kinetic Molecular Theory
## Foundation
We assume an understanding of the atomic molecular theory postulates, including that all matter is composed of discrete particles. The elements consist of identical atoms, and compounds consist of identical molecules, which are particles containing small whole number ratios of atoms. We also assume that we have determined a complete set of relative atomic weights, allowing us to determine the molecular formula for any compound. Finally, we assume a knowledge of the Ideal Gas Law, and the observations from which it is derived.
## Goals
Our continuing goal is to relate the properties of the atoms and molecules to the properties of the materials which they comprise. As simple examples, we compare the substances water, carbon dioxide, and nitrogen. Each of these is composed of molecules with few (two or three) atoms and low molecular weight. However, the physical properties of these substances are very different. Carbon dioxide and nitrogen are gases at room temperature, but it is well known that water is a liquid up to $$100^\text{o} \text{C}$$. To liquefy nitrogen, we must cool it to $$-196^\text{o} \text{C}$$, so the boiling temperatures of water and nitrogen differ by about $$300^\text{o} \text{C}$$. Water is a liquid for a rather large temperature range, freezing at $$0^\text{o} \text{C}$$. In contrast, nitrogen is a liquid for a very narrow range of temperatures, freezing at $$-210^\text{o} \text{C}$$. Carbon dioxide poses yet another very different set of properties. At atmospheric pressure, carbon dioxide gas cannot be liquefied at all: cooling the gas to $$-60^\text{o} \text{C}$$ converts it directly to solid "dry ice". As is commonly observed, warming dry ice does not produce any liquid, as the solid sublimes directly to gas.
Why should these materials, whose molecules do not seem all that different, behave so differently? What are the important characteristics of these molecules which produce these physical properties? It is important to keep in mind that these are properties of the bulk materials. At this point, it is not even clear that the concept of a molecule is useful in answering these questions about melting or boiling.
There are at least two principal questions that arise about the Ideal Gas Law. First, it is interesting to ask whether this law always holds true, or whether there are conditions under which the pressure of the gas cannot be calculated from $$\frac{nRT}{V}$$. We thus begin by considering the limitations of the validity of the Ideal Gas Law. We shall find that the ideal gas law is only approximately accurate and that there are variations which do depend upon the nature of the gas. Second, then, it is interesting to ask why the ideal gas law should ever hold true. In other words, why are the variations not the rule rather than the exception?
To answer these questions, we need a model which will allow us to relate the properties of bulk materials to the characteristics of individual molecules. We seek to know what happens to a gas when it is compressed into a smaller volume, and why it generates a greater resisting pressure when compressed. Perhaps most fundamentally of all, we seek to know what happens to a substance when it is heated. What property of a gas is measured by the temperature?
## Observation 1: Limitations of the Validity of the Ideal Gas Law
To design a systematic test for the validity of the Ideal Gas Law, we note that the value of $$\frac{PV}{nRT}$$, calculated, from the observed values of $$P$$, $$V$$, $$n$$, and $$T$$, should always be equal to 1, exactly. Deviation of $$\frac{PV}{nRT}$$ from 1 indicates a violation of the Ideal Gas Law. We thus measure the pressure for several different gases under a variety of conditions by varying $$n$$, $$V$$, and $$T$$, and we calculate the ratio $$\frac{PV}{nRT}$$ for these conditions.
In Figure 12.1, the value of this ratio is plotted for several gases as a function of the "particle density" of the gas in moles, $$\frac{n}{V}$$. To make the analysis of this plot more convenient, the particle density is given in terms of the particle density of an ideal gas at room temperature and atmospheric pressure (i.e. the density of air), which is $$0.04087 \: \frac{\text{mol}}{\text{L}}$$. In Figure 12.1, a particle density of 10 means that the particle density of the gas is 10 times the particle density of air at room temperature. The x-axis in the figure is thus unitless.
Figure 12.1: Validity of the Ideal Gas Law
Note that $$\frac{PV}{nRT}$$ on the y-axis is also unitless and has value exactly 1 for an ideal gas. We observe in the data in this figure that $$\frac{PV}{nRT}$$ is extremely close to 1 for particle densities which are close to that of normal air. Therefore, deviations from the Ideal Gas Law are not expected under "normal" conditions. This is not surprising, since Boyle's Law, Charles' Law, and the Law of Combining Volumes were all observed under normal conditions. Figure 12.1 also shows that, as the particle density increases above the normal range, the value of $$\frac{PV}{nRT}$$ starts to vary from 1, and the variation depends on the type of gas we are analyzing. However, even for particle densities 10 times greater than that of air at atmospheric pressure, the Ideal Gas Law is accurate to a few percent.
Thus, to observe any significant deviations from $$PV = nRT$$, we need to push the gas conditions to somewhat more extreme values. The results for such extreme conditions are shown in Figure 12.2. Note that the densities considered are large numbers corresponding to very high pressures. Under these conditions, we find substantial deviations from the Ideal Gas Law. In addition, we see that the pressure of the gas (and thus $$\frac{PV}{nRT}$$) does depend strongly on which type of gas we are examining. Finally, this figure shows that deviations from the Ideal Gas Law can generate pressures either greater than or less than that predicted by the Ideal Gas Law.
Figure 12.2: Deviations from the Ideal Gas Law
## Observation 2: Density and Compressibility of Gas
For low densities for which the Ideal Gas Law is valid, the pressure of a gas is independent of the nature of the gas, and is therefore independent of the characteristics of the particles of that gas. We can build on this observation by considering the significance of a low particle density. Even at the high particle densities considered in Figure 12.2, all gases have low density in comparison to the densities of liquids. To illustrate, we note that 1 gram of liquid water at its boiling point has a volume very close to 1 milliliter. In comparison, this same 1 gram of water, once evaporated into steam, has a volume of over 1700 milliliters. How does this expansion by a factor of 1700 occur? It is not credible that the individual water molecules suddenly increase in size by this factor. The only plausible conclusion is that the distance between gas molecules has increased dramatically.
Therefore, it is a characteristic of a gas that the molecules are far apart from one another. In addition, the lower the density of the gas the farther apart the molecules must be, since the same number of molecules occupies a larger volume at lower density.
We reinforce this conclusion by noting that liquids and solids are virtually incompressible, whereas gases are easily compressed. This is easily understood if the molecules in a gas are very far apart from one another, in contrast to the liquid and solid where the molecules are so close as to be in contact with one another.
We add this conclusion to the observations in Figures 12.1 and 12.2 that the pressure exerted by a gas depends only on the number of particles in the gas and is independent of the type of particles in the gas, provided that the density is low enough. This requires that the gas particles be far enough apart. We conclude that the Ideal Gas Law holds true because there is sufficient distance between the gas particles that the identity of the gas particles becomes irrelevant.
Why should this large distance be required? If gas particle A were far enough away from gas particle B that they experience no electrical or magnetic interaction, then it would not matter what types of particles A and B were. Nor would it matter what the sizes of particles A and B were. Finally, then, we conclude from this reasoning that the validity of the ideal gas law rests on the presumption that there are no interactions of any type between gas particles.
## Postulates of the Kinetic Molecular Theory
We recall at this point our purpose in these observations. Our primary concern in this study is attempting to relate the properties of individual atoms or molecules to the properties of mass quantities of the materials composed of these atoms or molecules. We now have extensive quantitative observations on some specific properties of gases, and we proceed with the task of relating these to the particles of these gases.
By taking an atomic molecule view of a gas, we can postulate that the pressure observed is a consequence of the collisions of the individual particles of the gas with the walls of the container. This presumes that the gas particles are in constant motion. The pressure is, by definition, the force applied per area, and there can be no other origin for a force on the walls of the container than that provided by the particles themselves. Furthermore, we observe easily that the pressure exerted by the gas is the same in all directions. Therefore, the gas particles must be moving equally in all directions, implying quite plausibly that the motions of the particles are random.
To calculate the force generated by these collisions, we must know something about the motions of the gas particles so that we know, for example, each particle's velocity upon impact with the wall. This is too much to ask: there are perhaps $$10^{20}$$ particles or more, and following the path of each particle is out of the question. Therefore, we seek a model which permits calculation of the pressure without this information.
Based on our observations and deductions, we take as the postulates of our model:
• A gas consists of individual particles in constant and random motion
• The individual particles have negligible volume.
• The individual particles do not attract or repel one another in any way.
• The pressure of the gas is due entirely to the force of the collisions of the gas particles with the walls of the container.
This model is the Kinetic Molecular Theory of Gases. We now look to see where this model leads.
## Derivation of Boyle's Law from the Kinetic Molecular Theory
To calculate the pressure generated by a gas of $$N$$ particles contained in a volume $$V$$, we must calculate the force $$F$$ generated per area $$A$$ by collisions against the walls. To do so, we begin by determining the number of collisions of particles with the walls. The number of collisions we observe depends on how long we wait. Let's measure the pressure for a period of time $$\Delta t$$ and calculate how many collisions occur in that time period. For a particle to collide with the wall within the time $$\Delta t$$, it must start close enough to the wall to impact it in that period of time. If the particle is traveling with speed $$v$$, then the particle must be within a distance $$v \Delta t$$ of the wall to hit it. Also, if we are measuring the force exerted on the area $$A$$, the particle must hit that area to contribute to our pressure measurement.
For simplicity, we can view the situation pictorially in Figure 12.3. We assume that the particles are moving perpendicularly to the walls. (This is clearly not true. However, very importantly, this assumption is only made to simplify the mathematics of our derivation. It is not necessary to make this assumption, and the result is not affected by the assumption.) In order for a particle to hit the area $$A$$ marked on the wall, it must lie within the cylinder shown, which is of length $$v \Delta t$$ and cross-sectional area $$A$$. The volume of this cylinder is $$A v \Delta t$$, so the number of particles contained in the cylinder is $$\left( A v \Delta t \right) \times \frac{N}{V}$$.
Figure 12.3: Collision of a Particle with a Wall within time $$\Delta t$$
Not all of these particles collide with the wall during $$\Delta t$$, though, since most of them are not traveling in the correct direction. There are six directions for a particle to go, corresponding to plus or minus direction in x, y, or z. Therefore, on average, the fraction of particles moving in the correct direction should be $$\frac{1}{6}$$, assuming as we have that the motions are all random. Therefore, the number of particles which impact the wall in time $$\Delta t$$ is $$\left( A v \Delta t \right) \times \frac{N}{6V}$$.
The force generated by these collisions is calculated from Newton's equation, $$F = ma$$, where $$a$$ is the acceleration due to the collisions. Consider first a single particle moving directly perpendicular to a wall with velocity $$v$$ as in Figure 12.3. We note that, when the particle collides with the wall, the wall does not move, so the collision must generally conserve the energy of the particle. Then the particle's velocity after the collision must be $$-v$$, since it is now traveling in the opposite direction. Thus, the change in velocity of the particle by the time $$\Delta t$$, we find that the total acceleration (change in velocity per time) is $$\frac{2A N v^2}{6V}$$, and the force imparted on the wall due to collisions is found by multiplying by the mass of the particles:
$F = \frac{2ANmv^2}{6V}$
To calculate the pressure, we divide by the area $$A$$, to find that
$P = \frac{Nmv^2}{3V}$
or, rearranged for comparison to Boyle's Law,
$PV = \frac{Nmv^2}{3}$
Since we have assumed that the particles travel with constant speed $$v$$, then the right side of this equation is a constant. Therefore the product of pressure times volume, $$PV$$, is a constant, in agreement with Boyle's Law. Furthermore, the product $$PV$$ is proportional to the number of particles, also in agreement with the Law of Combining Volumes. Therefore, the model we have developed to describe an ideal gas is consistent with our experimental observations.
We can draw two very important conclusions from this derivation. First, the inverse relationship observed between pressure and volume and the independence of this relationship on the type of gas analyzed are both due to the lack of interactions between gas particles. Second, the lack of interactions is in turn due to the great distances between gas particles, a fact which will be true provided that the density of the gas is low.
## Interpretation of Temperature
The absence of temperature in the above derivation is notable. The other gas properties have all been incorporated, yet we have derived an equation which omits temperature all together. The problem is that, as we discussed at length above, the temperature was somewhat arbitrarily defined. In fact, it is not precisely clear what has been measured by the temperature. We defined the temperature of a gas in terms of the volume of mercury in a glass tube in contact with the gas. It is perhaps then no wonder that such a quantity does not show up in a mechanical derivation of the gas properties.
On the other hand, the temperature does appear prominently in the Ideal Gas Law. Therefore, there must be a greater significance (and less arbitrariness) to the temperature than might have been expected. To discern this significance, we rewrite the last equation above in the form
$PV = \frac{2}{3} N \left( \frac{1}{2} mv^2 \right)$
The last quantity in parentheses can be recognized as the kinetic energy of an individual gas particle, and $$N \left( \frac{1}{2} mv^2 \right)$$ must be the total kinetic energy (KE) of the gas. Therefore
$PV = \frac{2}{3} KE$
Now we insert the Ideal Gas Law for $$PV$$ to find that
$KE = \frac{3}{2} nRT$
This is an extremely important conclusion, for it reveals the answer to the question of what property is measured by the temperature. We see now that the temperature is a measure of the total kinetic energy of the gas. Thus, when we heat a gas, elevating its temperature, we are increasing the average kinetic energy of the gas particles, causing them to move, on average, more rapidly.
## Analysis of Deviations from the Ideal Gas Law
We are at last in a position to understand the observations of deviations from the Ideal Gas Law. The most important assumption of our model of the behavior of an ideal gas is that the gas molecules do not interact. This allowed us to calculate the force imparted on the wall of the container due to a single particle collision without worrying about where the other particles were. In order for a gas to disobey the Ideal Gas Law, the conditions must be such that this assumption is violated.
What do the deviations from ideality tell us about the gas particles? Starting with very low density and increasing the density as in Figure 12.1, we find that, for many gases, the value of $$\frac{PV}{nRT}$$ falls below 1. One way to state this result is that, for a given value of $$V$$, $$n$$, and $$T$$, the pressure of the gas is less than it would have been for an ideal gas. This must be the result of the interactions of the gas particles. In order for the pressure to be reduced, the force of the collisions of the particles with the walls must be less than is predicted by our model of an ideal gas. Therefore, the effect of the interactions is to slow the particles as they approach the walls of the container. This means that an individual particle approaching a wall must experience a force acting to pull it back into the body of the gas. Hence, the gas particles are confined in closer proximity to one another. At this closer range, the attractions of individual particles become significant. It should not be surprising that these attractive forces depend on what the particles are. We note in Figure 12.1 that deviation from the Ideal Gas Law is greater for ammonia than for nitrogen, and greater for nitrogen than for helium. Therefore, the attractive interactions of ammonia molecules are greater than those of nitrogen molecules, which are in turn greater than those of helium atoms. We analyze this conclusion in more detail below.
Continuing to increase the density of the gas, we find in Figure 12.2 that the value of $$\frac{PV}{nRT}$$ begins to rise, eventually exceeding 1 and continuing to increase. Under these conditions, therefore, the pressure of the gas is greater than we would have expected from our model of non-interacting particles. What does this tell us? The gas particles are interacting in such a way as to increase the force of the collisions of the particles with the walls. This requires that the gas particles repel one another. As we move to higher density, the particles are forced into closer and closer proximity. We can conclude that gas particles at very close range experience strong repulsive forces away from one another.
Our model of the behavior of gases can be summarized as follows: at low density, the gas particles are sufficiently far apart that there are no interactions between them. In this case, the pressure of the gas is independent of the nature of the gas and agrees with the Ideal Gas Law. At somewhat higher densities, the particles are closer together and the interaction forces between the particles are attractive. The pressure of the gas now depends on the strength of these interactions and is lower than the value predicted by the Ideal Gas Law. At still higher densities, the particles are excessively close together, resulting in repulsive interaction forces. The pressure of the gas under these conditions is higher than the value predicted by the Ideal Gas Law.
## Observation 3: Boiling Points of simple hydrides
The postulates of the Kinetic Molecular Theory provide us a way to understand the relationship between molecular properties and the physical properties of bulk amounts of substance. As a distinct example of such an application, we now examine the boiling points of various compounds, focusing on hydrides of sixteen elements in the main group (Groups IV through VII). These are given in Table 12.1.
Table 12.1: Boiling Points of Hydrides of Groups IV to VII
Boiling Point $$\left( ^\text{o} \text{C} \right)$$ $$\ce{CH_4}$$ -164 $$\ce{NH_3}$$ -33 $$\ce{H_2O}$$ 100 $$\ce{HF}$$ 20 $$\ce{SiH_4}$$ -111.8 $$\ce{PH_3}$$ -87.7 $$\ce{H_2S}$$ -60.7 $$\ce{HCl}$$ -85 $$\ce{GeH_4}$$ -88.5 $$\ce{AsH_3}$$ -55 $$\ce{H_2Se}$$ -41.5 $$\ce{HBr}$$ -67 $$\ce{SnH_4}$$ -52 $$\ce{SbH_3}$$ -17.1 $$\ce{H_2Te}$$ -2.2 $$\ce{HI}$$ -35
In tabular form, there are no obvious trends here, and therefore no obvious connection to the structure or bonding in the molecules. The data in the table are displayed in a suggestive form, however, in Figure 12.4; the boiling point of each hydride is plotted according to which period (row) of the periodic table the main group element belongs. For example, the Period 2 hydrides ($$\ce{CH_4}$$, $$\ce{NH_3}$$, $$\ce{H_2O}$$, and $$\ce{HF}$$) are grouped in a column to the left of the figure, followed by a column for the Period 3 hydrides ($$\ce{SiH_4}$$, $$\ce{PH_3}$$, $$\ce{H_2S}$$, $$\ce{HCl}$$), etc.
Now a few trends are more apparent. First, the lowest boiling points in each period are associated with the Group IV hydrides ($$\ce{CH_4}$$, $$\ce{SiH_4}$$, $$\ce{GeH_4}$$, $$\ce{SnH_4}$$), and the highest boiling points in each period belong to the Group VI hydrides ($$\ce{H_2O}$$, $$\ce{H_2S}$$, $$\ce{H_2Se}$$, $$\ce{H_2Te}$$). For this reason, the hydrides belonging to a single group have been connected in Figure 12.4.
Figure 12.4: Boiling Points of Main Group Hydrides
Second, we notice that, with the exceptions of $$\ce{NH_3}$$, $$\ce{H_2O}$$, and $$\ce{HF}$$, the boiling points of the hydrides always increase in a single group as we go down the periodic table: for example, in Group IV, the boiling points increase in the order $$\ce{CH_4} < \ce{SiH_4} < \ce{GeH_4} < \ce{SnH_4}$$. Third, we can also say that the hydrides from Period 2 appear to have unusually high boiling points except for $$\ce{CH_4}$$, which as noted has the lowest boiling point of all.
We begin our analysis of these trends by assuming that there is a relationship between the boiling points of these compounds and the structure and bonding in their molecules. Recalling our kinetic molecular model of gases and liquids, we recognize that a primary difference between these two phases is that the strength of the interaction between the molecules in the liquid is much greater than that in the gas, due to the proximity of the molecules in the liquid. In order for a molecule to leave the liquid phase and enter into the gas phase, it must possess sufficient energy to overcome the interactions it has with other molecules in the liquid. Also recalling the kinetic molecular description, we recognize that, on average, the energies of molecules increase with increasing temperature. We can conclude from these two statements that a high boiling point implies that significant energy is required to overcome intermolecular interactions. Conversely, a substance with a low boiling point must have weak intermolecular interactions, surmountable even at low temperature.
In light of these conclusions, we can now look at Figure 12.4 as directly (though qualitatively) revealing the comparative strengths of intermolecular interactions of the various hydrides. For example, we can conclude that, amongst the hydrides considered here, the intermolecular interactions are greatest between $$\ce{H_2O}$$ molecules and weakest between $$\ce{CH_4}$$ molecules. We examine the three trends in this figure, described above, in light of the strength of intermolecular forces.
First, the most dominant trend in the boiling points is that, within a single group, the boiling points of the hydrides increase as we move down the periodic table. This is true in all four groups in Figure 12.4; the only exceptions to this trend are $$\ce{NH_3}$$, $$\ce{H_2O}$$, and $$\ce{HF}$$. We can conclude that, with notable exceptions, intermolecular interactions increase with increasing atomic number of the central atom in the molecule. This is true whether the molecules of the group considered have dipole moments (as in Groups V, VI, and VII) or not (as in Group IV). We can infer that the large intermolecular attractions for molecules with large central atoms arises from the large number of charged particles in these molecules.
This type of interaction arises from forces referred to as London forces or dispersion forces. These forces are believed to arise from the instantaneous interactions of the charged particles from one molecule with the charged particles in an adjacent molecule. Although these molecules may not be polar individually, the nuclei in one molecule may attract the electrons in a second molecule, thus inducing an instantaneous dipole in the second molecule. In turn, the second molecule induces a dipole in the first. Thus, two non-polar molecules can interact as if there were dipole-dipole attractions between them, with positive and negative charges interacting and attracting. The tendency of a molecule to have an induced dipole is called the polarizability of the molecule. The more charged particles there are in a molecule, the more polarizable a molecule is and the greater the attractions arising from dispersion forces will be.
Second, we note that, without exception, the Group IV hydrides must have the weakest intermolecular interactions in each period. As noted above, these are the only hydrides that have no dipole moment. Consequently, in general, molecules without dipole moments have weaker interactions than molecules which are polar. We must qualify this carefully, however, by noting that the nonpolar $$\ce{SnH_4}$$ has a higher boiling point than the polar $$\ce{PH_3}$$ and $$\ce{HCl}$$. We can conclude from these comparisons that the increased polarizability of molecules with heavier atoms can offset the lack of a molecular dipole.
Third, and most importantly, we note that the intermolecular attractions involving $$\ce{NH_3}$$, $$\ce{H_2O}$$, and $$\ce{HF}$$ must be uniquely and unexpectedly large, since their boiling points are markedly out of line with those of the rest of their groups. The common feature of these molecules is that they contain small atomic number atoms which are strongly electronegative, which have lone pairs, and which are bonded to hydrogen atoms. Molecules without these features do not have unexpectedly high boiling points. We can deduce from these observations that the hydrogen atoms in each molecule are unusually strongly attracted to the lone pair electrons on the strongly electronegative atoms with the same properties in other molecules. This intermolecular attraction of a hydrogen atom to an electronegative atom is referred to as hydrogen bonding. It is clear from our boiling point data that hydrogen bonding interactions are much stronger than either dispersion forces or dipole-dipole attractions.
## Review and Discussion Questions
Explain the significance to the development of the kinetic molecular model of the observation that the ideal gas law works well only at low pressure.
Explain the significance to the development of the kinetic molecular model of the observation that the pressure predicted by the ideal gas law is independent of the type of gas.
Sketch the value of $$\frac{PV}{nRT}$$ as a function of density for two gases, one with strong intermolecular attractions and one with weak intermolecular attractions but strong repulsions.
Give a brief molecular explanation for the observation that the pressure of a gas at fixed temperature increases proportionally with the density of the gas.
Give a brief molecular explanation for the observation that the pressure of a gas confined to a fixed volume increases proportionally with the temperature of the gas.
Give a brief molecular explanation for the observation that the volume of a balloon increases roughly proportionally with the temperature of the gas inside the balloon.
Explain why there is a correlation between high boiling point and strong deviation from the Ideal Gas Law.
Referring to Figure 12.4, explain why the hydride of the Group IV element always has the lowest boiling point in each period.
Explain why the Period 2 hydrides except $$\ce{CH_4}$$ all have high boiling points, and explain why $$\ce{CH_4}$$ is an exception. | 2022-05-20 00:31:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7523511052131653, "perplexity": 224.68463257843615}, "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/1652662530553.34/warc/CC-MAIN-20220519235259-20220520025259-00748.warc.gz"} |
https://www.skepticalcommunity.com/viewtopic.php?f=5&t=48933 | ## Nuclear vs Renewables
We are the Borg.
Pyrrho
Posts: 27829
Joined: Sat Jun 05, 2004 2:17 am
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Location: Division 6
### Nuclear vs Renewables
https://quillette.com/2019/02/27/why-re ... he-planet/
Germany’s carbon emissions have been flat since 2009, despite an investment of $580 billion by 2025 in a renewables-heavy electrical grid, a 50 percent rise in electricity cost. Meanwhile, France produces one-tenth the carbon emissions per unit of electricity as Germany and pays little more than half for its electricity. How? Through nuclear power. Then, under pressure from Germany, France spent$33 billion on renewables, over the last decade. What was the result? A rise in the carbon intensity of its electricity supply, and higher electricity prices, too.
The flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.
Rob Lister
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Location: Swimming in Lake Ed
### Re: Nuclear vs Renewables
Nuclear wins hands-down in all categories but love.
As a humorious aside.
Spoiler:
Abdul Alhazred
Posts: 75525
Joined: Mon Jun 07, 2004 1:33 pm
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Location: Chicago
### Re: Nuclear vs Renewables
In fact nuclear is (or can be) renewable (literally).
It can even be greenie weenie if were careful with it.
Any man writes a mission statement spends a night in the box.
-- our mission statement plappendale
sparks
Posts: 15167
Joined: Fri Oct 26, 2007 4:13 pm
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### Re: Nuclear vs Renewables
One more time on this and then I'll shut the fuck up: LFTR.
You can lead them to knowledge, but you can't make them think.
Abdul Alhazred
Posts: 75525
Joined: Mon Jun 07, 2004 1:33 pm
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Location: Chicago
### Re: Nuclear vs Renewables
Liquid fluoride?
You mean like the Commies put in our water?
And what if someone throws some into a child's face?
Why do you hate children?
Any man writes a mission statement spends a night in the box.
-- our mission statement plappendale
sparks
Posts: 15167
Joined: Fri Oct 26, 2007 4:13 pm
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### Re: Nuclear vs Renewables
Search again AA. If you can't come up with better than what you just posted, then go fuck yourself. And don't disgrace this thread with your bullshit again. You intolerable wanker.
You can lead them to knowledge, but you can't make them think.
Abdul Alhazred
Posts: 75525
Joined: Mon Jun 07, 2004 1:33 pm
Title: Yes, that one.
Location: Chicago
### Re: Nuclear vs Renewables
sparks wrote:
Thu Feb 28, 2019 11:16 pm
Search again AA. If you can't come up with better than what you just posted, then go fuck yourself. And don't disgrace this thread with your bullshit again. You intolerable wanker.
In fact I already know what it is without googling, but made a conscious deliberate moral decision to piss you off.
Any man writes a mission statement spends a night in the box.
-- our mission statement plappendale
sparks
Posts: 15167
Joined: Fri Oct 26, 2007 4:13 pm
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### Re: Nuclear vs Renewables
Abdul Alhazred wrote:
Thu Feb 28, 2019 11:25 pm
sparks wrote:
Thu Feb 28, 2019 11:16 pm
Search again AA. If you can't come up with better than what you just posted, then go fuck yourself. And don't disgrace this thread with your bullshit again. You intolerable wanker.
In fact I already know what it is without googling, but made a conscious deliberate moral decision to piss you off.
Thank you. You're the only one who cares these days.
You can lead them to knowledge, but you can't make them think.
Rob Lister
Posts: 20653
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Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: Nuclear vs Renewables
sparks wrote:
Thu Feb 28, 2019 9:41 pm
One more time on this and then I'll shut the fuck up: LFTR.
Or really just a handful of LFTRs that burn the 'waste' from the more conventional reactors. LFTR technology isn't a panacea, after all; it does have its issues.
Witness
Posts: 20714
Joined: Thu Sep 19, 2013 5:50 pm
### Re: Nuclear vs Renewables
From 2016:
Uranium Seawater Extraction Makes Nuclear Power Completely Renewable
America, Japan and China are racing to be the first nation to make nuclear energy completely renewable. The hurdle is making it economic to extract uranium from seawater, because the amount of uranium in seawater is truly inexhaustible.
And it seems America is in the lead. New technological breakthroughs from DOE’s Pacific Northwest (PNNL) and Oak Ridge (ORNL) national laboratories have made removing uranium from seawater within economic reach and the only question is - when will the source of uranium for our nuclear power plants change from mined ore to seawater extraction?
Nuclear fuel made with uranium extracted from seawater makes nuclear power completely renewable. It’s not just that the 4 billion tons of uranium in seawater now would fuel a thousand 1,000-MW nuclear power plants for a 100,000 years. It’s that uranium extracted from seawater is replenished continuously, so nuclear becomes as endless as solar, hydro and wind.
Specifically, this latest technology builds on work by researchers in Japan and uses polyethylene fibers coated with amidoxime to pull in and bind uranium dioxide from seawater (see figure above). In seawater, amidoxime attracts and binds uranium dioxide to the surface of the fiber braids, which can be on the order of 15 centimeters in diameter and run multiple meters in length depending on where they are deployed (see figure below).
After a month or so in seawater, the lengths are remotely released to the surface and collected. An acid treatment recovers the uranium in the form of a uranyl complex, regenerating the fibers that can be reused many times. The concentrated uranyl complex then can be enriched to become nuclear fuel.
https://www.forbes.com/sites/jamesconca ... renewable/
So where is it?
Rob Lister
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Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: Nuclear vs Renewables
Not much new under this sun. It's still a hundred times cheaper to mine it. When that changes, thorium will be the fuel of choice.
sparks
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### Re: Nuclear vs Renewables
I've read that the tailings from rare Earth mines are loaded with Thorium. Just a matter of refinement. Have to take exception with the statement that sea water extraction is renewable. It is a large reservoir of Uranium, but it's hardly inexhaustible. There is only so much source material to replace what we would remove from sea water. But then, one day the sun will run out of fuel too, so I realize I'm nit picking here. Wind, solar and hydro-electric are 'renewable' only so long as the sun shines you Greenmunists!
You can lead them to knowledge, but you can't make them think.
robinson
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Joined: Sat Aug 12, 2006 2:01 am
Title: Be good to each other
Location: USA
### Re: Nuclear vs Renewables
Nuclear is only affordable if you lay the real cost onto the next 500 generations.
You never know what's going to happen, then some shit happens nobody saw coming, then later somebody says they knew it was coming, then some new shit happens nobody saw coming, rinse and repeat
sparks
Posts: 15167
Joined: Fri Oct 26, 2007 4:13 pm
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### Re: Nuclear vs Renewables
That depends on how you do nuclear Robby. Our first attempt with solid fuel is fucked, no question about it. But Weinberg who patented his designs on those first attempts quickly realized that it was not the best way. At ORNL, he and some others built and operated the first Molten Salt Reactor as a test bed to calibrate what it would take to scale it up and eventually, use Thorium to make new fuel. All started as a part of a hair brained idea to run nuclear powered bombers from fucking fission. Never came to fruition of course because Von Braun, the fucking little Nazi, figured out ICBMs in the meantime. And then in the early 70s, our buddy Tricky Dick cancelled the program IIRC. That stupid sonofabitch.
Good night and good luck fuckers.
You can lead them to knowledge, but you can't make them think. | 2019-07-22 14:55:19 | {"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": 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": 0.3541533946990967, "perplexity": 7527.743648609394}, "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-2019-30/segments/1563195528037.92/warc/CC-MAIN-20190722133851-20190722155851-00343.warc.gz"} |
https://www.allaboutcircuits.com/video-tutorials/op-amp-basics-introduction-to-the-operational-amplifier/ | # Op-Amp Basics: Introduction to the Operational Amplifier
July 12, 2020 by Robert Keim
In this video, we will introduce a component that is an essential tool for anyone who designs analog or mixed-signal circuits. This component is called an operational amplifier, or op-amp for short.
Engineers currently have access to thousands of different op-amp ICs. It is somewhat misleading to identify all of these devices using the term “operational amplifier,” because in reality they form a diverse group of components. On the other hand, op-amps consistently exhibit various fundamental characteristics, and in this way they represent a fairly unified component category.
### Why Do We Use Op-Amps?
Before we begin exploring the defining electrical characteristics of op-amps, we need to understand why these components are so popular and effective.
• Op-amps have been produced in large quantities for decades, and consequently engineers have access to a large and diverse collection of parts that offer both low cost and high performance.
• Operational amplifiers are extremely versatile. It is difficult to think of an analog circuit that cannot be either implemented using an op-amp or improved by the addition of an op-amp.
• Designing circuits around op-amps is much easier than using discrete transistors. The electrical characteristics of operational amplifiers lead to simplifying assumptions, and in many applications, these assumptions do not introduce significant differences between the theoretical circuit and the actual circuit.
### The Op-Amp Circuit Symbol
The simplified, idealized op-amp is a three-terminal device.
The two terminals on the left are inputs, and the terminal on the right is the output. Note that the input terminals have different labels: the plus sign indicates the non-inverting input terminal, and the minus sign indicates the inverting input terminal.
A real op-amp needs at least five terminals—two inputs, one output, and two power-supply connections:
##### A dual-supply op-amp circuit (on the left) uses a positive supply voltage and a negative supply voltage. In a single-supply configuration (on the right), the negative supply terminal is connected to ground.
We often omit the power-supply terminals when we draw an op-amp because we assume that the device is connected to supply voltages that enable proper operation within the context of a given application. However, it’s important to remember that the op-amp’s output-voltage range is limited by its supply voltages.
### The Op-Amp Electrical Model
The idealized input-to-output relationship of a typical op-amp is conveyed by the following diagram:
Despite the complex circuitry that is present within a real operational amplifier, we can successfully carry out many op-amp-based design tasks by assuming that the op-amp is a voltage-controlled voltage source (VCVS). The controlling voltage is $(V_{IN+} - V_{IN-})$, and the factor of proportionality between the controlling voltage and the voltage generated by the VCVS is the op-amp’s gain, denoted by A:
$V_{OUT} = A(V_{IN+}-V_{IN-})$
Op-amps have very high gain, often above $10^5$ or even $10^6$ . As we’ll see in a future video, this high (ideally infinite) gain is extremely important—not because we frequently need to increase the amplitude of a signal by five or six orders of magnitude, but because an amplifier that combines high gain with a differential input stage provides a convenient means of exploiting the beneficial properties associated with negative feedback.
Let’s look at a few additional characteristics that are implied by the VCVS model shown above.
• An op-amp is a differential amplifier: it amplifies the difference between the two input voltages.
• It follows from the previous statement that op-amps exhibit common-mode rejection. The op-amp will reject (i.e., ignore) any voltage component, such as noise or a DC offset, that is present in both input signals.
• An op-amp has a differential input stage and a single-ended output; thus, it can be considered a differential-to-single-ended converter. However, it turns out that real-life op-amp applications are associated much more closely with single-ended input signals. In fact, we have another name for devices that are designed specifically for differential input signals: they’re called instrumentation amplifiers.
### Summary
• Operational amplifiers are used in countless analog and mixed-signal circuits.
• These extremely popular ICs are affordable, versatile, and easy to use.
• An idealized op-amp has three terminals: two inputs and one output.
• Op-amp functionality is similar to that of a voltage-controlled voltage source that applies an extremely high gain to the difference between the voltage at the non-inverting input terminal and the voltage at the inverting input terminal.
1 Comment
• B
Bernie Hutchins July 14, 2020
The video shows:
“ Pro-Tip
Op-amp gain (A) is typically 10^5 to 10^6 “ | 2021-10-22 12:39:02 | {"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": 0.5866216421127319, "perplexity": 1454.1889083140723}, "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/1634323585507.26/warc/CC-MAIN-20211022114748-20211022144748-00322.warc.gz"} |
https://www.physicsforums.com/threads/changing-rho-finding-resistance.774556/ | # Changing Rho, Finding Resistance
1. Oct 5, 2014
### Bluestribute
1. The problem statement, all variables and given/known data
A wire with cross-sectional radius 0.91 mm lying along the x axis from x=0 to x=0[PLAIN]http://loncapa.mines.edu/adm/jsMath/fonts/cmmi10/alpha/100/char3A.png90 [Broken] m is made of an alloy that varies with its length in such a way that the resistivity is given by [PLAIN]http://loncapa.mines.edu/adm/jsMath/fonts/cmmi10/alpha/100/char1A.png=6x^1 [Broken] [PLAIN]http://loncapa.mines.edu/adm/jsMath/fonts/cmr10/alpha/100/char0A.png[PLAIN]http://loncapa.mines.edu/adm/jsMath/fonts/cmsy10/alpha/100/char01.png [Broken] m, for x in meters. What is the resistance of this wire?
2. Relevant equations
dR=(rho)dL/A
3. The attempt at a solution
(I can't find the symbol window so bare with my explanation). Resistance is the integral of rho times length, divided by cross sectional area. I figured since L and A are constants, I could pull them out and simply do: rho, integrated from 0 to 0.9, times L/A, with everything converted to meters. Didn't work.
Then I tried not integrated and just plugging and chugging. Didn't work.
Then I came here.
Last edited by a moderator: May 7, 2017
2. Oct 5, 2014
### Staff: Mentor
What didn't work? Please show what you got and how you got it.
Also, is there a typo in $\rho$? It seems odd to me that it would be 6x1 ohms/m. I don't know why anyone would write x1 instead of just x.
Last edited by a moderator: May 7, 2017
3. Oct 5, 2014
### Bluestribute
So I'm on my phone, but I integrated the rho function and multipled by L/A, L being the difference in the bounds and A being the given mm amount (converted to meters).
It's to the first power cause the numbers are random, including the exponents
4. Oct 5, 2014
### Staff: Mentor
So what value did you get?
5. Oct 5, 2014
### Dick
The notation rho*dL/A doesn't mean integrate rho and then multiply by L/A. It means integrate rho WITH RESPECT to L and then divide by A. By all means, show your work. I think you are multiplying by an extra factor L. Check the units. Remember dL has units of m. Are they ohms or something else?
6. Oct 5, 2014
### RUber
$\int dR = \int_a^b \frac{\rho}{A} dL \implies R=\frac1A\int_a^b \rho dL$ You should not have to multiply by length, it is included in the integral.
7. Oct 6, 2014
### Bluestribute
Ok, I'm back and found out where I was going wrong. Also, the parameters change each time, so don't freak out with me inputting new numbers in my explanation.
So yes, I don't know why I was multiplying by length afterwards . . . Good to catch it now though. Anyways, I integrated the resistivity across the bounds: the integral of 4x from 0 to 0.5. Then I divided by the AREA of the given radius. That's where I was going wrong multiple times. They gave me a radius of 0.46 MM. For some reason I was using this as "A" instead of solving for A . . . So in short, integrated my resistivity function across my wire length and divided by the cross sectional area.
Got it instantly when I started writing out my variables here and realized it was radius, not area . . . | 2018-02-19 02:37:33 | {"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": 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": 0.7659829258918762, "perplexity": 1294.922874034692}, "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/1518891812306.16/warc/CC-MAIN-20180219012716-20180219032716-00576.warc.gz"} |
http://mathhelpforum.com/calculators/198931-ti-nspire-cas-trig-equations-print.html | # Ti-nspire cas trig equations
• May 17th 2012, 11:34 AM
noork85
Ti-nspire cas trig equations
How do i solve trig equations on this? And also factoring.
It seems the calc doesnt recignie cos^2_x keeps saying missing (
I want to factor and solve for x by putting in cot x cos^2x-2cot x
How do i put it in?
Thanks!
• May 17th 2012, 01:33 PM
Plato
Re: Ti-nspire cas trig equations
Quote:
Originally Posted by noork85
How do i solve trig equations on this? And also factoring.
It seems the calc doesnt recignie cos^2_x keeps saying missing (
I want to factor and solve for x by putting in cot x cos^2x-2cot x
With most CAS $cos^2(x)$ is written as $[cos(x)]^2.$
• May 17th 2012, 02:00 PM
noork85
Re: Ti-nspire cas trig equations
Thank u so much!
Also, how would i graph if im given,
Vertex (4,-1); point (2,3)
I know how to graph but smetimes i messup with signs and i screw up the whole graph.
• May 17th 2012, 02:33 PM
skeeter
Re: Ti-nspire cas trig equations
Quote:
Originally Posted by noork85
Thank u so much!
Also, how would i graph if im given,
Vertex (4,-1); point (2,3)
I know how to graph but smetimes i messup with signs and i screw up the whole graph.
is the graph a parabola?
• May 17th 2012, 02:35 PM
noork85
Re: Ti-nspire cas trig equations
Yes.
• May 17th 2012, 02:46 PM
skeeter
Re: Ti-nspire cas trig equations
you should be familiar with the vertex form of a parabola ...
$y = a(x-h)^2 + k$
sub in the given vertex coordinates $(h,k)$ and the coordinates of the given point $(x,y)$ ... solve for $a$ , then graph.
btw ... next time state what you are graphing. if you had not mentioned a vertex, I would've had no idea what you wanted graphed in the first place.
• May 17th 2012, 02:48 PM
noork85
Re: Ti-nspire cas trig equations
Yes, i realized that, sorry.
I know how to graph on paper, i just wanted to know how to graph on the calc so i can verify my ans.
Thanks.
• May 17th 2012, 03:00 PM
skeeter
Re: Ti-nspire cas trig equations
Quote:
Originally Posted by noork85
Yes, i realized that, sorry.
I know how to graph on paper, i just wanted to know how to graph on the calc so i can verify my ans.
Thanks.
have you taken a look at the owner's manual?
• May 17th 2012, 03:08 PM
noork85
Re: Ti-nspire cas trig equations
Yes, and its all very confusing.
• May 17th 2012, 03:08 PM
noork85
Re: Ti-nspire cas trig equations
Do u have a ti-nspire cas calc?
• May 17th 2012, 03:13 PM
skeeter
Re: Ti-nspire cas trig equations
no | 2017-06-24 05:40:12 | {"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": 0.9236204028129578, "perplexity": 4247.478122125021}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320226.61/warc/CC-MAIN-20170624050312-20170624070312-00084.warc.gz"} |
https://bioinformatics.stackexchange.com/tags/methylation/hot?filter=all | # Tag Info
## Hot answers tagged methylation
6 votes
Accepted
### How to understand methylation level?
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• 1,304
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• 2,099
2 votes
### Recommendations for missing value imputation - DNA methylation data
I think this is still an active field of research. I have heard of Phenix, which might be appropriate.
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• 12k
2 votes
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$\beta$ values are the ratio of methylated to unmethylated probe intensities. This means that $\beta \in [0, 1]$, so it can't be normally distributed, by definition. Its distribution may have sections ...
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1 vote
### ambiguous calls in nanopolish
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• 2,827
1 vote
### WGBS vs ONT, which one should I trust?
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• 7,980
1 vote
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1 vote
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• 1,222
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1 vote
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1 vote
### Error in createMAE function: non-unique values when setting 'row.names' in TCGA LIHC Data
The error message is pretty clear: R requires unique row names on data frames and some of the "values" that you are trying to set as row names are non-unique. You will need to provide a vector i) of ...
• 3,507
1 vote
Accepted
### DNA bioinformatic search and methylation analysis by qMSP
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• 1,222
1 vote
### Extract LINE-1 methylation status from WGBS data
Via BEDOPS: \$ bedmap —-echo —-echo-map-id-uniq <(bam2bed < reads.bam) <(rmsk2bed < repeats.rmsk) > answer.bed
• 3,065
1 vote
### Extract LINE-1 methylation status from WGBS data
After a more extensive search, I found a useful answer that I'll share in case anyone else wonders how to do this. I'm using MethylKit package for differential methylation expression and it is quite ...
• 395
1 vote
### Melt p-values for CpG sites mapping to the same gene
What is typically done in methylation analysis is to assess the islands of methylations. Check this workflow, in the section linked it uses some predefined islands for instance. I am no expert on ...
• 4,622
Only top scored, non community-wiki answers of a minimum length are eligible | 2022-09-29 10:31:07 | {"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": 0.39684805274009705, "perplexity": 6882.958907109464}, "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/1664030335350.36/warc/CC-MAIN-20220929100506-20220929130506-00003.warc.gz"} |
http://papers.nips.cc/paper/8569-scalable-gromov-wasserstein-learning-for-graph-partitioning-and-matching | # NIPS Proceedingsβ
## Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching
[PDF] [BibTeX] [Supplemental]
### Abstract
We propose a scalable Gromov-Wasserstein learning (S-GWL) method and establish a novel and theoretically-supported paradigm for large-scale graph analysis. The proposed method is based on the fact that Gromov-Wasserstein discrepancy is a pseudometric on graphs. Given two graphs, the optimal transport associated with their Gromov-Wasserstein discrepancy provides the correspondence between their nodes and achieves graph matching. When one of the graphs is a predefined graph with isolated but self-connected nodes ($i.e.$, disconnected graph), the optimal transport indicates the clustering structure of the other graph and achieves graph partitioning. Further, we extend our method to multi-graph partitioning and matching by learning a Gromov-Wasserstein barycenter graph for multiple observed graphs. Our method combines a recursive $K$-partition mechanism with a warm-start proximal gradient algorithm, whose time complexity is $\mathcal{O}(K(E+V)\log_K V)$ for graphs with $V$ nodes and $E$ edges. To our knowledge, our method is the first attempt to make Gromov-Wasserstein discrepancy applicable to large-scale graph analysis and unify graph partitioning and matching into the same framework. It outperforms state-of-the-art graph partitioning and matching methods, achieving a trade-off between accuracy and efficiency. | 2020-01-21 06:58:08 | {"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": 0.2743038237094879, "perplexity": 1172.130130820954}, "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-2020-05/segments/1579250601615.66/warc/CC-MAIN-20200121044233-20200121073233-00219.warc.gz"} |
https://math.stackexchange.com/questions/760499/question-from-russian-math-olympiad | # question from russian math olympiad
Let $f(x,y)=\frac{1}{2}(x+y-1)(x+y-2)$ be a function of two positive integers. Prove that for any positive integer $z$ there exists a single pair $x,y$ such that $f(x,y)=z$.
Isn't this clearly wrong? E.g. for $z=5$, there can be no successive pairs of integers $a,a'$ such that $\frac{1}{2}aa'=5$?
• Seems kinda easy for a olympiad question,from where did you get this problem? – kingW3 Apr 19 '14 at 15:03
• @kingW3 How easy? The question is wrong. It's from 1988. – user144248 Apr 19 '14 at 15:04
• Even if it was right,they don't give that type of questions.If it was written right it would still seem pretty easy.Anyway where did you get this problem? EDIT:Maybe there is minus or plus sign somewhere not intended – kingW3 Apr 19 '14 at 15:08
• @kingW3 It's from here andrej.fizika.org/ostalo/gimnazija/math/ruske_olimpijade/…, 1988, 51.9.4 – user144248 Apr 19 '14 at 15:12
Yes, the question as stated is incorrect.
• It's from 1988. Maybe it has been wrongly translated. – user144248 Apr 19 '14 at 14:47
From question we can form the equation $$z = 1/2(x+y-2)(x+y-1)$$
let $x+y = a$
$$2z = (a - 2)(a - 1)$$ $$2z = a^2 -3a + 2$$ $$a^2 -3a -2(z - 1)=0$$ apply $b^2 - 4ac$; $9+8(z-1)$
Now the discriminant have to be greater than or equal to $0$.
Let $z = 1$ the smallest positive integer. Since $9>0$ then any positive integer value for $z$ will satisfy and the $x +y$ can be found; hence proved; | 2019-09-15 20:09:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7483199238777161, "perplexity": 606.4945251284997}, "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/1568514572289.5/warc/CC-MAIN-20190915195146-20190915221146-00473.warc.gz"} |
https://qiskit.org/documentation/locale/pt_BR/stubs/qiskit.aqua.components.initial_states.Zero.html | qiskit.aqua.components.initial_states.Zero¶
class Zero(num_qubits)[código fonte]
The zero (null/vacuum) state.
This is suitable for those situations in which the all-zeros state is the desired state. This is the case for a vacuum state in physics or chemistry
Parâmetros
num_qubits (int) – Number of qubits, has a minimum value of 1.
__init__(num_qubits)[código fonte]
Parâmetros
num_qubits (int) – Number of qubits, has a minimum value of 1.
Methods
__init__(num_qubits) type num_qubits int construct_circuit([mode, register]) Construct the statevector of desired initial state.
Attributes
property bitstr
construct_circuit(mode='circuit', register=None)[código fonte]
Construct the statevector of desired initial state.
Parâmetros
• modevector or circuit. The vector mode produces the vector. While the circuit constructs the quantum circuit corresponding that vector.
• register – qubits for circuit construction.
Retorna
statevector.
Tipo de retorno
QuantumCircuit or numpy.ndarray
Levanta
AquaError – when mode is not ‘vector’ or ‘circuit’. | 2021-05-16 22:42:47 | {"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": 0.4441787004470825, "perplexity": 6908.897358393401}, "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-2021-21/segments/1620243989914.60/warc/CC-MAIN-20210516201947-20210516231947-00089.warc.gz"} |
https://mechcontent.com/how-venturimeter-used-flow-measuring/ | # How venturimeter is used for flow measuring? [with Pdf]
Hey friends, just now I have successfully performed flow measuring experiment on venturimeter setup in my college lab. Before performing this flow measurement practical I have done simple searches about the question “How venturimeter is used for flow measuring?” which I have mentioned below.
For the measurement of flow rate, the venturimeter is fitted between the pipe of which flow rate is to measure. The manometer attached before converging section and at the throat section shows the pressure difference between two section. Then by using Bernoulli’s equation and continuity equation we can easily calculate the flow rate through pipe.
Contents
## What is the venturimeter ?
Venturimeter is the device which works on principle of Bernoulli’s equation to measure the flow rate of fluid. It generally consists of following four components.
1. Converging section
2. Throat section
3. Diverging section
4. U-tube manometer
Let’s know in detail what is the role of each component.
1) Converging section:-
Converging section means the reduction in cross section area in the direction of flow.
as the liquid pass through the converging section the pressure energy of liquid gets converted into kinetic energy.
Hence because of converging section the pressure of the liquid decreases and the velocity of liquid increases.
P1>P2 and V2>V1
2) Throat section:-
Throat section has the constant cross-sectional area equal to the exit area of converging section.
The pressure and velocity across the throat section remains constant.
3) Diverging section:-
The diverging section means gradual increase in cross sectional area in the direction of flow.
Hence as the liquid passes through the the diverging section the kinetic energy of liquid is again gets converted into the pressure energy.
It means pressure across the diverging section again increases and velocity decreases.
4) U-tube manometer:-
U-tube manometer is used in venturimeter to record the pressure difference between pipe cross section before converging section and throat cross section.
The one end of a U-tube manometer is connected at the section 1 before converging section and at section 2 (throat).
## What is the Working principle of venturimeter?
• The venturimeter is based on Bernoulli’s principle.
• While passing through the venturimeter the fluid experience decrease in pressure energy in converging section and further it experience again increase in its pressure energy in divergent section.
• This change in pressure can be measured using U-tube manometer and then by using Bernoulli’s and continuity equation we can easily obtain value of flow rate.
## How venturimeter is used for flow measuring?
1) As shown in figure, liquid enters into the device at section-1 with initial velocity V1 and pressure P1.
2) While passing through the converging section some amount of pressure energy of fluid gets converted into kinetic energy.
3) Hence the velocity of liquid increase to the V2 and pressure decreases to P2.
4) In throat section there is no change in the velocity and pressure of liquid.
5) The U-tube manometer measures the pressure difference between two sections.
6) Further when liquid passes through the divergent section, the pressure of liquid gradually increases and velocity again reaches to its initial value.
7) Now as we have values of the inlet and throat area of Venturi meter and pressure difference we can easily calculate the flow rate using Bernoulli’s and continuity equation.
## How to calculate flow through venturimeter?
The notations to be used in venturimeter experiment are,
A1 = Area at section 1
A2= Area at section 2
P1= Inlet pressure at section 1
P2= Pressure at throat section
V1= Inlet velocity
V2= Velocity at throat section
h = Pressure head between two sections
By applying Bernoulli’s theorem across the section 1 and 2
\frac{P_{1}}{\rho g}+\frac{V_{1}^{2}}{2g}+Z_{1}= \frac{P_{2}}{\rho g}+\frac{V_{2}^{2}}{2g}+Z_{2}
For the horizontally mounted venturimeter,
Z_{1}=Z_{2}
Now the equation will become,
\frac{P_{1}}{\rho g}+\frac{V_{1}^{2}}{2g}= \frac{P_{2}}{\rho g}+\frac{V_{2}^{2}}{2g}
\frac{P_{1}-P_{2}}{\rho g}= \frac{V_{2}^{2}-V_{1}^{2}}{2g}
\frac{\Delta P}{\rho g}= \frac{V_{2}^{2}-V_{1}^{2}}{2g}
h= \frac{V_{2}^{2}-V_{1}^{2}}{2g}
2gh=V_{2}^{2}-V_{1}^{2}
………..Equation 1
By continuity equation,
A_{1}V_{1}=A_{2}V_{2}
V_{1}=\frac{A_{2}}{A_{1}}\times V_{2}
Now by putting this value the equation 1 become,
2gh=V_{2}^{2}- ( \frac{A_{2}}{A_{1}}\times V_{2} )^{2}
2gh=[ 1-(\frac{A_{2}}{A_{1}})^{2} ]\times V_{2}^{2}
2gh=\frac{A_{1}^{2}-A_{2}^{2}}{A_{1}^{2}}\times V_{2}^{2}
V_{2}=\frac{A_{1}}{\sqrt{A_{1}^{2}-A_{2}^{2}}}\times \sqrt{2gh}
Now, Flow rate is given by
Q=A_{2}V_{2}=\frac{A_{1}\times A_{2}}{\sqrt{A_{1}^{2}-A_{2}^{2}}}\times \sqrt{2gh}
This is the required equation for the flowrate through horizontally mounted venturimeter.
## Where we can use venturimeter?
The venturimeter is generally used in following applications
1) In industrial pipes to measure flow rate of liquid and slurry.
2) it gives higher accuracy and hence it is generally used where higher accuracy is required.
3) it is used in applications where high pressure recovery is necessary.
4) it can also use for slurry and dirt water.
5) Also used for incompressible as well as compressible fluids. | 2022-11-29 04:05:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7316736578941345, "perplexity": 2481.107875797576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00241.warc.gz"} |
https://publications.drdo.gov.in/ojs/index.php/dsj/article/download/295/4773 | Estimation of Life of an Elastomeric Component: A Stochastic Model
Life of equipment has been an area of great importance to engineers, however, largely unexplored. In the case of elastomers, this becomes more critical because of faster degradation in properties of the elastomers, and thereby performance of the elastomeric item, when compared to metals-the degradation referred to as aging of the elastomer. The present work focuses on the development of a stochastic model for estimating life of a vibration isolator, which finds many defence applications, to attenuate noise and vibration from the machinery. A majority of the vibration isolators use rubber for attenuation, and therefore, the life of the isolator invariably depends on the life of the rubber. The methodology of life estimation has been based on the Arrhenius theory of chemical kinetics and is applicable where the material degrades relatively faster at higher temperature so that the degradation rates can be studied. Statistical techniques have been applied to arrive at a reliable estimation. The method can be used for reliable estimation of storage life of elastomeric products, and thereby help to maintain cost-effective inventory.
Elastomeric products, like vibration isolators, rubber seals, etc., are commonly used for long-term applications in the industry/defence. A vibration isolator provides attenuation to the vibration and noise generated by the machinery and finds a number of applications in the defence sector, like onboard naval ships, missiles, etc. The key element in majority of the vibration isolators is the rubber, an elastomer, which dampens the vibration considerably, and thus provides attenuation. Similarly, seals of various types and other elastomeric products are used extensively in defence. When the elastomeric products are manufactured, these are supplied to the user in an optimum state for the desired performance. Before being put to service, many a times, these products are stored for a considerable period of time. However, the physical and dynamic properties of these products degrade with time and the performance deteriorates, mainly due to the degradation of the elastomeric material. The degradation of the elastomeric material is commonly referred to as ageing. Various factors contribute to the ageing of an elastomer, the important ones being oxygen, temperature, UV rays, ozone, etc. Therefore, it is imperative for the user to know the storage life, also called the shelf life, of these products. Interest in this area is high and a number of papers have been published on the estimation of shelf life as well as the service life of an elastomer by various authors, using the principle of accelerated ageing. These tests are carried out for two distinct purposes. Firstly, these are intended to measure changes in the elastomer at the (elevated) service temperature or, secondly, these can be used as accelerated tests to estimate the degree of change, which would take place over much longer times at normal ambient temperature1. A series of accelerated exposure tests were performed by Itoh2, et al. on various rubber materials including high damping rubber (HDR) to investigate the degradation due to different environmental factors. It was found that the thermal oxidation is the most predominant degradation factor affecting the HDR material.
Mainly two models exist in the literature on life estimation of an elastomer, namely, Arrhenius model and WLF model. There are also some variants to these main models. Le Huy3, et al. carried out research work on shelf life and service life estimation of rubber using Arrhenius and WLF models. They have concluded that WLF model is appropriate for lifetime predictions when polymer ageing is controlled by a viscoelastic process (e.g., relaxation, creep), as in the case of service life. On the other hand, when physio- chemical mechanisms are dominant, the Arrhenius model is the most widely used model for evaluating accelerated test results involving the effect of temperature. Mandel4, et al. studied the tread wear of tires, i.e. the service life. These works mainly indicate the methodologies to be followed for life estimation. The aim of the present work is to augment the work carried out by other researchers in the field by making the estimation of life reliable. The shelf life of nitrile rubber, used in various engineering applications, has been studied using the technique of accelerated ageing and subsequent application of Arrhenius theory. The estimate of shelf life has been rendered more reliable by developing a stochastic model, incorporating the statistical techniques.
The work gains significance because it being a reliable estimate of the shelf life of the elastomeric products, inventory of these products can be judiciously maintained. Not only this, as noted earlier, some of the products, like vibration isolators are used onboard naval ships, where it is required to be known whether a particular isolatormount can be stored safely for a certain duration of time before putting it into service.
The method of accelerated ageing involves first heating the elastomer samples at higher temperatures, so that the degradation of a property is faster. This can then be related to similar level of degradation at a lower temperature, but at scaled-up time. In other words, if the degradation of a property is studied at an elevated temperature, it is possible to predict the time at which a similar degradation can be expected at the actual temperature. Hence, this can be used to predict life, if cut-off level for the drop in property is defined. The underlying assumption is that the mechanism for the degradation of the rubber property remains the same at different elevated temperatures as that at the storage temperature.
The test methodology for life estimation of an elastomer is traceable to Arrhenius’ theory of thermal acceleration of reaction kinetics and the concept of the activation energy of a process5. With rise in temperature, the state of a chemical reaction increases. For many organic-chemical reactions, a temperature rise of 10 °C translates to about 2 to 3 times higher reaction rate6. The temperature dependence of chemical reactions is described by the Arrhenius equation6 as:
$K\left(T\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}A.\text{\hspace{0.17em}}{\text{e}}^{E/RT}$ (1)
where, K(T) is reaction rate constant (min-1)
A is pre-exponential factor (min-1)
E is activation energy (J/mol)
R is gas constant (8,314 J/mol K)
T is absolute temperature (K)
The following relation gives the state of a chemical reaction6:
${F}_{x}\text{\hspace{0.17em}}\left(T,t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}K\left(T\right).\text{\hspace{0.17em}}t$ (2)
where, Fx(T, t) is function of the state of the reaction x, and t is reaction time (min)
The state of the reaction, Fx(T, t) may be related to the value of any property of interest. Consequent to different reaction rates Ki for different temperatures Ti, the threshold value Fa of a reaction will be reached at different reaction times ti (equal-value times), e.g., t1 to t4 as shown in Fig. 1 (T1 > T2 > T3 > T4).
The threshold value is given by the equation
${F}_{a}\text{\hspace{0.17em}}\left({T}_{i},{t}_{i}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{K}_{i}\left({T}_{i}\right).\text{\hspace{0.17em}}{t}_{i}$ (3)
The Arrhenius equation Eqn (1) can be substituted in Eqn (3) as follows :
${F}_{a}\left({T}_{i},{t}_{i}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}A.{\text{e}}^{-E/RTi}.\text{\hspace{0.17em}}{t}_{i}$ (4)
or, in logarithmic form with the constant terms combined in one term B, we get
$\mathrm{ln}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}E/R{T}_{i}+B$ (5)
Hence, a plot of ln(t) versus 1/T gives a straightline with the slope being E/R, known as the Arrhenius plot, a sample plot shown in Fig. 2. Extrapolating this line to the temperature of interest (Ts) can indicate the estimated life (ts).
Therefore, if the degradation in a property of an elastomer is taken as thermally-activated molecular process with constant activation energy, the Arrhenius equation can be used to estimate long-term behaviour of the elastomer.
Among different degradation factors noted above, it was found by Yoshida7, et al. that thermal oxidation changes the HDR properties more than the other factors, resulting in an increase in HDR’s stiffness and a decrease in elongation at break as well as tensile strength. In the present work, too accelerated tests were performed on rubber samples, focusing on the most significant degradation factor, i.e. thermal oxidation.
4.1 Selection of Various Parameters
4.1.1 Selection of the Ageing Temperatures
The ageing temperatures were so chosen such that the chemical reaction at each temperature was identical to the one at the shelf temperature6. For nitrile rubber, beyond 110 °C-120 °C, there is a change in the type of reaction, which has implications. Hence this limits the maximum ageing temperature. The ageing temperatures should be chosen such that6:
• The time to attain the threshold values at the lowest ageing temperature is at least 1000 h.
• Likewise, the highest temperature should be chosen such that the time taken to attain the threshold is not <100 h.
In view of the above considerations, a range of 60 °C to 120 °C was chosen. The ageing temperatures selected thus were 120°C, 100°C, 90°C, 80°C, 70°C and 60°C.
4.1.2 Selection of Properties for Life Estimation
The properties chosen should be of significance. Their selection varies from product to product. In the present study, a number of static and dynamic properties were chosen. Under the static properties, the study was carried out for tensile strength, elongation at break, hardness, and stress relaxation in compression. In the dynamic category, dynamic modulus and damping factor were studied. A survey of the literature indicated that the elongation at break had been preferentially used for prediction of long-time behaviour of rubber. Results of the study for five rubbers have been given by Mandel4, et al. in which elongation at break was used as the measure of degradation. In the present work, though the study has been conducted for a number of properties, it was observed that the elongation at break gave the best representation of the state of degradation for studying the shelf life. This has been discussed further.
4.1.3 Selection of Threshold Value of Life
The threshold value should be chosen to suit the conditions of use. It was selected as 50 per cent of the initial value of the property6.
4.2 Experimental Procedure
The test specimens for different tests were made from the rubber blocks, which were used in the manufacture of the vibration isolator. The test specimens were prepared in line with applicable ASTM standards, indicated below. The ageing studies of the test specimens were carried out in air circulating ovens. Tests for tensile strength and elongation at break were conducted using the Universal Testing Machine, as per ASTM D 4128. The hardness test was performed on hardness tester, as per ASTM D 22409. Test for stress relaxation was carried out using stress relaxometer, in accordance with ASTM D 139010, while the dynamic tests for dynamic modulus and damping factor were performed using dynamic mechanical analyser, as per ASTM D 599211.
It may be noted that out of the six properties measured, four properties namely, tensile strength, elongation at break, hardness, and stress relaxation are related to the static performance of the mounts while dynamic modulus and damping are related to the dynamic performance of the mounts. Dynamic modulus and damping were measured to gain an inside view of the dynamic property of the rubber material. However, these two properties were not studied for shelf life estimation, as no dynamic loading is applicable during the storage period, and the data showed considerable scattering.
From the tests, it was found that the tensile strength data did not follow proper trend, it increased and decreased abnormally. This property, therefore, was not used for further study. The data pertaining to hardness and stress relaxation showed relatively less fluctuation, however the Arrhenius plots for these properties were observed to be deviating considerably from straight lines. Figures 3 and 4 show the respective Arrhenius plots. These properties too, therefore, were abandoned from the scope of study.
As indicated in Section 4.1.2, elongation at break has been used by various authors for life estimation study of elastomers. In the present case too, it was observed that the data pertaining to elongation at break were very much consistent at each temperature. Therefore, further study was carried out with this property.
The ageing data at 120 °C was found to be inconsistent when viewed with the data at other temperatures. It was observed that this temperature is too high for the present rubber sample where 50 per cent drop in the property was observed within just three days, and therefore, the test data at 120 °C were rejected for further analysis. On the other hand, degradation at 60 °C was observed to be very slow, and therefore, was not considered for further analysis.
In view of the observations made, analysis of test results as per Arrhenius model was restricted to four temperatures, i.e. 100 °C, 90 °C, 80 °C and 70 °C. The curves of elongation at break versus time data (log h) have been plotted and shown in Fig 5. The degradation curves for elongation at break at 70 °C, 80 °C and 90 °C were observed to be of similar pattern, except for that at 100°C. If the patterns of curves are the same at different temperatures, then identical reaction of the rubber may be considered, which is a precondition for Arrhenius model. Therefore, it was concluded that reaction at 100 °C was not the same as that at other temperatures. Thus, data at 100 °C were not considered for further analyses.
The Arrhenius plot based on the three remaining temperatures, namely 90 °C, 80 °C and 70 °C is shown in Fig. 6. On examining Fig. 6, it was observed that the three data points corresponding to 90 °C, 80 °C, and 70 °C were almost in a straight line, i.e., the best-fit line was passing very closely through the data points. This was reflected in the R2 value (= 0.992) also, which was very close to unity. The value of R2 quantifies goodness of fit, having a fractional value between 0.0 and 1.0. A higher value indicates that the model fits the data better. The straight line through the three data points was extrapolated to the storage temperature, i.e. 27 °C, and the life from the curve worked out to be 13.24 years for 50 per cent degradation as the threshold for life.
Historically, the vast majority of accelerated ageing studies have utilised the Arrhenius methodology. However, as pointed out by Wise12, et al., the estimation by this method may give little confidence due to non-Arrhenius behaviour. As experienced by the authors, the non-Arrhenius behaviour, however, can partly be avoided by discarding temperatures where the degradation pattern differs substantially from that at other temperatures, and also by selecting the ageing temperature closer to the shelf temperature or the temperature of use. In addition, the estimation can be rendered more reliable by incorporating statistical techniques, as elaborated here.
It may be noted that at each temperature, measurement of the property at a particular time was carried out on several samples. Therefore, for an ageing temperature, a set of values of the property, i.e. elongation at break, was available at a particular time. The mean of these property values was obtained at each time to generate the degradation plot of the property, i.e., the property vs time plot at that particular temperature. Thus, essentially the raw data points were replaced by corresponding averages. This may be said to be an exercise in point estimation. However, each datum point is actually representative of a data distribution. A sample raw data plot for a particular temperature is indicated in Fig. 7.
From the Fig. 7, it is evident that the raw data varies within some range; simple averageing may not be appropriate to estimate life and statistical techniques could be employed for a more reliable estimate. This has been discussed below wherein the concepts of sample/ population mean and confidence interval have been employed for working out a reliable model of shelf life estimation.
As noted above, each datum point is characterised by ageing temperature at which the relevant samples have been aged (T), the time duration for which the samples have been aged (t), and different sample values of the property. The values of the property at a particular temperature and time are theoretically expected to be identical for all the samples, but are observed to be different in practice even though enough care is taken in following the correct experimental procedure. This is in line with the normal experimental method. The difference is attributable to various causes, like difference in curing of different rubber samples, error in measurement, air flow variation at different locations within the oven, localised variations in temperature inside the oven, etc. Hence, the different experimental values can be treated as a statistical data distribution.
For cases in which the errors are likely to be equally positive as well as negative, the smooth distribution of an infinite number of readings coincides with the Gaussian or normal distribution13,14. The Gaussian distribution has been found to describe more real cases of experimental and instrument variability than any other distribution and is the one assumed in the present work.
The equation for the Gaussian distribution is
$f\left(X\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{\text{σ}\sqrt{2\text{π}}}{e}^{-{\left(X-\text{μ}\right)}^{2}/2{\text{σ}}^{{}^{2}}}$ (6)
where, f(X) dX is the probability that a single measurement of X will lie between X and X + dX, μ is the distribution mean and σ is the distribution standard deviation.
The above description assumes a Gaussian parent population, which would be well-described if infinite number of samples are taken. However, it is not possible to take infinite number of samples. Thus, the concept of sample population and sample standard deviation comes into the picture. The mean or average of the sample population is defined by
$\overline{X}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{N}\sum _{i=1}^{N}{X}_{i}$ (7)
where, N is the number of individual readings Xi. The sample standard deviation is given by
${S}_{X}=\text{\hspace{0.17em}}{\left[\frac{1}{N-1}\sum _{i=1}^{N}{\left({X}_{i}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\overline{X}\right)}^{2}\right]}^{1/2}$ (8)
According to the central limit theorem of statistics, irrespective of the shape of the distribution of the population or universe, the distribution of average values of samples drawn from that universe will tend toward a normal distribution as the sample size grows without bound15.
7.1 Confidence Intervals in Sample Populations
The confidence interval of the mean defines a band about the calculated sample mean within which the population mean is expected to lie for a given sample confidence/ probability level. This interval can be defined for each datum point, i.e. for each mean value of the property at a particular temperature and at a particular time. Mathematically, it is denoted as follows14:
$\text{α}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}N\left(p\right)×\text{\hspace{0.17em}}\text{σ}/\sqrt{n}$ (9)
where, N(p) is the value of normal function for a probability p, σ is the standard deviation of the sample and n is the sample, size.
From above, it may be deduced that one can define a standard normal distribution for each datum point. This will enable one to indicate the expectation for values of elongation at break at that duration and temperature of ageing, i.e., it is possible to have a maximum value, a minimum value, and a mean value of the expected population mean of elongation at break, aged at a temperature, T for duration t, for a particular confidence level, and thus three estimates of life can be made.
As brought out in section 7 three property degradation curves for each ageing temperature, pertaining to a specific confidence level can be defined. From the three curves for each ageing temperature, three durations of time, namely tmin, tmean and tmax were obtained at which the respective curves drop to the threshold level of elongation at break. The terms tmin and tmax represent the bounds of an interval within which the property is expected to drop to the threshold value of life with a defined probability. Plots of the three curves pertaining to the ageing temperature of 70 °C are shown in Fig. 8.
It is clear form the above, that three Arrhenius plots can be made respectively for tmin, tmean and tmax after obtaining all such values at all ageing temperatures.
The extrapolation of these three plots would give three estimates of life, i.e., lmin, lmean and lmax. Hence the interval (lminlmax) was obtained within which the life was expected to fall with a defined confidence. The three Arrhenius plots are shown in Fig. 9.
In the present work, life has been estimated within a range as against a single value. In statistical terms, interval estimation has been carried out as opposed to point estimation. The lower limit of the range can form a sound basis for the estimation of life. Analysis was carried out at the confidence level of 0.99. The result for 99 per cent confidence level is indicated in Table 1. It was observed that if the confidence level was increased further, no appreciable change in the lower limit value was obtained. However, accuracy of this model depends on the number of test samples used for ageing at different temperatures. More the number, more accurate will be the estimate of shelf life.
Arrhenius plots for the lower estimate are shown in Fig. 10.
9.1 Other Environmental Factors
The work described so far relates to estimation of shelf/ storage life considering oxidative ageing only, which has been found to be the maximum contributor for ageing, as noted earlier. Other environmental factors, like moisture, fluids, UV light, microorganisms, ozone, etc. have not been factored into this exercise. Even though the effect of these factors may not be of high order, further study is required to estimate the effect of these factors also.
The methodology for a reliable estimation of life of an elastomeric component/product, e.g., a vibration isolator, based on accelerated thermal ageing and the Arrhenius methodology is described. Life of an elastomeric product essentially means life of the elastomer because of faster degradation of the elastomer when compared to metals. Therefore, tests for life estimation are carried out on the elastomer. In the present study, the tests were carried out on a number of rubber samples, as per ISO/ ASTM guidelines. Several properties of the elastomers were selected for the study. However, it was found that the elongation at break best represents the degradation of the elastomer with time for shelf life study, and therefore, is best suitable for the study of shelf life estimation.
Accelerated ageing tests were carried out at different elevated temperatures. However, data pertaining to some of the temperatures had to be discarded since the application of Arrhenius theory demands that the pattern of degradation of the property should be the same for all the temperatures under consideration. For the discarded temperatures, it was observed that the data did not follow the regular patterns, as demonstrated by other data. Statistical techniques, based on Gaussian distribution, were incorporated to undertake a meaningful study, based on the variation in sample data for a particular temperature and time. This led to the development of a stochastic model for life estimation, wherein three estimates of life were obtained. Out of the three estimates of life, lower estimate can be taken as the more confident estimate of shelf/storage life and a practical decision on inventory of elastomeric items can be taken on the basis of this estimate.
The authors express their gratitude to Lt Cdr MA Khan (Retd); Shri AK Patalay, Asst Dir; Shri UP Thakur, Add Dir (Retd), and the Director, Defence Machinery Design Establishment, Secunderabad, for their support, encouragement, and permission to publish this work. The authors are also grateful to Shri PK Das, Dy Dir, Indian Rubber Manufacturers Research Association and Dr MS Banerjee, former Director, IRMRA for their valuable guidance during the course of this research.
1. Brown, Roger. Physical testing of rubber. Chapman & Hall, London, 1996. 342p.
2. Itoh, Y.; Gu, H.S.; Satoh, K. & Kutsuna, Y. Experimental investigation on ageing behaviours of rubbers used for bridge bearings. J. Struc. Mech. Earthquake Engg., JSCE, 2006, I-74(808), 17-31.
3. Le. Huy, M. & Evrand, G. Methodologies for life time prediction of rubber using Arrhenius and WLF models. Die Ange. Makro. Chrmie, 1998, 261-262, 135-42.
4. Mandel, J.; Roth, F.L.; Steel, M.N. & Stiehler, R.D. Measurement of the ageing of rubber vulcanizates. In Proceedings of International Rubber Conference, Washington D.C., November 1959, pp. 221.
5. Klingender, Robert C. Handbook of specialty elastomers. CRC Press, Florida, 2008. 576p.
6. International Organisation for Standardisation. Rubber, vulcanized or thermoplastic–Estimation of life time and maximum temperature of use from an Arrhenius plot. ISO-11346. Switzerland, 1997.
7. Yoshida, J.; Abe, M. & Fujino, Y. Constitutive model of high-damping rubber materials. J. Engg. Mech., 2004, 130(2), 129-41.
8. Standard test methods for vulcanized rubber and thermoplastic elastomers–Tension. ASTM D 412. Pennsylvania, 2006.
9. Test method for rubber property–durometer hardness. ASTM D 2240. Pennsylvania, 2006.
10. Test methods for rubber property–Stress relaxation in compression. ASTM D 1390. Pennsylvania, 2006.
11. American Standard for Testing of Materials. Standard guide for dynamic testing of vulcanized rubber and rubber-like materials using vibratory methods. ASTM D 5992. Pennsylvania, 2006.
12. Wise, J.; Gillen, K.T. & Cloug, R.L. Prediction of elastomer lifetimes from accelerated thermal ageing experiments. In Polymer durability. American Chemical Society, Washington, 1996. 728p.
13. Coleman, Hugh W. & Steele Glenn W. Experimentation, validation and uncertainty analysis for engineers. John Wiley & Sons, New Jersy, 1989. p.336.
14. Ayyub, Bilal M. & McCuen, Richard H. Probability, statistics and reliability for engineers. CRC Press, Florida, 1997. 528p.
15. Pyzdek, Thomas. Quality engineering handbook. Marcel Dekker, Inc., New York, 1999. 680p.
Mr Sachchidanand Das obtained his BTech from IIT Delhi. He is presently working as Asst Dir at DMDE, Secunderabad. His expertise includes life estimation of elastomers and finite element analysis. His current areas of interest include nonlinear finite element analysis, fluid-structure interaction, and noise and vibration studies. Mr A. Roy Chaudhuri obtained his MTech from IIT Madras, Chennai. He is presently working as Addl. Dir at Defence Machinery Design Establishment (DMDE), Secunderabad. His area of work includes shock, noise and vibration, finite element analysis, and design of various equipments for marine applications. | 2019-10-15 18:49:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "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": 0.5810757875442505, "perplexity": 1236.9440774010695}, "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/1570986660231.30/warc/CC-MAIN-20191015182235-20191015205735-00084.warc.gz"} |
https://mathematica.stackexchange.com/questions/111144/expanding-and-simplifying-expressions | # Expanding and simplifying expressions
I have some complicated expression, for example:
(-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) d c)/(b^2 (d c)^(3/2))
All variables are positive. I want to expand it and simplify each element of sum separately. So, I use the function Expand, then I apply the function Simplify to each element of the expanded sum and at the end I obtain a final result by copying and pasting partial results of each simplification. It looks like that:
In[80]:= Expand[(-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) d c)/(b^2 (d c)^(3/2))]
Out[80]= -((a^3 Sqrt[c d])/(c^2 d^2)) + (a^3 Sqrt[c d])/(b^2 c^2 d^2) - (6 a Sqrt[c d])/(c d) - (3 a b^2 Sqrt[c d])/(c d)
In[82]:= Simplify[-((a^3 Sqrt[c d])/(c^2 d^2)), {a > 0, b > 0, c > 0,
d > 0}]
Out[82]= -(a^3/(c d)^(3/2))
In[83]:= Simplify[(a^3 Sqrt[c d])/(b^2 c^2 d^2), {a > 0, b > 0, c > 0, d > 0}]
Out[83]= a^3/(b^2 (c d)^(3/2))
In[84]:= Simplify[-((6 a Sqrt[c d])/(c d)), {a > 0, b > 0, c > 0,
d > 0}]
Out[84]= -((6 a)/Sqrt[c d])
In[85]:= Simplify[-((3 a b^2 Sqrt[c d])/(c d)), {a > 0, b > 0, c > 0,
d > 0}]
Out[85]= -((3 a b^2)/Sqrt[c d])
The final result in this case is of course:
-(a^3/(c d)^(3/2)) + a^3/(b^2 (c d)^(3/2)) - (6 a)/Sqrt[c d] -((3 a b^2)/Sqrt[c d])
My question is: is it possible to do the same in more automatical way, without using copy-paste methods? Applying succesively Expand and Simplify does not work, because each of them removes effects of the other:
In[89]:= Simplify[
Expand[(-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) d c)/(b^2 (d c)^(3/2)), {a > 0, b > 0, c > 0, d > 0}], {a > 0, b > 0,
c > 0, d > 0}]
Out[89]= (-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) c d)/(b^2 (c d)^(3/2))
In[90]:= Expand[
Simplify[(-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) d c)/(b^2 (d c)^(3/2)), {a > 0, b > 0, c > 0, d > 0}], {a > 0, b > 0,
c > 0, d > 0}]
Out[90]= (-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) c d)/(b^2 (c d)^(3/2))
Simplify /@ Expand @ expr /. Sqrt[c d] / (c d)^2 -> 1/(c d)^(3/2),
where expr = (-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) d c)/(b^2 (d c)^(3/2))
Result
• Ok, but this works for this particular example. And I have many more expressions like that which need simplifying. Some of them are sums of hundreds elements. Therefore I'm looking for something more automatic. – wiktoria Mar 26 '16 at 20:24
• I want to simplify each element of a given sum separately. And I want to do it automatically, not by copying each element and applying to it the function Simplify. After applying the function Simplify to a whole sum it collapses to one expression with single denominator. – wiktoria Mar 26 '16 at 21:52
• @wiktoria, that is why I've used Map (/@). To apply the function to every element... – garej Mar 26 '16 at 21:54
• But you used also Sqrt[c d] / (c d)^2 -> 1/(c d)^(3/2) and this condition will be different for another sum, which contains another elements. If it is possible to accomodate the assumption that all variables are positive, Sqrt[c d] / (c d)^2 will be reduced automatically. – wiktoria Mar 26 '16 at 21:58
• The problem is that my expressions contain hundreds of elements... That's why I was looking for automatical solution. – wiktoria Mar 26 '16 at 22:05
Ok, I've found a solution:
expression = (-a^3 (-1 + b^2) - 3 a b^2 (2 + b^2) d c)/(b^2 (d c)^(3/2))
Assuming[{a > 0, b > 0, c > 0,}, Map[Simplify][Expand[expression]]]
• that is the same as Simplify[#, {a > 0, b > 0, c > 0}] & /@ Expand@expr – garej Mar 26 '16 at 22:32 | 2019-08-25 10:22:29 | {"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": 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": 0.47820284962654114, "perplexity": 2198.756190262051}, "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/1566027323246.35/warc/CC-MAIN-20190825084751-20190825110751-00010.warc.gz"} |
https://math.stackexchange.com/questions/1203598/a-few-questions-about-derivative-notation | # A few questions about derivative notation
$1)$ How do I denote derivative of $ax^2+b$ in terms of $ax^2$?
$(ax^2+b)'(ax^2)$ can easily be confused with $ax^2\cdot(ax^2+b)'$.
$2)$ How do I denote the derivative of $ax^2+b$ in terms of $ax^2$ at point $c$?
$3)$ How do I denote the derivative of $ax^2+b$ in terms of $x$ at point $c$?
$3)$ How do I denote the derivative of $f(g(x))$ in terms of $g(x)$ at point $a$?
I want to denote all of this without using $\text{d}$.
• Are you talking about applications of the chain rule? I'm not sure what you mean by "in terms of". Mar 24, 2015 at 1:22
• @GFauxPas Yes. If $ax^2+b$ is a function of $ax^2$, then I'm searching how to denote its derivative. Mar 24, 2015 at 1:24
• By "in term of $ax^2$" you mean "with respect to the variable $ax^2$"? Mar 24, 2015 at 1:26
• @DouglasFinamore If $f(x)=ax^2+b$ and $g(x)=ax^2$, then I'm searching how to denote $f'(g(x))$ without using $f$ and $g$. Mar 24, 2015 at 1:27
• Perhaps you mean "with respect to $x^2$"? Saying "with respect to $ax^2$" is unusual, if $a$ is a constant. Mar 24, 2015 at 1:28
1) How do I denote derivative of $ax^2 +b$ in terms of $ax^2$ ?
$(ax^2 +b) ′ (ax^2 )$ can easily be confused with $ax^2 \cdot (ax^2 +b) ′$ .
Ah. Where as the prime notation on a function symbol is taken as being with respect to the function's argument, the prime notation over an expression is taken as being with respect to the independent variable of the discussion (most usually that is either $x$ or $t$).
That is, if $f(x)=ax+b$, then $\;f'(ax^2) = \frac{\mathrm d f(ax^2)}{\mathrm d (ax^2)} = \frac{\mathrm d (ax^2+b)}{\mathrm d (ax^2)} \\[2ex] f(ax^2)' = (ax^2+b)' = \frac{\mathrm d (ax^2+b)}{\mathrm d x}$
So, you want to use the prime notation on an expression to indicate you are deriving with respect to another expression rather than the implicit variable itself.
$$[u\mapsto u+b]' (ax^2) = \left.\frac{\mathrm d u+b}{\mathrm d u}\right\vert_{u:=ax^2} = \frac{\mathrm d(ax^2+b)}{\mathrm d (ax^2)}$$
You could establish in your forward that you were using a subscripted dash notation.
$$(ax^2+b)'_{(ax^2)} = \frac{\mathrm d (ax^2+b)}{\mathrm d (ax^2)}$$
Or simply rely on the chain rule. $\frac{\mathrm d (ax^2+b)}{\mathrm d x}\frac{\mathrm d x}{\mathrm d (ax^2)} = \frac{(ax^2+b)'}{(ax^2)'}$
• Your idea seems to be to simply replace $f$ with $[x\to x+b]$, which is sort of cheating. Mar 24, 2015 at 11:00 | 2022-06-25 09:02:01 | {"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": 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": 0.981565535068512, "perplexity": 231.3776878473219}, "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-2022-27/segments/1656103034877.9/warc/CC-MAIN-20220625065404-20220625095404-00351.warc.gz"} |
https://www.vedantu.com/question-answer/valence-electrons-does-fluorine-have-class-11-chemistry-cbse-5fd79b96aef1aa270c092595 | Question
# How many valence electrons does Fluorine have?
Verified
91.5k+ views
Hint:In order to answer the number of valence electrons present in Fluorine, we must first know about the atomic number and electronic configuration of Fluorine. Fluorine is a chemical element which is represented by the chemical symbol ‘F’ and it will be having an atomic number of 9.
The electronic configuration of fluorine is given as $1{s^2}2{s^2}2{p^5}$ which can also be written as $[He]2{s^2}2{p^5}$. From the electronic configuration, we can say that the number of valence electrons present in the Fluorine is 7. Fluorine will be having an oxidation number of -1. | 2021-10-15 21:42:41 | {"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": 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": 0.6743820309638977, "perplexity": 483.8989179741212}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323583083.92/warc/CC-MAIN-20211015192439-20211015222439-00087.warc.gz"} |
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be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\utilities\gantrycache.class.php on line 70 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 321 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 324 Strict Standards: Non-static method JSite::getMenu() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 337
Strict Standards: Non-static method JApplication::getMenu() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\includes\application.php on line 523 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 337 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantrySessionParamOverride::populate() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 1237
Strict Standards: Non-static method GantryCookieParamOverride::populate() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 1238 Strict Standards: Non-static method GantryMenuItemParamOverride::populate() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 1241
Strict Standards: Non-static method GantryMenuItemParamOverride::_populateSingleItem() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\params\overrides\gantrymenuitemparamoverride.class.php on line 43 Strict Standards: Non-static method JSite::getMenu() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\params\overrides\gantrymenuitemparamoverride.class.php on line 54
Strict Standards: Non-static method JApplication::getMenu() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\includes\application.php on line 523 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\params\overrides\gantrymenuitemparamoverride.class.php on line 54 Strict Standards: Non-static method GantryUrlParamOverride::populate() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 1244
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\templates\rt_gantry\features\jstools.php on line 39
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\templates\rt_gantry\features\jstools.php on line 40
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Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\templates\rt_gantry\lib\gantry.php on line 30
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Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40
Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 81 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 82 ## Login Pop-up Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678
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Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_socialmedialinks\mod_socialmedialinks.php on line 14
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Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40
Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 81 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 82 +962 6 585 21 21 Login Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 81
Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 82
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\providers\RokMenuProviderJoomla16.php on line 23
Strict Standards: Non-static method JSite::getMenu() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\providers\RokMenuProviderJoomla16.php on line 55 Strict Standards: Non-static method JApplication::getMenu() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\includes\application.php on line 523
Strict Standards: Non-static method JSite::getMenu() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\providers\RokMenuProviderJoomla16.php on line 38 Strict Standards: Non-static method JApplication::getMenu() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\includes\application.php on line 523
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\AbstractJoomlaRokMenuFormatter.php on line 14
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 1051
Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_roknavmenu\lib\renderers\RokNavMenu2XRenderer.php on line 13
Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40
Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 81 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 82 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Non-static method GantryLoader::import() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\gantry.php on line 40 Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678
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Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\features\systemmessages.php on line 26
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Warning: Creating default object from empty value in D:\Hosting\11135813\html\elf-jo.com\components\com_k2\views\itemlist\view.html.php on line 162
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## The Best Way And Create Excellent Invention Options
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Strict Standards: Non-static method GantryModulesRenderer::display() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 678 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 81 Strict Standards: Only variables should be passed by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\renderers\gantrymodulesrenderer.class.php on line 82 ##### CONTACT DETAILS Elf Energy International Jordan, Amman, 7th Circle-314 T +962 6 585 11 21 Fax +962 6 585 21 21 Email: info@elf-energy.com Skype: elf.energy Map Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\modules\mod_contactdetails\mod_contactdetails.php on line 23 ##### JOIN OUR NEWSLETTER Strict Standards: Non-static method JINCFactory::jincimport() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\administrator\components\com_jinc\classes\factory.php on line 72 Strict Standards: Non-static method JINCFactory::jincimport() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\administrator\components\com_jinc\classes\factory.php on line 72 Strict Standards: Non-static method JINCFactory::jincimport() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\administrator\components\com_jinc\classes\factory.php on line 72 Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\administrator\components\com_jinc\classes\core\newsletter.php on line 37 Strict Standards: Non-static method NewsletterFactory::getInstance() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\modules\mod_jinc_subscription\mod_jinc_subscription.php on line 38 Strict Standards: Non-static method JINCFactory::jincimport() should not be called statically in D:\Hosting\11135813\html\elf-jo.com\administrator\components\com_jinc\classes\factory.php on line 72 Strict Standards: Non-static method ServiceLocator::getInstance() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\administrator\components\com_jinc\classes\core\newsletterfactory.php on line 71
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##### LIKE / PLUS / TWEET US
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Strict Standards: Only variables should be assigned by reference in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 487
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Strict Standards: Non-static method GantryCookieParamOverride::store() should not be called statically, assuming $this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\gantry.class.php on line 455 Strict Standards: Non-static method GantryCookieParamOverride::_flushOldCookies() should not be called statically, assuming$this from incompatible context in D:\Hosting\11135813\html\elf-jo.com\libraries\gantry\core\params\overrides\gantrycookieparamoverride.class.php on line 40 | 2017-11-23 22:06:26 | {"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": 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": 0.5177821516990662, "perplexity": 9379.25352287489}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806979.99/warc/CC-MAIN-20171123214752-20171123234752-00034.warc.gz"} |
https://harryrschwartz.com/2015/11/08/cad-paper-reading-group-notes.html | # Harry R. Schwartz
Software engineer, nominal scientist, gentleman of the internet.
Member, ←Hotline Webring→.
· 1B41 8F2C 23DE DD9C 807E A74F 841B 3DAE 25AE 721B
ovo-lacto vegetarian
he
him
his
Published .
Tags: computer-science.
I’ve recently gotten involved with a paper-reading group at the Institute down the street, where we’re reading some papers on CAD and CAM systems. These are some rough notes from the first meeting:
### Discussing Adrian Bowyer’s Computer-aided Design and Manufacturing.
About 14 people showed up! Half physical, half virtual.
### Questions & Discussion
Why is one good at calculating volume, and one at surface area? BREPs are easy for surface area, since you’re just summing up the surfaces. But volume is trickier, since topologies could be bizarre, and aren’t obvious from the representation. You could still do it, it’s just harder.
How does Solidworks represent CSG? Still ultimately BREPs in the background. Probably some scary stuff in there.
Two main B-Rep kernels in common use: Parasolid and ACIS.
Rhino uses B-Rep/NURBS. Not really for building things that are manufacturable.
Are there any open kernels out there (aside from Open CASCADE)? Apparently Open CASCADE is a mess to maintain, so it’s pretty limiting. It’s huge, not modular, limited unit testing.
What’s the deal with special purpose hardware and the depth buffer? We’ve had special-purpose hardware for rendering triangles for quite some time. That’s especially good for B-Reps—current hardware generally assumes that we’re using B-Rep.
The big priority shouldn’t be code quality, but testing. Not crashing would be good.
It’d be great to have a FOSS implementation. Apple and Intel would probably be interested.
Onshape makes a nice CAD platform that supports plugins well.
What are our goals here? Just learning? Creating a new kernel? Building new tools?
• Propriety CAD tools in the cloud for \$4,000 seems like a sad future.
• It’s hard to generate sensible models for a 5-axis router.
• Practical stuff. Functional representations in real time.
• Building tools to automate the design process.
• Helping to build high-quality FOSS CAD tools.
• Love/hate relationship with existing proprietary CAD tools.
• What directions can CAD tools take in the future?
• Enabling users to generate reusable tooling.
Any downsides to STEP and STL files?
See STEPcode for a set of tools built on ISO 10303. It’s BSD-licensed and used in a bunch of other tools.
It might be helpful to see the whole pipeline of how F-Rep/B-Rep models are rendered.
What’s involved in a B-Rep kernel? The most difficult part is handling intersections. Tons of other features, though: curved surfaces on the ends of models, making surfaces from point clouds, handling discontinuities, etc, etc.
You might like these related articles: | 2020-01-18 09:29:24 | {"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": 0.21787084639072418, "perplexity": 9297.66902498854}, "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/1579250592394.9/warc/CC-MAIN-20200118081234-20200118105234-00144.warc.gz"} |
https://tex.stackexchange.com/questions/482062/drawing-a-decision-diagram-with-tikz-and-layout-manager | # Drawing a Decision Diagram with Tikz and layout manager
I'm trying to draw decision diagrams (DD) in an automated fashion. A DD is a DAG, where all nodes are partitioned in layers. Edges typically go from a node in layer l to another node in layer l+1. Each edge has a label, and there can be multiple edges between the same pair of nodes. Two example DDs can be found below. These are drawn by manually specifying all coordinates of all nodes. Since I need to draw a large number of those, I'd like to automate this. Therefore I'm using the layout manager of tikz. Here's an example:
\RequirePackage{luatex85}
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{graphs,graphdrawing,quotes}
\usegdlibrary{layered}
\begin{document}
\begin{tikzpicture}[rounded corners]
\graph [layered layout,
edge quotes={fill=white,inner sep=1pt,font=\scriptsize,pos=0.2},
nodes={circle,draw,inner sep=.1,outer sep=0, minimum size=.35cm},
level sep=1cm
]
{
r ->["a"] 1,
r ->["b"] 0,
r ->["c"] 2,
r ->["d"] 3,
3 ->["e"] 4,
1 ->["f"] 4,
1 ->["g"] 6,
0 ->["h"] 4,
0 ->["i"] 5,
0 ->["j"] 6,
2 ->["k"] 6,
4 ->["l"] t,
5 ->["m"] t,
6 ->["n"] t;
};
\end{tikzpicture}
\end{document}
The above code works pretty well, especially since I can specify each arc individually. Next I'm looking for a way to do the following:
1. how to specify the vertical space between layers? Answer: use level sep=2cm
2. how to draw multiple edges between the same pair of nodes? In the right most figure, you see nodes with 2 edges, e.g. v2,v4. There can be more than 2 edges between the same pair of nodes. The edges must be drawn non-overlapping. E.g. what if I had 10 edges with different labels between the same pair of nodes?
3. Next to the left most graph, there are labels pi_1, pi_2. These labels are vertically aligned and are next to the edges. How do I add those?
Note: I'm not looking to exactly replicate the graphs in the figure. Something else that is aesthetically pleasing is good enough.
• For (2), it is pretty easy if you use pure TikZ: just draw one or two curves from a node to another. With positioning (3) is also not difficult to achieve. However, drawing such weighted graphs is very painful with pure TikZ. – user156344 Mar 29 '19 at 9:15
• @JouleV I've no idea how to get (3) done. The graph nodes are on their own layers. The labels pi_1, pi_2 etc are in between layers. – Joris Kinable Mar 29 '19 at 11:53
• @JorisKinable I have absolutely zero experience with your approach, so I can do nothing. I just said that using normal TikZ, we can have (3) done quite easily if we already have the graph. – user156344 Mar 29 '19 at 11:55
It depends a bit on what you want. If you just want to add extra bent arrows and pi's, that's easy.
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{graphs,graphdrawing,quotes}
\usegdlibrary{layered}
\begin{document}
\begin{tikzpicture}[rounded corners]
\begin{scope}[local bounding box=graph]
\graph [layered layout,
edge quotes={auto,inner sep=1pt,font=\scriptsize},
nodes={circle,draw,inner sep=.1,outer sep=0, minimum size=.35cm},
level sep=1cm
]
{
r ->["a"] 1,
r ->["b"] 0,
r ->["c"] 2,
r ->["d"] 3,
3 ->["e"] 4,
1 ->["f"] 4,
1 ->["g"] 6,
0 ->["h"] 4,
0 ->["i"] 5,
0 ->["j"] 6,
2 ->["k"] 6,
4 ->["l"] t,
5 ->["m"] t,
6 ->["n"] t;
};
\begin{scope}[nodes={auto,inner sep=1pt,font=\scriptsize}]
\draw[->] (r) to[bend left,"$c'$"] (2);
\end{scope}
\path (r) -- (3) coordinate[pos=0.5] (p1)
(3) -- (4) coordinate[pos=0.5] (p2) (4) -- (t) coordinate[pos=0.5] (p3);
\end{scope}
\foreach \X in {1,2,3}
{\node at ([xshift=-1cm]graph.west|-p\X) {$\pi_{\X}$}; }
\end{tikzpicture}
\end{document}
The nice thing about this is that the output is close to the screen shot, the not-so-nice part that you need to issue many commands of the sort \draw[->] (r) to[bend left,"$c'$"] (2); (which can be shortened somewhat using edges).
If you want a more automatic version, then the following may provide a starting point.
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{graphs,graphdrawing,quotes}
\usetikzlibrary{decorations.pathreplacing}
\usegdlibrary{layered}
\tikzset{extra bent arrow/.style={decoration={show path construction,
lineto code={
\draw (\tikzinputsegmentfirst) to[bend left=40,#1]
(\tikzinputsegmentlast);
}},postaction=decorate},
extra straight arrow/.style={decoration={show path construction,
curveto code={
\draw (\tikzinputsegmentfirst) -- node[pos=0.5,auto,font=\scriptsize]{#1}
(\tikzinputsegmentlast);
}},postaction=decorate}}
\begin{document}
\begin{tikzpicture}[rounded corners]
\begin{scope}[local bounding box=graph]
\graph [layered layout,
edge quotes={auto,inner sep=1pt,font=\scriptsize},
nodes={circle,draw,inner sep=.1,outer sep=0, minimum size=.35cm},
level sep=1cm
]
{
r ->["a"] 1,
r ->["b"] 0,
r ->[bend left,"c",extra straight arrow=c1] 2,
r ->["d"] 3,
3 ->["e"] 4,
1 ->["f"] 4,
1 ->["g"] 6,
0 ->["h"] 4,
0 ->["i"] 5,
0 ->["j"] 6,
2 ->["k"] 6,
4 ->["l"] t,
5 ->["m"] t,
6 ->["n"] t;
};
\path (r) -- (3) coordinate[pos=0.5] (p1)
(3) -- (4) coordinate[pos=0.5] (p2) (4) -- (t) coordinate[pos=0.5] (p3);
\end{scope}
\foreach \X in {1,2,3}
{\node at ([xshift=-1cm]graph.west|-p\X) {$\pi_{\X}$}; }
\end{tikzpicture}
\end{document}
I chose that output as it shows the open issues: the line does not connect as nicely in the above picture, and it is harder to control the edge labels. Both issues can be resolved but this will be more effort.
• In your first approach, what makes that pi_1, pi_2 etc are vertically aligned? – Joris Kinable Mar 29 '19 at 18:56
• @JorisKinable I place them relative to the local bounding box, which is, after all, a box with vertical edges. – user121799 Mar 29 '19 at 19:06
The following solution seems to work:
\RequirePackage{luatex85}
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{graphs,graphdrawing,quotes}
\usegdlibrary{layered}
\begin{document}
\begin{tikzpicture}[rounded corners]
\graph [layered layout,
edge quotes={fill=white,inner sep=1pt,font=\scriptsize},
nodes={circle,draw,inner sep=.1,outer sep=0, minimum size=.35cm},
level sep=1.5cm, %vertical distance between layers
sibling distance=2cm, %distance between nodes of the same connected component in the same layer
component sep=0cm %distance between connected components
]
{
{
[nodes={draw=none,text opacity=0}, edge={draw=none}]
l1 ->["$\pi_1$"] l2 ->["$\pi_2$"] l3 ->["$\pi_3$"] l4,
},
{
[edge={pos=.45}]
r ->["a"] 1,
r ->["b"] 0,
r ->["c"] 2,
r ->["c'",bend left=-20] 2,
r ->["c''",bend left=20] 2,
r ->["d"] 3,
3 ->["e"] 4,
1 ->["f"] 4,
1 ->["g"] 6,
0 ->["h"] 4,
0 ->["i"] 5,
0 ->["j"] 6,
2 ->["k"] 6,
4 ->["l"] t,
5 ->["m"] t,
6 ->["n"] t;
}
};
\end{tikzpicture}
\end{document}
The result:
This solution draws the pi labels on the left hand side as a 2nd graph. The labels are automatically aligned.
• That looks indeed very simple and good. +1 (I didn't try this mainly because I was afraid that drawing too many connections will distort the graph too much, but it is not too bad, and will get better once you add multiple connections at other locations.) – user121799 Mar 30 '19 at 0:37
• I also tried adding [same layer] tags, since I know beforehand which nodes are supposed to end up in the same layer. Oddly, this wasn't helpful at all: the graph got really distorted. Simply omitting those tags significantly improved the quality of the graph. – Joris Kinable Mar 30 '19 at 10:51 | 2020-10-22 20:18:33 | {"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": 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": 0.8066936731338501, "perplexity": 6563.884625366015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880038.27/warc/CC-MAIN-20201022195658-20201022225658-00379.warc.gz"} |
https://nrich.maths.org/7411 | ### Ball Bearings
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
### Air Routes
Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.
### Over-booking
The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?
# A Very Shiny Nose?
##### Age 16 to 18Challenge Level
Rudolph's nose glows because it is home to a species of bacteria, Aliivibrio Rudolphi, that luminesces when it reaches a certain population density. It detects the size of its population by a process known as quorum sensing.
Each bacterial cell releases a signal molecule, $X$, at a rate of $1$ molecule per minute and if the concentration of $X$ is greater than or equal to $10^{11}$ molecules per ml, the bacteria will glow. $X$ decays with a half-life of $10$ minutes but the bacteria divide every $30$ minutes.
Sadly, Rudolph catches a nasty cold, which, by the time he is better, has killed all of the bacterial cells in his nose except for one. Santa is worried: there are only $24$ hours left until they need to set off to deliver Christmas presents. Will Rudolph's nose be glowing again in time?
If you need any data that is not included, try to estimate it: Santa wants an answer now, so that he can make alternative plans if need be.
This might sound far fetched, but bioluminesent bacteria which emit light only when there is a sufficient number of them exist in the real world, and are often found in symbiotic relationships with marine animals. The first of these to be studied was Aliivibrio fischeri (sometimes called Vibrio fischeri) which can be grown in the lab and observed to produce light only when large numbers of bacteria are present.
Read about the Hawaiian Bobtail Squid and its relationship with Aliivibrio fischeri.
Watch "How bacteria talk", a TED talk by Bonnie Bassler, for a description of how quorum sensing works (which also features the Hawaiian Bobtail Squid and Aliivibro fischeri).
Find out more about reindeer noses here. | 2022-12-08 02:41:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.3908877968788147, "perplexity": 1708.5523510515184}, "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/1669446711232.54/warc/CC-MAIN-20221208014204-20221208044204-00559.warc.gz"} |
http://brink-suehs.de/how-to-solve-hexagon-number-puzzle.html | # How To Solve Hexagon Number Puzzle
4 x 4 Grid - Using shapes to solve. Required tools for solving a Sudoku puzzle. The main diagonal sums to 14; the off diagonal sums to 17. Solve Banana Clock Hexagon Puzzle - with Answer. The numbers in the shaded areas are the sum of the numbers in the 2 or 4 segments that they adjoin. We know 3 of the shapes equal 45, so a single shape equals 15. The number puzzles within the book include calcudoku, mental maths, kakuro, number hexagon, maths puzzle, symbols of value and wijuko. Drop the Number - Merge Game. Online Printable Hexagon Puzzle are a few from the most enjoyable things that you should use to pass the time, nevertheless they can also be perfect for getting an energetic role inside your puzzle resolving. Identified in the same way as the Killer Sudoku puzzle shown above, the hardest Calcudoku was a 9×9 puzzle published on the 2nd of April 2013, which only 9. The catch is that the touching cells across two hexagons must have the same number. Adding one to each exponent and multiplying we get (1 + 1) (1 + 1) = 2 × 2 = 4. Each puzzle is a number maze with a unique solution. It involves rolling a dice to work out the value of an algebraic expression. High quality World Cube Championship-inspired gifts and merchandise. Also known as. 1569 is a composite number. Who Am I? - SolveMe Puzzles. It is a system of teaching numeral and shape recognition through movement, music, and flash cards that contain a visual clue embedded into the numeral or shape. This page is a collection of some of my favourites that I have complied over the years. Inquisitive by nature. So if you've been looking for a solution to Document checked at the grocery store, we've got just the word for you to help you successfully complete your crossword. 6 days ago. When they solve enough puzzles to reach the central hexagon, the desert tile, they can submit answers to the meta for this island. Apr 9, 2020 - This Pin was discovered by Alla Redko. Struggling to solve one of our fiendishly difficult puzzles? Use our handy video solutions to those tricky metal and wooden puzzles and guide yourself towards puzzle-solving excellence! Can't find what you're looking for? Try looking on our YouTube channel! , UK Company Number: 04398865. Choose one from 40 similar games like Sudoku Solver Crossword Puzzle 2021. Here you’ll meet Barry, who knows Ruby and Celeste and also looks a lot like Gary. The sum of each row is to the right, and the sum of the columns is at the bottom. One Two Three Four Five Six Seven Eight Nine Sixteen Twenty-Five Thirty-Six This episode teaches tessellation using Thirty-Six's 6×6 grid. Rating: 8/10 - 2163 votes. If you then manage to solve it anyways, it's an extra bonus! 😀. The hexagon is an arrangement of 24 equilateral triangle. Cut out the pieces of cardboard using the templates. like these Easter-themed puzzles. For each test case output M on a single line, the minimum number of moves required to solve the puzzle. Just like the Square-1, it can adopt non-cubic shapes as it is twisted. Check, how smart you are. " As noted earlier, we've talked about using a clever trick to prove that a triangle's interior angles always add up to 180º. The Square-1 (previously called as Cube 21 and Back to Square One) is a shape-shifting three-layered twisty puzzle. Now let's use it to solve for the missing number in pentagon B, the bottom right. Puzzle #1: Blue, white, purple. A Genetic Algorithm for Solving Hidato Puzzles 3 Fig. Once you solve it, you get the blue memory card. A small program to solve the problem from this xkcd cartoon: Cryptarithmetic Two versions of the famous SEND+MORE=MONEY puzzle, and a more general solver for this kind of puzzle. Re-Size-It: to Solve the Brain Puzzle. For K-12 kids, teachers and parents. Subtle word play is often part of a plexer. That gives you a "center" number (19) for the triangle. Free UK shipping on orders over £10. 2 $\begingroup$ I recently stumbled upon this. The key to solving this puzzle is filling in the yellow shapes first!. 1 Synopsis 2 Characters 3 Trivia 4 Video Puzzle-master Thirty-Six challenges the others to find a way into Puzzle Square. Zero = nothing = empty ┐ (symbol) are equal to 6 (number). It helps them deal with failure if an attempt fails, and forces them to persevere and try new ways of solving the puzzle. In this room you will notice a champagne bottle on the floor close to the TV - click on it. Each number from 1 to 19 must be used exactly once. Finally solve the puzzle to make sure all the pieces fit as you can see in picture 7. A square based pyramid has ___ vertices 4. 4 x 4 Grid - Using the numbers 1,2,3,4 to solve. , 38 /2 = 19). Dec 4, 2016 - Solve this interesting Hexagon, Star, and Rectangle math puzzle. Rubik's mosaics are a fantastic way to take your Cube solving to the next level. I think I found a law for Sudoku Hoshi (also called Star Sudoku) puzzles but since I did the proof more or less in my head, please check. Solution: 9. It bears some resemblance to Sudoku, in that it's a logic puzzle with a single solution. Your task is to pick up and place Hexagon blocks on the board and to connect three or more blocks of the same number to merge them into one block of a higher number. Aristotle's Number Puzzle by Professor Puzzle is number puzzle involving 19 numbers arranged into a hexagon. Choose one from 40 similar games like Sudoku Solver Crossword Puzzle 2021. Altogether that is 15 edges equal to 15, so a single edge equals 1. Lily's Garden. Dino Sudokon - Historical Mysteries is a new puzzle game that will challenge your attention and logical thinking. In part, the reason is that it is difficult and Mathematics just love a challenge. Choose one from 40 similar games like Sudoku Number Puzzle Sodoku 2021. All Puzzle Solutions: The Witness takes place on an open world island and it's divided into a number of different areas, each with its own set of unique puzzles that you need to solve. And there are. Sep 28, 2014 · 3. A hexagon with consecutive sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle. Required tools for solving a Sudoku puzzle. When solving a Breach Protocol puzzle there are three things to keep an eye on: The alphanumeric grid on the left-hand side of the screen. That's a good sign… right? We're going to solve the top layer edges in two steps. Ask a question or add answers, watch video tutorials & submit own opinion about this game/app. Solve the Magic Hexagon - Code Golf Stack Exchange. An example of a number block puzzle created using Puzzle Master is given above. Boring, but effective; and can't be done by a human in a reasonable amount of time. com) There are 121,645,100,408,832,000 permutations for this. From traditional crosswords, word search, fill-in, and Sudoku puzzles to original brain teasers such as A Bloomin' Maze, Arrow Maze, Missing Vowels, Target Shoot, Cryptoquizzes, Picture Crosswords, Magic Number Squares, Vision Quest, and Hexagon Hunt - plus bonus coloring pages - this two-color 32-page booklet is jam-packed with a lively variety of activities for all ages. 5 days ago. The reports cover the time it took you to solve the puzzle and the number of moves as well. Rotate all the tiles so all the lines connect. Hexadecimal Sudoku Puzzles by Krazydad. The lampposts are the number 15, the snowmen are the number 4, and the trees are the number 3. Add the three hexagon chips to the puzzle. There is a simple logic behind solving this question. The objective of Drive Ya Nuts is to arrange the seven pieces so the numbers on each piece are situated next to the same number on the pieces that they touch. The key to solving this puzzle is filling in the yellow shapes first!. Answer & Explanation. The Madison School has a new cafeteria with very artistic, new tables for its 375 students. Daily Puzzles. Sep 07, 2021 · Proclaimed Rule: See image - any "hourglass" touching the hexagon (red) must contain the same number twice. As of 2009, it is sold by Uwe Mèffert in his puzzle shop, Meffert's. Maths and Logic Puzzles. Play the best free online Puzzle Games: we have a large collection of games where you have to solve puzzles. Solve Einstein's famous puzzle by dissecting his clever list of clues. Select one hexagon and put it in the center. Commercial Use: The puzzles generated using this tool are allowed for personal and commercial use. The number 7 has 3 little stripes (1 horizontal, 2 vertical), the number 8 has 7 stripes (3 horizontal and 4 vertical). It is a system of teaching numeral and shape recognition through movement, music, and flash cards that contain a visual clue embedded into the numeral or shape. Let's look at the master triangle again. Patterns and sequences are all aroudn us, in nature and in our school and work envrionment. These should be very obvious thank to the game highlighting them after you test the machine. Hive' puzzles. Who Am I? - SolveMe Puzzles. Conclusion. Wordplay is the place to discuss and learn more about crossword puzzles from The New York Times, written by Deb Amlen with contributions from Will Shortz, Caitlin Lovinger and Sam Ezersky. It is 12 in six triangles forming the wheel of a hexagon. Use unit square tiles to make rectangles for the given number of unit squares. ★ 28 simple & challenging match-3 games and puzzles of hexagons and numbers. The students eat lunch in 3 shifts, with an equal number of students dining during each lunch period. Solve this teddy flower ribbon puzzle and comment your answer. This page is a collection of some of my favourites that I have complied over the years. That gives you a “center” number (19) for the triangle. Each puzzle consists of five triangles (two small, one medium, and two large), a square, and a parallelogram. View answer. Instead of solving the puzzle in the fewest number of moves, these puzzles are similar to sliding-block puzzles in that there are specific places where coins. Maze games in form of different figures illustration set. 1569 is a composite number. Browse the best 1024 Puzzle Game - mobile logic Game - join the numbers alternatives for iPhone & iPad in 2021. Number walls, also called number pyramids or calculation walls, are solved with a specific method. Add A and B to get the next Fibonacci number 4. The object is to work out which numbers correspond to each letter. ★ 28 simple & challenging match-3 games and puzzles of hexagons and numbers. For example, take the top row of numbers. The idea is simple, but it's actually quite tricky to get right. Solve 3D Rubik's Cubes, challenge the computer in Tic-Tac-Toe, and cause chain reactions in our collection!. EUREKA MATH" Lesson 19: Number of unit squares = 20 Number of rectangles I made: Width Length 20. Robert Eisele - You can never have too many Caesar cipher decrypters. What can you learn? Many puzzles require you to go to the Internet to search for information needed to solve it. Hexagon Word Puzzle 1. You will need to match the colors correctly to solve this puzzle. 1570 is a composite number. Each number in the hexagon pyramid is made by adding up the 2 numbers below it. Computational Thinking: Number Hive Logic Puzzles is a booklet on computational thinking based on logic puzzles. ** Test your logical thinking skills with this top-notch brain teaser game for adults and solve the hexa puzzle in the shortest amount of time!. How big is the product? The Wood Hexagon Puzzle is 4. In part, the reason is that it is difficult and Mathematics just love a challenge. Learning how to solve cryptograms is fairly easy once you learn the decoding process. Activate all the squares of the irregular puzzles that contain hexagons with different shapes. Hence their sum is a triangular number , namely s = 1 2 ( 3 n 2 − 3 n + 1 ) ( 3 n 2 − 3 n + 2 ) = 9 n 4 − 18 n 3 + 18 n 2 − 9 n + 2 2 {\displaystyle s={1 \over {2}}(3n^{2}-3n+1)(3n^{2}-3n+2)={9n^{4}-18n^{3}+18n^{2}-9n+2 \over {2}}}. Hive' puzzles. The goal is to find a path of consecutive numbers in a honeycomb grid with hexagonal cells. Thus, it connects to the empty board without any symbol on it. Geometrical form. Barrel/Octagon 01/10/2007 05:07 PM. increment i with 1 i. Hexadecimal Sudokus (also known as 16x16 Sudoku) are a larger version of regular Sudoku that feature a 16 x 16 grid, and 16 hexadecimal digits. Woman who had 'affair' with zoo chimpanzee asked to 'change behaviour'. Maths and Logic Puzzles. This is de la Loubere's method for magic squares. For example, all the pieces of puzzle 1 has a 1 on one side. 1 Create an unlimited number of Sudoku, Kakuro, Futoshiki and Hexagon puzzles with Amigos Number Puzzles compendium software. Herewith, the game covers dozens of levels and this number keeps growing with each update s you'll never run out of gaming material. Thus, we will take the highest power of 2 which is less than the total number which is 64. We can effectively make any. i=i+1 and repeat step 3,4,5,6 with the last value of i=n(n is the no. You must draw a single continuous line that never crosses itself around the grid. 5 days ago. Put the next number in the top. The numbers in the 'Input' and 'Output' boxes are the sum of the numbers in the 3 triangular segments that they side with. Finally solve the puzzle to make sure all the pieces fit as you can see in picture 7. Active 4 years, 4 months ago. I think Gaussian elimination is just a method of reducing a set of equations by substituting one equation into the others to eliminate unnecessary variables. The answer is 10. Choose one from 40 similar games like Sudoku Solver Crossword Puzzle 2021. The total sides of a triangle, square and a hexagon 5. The first job when solving any matchstick puzzle is to count the total number of matchsticks. ** Test your logical thinking skills with this top-notch brain teaser game for adults and solve the hexa puzzle in the shortest amount of time!. Hexagon Word Puzzle 1. Embed this game. Tangrams Improve your spatial comprehension by solving these tangram puzzles. Crosswords. Then in the last equation, a light post with only one light would represent the number 5. Fascinating shapes fit together in this puzzle game for kids that helps to boost addition skills. Answer: The solution is posted below. Word Games. That gives you a “center” number (19) for the triangle. The Square-1 (previously called as Cube 21 and Back to Square One) is a shape-shifting three-layered twisty puzzle. 6% of the regular puzzlers at calcudoku. Active 4 years, 4 months ago. The objective of Drive Ya Nuts is to arrange the seven pieces so the numbers on each piece are situated next to the same number on the pieces that they touch. Instead of solving the puzzle in the fewest number of moves, these puzzles are similar to sliding-block puzzles in that there are specific places where coins. Choose one from 40 similar games like Sudoku Number Puzzle Sodoku 2021. Choose one from 40 similar games like Sudoku Solver Crossword Puzzle 2021. I have congruent sides. When you connect two shapes, you will hear a relaxing note. This is a fantastic puzzle for hand solving. Numbers and Texts. Mar 31, 2020 · There are exactly 19 points of intersection, which is precisely the number of plants. The puzzle consists of 7 hexagons with the numbers 1-6 on the edges of each hexagon. View answer. Desmos offers best-in-class calculators, digital math activities, and curriculum to help every student love math and love learning math. Tormented Souls: All Puzzles Guide. Place the other 6 hexagons around the center hexagon so that the numbers on the edges of the outside hexagons match the numbers on the edges of the adjoining hexagons. A square based pyramid has ___ vertices 4. Hexagon Puzzle is an awesome family game. Solution: 9. This page will document easy to miss puzzles in The Witness for those looking to. Dec 24, 2019 · 8. Hexagon Puzzle Fill Hexagon Solve Puzzle Hexagon Puzzle Fill Number. Inquisitive by nature. Press them to go up to the roof and head inside. Now, if you look closely you will notice 4 types of shape - clock(time varies in 3&4), banana, square in pentagon in hexagon, pentagon in hexagon. The first puzzle is simple. For example, take the top row of numbers. The best free and paid Sudoku Number Puzzle Sodoku alternatives for iPhone and iPad according to actual users and industry experts. Pick a Puzzle: This set includes a total of 10 different puzzles, which gives plenty of new challenges to test their skills. Answer: The solution is posted below. Your strategic skills and puzzle-solving abilities will be put to the test, as you try to complete one of thousands of different challenges! In our challenges, you can build bridges, match moving shapes, and define routes to win. It is inspired by puzzles by Japanese puzzle inventor Naoki Inaba, called 'Cut Blocks'. Apr 28, 2020 · Puzzles such as the 15 Sliding Puzzle and Lights Out can be solved using similar approaches or in some cases even using the same model. The number needs to be changed into a letter from the alphabet. So I have only got a few pages in and saw two grid of hexagons either blue or red and the questions goes: "The colour of each. Welcome to the Math Salamanders' Math Puzzles printable resource pages. Dec 4, 2016 - Solve this interesting Hexagon, Star, and Rectangle math puzzle. of terms which u wnt to generate Fibonacci no. Transcription: "Fit the numbers 1-6 once in every hexagon so that where the hexagons touch, the numbers are the same. The Witness is a big, open game, which can make it hard for completionists to achieve 100%. See more ideas about maths puzzles, math logic puzzles, 2nd grade math. Place one the seven hexagons in the center. An example of a number block puzzle created using Puzzle Master is given above. Rotate all the tiles so all the lines connect. If picture puzzles are more of your thing, see if you can find the hidden objects in these pictures. 1 Move the 2nd piece of the column just below the 1st piece. Write one number in every circle. Daily Puzzles. prepInsta is very useful freshers ,,you people are providing us everything which is required for placements. It includes many different hexa number games to practice and improve logic, brain training and deductive reasoning skills. Solve the puzzle in the room ahead by making a solid vertical line in the center Keep moving ahead, make note of the number across the broken bridge, collect the art piece on the left ledge, go right, and an obelisk should emerge. The boy has a wig. These puzzles usually don't require a. Fine a common, seven letter English word for something you might do on a trail. They need to investigate a hypothesis about numbers and find out whether it's true or false. In the bottom row depress the last four keys (10,10,1,10). labyrinth, riddle, mental activity concep. Where the hexagons touch, adjacent segments contain the same letter (the first clue has been done for you). For tangible and recurring puzzle types, see the list below. A small program to solve the problem from this xkcd cartoon: Cryptarithmetic Two versions of the famous SEND+MORE=MONEY puzzle, and a more general solver for this kind of puzzle. It bears some resemblance to Sudoku, in that it's a logic puzzle with a single solution. 1- Check if the numbers are not already present in that Block A. We're nearly there now. 2 The Proposed GA The proposed method is a steady-state genetic algorithm speci cally designed to solve hexagonal Hidato puzzles. Click on the puzzle to try and solve it yourself :-). EXAMPLE: UNIQUE SOLUTION: Rules of 'No Knight Step Sudoku'. It is inspired by puzzles by Japanese puzzle inventor Naoki Inaba, called 'Cut Blocks'. A= 1, B=2 etc. (I also suspect that the other hourglasses across the border (yellow) will contain "a lot" of same numbers, but as you see at the top, the numbers may be different. As of 2009, it is sold by Uwe Mèffert in his puzzle shop, Meffert's. Can you solve this topological logic puzzle involving hexagons and colors?. 1 Move the 2nd piece of the column just below the 1st piece. Select one hexagon and put it in the center. We all know good old sudoku brain teasers, but now you have to solve it with dinosaur pictures instead of digits! It's truly a new test for your eyes together with a classic logical task. Koshi Arai found the same three solutions (1, 2, 3) that I found. Now, here we have 100 people. The goal is to find a path of consecutive numbers in a honeycomb grid with hexagonal cells. Prime factorization: 1569 = 3 × 523. Download the generated patterns as PDF, PNG or SVG formats. The Square-1 (previously called as Cube 21 and Back to Square One) is a shape-shifting three-layered twisty puzzle. Your objective is to create a row of hexagons that will automatically disappear. Re-Size-It: to Solve the Brain Puzzle. I have congruent sides. 2) Solving The Eight Mysteries in The Witches Kitchen This section helps you to solve all the eight mysteries in The Witches Kitchen. With number and colors hexagon you can also enjoy hexagon that contains. Sometimes it's easier to solve these puzzles if you say your answers out loud. The numbers to the right an bottom are the sums of various rows, colums and diagonals of the 3-by-3 grid of numbers. 4 x 4 Grid - Using the numbers 1,2,3,4 to solve. By these missing number puzzles with answers, you have the full chance to improve your marks in the competitive exam. In order solve a HoneyComb Puzzle, you need to place a number (1 through 6) in each empty hexagon following these simple rules: 1. The number puzzles within the book include calcudoku, mental maths, kakuro, number hexagon, maths puzzle, symbols of value and wijuko. com Times Find the number that is the sum of 2 others 1st product Æ 63. A = 2, B = 5, and C = 3. The Wood Hexagon Puzzle is available in 1 color. How to solve Number Junctions by Matthew Budd - September 17, 2013. So I have only got a few pages in and saw two grid of hexagons either blue or red and the questions goes: "The colour of each. The object is to work out which numbers correspond to each letter. Instead of using only the digits 1 to 9, the hexadoku sudoku uses the 16 digits of the hexadecimal base:. We know 3 of the shapes equal 45, so a single shape equals 15. 5 or higher is required, but the latest 2. Thus, we will take the highest power of 2 which is less than the total number which is 64. • Click in one of the numbers of the left column. Rubik's mosaics are a fantastic way to take your Cube solving to the next level. In this room you will notice a champagne bottle on the floor close to the TV - click on it. It includes many different hexa number games to practice and improve logic, brain training and deductive reasoning skills. Solve this puzzle. increment i with 1 i. One stumbling block to solving this problem is the fact that, in general, you can’t solve it. A Cut Hive puzzle consists of a block of hexagons, with different areas marked out using thicker lines. The sum of each row is to the right, and the sum of the columns is at the bottom. If you have a Windows 7 operating system or higher we can start the process to. Inquisitive by nature. Slitherlinks are a type of puzzle I have been enjoying for a while. Koshi Arai found the same three solutions (1, 2, 3) that I found. com Times Find the number that is the sum of 2 others 1st product Æ 63. Thus, we will take the highest power of 2 which is less than the total number which is 64. Tory MP slammed for his analysis of the lorry driver shortage. The player also starts with 5 hints and receives extra ones when a level is completed to help solve the puzzles which are causing a challenge. This page is a collection of some of my favourites that I have complied over the years. Cassini, for instance, only took thermal images of the storm when it first arrived in 2004. This page allows you to enter the solution of a 16x16 sudoku grid. With 2 in place, this means that there is only one place for 4 that will get us to 6. Drive Ya Nuts is a puzzle game created by Milton Bradley in 1970. You don't need to have a degree in mathematics or English in order to solve common escape room puzzles, but you'll definitely be using numbers and words in your solutions. Almost all puzzles were told to us by a computer/math genius Vlad Mitlin. The objective is to fill in all of the empty white hexagons. Write one number in every circle. The key to solving this puzzle is filling in the yellow shapes first!. 1 Move the 2nd piece of the column just below the 1st piece. Oct 05, 2020 · Research may help solve puzzle of how Venus evolved Relatively little is known about the storm because the planet takes 30 years to orbit the sun, leaving either pole in darkness for that time. Each puzzle consists of five triangles (two small, one medium, and two large), a square, and a parallelogram. Iq puzzles stimulate you IQ Quite obvious that solving problems, raise your problem solving skills. These should be very obvious thank to the game highlighting them after you test the machine. 1569 has no exponents greater than 1 in its prime factorization, so √1569 cannot be simplified. That's a good sign… right? We're going to solve the top layer edges in two steps. To solve the puzzle do the following: In the top row, depress the number 25, and the last three keys (5,10,10). Oct 05, 2020 · Research may help solve puzzle of how Venus evolved Relatively little is known about the storm because the planet takes 30 years to orbit the sun, leaving either pole in darkness for that time. How fast will you manage to solve each puzzle? Filed under. Solve this hexagon puzzle. The Madison School has a new cafeteria with very artistic, new tables for its 375 students. To find out the required number of candies, take one in place of the least number (i. A = 2, B = 5, and C = 3. Amigos Number Puzzles for Mac OS v. The best free and paid Sudoku Solver Crossword Puzzle alternatives for iPhone and iPad according to actual users and industry experts. 2 has to be between 1 and 3. The Hexagon of the 19 numbers. The place for puzzles of all kinds including puzzle games. Welcome to the Math Salamanders' Math Puzzles printable resource pages. In this step will orient all of the pieces so that the yellow face is facing up for all of them. Easy to Miss Puzzles. Welcome to the Math Salamanders' Math Puzzles printable resource pages. Printable Hexagon Puzzle - Printable Hexagon Puzzle really are a preferred supply of enjoyment for people of any age. In order solve a HoneyComb Puzzle, you need to place a number (1 through 6) in each empty hexagon following these simple rules: 1. You must draw a single continuous line that never crosses itself around the grid. For example, all the pieces of puzzle 1 has a 1 on one side. Tower of Hanoi Solution How to solve Tower of Hanoi puzzles with any number of starting disks. Solving a puzzle. Q&A for work. Top 40 Games Like Sudoku Number Puzzle Sodoku - Best Picks for 2021. GCHQ is asking recruits to solve this puzzle - can you? Laurence Fox condemned as 'disgusting piece of work' over Mendy tweet. (a quote from Andrew Wiles in Fermat's Last Theorem, a BBC Horizon documentary). New polyforms and new puzzles can easily be defined and added. Use all 6 sums to create 1 ¶nal puzzle and solve. Tory MP slammed for his analysis of the lorry driver shortage. The numbers in the 'Input' and 'Output' boxes are the sum of the numbers in the 3 triangular segments that they side with. Thanks to Elliott Line for the suggestion! Elliott is the editor of a bimonthly puzzle magazine, Enigma, available exclusively to members of Mensa (a high IQ. Besides the big round tables, some tables are octagons, pentagons, hexagons or squares. The game tells you to get a score of 100 using three darts. Numbers Maze Puzzle is a logic puzzle game which is easy to understand but can be very hard to solve. Check out this awesome sudoku honeycomb and see if you have what it takes to solve this sweet puzzle in this fun sudoku game! number puzzle sudoku hexagon think board YOU MIGHT ALSO LIKE. If you then manage to solve it anyways, it's an extra bonus! 😀. Herewith, the game covers dozens of levels and this number keeps growing with each update s you'll never run out of gaming material. labyrinth, riddle, mental activity concep. Solve this puzzle. Come back regularly to find more new quizzes and conundrums. That way, = translates to 2 stripes and 3 translates to 5 stripes. Tangrams can be used to solve puzzles in which all seven pieces must be put together to create a specified shape. Puzzle #1: Blue, white, purple. I have congruent sides. 1 7 X 4---6 8 2 5---9 3 For a better explanation, please watch the video. (I also suspect that the other hourglasses across the border (yellow) will contain "a lot" of same numbers, but as you see at the top, the numbers may be different. To install it we will need a Windows 7 operating system or higher. Puzzled? In each 7-number hexagon, find the number that is the product of 2 other numbers. Solve Banana Clock Hexagon Puzzle - with Answer. 1 Move the 2nd piece of the column just below the 1st piece. KenKen: Number and logic puzzle. Aristotle's Number Puzzle by Professor Puzzle is number puzzle involving 19 numbers arranged into a hexagon. With number and colors hexagon you can also enjoy hexagon that contains. Adding one to each exponent and multiplying we get (1 + 1) (1 + 1) = 2 × 2 = 4. The initial Hidato grid. Solution to the matchstick puzzle: Remove 4 sticks to leave 3 triangles in hexagonal wheel: By Common stick analysis and Question analysis answer technique. This image shows a solution but I want a program that derives this result. Look for a small hidden container. Hexagon Word Puzzle 1. This guide will show how to complete all the puzzles quickly, which can prove helpful when doing the Speedrun, or simply for reference on a first playthrough. March 2, 2020. Now the arrow buttons work. Today we're having a go at solving the Hanayama Hexagon puzzle. Magic Squares Construct magic squares. 3- Check if it fits in the row. Tangrams Improve your spatial comprehension by solving these tangram puzzles. It is important to be able to convert between different number systems, and detect special properties of numbers such as that they are prime numbers. With 2 in place, this means that there is only one place for 4 that will get us to 6. This page will document easy to miss puzzles in The Witness for those looking to. How to solve Number Junctions by Matthew Budd - September 17, 2013. There are a total of 60 puzzles with four. That gives you a “center” number (19) for the triangle. This is a fantastic puzzle for hand solving. Magic Hexagon Task 190 Years 4 - 10 Summary As the history on the card suggests, this puzzle has been well know to mathematicians. The numbers in the shaded areas are the sum of the numbers in the 2 or 4 segments that they adjoin. No number can be repeated in a hexagon. There are many approaches to solve a Rubik’s cube. The goal is to arrange six of the. Magic Hexagon. Matchstick Puzzles With Answers. To install Hexagon Block Puzzle on a PC we are going to need an Android emulator. three strawberry and five pineapple candies) to it. So with this knowledge, we can actually solve a Sudoku-like puzzle on any graph we want by doing a 2-distant graph colouring in the minimum number of colours possible. So if you've been looking for a solution to Document checked at the grocery store, we've got just the word for you to help you successfully complete your crossword. The hexagon is an arrangement of 24 equilateral triangle. When you start a puzzle, you see a highlighted shape that you should fill with Triominos. symbols around the sides. Hexagon Puzzle Fill Hexagon Solve Puzzle Hexagon Puzzle Fill Number. Puzzled? In each 7-number hexagon, find the number that is the product of 2 other numbers. Total number of sides of 5 squares 2. Another interesting puzzle is here "The Fruits Puzzle". GCHQ is asking recruits to solve this puzzle - can you? Laurence Fox condemned as 'disgusting piece of work' over Mendy tweet. Each hexagon contains the 2nd to the 7th multiple of respectively 6, 5 and 4. 2 has to be between 1 and 3. Description Each row (not just horizontal, but also diagonal) add up to 38? There are 15 rows in all directions to complete, each made up of 3, 4 or 5 pieces. " I've done lots of these and I have been able to solve them using simple techniques. To install it we will need a Windows 7 operating system or higher. The Viral Banana Orange Pineapple Cherry Fruit Brainteaser Puzzles Question with correct answer. EXAMPLE: UNIQUE SOLUTION: Rules of 'No Knight Step Sudoku'. Tricky Matchstick Puzzles Only Brilliant Minds Can Solve Can You Solve The Diluted Wine Puzzle?MATHS PROBLEM STUMPING THE INTERNET - Bananas, Clock, Hexagon Algebra Problem Solved How To Solve The Maths Puzzle That Baffled The Loose Women WhatsApp Group | Loose Women The Incredible Missing Number Puzzle \"Only Page 2/18. Triominos offers a series of puzzles. Accept the challenge and solve this puzzle. 1569 has no exponents greater than 1 in its prime factorization, so √1569 cannot be simplified. 2:08:00 AM banana clock shapes puzzle , WhatsApp Forwards , WhatsApp picture puzzle , WhatsApp Puzzle , whatsapp puzzles. Also, try: Puzzle Time 2; Puzzle Time 3; Puzzle Time 4; Puzzle Time 5; Puzzle. The soundtrack features calming chimes that are triggered by your movement. This is a relaxing game with colorful hexagons placed against a dark background for easy-to-see visuals. Two types of tools are equally useful for solving and learning how to play the Sudoku puzzle. The place for puzzles of all kinds including puzzle games. Hexagon Puzzle Fill Hexagon Solve Puzzle Hexagon Puzzle Fill Number. It is a location on a plane. Hexagon Word PDF SAMPLE. Wood Block Puzzle - Free Classic Block Puzzle Game. • Click in one of the numbers of the left column. Then we ask two questions,. Learn how to solve Number (Cut) Hive puzzles, simple logic puzzles that involve filling…. Briefly, the aim is to draw a line the edges marked on a square grid, such that two constraints are met. New polyforms and new puzzles can easily be defined and added. Solution to puzzle 63: Cyclic hexagon. Tormented Souls contains many puzzles that you must solve to advance the story. Puzzle Square is the eighth episode of Season 8 of Numberblocks and the 113th episode in the series overall. The Super Square One is a 4-layer version of the Square-1. That way, = translates to 2 stripes and 3 translates to 5 stripes. Add A and B to get the next Fibonacci number 4. Each number in the hexagon pyramid is made by adding up the 2 numbers below it. One stumbling block to solving this problem is the fact that, in general, you can’t solve it. Sometimes it's easier to solve these puzzles if you say your answers out loud. In each brick the sum of the two bricks below is shown. The best free and paid Sudoku Number Puzzle Sodoku alternatives for iPhone and iPad according to actual users and industry experts. Printable Hexagon Puzzle - Printable Hexagon Puzzle really are a preferred supply of enjoyment for people of any age. Tricky Matchstick Puzzles Only Brilliant Minds Can Solve Can You Solve The Diluted Wine Puzzle?MATHS PROBLEM STUMPING THE INTERNET - Bananas, Clock, Hexagon Algebra Problem Solved How To Solve The Maths Puzzle That Baffled The Loose Women WhatsApp Group | Loose Women The Incredible Missing Number Puzzle \"Only Page 2/18. Maths and Logic Puzzles. The students eat lunch in 3 shifts, with an equal number of students dining during each lunch period. Free UK shipping on orders over £10. Game has number of different - 2 colors of hexagon. ★ All games have save button to save playing progress to resume it later. The answer keys to help maintain you from getting misplaced and. Tatiana Ayazo /Rd. Fill in the answers clockwise in the correct hexagon. There are various competitive examinations like CAT, MAT, XAT, SNAP, GRE, GMAT , Bank PO, SSC which are putting reasoning and puzzle questions for students to check their mental strength in order to prepare them for next level. If you see a single-cell cage with just a number and no operator, it means that the value in that cell is the target number. Æ 20 8 10 40 7 3 30 72 81 8 7 10 64 9 36 6 42 8 3 7 35 6 5 16 20 3 4 25 7 35 8 28 10 6 4 56 7 10 64 6 9 63 www. It also shows how to. I am not a quadrilateral. Solving (mean) Solving (max. In each brick the sum of the two bricks below is shown. Windows prior to version 7 will not work. Add the three hexagon chips to the puzzle. Top 40 Games Like Sudoku Number Puzzle Sodoku - Best Picks for 2021. Number Puzzles Puzzles Sum of 3 odd numbers equal to 30 | Find numbers from 1,3,5,7,9,11,13,15 to make sum 30 only genius can solve this flower teddy bow puzzle. Drop the Number - Merge Game. First, the rules of slitherlink for those who are unfamiliar with it. Finally solve the puzzle to make sure all the pieces fit as you can see in picture 7. Instructions. The three-dimensional puzzle star contains six pieces that you fit together to create the final product. Instead of solving the puzzle in the fewest number of moves, these puzzles are similar to sliding-block puzzles in that there are specific places where coins. Click on the image to see the animation. HangingHyena - Word game solvers, cryptography tools, Jumble solver and a guide to basic cryptography. Choose one from 40 similar games like Sudoku Solver Crossword Puzzle 2021. Game has number of different - 2 colors of hexagon. Another interesting puzzle is here "The Fruits Puzzle". Here you are just need to connect/match same color or same number hexagon to create larger number. We can effectively make any. The object is to work out which numbers correspond to each letter. 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The length of the first row must be the number of columns in the puzzle. We know 2 bunches of 4 plus the shape (equal to 15) equals 23. Number of the triangles. Otherwise you may end up with unfilled gaps in the star. The lampposts are the number 15, the snowmen are the number 4, and the trees are the number 3. A square based pyramid has ___ vertices 4. It's important to consider how words are written, the number of items, and the direction in which something appears. Level 17 Help baby duckling drink water. The goal is to put a number from 1 to 6 in each hexagon so that no number repeats. Avoid 2020 Connect - Hexagon Puzzle hack cheats for your own safety, choose our tips and advices confirmed by pro players, testers and users like you. The line forms exactly one loop. In other words, Pentagon B - Pentagon A = Pentagon C. To solve the third puzzle, click every circle once except the one in the bottom, left corner. Once you have gotten the cardboard pieces glue the side pieces to the base as you can see in pictures 4,5 and 6. Write the value of su to get next Fibonacci number in the series. Many come from the outstanding Numberplay Blog from the New York Times. Magic Hexagon. Instructions To play a puzzle game, simply use your mouse or finger to drag the piece onto the board. Before you cut out all the pieces, it's recommended you number each piece of each puzzle to make sure you don't accidentally mix all the puzzles up. can solve it! Puzzle: Using numbers between 2 and 20, can you completely fill out the figure below? Rule #1: The difference between the two adjacent hexagons should be more than four Rule #2: The number inside the yellow must be prime. Solve Einstein's famous puzzle by dissecting his clever list of clues. 8) And the numbers in the border around the star represent another number puzzle to solve. I have congruent sides. number(s) within a rebus puzzle You may find some Rebuses challenge you with the use of numbers and their frequently, due to the closeness in sound of 0, 1, 2, 4, 8, 10, etcetera. Instead of using only the digits 1 to 9, the hexadoku sudoku uses the 16 digits of the hexadecimal base:. Place the other 6 hexagons around the center hexagon so that the numbers on the edges of the outside hexagons match the numbers on the edges of the adjoining hexagons. | 2021-09-20 11:09:44 | {"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": 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": 0.324684202671051, "perplexity": 1176.7152895840904}, "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/1631780057036.89/warc/CC-MAIN-20210920101029-20210920131029-00449.warc.gz"} |
http://earthboundmisfit.space/existence-and-uniqueness/ | //Existence and Uniqueness
## Existence and Uniqueness
Nothing really. I just read something here, see something there and then try to relay the message to someone else. Something is always lost in translation. All the great articles, fantastic ideas of other people cease their uniqueness when they hit me. I don’t now why I’m being so hard on myself. I just get to feeling like I will never do anything of any importance, produce any Great Work. Of course, that makes me irritated at myself for feeling some stupid way about.
It’s the lousy weather, the impending maelstrom of the holiday season, the stress over college plans and financial worries. All these things distract me from whatever is supposed to be important. In Recovery Speak, we say that these things (weather, stress, plans, homework) are renting space inside my head. I keep kicking them out, much the way I have been kicked out so many times, but they keep coming back, showing up unannounced with some damn claim about squatters rights
Is there anything unique about being a failure? In being an alcoholic? In being thoroughly disappointed in myself? No. I can’t even be original in my dejection.
## Some new terms to wrap my head around as I delve deeper into differential equations, and trying become familiar with different means of publishing, but it’s so much easier to just write them by hand.
#### The Wronskian
$\mathit{The Wronskian} \\ \\ \mathit{W\left (y _{1},y_{2} \right )\left ( x \right )=\left | \frac{y_{1}\left ( x \right )}{y_{1}^{'}\left ( x \right )} \frac{y_{2}\left ( x \right )}{y_{2}^{'}\left ( x \right )} \right |=y_{1}\left ( x \right ) y_{2}^{'}\left ( x \right )-y_{1}^{'}\left ( x \right )y_{2}\left ( x \right )}$
#### Linear Independence
$\text{If}\; y_{1}\;\text{and } y_{2}\;\text{are two solutions of the equation} \\ y^{''}+p\left ( x \right )y^{'}+q\left ( x \right )y=0, then \\ W\left ( y_1,y_2 \right )\left ( x \right )=W\left ( y_1,y_2 \right )\left ( x \right )exp\left (- \int_{x_{0}}^{x} p\left ( t \right )dt \right ).$
#### Existence and Uniqueness
$\\ \text{Let}\; \mathbb{R}=\left \{ \left ( x,y \right ) \right;\left | x-x_{0} \right |\leq a,\left | y-y_{0} \right |\leq b \} \\ \text{Then there exists an Interval}\; \mathbb{I}=\left [ x_{0}-h,x_{0}+h \right ] \text {such that}\; \\ \left \{ y^{'}=f\left ( x,y \right ),y \left ( x_{0} \right )=y_{0}\right \} \text {has a unique solution y(x) on I.}$
There is something wrong with the LaTeX plugin that the above equation does not render. “I tried nothing and I’m all out of ideas.” Is there a unique solution to this problem? Yes, the lousy plugin does not support some commands the way a fully functioning LaTeX script console would. I’ll stick to inserting my equations as graphics when I’m doing it here, but fro typing up papers I will have to practice with the coding.
#### Variation of parameters
I’m still reading this section of the notes, but as soon as I understand so will you. my oft disappointed readers. Unlike other methods, which are largely algebraic, this method involves integration. The process, no doubt, will fit the general pattern of trying to alter the form of the equation in some way as to make a known method applicable. It is the exception rather than the rule that entirely new “outside the box” methods are called for.
## Is there a unique solution?
I am really enjoying this subject and everything I have been learning. It has been a while since I have tried to learn something totally new and I am glad that I am keeping up. Some new concept will arise and with it a solution of increasing complexity. It is a pattern that I recognize. What is the most exciting part is not the techniques of solving these incredibly useful and applicable equations, but rather my ability to anticipate the next challenge and it’s solution. An example is, while doing some extra work on problems involving undetermined coefficients it became apparent that it would be necessary to alter the form in order to account for initial value conditions. I even proposed the idea that we include the solution to the corresponding homogenous equation in our solution to the particular one. While my ideas lack the formalism and training of the mathematician, per se, I correctly anticipated both the issue and it’s logical (?) solution.
## One of a kind
All unique actually means is one of a kind. Nowhere in the definition does this imply that uniqueness is better, or worse, than commonness. For my part, I can wish things were this way or that and that and lament my sad state, but the truth is that everything I do and have ever done is unique. I’m pretty sure that no one has done things this way before, they would have to have been a damn fool. Can I predict my way out of this, can I anticipate the solution to the next challenge? In a way, yes. There’s some comfort in that.
## Misfit College Fund 2018
$56 of$1,975 raised
\$
Personal Info
They take apart their nightmares and they leave them by the door
Let me fall out of the window with confetti in my hair
Deal out Jacks or Better on a blanket by the stairs
I’ll tell you all my secrets, but I lie about my past
And send me off to bed for evermore | 2018-02-22 00:45:07 | {"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": 3, "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": 0.4944974482059479, "perplexity": 958.1283071063094}, "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/1518891813832.23/warc/CC-MAIN-20180222002257-20180222022257-00430.warc.gz"} |
http://mathoverflow.net/revisions/109149/list | MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
3 added 17 characters in body
Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:
(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?
(another member) Because it's false.
Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$. 0,\pm1$(Migotti, 1883). In other words, this holds true for$n=2^mp^kq^\ell$, where$p,q$are primes. On the other hand, it is false for$n=105=3\cdot5\cdot7$, because the coefficient of$X^7$(or of$X^{41}$as well) is$2$.-2$.
My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?
2 spelling
Let me begin with what looks like a joke. According to a Bourbaki member, the following conservation conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:
(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?
(another member) Because it's false.
Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$. In other words, this holds true for $n=2^mp^kq^\ell$, where $p,q$ are primes. On the other hand, it is false for $n=105=3\cdot5\cdot7$, because the coefficient of $X^7$ (or of $X^{41}$ as well) is $2$.
My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?
1
# Cyclotomic polynomials with coefficients $0,\pm1$
Let me begin with what looks like a joke. According to a Bourbaki member, the following conservation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:
(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?
(another member) Because it's false.
Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$. In other words, this holds true for $n=2^mp^kq^\ell$, where $p,q$ are primes. On the other hand, it is false for $n=105=3\cdot5\cdot7$, because the coefficient of $X^7$ (or of $X^{41}$ as well) is $2$.
My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property? | 2013-06-19 19:06:53 | {"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": 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": 0.7997828722000122, "perplexity": 264.18361903665294}, "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/1368709037764/warc/CC-MAIN-20130516125717-00073-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://msp.org/pjm/2022/316-2/p09.xhtml | #### Vol. 316, No. 2, 2022
Recent Issues Vol. 320: 1 Vol. 319: 1 2 Vol. 318: 1 2 Vol. 317: 1 2 Vol. 316: 1 2 Vol. 315: 1 2 Vol. 314: 1 2 Vol. 313: 1 2 Online Archive Volume: Issue:
The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
The structure of algebraic Baer $^*$-algebras
### Zsolt Szűcs and Balázs Takács
Vol. 316 (2022), No. 2, 431–452
##### Abstract
We describe when a general complex algebraic ${}^{\ast }$-algebra is pre-${C}^{\ast }$-normed, and investigate its structure when the ${}^{\ast }$-algebra is a Baer ${}^{\ast }$-ring in the presence of algebraicity. Our main result is that every complex algebraic Baer ${}^{\ast }$-algebra can be decomposed as a direct sum $M\oplus B$, where $M$ is a finite-dimensional Baer ${}^{\ast }$-algebra and $B$ is a commutative algebraic Baer ${}^{\ast }$-algebra. The summand $M$ is ${}^{\ast }$-isomorphic to a finite direct sum of full complex matrix algebras of size at least $2×2$. The commutative summand $B$ is ${}^{\ast }$-isomorphic to the linear span of the characteristic functions of the clopen sets in a Stonean topological space.
As an application we show that a group $G$ is finite exactly when the complex group algebra $ℂ\left[G\right]$ is an algebraic Baer ${}^{\ast }$-algebra.
##### Keywords
algebraic algebra, von Neumann regular algebra, Baer *-algebra
##### Mathematical Subject Classification
Primary: 16W10
Secondary: 22D15, 46L99 | 2022-11-26 11:51:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 18, "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": 0.7905932068824768, "perplexity": 2152.642970046685}, "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/1669446706291.88/warc/CC-MAIN-20221126112341-20221126142341-00774.warc.gz"} |
https://zbmath.org/?q=an:0662.46069 | ## Multiplication of certain non-commuting random variables.(English)Zbl 0662.46069
Let (A,$$\phi)$$ be a unital algebra over $${\mathbb{C}}$$ with a state $$\phi$$ :A$$\to {\mathbb{C}}$$, $$\phi (1)=1$$. A pair of subalgebras $$A_ 1$$ and $$A_ 2$$ is called free if for any $$a_ 1\in A_ 1$$, $$a_ 2\in A_ 2$$, $$\phi (a_ 1)=0$$ $$\phi (\alpha_ 2)=0$$ implies $$\phi (a_ 1a_ 2)=0$$. A pair of elements a, b is called free if subalgebras generated by a and b are free. An element a is considered as “random variable”, which generates the sequence of moments $$\mu (x^ n)=\phi (a^ n)$$. An analogue of the characteristic function $$S_ a(z)$$ is defined as follows: $$S_ a(z)$$ is a formal series such that $$S_ a(z)=\chi (z)z^{-1}(1+z)$$, where $$\chi (\psi (z))=z$$ and $$\psi (z)=\sum_{n\geq 1}\phi (a^ n)z^ n$$. It is shown that $$S_{ab}(z)=S_ a(z)\cdot S_ b(z)$$ if $$\phi$$ (a)$$\neq 0$$ and $$\phi$$ (b)$$\neq 0$$. The computation of the moments of ab with the help of moments of a and b in general situation is also considered.
Reviewer: V.I.Ovchinnikov
### MSC:
46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 60E10 Characteristic functions; other transforms | 2022-05-27 22:55:42 | {"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": 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": 0.8323736190795898, "perplexity": 259.0709030480211}, "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/1652663006341.98/warc/CC-MAIN-20220527205437-20220527235437-00192.warc.gz"} |
https://math.stackexchange.com/questions/1851140/binary-integer-variables-in-linear-programming | # Binary integer variables in linear programming
Could someone please explain the concept of switch variables (binary integer decision variables) in linear programming?
This example has two alternative constraints
$$\begin{array}{ll} \text{maximize} & 1.5x_1 + 2x_2\\ \text{subject to} & x_1, x_2 \leq 300\\ & x_1 = 0 \quad \mbox{XOR} \quad x_1 \geq 10\end{array}$$
I have seen examples of solutions for such tasks by applying something like following:
$$x_1+My_1 = 0\\x_1 - My_1 \geq 10+M$$
Does someone know and understand this approach and can explain it to me?
• Of course when $x_1=0,$ it cannot happen $x_1\ge10,$ so one can just use OR. – awllower Jul 6 '16 at 16:31
Note that
$$\begin{array}{rl} x_1 = 0 \lor x_1 \geq 10 &\equiv (x_1 \geq 0 \land x_1 \leq 0) \lor x_1 \geq 10\\\\ &\equiv x_1 \geq 0 \land (x_1 \leq 0 \lor x_1 \geq 10)\end{array}$$
We can handle the disjunction $$x_1 \leq 0 \lor x_1 \geq 10$$ using the Big M method. We introduce binary variables $$z_1, z_2 \in \{0,1\}$$ such that $$z_1 + z_2 = 1$$, i.e., either $$(z_1,z_2) = (1,0)$$ or $$(z_1,z_2) = (0,1)$$. We introduce also a large constant $$M \gg 10$$ so that we can write the disjunction in the form
$$x_1 \leq M z_1 \land x_1 \geq 10 - M z_2$$
If $$(z_1,z_2) = (1,0)$$, we have $$x_1 \leq M$$ and $$x_1 \geq 10$$, which is roughly "equivalent" to $$x_1 \geq 10$$. If $$(z_1,z_2) = (0,1)$$, we have $$x_1 \leq 0$$ and $$x_1 \geq 10 - M$$, which is roughly "equivalent" to $$x_1 \leq 0$$.
Thus, we have a mixed-integer linear program (MILP)
$$\begin{array}{ll} \text{maximize} & 1.5x_1 + 2x_2\\ \text{subject to} & x_1, x_2 \leq 300\\ & x_1 \geq 0\\ & x_1 - M z_1\leq 0\\ & x_1 + M z_2 \geq 10\\ & z_1 + z_2 = 1\\ & z_1, z_2 \in \{0,1\}\end{array}$$
For a quick overview of MILP, read Mixed-Integer Programming for Control by Arthur Richards and Jonathan How.
• If you set $M$ to $300$ in the third constraint and $M$ to $10$ in the fourth constraint, and replace $z_2$ by $1-z_1$, you end up with exactly Erwin Kalvelagen's solution (his $\delta$ is your $z_1$). – Kuifje Jul 7 '16 at 3:16
• Would it change anything if the first of the alternative constraints (x1 = 0) would have a value higher than zero at the right hand side? – Bastian Jul 7 '16 at 18:27
• @BastianSchoettle How much higher? Read my other answer to this question. If $x_1 = a$, where $a \in [10, 300]$, then the half-line is inside the polytope. Enlarging the feasible region cannot decrease the maximum. – Rodrigo de Azevedo Jul 7 '16 at 19:34
• @Kuifje The question alluded to the Big M method. Erwin's approach is much simpler, but it does not use any big M's. – Rodrigo de Azevedo Jul 7 '16 at 19:59
• @Rodrigo de Azevedo: Erwin's approach IS a big $M$ method, with specific values of $M$. As I mentioned above: if the first $M$ equals $300$, then you have $$x_1-300z_1\le 0$$ If the second equals $10$ then you have $$x_1+10z_2\ge 10$$ Now given that $z_1=1-z_1$, both equations are equivalent to $$10z_1\le x_1\le 300z_1$$ – Kuifje Jul 7 '16 at 21:47
Using an extra binary variable $\delta$ we can write: \begin{align} & 10 \delta \le x_1 \le 300 \delta \\ &\delta \in \{0,1\} \end{align} $x_1$ is called a semi-continuous variable and some solvers support this directly without the need for extra binary variables.
Note that $x_1 = 0$ and $x_1 \geq 10$ are mutually exclusive.
Writing the inequality constraints in Disjunctive Normal Form (DNF), we obtain
$$\begin{array}{rl} & (x_1 \leq 300 \land x_2 \leq 300) \land (x_1 = 0 \lor x_1 \geq 10) \equiv\\\\ \equiv& (x_1 = 0 \land x_2 \leq 300) \lor (10 \leq x_1 \leq 300 \land x_2 \leq 300)\end{array}$$
Thus, the feasible region is the union of a half-line and a polytope. Hence, we solve two linear programs, namely,
$$\begin{array}{ll} \text{maximize} & 1.5x_1 + 2x_2\\ \text{subject to} & x_1 = 0\\ & x_2 \leq 300\end{array}$$
and
$$\begin{array}{ll} \text{maximize} & 1.5x_1 + 2x_2\\ \text{subject to} & x_1, x_2 \leq 300\\ & x_1 \geq 10\end{array}$$
and then take the maximum of the maxima of each linear program:
• over the half-line, the maximum is $600$, which is attained at $(0,300)$.
• over the polytope, the maximum is $1050$, which is attained at $(300,300)$.
• Thank you for the formatting and your explanation but I need to solve this task in a single model. I've updated my answer accordingly. – Bastian Jul 6 '16 at 19:10 | 2019-10-15 16:47:14 | {"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": 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": 17, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9497273564338684, "perplexity": 424.2285617924114}, "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/1570986660067.26/warc/CC-MAIN-20191015155056-20191015182556-00070.warc.gz"} |
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# Construct a Regular Hexagon of Side 4 Cm. Construct a Circle Circumscribing the Hexagon. - Mathematics
Course
#### Question
Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.
#### Solution
Steps of constructions:
1) Draw a circle of radius 4 cm with centre O.
2) Since 360^@/6 = 60^@, draw radii OA and OB, such that ∠AOB = 60°
3) Cut off arcs BC, CD, DE, EF and each equal to arc AB on given circle.
4) Join AB, BC, CD, DE, EF and FA to get required regular hexagon ABCDEF in a given circle.
Is there an error in this question or solution? | 2020-08-13 15:07:24 | {"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": 0.41584646701812744, "perplexity": 1477.461920560388}, "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-2020-34/segments/1596439739046.14/warc/CC-MAIN-20200813132415-20200813162415-00284.warc.gz"} |