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https://link.springer.com/article/10.1007/s10474-018-0789-8 | # Weak square and stationary reflection
Article
## Abstract
It is well-known that the square principle $${\square_\lambda}$$ entails the existence of a non-reflecting stationary subset of λ+, whereas the weak square principle $${\square^{*} _\lambda}$$ does not. Here we show that if μcf(λ) < λ for all μ < λ, then $${\square^{*} _\lambda}$$ entails the existence of a non-reflecting stationary subset of $${E^{\lambda^+}_{{\rm cf}(\lambda)}}$$ in the forcing extension for adding a single Cohen subset of λ+.
It follows that indestructible forms of simultaneous stationary reflection entail the failure of weak square. We demonstrate this by settling a question concerning the subcomplete forcing axiom (SCFA), proving that SCFA entails the failure of $${\square^{*} _\lambda}$$ for every singular cardinal λ of countable cofinality.
## Key words and phrases
weak square simultaneous stationary reflection SCFA
## Mathematics Subject Classification
primary 03E35 secondary 03E57 03E05
## References
1. 1.
Cummings J., Foreman M., Magidor M.: Squares, scales and stationary reflection. J. Math. Log. 1, 35–98 (2001)
2. 2.
Cummings J., Magidor M.: Martin’s maximum and weak square. Proc. Amer. Math. Soc. 139, 3339–3348 (2011)
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Devlin K.J.: Variations on $${\diamondsuit}$$, J. Symbolic Logic. 44, 51–58 (1979)
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G. Fuchs, Hierarchies of forcing axioms, the continuum hypothesis and square principles, J. Symbolic Logic (to appear); preprint available at www.math.csi.cuny.edu/~fuchs/.
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G. Fuchs, Closure properties of parametric subcompleteness, submitted in 2017; preprint available at www.math.csi.cuny.edu/~fuchs/ .
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R. B. Jensen, Subcomplete forcing and $${\mathcal{L}}$$ -forcing, in: E-Recursion, Forcing and C*-Algebras, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 27, World Sci. Publ. (Hackensack, NJ, 2014), pp. 83–182.Google Scholar
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R. B. Jensen, Forcing axioms compatible with CH, Handwritten notes, available at www.mathematik.hu-berlin.de/~raesch/org/jensen.html (2009).
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R. B. Jensen, Subproper and subcomplete forcing, Handwritten notes, available at www.mathematik.hu-berlin.de/~raesch/org/jensen.html (2009).
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K. Kunen, Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co. (Amsterdam, 1983). An introduction to independence proofs, reprint of the 1980 original.Google Scholar
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Larson P.: Separating stationary reflection principles. J. Symbolic Logic. 65, 247–258 (2000)
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A. Rinot, A relative of the approachability ideal, diamond and non-saturation, J. Symbolic Logic, 75 (2010) 1035–1065.Google Scholar
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J. Silver, The independence of Kurepa’s conjecture and two-cardinal conjectures in model theory, in: Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), Amer. Math. Soc. (Providence, RI, 1971), pp. 383–390.Google Scholar
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S. Todorčević, Trees and linearly ordered sets, in: Handbook of Set-theoretic Topology, North-Holland (Amsterdam, 1984), pp. 235–293.Google Scholar | {"extraction_info": {"found_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": 0.9534235596656799, "perplexity": 4866.312883068599}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865928.45/warc/CC-MAIN-20180524053902-20180524073902-00454.warc.gz"} |
https://physics.stackexchange.com/questions/644458/do-black-holes-attract-more-in-gr-than-newtonian-gravity-beyond-their-event-hori | # Do black holes attract more in GR than Newtonian gravity beyond their event horizon? [duplicate]
I was once shown that if a black hole with the mass of our Sun replaced our Sun in the solar system (I believe it was in a simulation program), the attraction it has on the planets will be the same because the distance between the planets and centre of mass is the same, regardless of the density. How well does this hold up to more accurate mathematics of physics such as GR, because I was shown this in the context of Newtonian Mechanics?
## 4 Answers
Black holes attract slightly more than Newtonian point particles within a distance within a Schwarzschild radius of their event horizon. The reason is that the spacetime curvature makes freely falling particles behave as if there was an extra $$-1/r^3$$ potential in addition to the normal gravitational and centrifugal potential terms. This means that there is an innermost stable circular orbit, while in Newtonian mechanics one can just orbit arbitrarily close to the central mass. Particles falling close to the black hole can plunge in in a way that is different from the hyperbolic swing-bys you get with Newtonian trajectories.
In practice this effect is negligible a few radii away, so planets orbiting a star (distances hundreds of millions of km away) do not feel any noticeable effect since a stellar mass black hole has a radius on the order of a kilometre.
The metric outside a spherical body of matter is the same regardless if it is a fluid kept up by hydrostatic pressure or a black hole. The key difference is that material bodies typically have far greater radius than the Schwarzschild radius, so the above details do not come into play. Even for a neutron star the effects are small.
This claim is equally valid in Newtonian gravity, where it's called the shell theorem, and in General Relativity, where it's called Birkhoff's theorem. It's not actually precisely true in either theory, because (1) the sun is not precisely spherically symmetric, (2) the complete system is not static (because the planets are orbiting), and (3) there is matter involved.
My guess is that that third point would have the largest effect (though still a very small effect on anything like a human timescale). The planets actually raise tides on the sun, which transfers energy between those tides and the planets' orbital motion. The efficiency of this transfer depends on details about the sun's structure (usually expressed in terms of Love numbers) — which would change significantly if the sun was replaced by a black hole. This effect is actually a topic of active investigation for gravitational-wave detectors, as astronomers hope to be able to measure properties of neutron stars orbiting black holes by detecting this transfer. See here for a popular account of some of this research.
• Tides on the Sun actually transfer both momentum and energy from the Sun rotation to the orbital rotation of the planets. Sun rotates faster than any planet. Jun 10 '21 at 14:55
• @fraxinus Good point. Modified the text.
– Mike
Jun 10 '21 at 16:02
It's hard to make a precise comparison, because there is no single way to relate the radial distance $$R$$ in the Newtonian calculation to the parameter $$r$$ in the G.R. calculation.
A useful thing to note is the relationship between the angular frequency $$\omega$$ of a circular orbit and the distance from the central star. The period of the orbit is $$T = 2\pi/\omega$$. In the Newtonian calculation the relation is given by Kepler's third law: $$R^3 \omega^2 = G M$$ where $$M$$ is the mass of the central body. In the General Relativity calculation one gets a very similar result: $$r^3 \left( \frac{d\phi}{dt} \right)^2 = GM$$ I say "very similar" not "exactly the same" because the time parameter $$t$$ here is not quite the same as the Newtonian time, owing to time dilation, and the radial coordinate $$r$$ is not quite the same as the Newtonian $$R$$, but they become equal in the limit $$r \rightarrow \infty$$, in relative terms (i.e. $$(r/R) \rightarrow 1$$).
Another interesting result is for a particle falling in the radial direction. In the G.R. calculation one finds $$\frac{d^2 r}{d \tau^2} = - \frac{GM}{r^2} .$$ Again it looks a lot like the Newtonian result, but it is not quite the same because $$\tau$$ in this equation is proper time along the worldline.
So you see there is no single way to compare the Newtonian case with the G.R. case if you are looking for precision. You can't even assume that the Newtonian orbit of radius $$R$$ is necessarily comparable to the G.R. orbit at radial parameter $$r$$, because the distance between two points on a radial line is not equal to the difference between the two $$r$$ values.
• The best "comparison" I know about (there is still the "r" comparison issue) is equation 7.48 here: ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html. There is an extra cubic term in the GR effective potential that stops all particles with angular momentum "bouncing off" at small values of "r", as they would in the Newtonian case. Jun 10 '21 at 9:15
For non-spinning objects then Birkhoff's theorem applies and the potential outside a spherically symmetric mass distribution is the same (in General Relativity or Newtonian physics) for a non-spinning black hole or any other spherically symmetric mass distribution of the same total mass.
If the central object is spinning (like the Sun) then there is in principle a difference. For a given mass and angular momentum there is only one solution for a black hole - the Kerr metric. Thus the potential is uniquely determined by the mass and angular momentum.
This is no longer true when talking about masses that are not black holes. The potential then depends on exactly how the mass is distributed. I doubt this would produce observable effects at the distance of planets orbiting the Sun, but closer to any potential black hole there would certainly be a difference. For example there are a number of papers outlining the differences in the inner most stable circular orbit for spinning neutron stars vs black holes for example (Pappas & Apostolatos 2013.
The effective potential of a (non-spinning or spinning) black hole is however different in General Relativity (GR) and Newtonian physics. These differences become apparent for radial coordinates that are becoming comparable (but still greater) than the Schwarzschild radius.
The plot below shows the effective potential for a falling body (around a non-spinning black hole) with different amounts of specific angular momentum, $$L/m$$. In general, all orbiting/falling objects will have some angular momentum with respect to the black hole centre unless they are falling exactly radially.
The solid lines show the GR versions, whereas the Newtonian curves (for the same $$L/m$$) are the dashed lines of similar colour.
Now imagine some object approaching the black hole with a (kinetic) energy that places it above zero on the y-axis. For the Newtonian curves, the trajectory will always encounter the dashed line at some finite radius, with a larger radius for a larger value of $$L/m$$. The falling body will then veer away from the black hole on a hyperbolic trajectory.
However, a different fate could await the falling body in GR. If $$L/m < 4GM/c$$ then there is no potential barrier preventing the object falling straight (well, it would be a spiral actually) into the black hole. For larger values of $$L/m$$ then it is possible for a barrier to be encountered, somewhere between $$1.5< r/r_s<3$$ that will turn the falling body away, but there is always some value of (specific) kinetic energy that will allow this barrier to be overcome and allow the body to fall into the black hole.
So from that point of view it is somewhat easier, in the sense that there is a wider range of parameter space, for a falling body to enter the black hole. In fact, in Newtonian physics, unless the angular momentum is very small it is actually very hard to get the body to $$r unless it has an enormous amount of (specific) kinetic energy for its specific angular momentum.
You can also see that beyond about $$\sim 20r_s$$ in this plot, the GR and Newtonian potential become very similar and converge. This means that behaviours of bodies on orbits or trajectories that are a long way from the event horizon are not grossly affected by GR - there are just perturbations, such as the precession of periapsis in GR (famously for the orbit of Mercury) that is absent from a Newtonian treatment. This happens because the period with which it takes a planet in an elliptical orbit to move from its perihelion to aphelion and back again is slightly different (in GR) from the time it takes to execute a full angular orbit.
In terms of the orbital period of a planet for a given $$r$$ (though note that $$r$$ has a slightly different meaning in Newtonian vs GR), a distant observer would say that there is no difference. The circular orbital speed would be $$\sqrt{GM/r}$$ in each case. | {"extraction_info": {"found_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": 29, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8089579343795776, "perplexity": 255.6413492763141}, "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-2022-05/segments/1642320301309.22/warc/CC-MAIN-20220119094810-20220119124810-00045.warc.gz"} |
http://mathhelpforum.com/calculus/137517-error-bound-question.html | ## Error Bound question
The error bound formula for Simpson’s rule is
EBSN = K(b − a)^5/180N^4
where N is the number of subintervals used and K = 12 specifically for our h(x) on the interval [0, 1]. However, it can be shown that the error bound using an nth degree polynomial approximation is given by
EBTn = 1/((2n + 3)(n + 1)!
(The “T” is for “Taylor” after whom these polynomials are named.) Our example above used n = 5. Discuss and compare the errors predicted by these formula for n = 5. If accurate estimates are needed, say to 12 decimal places (i.e. an error less than 10−12 ), which method would you use? How many terms (or subintervals) are necessary for 20 decimal places of accuracy with each method? (Standard encoding of floating point numbers are capable of an accuracy of approximately 16 decimal places.) Is one method always more accurate than the other? Other than accuracy, are there other advantages or disadvantages to using one form of approximation over the other?
Thanks in advance | {"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": 0.9514173269271851, "perplexity": 898.747279641606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469258948913.96/warc/CC-MAIN-20160723072908-00058-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://support.esri.com/en/technical-article/000011814 | English
# How To: Package a published map that has hyperlinked files
## Summary
The instructions provided describe how to package a published map that has hyperlinked files. When packaging data for a map published using the Publisher extension for ArcGIS for Desktop, the packaging process does not automatically include files on disk that are associated with feature layers through hyperlinks. It is up to the user to include those files in the package. However, since the paths of the hyperlinked files must be specified in the attribute table of layers having hyperlinks, confusion can arise about how to organize the locations of the hyperlinked files so that the Published Map File (PMF) can find the referenced files.
## Procedure
Below is an example of a way to organize maps and their hyperlinked files. In this example, the hyperlinked files are .jpg images, and all the .jpg files are stored in a single folder called '\Images'. The map is in a folder called '\Map' at the same directory level as the '\Data' folder where the map's data sources are stored. When the map is published, the PMF is placed in the same folder as the map. For example:
`Code:D:\Temp\1160497\Data\Images\a.jpg b.jpg c.jpg d.jpgD:\Temp\1160497\Data\FGDB.gdbD:\Temp\1160497\Map\Test1.mxd Test1.pmf`
The hyperlink base is the '..\Data\Images' folder, where the '..\' notation means to travel up the directory tree one level from where the map resides, and then down to the 'Data' folder.
In the attribute table for the layer, the field called 'Image1' contains file names in the format 'a.jpg,' 'b.jpg,' and so forth.
Follow the steps below to package a published map that has hyperlinked files:
1. In the Layer Properties dialog box, under the Display tab, check the 'Support Hyperlinks using field' check box. Click the drop-down arrow for the Hyperlink field and select Image1. Click the Document radio button.
As a result of setting these options, when the MXD or PMF is opened, the document successfully links to its data sources and hyperlinked files.
2. Run the Create Data Package tool. An output folder is specified, such as D:\Temp\1160497_pkg. In this example, the data format is 'Compressed and Locked File Geodatabase.'
When packaging, the '\data' and '\pmf' folders are created in an output folder. Any other folders or files are automatically removed from the target folder. The output directory looks like this:
A)
`Code:D:\Temp\1160497_pkg\data\dp00.gdbD:\Temp\1160497_pkg\pmf\Test1.pmf`
B)
3. Copy the '\Images' folder to the '\data' folder:
A)
`Code:D:\Temp\1160497_pkg\data\Images\a.jpg b.jpg c.jpg d.jpgD:\Temp\1160497_pkg\data\dp00.gdbD:\Temp\1160497_pkg\pmf\Test1.pmf`
B)
4. Copy the packaged output folder (in this case, \1160497_pkg) to the intended target location. Open the PMF in the packaged folder. The hyperlinks should be active and reference the corresponding files in the package. | {"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": 0.8542027473449707, "perplexity": 4280.506349138595}, "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/1593655933254.67/warc/CC-MAIN-20200711130351-20200711160351-00084.warc.gz"} |
https://www.physicsforums.com/threads/sterngerlach-experiments-problems-please-help.89879/ | 1. Sep 19, 2005
belleamie
HI there, I was assigned 7 hw problems but there were three I didnt know how to answer...
#3 the state of spin-1/2 particle that is spin up along the axis whose direction is specified by the unit vector n=sin (theta) cos (phi) i+sin (theta) sin (phi)j+cos (theta)k, with theata and phi shown in attachment given by
|+n> = cos (theta/2)|+z>+e^(i*theta) sin (theta/2)|-z>
a) Verify that the state |+n> reduces to the states |+x> and |+y> for angles theta and phi
b)Suppose that a measurement of S(sub z) is carried out on a particle in the state |+n> What is the probability that the measurement yields ((hbar)/2)? and ((-hbar)/2))
c) Determine the uncertainty (change of S(subz))of your measurements
#7 a) what is the amp to find a particle that is in the state |+n> from problem #3 with S(sub y)=hbar/2? what is the probability? check result by evaluating he probability for an appropriate chocice of hte angles phi and theta
b)What is the amp to find a particle that is in the state |+y> with S(sub n)=hbar/2? What is hte probabtility?
#8 Show that the state
|+n> = sin(theta/2)|+z>-e^(i(theta)) cos (theta/2)|-z>
satisfies <+n|-n>=0, where the state |+n> is given from #3 Verify that <-n|-n>=1
Attached Files:
• physics hw problem 3.JPG
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Last edited: Sep 20, 2005
2. Sep 20, 2005
Norman
First of all, you need to read the thread at the top of the page (of the homework help section) about guidelines for posting homework help. With that said I have these questions for you: What have you done so far? Are we supposed to answer these questions straight away for you? I sincerely doubt anyone will. Either way, we cannot help you unless we understand why you don't understand the problems and where you are getting stuck. Please post what you have done so far. Also make sure that problem number eight is written correctly.
Cheers,
Ryan
edit: didn't realize what forum I was in- my apologies | {"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": 0.9333044290542603, "perplexity": 2050.2861058430485}, "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-2018-17/segments/1524125948089.47/warc/CC-MAIN-20180426051046-20180426071046-00467.warc.gz"} |
https://www.physicsforums.com/threads/simple-harmonic-motion-rate-of-changes.635444/ | # Simple harmonic motion + rate of changes
1. Sep 12, 2012
### Carnauba
A bucket of mass 2.0 kg containing 10 kg of water is hanged on a vertical ideal spring with constant 125 N/m, oscillating up and down with an amplitude equal to 3.0 cm.
Suddenly arises a leaky in the bottom of the bucket so that the water flows at constant rate of 2.0 grams/s. When the bucket is half full:
a) determine the period of oscillation and the rate at which the period varies with time.
b) What is the shortest period that this system can have?
2. Sep 12, 2012
### Simon Bridge
Welcome to PF:
To get the best out of this forum, please show us how you have attempted the problem yourself. Do you, for instance, know the equation for the period of a mass on a spring?
Your English is pretty good! Just a few tips...
the bucket is "hung" not "hanged" ... "hanged" means you've killed it via a popular execution method and is usually reserved for humans. "hung" is fine for inanimate objects and lumps of meat, thus: "The man was hanged, then he hung there."
"the bucket springs a leak" or "the bucket starts leaking" ... this is a tricky one since it relied on idiom. It is reasonable to say "the bucket becomes leaky" or "a leak arises in the bottom" but pretty unusual.
Similar Discussions: Simple harmonic motion + rate of changes | {"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": 0.9150890111923218, "perplexity": 1500.8285922550479}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814292.75/warc/CC-MAIN-20180222220118-20180223000118-00797.warc.gz"} |
https://www.physicsforums.com/threads/need-help-with-inverse-laplace-transformation.275458/ | # Need help with inverse laplace transformation
• Start date
• #1
12
0
## Homework Statement
(s^3+6s^2-18s+13)/(s(s+1)(s^2-4s+13)
## The Attempt at a Solution
I am so lost on this problem. I have tried it several times and just keep confusing myself. I think that I am messing up jsut setting up the problem. Can someone please help me out? How do I get this into partial fraction and then where do I go from there.
• #2
156
0
Hi Karmel,
Assuming you want to take the inverse Laplace transform of F(s) = (s^3+6s^2-18s+13)/(s(s+1)(s^2-4s+13)), the first step is to decompose it into partial fractions, which I'm sure you would have come across before. There are three such partial fractions, of the form A/s, B/(s+1) and (Cs+D)/(s^2-4s+13), with A, B, C and D constants to be determined.
Show us your work so we know where you're getting stuck.
• #3
12
0
I would love to show my work but after I set it up like showed to me above with the partial fraction I dont know what to do. The book I am using is so unclear on partial fractions. What are the steps?
• #4
12
0
so after doing the partial fractions if I am doing it right I get teh final answer to be
1-e^-s+(s+4)(sin(3s)(e2s)/3)
I really hope that is right cause I have tried it a thousand ways and Iam getting no where....
• #5
Defennder
Homework Helper
2,591
5
I don't know if that link would expire over time, but if it does, just head to the main page and look for the online Fourier-Laplace calculator. Apart from that I'm kind of lazy to work it out.
Last edited by a moderator:
• #6
3
0
(s^2+2s+1)/(s^3+2s^2+4s+2)
• #7
3
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(s^2+2s+1)/(s^3)+2(s^2)+4s+2)
• #8
3
0
(s^2+2s+1)/((s^3)+2(s^2)+4s+2)
• #9
1
0
can anyone help with this
s((s-a)^0.5)
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4K | {"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": 0.801888108253479, "perplexity": 761.9343126615389}, "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/1634323585916.29/warc/CC-MAIN-20211024081003-20211024111003-00221.warc.gz"} |
https://www.ncgeo.nl/index.php/en/publicatiesgb/publications-on-geodesy/item/2573-pog-81-miguel-caro-cuenca-improving-radar-interferometry-for-monitoring-fault-related-surface-deformation | ## Applications for the Roer Valley Graben and coal mine induced displacements in the southern Netherlands
###### Miguel Caro Cuenca
Publications on Geodesy 81
Delft, 2013. 166 pagina's. ISBN: 978 90 6132 341 9. Alleen verkrijgbaar als pdf.
## Summary
Radar interferometry (InSAR) is a valuable tool to measure surface motion. Applying time series techniques such as Persistent Scatterer Interferometry (PSI), InSAR is able to provide surface displacements maps with mm-precision. However, InSAR can still be further optimized, e.g. by exploiting spatial characteristics of the signal of interest. This study addresses surface deformation associated with geological faults. In principle, this signal is generally spatially smooth but with significant-to-large gradients at fault locations.
We first focus on optimizing InSAR time series analysis, in particular the PSI method, for this specific class of ground deformation processes. Secondly, we apply the improved technique to study fault-related motion in the southern Netherlands, with special interest in detecting neotectonic motion in the Roer Valley Graben and deformation in the abandoned mines of South Limburg.
The proposed optimization adapts PSI to analyze in an iterative manner the signal of interest to estimate spatially the probability density function of displacements. Since the signal is expected to change quickly near faults, we do not restrict this distribution to be unimodal but we allow it to have any shape. Finally, we use the determined distribution to constrain, through Bayesian inference, phase unwrapping (the operation of unfolding the phase outside its natural range of (-phi, phi] radians). We demonstrate a substantial benefit of the Bayesian approach as we show a decrease in the number of unwrapping errors. This thesis also suggests a method that analyzes interferometric phases to estimate noise variance. It is built upon the assumption that coherence can be spatially correlated. The estimated stochastic parameters are used in phase unwrapping by assigning lower weights to noisy observations.
The improved methodology is applied to study fault-related motion in the southern Netherlands, exploiting data from three satellite missions: ERS-1, ERS-2 and Envisat. In particular, we focus on two main areas: the Roer Valley Graben and the abandoned mines of south Limburg.
In the Roer Valley Graben area, a deformation signal associated with geological faults is detected. However, we do not observe any significant indication to attribute a tectonic origin to this signal for two main reasons. First, during large part of the studied period the most of the graben uplifts with respect to adjacent horsts at rates of ~1 mm/yr, behaving opposite to predicted by tectonics. Second, the deformation signal in this area appears to be largely related to water pumping. For example, we observe an uplift signal of about +4 mm/yr that matches in time and space with the cease of pumping in the Erkelenz Coal District, which is located in the Peel horst, adjacent to the Roer Valley Graben.
Concerning the mines of South Limburg, we detect strong surface displacements (uplift) which appear to be centered on the old mines and constrained by tectonic faults. The signal is variable in space and time, with uplift rates up to 20 cm in 18 years, and relatively large gradients across faults (~5 cm/km), in the same time span. Laterally the uplift signal propagates towards the west in this period. The comparison of surface displacements with rising groundwater levels reveal a strong correlation between the two, suggesting the groundwater to be the cause of the uplift.
Assuming that rising ground water levels in the abandoned mines are responsible for the uplift, we estimate the relation between the groundwater and the associated uplift. The skeletal storage coefficient, which directly depends on porosity, is on average 0.5±0.1·10−3, implying that 1 m of water level increase produces 0.5 mm uplift. As we expect that the water may rise many tens of meters, especially in the western side, this may result in everal additional centimeters of future uplift.
Essentially, the surface displacements that we observe in the southern Netherlands seem to be mainly caused by fluctuations in groundwater flow, which appear to be constrained by faults.
Preface … iii
Acknowledgements … v
Summary … vii
Samenvatting (summary in Dutch) … x
Nomenclature … xi
1. Introduction … 1
2. Ground Deformation in the Southern Netherlands … 5
3. Time Series InSAR Analysis: The Persistent Scatterer Interferometry Approach … 23
4. Improvements to Persistent Scatterer Interferometry … 49
5. Surface Deformation in the Roer Valley Rift System Observed by PSI … 77
6. Surface Deformation in the Dutch Coal Field Observed by PSI … 97
7. Conclusions and Recommendations … 115
A. Used software tools … 121
B. Used SAR data … 123 | {"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": 0.8237658739089966, "perplexity": 3280.915157185586}, "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-2020-34/segments/1596439737050.56/warc/CC-MAIN-20200807000315-20200807030315-00145.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/search.php?author_id=14993&sr=posts | ## Search found 25 matches
Sun Feb 09, 2020 1:57 pm
Forum: Concepts & Calculations Using First Law of Thermodynamics
Topic: extensive property
Replies: 12
Views: 118
### Re: extensive property
DHavo_1E wrote:Hello,
What would be an example of an intensive property? Thank you!
Intensive properties are properties that don't rely on the amount of substances but the type of matter. Examples would include temperature, color, and density.
Sun Feb 09, 2020 1:50 pm
Forum: Heat Capacities, Calorimeters & Calorimetry Calculations
Topic: Calorimeter and Bomb Calorimeter
Replies: 4
Views: 60
### Calorimeter and Bomb Calorimeter
I don't know what exactly calorimeters are and I don't know what the differences are either. Can someone please explain how we're also going to use them in problems as well?
Sun Feb 09, 2020 1:45 pm
Forum: Heat Capacities, Calorimeters & Calorimetry Calculations
Topic: Topics on the Midterm
Replies: 22
Views: 229
### Re: Topics on the Midterm
I believe we have to review everything from outlines 1-4 including gibbs free energy but anything after that won't be covered in the test.
Sun Feb 09, 2020 1:40 pm
Forum: Non-Equilibrium Conditions & The Reaction Quotient
Topic: solids and liquids in the rxn quotient
Replies: 8
Views: 109
### Re: solids and liquids in the rxn quotient
We never include solvents/liquids and solids for K and Q. Only aq and g are accepted.
Tue Feb 04, 2020 11:55 am
Forum: Third Law of Thermodynamics (For a Unique Ground State (W=1): S -> 0 as T -> 0) and Calculations Using Boltzmann Equation for Entropy
Topic: Lecture on Feb.3
Replies: 1
Views: 22
### Lecture on Feb.3
Can anyone send me notes or tell me what was being taught on February 3? I had to miss class due to an appointment. Thanks in advance!
Sun Feb 02, 2020 8:01 am
Forum: Thermodynamic Definitions (isochoric/isometric, isothermal, isobaric)
Replies: 10
Views: 441
According to the textbook, it just means that there's o heat transfer between the system and its surroundings even if there's a temperature difference.
Sun Feb 02, 2020 7:50 am
Forum: Thermodynamic Definitions (isochoric/isometric, isothermal, isobaric)
Topic: Reversible Expansion
Replies: 2
Views: 36
### Reversible Expansion
Could someone please explain the concept of this and give an example? Also, are we supposed to know/understand equation involving the integral(for the work of a reversible, isothermal expansion of an ideal gas) that Lavelle showed during class? Or was that just a demonstration of how we got to w = -...
Sat Feb 01, 2020 8:03 pm
Forum: Thermodynamic Systems (Open, Closed, Isolated)
Topic: Closed vs isolated systems
Replies: 24
Views: 198
### Re: Closed vs isolated systems
A way to understand the difference between a closed and isolated system could be used by some examples. For a closed system, think of an ice pack. A closed system is defined as a system that exchanges energy with its surroundings but no matter comes out of it. An ice pack is closed but can still exc...
Sat Feb 01, 2020 7:22 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Self-Test 6A.3A
Replies: 2
Views: 36
### Re: Self-Test 6A.3A
Basically, because hydroiodic acid is a strong acid, it dissociates completely in an aqueous solution. This means that the hydronium concentration is equivalent to the initial molarity of the acid. The back of the book represented this value in micromoles, so instead of just writing it as 6.0x10^-5...
Sat Feb 01, 2020 9:57 am
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Self-Test 6A.3A
Replies: 2
Views: 36
### Self-Test 6A.3A
Estimate the concentrations of (a) H3O+ and (b) OH- at 25 C in 6.0 × 10^-5M HI(aq).
The answer for a is 60.μmol⋅L^-1 and for b it's 0.17nmol⋅L^-1. Can someone tell me how to approach this problem?
Sat Jan 25, 2020 12:13 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Assuming X
Replies: 3
Views: 35
### Re: Assuming X
I think for all questions concerning ICE tables, it’s okay to assume that x is an extremely small number if K(Ka, KB, etc) if is less than 10^-3. For this specific type of problem, if x is less than 5% of the initial acid then the approximation should be fine.
Sat Jan 25, 2020 12:06 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Enthalpy
Replies: 2
Views: 25
### Enthalpy
I’m still having a hard time comprehending what enthalpy is. Could someone please explain to me the concept is in simple terms and possibly even provide an analogy/example?
Sat Jan 25, 2020 12:03 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: H3O+ < 10^-7 Neutral?
Replies: 1
Views: 27
### H3O+ < 10^-7 Neutral?
In lecture, Lavelle had on his slide the following:
Note: If [H3O+] < 10^-7 then the solution is neutral because we know autoprotolysis makes 10^-7 mol/L H3O+.
I don’t understand how this is possible, can someone please explain this to me? Wouldn’t the solution be more basic?
Fri Jan 24, 2020 1:25 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Delta H
Replies: 10
Views: 360
### Re: Delta H
Positive delta H is endothermic(since there’s more energy in the products) whereas negative delta H is exothermic(there’s less energy in the products).
Wed Jan 22, 2020 7:39 pm
Forum: Phase Changes & Related Calculations
Topic: State Property
Replies: 6
Views: 47
### State Property
Can someone explain to me the concept of state property? During lecture, I've written down in my notes that state properties can be added or subtracted which I don't understand. Additionally, I've written down that work and heart depend on a path taken and they're not state properties. Can someone p...
Sat Jan 18, 2020 6:59 pm
Forum: Ideal Gases
Topic: Le Chatelier's Principle
Replies: 7
Views: 201
### Re: Le Chatelier's Principle
The principle states that reactions adjust to reduce the effects of a changing system. It will basically try to go into equilibrium state. For instance, think about how we can create more products without adding more reactants. We are able to get more products by actually removing them in which the ...
Sat Jan 18, 2020 6:51 pm
Forum: Ideal Gases
Topic: Solids and liquids
Replies: 4
Views: 58
### Re: Solids and liquids
Solids and liquids aren't included in the equation because the equation takes into account concentration. It's common sense that solids don't have concentration whereas for liquids, since most are solvents in large excess, it would not really affect the reaction. Therefore, we can just remove solids...
Sat Jan 18, 2020 6:44 pm
Forum: Ideal Gases
Topic: Kp given instead of Kc
Replies: 8
Views: 98
### Re: Kp given instead of Kc
First, if given grams, then convert it to moles. Afterwards, input what you have as moles in the ideal gas law equation but rearrange it to get P alone.
Sat Jan 18, 2020 6:30 pm
Forum: Phase Changes & Related Calculations
Topic: Autoprotolysis
Replies: 15
Views: 144
### Re: Autoprotolysis
It basically means that a proton is transferred between the same molecule. For example, 2H2O(l) <-> H30+(aq) + OH-(aq).
Sat Jan 18, 2020 6:09 pm
Forum: Ideal Gases
Topic: Topics on Test 1
Replies: 37
Views: 417
### Re: Topics on Test 1
The test will cover everything we learned from outlines 1 and 2 (from chemical equilibrium to acids and bases). Anything we learn after that(from week 3) won't appear on the test.
Wed Jan 08, 2020 4:36 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Reaction quotient (Q)
Replies: 3
Views: 42
### Re: Reaction quotient (Q)
What happens when Q>K, Q<K, and Q=K, and why? When Q>K: The reaction is favoring the reverse reaction because there’s more products formed. When Q<K: The reaction is favoring the forward reaction because more reactants are made. As a result, it wants to create more products with the excess reactant...
Wed Jan 08, 2020 4:26 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Chemical Equilibrium Part 2 Post Module #12
Replies: 2
Views: 35
### Re: Chemical Equilibrium Part 2 Post Module #12
Consider the following reaction at 1200 K, for which you know Kc = 1.7 x 10-3. Br2 (g) ⇌ 2 Br (g) Your experimental setup is able to measure the equilibrium concentration of Br2 based on its color, but you are unable to measure the concentration of Br directly. If you measure at equilibrium [Br2] t...
Wed Jan 08, 2020 4:10 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Small K value and Large K value
Replies: 10
Views: 107
### Re: Small K value and Large K value
Because K= Products/Reactants, when K > 10^3, this means that products are being favored since K would be large(thus, shifting to the right). When K<10^-3, the reaction favors reactants because K would be small(shifting to the left).
Wed Jan 08, 2020 4:04 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: K vs Kp vs Kc
Replies: 9
Views: 97
### Re: K vs Kp vs Kc
Kp is only used for partial pressure(for gases) whereas Kc is just the equilibrium constant regarding molar concentrations.
Wed Jan 08, 2020 3:54 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Are Both L and Aq Excluded From Equilibrium Constant Expressions?
Replies: 4
Views: 52
### Are Both L and Aq Excluded From Equilibrium Constant Expressions?
So, I know that solids and liquids are not supposed to be included in Kc but I was wondering if Aq is excluded as well. Isn’t Aq liquid too? Also, I still don’t really understand why these things are excluded from the expression so can someone please explain to me why. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8401917219161987, "perplexity": 4320.588751809291}, "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-20201022195658-20201022225658-00456.warc.gz"} |
https://www.jiskha.com/display.cgi?id=1213322414 | # Math/117
posted by PurplePeople
Graph using the slope and y-intercept..
x + 3y = 9.
1. Reiny
x + 3y = 9.
3y = -x + 9
y = (-1/3)x + 3
the y-intercept is 3, so plot the point (0,3) on the y-axis
the slope is -1/3, so the "rise" is -1 and the "run" is 3
from the y-intercept count 3 units to the right and 1 unit down for the point (3,2)
Join the two points with a line, done!
(notice the points (0,3) and (3,2) both satisfy your equation, so we are right)
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Given y = 2x – 4, choose the statement that accurately describes this line. The slope is 2 and the y intercept is 4. The slope is 2 and the y intercept is -4. The slope is 4 and the y intercept is 2. The slope is -4 and they y intercept …
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Write the equation of the line with given slope and y-intercept. Then graph each line using the slope and y-intercept. Problem #1: Slope: -2; y-intercept: (0,4) my answer: The points I used to graph were: (0, 4) and (2, 0), the equation …
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Please help I am a little confused!!!! Graph using the slope and y-intercept.. x + 3y = 9.
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Graph the following equation: clculte the slope, x-intercept, and y-intercept, and label the the intercelts on the graph. problem: Find an equationin slope-intercept form passing through the points (2,5) and ( 7,-3). (2+5)/(7+ -3) … | {"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": 0.8584917783737183, "perplexity": 1593.464682046168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867041.69/warc/CC-MAIN-20180525043910-20180525063910-00414.warc.gz"} |
https://peeterjoot.wordpress.com/2012/04/01/channel-flow-with-step-pressure-gradient/ | # Peeter Joot's (OLD) Blog.
• 284,779
## Channel flow with step pressure gradient
Posted by peeterjoot on April 1, 2012
[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Motivation and statement.
This is problem 2.5 from [1].
Viscous fluid is at rest in a two-dimensional channel between stationary rigid walls with $y = \pm h$. For $t \ge 0$ a constant pressure gradient $P = -dp/dx$ is imposed. Show that $u(y, t)$ satifies
\begin{aligned}\frac{\partial {u}}{\partial {t}} = \nu \frac{\partial^2 {{u}}}{\partial {{y}}^2} + \frac{P}{\rho},\end{aligned} \hspace{\stretch{1}}(1.1)
and give suitable initial and boundary conditions. Find $u(y, t)$ in form of a Fourier series, and show that the flow approximates to steady channel flow when $t \gg h^2/\nu$.
# Solution
With only horizontal components to the flow, the Navier-Stokes equations for incompressible flow are
\begin{subequations}
\begin{aligned}\frac{\partial {u}}{\partial {t}} + u \frac{\partial {u}}{\partial {x}} = -\frac{1}{{\rho}} \frac{\partial {p}}{\partial {x}} + \nu \left( \frac{\partial^2 {{}}}{\partial {{x}}^2}+\frac{\partial^2 {{}}}{\partial {{y}}^2}\right)u\end{aligned} \hspace{\stretch{1}}(2.2a)
\begin{aligned}\frac{\partial {u}}{\partial {x}} = 0\end{aligned} \hspace{\stretch{1}}(2.2b)
\end{subequations}
Substitution of 2.2b into 2.2a gives us
\begin{aligned}\frac{\partial {u}}{\partial {t}} = -\frac{1}{{\rho}} \frac{\partial {p}}{\partial {x}} + \nu \frac{\partial^2 {{u}}}{\partial {{y}}^2}.\end{aligned} \hspace{\stretch{1}}(2.3)
Our equation to solve is therefore
\begin{aligned}\boxed{\frac{\partial {u}}{\partial {t}} = \Theta(t) \frac{P}{\rho} + \nu \frac{\partial^2 {{u}}}{\partial {{y}}^2}.}\end{aligned} \hspace{\stretch{1}}(2.4)
This equation, for $t < 0$, allows for solutions
\begin{aligned}u = A y + B\end{aligned} \hspace{\stretch{1}}(2.5)
but the problem states that the fluid is at rest initially, so we don’t really have to solve anything (i.e. $A = B = 0$).
The no-slip conditions introduce boundary value conditions $u(\pm h, t) = 0$.
For $t \ge 0$ we have
\begin{aligned}\frac{\partial {u}}{\partial {t}} = \frac{P}{\rho} + \nu \frac{\partial^2 {{u}}}{\partial {{y}}^2}.\end{aligned} \hspace{\stretch{1}}(2.6)
If we attempt separation of variables with $u(y, t) = Y(y) T(t)$, our equation takes the form
\begin{aligned}T' Y = \frac{P}{\rho} + \nu T Y''.\end{aligned} \hspace{\stretch{1}}(2.7)
We see that the non-homogenous term prevents successful application of separation of variables. Let’s modify our problem by attempting to recast our equation into a homogenous form by adding a particular solution for the steady state flow problem. That problem was the solution of
\begin{aligned}\frac{\partial^2 {{}}}{\partial {{y}}^2} u_s(y, 0) = -\frac{P}{\rho \nu}\end{aligned} \hspace{\stretch{1}}(2.8)
which has solution
\begin{aligned}u_s(y, 0) = \frac{P}{2 \rho \nu} \left( h^2 - y^2 \right) + A y + B.\end{aligned} \hspace{\stretch{1}}(2.9)
The freedom to incoporate an $h^2$ constant into the equation as an integration constant has been employed, knowing that it will kill the $y^2$ contributions at $y = \pm h$ to make the boundary condition matching easier. Our no-slip conditions give us
\begin{aligned}0 &= A h + B \\ 0 &= -A h + B.\end{aligned} \hspace{\stretch{1}}(2.10)
Adding this we have $2 B = 0$, and subtracting gives us $2 A h = 0$, so a specific solution that matches our required boundary value (and initial value) conditions is just the steady state channel flow solution we are familiar with
\begin{aligned}u_s(y, 0) = \frac{P}{2 \rho \nu} \left( h^2 - y^2 \right).\end{aligned} \hspace{\stretch{1}}(2.12)
Let’s now assume that our general solution has the form
\begin{aligned}u(y, t) = u_H(y, t) + u_s(y, 0).\end{aligned} \hspace{\stretch{1}}(2.13)
Applying the Navier-Stokes equation to this gives us
\begin{aligned}\frac{\partial {u_H}}{\partial {t}} = \frac{P}{\rho} + \nu \frac{\partial^2 {{u_H}}}{\partial {{y}}^2} + \nu \frac{\partial^2 {{u_s}}}{\partial {{y}}^2}.\end{aligned} \hspace{\stretch{1}}(2.14)
But from 2.8, we see that all we have left is a homogenous problem in $u_H$
\begin{aligned}\frac{\partial {u_H}}{\partial {t}} = \nu \frac{\partial^2 {{u_H}}}{\partial {{y}}^2},\end{aligned} \hspace{\stretch{1}}(2.15)
where our boundary value conditions are now given by
\begin{aligned}0 &= u_H(\pm h, t) + u_s(\pm h) \\ &= u_H(\pm h, t),\end{aligned}
and
\begin{aligned}0 &= u(y, 0) \\ &= u_H(y, 0) + \frac{P}{2 \rho \nu} \left( h^2 - y^2 \right),\end{aligned}
or
\begin{subequations}
\begin{aligned}u_H(\pm h, t) = 0\end{aligned} \hspace{\stretch{1}}(2.16a)
\begin{aligned}u_H(y, 0) = -\frac{P}{2 \rho \nu} \left( h^2 - y^2 \right).\end{aligned} \hspace{\stretch{1}}(2.16b)
\end{subequations}
Now we can apply separation of variables with $u_H = T(t) Y(y)$, yielding
\begin{aligned}T' Y = \nu T Y'',\end{aligned} \hspace{\stretch{1}}(2.17)
or
\begin{aligned}\frac{T'}{T} = \nu \frac{Y''}{Y} = \text{constant} = - \nu \alpha^2.\end{aligned} \hspace{\stretch{1}}(2.18)
Here a positive constant $\nu \alpha^2$ has been used assuming that we want a solution that is damped with time.
Our solutions are
\begin{aligned}T &\propto e^{- \nu \alpha^2 t} \\ Y &= A \sin \alpha y + B \cos\alpha y,\end{aligned} \hspace{\stretch{1}}(2.19)
or
\begin{aligned}u_H(y, t) = \sum_\alpha e^{-\alpha^2 \nu t} \left( A_\alpha \sin \alpha y + B_\alpha \cos\alpha y \right).\end{aligned} \hspace{\stretch{1}}(2.21)
We have constraints on $\alpha$ due to our boundary value conditions. For our sin terms to be solutions we require
\begin{aligned}\sin (\alpha (\pm h)) = \sin n \pi\end{aligned} \hspace{\stretch{1}}(2.22)
and for our cosine terms to be solutions we require
\begin{aligned}\cos (\alpha (\pm h)) = \cos \left( \frac{\pi}{2} + n \pi \right)\end{aligned} \hspace{\stretch{1}}(2.23)
\begin{aligned}\alpha &= \frac{2 n \pi}{2 h} \\ \alpha &= \frac{2 n + 1 \pi}{2 h}\end{aligned} \hspace{\stretch{1}}(2.24)
respectively.
Our homogeneous solution therefore takes the form
\begin{aligned}u_H(y, t) = C_0 + \sum_{m > 0} C_m e^{ -(m \pi/2h)^2 \nu t } \left\{\begin{array}{l l}\sin \left( \frac{ m \pi y }{2 h} \right) & \quad \mbox{latex meven} \\ \cos \left( \frac{ m \pi y }{2 h} \right) & \quad \mbox{$m$ odd} \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.26)
Our undetermined constants should be provided by the boundary value constraint at $t = 0$ 2.16b, leaving us to solve the Fourier problem
\begin{aligned}-\frac{P}{2 \mu} \left( h^2 - y^2 \right)=\sum_{m \ge 0} C_m \left\{\begin{array}{l l}\sin \left( \frac{ m \pi y }{2 h} \right) & \quad \mbox{latex meven} \\ \cos \left( \frac{ m \pi y }{2 h} \right) & \quad \mbox{$m$ odd} \\ \end{array}\right.\end{aligned} \hspace{\stretch{1}}(2.27)
Multiplying by a sine and integrating will clearly give zero (even times odd function over a symmetric interval). Let’s see if there’s any scaling required to select out the $C_m$ term
\begin{aligned}\int_{-h}^h \cos \left( \frac{ m \pi y }{2 h} \right) \cos \left( \frac{ n \pi y }{2 h} \right) dy&=\frac{2h}{\pi} \int_{-h}^h \cos \left( \frac{ m \pi y }{2 h} \right) \cos \left( \frac{ n \pi y }{2 h} \right) \pi dy/2h \\ &=\frac{2h}{\pi} \int_{-\pi/2}^{\pi/2}\cos m x \cos n x dx \\ &=\frac{h}{\pi} \int_{-\pi/2}^{\pi/2}\left( \cos( (m - n) \pi/2 ) +\cos( (m + n) \pi/2 ) \right) dx\end{aligned}
Note that since $m$ and $n$ must be odd, $m \pm n = 2 c$ for some integer $c$, so this integral is zero unless $m = n$ (consider $m = 2 a + 1, n = 2 b + 1$). For the $m = n$ term we have
\begin{aligned}\int_{-h}^h \cos \left( \frac{ m \pi y }{2 h} \right) \cos \left( \frac{ n \pi y }{2 h} \right) dy&=\frac{h}{\pi} \int_{-\pi/2}^{\pi/2}\left( 1+\cos( m \pi ) \right) dx \\ &=h\end{aligned}
Therefore, our constants $C_m$ (for odd $m$) are given by
\begin{aligned}C_m &= -\frac{P h }{2 \mu} \int_{-h}^h\left( 1 - \left( \frac{y}{h}\right)^2 \right)\cos \left( \frac{ m \pi y }{2 h} \right) dy \\ &=-\frac{P h^2 }{2 \mu} \int_{-1}^1\left( 1 - x^2 \right)\cos \left( \frac{ m \pi x }{2} \right) x \\ \end{aligned}
With $m = 2 n + 1$, we have
\begin{aligned}C_{2 n + 1} = -\frac{16 P h^2 (-1)^n}{\mu \pi^3 (2 n + 1)^3}.\end{aligned} \hspace{\stretch{1}}(2.28)
Our complete solution is
\begin{aligned}u(y, t) =\frac{P h^2}{2 \mu} \left( 1 - \left( \frac{y}{h} \right)^2 \right)- \frac{16 P h^2 }{\mu \pi^3 }\sum_{n = 0}^\infty \frac{(-1)^n}{(2 n + 1)^3}e^{ -((2 n + 1) \pi/2h)^2 \nu t } \cos \left( \frac{ (2 n + 1) \pi y }{2 h} \right) .\end{aligned} \hspace{\stretch{1}}(2.29)
The largest of the damped exponentials above is the $n = 0$ term which is
\begin{aligned}e^{ - \pi^2 \nu t /h^2 },\end{aligned} \hspace{\stretch{1}}(2.30)
so if $\nu t >> h^2$ these terms all die off, leaving us with just the steady state.
Rather remarkably, this Fourier series is actually a very good fit even after only a single term. Using the viscosity and density of water, $h = 1 \text{cm}$, and $P = 3 \times \mu_{\text{water}} \times (2 \text{cm}/{s})/ h^2$ (parameterizing the pressure gradient by the average velocity it will induce), a plot of the parabola that we are fitting to and the difference of that from the first Fourier term is shown in figure (1).
Figure 1: Parabolic channel flow steady state, and difference from first Fourier term.
The higher order corrections are even smaller. Even the first order deviations from the parabola that we are fitting to is a correction on the scale of $1/100$ of the height of the parabola. This is illustrated in figure (2) where the magnitude of the first 5 deviations from the steady state are plotted.
Figure 2: Difference from the steady state for the first five Fourier terms.
An animation of the time evolution above can be found in figure (3). If this animation is unavailable, it can also be found at http://youtu.be/0vZuv9HBtmo.
Figure 3: Time evolution of channel flow velocity profile after turning on a constant pressure gradient.
It’s also interesting to look at the very earliest part of the time evolution. Observe in figure (4) (or http://youtu.be/dDkx8iLwOew) the oscillatory phenomina. Could some of that be due to not running with enough Fourier terms in this early part of the evolution when more terms are probably significant?
Figure 4: Early time evolution of channel flow velocity profile after turning on a constant pressure gradient.
# References
[1] D.J. Acheson. Elementary fluid dynamics. Oxford University Press, USA, 1990. | {"extraction_info": {"found_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": 73, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999406337738037, "perplexity": 2778.5535031345576}, "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-2017-13/segments/1490218188926.39/warc/CC-MAIN-20170322212948-00267-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/trig-proof.193763/ | # Trig proof
1. Oct 25, 2007
### ehrenfest
1. The problem statement, all variables and given/known data
This always seemed intuitive to me, but when I tried to prove it I got stuck:
sin(x +pi/2) = cos(x)
It is easy with the angle addition formula, but is there another way?
2. Relevant equations
3. The attempt at a solution
2. Oct 25, 2007
### mjsd
how else do u want to prove it?
consider
$$e^{i(\theta + \pi/2)} = \cos (\theta + \pi/2) +i \sin (\theta + \pi/2) \qquad\quad (1)$$
$$e^{i(\theta + \pi/2)} = e^{i\theta} e^{i\pi/2} = i e^{i\theta} = -\sin (\theta) + i \cos (\theta) \quad (2)$$
equating Re and Im part of (1) and (2) to get two relationships between sin and cos.
3. Oct 25, 2007
### BerryBoy
I love the beauty of this mathematical proof:
Using Taylor expansion about $x=0$:
$$\cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + ...$$
$$\sin(x) = x - \frac{1}{6}x^3 + \frac{1}{120}x^5 + ...$$
and:
$$e^{ix} = 1 + ix - \frac{1}{2}x^2 - i\frac{1}{6}x^3 + ... + \frac{i^n}{n!}x^n$$
So
$$e^{ix} = \cos(x) + i\sin(x)$$
It still amazes me, absolutely incredible :P
Sam
4. Oct 26, 2007
### dynamicsolo
There are many ways. The analytical approaches are very nice, but awfully sophisticated and high-powered.
This trig relation just came up in the work I'm doing with students on torque. Here's a trigonometric proof (which I'll describe rather than scanning and uploading a drawing):
Draw a right triangle and mark one of the non-right angles, theta. The other angle is of course complimentary, so it's (90º - theta); the sine of this angle will be the cosine of the other angle, which is the familiar "co-relation"
$$sin(90º - \theta) = cos \theta$$.
Now extend the side of the triangle adjacent to the complimentary angle outward away from the right angle. The angle between that ray and the hypotenuse is supplementary to the angle (90º - theta), so its measure is 180º - (90º - theta) = 90º + theta . But the sine of a supplementary angle is the same as the sine of the angle itself:
$$sin(180º - \theta) = sin \theta$$ , so
$$sin(90º + \theta) = sin(90º - \theta) = cos \theta$$. Q.E.D.
P.S. *heh* I just thought of a graphical way to prove it. The graph of sin x looks like the graph of cos x translated to the right by $$\frac{\pi}{2}$$. So if you shift the graph of sin x to the left by the same amount, you have $$sin( x + \frac{\pi}{2}) = cos x$$.
Last edited: Oct 26, 2007
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https://www.investopedia.com/articles/optioninvestor/07/gamm_delta_neutral.asp | Have you found strategies that make use of the decay of an option's theta that are attractive but you can't stand the associated risk? At the same time, conservative strategies such as covered-call writing or synthetic covered-call writing can be too restrictive. The gamma-delta neutral spread may be the best middle ground when searching for a way to exploit time decay while neutralizing the effect of price actions on your position's value. In this article, we'll introduce you to this strategy.
## Options "Greeks"
To understand the application of this strategy, knowledge of the basic Greek measures is essential. This means that the reader must also be familiar with options and their characteristics.
## Theta
Theta is the decay rate in an option's value that can be attributed to the passage of one day's time. With this spread, we will exploit the decay of theta to our advantage to extract a profit from the position. Of course, many other spreads do this; but as you'll discover, by hedging the net gamma and net delta of our position, we can safely keep our position direction neutral.
## Strategy
For our purposes, we will use a ratio call write strategy as our core position. In these examples, we will buy options at a lower strike price than that at which they are sold. For example, if we buy the calls with a $30 strike price, we will sell the calls at a$35 strike price. We will perform a regular ratio call write strategy and adjust the ratio at which we buy and sell options to materially eliminate the net gamma of our position.
We know that in a ratio write options strategy, more options are written than are purchased. This means that some options are sold "naked." This is inherently risky. The risk here is that if the stock rallies enough, the position will lose money as a result of the unlimited exposure to the upside with the naked options. By reducing the net gamma to a value close to zero, we eliminate the risk that the delta will shift significantly (assuming only a very short time frame).
## Neutralizing the Gamma
To effectively neutralize the gamma, we first need to find the ratio at which we will buy and write. Instead of going through a system of equation models to find the ratio, we can quickly figure out the gamma neutral ratio by doing the following:
1. Find the gamma of each option.
2. To find the number you will buy, take the gamma of the option you are selling, round it to three decimal places and multiply it by 100.
3. To find the number you will sell, take the gamma of the option you are buying, round it to three decimal places and multiply it by 100.
For example, if we have our $30 call with a gamma of 0.126 and our$35 call with a gamma of 0.095, we would buy 95 $30 calls and sell 126$35 calls. Remember this is per share, and each option represents 100 shares.
• Buying 95 calls with a gamma of 0.126 is a gamma of 1,197, or: \begin{aligned} &95 \times ( 0.126 \times 100 ) \\ \end{aligned}
• Selling 126 calls with a gamma of -0.095 (negative because we're selling them) is a gamma of -1,197, or: \begin{aligned} &126 \times ( -0.095 \times 100 ) \\ \end{aligned}
This adds up to a net gamma of 0. Because the gamma is usually not nicely rounded to three decimal places, your actual net gamma might vary by about 10 points around zero. But because we are dealing with such large numbers, these variations of actual net gamma are not material and will not affect a good spread.
## Neutralizing the Delta
Now that we have the gamma neutralized, we will need to make the net delta zero. If our $30 calls have a delta of 0.709 and our$35 calls have a delta of 0.418, we can calculate the following.
• 95 calls bought with a delta of 0.709 is 6,735.5, or: \begin{aligned} &95 \times ( 0.709 \times 100 ) \\ \end{aligned}
• 126 calls sold with a delta of -0.418 (negative because we're selling them) is -5,266.8, or: \begin{aligned} &126 \times ( -0.418 \times 100 ) \\ \end{aligned}
This results in a net delta of positive 1,468.7. To make this net delta very close to zero, we can short 1,469 shares of the underlying stock. This is because each share of stock has a delta of 1. This adds -1,469 to the delta, making it -0.3, very close to zero. Because you cannot short parts of a share, -0.3 is as close as we can get the net delta to zero. Again, as we stated in the gamma because we are dealing with large numbers, this will not be materially large enough to affect the outcome of a good spread.
## Examining the Theta
Now that we have our position effectively price neutral, let's examine its profitability. The $30 calls have a theta of -0.018 and the$35 calls have a theta of -0.027. This means:
• 95 calls bought with a theta of -0.018 is -171, or: \begin{aligned} &95 \times ( -0.018 \times 100 ) \\ \end{aligned}
• 126 calls sold with a theta of 0.027 (positive because we're selling them) is 340.2, or: \begin{aligned} &126 \times ( 0.027 \times 100 ) \\ \end{aligned}
## Drawbacks
A few risks are associated with this strategy. First, you'll need low commissions to make a profit. This is why it is important to have a very low commission broker. Very large price moves can also throw this out of whack. If held for a week, a required adjustment to the ratio and the delta hedge is not probable; if held for a longer time, the price of the stock will have more time to move in one direction.
Changes in implied volatility, which are not hedged here, can result in dramatic changes in the position's value. Although we have eliminated the relative day-to-day price movements, we are faced with another risk: increased exposure to changes in implied volatility. Over the short time horizon of a week, changes in volatility should play a small role in your overall position.
## The Bottom Line
The risk of ratio writes can be brought down by mathematically hedging certain characteristics of the options, along with adjusting our position in the underlying common stock. By doing this, we can profit from the theta decay in the written options. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9760961532592773, "perplexity": 1370.3563382674824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107912807.78/warc/CC-MAIN-20201031032847-20201031062847-00635.warc.gz"} |
https://mathoverflow.net/questions/48409/homotopical-descent-information-contained-in-the-dwyer-kan-function-complexes-of | # Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?
Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the system of local isomorphisms generated by the Grothendieck topology (one way to describe these is as the morphisms that become isomorphisms under sheafification, but there are other, more direct methods to deduce what they are).
Recall that given any pair $(C,W)$ of a category and a class of morphisms, we can apply any of the methods of Dwyer-Kan simplicial localization to enrich $C$ to a simplicial category such that $\pi_0$ of the function complexes are exactly the hom-sets in the classical localization of the category.
Hammock localization is probably the best choice in this case, since local isomorphisms arising from a Grothendieck topology admit a calculus of left fractions, which means that the resulting function complexes are exactly the nerves of certain cospan categories.
Anyhow, is there any interesting descent-related information contained in the higher homotopical data of the function complexes? Is there any relation between this sort of thing and cohomological descent?
- | {"extraction_info": {"found_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": 0.8938605785369873, "perplexity": 250.94163469479753}, "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/1440644064160.12/warc/CC-MAIN-20150827025424-00352-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://codeforces.com/contest/1540/problem/B | Please subscribe to the official Codeforces channel in Telegram via the link https://t.me/codeforces_official. ×
B. Tree Array
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
You are given a tree consisting of $n$ nodes. You generate an array from the tree by marking nodes one by one.
Initially, when no nodes are marked, a node is equiprobably chosen and marked from the entire tree.
After that, until all nodes are marked, a node is equiprobably chosen and marked from the set of unmarked nodes with at least one edge to a marked node.
It can be shown that the process marks all nodes in the tree.
The final array $a$ is the list of the nodes' labels in order of the time each node was marked.
Find the expected number of inversions in the array that is generated by the tree and the aforementioned process.
The number of inversions in an array $a$ is the number of pairs of indices $(i, j)$ such that $i < j$ and $a_i > a_j$. For example, the array $[4, 1, 3, 2]$ contains $4$ inversions: $(1, 2)$, $(1, 3)$, $(1, 4)$, $(3, 4)$.
Input
The first line contains a single integer $n$ ($2 \le n \le 200$) — the number of nodes in the tree.
The next $n - 1$ lines each contains two integers $x$ and $y$ ($1 \le x, y \le n$; $x \neq y$), denoting an edge between node $x$ and $y$.
It's guaranteed that the given edges form a tree.
Output
Output the expected number of inversions in the generated array modulo $10^9+7$.
Formally, let $M = 10^9+7$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.
Examples
Input
3
1 2
1 3
Output
166666669
Input
6
2 1
2 3
6 1
1 4
2 5
Output
500000009
Input
5
1 2
1 3
1 4
2 5
Output
500000007
Note
This is the tree from the first sample:
For the first sample, the arrays are almost fixed. If node $2$ is chosen initially, then the only possible array is $[2, 1, 3]$ ($1$ inversion). If node $3$ is chosen initially, then the only possible array is $[3, 1, 2]$ ($2$ inversions). If node $1$ is chosen initially, the arrays $[1, 2, 3]$ ($0$ inversions) and $[1, 3, 2]$ ($1$ inversion) are the only possibilities and equiprobable. In total, the expected number of inversions is $\frac{1}{3}\cdot 1 + \frac{1}{3} \cdot 2 + \frac{1}{3} \cdot (\frac{1}{2} \cdot 0 + \frac{1}{2} \cdot 1) = \frac{7}{6}$.
$166666669 \cdot 6 = 7 \pmod {10^9 + 7}$, so the answer is $166666669$.
This is the tree from the second sample:
This is the tree from the third sample: | {"extraction_info": {"found_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": 0.8058757185935974, "perplexity": 451.79075243866123}, "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-2021-39/segments/1631780056900.32/warc/CC-MAIN-20210919190128-20210919220128-00228.warc.gz"} |
https://www.chemeurope.com/en/encyclopedia/Troposphere.html | My watch list
my.chemeurope.com
# Troposphere
The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 75% of the atmosphere's mass and almost all of its water vapor and aerosols.
The average depth of the troposphere is about 11 km in the middle latitudes. It is deeper in the tropical regions (up to 20 km) and shallower near the poles (about 7 km in summer, indistinct in winter). The lowest part of the troposphere, where friction with the Earth's surface influences air flow, is the planetary boundary layer. This layer is typically a few hundred meters to 2 km deep depending on the landform and time of day. The border between the troposphere and stratosphere, called the tropopause, is a temperature inversion.[1]
The word troposphere derives from the Greek "tropos" for "turning" or "mixing," reflecting the fact that turbulent mixing plays an important role in the troposphere's structure and behavior. Most of the phenomena we associate with day-to-day weather occur in the troposphere.[1]
## Pressure and temperature structure
### Composition
The chemical composition of the troposphere is essentially uniform, with the notable exception of water vapor. The source of water vapor is at the surface through the processes of evaporation and transpiration. Furthermore the temperature of the troposphere decreases with height, and saturation vapor pressure decreases strongly with temperature, so the amount of water vapor that can exist in the atmosphere decreases strongly with height. Thus the proportion of water vapor is normally greatest near the surface and decreases with height.
### Pressure
The pressure of the atmosphere is maximum at the surface and decreases with higher altitude. This is because the atmosphere is very nearly in hydrostatic equilibrium, so that the pressure is equal to the weight of air above a given point. The change in pressure with height therefore can be equated to the density with this hydrostatic equation:[2]
$\frac{dp}{dz} = -\rho g_n = - \frac {mpg}{RT}$
where:
• gn stands for the standard gravity
• ρ stands for density
• z stands for height
• p stands for pressure
• R stands for the gas constant
• T stands for temperature in kelvins
• m stands for the molar mass
Since temperature in principle also depends on altitude, one needs a second equation to determine the pressure as a function of height, as discussed in the next section.
### Temperature
Main article: Lapse rate
The temperature of the troposphere generally decreases with altitude. The rate at which the temperature decreases, dT / dz, is called the lapse rate. The reason for this decrease is as follows. When the air is stirred by convection, and a parcel of air rises, it expands, because the pressure is lower at higher altitudes. As the air parcel expands, it pushes on the air around it, doing work; but generally it does not gain heat in exchange from its environment, because its thermal conductivity is low (such a process is called adiabatic). Since the parcel does work and gains no heat, it loses energy, and so its temperature decreases. (The reverse, of course, will be true for a sinking parcel of air.) [1]
Since the heat exchanged dQ is related to the entropy change dS by dQ=T dS, the equation governing the temperature as a function of height for a thoroughly mixed atmosphere is
$\frac{dS}{dz} = 0$
where S is the entropy. The rate at which temperature decreases with height under such conditions is called the adiabatic lapse rate.
For dry air, which is approximately an ideal gas, we can proceed further. The adiabatic equation for an ideal gas is [3]
$p(z)T(z)^{-\frac{\gamma}{\gamma-1}}=constant$
where γ is the heat capacity ratio (γ=7/5, for air). Combining with the equation for the pressure, one arrives at the dry adiabatic lapse rate,[4]
$\frac{dT}{dz}=- \frac{mg}{R} \frac{\gamma-1}{\gamma}=-9.8^{\circ}\mathrm{C}/\mathrm{km}$
If the air contains water vapor, then cooling of the air can cause the water to condense, and the behavior is no longer that of an ideal gas. If the air is at the saturated vapor pressure, then the rate at which temperature drops with height is called the saturated adiabatic lapse rate. More generally, the actual rate at which the temperature drops with altitude is called the environmental lapse rate. In the troposphere, the average environmental lapse rate is a drop of about 6.5 °C for every 1 km (1000 meters) increase in height. [1]
Depending on the weather conditions, one may find that the environmental lapse rate (the actual rate at which temperature drops with height, dT / dz) is not equal to the adiabatic lapse rate (or correspondingly, that $dS/dz \ne 0$). If the upper air is warmer than predicted by the adiabatic lapse rate (dS / dz > 0), then when a parcel of air rises and expands, it will arrive at the new height at a lower temperature than its surroundings. In this case, the air parcel is denser than its surroundings, so it sinks back to its original height, and the air is stable against being stirred. Such a situation is called temperature inversion, and can lead to the trapping of air pollution in basins such as that of Los Angeles. If, on the contrary, the upper air is cooler than predicted by the adiabatic lapse rate, then when the air parcel rises to its new height it will have a higher temperature and a lower density than its surroundings, and will float. Such a process can happen spontaneously, and under such conditions, the air will be stirred by spontaneous convection currents.[1][2]
Temperatures decrease at middle latitudes from an average of 15°C at sea level to about -55°C at the beginning of the tropopause. At the poles, the troposphere is thinner and the temperature only decreases to -45°C, while at the equator the temperature at the top of the troposphere can reach -75°C.[citation needed]
### Tropopause
Main article: Tropopause
The tropopause is the boundary region between the troposphere and the stratosphere.
Measuring the temperature change with height through the troposphere and the stratosphere identifies the location of the tropopause. In the troposphere, temperature decreases with altitude. In the stratosphere, however, the temperature remains constant for a while and then increases with altitude. The region of the atmosphere where the lapse rate changes from positive (in the troposphere) to negative (in the stratosphere), is defined as the tropopause.[1] Thus, the tropopause is an inversion layer, and there is little mixing between the two layers of the atmosphere.
## Atmospheric circulation
Main article: Atmospheric circulation
The basic structure of large-scale circulation in the troposphere remains fairly constant. There are three convection cells in each hemisphere: the Hadley cell, the Ferrel cell, and the Polar cell, which guide the prevailing winds, thereby transporting heat from the equator to the poles.[1] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9329360723495483, "perplexity": 655.6689474874995}, "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-20190625072148-20190625094148-00123.warc.gz"} |
https://www.dm.unipi.it/eventi/limiting-behaviour-of-rescaled-nonlocal-perimeters-and-of-their-first-variations-valerio-pagliari-universita-di-pisa/ | # Limiting behaviour of rescaled nonlocal perimeters and of their first variations. – Valerio Pagliari (Universita’ di Pisa)
#### Abstract
We introduce a class of integral functionals known as nonlocal perimeters, which can be thought as interactions between a set and its complement that are weighted by a positive kernel. In the first part of the talk, we summarise the main features of these functionals and then we study the asymptotic behaviour of the family associated with mass-preserving rescalings of a given kernel. Namely, we prove that when the scaling parameter approaches $0$, the rescaled non local perimeters $Gamma$-converge to De Giorgi’s perimeter, up to a multiplicative constant. In the second part of the talk, we show that a similar result holds for nonlocal curvatures, i.e. for the first variations of the nonlocal perimeters; time permitting, we shall hint at possible applications of this to dislocation dynamics.
Torna in cima | {"extraction_info": {"found_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": 0.951564371585846, "perplexity": 588.6106580726479}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943750.71/warc/CC-MAIN-20230322051607-20230322081607-00302.warc.gz"} |
https://en.wikipedia.org/wiki/Perspectivity | # Perspectivity
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.
## Graphics
The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura (1435).[1] In English, Brook Taylor presented his Linear Perspective in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry".[2] In a second book, New Principles of Linear Perspective (1719), Taylor wrote
When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure. The Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone[3]
## Projective Geometry
A perspectivity:
$ABCD \doublebarwedge A'B'C'D',$
In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.
Given two lines $\ell$ and $m$ in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of $\ell$ and the range of $m$ determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P).[4] A special symbol has been used to show that points X and Y are related by a perspectivity; $X \doublebarwedge Y .$ In this notation, to show that the center of perspectivity is P, write $X \ \overset {P}{\doublebarwedge} \ Y.$ Using the language of functions, a central perspectivity with center P is a function $f_P \colon [\ell] \mapsto [m]$ (where the square brackets indicate the projective range of the line) defined by $f_P (X) = Y \text{ whenever } P \in XY$.[5] This map is an involution, that is, $f_P (f_P (X)) = X \text{ for all }X \in [\ell]$.
The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range.
### Projectivity
The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a projectivity (projective transformation, projective collineation and homography are synonyms).
There are several results concerning projectivities and perspectivities which hold in any pappian projective plane:[6]
Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities.
Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities.
Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity.
### Higher-dimensional perspectivities
The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities.
Let Sm and Tm be two distinct m-dimensional projective spaces contained in an n-dimensional projective space Rn. Let Pn-m-1 be an (n-m-1) - dimensional subspace of Rn with no points in common with either Sm or Tm. For each point X of Sm, the space L spanned by X and Pn-m-1 meets Tm in a point Y = fP(X). This correspondence fP is also called a perspectivity.[7] The central perspectivity described above is the case with n = 2 and m = 1.
### Perspective collineations
Let S2 and T2 be two distinct projective planes in a projective 3-space R3. With O and O* being points of R3 in neither plane, use the construction of the last section to project S2 onto T2 by the perspectivity with center O followed by the projection of T2 back onto S2 with the perspectivity with center O*. This composition is a bijective map of the points of S2 onto itself which preserves collinear points and is called a perspective collineation (central collineation in more modern terminology).[8] Let φ be a perspective collineation of S2. Each point of the line of intersection of S2 and T2 will be fixed by φ and this line is called the axis of φ. Let point P be the intersection of line OO* with the plane S2. P is also fixed by φ and every line of S2 that passes through P is stabilized by φ (fixed, but not necessarily pointwise fixed). P is called the center of φ. The restriction of φ to any line of S2 not passing through P is the central perspectivity in S2 with center P between that line and the line which is its image under φ.
## Notes
1. ^ Kirsti Andersen (2007) The Geometry of an Art, page 1,Springer ISBN 978-0-387-25961-1
2. ^ Andersen 1992, p. 75
3. ^ Andersen 1992, p. 163
4. ^ Coxeter 1969, p. 242
5. ^ Pedoe 1988, p. 281
6. ^ Fishback 1969, pp. 65-66
7. ^ Pedoe 1988, pp. 282-3
8. ^ Young 1930, p. 116 | {"extraction_info": {"found_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": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8631288409233093, "perplexity": 947.379066453638}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988061.16/warc/CC-MAIN-20150728002308-00101-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://link.springer.com/article/10.1007%2FBF00573899 | , Volume 26, Issue 1, pp 35-44
# Analysis of the gain and noise characteristics of fibre Brillouin amplifiers
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## Abstract
A theoretical analysis is performed of the transfer characteristics, gain, amplified spontaneous emission noise, on/off ratio and nonlinear phase shift of a fibre Brillouin amplifier, taking into account a possible signal detuning from the line centre and the nonlinear pump power depletion effect. Among other features, it is shown that the amplified spontaneous emission noise is reduced at higher signal levels and that detuning the signal reduces the amplifier gain in the linear regime and increases the spontaneous noise power in the saturation regime. As a consequence, the on/off ratio is significantly degraded by such an effect. | {"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": 0.8211032748222351, "perplexity": 1353.096097396655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663365.9/warc/CC-MAIN-20140930004103-00105-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/occasionally-huge-icebergs-found-floating-ocean-s-currents-suppose-one-iceberg-114-km-long-q2464149 | Occasionally, huge icebergs are found floating on the ocean's currents. Suppose one such iceberg is 114 km long, 32.7 km wide, and 155 m thick. (a) How much heat in joules would be required to melt this iceberg (assumed to be at 0 �C) into liquid water at 0 �C? The density of ice is 917 kg/m3. (b) The annual energy consumption by the United States in 1994 was 9.3 x 1019 J. If this energy were delivered to the iceberg every year, how many years would it take before the ice melted? | {"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": 0.8803935647010803, "perplexity": 1323.652537850515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824226.33/warc/CC-MAIN-20160723071024-00252-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/403682/about-the-experimental-measurement-of-neutrino-magnetic-moment-and-its-value | # About the experimental measurement of neutrino magnetic moment and its value
Massive neutrinos, be it Dirac or majorana, have magnetic moment. See here, here and here.
$\bullet$ How is it measured (or can be measured) in real experiments? What are the (upper and lower) bounds on the value of the neutrino magnetic moment from direct measurements?
$\bullet$ Does the measured values of neutrino observables i.e., the oscillation angles, Dirac CP phase and bound on the sum of neutrino masses constrain the value of the neutrino magnetic moment? | {"extraction_info": {"found_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": 0.995520830154419, "perplexity": 1070.7609551191667}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315809.69/warc/CC-MAIN-20190821043107-20190821065107-00187.warc.gz"} |
https://brilliant.org/problems/too-easy-aint-it/ | # So Last Year
The sum of the digits of the number $$1000^{2014}-2014$$ when expressed in decimal notation is?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9922435283660889, "perplexity": 618.0371275108214}, "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/1495463608684.93/warc/CC-MAIN-20170526203511-20170526223511-00014.warc.gz"} |
http://mathhelpforum.com/calculus/81520-application-integral-word-problem.html | # Thread: application integral word problem
1. ## application integral word problem
I need some help on this word problem.
A water tank is in the shape of a right circular cone of altitude 10 feet and base radius 5 feet, with it's vertex at the ground. If the tank is full, find the work done in pumping all of the water out the top of the tank.
Note: The weight of water is not given so I assume it is the standard 62.4pi.
Would I need to slice the interval [0,10] or do I need to factor in the base radius?
Is this close $\frac{62.4\pi}{4}\int(y^2)(10-y)dy$
2. Originally Posted by gammaman
I need some help on this word problem.
A water tank is in the shape of a right circular cone of altitude 10 feet and base radius 5 feet, with it's vertex at the ground. If the tank is full, find the work done in pumping all of the water out the top of the tank.
Note: The weight of water is not given so I assume it is the standard 62.4pi.
Would I need to slice the interval [0,10] or do I need to factor in the base radius?
Is this close $\frac{62.4\pi}{4}\int(y^2)(10-y)dy$
sketch the lines $y = 2x$ and $y = -2x$ starting at the origin, up to the points $(5,10)$ and $(-5,10)$
weight of a representative slice is ...
$62.4 \, dV = 62.4 \pi \cdot x^2 \, dy = 62.4 \pi \cdot \frac{y^2}{4} \, dy
$
the slice needs to be lifted a distance $(10 - y)$ ... work in raising the slice is
$dW = 62.4 \pi \cdot \frac{y^2}{4}(10 - y) \, dy$
total work to raise all slices ...
$W = 15.6 \pi \int_0^{10} y^2(10 - y) \, dy$
3. sketch the lines and starting at the origin, up to the points and
why are we doing this?
4. Originally Posted by gammaman
why are we doing this?
it give you a side view of the cone so that you may determine dV and the limits of integration.
5. Where does the base radius come into play? Also I am confused why (y^2) is. Does it just come from the formula for a cone 1/3pi*r^2h. If so why do my notes say to divide by 4?
6. Originally Posted by gammaman
Where does the base radius come into play? Also I am confused why (y^2) is. Does it just come from the formula for a cone 1/3pi*r^2h. If so why do my notes say to divide by 4?
another "why" for sketching a diagram ...
a horizontal slice of the cone's liquid is a cylinder with radius $x$ and thickness $dy$.
since $y = 2x$ , $x = \frac{y}{2}$
$dV = \pi x^2 \, dy = \pi \left(\frac{y}{2}\right)^2 \, dy = \frac{\pi}{4} y^2 \, dy$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8602033257484436, "perplexity": 384.6259953768753}, "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-2017-13/segments/1490218190295.65/warc/CC-MAIN-20170322212950-00367-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.pdfstall.online/2019/08/pocket-book-of-electrical-engineering.html | ## Thursday, August 15, 2019
### Pocket Book of Electrical Engineering Formulas (Free PDF)
File Size: 2.75 Mb
Description
The purpose of this book is to serve the reference needs of electrical engineers. The material has been compiled so that it may serve the needs of students and professionals who wish to have a ready reference to formulas, equations, methods, concepts, and their mathematical formulation.
The contents and size make it especially convenient and portable. The widespread availability and low price of scientiflc calculators have greatly reduced the need for many numerical tables. Accordingly, this book contains the informaton required by electrical engineers. Sections 1 through 13 cover the key mathematical concepts and formulas used by most electrical engineers. Sections 14 through 31 cover the wide range of subjects normally included as the basics of electrical engineering.
The size of the book is comparable to that of many calculators and it is really very much a companion to the calculator and the computer as a source of information for writing one’s own programs. To facilitate such use, the authors and the publisher have worked together to make the format attractive and clear.
Content:-
1. Elementary Algebra and Geometry
2. Determinants, Matrices, and Linear Systems of Equations
3. Trigonometry
4. Analytic Geometry
5. Seríes
6. Differential Calculus
7. Integral Calculus
8. Vector Analysis
9. Special Functions
10. Differential Equations
11. Statistics
12. Table of Derivatives
13. Table of Integrals
14. Resistor Circuits
15. Circuits with Energy Storage Elements
16. AC Circuits
17. T and II and Two-Port Networks
18. Operational Amplifier Circuits
19. Electric Signals
20. Feedback Systems
21. Frequency Response
22. System Response
23. Fourier Series
24. Fourier Transform
25. Paresval’s Theorem
26. Static Electric Fields
27. Static Magnetic Fields
28. Maxwell’s Equations
29. Semiconductors
30. Digital Logic
31. Communication Systems
Index
Author Details
"Richard C. Dorf"
"Ronald J. Tallarida" | {"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": 0.8919017910957336, "perplexity": 3583.6032567620573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038061820.19/warc/CC-MAIN-20210411085610-20210411115610-00195.warc.gz"} |
https://t-redactyl.io/blog/2021/04/coprime-numbers.html | # Coprime numbers
written in
In the last blog post, we talked about divisibility and the greatest common divisor, or $$\gcd$$. You might have been wondering already in that post what happens when two integers don’t have any common divisors? Well, in this case, as 1 divides every divisor, their $$\gcd$$ is equal to 1. Such pairs of numbers are called coprime, and they have applications in other areas of mathematics (such as modular arithmetic). In this post, we’ll have a look at a couple of ways of determining if pairs of numbers are coprime, and we’ll also have a look at a proof involving coprime integers.
## Definition of a coprime pair
From the previous post, we know that when we have $$d = \gcd(a, b)$$, then we can represent $$d$$ as a linear combination of $$a$$ and $$b$$, such that $$am + bn = d$$. This holds true also for coprime pairs, however, this time, $$am + bn = 1$$. But how do we connect these two facts? We can use the following elegant proof below.
Let’s say that we have a pair of natural numbers, $$a$$ and $$b$$, and integers $$m$$ and $$n$$. We know that $$am + bn = 1$$. What we need to now show is that these numbers are coprime, that is, that the $$\gcd(a, b) = 1$$. As we’re trying to find the $$\gcd(a, b)$$, let’s suppose there is some common divisor of the two which we’ll call $$c$$ (where $$c \in \mathbb{Z}$$). As $$c$$ is a divisor of both $$a$$ and $$b$$, this means that $$c \mid a$$ and $$c \mid b$$, and by the definition of divisibility, $$a = cd$$ and $$b = cf$$ for some $$d, f \in \mathbb{Z}$$. We now have some new definitions of $$a$$ and $$b$$, and we can substitute them into the linear equation we were given above:
\begin{aligned} am + bn &= 1 \\ cdm + cfn &= 1 \\ c(dm + fn) &= 1 \end{aligned}
As $$dm + fn$$ is a linear combination of integers, it is also an integer, so let’s redefine it as $$g = dm + fn$$, where $$g \in \mathbb{Z}$$. So our equation above becomes $$1 = cg$$, indicating that $$c \mid 1$$. As $$c \in \mathbb{Z}$$, then the only possible values of $$c$$ are $$\pm 1$$. However, as $$a$$ and $$b$$ are both in $$\mathbb{N}$$, $$c$$ cannot be a negative number, so $$c = 1$$. This means that as the only common divisor of $$a$$ and $$b$$, $$c$$ is also the $$\gcd(a, b)$$, meaning $$\gcd(a, b) = 1$$ and that $$a$$ and $$b$$ are coprime.
## Another way of defining coprime pairs
Let’s move onto a more complicated example. Let’s say that we’ve been asked to show that for all $$n \in \mathbb{N}$$, the numbers $$9n + 8$$ and $$6n + 5$$ are coprime. This time the simplest path forward is to revisit the Euclidean algorithm that we discussed in the last post. We start with the idea that we’re trying to find the $$\gcd(9n + 8, 6n + 5)$$. We start as per usual with the Euclidean algorithm, where we divide the larger number $$9n + 8$$ by the smaller number $$6n + 5$$:
$$9n + 8 = 6n + 5 \times 1 + 3n + 3$$
We now have $$\gcd(b, r)$$, in this case $$\gcd(6n + 5, 3n + 3)$$. We can proceed as normal with the Euclidean algorithm, using the fact that $$\gcd(a, b) = \gcd(b, r)$$ to seed each successive line until we get to our common divisor:
\begin{aligned} 9n + 8 &= 6n + 5 \times 1 + 3n + 3 &\Rightarrow gcd(9n + 8, 6n + 5) &= gcd(6n + 5, 3n + 3) \\ 6n + 5 &= 3n + 3 \times 1 + 3n + 2 &\Rightarrow gcd(6n + 5, 3n + 3) &= gcd(3n + 3, 3n + 2) \\ 3n + 3 &= 3n + 2 \times 1 + 1 &\Rightarrow gcd(3n + 3, 3n + 2) &= gcd(3n + 2, 1) \end{aligned}
Quickly ducking out of our Euclidean algorithm, we can now simplify the equation to get rid of $$3n$$ from both sides:
\begin{aligned} 3n + 3 = 3n + 2 \times 1 + 1 \\ 3 = 2 \times 1 + 1 \end{aligned}
Continuing on, we get:
\begin{aligned} 3 &= 2 \times 1 + 1 &\Rightarrow gcd(3, 2) &= gcd(2, 1) \\ 2 &= 1 \times 1 + 1 &\Rightarrow gcd(2, 1) &= gcd(1, 1) \\ 1 &= 1 \times 1 + 0 &\Rightarrow gcd(1, 1) &= gcd(1, 0) \end{aligned}
And with this, we’ve found that the $$\gcd(9n + 8, 6n + 5) = 1$$, meaning that these integers are coprime for every $$n \in \mathbb{N}$$.
## A proof using coprime pairs
We’ll end this post by showing how we can exploit what we know about coprime pairs to complete a proof. Suppose that $$a, b \in \mathbb{N}$$ and that $$gcd(a, b) = 1$$. Suppose also that $$a \mid r$$ and $$b \mid r$$. What we need to show is that $$ab \mid r$$.
Given that $$a$$ and $$b$$ are coprime, we know right away that $$am + bn = 1$$ for some $$m, n \in \mathbb{Z}$$. We also know that $$a \mid r$$ and $$b \mid r$$, so by the definition of divisibility, $$r = ax$$ and $$r = by$$ for some $$x, y \in \mathbb{Z}$$. We’re trying to show that $$ab \mid r$$, so ideally we want some equation where $$r = abz$$, where $$z$$ is some integer. We can exploit the fact that if we multiply $$am + bn = 1$$ by some value, the righthand side will now equal that value. Let’s try this with $$r$$:
\begin{aligned} r(am + bn) &= r \\ ram + rbn &= r \end{aligned}
Ah ha! We have half of the problem solved, where we now have $$r$$ on one side of the equation. How do we get $$ab$$ as a common factor on the lefthand side? Well, we still have the fact that $$r = ax$$ and $$r = by$$. We can subsitute these into our lefthand side, but we can be a little clever and use the definition that involves $$a$$ in with $$bn$$ and the definition that involves $$b$$ with $$am$$:
\begin{aligned} ram + rbn &= r \\ by \cdot am + ax \cdot bn &= r \\ abmy + abnx &= r \\ ab(my + nx) &= r \end{aligned}
We’re almost there. The last step is to define $$z = my + nx$$, where we know that $$my + nx$$ is in $$\mathbb{Z}$$ as a linear combination of integers, giving us $$r = abz$$. Rewriting this, $$ab \mid r$$, as required. | {"extraction_info": {"found_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": 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": 0.9999922513961792, "perplexity": 132.13817638944303}, "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/1627046153854.42/warc/CC-MAIN-20210729074313-20210729104313-00695.warc.gz"} |
https://enacademic.com/dic.nsf/enwiki/26085 | 3 Geostationary orbit
# Geostationary orbit
Geostationary orbit.To an observer on the rotating Earth (fixed point on the Earth), the satellite appears stationary in the sky. A red satellite is also geostationary above its own point on Earth. Top Down View
Geostationary orbit.To an observer on the rotating Earth (green dot on the blue sphere), the magenta satellite appears stationary in the sky. A red satellite is also geostationary above its own point on the blue sphere
Side view of Geostationary 3D of 2 satellites
Side view of Geostationary 3D of 2 satellites of Earth
A 5 x 6 degrees view of a part of the geostationary belt, showing several geostationary satellites. Those with inclination 0 degrees form a diagonal belt across the image: a few objects with small inclinations to the equator are visible above this line. Note how the satellites are pinpoint, while stars have created small trails due to the Earth's rotation.
A geostationary orbit (or Geostationary Earth Orbit - GEO) is a geosynchronous orbit directly above the Earth's equator (0° latitude), with a period equal to the Earth's rotational period and an orbital eccentricity of approximately zero. An object in a geostationary orbit appears motionless, at a fixed position in the sky, to ground observers. Communications satellites and weather satellites are often given geostationary orbits, so that the satellite antennas that communicate with them do not have to move to track them, but can be pointed permanently at the position in the sky where they stay. Due to the constant 0° latitude and circularity of geostationary orbits, satellites in GEO differ in location by longitude only.
The notion of a geosynchronous satellite for communication purposes was first published in 1928 (but not widely so) by Herman Potočnik.[1] The idea of a geostationary orbit was first disseminated on a wide scale in a 1945 paper entitled "Extra-Terrestrial Relays — Can Rocket Stations Give Worldwide Radio Coverage?" by British science fiction writer Arthur C. Clarke, published in Wireless World magazine. The orbit, which Clarke first described as useful for broadcast and relay communications satellites,[2] is sometimes called the Clarke Orbit.[3] Similarly, the Clarke Belt is the part of space about 35,786 km (22,000 mi) above sea level, in the plane of the equator, where near-geostationary orbits may be implemented. The Clarke Orbit is about 265,000 km (165,000 mi) long.
Geostationary orbits are useful because they cause a satellite to appear stationary with respect to a fixed point on the rotating Earth, allowing a fixed antenna to maintain a link with the satellite. The satellite orbits in the direction of the Earth's rotation, at an altitude of 35,786 km (22,236 mi) above ground, producing an orbital period equal to the Earth's period of rotation, known as the sidereal day.
## Introduction
A geostationary transfer orbit is used to move a satellite from low Earth orbit (LEO) into a geostationary orbit.
A worldwide network of operational geostationary meteorological satellites is used to provide visible and infrared images of Earth's surface and atmosphere. These satellite systems include:
Most commercial communications satellites, broadcast satellites and SBAS satellites operate in geostationary orbits. (Russian television satellites have used elliptical Molniya and Tundra orbits due to the high latitudes of the receiving audience.) The first satellite placed into a geostationary orbit was the Syncom-3, launched by a Delta-D rocket in 1964.
A statite, a hypothetical satellite that uses a solar sail to modify its orbit, could theoretically hold itself in a "geostationary" orbit with different altitude and/or inclination from the "traditional" equatorial geostationary orbit.
## Derivation of geostationary altitude
In any circular orbit, the centripetal force required to maintain the orbit is provided by the gravitational force on the satellite. To calculate the geostationary orbit altitude, one begins with this equivalence, and uses the fact that the orbital period is one sidereal day.
$\mathbf{F}_\text{c} = \mathbf{F}_\text{g}$
By Newton's second law of motion, we can replace the forces F with the mass m of the object multiplied by the acceleration felt by the object due to that force:
$m \mathbf{a}_\text{c} = m \mathbf{g}$
We note that the mass of the satellite m appears on both sides — geostationary orbit is independent of the mass of the satellite.[4] So calculating the altitude simplifies into calculating the point where the magnitudes of the centripetal acceleration required for orbital motion and the gravitational acceleration provided by Earth's gravity are equal.
The centripetal acceleration's magnitude is:
$|\mathbf{a}_\text{c}| = \omega^2 r$
where ω is the angular speed, and r is the orbital radius as measured from the Earth's center of mass.
The magnitude of the gravitational acceleration is:
$|\mathbf{g}| = \frac{G M}{r^2}$
where M is the mass of Earth, 5.9736 × 1024 kg, and G is the gravitational constant, 6.67428 ± 0.00067 × 10−11 m3 kg−1 s−2.
Equating the two accelerations gives:
$r^3 = \frac{G M}{\omega^2} \to r = \sqrt[3]{\frac{G M}{\omega^2}}$
The product GM is known with much greater precision than either factor alone; it is known as the geocentric gravitational constant μ = 398,600.4418 ± 0.0008 km3 s−2:
$r = \sqrt[3]{\frac\mu{\omega^2}}$
The angular speed ω is found by dividing the angle travelled in one revolution (360° = 2π rad) by the orbital period (the time it takes to make one full revolution: one sidereal day, or 86,164.09054 seconds).[5] This gives:
$\omega \approx \frac{2 \mathrm\pi~\mathrm{rad}} {86\,164~\mathrm{s}} \approx 7.2921 \times 10^{-5}~\mathrm{rad} / \mathrm{s}$
The resulting orbital radius is 42,164 kilometres (26,199 mi). Subtracting the Earth's equatorial radius, 6,378 kilometres (3,963 mi), gives the altitude of 35,786 kilometres (22,236 mi).
Orbital speed (how fast the satellite is moving through space) is calculated by multiplying the angular speed by the orbital radius:
$v = \omega r \approx 3.0746~\mathrm{km}/\mathrm{s} \approx 11\,068~\mathrm{km}/\mathrm{h} \approx 6877.8~\mathrm{mph}\text{.}$
Now, by the same formula, let us find the geostationary-type orbit of an object in relation to Mars (this type of orbit above is referred to as an areostationary orbit if it is above Mars). The geocentric gravitational constant GM (which is μ) for Mars has the value of 42,828 km3s-2, and the known rotational period (T) of Mars is 88,642.66 seconds. Since ω = 2π/T, using the formula above, the value of ω is found to be approx 7.088218×10-5 s-1. Thus, r3 = 8.5243×1012 km3, whose cube root is 20,427 km; subtracting the equatorial radius of Mars (3396.2 km) we have 17,031 km.
## Practical limitations
A geostationary orbit can only be achieved at an altitude very close to 35,786 km (22,236 mi), and directly above the equator. This equates to an orbital velocity of 3.07 km/s (1.91 mi/s) or a period of 1,436 minutes, which equates to almost exactly one sidereal day or 23.934461223 hours. This makes sense considering that the satellite must be locked to the Earth's rotational period in order to have a stationary footprint on the ground. In practice, this means that all geostationary satellites have to exist on this ring, which poses problems for satellites that will be decommissioned at the end of their service lives (e.g., when they run out of thruster fuel). Such satellites will either continue to be used in inclined orbits (where the orbital track appears to follow a figure-eight loop centered on the equator), or else be elevated to a "graveyard" disposal orbit.
A combination of lunar gravity, solar gravity, and the flattening of the Earth at its poles is causing a precession motion of the orbit plane of any geostationary object with a period of about 53 years and an initial inclination gradient of about 0.85 degrees per year, achieving a maximum inclination of 15 degrees after 26.5 years. To correct for this orbital perturbation, regular orbital stationkeeping maneuvers are necessary, amounting to a delta-v of approximately 50 m/s per year.
The second effect to be taken into account is the longitude drift, caused by the asymmetry of the earth - the equator is slightly elliptical. There are two stable (at 75.3°E, and at 104.7°W) and two unstable (at 165.3°E, and at 14.7°W) equilibrium points. Any geostationary object placed between the equilibrium points would (without any action) be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation. The correction of this effect requires orbit control maneuvers with a maximum delta-v of about 2 m/s per year, depending on the desired longitude.
In the absence of servicing missions from the Earth, the consumption of thruster propellant for station-keeping places a limitation on the lifetime of the satellite.
### Communications
Satellites in geostationary orbits are far enough away from Earth that communication latency becomes very high — about a quarter of a second for a one-way trip from one ground based transmitter to another via the geostationary satellite; close to half a second for round-trip communication between two earth stations.
For example, for ground stations at latitudes of φ=±45° on the same meridian as the satellite, the one-way delay can be computed by using the cosine rule, given the above derived geostationary orbital radius r, the Earth's radius R and the speed of light c, as
$2 \frac {\sqrt{R^2+r^2-2 R r \cos\varphi}} c \approx253\,\mathrm{ms}$
This presents problems for latency-sensitive applications such as voice communication or online gaming.[6]
## Orbit allocation
Satellites in geostationary orbit must all occupy a single ring above the equator. The requirement to space these satellites apart to avoid harmful radio-frequency interference during operations means that there are a limited number of orbital "slots" available, thus only a limited number of satellites can be operated in geostationary orbit. This has led to conflict between different countries wishing access to the same orbital slots (countries at the same longitude but differing latitudes) and radio frequencies. These disputes are addressed through the International Telecommunication Union's allocation mechanism.[7] Countries located at the Earth's equator have also asserted their legal claim to control the use of space above their territory.[8]
## Notes and references
1. ^ Noordung, Hermann; et al. (1995) [1929]. The Problem With Space Travel. Translation from original German. DIANE Publishing. pp. 72. ISBN 978-0788118494.
2. ^ "Extra-Terrestrial Relays — Can Rocket Stations Give Worldwide Radio Coverage?". Arthur C. Clark. October 1945. Retrieved 2009-03-04.
3. ^ "Basics of Space Flight Section 1 Part 5, Geostationary Orbits". NASA. Retrieved 2009-06-21.
4. ^ In the small body approximation, the geostationary orbit is independent of the satellite's mass. For satellites having a mass less than M μerr/μ≈1015 kg, that is, over a billion times that of the ISS, the error due to the approximation is smaller than the error on the universal geocentric gravitational constant (and thus negligible).
5. ^ Edited by P. Kenneth Seidelmann, "Explanatory Supplement to the Astronomical Almanac", University Science Books,1992, pp. 700
6. ^ The Teledesic Network: Using Low-Earth-Orbit Satellites to Provide Broadband, Wireless, Real-Time Internet Access Worldwide
7. ^ http://www.itu.int/ITU-R/conferences/seminars/mexico-2001/docs/06-procedure-mechanism.doc
8. ^ Oduntan, Gbenga. Hertfordshire Law Journal, 1(2), p. 75.
This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188).
Wikimedia Foundation. 2010.
### Look at other dictionaries:
• geostationary orbit — a circular orbit 35,785 km (22,236 miles) above Earth s (Earth) Equator in which a satellite s (satellite) orbital period is equal to Earth s rotation period of 23 hours and 56 minutes. A spacecraft in this orbit appears to an observer on… … Universalium
• geostationary orbit — noun a geosynchronous orbit that is fixed with respect to a position on the Earth • Hypernyms: ↑geosynchronous orbit * * * geostationary orbit, the orbit of a synchronous satellite; an orbit in which an artificial satellite moves at the same rate … Useful english dictionary
• Geostationary Orbit — A spaceship or satellite in orbit 35,900 kilometers above the equator with an orbital period of 24 hours. This orbit keeps the object over a specific Earth location at all times. This type of orbit is used most often with communication… … The writer's dictionary of science fiction, fantasy, horror and mythology
• geostationary orbit — an orbit path that keeps a satellite over the exact same point on the earth surface at all times … Geography glossary
• geostationary orbit — /dʒioʊˌsteɪʃənri ˈɔbət/ (say jeeoh.stayshuhnree awbuht) noun → synchronous equatorial orbit … Australian English dictionary
• Geostationary transfer orbit — A Geosynchronous Transfer Orbit or Geostationary Transfer Orbit (GTO) is a Hohmann transfer orbit around the Earth between a low Earth orbit (LEO) and a geosynchronous orbit (GEO). It is an ellipse where the perigee is a point on a LEO and the… … Wikipedia
• Geostationary ring — The geostationary ring is a volume segment around the geostationary orbit defined by variations in altitude and declination that can occur for uncontrolled objects left in the geostationary orbit.The geostationary orbit is subject to orbit… … Wikipedia
• Geostationary Operational Environmental Satellite — The Geostationary Operational Environmental Satellite (or GOES) program is a key element in United States National Weather Service (NWS) operations. GOES weather imagery and quantitative sounding data are a continuous and reliable stream of… … Wikipedia
• geostationary — /jee oh stay sheuh ner ee/, adj. of or pertaining to a satellite traveling in an orbit 22,300 miles (35,900 km) above the earth s equator: at this altitude, the satellite s period of rotation, 24 hours, matches the earth s and the satellite… … Universalium
• geostationary — Also known as geosynchronous. The type of orbit required to keep a communications satellite in a fixed position relative to the earth. The satellite s angular rate and direction of rotation are matched to those of the earth, and the… … Dictionary of networking | {"extraction_info": {"found_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": 0.9083361625671387, "perplexity": 1902.4876684345807}, "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/1637964360881.12/warc/CC-MAIN-20211201173718-20211201203718-00173.warc.gz"} |
http://math.stackexchange.com/questions/405599/matrix-operations-equivalent-operation-for-a-given-operation | # Matrix operations - equivalent operation for a given operation
This is the given problem, I need to write a code for this:
$(M*Q) \circ (N*Q)$
where $M,Q,N$ are known matrices, "$\ast$" denotes matrix multiplication and "$\circ$" denotes elementwise division.
Dimensions of the matrices: \begin{align*} M: &a\times l,\\ N: &a\times l,\\ Q: &l\times1. \end{align*}
By element-wise division, I mean this:
$A \circ B = C$ where $C_{ij} = A_{ij}/B_{ij} \ \ \forall\ i,j$.
I want to convert this above problem to this:
Find $K$ such that $K*Q = (M*Q) \circ (N*Q)$.
-
In general, this is impossible because $(M*Q) \circ (N*Q)$ is a rational function in the entries of $Q$ but $K*Q$ is linear. For instance, consider $M=(1,0),N=(0,1)$ and $Q=\pmatrix{x\\ y}$. Then $(M*Q) \circ (N*Q)=\frac xy$. Surely it is not a linear function in $x$ and $y$. | {"extraction_info": {"found_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": 0.9981250762939453, "perplexity": 522.7698675195743}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510270577.0/warc/CC-MAIN-20140728011750-00451-ip-10-146-231-18.ec2.internal.warc.gz"} |
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# 3.7: Absolute Value Functions
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Learning Objectives
• Graph an absolute value function.
• Solve an absolute value equation.
Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions.
## Understanding Absolute Value
Recall that in its basic form $$f(x)=|x|$$, the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.
Absolute value function
The absolute value function can be defined as a piecewise function
$f(x)=|x|= \begin{cases} x & \text{ if }x{\geq}0 \\ -x & \text{ if } x<0 \end{cases}$
Example $$\PageIndex{1}$$: Determine a Number within a Prescribed Distance
Describe all values $$x$$ within or including a distance of 4 from the number 5.
Solution
We want the distance between $$x$$ and 5 to be less than or equal to 4. We can draw a number line, such as the one in , to represent the condition to be satisfied.
The distance from $$x$$ to 5 can be represented using the absolute value as $$|x−5|$$. We want the values of $$x$$ that satisfy the condition $$| x−5 |\leq4$$.
Analysis
Note that
\begin{align} -4&{\leq}x-5 & x-5&\leq4 \\1&{\leq}x & x&{\leq}9 \end{align}
So $$|x−5|\leq4$$ is equivalent to $$1{\leq}x\leq9$$.
However, mathematicians generally prefer absolute value notation.
Exercise $$\PageIndex{1}$$
Describe all values $$x$$ within a distance of 3 from the number 2.
Solution
$$|x−2|\leq3$$
Example $$\PageIndex{2}$$: Resistance of a Resistor
Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%, ±5%, or ±10%.
Suppose we have a resistor rated at 680 ohms, ±5%. Use the absolute value function to express the range of possible values of the actual resistance.
Solution
5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance $$R$$ in ohms,
$|R−680|\leq34$
Exercise $$\PageIndex{2}$$
Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.
Solution
Using the variable $$p$$ for passing, $$| p−80 |\leq20$$
## Graphing an Absolute Value Function
The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure $$\PageIndex{3}$$.
Figure 1.6.4 shows the graph of $$y=2|x–3|+4$$. The graph of $$y=|x|$$ has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at $$(3,4)$$ for this transformed function.
Example $$\PageIndex{3}$$: Writing an Equation for an Absolute Value Function
Write an equation for the function graphed in Figure 1.6.5.
Solution
The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure $$\PageIndex{6}$$.
We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure $$\PageIndex{7}$$.
From this information we can write the equation
\begin{align} f(x)&=2|x-3|-2, \;\;\;\;\;\; \text{treating the stretch as a vertial stretch, or} \\ f(x)&=|2(x-3)|-2, \;\;\; \text{treating the stretch as a horizontal compression.} \end{align}
Analysis
Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.
If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?
Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for $$x$$ and $$f(x)$$.
$f(x)=a|x−3|−2$
Now substituting in the point $$(1, 2)$$
\begin{align} 2&=a|1-3|-2 \\ 4&=2a \\ a&=2 \end{align}
Exercise $$\PageIndex{3}$$
Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.
Solution
$$f(x)=−| x+2 |+3$$
Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?
Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.
No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure $$\PageIndex{8}$$).
## Solving an Absolute Value Equation
Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as $$8=|2x−6|$$, we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.
$2x-6=8 \text{ or } 2x-6=-8$
\begin{align} 2x&=14 & 2x&=-2 \\x&=7 & x&=-1 \end{align}
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.
An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,
$|x|=4,$
$|2x−1|=3,$
$|5x+2|−4=9,$
Solutions to Absolute Value Equations
For real numbers $$A$$ and $$B$$, an equation of the form $$|A|=B$$, with $$B\geq0$$, will have solutions when $$A=B$$ or $$A=−B$$. If $$B<0$$, the equation $$|A|=B$$ has no solution.
Given the formula for an absolute value function, find the horizontal intercepts of its graph.
1. Isolate the absolute value term.
2. Use $$|A|=B$$ to write $$A=B$$ or $$−A=B$$, assuming $$B>0$$.
3. Solve for $$x$$.
Example $$\PageIndex{4}$$: Finding the Zeros of an Absolute Value Function
For the function $$f(x)=|4x+1|−7$$, find the values of $$x$$ such that $$f(x)=0$$.
Solution
\begin{align} 0&=|4x+1|-7 & & &\text{Substitute 0 for f(x).} \\ 7&=|4x+1| & & &\text{Isolate the absolute value on one side of the equation.} \\ 7&=4x+1 &\text{or} -7&=4x+1 &\text{Break into two separate equations and solve.} \\ 6&=4x & -8&=4x & \\ x&=\frac{6}{4}=1.5 & x&=\frac{-8}{4}=-2 \end{align}
The function outputs 0 when $$x=1.5$$ or $$x=−2$$. See Figure $$\PageIndex{9}$$.
Exercise $$\PageIndex{4}$$
For the function $$f(x)=|2x−1|−3$$,find the values of $$x$$ such that $$f(x)=0$$.
Solution
$$x=−1$$ or $$x=2$$
Should we always expect two answers when solving $$|A|=B$$?
No. We may find one, two, or even no answers. For example, there is no solution to $$2+|3x−5|=1$$.
Given an absolute value equation, solve it.
1. Isolate the absolute value term.
2. Use $$|A|=B$$ to write $$A=B$$ or $$A=−B$$.
3. Solve for $$x$$.
Example $$\PageIndex{5}$$: Solving an Absolute Value Equation
Solve $$1=4|x−2|+2$$.
Solution
Isolating the absolute value on one side of the equation gives the following.
\begin{align} 1&=4|x-2|+2 \\ -1&=4|x-2| \\ -\frac{1}{4}&=|x-2| \end{align}
The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.
In example 1.6.3, if $$f(x)=1$$ and $$g(x)=4|x−2|+2$$ were graphed on the same set of axes, would the graphs intersect?
No. The graphs of $$f$$ and $$g$$ would not intersect, as shown in Figure $$\PageIndex{10}$$. This confirms, graphically, that the equation $$1=4|x−2|+2$$ has no solution.
Find where the graph of the function $$f(x)=−| x+2 |+3$$ intersects the horizontal and vertical axes.
$$f(0)=1$$, so the graph intersects the vertical axis at $$(0,1)$$. $$f(x)=0$$ when $$x=−5$$ and $$x=1$$ so the graph intersects the horizontal axis at $$(−5,0)$$ and $$(1,0)$$.
## Solving an Absolute Value Inequality
Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form
$|A|<B,\;|A|{\leq}B,|A|>B, \text{ or } |A|{\geq}B$,
where an expression $$A$$ (and possibly but not usually $$B$$) depends on a variable $$x$$. Solving the inequality means finding the set of all $$x$$ that satisfy the inequality. Usually this set will be an interval or the union of two intervals.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.
For example, we know that all numbers within 200 units of 0 may be expressed as
$|x|<200 \text{ or } −200<x<200$
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of$600. We can solve algebraically for the set of values $$x$$ such that the distance between $$x$$ and 600 is less than 200. We represent the distance between $$x$$ and 600 as $$|x−600|$$.
$|x−600|<200 \text{ or } −200<x−600<200$
$−200+600<x−600+600<200+600$
$400<x<800$
This means our returns would be between $400 and$800.
Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.
Given an absolute value inequality of the form $$|x−A|{\leq}B$$ for real numbers $$a$$ and $$b$$ where $$b$$ is positive, solve the absolute value inequality algebraically.
1. Find boundary points by solving $$|x−A|=B$$.
2. Test intervals created by the boundary points to determine where $$|x−A|{\leq}B$$.
3. Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.
Example $$\PageIndex{6}$$: Solving an Absolute Value Inequality
Solve $$|x −5|{\leq}4$$.
Solution
With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where $$|x−5|=4$$. We do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve $$|x−5|=4$$.
\begin{align} x−5&=4 &\text{ or }\;\;\;\;\;\;\;\; x&=9 \\ x−5&=−4 & x&=1\end{align}
After determining that the absolute value is equal to 4 at $$x=1$$ and $$x=9$$, we know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:
$x<1,\; 1<x<9, \text{ and } x>9.$
To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in Table $$\PageIndex{1}$$.
Interval test $$x$$ $$f(x)$$ $$<4$$ or $$>4$$ $$x<1$$ 0 $$|0-5|=5$$ Greater than $$19$$ 11 $$|11-5|=6$$ Greater than
Because $$1{\leq}x{\leq}9$$ is the only interval in which the output at the test value is less than 4, we can conclude that the solution to $$|x−5|{\leq}4$$ is $$1{\leq}x{\leq}9$$, or $$[1,9]$$.
To use a graph, we can sketch the function $$f(x)=|x−5|$$. To help us see where the outputs are 4, the line $$g(x)=4$$ could also be sketched as in Figure $$\PageIndex{11}$$.
We can see the following:
• The output values of the absolute value are equal to 4 at $$x=1$$ and $$x=9$$.
• The graph of $$f$$ is below the graph of $$g$$ on $$1<x<9$$. This means the output values of $$f(x)$$ are less than the output values of $$g(x)$$.
• The absolute value is less than or equal to 4 between these two points, when $$1{\leq}x\leq9$$. In interval notation, this would be the interval $$[1,9]$$.
Analysis
For absolute value inequalities,
$|x−A|<C,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; |x−A|>C, \\−C<x−A<C,\;\;\;\; x−A<−C \text{ or } x−A>C.$
The $$<$$ or $$>$$ symbol may be replaced by $$\leq$$ or $$\geq$$.
So, for this example, we could use this alternative approach.
\begin{align} |x−5|&{\leq}4 \\ −4&{\leq}x−5{\leq}4 &\text{Rewrite by removing the absolute value bars.} \\ −4+5&{\leq}x−5+5{\leq}4+5 &\text{Isolate the x.} \\ 1&{\leq}x\leq9 \end{align}
$$\PageIndex{5}$$: Solve $$|x+2|\leq6$$.
Solution
$$4{\leq}x\leq8$$
Given an absolute value function, solve for the set of inputs where the output is positive (or negative).
1. Set the function equal to zero, and solve for the boundary points of the solution set.
2. Use test points or a graph to determine where the function’s output is positive or negative.
Example $$\PageIndex{7}$$: Using a Graphical Approach to Solve Absolute Value Inequalities
Given the function $$f(x)=−\frac{1}{2}|4x−5|+3$$, determine the $$x$$-values for which the function values are negative.
Solution
We are trying to determine where $$f(x)<0$$, which is when $$−\frac{1}{2}|4x−5|+3<0$$. We begin by isolating the absolute value.
\begin{align} -\frac{1}{2}|4x−5|&<−3 \;\;\; \text{Multiply both sides by –2, and reverse the inequality.} \\ |4x−5|&>6\end{align}
Next we solve for the equality $$|4x−5|=6$$.
\begin{align} 4x-5&=6 & 4x-5&=-6 \\ 4x-6&=6 \;\; &\text{or } \;\;\; 4x&=-1 \\ x&=\frac{11}{4} & x&=-\frac{1}{4} \end{align}
Now, we can examine the graph of $$f$$ to observe where the output is negative. We will observe where the branches are below the $$x$$-axis. Notice that it is not even important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at $$x=−\frac{1}{4}$$ and $$x=\frac{11}{4}$$ and that the graph has been reflected vertically. See Figure $$\PageIndex{12}$$.
We observe that the graph of the function is below the $$x$$-axis left of $$x=−\frac{1}{4}$$ and right of $$x=\frac{11}{4}$$. This means the function values are negative to the left of the first horizontal intercept at $$x=−\frac{1}{4}$$, and negative to the right of the second intercept at $$x=\frac{11}{4}$$. This gives us the solution to the inequality.
$x<−\frac{1}{4} \text{ or } x>1\frac{1}{4}$
In interval notation, this would be $$( −\infty,−0.25 )\cup( 2.75,\infty)$$.
$$\PageIndex{6}$$: Solve $$−2|k−4|\leq−6$$.
Solution
$$k\leq1$$ or $$k\leq7$$; in interval notation, this would be $$(-\leq 7)$$; in interval notation, this would be $$\left(−\infty,1\right]\cup\left[7,\infty\right)$$
## Key Concepts
• The absolute value function is commonly used to measure distances between points.
• Applied problems, such as ranges of possible values, can also be solved using the absolute value function.
• The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.
• In an absolute value equation, an unknown variable is the input of an absolute value function.
• If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.
• An absolute value equation may have one solution, two solutions, or no solutions.
• An absolute value inequality is similar to an absolute value equation but takes the form | A |<B, | A |≤B, | A |>B, or | A |≥B.It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.
• Absolute value inequalities can also be solved graphically.
## Glossary
absolute value equation
an equation of the form $$|A|=B$$, with $$B\geq0$$; it will have solutions when $$A=B$$ or $$A=−B$$
absolute value inequality
a relationship in the form $$|A|<B$$, $$|A|{\leq}B$$, $$|A|>B$$, or $$|A|{\geq}B$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 10, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9785950183868408, "perplexity": 296.4594285440994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145713.39/warc/CC-MAIN-20200222180557-20200222210557-00251.warc.gz"} |
https://www.nature.com/articles/s41534-018-0097-8?error=cookies_not_supported&code=9093a21c-2e9b-4b82-a9e2-f77bae237980 | ## Introduction
Electronic spins of lattice defects in wide-bandgap semiconductors have come forward as an important platform for quantum technologies,1 in particular for applications that require both manipulation of long-coherent spin and spin-photon interfacing via bright optical transitions. In recent years this field showed strong development, with demonstrations of distribution and storage of non-local entanglement in networks for quantum communication2,3,4,5,6 and quantum-enhanced field-sensing.7,8,9,10,11 The nitrogen-vacancy defect in diamond is the material system that is most widely used12,13 and best characterized14,15,16 for these applications. However, its zero-phonon-line (ZPL) transition wavelength (637 nm) is not optimal for integration in standard telecom technology, which uses near-infrared wavelength bands where losses in optical fibers are minimal. A workaround could be to convert photon energies between the emitter-resonance and telecom values,17,18,19 but optimizing these processes is very challenging.
This situation has been driving a search for similar lattice defects that do combine favorable spin properties with bright emission directly at telecom wavelength. It was shown that both diamond and silicon carbide (SiC) can host many other spin-active color centers that could have suitable properties20,21,22,23 (where SiC is also an attractive material for its established position in the semiconductor device industry24,25). However, for many of these color centers detailed knowledge about the spin and optical properties is lacking. In SiC the divacancy26,27,28 and silicon vacancy10,29,30,31 were recently explored, and these indeed show millisecond homogeneous spin coherence times with bright ZPL transitions closer to the telecom band.
We present here a study of transition-metal impurity defects in SiC, which exist in great variety.32,33,34,35,36,37 There is at least one case (the vanadium impurity) that has ZPL transitions at telecom wavelengths,33 around 1300 nm, but we focus here (directed by availability of lasers in our lab) on the molybdenum impurity with ZPL transitions at 1076 nm (in 4H-SiC) and 1121 nm (in 6H-SiC), which turns out to be a highly analogous system. Theoretical investigations,38 early electron paramagnetic resonance33,39 (EPR), and photoluminescence (PL) studies40,41,42 indicate that these transition-metal impurities have promising properties. These studies show that they are deep-level defects that can be in several stable charge states, each with a distinctive value for its electronic spin S and near-infrared optical transitions. Further tuning and engineering possibilities come from the fact that these impurities can be embedded in a variety of SiC polytypes (4H, 6H, etc., Fig. 1a). Recent work by Koehl et al.37 studied chromium impurities in 4H-SiC using optically detected magnetic resonance. They identified efficient ZPL (little phonon-sideband) emission at 1042 nm and 1070 nm, and their charge state as neutral with an electronic spin S = 1 for the ground state.
Our work is an all-optical study of ensembles of molybdenum impurities in p-type 4H-SiC and 6H-SiC material. The charge and spin configuration of these impurities, and the defect configuration in the SiC lattice that is energetically favored, was until our work not yet identified with certainty. Our results show that these Mo impurities are in the Mo5+ (4d1) charge state (we follow here conventional notation:33 the label 5+ indicates that of an original Mo atom 4 electrons participate in bonds with SiC and that 1 electron is transferred to the p-type lattice environment). The single remaining electron in the 4d shell gives spin S = 1/2 for the ground state and optically excited state that we address. While we will show later that this can be concluded from our measurements, we assume it as a fact from the beginning since this simplifies the explanation of our experimental approach.
In addition to this identification of the impurity properties, we explore whether ground-state spin coherence is compatible with optical control. Using a two-laser magneto-spectroscopy method,28,43,44 we identify the spin Hamiltonian of the S = 1/2 ground state and optically excited state, which behave as doublets with highly anisotropic Landé g-factors. This gives insight in how a situation with only spin-conserving transitions can be broken, and we find that we can use a weak magnetic field to enable optical transitions from both ground-state spin levels to a common excited-state level (Λ level scheme). Upon two-laser driving of such Λ schemes, we observe coherent population trapping (CPT, all-optical control of ground-state spin coherence and fundamental to operating quantum memories45,46). The observed CPT reflects inhomogeneous spin dephasing times comparable to that of the SiC divacancy28,47 (near 1 μs).
In what follows, we first present our methods and results of single-laser spectroscopy performed on ensembles of Mo impurities in both SiC polytypes. Next, we discuss a two-laser method where optical spin pumping is detected. This allows for characterizing the spin sublevels in the ground and excited state, and we demonstrate how this can be extended to controlling spin coherence.
Both the 6H-SiC and 4H-SiC (Fig. 1a) samples were intentionally doped with Mo. There was no further intentional doping, but near-band-gap photoluminescence revealed that both materials had p-type characteristics. The Mo concentrations in the 4H and 6H samples were estimated41,42 to be in the range 1015–1017 cm−3 and 1014–1016 cm−3, respectively. The samples were cooled in a liquid-helium flow cryostat with optical access, which was equipped with a superconducting magnet system. The set-up geometry is depicted in Fig. 1b. The angle ϕ between the direction of the magnetic field and the c-axis of the crystal could be varied, while both of these directions were kept orthogonal to the propagation direction of excitation laser beams. In all experiments where we resonantly addressed ZPL transitions the laser fields had linear polarization, and we always kept the direction of the linear polarization parallel to the c-axis. Earlier studies38,41,42 of these materials showed that the ZPL transition dipoles are parallel to the c-axis. For our experiments we confirmed that the photoluminescence response was clearly the strongest for excitation with linear polarization parallel to the c-axis, for all directions and magnitudes of the magnetic fields that we applied. All results presented in this work come from photoluminescence (PL) or photoluminescence excitation (PLE) measurements. The excitation lasers were focused to a ~100 μm spot in the sample. PL emission was measured from the side. A more complete description of experimental aspects is presented in Methods section.
## Results
For initial characterization of Mo transitions in 6H-SiC and 4H-SiC we used PL and PLE spectroscopy (Methods). Figure 1c shows the PL emission spectrum of the 6H-SiC sample at 3.5 K, measured using an 892.7 nm laser for excitation. The ZPL transition of the Mo defect visible in this spectrum will be studied in detail throughout this work. The shaded region indicates the emission of phonon replicas related to this ZPL.41,42 While we could not perform a detailed analysis, the peak area of the ZPL in comparison with that of the phonon replicas indicates that the ZPL carries clearly more than a few percent of the full PL emission. Similar PL data from Mo in the 4H-SiC sample, together with a study of the temperature dependence of the PL, can be found in Supplementary Information (Fig. S1).
For a more detailed study of the ZPL of the Mo defects, PLE was used. In PLE measurements, the photon energy of a narrow-linewidth excitation laser is scanned across the ZPL part of the spectrum, while resulting PL of phonon-sideband (phonon-replica) emission is detected (Fig. 1b, we used filters to keep light from the excitation laser from reaching the detector, Methods). The inset of Fig. 1c shows the resulting ZPL for Mo in 6H-SiC at 1.1057 eV (1121.3 nm). For 4H-SiC we measured the ZPL at 1.1521 eV (1076.2 nm, Supplementary Information). Both are in close agreement with literature.41,42 Temperature dependence of the PLE from the Mo defects in both 4H-SiC and 6H-SiC can be found in Supplementary Information (Fig. S2).
The width of the ZPL is governed by the inhomogeneous broadening of the electronic transition throughout the ensemble of Mo impurities, which is typically caused by non-uniform strain in the crystal. For Mo in 6H-SiC we observe a broadening of 24 ± 1 GHz FWHM, and 23 ± 1 GHz for 4H-SiC. This inhomogeneous broadening is larger than the anticipated electronic spin splittings,33 and it thus masks signatures of spin levels in optical transitions between the ground and excited state.
In order to characterize the spin-related fine structure of the Mo defects, a two-laser spectroscopy technique was employed.28,43,44 We introduce this for the four-level system sketched in Fig. 2a. A laser fixed at frequency f0 is resonant with one possible transition from ground to excited state (for the example in Fig. 2a |g2〉 to |e2〉), and causes PL from a sequence of excitation and emission events. However, if the system decays from the state |e2〉 to |g1〉, the laser field at frequency f0 is no longer resonantly driving optical excitations (the system goes dark due to optical pumping). In this situation, the PL is limited by the (typically long) lifetime of the |g1〉 state. Addressing the system with a second laser field, in frequency detuned from the first by an amount δ, counteracts optical pumping into off-resonant energy levels if the detuning δ equals the splitting Δg between the ground-state sublevels. Thus, for specific two-laser detuning values corresponding to the energy spacings between ground-state and excited-state sublevels the PL response of the ensemble is greatly increased. Notably, this technique gives a clear signal for sublevel splittings that are smaller than the inhomogeneous broadening of the optical transition, and the spectral features now reflect the homogeneous linewidth of optical transitions.28,47
In our measurements a 200 μW continuous-wave control and probe laser were made to overlap in the sample. For investigating Mo in 6H-SiC the control beam was tuned to the ZPL at 1121.32 nm (fcontrol = f0 = 267.3567 THz), the probe beam was detuned from f0 by a variable detuning δ (i.e., fprobe = f0 + δ). In addition, a 100 μW pulsed 770 nm re-pump laser was focused onto the defects to counteract bleaching of the Mo impurities due to charge-state switching28,48,49 (which we observed to only occur partially without re-pump laser). All three lasers were parallel to within 3° inside the sample. A magnetic field was applied to ensure that the spin sublevels were at non-degenerate energies. Finally, we observed that the spectral signatures due to spin disappear in a broad background signal above a temperature of ~10 K (Fig. S4), and we thus performed measurements at 4 K (unless stated otherwise).
Figure 2b shows the dependence of the PLE on the two-laser detuning for the 6H-SiC sample (4H-SiC data in Supplementary Information Fig. S6), for a range of magnitudes of the magnetic field (here aligned close to parallel with the c-axis, ϕ = 1°). Two emission peaks can be distinguished, labeled line L1 and L2. The emission (peak height) of L2 is much stronger than that of L1. Figure 2c shows the results of a similar measurement with the magnetic field nearly orthogonal to the crystal c-axis (ϕ = 87°), where four spin-related emission signatures are visible, labeled as lines L1 through L4 (a very small peak feature left from L1, at half its detuning, is an artifact that occurs due to a leakage effect in the spectral filtering that is used for beam preparation, see Methods). The two-laser detuning frequencies corresponding to all four lines emerge from the origin (B = 0, δ = 0) and evolve linearly with magnetic field (we checked this up to 1.2 T). The slopes of all four lines (in Hertz per Tesla) are smaller in Fig. 2c than in Fig. 2b. In contrast to lines L1, L2, and L4, which are peaks in the PLE spectrum, L3 shows a dip.
In order to identify the lines at various angles ϕ between the magnetic field and the c-axis, we follow how each line evolves with increasing angle. Figure 2d shows that as ϕ increases, L1, L3, and L4 move to the left, whereas L2 moves to the right. Near 86°, L2 and L1 cross. At this angle, the left-to-right order of the emission lines is swapped, justifying the assignment of L1, L2, L3, and L4 as in Fig. 2b, c. Supplementary Information presents further results from two-laser magneto-spectroscopy at intermediate angles ϕ (section 2a).
We show below that the results in Fig. 2 indicate that the Mo impurities have electronic spin S = 1/2 for the ground and excited state. This contradicts predictions and interpretations of initial results.33,38,41,42 Theoretically, it was predicted that the defect associated with the ZPL under study here is a Mo impurity in the asymmetric split-vacancy configuration (Mo impurity asymmetrically located inside a Si–C divacancy), where it would have a spin S = 1 ground state with zero-field splittings of about 3–6 GHz.33,38,41,42 However, this would lead to the observation of additional emission lines in our measurements. Particularly, in the presence of a zero-field splitting, we would expect to observe two-laser spectroscopy lines emerging from a non-zero detuning.28 We have measured near zero fields and up to 1.2 T, as well as from 100 MHz to 21 GHz detuning (Supplementary Information section 2c), but found no more peaks than the four present in Fig. 2c. A larger splitting would have been visible as a splitting of the ZPL in measurements as presented in the inset of Fig. 1c, which was not observed in scans up to 1000 GHz. Additionally, a zero-field splitting and corresponding avoided crossings at certain magnetic fields would result in curved behavior for the positions of lines in magneto-spectroscopy. Thus, our observations rule out that there is a zero-field splitting for the ground-state and excited-state spin sublevels. In this case the effective spin-Hamiltonian50 can only take the form of a Zeeman term
$$H_{g(e)} = \mu _Bg_{g(e)}{\mathbf{B}} \cdot {\tilde{\mathbf S}},$$
(1)
where gg(e) is the g-factor for the electronic ground (excited) state (both assumed positive), μB the Bohr magneton, B the magnetic field vector of an externally applied field, and $${\tilde{\mathbf S}}$$ the effective spin vector. The observation of four emission lines can be explained, in the simplest manner, by a system with spin S = 1/2 (doublet) in both the ground and excited state.
For such a system, Fig. 3 presents the two-laser optical pumping schemes that correspond to the observed emission lines L1 through L4. Addressing the system with the V-scheme excitation pathways from Fig. 3c leads to increased pumping into a dark ground-state sublevel, since two excited states contribute to decay into the off-resonant ground-state energy level while optical excitation out of the other ground-state level is enhanced. This results in reduced emission observed as the PLE dip feature of L3 in Fig. 2c (for details see Supplementary Information section 5).
We find that for data as in Fig. 2c the slopes of the emission lines are correlated by a set of sum rules
$$\Theta _{L3} = \Theta _{L1} + \Theta _{L2},$$
(2)
$$\Theta _{L4} = 2\Theta _{L1} + \Theta _{L2},$$
(3)
Here ΘLn denotes the slope of emission line Ln in Hertz per Tesla. The two-laser detuning frequencies for the pumping schemes in Fig. 3a–d are related in the same way, which justifies the assignment of these four schemes to the emission lines L1 through L4, respectively. These schemes and equations directly yield the g-factor values gg and ge for the ground and excited state (Supplementary Information section 2).
We find that the g-factor values gg and ge strongly depend on ϕ, that is, they are highly anisotropic. While this is in accordance with earlier observations for transition metal defects in SiC,33 we did not find a comprehensive report on the underlying physical picture. In Supplementary Information section 7, we present a group-theoretical analysis that explains the anisotropy gg ≈ 1.7 for ϕ = 0° and gg = 0 for ϕ = 90°, and similar behavior for ge (which we also use to identify the orbital character of the ground and excited state). In this scenario the effective Landé g-factor50 is given by
$$g(\phi ) = \sqrt {\left( {g_{||}{\mathrm{cos}}\phi } \right)^2 + \left( {g_ \bot {\mathrm{sin}}\phi } \right)^2},$$
(4)
where g|| represents the component of g along the c-axis of the silicon carbide structure and g the component in the basal plane. Figure 4 shows the ground and excited state effective g-factors extracted from our two-laser magneto-spectroscopy experiments for 6H-SiC and 4H-SiC (additional experimental data can be found in Supplementary Information). The solid lines represent fits to the Eq. (4) for the effective g-factor. The resulting g|| and g parameters are given in Table 1.
The reason why diagonal transitions (in Fig. 3a, c), and thus the Λ and V scheme are allowed, lies in the different behavior of ge and gg. When the magnetic field direction coincides with the internal quantization axis of the defect, the spin states in both the ground and excited state are given by the basis of the Sz operator, where the z-axis is defined along the c-axis. This means that the spin-state overlap for vertical transitions, e.g., from |g1〉 to |e1〉, is unity. In such cases, diagonal transitions are forbidden as the overlap between e.g., |g1〉 and |e2〉 is zero. Tilting the magnetic field away from the internal quantization axis introduces mixing of the spin states. The amount of mixing depends on the g-factor, such that it differs for the ground and excited state. This results in a tunable non-zero overlap for all transitions, allowing all four schemes to be observed (as in Fig. 2b where ϕ = 87°). This reasoning also explains the suppression of all emission lines except L2 in Fig. 2b, where the magnetic field is nearly along the c-axis. A detailed analysis of the relative peak heights in Fig. 2b, c compared to wave function overlap can be found in Supplementary Information (section 4).
The Λ driving scheme depicted in Fig. 3a, where both ground states are coupled to a common excited state, is of particular interest. In such cases it is possible to achieve all-optical coherent population trapping (CPT),45 which is of great significance in quantum-optical operations that use ground-state spin coherence. This phenomenon occurs when two lasers address a Λ system at exact two-photon resonance, i.e., when the two-laser detuning matches the ground-state splitting. The ground-state spin system is then driven toward a superposition state that approaches $$\left| {\Psi _{{\rm CPT}}} \right\rangle \propto {\mathrm{\Omega }}_2\left| {g_1} \right\rangle - {\mathrm{\Omega }}_1\left| {g_2} \right\rangle$$ for ideal spin coherence. Here $$\left| {{\mathrm{\Omega }}_n} \right\rangle$$ is the Rabi frequency for the driven transition from the $$\left| {g_n} \right\rangle$$ state to the common excited state. Since the system is now coherently trapped in the ground state, the photoluminescence decreases.
In order to study the occurrence of CPT, we focus on the two-laser PLE features that result from a Λ scheme. A probe field with variable two-laser detuning relative to a fixed control laser was scanned across this line in frequency steps of 50 kHz, at 200 μW. The control laser power was varied between 200 μW and 5 mW. This indeed yields signatures of CPT, as presented in Fig. 5. A clear power dependence is visible: when the control beam power is increased, the depth of the CPT dip increases (and can fully develop at higher laser powers or by concentrating laser fields in SiC waveguides47). This observation of CPT confirms our earlier interpretation of lines L1L4, in that it confirms that L1 results from a Λ scheme. It also strengthens the conclusion that this system is S = 1/2, since otherwise optical spin-pumping into the additional (dark) energy levels of the ground state would be detrimental for the observation of CPT.
Using a standard model for CPT,45 adapted to account for strong inhomogeneous broadening of the optical transitions47 (see also Supplementary Information section 6) we extract an inhomogeneous spin dephasing time $$T_2^ \ast$$ of 0.32 ± 0.08 μs and an optical lifetime of the excited state of 56 ± 8 ns. The optical lifetime is about a factor two longer than that of the nitrogen-vacancy defect in diamond,12,51 indicating that the Mo defects can be applied as bright emitters (although we were not able to measure their quantum efficiency). The value of $$T_2^ \ast$$ is relatively short but sufficient for applications based on CPT.45 Moreover, the EPR studies by Baur et al.33 on various transition-metal impurities show that the inhomogeneity probably has a strong static contribution from an effect linked to the spread in mass for Mo isotopes in natural abundance (nearly absent for the mentioned vanadium case), compatible with elongating spin coherence via spin-echo techniques. In addition, their work showed that the hyperfine coupling to the impurity nuclear spin can be resolved. There is thus clearly a prospect for storage times in quantum memory applications that are considerably longer than $$T_2^ \ast$$.
## Discussion
The anisotropic behavior of the g-factor that we observed for Mo was also observed for vanadium and titanium in the EPR studies by Baur et al.33 (they observed g|| ≈ 1.7 and g = 0 for the ground state). In these cases the transition metal has a single electron in its 3d orbital and occupies the hexagonal (h) Si substitutional site. We show in Supplementary Information section 7 that the origin of this behavior can be traced back to a combination of a crystal field with C3v symmetry and spin-orbit coupling for the specific case of an ion with one electron in its d-orbital.
The correspondence of this behavior with what we observe for the Mo impurity identifies that our materials have Mo impurities present as Mo5+ (4d1) systems residing on a hexagonal h silicon substitutional site. In this case of a hexagonal (h) substitutional site, the molybdenum is bonded in a tetrahedral geometry, sharing four electrons with its nearest neighbors. For Mo5+ (4d1) the defect is then in a singly ionized +|e| charge state (e denotes the elementary charge), due to the transfer of one electron to the p-type SiC host material.
An alternative scenario for our type of Mo impurities was recently proposed by Ivády et al.35. They proposed, based on theoretical work,35 the existence of the asymmetric split-vacancy (ASV) defect in SiC. An ASV defect in SiC occurs when an impurity occupies the interstitial site formed by adjacent silicon and carbon vacancies. The local symmetry of this defect is a distorted octahedron with a threefold symmetry axis in which the strong g-factor anisotropy (g = 0) may also be present for the S = 1/2 state.50 Considering six shared electrons for this divacancy environment, the Mo5+ (4d1) Mo configuration occurs for the singly charged −|e| state. For our observations this is a highly improbable scenario as compared to one based on the +|e| state, given the p-type SiC host material used in our work. We thus conclude that this scenario by Ivády et al. does not occur in our material. Interestingly, niobium defects have been shown to grow in this ASV configuration,52 indicating there indeed exist large varieties in the crystal symmetries involved with transition metal defects in SiC. This defect displays S = 1/2 spin with several optical transitions between 892 and 897 nm in 4H-SiC and between 907 and 911 nm in 6H-SiC.52
Another defect worth comparing to is the aforementioned chromium defect, studied by Koehl et al.37 Like Mo in SiC, the Cr defect is located at a silicon substitutional site, thus yielding a 3d2 configuration for this defect in its neutral charge state. The observed S = 1 spin state has a zero-field splitting parameter of 6.7 GHz.37 By employing optically detected magnetic resonance techniques they measured an inhomogeneous spin coherence time $$T_2^ \ast$$ of 37 ns,37 which is considerably shorter than observed for molybdenum in the present work. Regarding spin-qubit applications, the exceptionally low phonon-sideband emission of Cr seems favorable for optical interfacing. However, the optical lifetime for this Cr configuration (146 μs37) is much longer than that of the Mo defect we studied, hampering its application as a bright emitter. It is clear that there is a wide variety in optical and spin properties throughout transition-metal impurities in SiC, which makes up a useful library for engineering quantum technologies with spin-active color centers.
We have studied ensembles of molybdenum defect centers in 6H and 4H silicon carbide with 1.1521 eV and 1.1057 eV transition energies, respectively. The ground-state and excited-state spin of both defects was determined to be S = 1/2 with large g-factor anisotropy. Since this is allowed in hexagonal symmetry, but forbidden in cubic, we find this to be consistent with theoretical descriptions that predict that Mo resides at a hexagonal lattice site in 4H-SiC and 6H-SiC,35,38 and our p-type host environment strongly suggests that this occurs for Mo at a silicon substitutional site. We used the measured insight in the S = 1/2 spin Hamiltonians for tuning control schemes where two-laser driving addresses transitions of a Λ system, and observed CPT for such cases. This demonstrates that the Mo defect and similar transition-metal impurities are promising for quantum information technology. In particular for the highly analogous vanadium color center, engineered to be in SiC material where it stays in its neutral V4+ (3d1) charge state, this holds promise for combining S = 1/2 spin coherence with operation directly at telecom wavelengths.
## Methods
### Materials
The samples used in this study were ~1 mm thick epilayers grown with chemical vapor deposition, and they were intentionally doped with Mo during growth. The PL signals showed that a relatively low concentration of tungsten was present due to unintentional doping from metal parts of the growth setup (three PL peaks near 1.00 eV, outside the range presented in Fig. 1a). The concentration of various types of (di)vacancies was too low to be observed in the PL spectrum that was recorded. For more details see ref.42
### Cryostat
During all measurements, the sample was mounted in a helium flow cryostat with optical access through four windows and equipped with a superconducting magnet system.
### Photoluminescence (PL)
The PL spectrum of the 6H-SiC sample was measured by exciting the material with an 892.7 nm laser, and using a double monochromator equipped with infrared-sensitive photomultiplier. For the 4H-SiC sample, we used a 514.5 nm excitation laser and an FTIR spectrometer.
### Photoluminescence Excitation (PLE)
The PLE spectrum was measured by exciting the defects using a CW diode laser tunable from 1050 nm to 1158 nm with linewidth below 50 kHz, stabilized within 1 MHz using feedback from a HighFinesse WS-7 wavelength meter. The polarization was linear along the sample c-axis. The laser spot diameter was ~100 μm at the sample. The PL exiting the sample sideways was collected with a high-NA lens, and detected by a single-photon counter. The peaks in the PLE data were typically recorded at a rate of about 10 kcounts/s by the single-photon counter. We present PLE count rates in arb.u. since the photon collection efficiency was not well defined, and it varied with changing the angle ϕ. For part of the settings we placed neutral density filters before the single-photon counter to keep it from saturating. The excitation laser was filtered from the PLE signals using a set of three 1082 nm (for the 4H-SiC case) or 1130 nm (for the 6H-SiC case) longpass interference filters. PLE was measured using an ID230 single-photon counter. Additionally, to counter charge state switching of the defects, a 770 nm re-pump beam from a tunable pulsed Ti:sapphire laser was focused at the same region in the sample. Laser powers as mentioned in the main text.
### Two-laser characterization
The PLE setup described above was modified by focusing a detuned laser beam to the sample, in addition to the present beams. The detuned laser field was generated by splitting off part of the stabilized diode laser beam. This secondary beam was coupled into a single-mode fiber and passed through an electro-optic phase modulator in which an RF signal (up to ~5 GHz) modulated the phase. Several sidebands were created next to the fundamental laser frequency, the spacing of these sidebands was determined by the RF frequency. Next, a Fabry–Pérot interferometer was used to select one of the first-order sidebands (and it was locked to the selected mode). The resulting beam was focused on the same region in the sample as the original PLE beams (diode laser and re-pump) with similar spot size and polarization along the sample c-axis. Laser powers were as mentioned in the main text. Small rotations of the c-axis with respect to the magnetic field were performed using a piezo-actuated goniometer with 7.2 degrees travel.
### Data processing
For all graphs with PLE data a background count rate is subtracted from each line, determined by the minimum value of the PLE in that line (far away from resonance features). After this a fixed vertical offset is added for clarity. For each graph, the scaling is identical for all lines within that graph. | {"extraction_info": {"found_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": 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": 0.832267165184021, "perplexity": 1819.9536433183803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943555.25/warc/CC-MAIN-20230320175948-20230320205948-00480.warc.gz"} |
http://mathhelpforum.com/differential-geometry/154786-complex-analysis.html | 1. ## complex analysis
Help me please to prove this one:
Let us $f(z)\; and \; z^{5}\ \overline{f}(z)$ both are entire functions.
Prove that $f(z)$ is constant function necessarily.
Thanks
2. The only way I can see this happening is to show that f is bounded, the result then follows from Liouville's theorem | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.896065890789032, "perplexity": 267.0307992166178}, "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/1495463610374.3/warc/CC-MAIN-20170528181608-20170528201608-00472.warc.gz"} |
http://mathhelpforum.com/algebra/199995-parametric-system-equation-problem.html | # Math Help - Parametric system of equation problem
1. ## Parametric system of equation problem
x^2+y^2=2a+4
x+y=a+3
find the meaning for "a" when system of equation has only 1 root
2. ## Re: Parametric system of equation problem
Originally Posted by Telo
x^2+y^2=2a+4
x+y=a+3
find the meaning for "a" when system of equation has only 1 root
Solve $x^2+(a+3-x)^2=2a+4$ for $x$.
3. ## Re: Parametric system of equation problem
thanks for answer but i didnt get it, i have this: 2x^2-2ax+a^2+9=2a+4 what do i do next? how do i find out what meaning must "a" have? for 1 root of x
4. ## Re: Parametric system of equation problem
Do you not know how to solve a quadratic equation?
But more to the point, that equation can be written $2x^2- 2ax+ (a^2-2a+ 5)= 0$ Now, an equation like that will have only one root provided it is a double root. That is, if it is of the form $2(x- x_0)^2= 0$.
5. ## Re: Parametric system of equation problem
Expanding on HallsofIvy's post, you have a double root if and only if the discriminant is zero, i.e.
$(-2a)^2 - 4(2)(a^2 - 2a + 5) = 0$
6. ## Re: Parametric system of equation problem
thank you very much! I've got a=-1
7. ## Re: Parametric system of equation problem
Hmm I'm not sure if -1 is correct...according to the quadratic I posted, it has two non-real roots for a.
8. ## Re: Parametric system of equation problem
Originally Posted by richard1234
Hmm I'm not sure if -1 is correct...according to the quadratic I posted, it has two non-real roots for a.
-1 should be correct. I get a single solution for the resulting system.
My work disagrees slightly with HallsofIvy, however:
\begin{align*}x^2+y^2&=2a+4\\x+y&=a+3\end{align*}
$\Rightarrow x^2 + \left(a-x+3\right)^2=2a+4$
$\Rightarrow2x^2-2ax-6x+a^2+6a+9=2a+4$
$\Rightarrow2x^2-2(a+3)x+\left(a^2+4a+5\right)=0$
For this quadratic to have a single root, the discriminant $b^2-4ac$ must be 0:
$4(a+3)^2-8\left(a^2+4a+5\right)=0$
$\Rightarrow a^2+2a+1=0$
$\Rightarrow a=-1.$
Substituting this back into the original equations gives the system
\begin{align*}x^2+y^2&=2\\x+y&=2\end{align*}
So
$x^2 + (2-x)^2=2$
$\Rightarrow x^2-2x+1=0$
$\Rightarrow(x-1)^2=0$
$\Rightarrow x=1$
$\Rightarrow y=1$
A single solution, as desired. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9491369724273682, "perplexity": 1136.9547118009855}, "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-2015-14/segments/1427131297689.58/warc/CC-MAIN-20150323172137-00241-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.varsitytutors.com/hotmath/hotmath_help/topics/quadratic-regression.html | A quadratic regression is the process of finding the equation of the parabola that best fits a set of data. As a result, we get an equation of the form:
$y=a{x}^{2}+bx+c$ where $a\ne 0$ .
The best way to find this equation manually is by using the least squares method. That is, we need to find the values of $a,b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c$ such that the squared vertical distance between each point $\left({x}_{i},{y}_{i}\right)$ and the quadratic curve $y=a{x}^{2}+bx+c$ is minimal.
The matrix equation for the quadratic curve is given by:
$\left[\begin{array}{ccc}\sum {x}_{i}{}^{4}& \sum {x}_{i}{}^{3}& \sum {x}_{i}{}^{2}\\ \sum {x}_{i}{}^{3}& \sum {x}_{i}{}^{2}& \sum {x}_{i}\\ \sum {x}_{i}{}^{2}& \sum {x}_{i}& n\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}\sum {x}_{i}{}^{2}{y}_{i}\\ \sum {x}_{i}{y}_{i}\\ \sum {y}_{i}\end{array}\right]$
The relative predictive power of a quadratic model is denoted by ${R}^{2}$ .
This can be obtained using the formula:
${R}^{2}=1-\frac{\text{SSE}}{\text{SST}}$ where $\text{SSE}=\sum {\left({y}_{i}-a{x}_{i}{}^{2}-b{x}_{i}-c\right)}^{2}$ and $\text{SST}={\sum \left({y}_{i}-\stackrel{¯}{y}\right)}^{2}$
The value of ${R}^{2}$ varies between $0$ and $1$ . The closer the value is to $1$ , the more accurate the model is.
But these are very tedious calculations. So, we will use a graphing calculator to automatically calculate the curve.
Example 1:
Consider the set of data. Determine the quadratic regression for the set.
$\left(-3,7.5\right),\left(-2,3\right),\left(-1,0.5\right),\left(0,1\right),\left(1,3\right),\left(2,6\right),\left(3,14\right)$
Enter the $x$ -coordinates and $y$ -coordinates in your calculator and do a quadratic regression. The equation of the parabola that best approximates the points is
$y=1.1071{x}^{2}+x+0.5714$
Plot the graph. You should get a graph like this.
You can see that the ${R}^{2}$ value for the data is $0.9942$ . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 20, "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": 0.886554479598999, "perplexity": 144.24040465756224}, "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/1521257645550.13/warc/CC-MAIN-20180318071715-20180318091715-00766.warc.gz"} |
https://www.physicsforums.com/threads/calculating-uncertainties-of-measured-quantities-physics.722090/ | # Calculating Uncertainties of Measured quantities (Physics)
1. Nov 11, 2013
### Joystar77
1. The problem statement, all variables and given/known data
d1 = 2.53 cm +/- .05 cm
d2 = 1.753 m +/- .001 m
0 = 23.5 degrees +/- .5 degrees
v1 = 1.55 m/s +/- .15 m/s
Using the measured quantities above, calculate the following. Express the uncertainty calculated value.
2. Relevant equations
d3 = 4 ( d1 + d2)
3. The attempt at a solution
d3 = 4 (2.53 cm +/- .05 cm) + (1.753 m +/- .001 m)
d3 = 10.12 +/- .2 cm + 7.012 m +/- .004 m
d3 = 10.32 cm + 7.016 m
I don't understand this problem so I would appreciate some help.
2. Nov 11, 2013
### Staff: Mentor
How did you get that?
Try to convert both numbers to the same unit, either cm or m.
Do you know how to combine uncertainties from multiple sources? This is certainly something you had in class and it is written in your textbook, too.
3. Nov 11, 2013
### iRaid
Is there an equation involved in this problem? Like what do d1, d2, 0, v all have in common?
4. Nov 11, 2013
### BOAS
d3 = 4 ( d1 + d2)
σd32 = σd12 + σd22
You need to perform the calculation with just the values, and then calculate the error giving your answer in the form
d3 +/- σd3
ETA oops, you're not the question asker.
Last edited: Nov 11, 2013
5. Nov 11, 2013
### Joystar77
response to uncertainties
My textbook isn't very good and doesn't give an explanation about combining uncertainties. As for in class, my instructor doesn't explain the steps and there aren't any example given about the uncertainties. All it mentions is the definition of an uncertainty and the basic rules for uncertainties. I still don't understand how to do the problem.
6. Nov 11, 2013
### Joystar77
I added the numbers together and that is how I got it. I don't know how to convert to centimeters or meters. No, I don't know how to combine uncertainties from multiple sources.
7. Nov 11, 2013
### Joystar77
In response to your question, I am not sure what they have in common. I don't understand how to do the problem and this is how come I am asking for help.
8. Nov 11, 2013
### Staff: Mentor
Good, you'll just need the most basic rule.
How old is the captain?
Then you can look that up.
9. Nov 11, 2013
### Joystar77
The following error propagation (sample calculations) consists of the ‘simple’ methods
outlined in lab appendix (pages A7-A9). This method yields uncertainties which are
slightly high, but still gives ‘reasonably good values’.
For added/subtracted quantities, the uncertainties are obtained (propagated) by simply
• Write correct significant figures based on the final uncertainty.
For multiplied/divided quantities, the uncertainties are obtained by 1) converted to
percent uncertainties (i.e., fractional uncertainties), and 2) the percent uncertainties are
• Convert from percent to absolute uncertainties (to get correct significant figures for
Important note for uncertainty calculations –Keep extra significant figures in
uncertainties when doing computations. Convert to one significant figure in the final
This is what I have as a basic rule for uncertainties, but this doesn't mention the fact of showing me how to convert centimeters or meters in uncertainties. Please let me know if these are correct!
d1 = 2.53 cm +/- .05 cm
d2 = 1.753 m +/- .001 m
0 = 23.5 degrees +/- .5 degrees
v1 = 1.55 m/s +/- .15 m/s
Using the measured quantities above, calculate the following. Express the uncertainty calculated value.
1. d3 = 4 (d1 + d2)
delta d3 = 4 * (0.05 + 0.001) = 0.204
2. a = 4 v1^2 / d2
delta a = 4 * (2 * 0.15 - 0.001) = 1.196
3. d1 (tan (0))
0
4. Z = 4d1 (cos (0)) ^2
4 * 0.05 = 0.2
Are these right or are they still wrong? I did try to work on them. Please let me know as soon as you can!
10. Nov 11, 2013
### BOAS
1 cm = 0.01m
You should be able to figure the rest out from this.
I shall give you an example of error propagation and you should be able to apply this to your problems.
Two measured lengths, a and b.
a = 106.0 ± 0.3 mm
b = 58.3 ± 0.4 mm
When we add these together we expect to see the final error is bigger than either of the contributing errors, but not larger than the sum of the errors.
if x = a + b
σ2x = σ2a + σ2b
(where σ means uncertainty)
a + b = 164.3 ± (0.32 + 0.42)1/2
= 164.3 ± 0.5 mm
All i've done is taken the square root of the errors added in quadrature to find the error in x.
11. Nov 12, 2013
### Staff: Mentor
And this is all you need.
It also does not mention that "5" is the number that follows "4". It is expected that you know the basics, or know where to find them.
You just randomly add numbers that appear somewhere. That does not work.
Draft saved Draft deleted
Similar Discussions: Calculating Uncertainties of Measured quantities (Physics) | {"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": 0.868493914604187, "perplexity": 2097.8032449164607}, "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/1510934804965.9/warc/CC-MAIN-20171118132741-20171118152741-00448.warc.gz"} |
http://math.stackexchange.com/questions/293892/calculus-integration-problem-int-sin5-x-cos2-x-dx | # Calculus integration problem: $\int \sin^5 (x) \cos^2 (x)\,dx$
What's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$
-
Share with us what you've tried so we can help you along. – JohnD Feb 3 '13 at 19:53
Similar integral: math.stackexchange.com/questions/902722/… – Martin Sleziak Aug 19 '14 at 11:01
Hint: Write $$\sin^5(x)\cos^2(x)=(\sin^2(x))^2\cos^2(x)\sin(x).$$
Now use $\cos^2(x)+\sin^2(x)=1$ and do the appropriate change of variable.
This is the general method to integrate functions of the type $$\cos^n(x)\sin^m(x)$$ when one of the integers $n,m$ is odd.
-
$$\int \sin^5 (x) \cos^2(x) dx$$ $$= \int(\sin^2(x))^2 \cos^2(x) \sin(x) dx$$ $$=-\int(1 - \cos^2(x))^2 cos^2(x) (-sin(x) dx)$$
Let $u = \cos(x)$ $\implies du = -\sin(x) dx$
$$= -\int(1 - u^2)² u² (du)$$ $$= -\int(1 - 2u^2 + u^4) u^2 du$$ $$= -\int(u^2 - 2u^4+ u^6) du$$ $$= -\left(\frac{u^3}{3} - \frac{2u^5}{5} + \frac{u^7}{7}\right) + C$$ $$= -u^3\left(\frac{1}{3} - \frac{2u^2}{5} +\frac{ u^4}{7}\right) + C$$ $$= -\cos^3(x) \left(\frac{1}{3} - \frac{2\cos^2(x)}{5} + \frac{\cos^4(x)}{7}\right) + C$$ $$= -\cos^3(x)\frac{15\cos^4(x) - 42\cos^2(x) + 35}{105} + C$$
-
For some basic information about writing math at this site see e.g. here, here, here and here. – SBareS Oct 21 '15 at 14:46
Using trig identities, you can show that: $$\sin ^5(x) \cos ^2(x)=\frac{5 \sin (x)}{64}+\frac{1}{64} \sin (3 x)-\frac{3}{64} \sin (5 x)+\frac{1}{64} \sin (7 x)$$
To do this, first use the "Power-reduction formulas" to reduce to get: $$\sin^5(x)=\frac{10 \sin x - 5 \sin 3 x+ \sin 5 x}{16}$$ $$\cos^2(x)=\frac{1 + \cos (2 x)}{2}$$ And then use: $$\cos (2 x) \sin (nx) = {{\sin((n+2)x) - \sin((n-2)x)} \over 2}$$
-
Could you show how? As I see it now it may as well just be reverse-engineered from the Wolfram Alpha solution. I'm not saying you did this, but I don't know what identities you used to come to this. – user50407 Feb 3 '13 at 20:07
Is it really how you solve the question in general, or is it just to provide a different answer? Just curious. – 1015 Feb 3 '13 at 20:21
@MichaelCorleone - fixed. – nbubis Feb 3 '13 at 20:21
Yes I saw it and upvoted it. Thanks – user50407 Feb 3 '13 at 20:22
@julien - just to provide a different answer of course :) It's not a very practical way, but it's helpful if the powers are very large, so that you can use general power reduction rules. – nbubis Feb 3 '13 at 20:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9191955924034119, "perplexity": 394.34929246208196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824230.71/warc/CC-MAIN-20160723071024-00209-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/114798?ln=fr | A truncated Fourier/finite element discretization of the Stokes equations in an axisymmetric domain
We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.
Publié dans:
Mathematical Models and Methods in Applied Sciences (M3AS), 16, 2, 233-263
Année
2006
Mots-clefs:
Laboratoires:
Notice créée le 2007-12-13, modifiée le 2019-01-11
Lien externe:
URL
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(Pas encore évalué) | {"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": 0.9490801692008972, "perplexity": 662.587191081925}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583792338.50/warc/CC-MAIN-20190121111139-20190121133139-00019.warc.gz"} |
http://mathonline.wikidot.com/dense-and-nowhere-dense-sets-in-a-topological-space | Dense and Nowhere Dense Sets in a Topological Space
# Dense and Nowhere Dense Sets in a Topological Space
## Dense Sets in a Topological Space
Definition: Let $(X, \tau)$ be a topological space. The set $A \subseteq X$ is said to be Dense in $X$ if the intersection of every nonempty open set with $A$ is nonempty, that is, $A \cap U \neq \emptyset$ for all $U \in \tau \setminus \{ \emptyset \}$.
Given any topological space $(X, \tau)$ it is important to note that $X$ is dense in $X$ because every $U \in \tau$ is such that $U \subseteq X$, and so $X \cap U = U \neq \emptyset$ for all $U \in \tau \setminus \{ \emptyset \}$.
For another example, consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology of open intervals. Then the set of rational numbers $\mathbb{Q} \subset \mathbb{R}$ is dense in $\mathbb{R}$. If not, then there exists an $U \in \tau \setminus \{ \emptyset \}$ such that $\mathbb{Q} \cap U = \emptyset$.
Since $U \in \tau$ we have that $(a, b) \subseteq U$ for some open interval $(a, b)$ with $a, b \in \mathbb{R}$ and $a < b$. Suppose that $\mathbb{Q} \setminus U = \emptyset$. Then we must also have that:
(1)
\begin{align} \quad \mathbb{Q} \cap U = \mathbb{Q} \cap (a, b) = \emptyset \end{align}
The intersection above implies that there exists no rational numbers in the interval $(a, b)$, i.e., there exists no $q \in \mathbb{Q}$ such that $a < q < b$. But this is a contradiction since for all $a, b \in \mathbb{R}$ with $a < b$ there ALWAYS exists a rational number $q \in \mathbb{Q}$ such that $a < q < b$, i.e., $q \in (a, b)$. So $\mathbb{Q} \cap (a, b) \neq \emptyset$ for all $U \in \tau \setminus \{ \emptyset \}$. Thus, $\mathbb{Q}$ is dense in $\mathbb{R}$.
We will now look at a very important theorem which will give us a way to determine whether a set $A \subseteq X$ is dense in $X$ or not.
Theorem 1: Let $(X, \tau)$ be a topological space and let $A \subseteq X$. Then $A$ is dense in $X$ if and only if $\bar{A} = X$.
• Proof: $\Rightarrow$ Suppose that $A$ is dense in $X$. Then for all $U \in \tau \setminus \{ \emptyset \}$ we have that $A \cap U = \emptyset$. Clearly $\bar{A} \subseteq X$ so we only need to show that $X \subseteq \bar{A}$.
## Nowhere Dense Sets in a Topological Space
Definition: Let $(X, \tau)$ be a topological space. A set $A \subseteq X$ is said to be Nowhere Dense in $X$ if the interior of the closure of $A$ is empty, that is, $\mathrm{int} (\bar{A}) = \emptyset$.
For example, consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usually topology of open intervals on $\mathbb{R}$, and consider the set of integers $\mathbb{Z}$. The closure of $\mathbb{Z}$, $\bar{\mathbb{Z}}$ is the smallest closed set containing $\mathbb{Z}$. The smallest closed set containing $\mathbb{Z}$ is $\mathbb{Z}$ since $\mathbb{Z}^c$ is open as $\mathbb{Z}^c$ is an arbitrary union of open sets:
(2)
\begin{align} \quad \mathbb{Z}^c = ... (-2, -1) \cup (-1, 0) \cup (0, 1) \cup (1, 2) \cup ... \end{align}
So what is the interior of $\bar{\mathbb{Z}} = \mathbb{Z}$? It is the largest open set contained in $\bar{\mathbb{Z}} = \mathbb{Z}$. All open sets of $\mathbb{R}$ with respect to this topology $\tau$ are either the empty set, an open interval, a union of open intervals, or the whole set (the union of all open intervals). But no open intervals are contained in $\mathbb{Z}$ and so:
(3)
\begin{align} \quad \mathrm{int} (\bar{\mathbb{Z}}) = \emptyset \end{align}
Therefore $\mathbb{Z}$ is a nowhere dense set in $\mathbb{R}$ with respect to the usual topology $\tau$ on $\mathbb{R}$. | {"extraction_info": {"found_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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9954931735992432, "perplexity": 38.01417659618716}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541995.74/warc/CC-MAIN-20161202170901-00082-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.scribd.com/document/15769708/PC-Intro-to-Seq-and-Series | You are on page 1of 22
# Writin g a nd
## MATH PRO JECT
USING AND WRITING SEQUENCES
## You can think of a sequence as a function whose domain
is a set of consecutive integers. If a domain is not
specified, it is understood that the domain starts with 1.
USING AND WRITING SEQUENCES
DOMAIN:
n 1 2 3 4 5 The domain gives
the relative position
of each term.
The range gives the
an
RANGE: 3 6 9 12 15 terms of the sequence.
## This is a finite sequence having the rule
an = 3n,
where an represents the nth term of the sequence.
Writing Terms of Sequences
## Write the first six terms of the sequence an = 2n + 3.
SOLUTION
a 1 = 2(1) + 3 = 5 1st term
a 2 = 2(2) + 3 = 7 2nd term
a 3 = 2(3) + 3 = 9 3rd term
a 4 = 2(4) + 3 = 11 4th term
a 5 = 2(5) + 3 = 13 5th term
a 6 = 2(6) + 3 = 15 6th term
Writing Terms of Sequences
SOLUTION
## f (6) = (–2) 6 – 1 = – 32 6th term
Writing Rules for Sequences
## If the terms of a sequence have a recognizable pattern,
then you may be able to write a rule for the n th term
of the sequence.
## Describe the pattern, write the next term, and write
a rule for the n th term of the sequence
– _1 , _1 , – __
1 , __
1 , ….
3 9 27 81
Writing Rules for Sequences
SOLUTION
n 1 2 3 4 5
1 , 1 , 1 − 1
terms − − ,
3 9 1 81 243
2
1 2 7 3 4 5
rewrite 1 , 1 , 1 , 1 1
terms − − − − −
3 3 3 3 3
n
1
A rule for the nth term is an = −
3
Writing Rules for Sequences
## Describe the pattern, write the next term, and write
a rule for the n th term of the sequence.
SOLUTION 2, 6, 12 , 20,….
n 1 2 3 4 5
terms 2 6 12 20 30
rewrite
terms 1(1 +1) 2(2 +1) 3(3 +1) 4(4 +1) 5(5 +1)
## A rule for the nth term is f (n) = n (n+1).
Graphing a Sequence
## You can graph a sequence by letting the horizontal axis
represent the position numbers (the domain) and the
vertical axis represent the terms (the range).
Graphing a Sequence
## You work in the produce
department of a grocery store
and are stacking oranges in
the shape of square pyramid
with ten layers.
## • Write a rule for the number
of oranges in each layer.
## • Graph the sequence.
Graphing a Sequence
SOLUTION
## The diagram below shows the first three layers of the
stack. Let an represent the number of oranges in layer n.
n 1 2 3
an 1 = 12 4=22 9 = 32
2
From the diagram, you can see that an = n
Graphing a Sequence
an = n2
## Plot the points (1, 1), (2, 4),
(3, 9), . . . , (10, 100).
USING SERIES
## When the terms of a sequence are added, the resulting
expression is a series. A series can be finite or infinite.
FINITE SEQUENCE INFINITE SEQUENCE
## 3, 6, 9, 12, 15 3, 6, 9, 12, 15, . . .
FINITE SERIES INFINITE SERIES
3 + 6 + 9 + 12 + 15 3 + 6 + 9 + 12 + 15 + . . . .
## You can use summation notation to write a series. For
example, for the finite series shown above, you can write
5
3 + 6 + 9 + 12 + 15 = Σ 3i
i=1
USING SERIES
upper limit of Is read as
summation “the sum from i equals 1 to 5 of 3i.”
5
Σ 3i 5
3 + 6 + 9 + 12 + 15 = Σ 3i
i=1
i=1
index of summation lower limit of summation
USING SERIES
## Summation notation is also called sigma notation
because it uses the uppercase Greek letter sigma,
written Σ.
## Summation notation for an infinite series is similar
to that for a finite series. For example, for the infinite
series shown earlier, you can write:
3 + 6 + 9 + 12 + 15 + . . . = Σ 3i
i=1
## The infinity symbol, ∞ , indicates that the series
continues without end.
USING SERIES
## The index of summation does not have to be i.
Any letter can be used. Also, the index does not
have to begin at 1.
Writing Series with Summation Notation
## Write the series with summation notation.
5 + 10 + 15 + . . . + 100
SOLUTION
## Notice that the first term is 5 (1), the second is 5 (2),
the third is 5 (3), and the last is 5 (20). So the terms
of the series can be written as:
ai = 5i where i = 1, 2, 3, . . . , 20
20
The summation notation is Σ 5i.
i=1
Writing Series with Summation Notation
## Write the series with summation notation.
1 + 2 + 3 + 4 +...
2 3 4 5
SOLUTION
## Notice that for each term the denominator of the fraction
is 1 more than the numerator. So, the terms of the series
can be written as:
i
ai = where i = 1, 2, 3, 4 . . .
i+1
The summation notation for the series is Σ i .
i=1i+1
Writing Series with Summation Notation
## The sum of the terms of a finite sequence can be found
by simply adding the terms. For sequences with many
terms, however, adding the terms can be tedious.
Formulas for finding the sum of the terms of three
special types of sequences are shown next.
Writing Series with Summation Notation
CONCEPT
SUMMARY
FORMULAS FOR SPECIAL SERIES
n
1 Σ 1 =n gives the sum of n 1’s .
i=1
n n (n + 1) gives the sum of positive
2
Σ i=
integers from 1 to n .
i=1 2
n n (n + 1)(2 n + 1) gives the sum of squares
3
Σ i 2=
of positive integers from
i=1 6
1 to n.
Using a Formula for a Sum
## RETAIL DISPLAYS How many oranges are in a
square pyramid 10 layers high?
Using a Formula for a Sum
SOLUTION
## You know from the earlier example that the i th term of
the series is given by ai = i 2, where i = 1, 2, 3, . . . , 10.
10
Σ i 2 = 12+ 22 + . . . + 102
i=1
= 10(10 + 1)(2 • 10 + 1)
6
10(11)(21)
=
6
= 385 | {"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": 0.9584331512451172, "perplexity": 365.0569299778804}, "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/1566027315558.25/warc/CC-MAIN-20190820180442-20190820202442-00544.warc.gz"} |
https://infoscience.epfl.ch/record/166520 | Infoscience
Journal article
# A stabilized finite volume element formulation for sedimentation-consolidation processes
A model of sedimentation-consolidation processes in so-called clarifier-thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the Stokes system for an incompressible fluid describing the motion of the mixture. In cylindrical coordinates, and if an axially symmetric solution is assumed, the original problem reduces to two space dimensions. This poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted Sobolev spaces. A novel finite volume element method is introduced for the spatial discretization, where the velocity field and the solids concentration are discretized on two different dual meshes. The method is based on a stabilized discontinuous Galerkin formulation for the concentration field, and a multiscale stabilized pair of $\mathbb{P}_1$-$\mathbb{P}_1$ elements for velocity and pressure, respectively. Numerical experiments illustrate properties of the model and the satisfactory performance of the proposed method. | {"extraction_info": {"found_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": 0.8263119459152222, "perplexity": 339.8098822013487}, "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/1484560285337.76/warc/CC-MAIN-20170116095125-00109-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://testbook.com/question-answer/with-reference-tomucormycosis-consider-the--61adbc6bccfc5ade55ceb099 | # With reference to Mucormycosis, consider the following statements:1. It is an aggressive and invasive algal infection caused by a group of moulds called Mucormycetes.2. It is also known by the name ‘white fungus’.Which of the above statements is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Option 4 : Neither 1 nor 2
## Detailed Solution
The correct answer is Neither 1 nor 2.
Key Points
• Mucormycosis:
• Mucormycosis is an aggressive and invasive fungal infection caused by a group of moulds called Mucormycetes. Hence statement 1 is incorrect.
• It is also known by the name ‘black fungus’. Hence statement 2 is incorrect.
• Effect:
• It most commonly affects the sinuses or the lungs after inhaling fungal spores from the air.
• It can also happen on the skin after a burn, cut or other type of skin wound through which the fungus enters the skin.
• It can also affect the brain.
• Treatment:
• The main line of treatment is an anti-fungal drug called amphotericin B, which is given over an extended period of time (4-6 weeks) under the strict observation of a physician. | {"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": 0.8801118731498718, "perplexity": 3624.527398704311}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302740.94/warc/CC-MAIN-20220121071203-20220121101203-00687.warc.gz"} |
http://parasys.net/error-on/error-on-mean.php | # parasys.net
Home > Error On > Error On Mean
# Error On Mean
## Contents
It can only be calculated if the mean is a non-zero value. Student approximation when σ value is unknown Further information: Student's t-distribution §Confidence intervals In many practical applications, the true value of σ is unknown. Notice that s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} is only an estimate of the true standard error, σ x ¯ = σ n This often leads to confusion about their interchangeability.
With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. In other words, it is the standard deviation of the sampling distribution of the sample statistic. BREAKING DOWN 'Standard Error' The term "standard error" is used to refer to the standard deviation of various sample statistics such as the mean or median. Repeating the sampling procedure as for the Cherry Blossom runners, take 20,000 samples of size n=16 from the age at first marriage population. https://en.wikipedia.org/wiki/Standard_error
## Percent Error Mean
In this scenario, the 2000 voters are a sample from all the actual voters. The proportion or the mean is calculated using the sample. Gurland and Tripathi (1971)[6] provide a correction and equation for this effect. However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and
and Keeping, E.S. (1963) Mathematics of Statistics, van Nostrand, p. 187 ^ Zwillinger D. (1995), Standard Mathematical Tables and Formulae, Chapman&Hall/CRC. It represents the standard deviation of the mean within a dataset. Read More »
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Exam Prep Series 7 Exam Error Average The concept of a sampling distribution is key to understanding the standard error.
The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean. Error Median If σ is not known, the standard error is estimated using the formula s x ¯ = s n {\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}} where s is the sample The graph shows the ages for the 16 runners in the sample, plotted on the distribution of ages for all 9,732 runners. http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/tests-of-means/what-is-the-standard-error-of-the-mean/ The standard error estimated using the sample standard deviation is 2.56.
The standard deviation of all possible sample means of size 16 is the standard error. Error Variance For the purpose of hypothesis testing or estimating confidence intervals, the standard error is primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. Now click on the fx symbol again. Choose Statistical on the left hand menu, and then COUNT on the right hand menu. 7. Next, consider all possible samples of 16 runners from the population of 9,732 runners.
## Error Median
n is the size (number of observations) of the sample. useful source The standard error of the mean estimates the variability between samples whereas the standard deviation measures the variability within a single sample. Percent Error Mean In an example above, n=16 runners were selected at random from the 9,732 runners. Error Standard Deviation In cases where the standard error is large, the data may have some notable irregularities.Standard Deviation and Standard ErrorThe standard deviation is a representation of the spread of each of the
The graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16. The unbiased standard error plots as the ρ=0 diagonal line with log-log slope -½. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean describes bounds on a random sampling process. The proportion or the mean is calculated using the sample. Error Range
Click on the spreadsheet picture in the pop-up box, and then highlight the list of numbers you averaged. Hit enter and OK as before. 8. doi:10.2307/2340569. Scenario 2. Notice that the population standard deviation of 4.72 years for age at first marriage is about half the standard deviation of 9.27 years for the runners.
The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean. Error On Mean Value The true standard error of the mean, using σ = 9.27, is σ x ¯ = σ n = 9.27 16 = 2.32 {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt Minitab uses the standard error of the mean to calculate the confidence interval, which is a range of values likely to include the population mean.Minitab.comLicense PortalStoreBlogContact UsCopyright © 2016 Minitab Inc.
## Because the 5,534 women are the entire population, 23.44 years is the population mean, μ {\displaystyle \mu } , and 3.56 years is the population standard deviation, σ {\displaystyle \sigma }
This formula does not assume a normal distribution. See unbiased estimation of standard deviation for further discussion. Scenario 2. What Does Se Stand For In Statistics Moreover, this formula works for positive and negative ρ alike.[10] See also unbiased estimation of standard deviation for more discussion.
The mean age was 23.44 years. v t e Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Moments The sample mean will very rarely be equal to the population mean. ISBN 0-8493-2479-3 p. 626 ^ a b Dietz, David; Barr, Christopher; Çetinkaya-Rundel, Mine (2012), OpenIntro Statistics (Second ed.), openintro.org ^ T.P.
For the runners, the population mean age is 33.87, and the population standard deviation is 9.27. Despite the small difference in equations for the standard deviation and the standard error, this small difference changes the meaning of what is being reported from a description of the variation It will be shown that the standard deviation of all possible sample means of size n=16 is equal to the population standard deviation, σ, divided by the square root of the When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9]
The formula for the standard error of the mean is: where σ is the standard deviation of the original distribution and N is the sample size (the number of scores each Retrieved 17 July 2014. Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution.
Greek letters indicate that these are population values. They may be used to calculate confidence intervals. The graphs below show the sampling distribution of the mean for samples of size 4, 9, and 25. American Statistical Association. 25 (4): 30–32.
The smaller standard deviation for age at first marriage will result in a smaller standard error of the mean. The margin of error of 2% is a quantitative measure of the uncertainty – the possible difference between the true proportion who will vote for candidate A and the estimate of Roman letters indicate that these are sample values. Of the 2000 voters, 1040 (52%) state that they will vote for candidate A.
The standard error of a proportion and the standard error of the mean describe the possible variability of the estimated value based on the sample around the true proportion or true This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Standard error functions more as a way to determine the accuracy of the sample or the accuracy of multiple samples by analyzing deviation within the means. If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of
The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean. The standard deviation is used to help determine validity of the data based the number of data points displayed within each level of standard deviation. | {"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": 0.9808162450790405, "perplexity": 712.8651516892327}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156305.13/warc/CC-MAIN-20180919200547-20180919220547-00353.warc.gz"} |
https://worldwidescience.org/topicpages/a/amplitude+acoustic+wave.html | #### Sample records for amplitude acoustic wave
1. Small amplitude electron acoustic solitary waves in a magnetized superthermal plasma
Science.gov (United States)
Devanandhan, S.; Singh, S. V.; Lakhina, G. S.; Bharuthram, R.
2015-05-01
The propagation of electron acoustic solitary waves in a magnetized plasma consisting of fluid cold electrons, electron beam and superthermal hot electrons (obeying kappa velocity distribution function) and ion is investigated in a small amplitude limit using reductive perturbation theory. The Korteweg-de-Vries-Zakharov-Kuznetsov (KdV-ZK) equation governing the dynamics of electron acoustic solitary waves is derived. The solution of the KdV-ZK equation predicts the existence of negative potential solitary structures. The new results are: (1) increase of either the beam speed or temperature of beam electrons tends to reduce both the amplitude and width of the electron acoustic solitons, (2) the inclusion of beam speed and temperature pushes the allowed Mach number regime upwards and (3) the soliton width maximizes at certain angle of propagation (αm) and then decreases for α >αm . In addition, increasing the superthermality of the hot electrons also results in reduction of soliton amplitude and width. For auroral plasma parameters observed by Viking, the obliquely propagating electron-acoustic solitary waves have electric field amplitudes in the range (7.8-45) mV/m and pulse widths (0.29-0.44) ms. The Fourier transform of these electron acoustic solitons would result in a broadband frequency spectra with peaks near 2.3-3.5 kHz, thus providing a possible explanation of the broadband electrostatic noise observed during the Burst a.
2. High amplitude nonlinear acoustic wave driven flow fields in cylindrical and conical resonators.
Science.gov (United States)
Antao, Dion Savio; Farouk, Bakhtier
2013-08-01
A high fidelity computational fluid dynamic model is used to simulate the flow, pressure, and density fields generated in a cylindrical and a conical resonator by a vibrating end wall/piston producing high-amplitude standing waves. The waves in the conical resonator are found to be shock-less and can generate peak acoustic overpressures that exceed the initial undisturbed pressure by two to three times. A cylindrical (consonant) acoustic resonator has limitations to the output response observed at one end when the opposite end is acoustically excited. In the conical geometry (dissonant acoustic resonator) the linear acoustic input is converted to high energy un-shocked nonlinear acoustic output. The model is validated using past numerical results of standing waves in cylindrical resonators. The nonlinear nature of the harmonic response in the conical resonator system is further investigated for two different working fluids (carbon dioxide and argon) operating at various values of piston amplitude. The high amplitude nonlinear oscillations observed in the conical resonator can potentially enhance the performance of pulse tube thermoacoustic refrigerators and these conical resonators can be used as efficient mixers.
3. Nonlinear ionospheric responses to large-amplitude infrasonic-acoustic waves generated by undersea earthquakes
Science.gov (United States)
Zettergren, M. D.; Snively, J. B.; Komjathy, A.; Verkhoglyadova, O. P.
2017-02-01
Numerical models of ionospheric coupling with the neutral atmosphere are used to investigate perturbations of plasma density, vertically integrated total electron content (TEC), neutral velocity, and neutral temperature associated with large-amplitude acoustic waves generated by the initial ocean surface displacements from strong undersea earthquakes. A simplified source model for the 2011 Tohoku earthquake is constructed from estimates of initial ocean surface responses to approximate the vertical motions over realistic spatial and temporal scales. Resulting TEC perturbations from modeling case studies appear consistent with observational data, reproducing pronounced TEC depletions which are shown to be a consequence of the impacts of nonlinear, dissipating acoustic waves. Thermospheric acoustic compressional velocities are ˜±250-300 m/s, superposed with downward flows of similar amplitudes, and temperature perturbations are ˜300 K, while the dominant wave periodicity in the thermosphere is ˜3-4 min. Results capture acoustic wave processes including reflection, onset of resonance, and nonlinear steepening and dissipation—ultimately leading to the formation of ionospheric TEC depletions "holes"—that are consistent with reported observations. Three additional simulations illustrate the dependence of atmospheric acoustic wave and subsequent ionospheric responses on the surface displacement amplitude, which is varied from the Tohoku case study by factors of 1/100, 1/10, and 2. Collectively, results suggest that TEC depletions may only accompany very-large amplitude thermospheric acoustic waves necessary to induce a nonlinear response, here with saturated compressional velocities ˜200-250 m/s generated by sea surface displacements exceeding ˜1 m occurring over a 3 min time period.
4. Small amplitude nonlinear electron acoustic solitary waves in weakly magnetized plasma
Energy Technology Data Exchange (ETDEWEB)
Dutta, Manjistha; Khan, Manoranjan [Department of Instrumentation Science, Jadavpur University, Kolkata-700 032 (India); Ghosh, Samiran [Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata-700 009 (India); Roychoudhury, Rajkumar [Indian Statistical Institute, Kolkata-700 108 (India); Chakrabarti, Nikhil [Saha Institute of Nuclear Physics, 1/AF Bidhannagar Kolkata-700 064 (India)
2013-01-15
Nonlinear propagation of electron acoustic waves in homogeneous, dispersive plasma medium with two temperature electron species is studied in presence of externally applied magnetic field. The linear dispersion relation is found to be modified by the externally applied magnetic field. Lagrangian transformation technique is applied to carry out nonlinear analysis. For small amplitude limit, a modified KdV equation is obtained, the modification arising due to presence of magnetic field. For weakly magnetized plasma, the modified KdV equation possesses stable solitary solutions with speed and amplitude increasing temporally. The solutions are valid upto some finite time period beyond which the nonlinear wave tends to wave breaking.
5. Dependence of oscillational instabilities on the amplitude of the acoustic wave in single-axis levitators
DEFF Research Database (Denmark)
Orozco-Santillán, Arturo; Ruiz-Boullosa, Ricardo; Cutanda Henríquez, Vicente
2007-01-01
published on the topic predicts that these instabilities appear when the levitator is driven with a frequency above the resonant frequency of the empty device. The theory also shows that the instabilities can either saturate to a state with constant amplitude, or they can grow without limit until the object...... pressure amplitude in the cavity because of the presence of the sample. The theory predicts that the phase difference depends on the speed of the oscillating object. In this paper, we give for the first time experimental evidence that shows the existence of the phase difference, and that it is negatively...... proportional to the oscillation frequency of the levitated sample. We also present experimental results that show that the oscillational instabilities can be reduced if the amplitude of the acoustic wave is increased; as a result, stable conditions can be obtained where the oscillations of the sphere...
6. Arbitrary amplitude ion-acoustic solitary waves in electronegative plasmas with electrons featuring Tsallis distribution
Science.gov (United States)
Ghebache, Siham; Tribeche, Mouloud
2017-10-01
The problem of arbitrary amplitude ion-acoustic solitary waves (IASWs), which accompany electronegative plasmas having positive ions, negative ions, and nonextensive electrons is addressed. The energy integral equation with a new Sagdeev potential is analyzed to examine the existence regions of the IASWs. Different types of electronegative plasmas inspired from the experimental studies of Ichiki et al. (2001) are discussed. Our results show that in such plasmas IASWs, the amplitude and nature of which depend sensitively on the mass and density ratio of the positive and negative ions as well as the q-nonextensive parameter, can exist. Interestingly, one finds that our plasma model supports the coexistence of smooth rarefactive and spiky compressive IASWs. Our results complement and provide new insights on previously published findings on this problem.
7. a Finite Difference Numerical Model for the Propagation of Finite Amplitude Acoustical Blast Waves Outdoors Over Hard and Porous Surfaces
Science.gov (United States)
Sparrow, Victor Ward
1990-01-01
This study has concerned the propagation of finite amplitude, i.e. weakly non-linear, acoustical blast waves from explosions over hard and porous media models of outdoor ground surfaces. The nonlinear acoustic propagation effects require a numerical solution in the time domain. To model a porous ground surface, which in the frequency domain exhibits a finite impedance, the linear phenomenological porous model of Morse and Ingard was used. The phenomenological equations are solved in the time domain for coupling with the time domain propagation solution in the air. The numerical solution is found through the method of finite differences. The second-order in time and fourth -order in space MacCormack method was used in the air, and the second-order in time and space MacCormack method was used in the porous medium modeling the ground. Two kinds of numerical absorbing boundary conditions were developed for the air propagation equations to truncate the physical domain for solution on a computer. Radiation conditions first were used on those sides of the domain where there were outgoing waves. Characteristic boundary conditions secondly are employed near the acoustic source. The numerical model agreed well with the Pestorius algorithm for the propagation of electric spark pulses in the free field, and with a result of Pfriem for normal plane reflection off a hard surface. In addition, curves of pressure amplification versus incident angle for waves obliquely incident on the hard and porous surfaces were produced which are similar to those in the literature. The model predicted that near grazing finite amplitude acoustic blast waves decay with distance over hard surfaces as r to the power -1.2. This result is consistent with the work of Reed. For propagation over the porous ground surface, the model predicted that this surface decreased the decay rate with distance for the larger blasts compared to the rate expected in the linear acoustics limit.
8. Magneto-Acoustic Waves of Small Amplitude in Optically Thin Quasi-Isentropic Plasmas
CERN Document Server
Nakariakov, V M; Ibáñez, M H; Nakariakov, Valery M.; Mendoza-Briceno, Cesar A.
1999-01-01
The evolution of quasi-isentropic magnetohydrodynamic waves of small but finite amplitude in an optically thin plasma is analyzed. The plasma is assumed to be initially homogeneous, in thermal equilibrium and with a straight and homogeneous magnetic field frozen in. Depending on the particular form of the heating/cooling function, the plasma may act as a dissipative or active medium for magnetoacoustic waves, while Alfven waves are not directly affected. An evolutionary equation for fast and slow magnetoacoustic waves in the single wave limit, has been derived and solved, allowing us to analyse the wave modification by competition of weakly nonlinear and quasi-isentropic effects. It was shown that the sign of the quasi-isentropic term determines the scenario of the evolution, either dissipative or active. In the dissipative case, when the plasma is first order isentropically stable the magnetoacoustic waves are damped and the time for shock wave formation is delayed. However, in the active case when the plasm...
9. Vector network analyzer measurement of the amplitude of an electrically excited surface acoustic wave and validation by X-ray diffraction
Science.gov (United States)
Camara, I. S.; Croset, B.; Largeau, L.; Rovillain, P.; Thevenard, L.; Duquesne, J.-Y.
2017-01-01
Surface acoustic waves are used in magnetism to initiate magnetization switching, in microfluidics to control fluids and particles in lab-on-a-chip devices, and in quantum systems like two-dimensional electron gases, quantum dots, photonic cavities, and single carrier transport systems. For all these applications, an easy tool is highly needed to measure precisely the acoustic wave amplitude in order to understand the underlying physics and/or to optimize the device used to generate the acoustic waves. We present here a method to determine experimentally the amplitude of surface acoustic waves propagating on Gallium Arsenide generated by an interdigitated transducer. It relies on Vector Network Analyzer measurements of S parameters and modeling using the Coupling-Of-Modes theory. The displacements obtained are in excellent agreement with those measured by a very different method based on X-ray diffraction measurements.
10. New analytical solutions for propagation of small but finite amplitude ion-acoustic waves in a dense plasma
Science.gov (United States)
Hafez, M. G.; Talukder, M. R.; Ali, M. Hossain
2016-01-01
The theoretical and numerical studies have been investigated on the nonlinear propagation of electrostatic ion-acoustic waves (IAWs) in an un-magnetized Thomas-Fermi plasma system consisting of electron, positrons, and positive ions for both of ultra-relativistic and non-relativistic degenerate electrons. Korteweg-de Vries (K-dV) equation is derived from the model equations by using the well-known reductive perturbation method. This equation is solved by employing the generalized Riccati equation mapping method. The hyperbolic functions type solutions to the K-dV equation are only considered for describing the effect of plasma parameters on the propagation of electrostatic IAWs for both of ultra-relativistic and non-relativistic degenerate electrons. The obtained results may be helpful in proper understanding the features of small but finite amplitude localized IAWs in degenerate plasmas and provide the mathematical foundation in plasma physics.
11. Electron Acoustic Waves and the Search for a Truly Self-Consistent Large-Amplitude Plasma Response
Science.gov (United States)
Johnston, Tudor; Afeyan, Bedros
2003-10-01
We examine some theoretical nonlinear Vlasov work invoked in connection with recent laser-plasma experiments [1] on Electron Acoustic Waves and their stimulated scatter (SEAS). Earlier work discussed [2-5] is then related to more recent theory [6] used to interpret [1]. All this [2-6] is then related the recent Vlasov-Poisson findings of Afeyan et al. [7] on Kinetic Electrostatic Electron Nonlinear (KEEN) waves. (Part of this work was performed under the auspices of the U.S. Department of Energy under grant number DE-FG03-NA00059.) [1] D.S. Montgomery et al., Phys. Rev. Lett. 87, 155001 (2001), Phys. Plasmas 9, 2311(2002). [2] I.B. Bernstein et al., Phys.Rev. 108, 546 (1957). [3] W.P. Allis, paper no. 3 pp. 21-42, in In Honor of Philip M. Morse, ed. H. Feshbach and K. Ingard, MIT Press, Cambridge, MA, (1969). (Source for V.B. Krapchev and A.K. Ram, Phys. Rev. A, 22, 1229 (1980)). [4] H. Schamel, Phys. Scr. 20, 336 (1979), Phys. Rep. 140, 161 (1986), Phys. Plasmas 7, 4831 (2000). [5] J.P. Holloway and J.J. Dorning, Phys. Lett. A 138, 279 (1989) and Phys. Rev. A 44, 3856 (1991). [6] H. A. Rose and D. A. Russell, Phys. Plasmas 8, 4784 (2001). [7] B. B. Afeyan et al., "Optical Mixing Generated Kinetic Electrostatic Electron Nonlinear (KEEN) Waves", manuscript in preparation and poster at this conference.
12. Acoustics waves and oscillations
CERN Document Server
Sen, S.N.
2013-01-01
Parameters of acoustics presented in a logical and lucid style Physical principles discussed with mathematical formulations Importance of ultrasonic waves highlighted Dispersion of ultrasonic waves in viscous liquids explained This book presents the theory of waves and oscillations and various applications of acoustics in a logical and simple form. The physical principles have been explained with necessary mathematical formulation and supported by experimental layout wherever possible. Incorporating the classical view point all aspects of acoustic waves and oscillations have been discussed together with detailed elaboration of modern technological applications of sound. A separate chapter on ultrasonics emphasizes the importance of this branch of science in fundamental and applied research. In this edition a new chapter ''Hypersonic Velocity in Viscous Liquids as revealed from Brillouin Spectra'' has been added. The book is expected to present to its readers a comprehensive presentation of the subject matter...
13. Non-Linear Excitation of Ion Acoustic Waves
DEFF Research Database (Denmark)
Michelsen, Poul; Hirsfield, J. L.
1974-01-01
The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation.......The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation....
14. Numerical investigation of amplitude-dependent dynamic response in acoustic metamaterials with nonlinear oscillators.
Science.gov (United States)
Manimala, James M; Sun, C T
2016-06-01
The amplitude-dependent dynamic response in acoustic metamaterials having nonlinear local oscillator microstructures is studied using numerical simulations on representative discrete mass-spring models. Both cubically nonlinear hardening and softening local oscillator cases are considered. Single frequency, bi-frequency, and wave packet excitations at low and high amplitude levels were used to interrogate the models. The propagation and attenuation characteristics of harmonic waves in a tunable frequency range is found to correspond to the amplitude and nonlinearity-dependent shifts in the local resonance bandgap for such nonlinear acoustic metamaterials. A predominant shift in the propagated wave spectrum towards lower frequencies is observed. Moreover, the feasibility of amplitude and frequency-dependent selective filtering of composite signals consisting of individual frequency components which fall within propagating or attenuating regimes is demonstrated. Further enrichment of these wave manipulation mechanisms in acoustic metamaterials using different combinations of nonlinear microstructures presents device implications for acoustic filters and waveguides.
15. Amplitude Modulations of Acoustic Communication Signals
Science.gov (United States)
Turesson, Hjalmar K.
2011-12-01
In human speech, amplitude modulations at 3 -- 8 Hz are important for discrimination and detection. Two different neurophysiological theories have been proposed to explain this effect. The first theory proposes that, as a consequence of neocortical synaptic dynamics, signals that are amplitude modulated at 3 -- 8 Hz are propagated better than un-modulated signals, or signals modulated above 8 Hz. This suggests that neural activity elicited by vocalizations modulated at 3 -- 8 Hz is optimally transmitted, and the vocalizations better discriminated and detected. The second theory proposes that 3 -- 8 Hz amplitude modulations interact with spontaneous neocortical oscillations. Specifically, vocalizations modulated at 3 -- 8 Hz entrain local populations of neurons, which in turn, modulate the amplitude of high frequency gamma oscillations. This suggests that vocalizations modulated at 3 -- 8 Hz should induce stronger cross-frequency coupling. Similar to human speech, we found that macaque monkey vocalizations also are amplitude modulated between 3 and 8 Hz. Humans and macaque monkeys share similarities in vocal production, implying that the auditory systems subserving perception of acoustic communication signals also share similarities. Based on the similarities between human speech and macaque monkey vocalizations, we addressed how amplitude modulated vocalizations are processed in the auditory cortex of macaque monkeys, and what behavioral relevance modulations may have. Recording single neuron activity, as well as, the activity of local populations of neurons allowed us to test both of the neurophysiological theories presented above. We found that single neuron responses to vocalizations amplitude modulated at 3 -- 8 Hz resulted in better stimulus discrimination than vocalizations lacking 3 -- 8 Hz modulations, and that the effect most likely was mediated by synaptic dynamics. In contrast, we failed to find support for the oscillation-based model proposing a
16. Reflection and Refraction of Acoustic Waves by a Shock Wave
Science.gov (United States)
Brillouin, J.
1957-01-01
The presence of sound waves in one or the other of the fluid regions on either side of a shock wave is made apparent, in the region under superpressure, by acoustic waves (reflected or refracted according to whether the incident waves lie in the region of superpressure or of subpressure) and by thermal waves. The characteristics of these waves are calculated for a plane, progressive, and uniform incident wave. In the case of refraction, the refracted acoustic wave can, according to the incidence, be plane, progressive, and uniform or take the form of an 'accompanying wave' which remains attached to the front of the shock while sliding parallel to it. In all cases, geometrical constructions permit determination of the kinematic characteristics of the reflected or refractive acoustic waves. The dynamic relationships show that the amplitude of the reflected wave is always less than that of the incident wave. The amplitude of the refracted wave, whatever its type, may in certain cases be greater than that of the incident wave.
17. Identification of rocket-induced acoustic waves in the ionosphere
Science.gov (United States)
Mabie, Justin; Bullett, Terence; Moore, Prentiss; Vieira, Gerald
2016-10-01
Acoustic waves can create plasma disturbances in the ionosphere, but the number of observations is limited. Large-amplitude acoustic waves generated by energetic sources like large earthquakes and tsunamis are more readily observed than acoustic waves generated by weaker sources. New observations of plasma displacements caused by rocket-generated acoustic waves were made using the Vertically Incident Pulsed Ionospheric Radar (VIPIR), an advanced high-frequency radar. Rocket-induced acoustic waves which are characterized by low amplitudes relative to those induced by more energetic sources can be detected in the ionosphere using the phase data from fixed frequency radar observations of a plasma layer. This work is important for increasing the number and quality of observations of acoustic waves in the ionosphere and could help improve the understanding of energy transport from the lower atmosphere to the thermosphere.
18. Finite amplitude wave interaction with premixed laminar flames
Science.gov (United States)
2014-11-01
The physics underlying combustion instability is an active area of research because of its detrimental impact in many combustion devices, such as turbines, jet engines, and liquid rocket engines. Pressure waves, ranging from acoustic waves to strong shocks, are potential sources of these disturbances. Literature on flame-disturbance interactions are primarily focused on either acoustics or strong shock wave interactions, with little information about the wide spectrum of behaviors that may exist between these two extremes. For example, the interaction between a flame and a finite amplitude compression wave is not well characterized. This phenomenon is difficult to study numerically due to the wide range of scales that need to be captured, requiring powerful and efficient numerical techniques. In this work, the interaction of a perturbed laminar premixed flame with a finite amplitude compression wave is investigated using the Parallel Adaptive Wavelet Collocation Method (PAWCM). This method optimally solves the fully compressible Navier-Stokes equations while capturing the essential scales. The results show that depending on the amplitude and duration of a finite amplitude disturbance, the interaction between these waves and premixed flames can produce a broad range of responses.
19. Surface Acoustic Wave Devices
DEFF Research Database (Denmark)
Dühring, Maria Bayard
of a Mach-Zehnder interferometer (MZI). This is an optical device consisting if one waveguide that is split into two waveguide arms which are assembled again later on. By applying the mechanical field from a SAW the light in the two arms can be modulated and interfere constructively and destructively......The work of this project is concerned with the simulation of surface acoustic waves (SAW) and topology optimization of SAW devices. SAWs are elastic vibrations that propagate along a material surface and are extensively used in electromechanical filters and resonators in telecommunication. A new...... application is modulation of optical waves in waveguides. This presentation elaborates on how a SAW is generated by interdigital transducers using a 2D model of a piezoelectric, inhomogeneous material implemented in the high-level programming language Comsol Multiphysics. The SAW is send through a model...
20. Dynamic Nonlinear Focal Shift in Amplitude Modulated Moderately Focused Acoustic Beams
CERN Document Server
Jiménez, Noé; González-Salido, Nuria
2016-01-01
The phenomenon of the displacement of the position of the pressure, intensity and acoustic radiation force maxima along the axis of focused acoustic beams under increasing driving amplitudes (nonlinear focal shift) is studied for the case of a moderately focused beam excited with continuous and 25 kHz amplitude modulated signals, both in water and tissue. We prove that in amplitude modulated beams the linear and nonlinear propagation effects coexist in a semi-period of modulation, giving place to a complex dynamic behaviour, where the singular points of the beam (peak pressure, rarefaction, intensity and acoustic radiation force) locate at different points on axis as a function of time. These entire phenomena are explained in terms of harmonic generation and absorption during the propagation in a lossy nonlinear medium both, for a continuous and an amplitude modulated beam. One of the possible applications of the acoustic radiation force displacement is the generation of shear waves at different locations by ...
1. Analytical Interaction of the Acoustic Wave and Turbulent Flame
Institute of Scientific and Technical Information of China (English)
TENG Hong-Hui; JIANG Zong-Lin
2007-01-01
A modified resonance model of a weakly turbulent flame in a high-frequency acoustic wave is derived analytically.Under the mechanism of Darrieus-Landau instability, the amplitude of flame wrinkles, which is as functions of turbulence. The high perturbation wave number makes the resonance easier to be triggered but weakened with respect to the extra acoustic wave. In a closed burning chamber with the acoustic wave induced by the flame itself, the high perturbation wave number is to restrain the resonance for a realistic flame.
2. Surface acoustic wave microfluidics.
Science.gov (United States)
Ding, Xiaoyun; Li, Peng; Lin, Sz-Chin Steven; Stratton, Zackary S; Nama, Nitesh; Guo, Feng; Slotcavage, Daniel; Mao, Xiaole; Shi, Jinjie; Costanzo, Francesco; Huang, Tony Jun
2013-09-21
The recent introduction of surface acoustic wave (SAW) technology onto lab-on-a-chip platforms has opened a new frontier in microfluidics. The advantages provided by such SAW microfluidics are numerous: simple fabrication, high biocompatibility, fast fluid actuation, versatility, compact and inexpensive devices and accessories, contact-free particle manipulation, and compatibility with other microfluidic components. We believe that these advantages enable SAW microfluidics to play a significant role in a variety of applications in biology, chemistry, engineering and medicine. In this review article, we discuss the theory underpinning SAWs and their interactions with particles and the contacting fluids in which they are suspended. We then review the SAW-enabled microfluidic devices demonstrated to date, starting with devices that accomplish fluid mixing and transport through the use of travelling SAW; we follow that by reviewing the more recent innovations achieved with standing SAW that enable such actions as particle/cell focusing, sorting and patterning. Finally, we look forward and appraise where the discipline of SAW microfluidics could go next.
3. Surface acoustic wave propagation in graphene film
Energy Technology Data Exchange (ETDEWEB)
Roshchupkin, Dmitry, E-mail: [email protected]; Plotitcyna, Olga; Matveev, Viktor; Kononenko, Oleg; Emelin, Evgenii; Irzhak, Dmitry [Institute of Microelectronics Technology and High-Purity Materials Russian Academy of Sciences, Chernogolovka 142432 (Russian Federation); Ortega, Luc [Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, 91405 Orsay Cedex (France); Zizak, Ivo; Erko, Alexei [Institute for Nanometre Optics and Technology, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein Strasse 15, 12489 Berlin (Germany); Tynyshtykbayev, Kurbangali; Insepov, Zinetula [Nazarbayev University Research and Innovation System, 53 Kabanbay Batyr St., Astana 010000 (Kazakhstan)
2015-09-14
Surface acoustic wave (SAW) propagation in a graphene film on the surface of piezoelectric crystals was studied at the BESSY II synchrotron radiation source. Talbot effect enabled the visualization of the SAW propagation on the crystal surface with the graphene film in a real time mode, and high-resolution x-ray diffraction permitted the determination of the SAW amplitude in the graphene/piezoelectric crystal system. The influence of the SAW on the electrical properties of the graphene film was examined. It was shown that the changing of the SAW amplitude enables controlling the magnitude and direction of current in graphene film on the surface of piezoelectric crystals.
4. Surface acoustic wave propagation in graphene film
Science.gov (United States)
Roshchupkin, Dmitry; Ortega, Luc; Zizak, Ivo; Plotitcyna, Olga; Matveev, Viktor; Kononenko, Oleg; Emelin, Evgenii; Erko, Alexei; Tynyshtykbayev, Kurbangali; Irzhak, Dmitry; Insepov, Zinetula
2015-09-01
Surface acoustic wave (SAW) propagation in a graphene film on the surface of piezoelectric crystals was studied at the BESSY II synchrotron radiation source. Talbot effect enabled the visualization of the SAW propagation on the crystal surface with the graphene film in a real time mode, and high-resolution x-ray diffraction permitted the determination of the SAW amplitude in the graphene/piezoelectric crystal system. The influence of the SAW on the electrical properties of the graphene film was examined. It was shown that the changing of the SAW amplitude enables controlling the magnitude and direction of current in graphene film on the surface of piezoelectric crystals.
5. Nonlinear acoustic waves in a collisional self-gravitating dusty plasma
Institute of Scientific and Technical Information of China (English)
Guo Zhi-Rong; Yang Zeng-Qiang; Yin Bao-Xiang; Sun Mao-Zhu
2010-01-01
Using the reductive perturbation method,we investigate the small amplitude nonlinear acoustic wave in a collisional self-gravitating dusty plasma.The result shows that the small amplitude dust acoustic wave can be expressed by a modified Korteweg-de Vries equation,and the nonlinear wave is instable because of the collisions between the neutral gas molecules and the charged particles.
6. Canonical Acoustics and Its Application to Surface Acoustic Wave on Acoustic Metamaterials
Science.gov (United States)
Shen, Jian Qi
2016-08-01
In a conventional formalism of acoustics, acoustic pressure p and velocity field u are used for characterizing acoustic waves propagating inside elastic/acoustic materials. We shall treat some fundamental problems relevant to acoustic wave propagation alternatively by using canonical acoustics (a more concise and compact formalism of acoustic dynamics), in which an acoustic scalar potential and an acoustic vector potential (Φ ,V), instead of the conventional acoustic field quantities such as acoustic pressure and velocity field (p,u) for characterizing acoustic waves, have been defined as the fundamental variables. The canonical formalism of the acoustic energy-momentum tensor is derived in terms of the acoustic potentials. Both the acoustic Hamiltonian density and the acoustic Lagrangian density have been defined, and based on this formulation, the acoustic wave quantization in a fluid is also developed. Such a formalism of acoustic potentials is employed to the problem of negative-mass-density assisted surface acoustic wave that is a highly localized surface bound state (an eigenstate of the acoustic wave equations). Since such a surface acoustic wave can be strongly confined to an interface between an acoustic metamaterial (e.g., fluid-solid composite structures with a negative dynamical mass density) and an ordinary material (with a positive mass density), it will give rise to an effect of acoustic field enhancement on the acoustic interface, and would have potential applications in acoustic device design for acoustic wave control.
7. Oscillating nonlinear acoustic shock waves
DEFF Research Database (Denmark)
Gaididei, Yuri; Rasmussen, Anders Rønne; Christiansen, Peter Leth
2016-01-01
We investigate oscillating shock waves in a tube using a higher order weakly nonlinear acoustic model. The model includes thermoviscous effects and is non isentropic. The oscillating shock waves are generated at one end of the tube by a sinusoidal driver. Numerical simulations show...... that at resonance a stationary state arise consisting of multiple oscillating shock waves. Off resonance driving leads to a nearly linear oscillating ground state but superimposed by bursts of a fast oscillating shock wave. Based on a travelling wave ansatz for the fluid velocity potential with an added 2'nd order...... polynomial in the space and time variables, we find analytical approximations to the observed single shock waves in an infinitely long tube. Using perturbation theory for the driven acoustic system approximative analytical solutions for the off resonant case are determined....
8. Modulation of a quantum positron acoustic wave
Science.gov (United States)
Amin, M. R.
2015-09-01
Amplitude modulation of a positron acoustic wave is considered in a four-component electron-positron plasma in the quantum magnetohydrodynamic regime. The important ingredients of this study are the inclusion of the particle exchange-correlation potential, quantum diffraction effects via the Bohm potential, and dissipative effect due to viscosity in the momentum balance equation of the charged carriers. A modified nonlinear Schrödinger equation is derived for the evolution of the slowly varying amplitude of the quantum positron acoustic wave by employing the standard reductive perturbation technique. Detailed analysis of the linear and nonlinear dispersions of the quantum positron acoustic wave is presented. For a typical parameter range, relevant to some dense astrophysical objects, it is found that the quantum positron acoustic wave is modulationally unstable above a certain critical wavenumber. Effects of the exchange-correlation potential and the Bohm potential in the wave dynamics are also studied. It is found that the quantum effect due to the particle exchange-correlation potential is significant in comparison to the effect due to the Bohm potential for smaller values of the carrier wavenumber. However, for comparatively larger values of the carrier wavenumber, the Bohm potential effect overtakes the effect of the exchange-correlation potential. It is found that the critical wavenumber for the modulation instability depends on the ratio of the equilibrium hot electron number density and the cold positron number density and on the ratio of the equilibrium hot positron number density and the cold positron number density. A numerical result on the growth rate of the modulation instability is also presented.
9. Ion-acoustic cnoidal waves in a quantum plasma
CERN Document Server
2016-01-01
Nonlinear ion-acoustic cnoidal wave structures are studied in an unmagnetized quantum plasma. Using the reductive perturbation method, a Korteweg-de Vries equation is derived for appropriate boundary conditions and nonlinear periodic wave solutions are obtained. The corresponding analytical solution and numerical plots of the ion-acoustic cnoidal waves and solitons in the phase plane are presented using the Sagdeev pseudo-potential approach. The variations in the nonlinear potential of the ion-acoustic cnoidal waves are studied at different values of quantum parameter $H_{e}$ which is the ratio of electron plasmon energy to electron Fermi energy defined for degenerate electrons. It is found that both compressive and rarefactive ion-acoustic cnoidal wave structures are formed depending on the value of the quantum parameter. The dependence of the wavelength and frequency on nonlinear wave amplitude is also presented.
10. Organizing Filament of Small Amplitude Scroll Waves
Institute of Scientific and Technical Information of China (English)
ZHOU TianShou; ZHANG SuoChun
2001-01-01
We theoretically analyze the organizing filament of small amplitude scroll waves in general excitable media by perturbation method and explicitly give the expressions of coefficients in Keener theory. In particular for the excitable media with equal diffusion, we obtain a close system for the motion of the filament. With an example of the Oregonator model, our results are in good agreement with those simulated by Winfree.
11. Acoustic spin pumping in magnetoelectric bulk acoustic wave resonator
Directory of Open Access Journals (Sweden)
N. I. Polzikova
2016-05-01
Full Text Available We present the generation and detection of spin currents by using magnetoelastic resonance excitation in a magnetoelectric composite high overtone bulk acoustic wave (BAW resonator (HBAR formed by a Al-ZnO-Al-GGG-YIG-Pt structure. Transversal BAW drives magnetization oscillations in YIG film at a given resonant magnetic field, and the resonant magneto-elastic coupling establishes the spin-current generation at the Pt/YIG interface. Due to the inverse spin Hall effect (ISHE this BAW-driven spin current is converted to a dc voltage in the Pt layer. The dependence of the measured voltage both on magnetic field and frequency has a resonant character. The voltage is determined by the acoustic power in HBAR and changes its sign upon magnetic field reversal. We compare the experimentally observed amplitudes of the ISHE electrical field achieved by our method and other approaches to spin current generation that use surface acoustic waves and microwave resonators for ferromagnetic resonance excitation, with the theoretically expected values.
12. Acoustic spin pumping in magnetoelectric bulk acoustic wave resonator
Energy Technology Data Exchange (ETDEWEB)
Polzikova, N. I., E-mail: [email protected]; Alekseev, S. G.; Pyataikin, I. I.; Kotelyanskii, I. M.; Luzanov, V. A.; Orlov, A. P. [Kotel’nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Mokhovaya 11, building 7, Moscow, 125009 (Russian Federation)
2016-05-15
We present the generation and detection of spin currents by using magnetoelastic resonance excitation in a magnetoelectric composite high overtone bulk acoustic wave (BAW) resonator (HBAR) formed by a Al-ZnO-Al-GGG-YIG-Pt structure. Transversal BAW drives magnetization oscillations in YIG film at a given resonant magnetic field, and the resonant magneto-elastic coupling establishes the spin-current generation at the Pt/YIG interface. Due to the inverse spin Hall effect (ISHE) this BAW-driven spin current is converted to a dc voltage in the Pt layer. The dependence of the measured voltage both on magnetic field and frequency has a resonant character. The voltage is determined by the acoustic power in HBAR and changes its sign upon magnetic field reversal. We compare the experimentally observed amplitudes of the ISHE electrical field achieved by our method and other approaches to spin current generation that use surface acoustic waves and microwave resonators for ferromagnetic resonance excitation, with the theoretically expected values.
13. Piezoelectric Film Waveguides for Surface Acoustic Waves
Directory of Open Access Journals (Sweden)
M.F. Zhovnir
2016-11-01
Full Text Available The paper presents results of mathematical modeling of piezoelectric film waveguide structures for surface acoustic waves (SAW. Piezoelectric ZnO film is supposed to be placed on a fused quartz substrate. The analytical ratios and numerical results allow to determine the design parameters of the waveguide structures to provide a single-mode SAW propagation mode. The results of amplitude and phase experimental studies of the SAW in the waveguide structures that were carried out on the laser optical sensing set up confirm the theoretical calculations.
14. Dynamic nonlinear focal shift in amplitude modulated moderately focused acoustic beams.
Science.gov (United States)
Jiménez, Noé; Camarena, Francisco; González-Salido, Nuria
2017-03-01
The phenomenon of the displacement of the position of the pressure, intensity and acoustic radiation force maxima along the axis of focused acoustic beams under increasing driving amplitudes (nonlinear focal shift) is studied for the case of a moderately focused beam excited with continuous and 25kHz amplitude modulated signals, both in water and tissue. We prove that in amplitude modulated beams the linear and nonlinear propagation effects coexist in a semi-period of modulation, giving place to a complex dynamic behavior, where the singular points of the beam (peak pressure, rarefaction, intensity and acoustic radiation force) locate at different points on axis as a function of time. These entire phenomena are explained in terms of harmonic generation and absorption during the propagation in a lossy nonlinear medium both for a continuous and an amplitude modulated beam. One of the possible applications of the acoustic radiation force displacement is the generation of shear waves at different locations by using a focused mono-element transducer excited by an amplitude modulated signal.
15. Nonlinear ion acoustic waves scattered by vortexes
Science.gov (United States)
Ohno, Yuji; Yoshida, Zensho
2016-09-01
The Kadomtsev-Petviashvili (KP) hierarchy is the archetype of infinite-dimensional integrable systems, which describes nonlinear ion acoustic waves in two-dimensional space. This remarkably ordered system resides on a singular submanifold (leaf) embedded in a larger phase space of more general ion acoustic waves (low-frequency electrostatic perturbations). The KP hierarchy is characterized not only by small amplitudes but also by irrotational (zero-vorticity) velocity fields. In fact, the KP equation is derived by eliminating vorticity at every order of the reductive perturbation. Here, we modify the scaling of the velocity field so as to introduce a vortex term. The newly derived system of equations consists of a generalized three-dimensional KP equation and a two-dimensional vortex equation. The former describes 'scattering' of vortex-free waves by ambient vortexes that are determined by the latter. We say that the vortexes are 'ambient' because they do not receive reciprocal reactions from the waves (i.e., the vortex equation is independent of the wave fields). This model describes a minimal departure from the integrable KP system. By the Painlevé test, we delineate how the vorticity term violates integrability, bringing about an essential three-dimensionality to the solutions. By numerical simulation, we show how the solitons are scattered by vortexes and become chaotic.
16. Ion Acoustic Travelling Waves
CERN Document Server
Webb, G M; Ao, X; Zank, G P
2013-01-01
Models for travelling waves in multi-fluid plasmas give essential insight into fully nonlinear wave structures in plasmas, not readily available from either numerical simulations or from weakly nonlinear wave theories. We illustrate these ideas using one of the simplest models of an electron-proton multi-fluid plasma for the case where there is no magnetic field or a constant normal magnetic field present. We show that the travelling waves can be reduced to a single first order differential equation governing the dynamics. We also show that the equations admit a multi-symplectic Hamiltonian formulation in which both the space and time variables can act as the evolution variable. An integral equation useful for calculating adiabatic, electrostatic solitary wave signatures for multi-fluid plasmas with arbitrary mass ratios is presented. The integral equation arises naturally from a fluid dynamics approach for a two fluid plasma, with a given mass ratio of the two species (e.g. the plasma could be an electron pr...
17. Amplitude Equations for Electrostatic Waves multiple species
CERN Document Server
Crawford, J D; Crawford, John David; Jayaraman, Anandhan
1997-01-01
The amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude $A(t)$. In the limit of weak instability, i.e. $\\gamma\\to 0^+$ where $\\gamma$ is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of $A(t)$ on $\\gamma$. Generically the scaling $|A(t)|=\\gamma^{5/2}r(\\gamma t)$ as orders. This result predicts the electric field scaling $|E_k|\\sim\\gamma^{5/2}$ will hold universally for these instabilities (including beam-plasma and two-stream configurations) throughout the dynamical evolution and in the time-asymptotic state. In exceptional cases, such as infinitely massive ions, the coefficients are less singular and the more familiar trapping scaling $|E_k|\\sim\\gamma^2$ is recovered.
18. Simulating acoustic waves in spotted stars
CERN Document Server
Papini, Emanuele; Gizon, Laurent; Hanasoge, Shravan M
2015-01-01
Acoustic modes of oscillation are affected by stellar activity, however it is unclear how starspots contribute to these changes. Here we investigate the non-magnetic effects of starspots on global modes with angular degree $\\ell \\leq 2$ in highly active stars, and characterize the spot seismic signature on synthetic light curves. We perform 3D time-domain simulations of linear acoustic waves to study their interaction with a model starspot. We model the spot as a 3D change in the sound speed stratification with respect to a convectively stable stellar background, built from solar Model S. We perform a parametric study by considering different depths and perturbation amplitudes. Exact numerical simulations allow investigation of the wavefield-spot interaction beyond first order perturbation theory. The interaction of the axisymmetric modes with the starspot is strongly nonlinear. As mode frequency increases, the frequency shifts for radial modes exceed the value predicted by linear theory, while the shifts for...
19. Electron Acoustic Waves in Pure Ion Plasmas
Science.gov (United States)
Anderegg, F.; Affolter, M.; Driscoll, C. F.; O'Neil, T. M.; Valentini, F.
2012-10-01
Electron Acoustic Waves (EAWs) are the low-frequency branch of near-linear Langmuir (plasma) waves: the frequency is such that the complex dielectric function (Dr, Di) has Dr= 0; and flattening'' of f(v) near the wave phase velocity vph gives Di=0 and eliminates Landau damping. Here, we observe standing axisymmetric EAWs in a pure ion column.footnotetextF. Anderegg, et al., Phys. Rev. Lett. 102, 095001 (2009). At low excitation amplitudes, the EAWs have vph˜1.4 v, in close agreement with near-linear theory. At moderate excitation strengths, EAW waves are observed over a range of frequencies, with 1.3 v vphvph.footnotetextF. Valentini et al., arXiv:1206.3500v1. Large amplitude EAWs have strong phase-locked harmonic content, and experiments will be compared to same-geometry simulations, and to simulations of KEENfootnotetextB. Afeyan et al., Proc. Inertial Fusion Sci. and Applications 2003, A.N.S. Monterey (2004), p. 213. waves in HEDLP geometries.
20. Quasi-periodic behavior of ion acoustic solitary waves in electron-ion quantum plasma
Energy Technology Data Exchange (ETDEWEB)
Sahu, Biswajit [Department of Mathematics, West Bengal State University Barasat, Kolkata-700126 (India); Poria, Swarup [Department of Applied Mathematics, University of Calcutta Kolkata-700009 (India); Narayan Ghosh, Uday [Department of Mathematics, Siksha Bhavana, Visva Bharati University Santiniketan (India); Roychoudhury, Rajkumar [Physics and Applied Mathematics Unit, Indian Statistical Institute Kolkata-700108 (India)
2012-05-15
The ion acoustic solitary waves are investigated in an unmagnetized electron-ion quantum plasmas. The one dimensional quantum hydrodynamic model is used to study small as well as arbitrary amplitude ion acoustic waves in quantum plasmas. It is shown that ion temperature plays a critical role in the dynamics of quantum electron ion plasma, especially for arbitrary amplitude nonlinear waves. In the small amplitude region Korteweg-de Vries equation describes the solitonic nature of the waves. However, for arbitrary amplitude waves, in the fully nonlinear regime, the system exhibits possible existence of quasi-periodic behavior for small values of ion temperature.
1. Millimeter waves: acoustic and electromagnetic.
Science.gov (United States)
Ziskin, Marvin C
2013-01-01
This article is the presentation I gave at the D'Arsonval Award Ceremony on June 14, 2011 at the Bioelectromagnetics Society Annual Meeting in Halifax, Nova Scotia. It summarizes my research activities in acoustic and electromagnetic millimeter waves over the past 47 years. My earliest research involved acoustic millimeter waves, with a special interest in diagnostic ultrasound imaging and its safety. For the last 21 years my research expanded to include electromagnetic millimeter waves, with a special interest in the mechanisms underlying millimeter wave therapy. Millimeter wave therapy has been widely used in the former Soviet Union with great reported success for many diseases, but is virtually unknown to Western physicians. I and the very capable members of my laboratory were able to demonstrate that the local exposure of skin to low intensity millimeter waves caused the release of endogenous opioids, and the transport of these agents by blood flow to all parts of the body resulted in pain relief and other beneficial effects. Copyright © 2012 Wiley Periodicals, Inc.
2. Nonlinear ion acoustic waves scattered by vortexes
CERN Document Server
Ohno, Yuji
2015-01-01
The Kadomtsev--Petviashvili (KP) hierarchy is the archetype of infinite-dimensional integrable systems, which describes nonlinear ion acoustic waves in two-dimensional space. This remarkably ordered system resides on a singular submanifold (leaf) embedded in a larger phase space of more general ion acoustic waves (low-frequency electrostatic perturbations). The KP hierarchy is characterized not only by small amplitudes but also by irrotational (zero-vorticity) velocity fields. In fact, the KP equation is derived by eliminating vorticity at every order of the reductive perturbation. Here we modify the scaling of the velocity field so as to introduce a vortex term. The newly derived system of equations consists of a generalized three-dimensional KP equation and a two-dimensional vortex equation. The former describes scattering' of vortex-free waves by ambient vortexes that are determined by the latter. We say that the vortexes are ambient' because they do not receive reciprocal reactions from the waves (i.e.,...
3. Theoretical and Experimental Study on the Acoustic Wave Energy After the Nonlinear Interaction of Acoustic Waves in Aqueous Media
Institute of Scientific and Technical Information of China (English)
兰朝凤; 李凤臣; 陈欢; 卢迪; 杨德森; 张梦
2015-01-01
Based on the Burgers equation and Manley-Rowe equation, the derivation about nonlinear interaction of the acoustic waves has been done in this paper. After nonlinear interaction among the low-frequency weak waves and the pump wave, the analytical solutions of acoustic waves’ amplitude in the field are deduced. The relationship between normalized energy of high-frequency and the change of acoustic energy before and after the nonlinear interaction of the acoustic waves is analyzed. The experimental results about the changes of the acoustic energy are presented. The study shows that new frequencies are generated and the energies of the low-frequency are modulated in a long term by the pump waves, which leads the energies of the low-frequency acoustic waves to change in the pulse trend in the process of the nonlinear interaction of the acoustic waves. The increase and decrease of the energies of the low-frequency are observed under certain typical conditions, which lays a foundation for practical engineering applications.
4. Magneto-acoustic imaging by continuous-wave excitation.
Science.gov (United States)
Shunqi, Zhang; Zhou, Xiaoqing; Tao, Yin; Zhipeng, Liu
2016-07-01
The electrical characteristics of tissue yield valuable information for early diagnosis of pathological changes. Magneto-acoustic imaging is a functional approach for imaging of electrical conductivity. This study proposes a continuous-wave magneto-acoustic imaging method. A kHz-range continuous signal with an amplitude range of several volts is used to excite the magneto-acoustic signal and improve the signal-to-noise ratio. The magneto-acoustic signal amplitude and phase are measured to locate the acoustic source via lock-in technology. An optimisation algorithm incorporating nonlinear equations is used to reconstruct the magneto-acoustic source distribution based on the measured amplitude and phase at various frequencies. Validation simulations and experiments were performed in pork samples. The experimental and simulation results agreed well. While the excitation current was reduced to 10 mA, the acoustic signal magnitude increased up to 10(-7) Pa. Experimental reconstruction of the pork tissue showed that the image resolution reached mm levels when the excitation signal was in the kHz range. The signal-to-noise ratio of the detected magneto-acoustic signal was improved by more than 25 dB at 5 kHz when compared to classical 1 MHz pulse excitation. The results reported here will aid further research into magneto-acoustic generation mechanisms and internal tissue conductivity imaging.
5. Symmetric waves are traveling waves for a shallow water equation for surface waves of moderate amplitude
OpenAIRE
Geyer, Anna
2016-01-01
Following a general principle introduced by Ehrnstr\\"{o}m et.al. we prove that for an equation modeling the free surface evolution of moderate amplitude waves in shallow water, all symmetric waves are traveling waves.
6. Symmetric waves are traveling waves for a shallow water equation for surface waves of moderate amplitude
OpenAIRE
Geyer, Anna
2016-01-01
Following a general principle introduced by Ehrnstr\\"{o}m et.al. we prove that for an equation modeling the free surface evolution of moderate amplitude waves in shallow water, all symmetric waves are traveling waves.
7. Propagation behavior of acoustic wave in wood
Institute of Scientific and Technical Information of China (English)
Huadong Xu; Guoqi Xu; Lihai Wang; Lei Yu
2014-01-01
We used acoustic tests on a quarter-sawn poplar timbers to study the effects of wood anisotropy and cavity defects on acoustic wave velocity and travel path, and we investigated acoustic wave propagation behavior in wood. The timber specimens were first tested in unmodified condition and then tested after introduction of cavity defects of varying sizes to quantify the transmitting time of acoustic waves in laboratory conditions. Two-dimensional acoustic wave contour maps on the radial section of specimens were then simulated and analyzed based on the experimental data. We tested the relationship between wood grain and acoustic wave velocity as waves passed in various directions through wood. Wood anisotropy has significant effects on both velocity and travel path of acoustic waves, and the velocity of waves passing longitudinally through timbers exceeded the radial velocity. Moreover, cavity defects altered acoustic wave time contours on radial sections of timbers. Acous-tic wave transits from an excitation point to the region behind a cavity in defective wood more slowly than in intact wood.
8. Ion Acoustic Waves in the Presence of Electron Plasma Waves
DEFF Research Database (Denmark)
Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens
1977-01-01
Long-wavelength ion acoustic waves in the presence of propagating short-wavelength electron plasma waves are examined. The influence of the high frequency oscillations is to decrease the phase velocity and the damping distance of the ion wave.......Long-wavelength ion acoustic waves in the presence of propagating short-wavelength electron plasma waves are examined. The influence of the high frequency oscillations is to decrease the phase velocity and the damping distance of the ion wave....
9. Manipulate acoustic waves by impedance matched acoustic metasurfaces
Science.gov (United States)
Wu, Ying; Mei, Jun; Aljahdali, Rasha
We design a type of acoustic metasurface, which is composed of carefully designed slits in a rigid thin plate. The effective refractive indices of different slits are different but the impedances are kept the same as that of the host medium. Numerical simulations show that such a metasurface can redirect or reflect a normally incident wave at different frequencies, even though it is impedance matched to the host medium. We show that the underlying mechanisms can be understood by using the generalized Snell's law, and a unified analytic model based on mode-coupling theory. We demonstrate some simple realization of such acoustic metasurface with real materials. The principle is also extended to the design of planar acoustic lens which can focus acoustic waves. Manipulate acoustic waves by impedance matched acoustic metasurfaces.
10. PROGRESS OF ACOUSTIC WAVE TECHNIQUE AND ITS APPLICATION IN UNDERGROUND PRESSURE MEASUREMENT
Institute of Scientific and Technical Information of China (English)
周楚良; 李新元; 张晓龙
1994-01-01
This paper carries out the experiment study on the correlation between full stress-strain process of rock samples and the acoustic parameter change of rock by using the measurement system of KS acoustic wave data processing device. On the spot, the stability of surrounding rock is studied by means of experiments on the relationship between the change process (from elastic to plastic failure zone) in surrounding rock of roadway and the change law of acoustic parameters of rock. These acoustic parameters include wave amplitude, spectral amplitude, spectrum area, spectral density, wave velocity and attenuation coefficient etc.
11. Modulational instability of ion-acoustic waves in a warm plasma
Institute of Scientific and Technical Information of China (English)
薛具奎; 段文山; 郎和
2002-01-01
Using the standard reductive perturbation technique, a nonlinear Schrodinger equation is derived to study themodulational instability of finite-amplitude ion-acoustic waves in a non-magnetized warm plasma. It is found thatthe inclusion of ion temperature in the equation modifies the nature of the ion-acoustic wave stability and the solitonstructures. The effects of ion plasma temperature on the modulational stability and ion-acoustic wave properties areinvestigated in detail.
12. Surface Acoustic Wave Frequency Comb
CERN Document Server
Savchenkov, A A; Ilchenko, V S; Seidel, D; Maleki, L
2011-01-01
We report on realization of an efficient triply-resonant coupling between two long lived optical modes and a high frequency surface acoustic wave (SAW) mode of the same monolithic crystalline whispering gallery mode resonator. The coupling results in an opto-mechanical oscillation and generation of a monochromatic SAW. A strong nonlinear interaction of this mechanical mode with other equidistant SAW modes leads to mechanical hyper-parametric oscillation and generation of a SAW pulse train and associated frequency comb in the resonator. We visualized the comb observing the modulation of the modulated light escaping the resonator.
13. Quantum ion-acoustic solitary waves in weak relativistic plasma
Biswajit Sahu
2011-06-01
Small amplitude quantum ion-acoustic solitary waves are studied in an unmagnetized twospecies relativistic quantum plasma system, comprised of electrons and ions. The one-dimensional quantum hydrodynamic model (QHD) is used to obtain a deformed Korteweg–de Vries (dKdV) equation by reductive perturbation method. A linear dispersion relation is also obtained taking into account the relativistic effect. The properties of quantum ion-acoustic solitary waves, obtained from the deformed KdV equation, are studied taking into account the quantum mechanical effects in the weak relativistic limit. It is found that relativistic effects significantly modify the properties of quantum ion-acoustic waves. Also the effect of the quantum parameter on the nature of solitary wave solutions is studied in some detail.
14. Adiabatic trapping in coupled kinetic Alfven-acoustic waves
Energy Technology Data Exchange (ETDEWEB)
Shah, H. A.; Ali, Z. [Department of Physics, G.C. University, 54000 Lahore (Pakistan); Masood, W. [COMSATS, Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000 (Pakistan); National Centre for Physics (NCP), Shahdara Valley Road, 44000 Islamabad (Pakistan); Theoretical Plasma Physics Division, P. O. Nilore, Islamabad (Pakistan)
2013-03-15
In the present work, we have discussed the effects of adiabatic trapping of electrons on obliquely propagating Alfven waves in a low {beta} plasma. Using the two potential theory and employing the Sagdeev potential approach, we have investigated the existence of arbitrary amplitude coupled kinetic Alfven-acoustic solitary waves in both the sub and super Alfvenic cases. The results obtained have been analyzed and presented graphically and can be applied to regions of space where the low {beta} assumption holds true.
15. Robust acoustic wave manipulation of bubbly liquids
Energy Technology Data Exchange (ETDEWEB)
Gumerov, N. A., E-mail: [email protected] [Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742 (United States); Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Akhatov, I. S. [Center for Design, Manufacturing and Materials, Skolkovo Institute of Science and Technology, Moscow 143026 (Russian Federation); Ohl, C.-D. [Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 (Singapore); Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Sametov, S. P. [Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Khazimullin, M. V. [Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Institute of Molecule and Crystal Physics, Ufa Research Center of Russian Academy of Sciences, Ufa 450054 (Russian Federation); Gonzalez-Avila, S. R. [Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 (Singapore)
2016-03-28
Experiments with water–air bubbly liquids when exposed to acoustic fields of frequency ∼100 kHz and intensity below the cavitation threshold demonstrate that bubbles ∼30 μm in diameter can be “pushed” away from acoustic sources by acoustic radiation independently from the direction of gravity. This manifests formation and propagation of acoustically induced transparency waves (waves of the bubble volume fraction). In fact, this is a collective effect of bubbles, which can be described by a mathematical model of bubble self-organization in acoustic fields that matches well with our experiments.
16. An analysis of beam parameters on proton-acoustic waves through an analytic approach.
Science.gov (United States)
Aytac Kipergil, Esra; Erkol, Hakan; Kaya, Serhat; Gulsen, Gultekin; Unlu, Mehmet
2017-03-02
It has been reported that acoustic waves are generated when a high energy pulsed proton beam is deposited in a small volume within tissue. One possible application of the proton induced acoustics is to get a real-time feedback for intratreatment adjustments by monitoring such acoustic waves. High spatial resolution in ultrasound imaging may reduce proton range uncertainty. Thus, it is crucial to understand the dependence of the acoustic waves on the proton beam characteristics. In this manuscript, firstly, an analytic solution to the proton induced acoustic wave is presented to reveal the dependence of signal on beam parameters, and then combined with an analytic approximation of the Bragg curve. The influence of the beam energy, pulse duration, and beam diameter variation on the acoustic waveform are investigated. Further analysis is performed regarding the Fourier decomposition of proton-acoustic signals. Our results show that smaller spill time of proton beam upsurges the amplitude of acoustic wave for constant number of protons, and hence beneficial for dose monitoring. The increase in the energy of each individual proton in the beam leads to spatial broadening of the Bragg curve, which also yields acoustic waves of greater amplitude. The pulse duration and the beam width of the proton beam do not affect the central frequency of the acoustic wave, but they change the amplitude of the spectral components.
17. Electron acoustic solitary waves with kappa-distributed electrons
Energy Technology Data Exchange (ETDEWEB)
Devanandhan, S; Singh, S V; Lakhina, G S, E-mail: [email protected] [Indian Institute of Geomagnetism, New Panvel (West), Navi Mumbai (India)
2011-08-01
Electron acoustic solitary waves are studied in a three-component, unmagnetized plasma composed of hot electrons, fluid cold electrons and ions having finite temperatures. Hot electrons are assumed to have kappa distribution. The Sagdeev pseudo-potential technique is used to study the arbitrary amplitude electron-acoustic solitary waves. It is found that inclusion of cold electron temperature shrinks the existence regime of the solitons, and soliton electric field amplitude decreases with an increase in cold electron temperature. A decrease in spectral index, {kappa}, i.e. an increase in the superthermal component of hot electrons, leads to a decrease in soliton electric field amplitude as well as the soliton velocity range. The soliton solutions do not exist beyond T{sub c}/T{sub h}>0.13 for {kappa}=3.0 and Mach number M=0.9 for the dayside auroral region parameters.
18. On Collisionless Damping of Ion Acoustic Waves
DEFF Research Database (Denmark)
Jensen, Vagn Orla; Petersen, P.I.
1973-01-01
Exact theoretical treatments show that the damping of ion acoustic waves in collisionless plasmas does not vanish when the derivative of the undisturbed distribution function at the phase velocity equals zero.......Exact theoretical treatments show that the damping of ion acoustic waves in collisionless plasmas does not vanish when the derivative of the undisturbed distribution function at the phase velocity equals zero....
19. Acoustic tweezing of particles using decaying opposing travelling surface acoustic waves (DOTSAW).
Science.gov (United States)
Ng, Jia Wei; Devendran, Citsabehsan; Neild, Adrian
2017-09-20
Surface acoustic waves offer a versatile and biocompatible method of manipulating the location of suspended particles or cells within microfluidic systems. The most common approach uses the interference of identical frequency, counter propagating travelling waves to generate a standing surface acoustic wave, in which particles migrate a distance less than half the acoustic wavelength to their nearest pressure node. The result is the formation of a periodic pattern of particles. Subsequent displacement of this pattern, the prerequisite for tweezing, can be achieved by translation of the standing wave, and with it the pressure nodes; this requires changing either the frequency of the pair of waves, or their relative phase. Here, in contrast, we examine the use of two counterpropagating traveling waves of different frequency. The non-linearity of the acoustic forces used to manipulate particles, means that a small frequency difference between the two waves creates a substantially different force field, which offers significant advantages. Firstly, this approach creates a much longer range force field, in which migration takes place across multiple wavelengths, and causes particles to be gathered together in a single trapping site. Secondly, the location of this single trapping site can be controlled by the relative amplitude of the two waves, requiring simply an attenuation of one of the electrical drive signals. Using this approach, we show that by controlling the powers of the opposing incoherent waves, 5 μm particles can be migrated laterally across a fluid flow to defined locations with an accuracy of ±10 μm.
20. Parametric instabilities of large-amplitude parallel propagating Alfven waves: 2-D PIC simulation
CERN Document Server
2008-01-01
We discuss the parametric instabilities of large-amplitude parallel propagating Alfven waves using the 2-D PIC simulation code. First, we confirmed the results in the past study [Sakai et al, 2005] that the electrons are heated due to the modified two stream instability and that the ions are heated by the parallel propagating ion acoustic waves. However, although the past study argued that such parallel propagating longitudinal waves are excited by transverse modulation of parent Alfven wave, we consider these waves are more likely to be generated by the usual, parallel decay instability. Further, we performed other simulation runs with different polarization of the parent Alfven waves or the different ion thermal velocity. Numerical results suggest that the electron heating by the modified two stream instability due to the large amplitude Alfven waves is unimportant with most parameter sets.
1. Drops subjected to surface acoustic waves: flow dynamics
Science.gov (United States)
Brunet, Philippe; Baudoin, Michael; Bou Matar, Olivier; Dynamique Des Systèmes Hors Equilibre Team; Aiman-Films Team
2012-11-01
Ultrasonic acoustic waves of frequency beyond the MHz are known to induce streaming flow in fluids that can be suitable to perform elementary operations in microfluidics systems. One of the currently appealing geometry is that of a sessile drop subjected to surface acoustic waves (SAW). Such Rayleigh waves produce non-trival actuation in the drop leading to internal flow, drop displacement, free-surface oscillations and atomization. We recently carried out experiments and numerical simulations that allowed to better understand the underlying physical mechanisms that couple acoustic propagation and fluid actuation. We varied the frequency and amplitude of actuation, as well as the properties of the fluid, and we measured the effects of these parameters on the dynamics of the flow. We compared these results to finite-elements numerical simulations.
2. Acoustic wave science realized by metamaterials.
Science.gov (United States)
Lee, Dongwoo; Nguyen, Duc Minh; Rho, Junsuk
2017-01-01
Artificially structured materials with unit cells at sub-wavelength scale, known as metamaterials, have been widely used to precisely control and manipulate waves thanks to their unconventional properties which cannot be found in nature. In fact, the field of acoustic metamaterials has been much developed over the past 15 years and still keeps developing. Here, we present a topical review of metamaterials in acoustic wave science. Particular attention is given to fundamental principles of acoustic metamaterials for realizing the extraordinary acoustic properties such as negative, near-zero and approaching-infinity parameters. Realization of acoustic cloaking phenomenon which is invisible from incident sound waves is also introduced by various approaches. Finally, acoustic lenses are discussed not only for sub-diffraction imaging but also for applications based on gradient index (GRIN) lens.
3. Unidirectional propagation of designer surface acoustic waves
CERN Document Server
Lu, Jiuyang; Ke, Manzhu; Liu, Zhengyou
2014-01-01
We propose an efficient design route to generate unidirectional propagation of the designer surface acoustic waves. The whole system consists of a periodically corrugated rigid plate combining with a pair of asymmetric narrow slits. The directionality of the structure-induced surface waves stems from the destructive interference between the evanescent waves emitted from the double slits. The theoretical prediction is validated well by simulations and experiments. Promising applications can be anticipated, such as in designing compact acoustic circuits.
4. Turbulence beneath finite amplitude water waves
Energy Technology Data Exchange (ETDEWEB)
Beya, J.F. [Universidad de Valparaiso, Escuela de Ingenieria Civil Oceanica, Facultad de Ingenieria, Valparaiso (Chile); The University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney, NSW (Australia); Peirson, W.L. [The University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney, NSW (Australia); Banner, M.L. [The University of New South Wales, School of Mathematics and Statistics, Sydney, NSW (Australia)
2012-05-15
Babanin and Haus (J Phys Oceanogr 39:2675-2679, 2009) recently presented evidence of near-surface turbulence generated below steep non-breaking deep-water waves. They proposed a threshold wave parameter a {sup 2}{omega}/{nu} = 3,000 for the spontaneous occurrence of turbulence beneath surface waves. This is in contrast to conventional understanding that irrotational wave theories provide a good approximation of non-wind-forced wave behaviour as validated by classical experiments. Many laboratory wave experiments were carried out in the early 1960s (e.g. Wiegel 1964). In those experiments, no evidence of turbulence was reported, and steep waves behaved as predicted by the high-order irrotational wave theories within the accuracy of the theories and experimental techniques at the time. This contribution describes flow visualisation experiments for steep non-breaking waves using conventional dye techniques in the wave boundary layer extending above the wave trough level. The measurements showed no evidence of turbulent mixing up to a value of a {sup 2}{omega}/{nu} = 7,000 at which breaking commenced in these experiments. These present findings are in accord with the conventional understandings of wave behaviour. (orig.)
5. Attenuation compensation in least-squares reverse time migration using the visco-acoustic wave equation
KAUST Repository
Dutta, Gaurav
2013-08-20
Attenuation leads to distortion of amplitude and phase of seismic waves propagating inside the earth. Conventional acoustic and least-squares reverse time migration do not account for this distortion which leads to defocusing of migration images in highly attenuative geological environments. To account for this distortion, we propose to use the visco-acoustic wave equation for least-squares reverse time migration. Numerical tests on synthetic data show that least-squares reverse time migration with the visco-acoustic wave equation corrects for this distortion and produces images with better balanced amplitudes compared to the conventional approach. © 2013 SEG.
6. Surface acoustic wave (SAW) vibration sensors.
Science.gov (United States)
Filipiak, Jerzy; Solarz, Lech; Steczko, Grzegorz
2011-01-01
In the paper a feasibility study on the use of surface acoustic wave (SAW) vibration sensors for electronic warning systems is presented. The system is assembled from concatenated SAW vibration sensors based on a SAW delay line manufactured on a surface of a piezoelectric plate. Vibrations of the plate are transformed into electric signals that allow identification of the sensor and localization of a threat. The theoretical study of sensor vibrations leads us to the simple isotropic model with one degree of freedom. This model allowed an explicit description of the sensor plate movement and identification of the vibrating sensor. Analysis of frequency response of the ST-cut quartz sensor plate and a damping speed of its impulse response has been conducted. The analysis above was the basis to determine the ranges of parameters for vibrating plates to be useful in electronic warning systems. Generally, operation of electronic warning systems with SAW vibration sensors is based on the analysis of signal phase changes at the working frequency of delay line after being transmitted via two circuits of concatenated four-terminal networks. Frequencies of phase changes are equal to resonance frequencies of vibrating plates of sensors. The amplitude of these phase changes is proportional to the amplitude of vibrations of a sensor plate. Both pieces of information may be sent and recorded jointly by a simple electrical unit.
7. Surface Acoustic Wave (SAW Vibration Sensors
Directory of Open Access Journals (Sweden)
Jerzy Filipiak
2011-12-01
Full Text Available In the paper a feasibility study on the use of surface acoustic wave (SAW vibration sensors for electronic warning systems is presented. The system is assembled from concatenated SAW vibration sensors based on a SAW delay line manufactured on a surface of a piezoelectric plate. Vibrations of the plate are transformed into electric signals that allow identification of the sensor and localization of a threat. The theoretical study of sensor vibrations leads us to the simple isotropic model with one degree of freedom. This model allowed an explicit description of the sensor plate movement and identification of the vibrating sensor. Analysis of frequency response of the ST-cut quartz sensor plate and a damping speed of its impulse response has been conducted. The analysis above was the basis to determine the ranges of parameters for vibrating plates to be useful in electronic warning systems. Generally, operation of electronic warning systems with SAW vibration sensors is based on the analysis of signal phase changes at the working frequency of delay line after being transmitted via two circuits of concatenated four-terminal networks. Frequencies of phase changes are equal to resonance frequencies of vibrating plates of sensors. The amplitude of these phase changes is proportional to the amplitude of vibrations of a sensor plate. Both pieces of information may be sent and recorded jointly by a simple electrical unit.
8. Dynamics of coupled light waves and electron-acoustic waves.
Science.gov (United States)
Shukla, P K; Stenflo, L; Hellberg, M
2002-08-01
The nonlinear interaction between coherent light waves and electron-acoustic waves in a two-electron plasma is considered. The interaction is governed by a pair of equations comprising a Schrödinger-like equation for the light wave envelope and a driven (by the light pressure) electron-acoustic wave equation. The newly derived nonlinear equations are used to study the formation and dynamics of envelope light wave solitons and light wave collapse. The implications of our investigation to space and laser-produced plasmas are pointed out.
9. Microfabricated bulk wave acoustic bandgap device
Science.gov (United States)
Olsson, Roy H.; El-Kady, Ihab F.; McCormick, Frederick; Fleming, James G.; Fleming, Carol
2010-06-08
A microfabricated bulk wave acoustic bandgap device comprises a periodic two-dimensional array of scatterers embedded within the matrix material membrane, wherein the scatterer material has a density and/or elastic constant that is different than the matrix material and wherein the periodicity of the array causes destructive interference of the acoustic wave within an acoustic bandgap. The membrane can be suspended above a substrate by an air or vacuum gap to provide acoustic isolation from the substrate. The device can be fabricated using microelectromechanical systems (MEMS) technologies. Such microfabricated bulk wave phononic bandgap devices are useful for acoustic isolation in the ultrasonic, VHF, or UHF regime (i.e., frequencies of order 1 MHz to 10 GHz and higher, and lattice constants of order 100 .mu.m or less).
10. Kinematics and amplitude evolution of global coronal extreme ultraviolet waves
Institute of Scientific and Technical Information of China (English)
Ting Li; Jun Zhang; Shu-Hong Yang; Wei Liu
2012-01-01
With the observations of the Solar-Terrestrial Relations Observatory (STEREO) and the Solar Dynamics Observatory (SDO),we analyze in detail the kinematics of global coronal waves together with their intensity amplitudes (so-called "perturbation profiles").We use a semi-automatic method to investigate the perturbation profiles of coronal waves.The location and amplitude of the coronal waves are calculated over a 30° sector on the sphere,where the wave signal is strongest.The position with the strongest perturbation at each time is considered as the location of the wave front.In all four events,the wave velocities vary with time for most of their lifetime,up to 15 min,while in the event observed by the Atmospheric Imaging Assembly there is an additional early phase with a much higher velocity.The velocity varies greatly between different waves from 216 to 440 km s-1.The velocity of the two waves initially increases,subsequently decreases,and then increases again.Two other waves show a deceleration followed by an acceleration.Three categories of amplitude evolution of global coronal waves are found for the four events.The first is that the amplitude only shows a decrease.The second is that the amplitude initially increases and then decreases,and the third is that the amplitude shows an orderly increase,a decrease,an increase again and then a decrease.All the extreme ultraviolet waves show a decrease in amplitude while propagating farther away,probably because the driver of the global coronal wave (coronal mass ejection) is moving farther away from the solar surface.
11. Seismic wave imaging in visco-acoustic media
Institute of Scientific and Technical Information of China (English)
WANG Huazhong; ZHANG Libin; MA Zaitian
2004-01-01
Realistic representation of the earth may be achieved by combining the mechanical properties of elastic solids and viscousliquids. That is to say, the amplitude will be attenuated withdifferent frequency and the phase will be changed in the seismicdata acquisition. In the seismic data processing, this effect mustbe compensated. In this paper, we put forward a visco-acoustic wavepropagator which is of better calculating stability and tolerablecalculating cost (little more than an acoustic wave propagator).The quite good compensation effect is demonstrated by thenumerical test results with synthetic seismic data and real data.
12. Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas
Directory of Open Access Journals (Sweden)
S. S. Ghosh
2004-01-01
Full Text Available The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.
13. Generation of Nanometer Wavelength Acoustic Waves
Directory of Open Access Journals (Sweden)
O.Yu. Komina
2016-11-01
Full Text Available The possibility of acoustic wave generation of nanometer range in plates is shown. The experimental results that show the possible reconfiguring of the generator frequency in YFeO3 with a constant magnetic field are given.
14. Surface Wave Amplitude Anomalies in the Western United States
Science.gov (United States)
Eddy, C.; Ekstrom, G.
2011-12-01
We determine maps of local surface wave amplitude factors across the Western United States for Rayleigh and Love waves at discrete periods between 25 and 125s. Measurements of raw amplitude anomalies are made from data recorded at 1161 USArray stations for minor arc arrivals of earthquakes with Mw>5.5 occurring between 2006 and 2010. We take the difference between high-quality amplitude anomaly measurements for events recorded on station pairs less than 2 degrees apart. The mean of these differences for each station pair is taken as the datum. Surface wave amplitudes are controlled by four separate mechanisms: focusing due to elastic structure, attenuation due to anelastic structure, source effects, and receiver effects. By taking the mean of the differences of amplitude anomalies for neighboring stations, we reduce the effects of focusing, attenuation, and the seismic source, thus isolating amplitude anomalies due to near-receiver amplitude effects. We determine local amplitude factors for each USArray station by standard linear inversion of the differential data set. The individual station amplitude factors explain the majority of the variance of the data. For example, derived station amplitude factors for 50s Rayleigh waves explain 92% of the variance of the data. We explore correlations between derived station amplitude factors and local amplitude factors predicted by crust and upper mantle models. Maps of local amplitude factors show spatial correlation with topography and geologic structures in the Western United States, particularly for maps derived from Rayleigh wave amplitude anomalies. A NW-SE trending high in amplitude factors in Eastern California is evident in the 50s map, corresponding to the location of the Sierra Nevada Mountains. High amplitude factors are observed in Colorado and New Mexico in the 50s-125s maps in the location of the highest peaks of the Rocky Mountains. High amplitude factors are also seen in Southern Idaho and Eastern Wyoming in
15. Relativistic degeneracy effect on propagation of arbitrary amplitude ion-acoustic solitons in Thomas-Fermi plasmas
CERN Document Server
Esfandyari-Kalejahi, Abdolrasoul; Saberian, Ehsan; 10.1585/pfr.5.045
2011-01-01
Arbitrary amplitude ion-acoustic solitary waves (IASWs) are studied using Sagdeev-Potential approach in electron-positron-ion plasma with ultra-relativistic or non-relativistic degenerate electrons and positrons and the matching criteria of existence of such solitary waves are numerically investigated. It has been shown that the relativistic degeneracy of electrons and positrons has significant effects on the amplitude and the Mach-number range of IASWs. Also it is remarked that only compressive IASWs can propagate in both non-relativistic and ultra-relativistic degenerate plasmas.
16. Acoustic holography based on composite metasurface with decoupled modulation of phase and amplitude
Science.gov (United States)
Tian, Ye; Wei, Qi; Cheng, Ying; Liu, Xiaojun
2017-05-01
Acoustic holography has extensive possibilities in acoustic sensing, acoustic illusion, contactless particle manipulation, and medical imaging. Based on coating unit cells and perforated panels, an acoustic composite metasurface is constructed with a decoupled modulation of phase and amplitude, which has been used to design acoustic holography. This proposal not only has lower complexity than conventional acoustic holography of active arrays due to the avoidance of complex structures and circuits but also provides more flexibility than acoustic holography based on the acoustic metasurface with phase-only modulation benefitting from the efficient decoupled modulation of phase and amplitude. We have further demonstrated three acoustic holographic applications, such as multi-directional transmission, multi-focal focusing, and holographic imaging. Due to the low complexity and the great flexibility, this proposal has potential to achieve the high-quality holograms with high information content, fine resolution, and large scale.
17. Focusing of Acoustic Waves through Acoustic Materials with Subwavelength Structures
KAUST Repository
Xiao, Bingmu
2013-05-01
In this thesis, wave propagation through acoustic materials with subwavelength slits structures is studied. Guided by the findings, acoustic wave focusing is achieved with a specific material design. By using a parameter retrieving method, an effective medium theory for a slab with periodic subwavelength cut-through slits is successfully derived. The theory is based on eigenfunction solutions to the acoustic wave equation. Numerical simulations are implemented by the finite-difference time-domain (FDTD) method for the two-dimensional acoustic wave equation. The theory provides the effective impedance and refractive index functions for the equivalent medium, which can reproduce the transmission and reflection spectral responses of the original structure. I analytically and numerically investigate both the validity and limitations of the theory, and the influences of material and geometry on the effective spectral responses are studied. Results show that large contrasts in impedance and density are conditions that validate the effective medium theory, and this approximation displays a better accuracy for a thick slab with narrow slits in it. Based on the effective medium theory developed, a design of a at slab with a snake shaped" subwavelength structure is proposed as a means of achieving acoustic focusing. The property of focusing is demonstrated by FDTD simulations. Good agreement is observed between the proposed structure and the equivalent lens pre- dicted by the theory, which leads to robust broadband focusing by a thin at slab.
18. Damping and Frequency Shift of Large Amplitude Electron Plasma Waves
DEFF Research Database (Denmark)
Thomsen, Kenneth; Juul Rasmussen, Jens
1983-01-01
The initial evolution of large-amplitude one-dimensional electron waves is investigated by applying a numerical simulation. The initial wave damping is found to be strongly enhanced relative to the linear damping and it increases with increasing amplitude. The temporal evolution of the nonlinear...... damping rate γ(t) shows that it increases with time within the initial phase of propagation, t≲π/ωB (ωB is the bounce frequency), whereafter it decreases and changes sign implying a regrowth of the wave. The shift in the wave frequency δω is observed to be positive for t≲π/ωB; then δω changes sign...
19. Investigation into Mass Loading Sensitivity of Sezawa Wave Mode-Based Surface Acoustic Wave Sensors
OpenAIRE
N. Ramakrishnan; Parthiban, R.; Sawal Hamid Md Ali; Md. Shabiul Islam; Ajay Achath Mohanan
2013-01-01
In this work mass loading sensitivity of a Sezawa wave mode based surface acoustic wave (SAW) device is investigated through finite element method (FEM) simulation and the prospects of these devices to function as highly sensitive SAW sensors is reported. A ZnO/Si layered SAW resonator is considered for the simulation study. Initially the occurrence of Sezawa wave mode and displacement amplitude of the Rayleigh and Sezawa wave mode is studied for lower ZnO film thickness. Further, a thin film...
20. Analytical description of nonlinear acoustic waves in the solar chromosphere
Science.gov (United States)
Litvinenko, Yuri E.; Chae, Jongchul
2017-02-01
Aims: Vertical propagation of acoustic waves of finite amplitude in an isothermal, gravitationally stratified atmosphere is considered. Methods: Methods of nonlinear acoustics are used to derive a dispersive solution, which is valid in a long-wavelength limit, and a non-dispersive solution, which is valid in a short-wavelength limit. The influence of the gravitational field on wave-front breaking and shock formation is described. The generation of a second harmonic at twice the driving wave frequency, previously detected in numerical simulations, is demonstrated analytically. Results: Application of the results to three-minute chromospheric oscillations, driven by velocity perturbations at the base of the solar atmosphere, is discussed. Numerical estimates suggest that the second harmonic signal should be detectable in an upper chromosphere by an instrument such as the Fast Imaging Solar Spectrograph installed at the 1.6-m New Solar Telescope of the Big Bear Observatory.
1. Dust-acoustic solitary waves in a dusty plasma with two-temperature nonthermal ions
Zhi-Jian Zhou; Hong-Yan Wang; Kai-Biao Zhang
2012-01-01
By using reductive perturbation method, the nonlinear propagation of dust-acoustic waves in a dusty plasma (containing a negatively charged dust fluid, Boltzmann distributed electrons and two-temperature nonthermal ions) is investigated. The effects of two-temperature nonthermal ions on the basic properties of small but finite amplitude nonlinear dust-acoustic waves are examined. It is found that two-temperature nonthermal ions affect the basic properties of the dust-acoustic solitary waves. It is also observed that only compressive solitary waves exist in this system.
2. Turbulent Flow over Small Amplitude Solid Waves
Science.gov (United States)
1984-01-01
7. Annubar flow meter 8. Butterfly throttling valve 9. Removable blanking plate 10. Diaphragm valve 11. Small pump 12. By pass diaphragm valve...monitored by using an annubar connected to either a mercury or mirriam oil filled manometer. (b) Test Section 2 a A wave with a ratio of 0.014
3. Scattering of Acoustic Waves from Ocean Boundaries
Science.gov (United States)
2015-09-30
of Acoustic Waves from Ocean Boundaries Marcia...J. Isakson Applied Research Laboratories The University of Texas at Austin, TX 78713-8029 phone: (512)835-3790 fax: (512)835-3259 email...plane wave integral transform method which assumes invariance in one spatial dimension of the waveguide. In this case, the dimension is
4. Finite-amplitude waves in cylindrical lined ducts
Science.gov (United States)
Nayfeh, A. H.; Tsai, M.-S.
1974-01-01
A second-order uniformly valid expansion is obtained for nonlinear waves propagating in a cylindrical duct lined with a point-reacting acoustic material that consists of a porous sheet followed by honey-comb cavities and backed by the impervious walls of the duct. The effect of the liner is taken into account by coupling the waves in the duct with those in the liner. As in the two-dimensional case, the nonlinearity increases the attenuation rate at all frequencies except in narrow bandwidths around the resonant frequencies, irrespective of the geometrical dimensions of the liner or the acoustic properties of the porous sheet.
5. Nonlinear electron acoustic waves in presence of shear magnetic field
Energy Technology Data Exchange (ETDEWEB)
Dutta, Manjistha; Khan, Manoranjan [Department of Instrumentation Science, Jadavpur University, Kolkata 700 032 (India); Ghosh, Samiran [Department of Applied Mathematics, University of Calcutta 92, Acharya Prafulla Chandra Road, Kolkata 700 009 (India); Chakrabarti, Nikhil [Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064 (India)
2013-12-15
Nonlinear electron acoustic waves are studied in a quasineutral plasma in the presence of a variable magnetic field. The fluid model is used to describe the dynamics of two temperature electron species in a stationary positively charged ion background. Linear analysis of the governing equations manifests dispersion relation of electron magneto sonic wave. Whereas, nonlinear wave dynamics is being investigated by introducing Lagrangian variable method in long wavelength limit. It is shown from finite amplitude analysis that the nonlinear wave characteristics are well depicted by KdV equation. The wave dispersion arising in quasineutral plasma is induced by transverse magnetic field component. The results are discussed in the context of plasma of Earth's magnetosphere.
6. Shoaling Large Amplitude Internal Solitary Waves in a Laboratory Tank
Science.gov (United States)
Allshouse, Michael; Larue, Conner; Swinney, Harry
2014-11-01
The shoaling of internal solitary waves onto the continental shelf can change both the wave dynamics and the state of the environment. Previous observations have demonstrated that these waves can trap fluid and transport it over long distances. Through the use of a camshaft-based wavemaker, we produce large amplitude shoaling waves in a stratified fluid in a laboratory tank. Simulations of solitary waves are used to guide the tuning of the wave generator to approximate solitary waves; thus nonlinear waves can be produced within the 4m long tank. PIV and synthetic schlieren measurements are made to study the transport of fluid by the wave as it moves up a sloping boundary. The results are then compared to numerical simulations and analyzed using finite time Lyapunov exponent calculations. This Lagrangian analysis provides an objective measure of barriers surrounding trapped regions in the flow. Supported by ONR MURI Grant N000141110701 (WHOI).
7. Finite Amplitude Electron Plasma Waves in a Cylindrical Waveguide
DEFF Research Database (Denmark)
Juul Rasmussen, Jens
1978-01-01
The nonlinear behaviour of the electron plasma wave propagating in a cylindrical plasma waveguide immersed in an infinite axial magnetic field is investigated using the Krylov-Bogoliubov-Mitropolsky perturbation method, by means of which is deduced the nonlinear Schrodinger equation governing...... the long-time slow modulation of the wave amplitude. From this equation the amplitude-dependent frequency and wavenumber shifts are calculated, and it is found that the electron waves with short wavelengths are modulationally unstable with respect to long-wavelength, low-frequency perturbations...
8. Synchronization of self-excited dust acoustic waves
Science.gov (United States)
Suranga Ruhunusiri, W. D.; Goree, John
2012-10-01
Synchronization is a nonlinear phenomenon where a self-excited oscillation, like a wave in a plasma, interacts with an external driving, resulting in an adjustment of the oscillation frequency. Dust acoustic wave synchronization has been experimentally studied previously in laboratory and in microgravity conditions, e.g. [Pilch PoP 2009] and [Menzel PRL 2010]. We perform a laboratory experiment to study synchronization of self-excited dust acoustic waves. An rf glow discharge argon plasma is formed by applying a low power radio frequency voltage to a lower electrode. A 3D dust cloud is formed by levitating 4.83 micron microspheres inside a glass box placed on the lower electrode. Dust acoustic waves are self-excited with a natural frequency of 22 Hz due to an ion streaming instability. A cross section of the dust cloud is illuminated by a vertical laser sheet and imaged from the side with a digital camera. To synchronize the waves, we sinusoidally modulate the overall ion density. Differently from previous experiments, we use a driving electrode that is separate from the electrode that sustains the plasma, and we characterize synchronization by varying both driving amplitude and frequency.
9. Imaging of Acoustic Waves in Sand
Energy Technology Data Exchange (ETDEWEB)
Deason, Vance Albert; Telschow, Kenneth Louis; Watson, Scott Marshall
2003-08-01
There is considerable interest in detecting objects such as landmines shallowly buried in loose earth or sand. Various techniques involving microwave, acoustic, thermal and magnetic sensors have been used to detect such objects. Acoustic and microwave sensors have shown promise, especially if used together. In most cases, the sensor package is scanned over an area to eventually build up an image or map of anomalies. We are proposing an alternate, acoustic method that directly provides an image of acoustic waves in sand or soil, and their interaction with buried objects. The INEEL Laser Ultrasonic Camera utilizes dynamic holography within photorefractive recording materials. This permits one to image and demodulate acoustic waves on surfaces in real time, without scanning. A video image is produced where intensity is directly and linearly proportional to surface motion. Both specular and diffusely reflecting surfaces can be accomodated and surface motion as small as 0.1 nm can be quantitatively detected. This system was used to directly image acoustic surface waves in sand as well as in solid objects. Waves as frequencies of 16 kHz were generated using modified acoustic speakers. These waves were directed through sand toward partially buried objects. The sand container was not on a vibration isolation table, but sat on the lab floor. Interaction of wavefronts with buried objects showed reflection, diffraction and interference effects that could provide clues to location and characteristics of buried objects. Although results are preliminary, success in this effort suggests that this method could be applied to detection of buried landmines or other near-surface items such as pipes and tanks.
10. T-wave amplitude is related to physical fitness status.
Science.gov (United States)
Arbel, Yaron; Birati, Edo Y; Shapira, Itzhak; Topilsky, Yan; Wirguin, Michal; Canaani M D, Jonathan
2012-07-01
Abnormalities in repolarization may reflect underlying myocardial pathology and play a prominent role in arrhythmogenesis The T-wave amplitude has been associated with cardiovascular outcome in patients with acute myocardial infarction (MI) Additionally, T-wave amplitude is considered a predictor of arrhythmias, as well as being related to an individual's inflammatory status. The combined influence of different variables, such as inflammation, cardiovascular risk factors and physical fitness status, on the T-wave amplitude has not been evaluated to date. The aim of this study was to identify factors that affect the T-wave amplitude. Data from 255 consecutive apparently healthy individuals included in the Tel Aviv Medical Center Inflammation Survey (TAMCIS) were reviewed. All patients had undergone a physical examination and an exercise stress test, and different inflammatory and metabolic biomarkers (fibrinogen, potassium, and high-sensitivity C-reactive protein) were measured. Multivariate stepwise analysis revealed that the body mass index and the resting heart rate were significantly associated with the T-wave amplitude (β=-0.34, P physical fitness and not to his/her inflammatory status. ©2012, Wiley Periodicals, Inc.
11. Active micromixer using surface acoustic wave streaming
Science.gov (United States)
Branch; Darren W. , Meyer; Grant D. , Craighead; Harold G.
2011-05-17
An active micromixer uses a surface acoustic wave, preferably a Rayleigh wave, propagating on a piezoelectric substrate to induce acoustic streaming in a fluid in a microfluidic channel. The surface acoustic wave can be generated by applying an RF excitation signal to at least one interdigital transducer on the piezoelectric substrate. The active micromixer can rapidly mix quiescent fluids or laminar streams in low Reynolds number flows. The active micromixer has no moving parts (other than the SAW transducer) and is, therefore, more reliable, less damaging to sensitive fluids, and less susceptible to fouling and channel clogging than other types of active and passive micromixers. The active micromixer is adaptable to a wide range of geometries, can be easily fabricated, and can be integrated in a microfluidic system, reducing dead volume. Finally, the active micromixer has on-demand on/off mixing capability and can be operated at low power.
12. Finite difference solutions to shocked acoustic waves
Science.gov (United States)
Walkington, N. J.; Eversman, W.
1983-01-01
The MacCormack, Lambda and split flux finite differencing schemes are used to solve a one dimensional acoustics problem. Two duct configurations were considered, a uniform duct and a converging-diverging nozzle. Asymptotic solutions for these two ducts are compared with the numerical solutions. When the acoustic amplitude and frequency are sufficiently high the acoustic signal shocks. This condition leads to a deterioration of the numerical solutions since viscous terms may be required if the shock is to be resolved. A continuous uniform duct solution is considered to demonstrate how the viscous terms modify the solution. These results are then compared with a shocked solution with and without viscous terms. Generally it is found that the most accurate solutions are those obtained using the minimum possible viscosity coefficients. All of the schemes considered give results accurate enough for acoustic power calculations with no one scheme performing significantly better than the others.
13. Non-Linear High Amplitude Oscillations in Wave-shaped Resonators
Science.gov (United States)
Antao, Dion; Farouk, Bakhtier
2011-11-01
A numerical and experimental study of non-linear, high amplitude standing waves in wave-shaped'' resonators is reported here. These waves are shock-less and can generate peak acoustic overpressures that can exceed the ambient pressure by three/four times its nominal value. A high fidelity compressible axisymmetric computational fluid dynamic model is used to simulate the phenomena in cylindrical and arbitrarily shaped axisymmetric resonators. Working fluids (Helium, Nitrogen and R-134a) at various operating pressures are studied. The experiments are performed in a constant cross-section cylindrical resonator in atmospheric pressure nitrogen and helium to provide model validation. The high amplitude non-linear oscillations demonstrated can be used as a prime mover in a variety of applications including thermoacoustic cryocooling. The work reported is supported by the US National Science Foundation under grant CBET-0853959.
14. Effect of adiabatic variation of dust charges on dust acoustic solitary waves in magnetized dusty plasmas
Institute of Scientific and Technical Information of China (English)
Duan Wen-Shan
2004-01-01
The effect of dust charging and the influence of its adiabatic variation on dust acoustic waves is investigated. By employing the reductive perturbation technique we derived a Zakharov-Kuznetsov (ZK) equation for small amplitude dust acoustic waves. We have analytically verified that there are only rarefactive solitary waves for this system. The instability region for one-dimensional solitary wave under transverse perturbations has also been obtained. The obliquely propagating solitary waves to the z-direction for the ZK equation are given in this paper as well.
15. Intermittent large amplitude internal waves observed in Port Susan, Puget Sound
Science.gov (United States)
Harris, J. C.; Decker, L.
2017-07-01
A previously unreported internal tidal bore, which evolves into solitary internal wave packets, was observed in Port Susan, Puget Sound, and the timing, speed, and amplitude of the waves were measured by CTD and visual observation. Acoustic Doppler current profiler (ADCP) measurements were attempted, but unsuccessful. The waves appear to be generated with the ebb flow along the tidal flats of the Stillaguamish River, and the speed and width of the resulting waves can be predicted from second-order KdV theory. Their eventual dissipation may contribute significantly to surface mixing locally, particularly in comparison with the local dissipation due to the tides. Visually the waves appear in fair weather as a strong foam front, which is less visible the farther they propagate.
16. Generation of acoustic rogue waves in dusty plasmas through three-dimensional particle focusing by distorted waveforms
Science.gov (United States)
Tsai, Ya-Yi; Tsai, Jun-Yi; I, Lin
2016-06-01
Rogue waves--rare uncertainly emerging localized events with large amplitudes--have been experimentally observed in many nonlinear wave phenomena, such as water waves, optical waves, second sound in superfluid He II (ref. ) and ion acoustic waves in plasmas. Past studies have mainly focused on one-dimensional (1D) wave behaviour through modulation instabilities, and to a lesser extent on higher-dimensional behaviour. The question whether rogue waves also exist in nonlinear 3D acoustic-type plasma waves, the kinetic origin of their formation and their correlation with surrounding 3D waveforms are unexplored fundamental issues. Here we report the direct experimental observation of dust acoustic rogue waves in dusty plasmas and construct a picture of 3D particle focusing by the surrounding tilted and ruptured wave crests, associated with the higher probability of low-amplitude holes for rogue-wave generation.
17. Relativistic electron scattering by magnetosonic waves: Effects of discrete wave emission and high wave amplitudes
Energy Technology Data Exchange (ETDEWEB)
Artemyev, A. V., E-mail: [email protected] [Space Research Institute, RAS, Moscow (Russian Federation); Mourenas, D.; Krasnoselskikh, V. V. [LPC2E/CNRS - University of Orleans, Orleans (France); Agapitov, O. V. [Space Sciences Laboratory, University of California, Berkeley, California 94720 (United States)
2015-06-15
In this paper, we study relativistic electron scattering by fast magnetosonic waves. We compare results of test particle simulations and the quasi-linear theory for different spectra of waves to investigate how a fine structure of the wave emission can influence electron resonant scattering. We show that for a realistically wide distribution of wave normal angles θ (i.e., when the dispersion δθ≥0.5{sup °}), relativistic electron scattering is similar for a wide wave spectrum and for a spectrum consisting in well-separated ion cyclotron harmonics. Comparisons of test particle simulations with quasi-linear theory show that for δθ>0.5{sup °}, the quasi-linear approximation describes resonant scattering correctly for a large enough plasma frequency. For a very narrow θ distribution (when δθ∼0.05{sup °}), however, the effect of a fine structure in the wave spectrum becomes important. In this case, quasi-linear theory clearly fails in describing accurately electron scattering by fast magnetosonic waves. We also study the effect of high wave amplitudes on relativistic electron scattering. For typical conditions in the earth's radiation belts, the quasi-linear approximation cannot accurately describe electron scattering for waves with averaged amplitudes >300 pT. We discuss various applications of the obtained results for modeling electron dynamics in the radiation belts and in the Earth's magnetotail.
18. River plumes as a source of large-amplitude internal waves in the coastal ocean
Science.gov (United States)
Nash, Jonathan D.; Moum, James N.
2005-09-01
Satellite images have long revealed the surface expression of large amplitude internal waves that propagate along density interfaces beneath the sea surface. Internal waves are typically the most energetic high-frequency events in the coastal ocean, displacing water parcels by up to 100m and generating strong currents and turbulence that mix nutrients into near-surface waters for biological utilization. While internal waves are known to be generated by tidal currents over ocean-bottom topography, they have also been observed frequently in the absence of any apparent tide-topography interactions. Here we present repeated measurements of velocity, density and acoustic backscatter across the Columbia River plume front. These show how internal waves can be generated from a river plume that flows as a gravity current into the coastal ocean. We find that the convergence of horizontal velocities at the plume front causes frontal growth and subsequent displacement downward of near-surface waters. Individual freely propagating waves are released from the river plume front when the front's propagation speed decreases below the wave speed in the water ahead of it. This mechanism generates internal waves of similar amplitude and steepness as internal waves from tide-topography interactions observed elsewhere, and is therefore important to the understanding of coastal ocean mixing.
19. Conjugate flows and amplitude bounds for internal solitary waves
Directory of Open Access Journals (Sweden)
N. I. Makarenko
2009-03-01
Full Text Available Amplitude bounds imposed by the conservation of mass, momentum and energy for strongly nonlinear waves in stratified fluid are considered. We discuss the theoretical scheme which allows to determine broadening limits for solitary waves in the terms of a given upstream density profile. Attention is focused on the continuously stratified flows having multiple broadening limits. The role of the mean density profile and the influence of fine-scale stratification are analyzed.
20. Gas sensing with surface acoustic wave devices
Science.gov (United States)
Martin, S. J.; Schweizer, K. S.; Ricco, A. J.; Zipperian, T. E.
1985-03-01
The use of a ZnO-on-Si surface acoustic wave (SAW) resonator as a gas sensor is discussed. In particular, the sensitivity of the device to organic vapors is examined. The planar nature of the SAW device, in which the acoustic energy is confined to within roughly one acoustic wavelength of the surface, makes the device extremely sensitive to surface perturbations. This characteristic has been exploited in the construction of SAW gas sensors in which the surface wave propagation characteristics are altered by species adsorbed from the ambient gas. The porous nature of the sputtered ZnO film, in conjunction with the microbalance capability of the SAW device, gives the sensor the ability to distinguish molecules on the basis of both size and mass.
1. Vibration of a single microcapsule with a hard plastic shell in an acoustic standing wave field.
Science.gov (United States)
Koyama, Daisuke; Kotera, Hironori; Kitazawa, Natsuko; Yoshida, Kenji; Nakamura, Kentaro; Watanabe, Yoshiaki
2011-04-01
Observation techniques for measuring the small vibration of a single microcapsule of tens of nanometers in an acoustic standing wave field are discussed. First, simultaneous optical observation of a microbubble vibration by two methods is investigated, using a high-speed video camera, which permits two-dimensional observation of the bubble vibration, and a laser Doppler vibrometer (LDV), which can observe small bubble vibration amplitudes at high frequency. Bubbles of tens of micrometers size were trapped at the antinode of an acoustic standing wave generated in an observational cell. Bubble vibration at 27 kHz could be observed and the experimental results for the two methods showed good agreement. The radial vibration of microcapsules with a hard plastic shell was observed using the LDV and the measurement of the capsule vibration with radial oscillation amplitude of tens of nanometers was successful. The acoustic radiation force acting on microcapsules in the acoustic standing wave was measured from the trapped position of the standing wave and the radial oscillation amplitude of the capsules was estimated from the theoretical equation of the acoustic radiation force, giving results in good agreement with the LDV measurements. The radial oscillation amplitude of a capsule was found to be proportional to the amplitude of the driving sound pressure. A larger expansion ratio was observed for capsules closer to the resonance condition under the same driving sound pressure and frequency.
2. Amplitude-dependent contraction/elongation of nonlinear Lamb waves
Science.gov (United States)
Packo, Pawel; Staszewski, Wieslaw J.; Uhl, Tadeusz; Leamy, Michael J.
2016-04-01
Nonlinear elastic guided waves find application in various disciplines of science and engineering, such as non- destructive testing and structural health monitoring. Recent recognition and quantification of their amplitude- dependent changes in spectral properties has contributed to the development of new monitoring concepts for mechanical structures. The focus of this work is to investigate and predict amplitude-dependent shifts in Lamb wave dispersion curves. The theory for frequency/wavenumber shifts for plate waves, based on a Lindstedt-Poincaré perturbation approach, was presented by the authors in previous years. Equivalently, spectral properties changes can be seen as wavelength contraction/elongation. Within the proposed framework, the wavelength of a Lamb wave depends on several factors; e.g., wave amplitude and second-, third- and fourth-order elastic constants, and others. Various types of nonlinear effects are considered in presented studies. Sensitivity studies for model parameters, i.e. higher-order elastic constants, are performed to quantify their influence on Lamb wave frequency/wavenumber shifting, and to identify the key parameters governing wavelength tuning.
3. Some Applications of Surface Acoustic Wave Sensors
Institute of Scientific and Technical Information of China (English)
2000-01-01
The paper describes the evaluation of thin amorphous magnetic film by using of surface acoustic waves on piezo electric substrate. The obtained experimental data show strong dependence of material parameters on the annealing temperature. The mixed ferromagnetic/SAW devices for electronic applications will be also discussed.
4. Abstract wave equations with acoustic boundary conditions
CERN Document Server
Mugnolo, Delio
2010-01-01
We define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of ${\\bf R}^n$. We prove a well-posedness result and develop a spectral theory which also allows to prove a conjecture proposed in (Gal-Goldstein-Goldstein, J. Evol. Equations 3 (2004), 623-636). Concrete problems are also discussed.
5. Dust-acoustic waves modulational instability and rogue waves in a polarized dusty plasma
Energy Technology Data Exchange (ETDEWEB)
Bouzit, Omar; Tribeche, Mouloud [Faculty of Physics, Theoretical Physics Laboratory, Plasma Physics Group, University of Bab-Ezzouar, USTHB, B.P. 32, El Alia, Algiers 16111 (Algeria)
2015-10-15
The polarization force-induced changes in the dust-acoustic waves (DAWs) modulational instability (MI) are examined. Using the reductive perturbation method, the nonlinear Schrödinger equation that governs the MI of the DAWs is obtained. It is found that the effect of the polarization term R is to narrow the wave number domain for the onset of instability. The amplitude of the wave envelope decreases as R increases, meaning that the polarization force effects render weaker the associated DA rogue waves. The latter may therefore completely damp in the vicinity of R ∼ 1, i.e., as the polarization force becomes close to the electrostatic one (the net force acting on the dust particles becomes vanishingly small). The DA rogue wave profile is very sensitive to any change in the restoring force acting on the dust particles. It turns out that the polarization effects may completely smear out the DA rogue waves.
6. Effect of excess superthermal hot electrons on finite amplitude ion-acoustic solitons and supersolitons in a magnetized auroral plasma
Energy Technology Data Exchange (ETDEWEB)
Rufai, O. R., E-mail: [email protected] [Council for Scientific and Industrial Research, Pretoria (South Africa); Bharuthram, R., E-mail: [email protected] [University of the Western Cape, Bellville (South Africa); Singh, S. V., E-mail: [email protected]; Lakhina, G. S., E-mail: [email protected] [Indian Institute of Geomagnetism, New Panvel (W), Navi, Mumbai-410218 (India)
2015-10-15
The effect of excess superthermal electrons is investigated on finite amplitude nonlinear ion-acoustic waves in a magnetized auroral plasma. The plasma model consists of a cold ion fluid, Boltzmann distribution of cool electrons, and kappa distributed hot electron species. The model predicts the evolution of negative potential solitons and supersolitons at subsonic Mach numbers region, whereas, in the case of Cairn's nonthermal distribution model for the hot electron species studied earlier, they can exist both in the subsonic and supersonic Mach number regimes. For the dayside auroral parameters, the model generates the super-acoustic electric field amplitude, speed, width, and pulse duration of about 18 mV/m, 25.4 km/s, 663 m, and 26 ms, respectively, which is in the range of the Viking spacecraft measurements.
7. Finite amplitude waves in two-dimensional lined ducts
Science.gov (United States)
Nayfeh, A. H.; Tsai, M.-S.
1974-01-01
A second-order uniform expansion is obtained for nonlinear wave propagation in a two-dimensional duct lined with a point-reacting acoustic material consisting of a porous sheet followed by honeycomb cavities and backed by the impervious wall of the duct. The waves in the duct are coupled with those in the porous sheet and the cavities. An analytical expression is obtained for the absorption coefficient in terms of the sound frequency, the physical properties of the porous sheet, and the geometrical parameters of the flow configuration. The results show that the nonlinearity flattens and broadens the absorption vs. frequency curve, irrespective of the geometrical dimensions or the porous material acoustic properties, in agreement with experimental observations.
8. Circuit Design of Surface Acoustic Wave Based Micro Force Sensor
Directory of Open Access Journals (Sweden)
Yuanyuan Li
2014-01-01
Full Text Available Pressure sensors are commonly used in industrial production and mechanical system. However, resistance strain, piezoresistive sensor, and ceramic capacitive pressure sensors possess limitations, especially in micro force measurement. A surface acoustic wave (SAW based micro force sensor is designed in this paper, which is based on the theories of wavelet transform, SAW detection, and pierce oscillator circuits. Using lithium niobate as the basal material, a mathematical model is established to analyze the frequency, and a peripheral circuit is designed to measure the micro force. The SAW based micro force sensor is tested to show the reasonable design of detection circuit and the stability of frequency and amplitude.
9. Quantum corrections to nonlinear ion acoustic wave with Landau damping
Energy Technology Data Exchange (ETDEWEB)
Mukherjee, Abhik; Janaki, M. S. [Saha Institute of Nuclear Physics, Calcutta (India); Bose, Anirban [Serampore College, West Bengal (India)
2014-07-15
Quantum corrections to nonlinear ion acoustic wave with Landau damping have been computed using Wigner equation approach. The dynamical equation governing the time development of nonlinear ion acoustic wave with semiclassical quantum corrections is shown to have the form of higher KdV equation which has higher order nonlinear terms coming from quantum corrections, with the usual classical and quantum corrected Landau damping integral terms. The conservation of total number of ions is shown from the evolution equation. The decay rate of KdV solitary wave amplitude due to the presence of Landau damping terms has been calculated assuming the Landau damping parameter α{sub 1}=√(m{sub e}/m{sub i}) to be of the same order of the quantum parameter Q=ℏ{sup 2}/(24m{sup 2}c{sub s}{sup 2}L{sup 2}). The amplitude is shown to decay very slowly with time as determined by the quantum factor Q.
10. Acoustic wave-equation-based earthquake location
Science.gov (United States)
Tong, Ping; Yang, Dinghui; Liu, Qinya; Yang, Xu; Harris, Jerry
2016-04-01
We present a novel earthquake location method using acoustic wave-equation-based traveltime inversion. The linear relationship between the location perturbation (δt0, δxs) and the resulting traveltime residual δt of a particular seismic phase, represented by the traveltime sensitivity kernel K(t0, xs) with respect to the earthquake location (t0, xs), is theoretically derived based on the adjoint method. Traveltime sensitivity kernel K(t0, xs) is formulated as a convolution between the forward and adjoint wavefields, which are calculated by numerically solving two acoustic wave equations. The advantage of this newly derived traveltime kernel is that it not only takes into account the earthquake-receiver geometry but also accurately honours the complexity of the velocity model. The earthquake location is obtained by solving a regularized least-squares problem. In 3-D realistic applications, it is computationally expensive to conduct full wave simulations. Therefore, we propose a 2.5-D approach which assumes the forward and adjoint wave simulations within a 2-D vertical plane passing through the earthquake and receiver. Various synthetic examples show the accuracy of this acoustic wave-equation-based earthquake location method. The accuracy and efficiency of the 2.5-D approach for 3-D earthquake location are further verified by its application to the 2004 Big Bear earthquake in Southern California.
11. Small amplitude ion-acoustic double layers with cold electron beam and q-nonextensive electrons
Energy Technology Data Exchange (ETDEWEB)
Ali Shan, S., E-mail: [email protected] [Theoretical Plasma Physics Division, PINSTECH, Nilore, 44000 Islamabad (Pakistan); National Centre for Physics (NCP), Shahdra Valley Road, 44000 Islamabad (Pakistan); Department of Mathematics and Applied Physics (DPAM), PIEAS, Islamabad (Pakistan); Saleem, H., E-mail: [email protected] [National Centre for Physics (NCP), Shahdra Valley Road, 44000 Islamabad (Pakistan); Department of Mathematics and Applied Physics (DPAM), PIEAS, Islamabad (Pakistan)
2014-02-01
Small amplitude ion-acoustic double layers in an unmagnetized and collisionless plasma consisting of cold positive ions, q-nonextensive electrons, and a cold electron beam are investigated. Small amplitude double layer solution is obtained by expanding the Sagdeev potential truncated method. The effects of entropic index q, speed and density of cold electron beam on double layer structures are discussed.
12. Nonextensive dust acoustic waves in a charge varying dusty plasma
Science.gov (United States)
Bacha, Mustapha; Tribeche, Mouloud
2012-01-01
Our recent analysis on nonlinear nonextensive dust-acoustic waves (DA) [Amour and Tribeche in Phys. Plasmas 17:063702, 2010] is extended to include self-consistent nonadiabatic grain charge fluctuation. The appropriate nonextensive electron charging current is rederived based on the orbit-limited motion theory. Our results reveal that the amplitude, strength and nature of the nonlinear DA waves (solitons and shocks) are extremely sensitive to the degree of ion nonextensivity. Stronger is the electron correlation, more important is the charge variation induced nonlinear wave damping. The anomalous dissipation effects may prevail over that dispersion as the electrons evolve far away from their Maxwellian equilibrium. Our investigation may be of wide relevance to astronomers and space scientists working on interstellar dusty plasmas where nonthermal distributions are turning out to be a very common and characteristic feature.
13. Extraordinary transmission of gigahertz surface acoustic waves
Science.gov (United States)
Mezil, Sylvain; Chonan, Kazuki; Otsuka, Paul H.; Tomoda, Motonobu; Matsuda, Osamu; Lee, Sam H.; Wright, Oliver B.
2016-09-01
Extraordinary transmission of waves, i.e. a transmission superior to the amount predicted by geometrical considerations of the aperture alone, has to date only been studied in the bulk. Here we present a new class of extraordinary transmission for waves confined in two dimensions to a flat surface. By means of acoustic numerical simulations in the gigahertz range, corresponding to acoustic wavelengths λ ~ 3–50 μm, we track the transmission of plane surface acoustic wave fronts between two silicon blocks joined by a deeply subwavelength bridge of variable length with or without an attached cavity. Several resonant modes of the structure, both one- and two-dimensional in nature, lead to extraordinary acoustic transmission, in this case with transmission efficiencies, i.e. intensity enhancements, up to ~23 and ~8 in the two respective cases. We show how the cavity shape and bridge size influence the extraordinary transmission efficiency. Applications include new metamaterials and subwavelength imaging.
14. Extraordinary transmission of gigahertz surface acoustic waves.
Science.gov (United States)
Mezil, Sylvain; Chonan, Kazuki; Otsuka, Paul H; Tomoda, Motonobu; Matsuda, Osamu; Lee, Sam H; Wright, Oliver B
2016-09-19
Extraordinary transmission of waves, i.e. a transmission superior to the amount predicted by geometrical considerations of the aperture alone, has to date only been studied in the bulk. Here we present a new class of extraordinary transmission for waves confined in two dimensions to a flat surface. By means of acoustic numerical simulations in the gigahertz range, corresponding to acoustic wavelengths λ ~ 3-50 μm, we track the transmission of plane surface acoustic wave fronts between two silicon blocks joined by a deeply subwavelength bridge of variable length with or without an attached cavity. Several resonant modes of the structure, both one- and two-dimensional in nature, lead to extraordinary acoustic transmission, in this case with transmission efficiencies, i.e. intensity enhancements, up to ~23 and ~8 in the two respective cases. We show how the cavity shape and bridge size influence the extraordinary transmission efficiency. Applications include new metamaterials and subwavelength imaging.
15. Extraordinary transmission of gigahertz surface acoustic waves
Science.gov (United States)
Mezil, Sylvain; Chonan, Kazuki; Otsuka, Paul H.; Tomoda, Motonobu; Matsuda, Osamu; Lee, Sam H.; Wright, Oliver B.
2016-01-01
Extraordinary transmission of waves, i.e. a transmission superior to the amount predicted by geometrical considerations of the aperture alone, has to date only been studied in the bulk. Here we present a new class of extraordinary transmission for waves confined in two dimensions to a flat surface. By means of acoustic numerical simulations in the gigahertz range, corresponding to acoustic wavelengths λ ~ 3–50 μm, we track the transmission of plane surface acoustic wave fronts between two silicon blocks joined by a deeply subwavelength bridge of variable length with or without an attached cavity. Several resonant modes of the structure, both one- and two-dimensional in nature, lead to extraordinary acoustic transmission, in this case with transmission efficiencies, i.e. intensity enhancements, up to ~23 and ~8 in the two respective cases. We show how the cavity shape and bridge size influence the extraordinary transmission efficiency. Applications include new metamaterials and subwavelength imaging. PMID:27640998
16. Ion acoustic solitary waves in plasmas with nonextensive electrons, Boltzmann positrons and relativistic thermal ions
Science.gov (United States)
Hafez, M. G.; Talukder, M. R.
2015-09-01
This work investigates the theoretical and numerical studies on nonlinear propagation of ion acoustic solitary waves (IASWs) in an unmagnetized plasma consisting of nonextensive electrons, Boltzmann positrons and relativistic thermal ions. The Korteweg-de Vries (KdV) equation is derived by using the well known reductive perturbation method. This equation admits the soliton like solitary wave solution. The effects of phase velocity, amplitude of soliton, width of soliton and electrostatic nonlinear propagation of weakly relativistic ion-acoustic solitary waves have been discussed with graphical representation found in the variation of the plasma parameters. The obtained results can be helpful in understanding the features of small but finite amplitude localized relativistic ion-acoustic waves for an unmagnetized three component plasma system in astrophysical compact objects.
17. Absorption of surface acoustic waves by graphene
Directory of Open Access Journals (Sweden)
S. H. Zhang
2011-06-01
Full Text Available We present a theoretical study on interactions of electrons in graphene with surface acoustic waves (SAWs. We find that owing to momentum and energy conservation laws, the electronic transition accompanied by the SAW absorption cannot be achieved via inter-band transition channels in graphene. For graphene, strong absorption of SAWs can be observed in a wide frequency range up to terahertz at room temperature. The intensity of SAW absorption by graphene depends strongly on temperature and can be adjusted by changing the carrier density. This study is relevant to the exploration of the acoustic properties of graphene and to the application of graphene as frequency-tunable SAW devices.
18. Acoustic Remote Sensing of Rogue Waves
Science.gov (United States)
2016-04-01
We propose an early warning system for approaching rogue waves using the remote sensing of acoustic-gravity waves (AGWs) - progressive sound waves that propagate at the speed of sound in the ocean. It is believed that AGWs are generated during the formation of rogue waves, carrying information on the rogue waves at near the speed of sound, i.e. much faster than the rogue wave. The capability of identifying those special sound waves would enable detecting rogue waves most efficiently. A lot of promising work has been reported on AGWs in the last few years, part of which in the context of remote sensing as an early detection of tsunami. However, to our knowledge none of the work addresses the problem of rogue waves directly. Although there remains some uncertainty as to the proper definition of a rogue wave, there is little doubt that they exist and no one can dispute the potential destructive power of rogue waves. An early warning system for such extreme waves would become a demanding safety technology. A closed form expression was developed for the pressure induced by an impulsive source at the free surface (the Green's function) from which the solution for more general sources can be developed. In particular, we used the model of the Draupner Wave of January 1st, 1995 as a source and calculated the induced AGW signature. In particular we studied the AGW signature associated with a special feature of this wave, and characteristic of rogue waves, of the absence of any local set-down beneath the main crest and the presence of a large local set-up.
19. Large-amplitude ULF waves at high latitudes
Science.gov (United States)
Guido, T.; Tulegenov, B.; Streltsov, A. V.
2014-11-01
We present results from the statistical study of ULF waves detected by the fluxgate magnetometer in Gakona, Alaska during several experimental campaigns conducted at the High Frequency Active Auroral Research Program (HAARP) facility in years 2011-2013. We analyzed frequencies of ULF waves recorded during 26 strongly disturbed geomagnetic events (substorms) and compared them with frequencies of ULF waves detected during magnetically quiet times. Our analysis demonstrates that the frequency of the waves carrying most of the power in almost all these events is less than 1 mHz. We also analyzed data from the ACE satellite, measuring parameters of the solar wind in the L1 Lagrangian point between Earth and Sun, and found that in several occasions there is a strong correlation between oscillations of the magnetic field in the solar wind and oscillations detected on the ground. We also found several cases when there is no correlation between signals detected on ACE and on the ground. This finding suggests that these frequencies correspond to the fundamental eigenfrequency of the coupled magnetosphere-ionosphere system, and the amplitude of these waves can reach significant magnitude when the system is driven by the external driver (for example, the solar wind) with this particular frequency. When the frequency of the driver does not match the frequency of the system, the waves still are observed, but their amplitudes are much smaller.
20. Acoustic measurements above a plate carrying Lamb waves
CERN Document Server
Talberg, Andreas Sørbrøden
2016-01-01
This article presents a set of acoustic measurements conducted on the Statoil funded Behind Casing Logging Set-Up, designed by SINTEF Petroleum Research to resemble an oil well casing. A set of simple simulations using COMSOL Multiphysics were also conducted and the results compared with the measurements. The experiments consists of measuring the pressure wave radiated of a set of Lamb waves propagating in a 3 mm thick steel plate, using the so called pitch-catch method. The Lamb waves were excited by a broadband piezoelectric immersion transducer with center frequency of 1 MHz. Through measurements and analysis the group velocity of the fastest mode in the plate was found to be 3138.5 m/s. Measuring the wave radiated into the water in a grid consisting of 8x33 measuring points, the spreading of the plate wave normal to the direction of propagation was investigated. Comparing the point where the amplitude had decreased 50 % relative to the amplitude measured at the axis pointing straight forward from the tran...
1. Acoustoelectric effects in reflection of leaky-wave-radiated bulk acoustic waves from piezoelectric crystal-conductive liquid interface.
Science.gov (United States)
Rimeika, Romualdas; Čiplys, Daumantas; Jonkus, Vytautas; Shur, Michael
2016-01-01
The leaky surface acoustic wave (SAW) propagating along X-axis of Y-cut lithium tantalate crystal strongly radiates energy in the form of an obliquely propagating narrow bulk acoustic wave (BAW) beam. The reflection of this beam from the crystal-liquid interface has been investigated. The test liquids were solutions of potassium nitrate in distilled water and of lithium chloride in isopropyl alcohol with the conductivity varied by changing the solution concentration. The strong dependences of the reflected wave amplitude and phase on the liquid conductivity were observed and explained by the acoustoelectric interaction in the wave reflection region. The novel configuration of an acoustic sensor for liquid media featuring important advantages of separate measuring and sensing surfaces and rigid structure has been proposed. The application of leaky-SAW radiated bulk waves for identification of different brands of mineral water has been demonstrated.
2. Experimental study of nonlinear dust acoustic solitary waves in a dusty plasma
CERN Document Server
Bandyopadhyay, P; Sen, A; Kaw, P K
2008-01-01
The excitation and propagation of finite amplitude low frequency solitary waves are investigated in an Argon plasma impregnated with kaolin dust particles. A nonlinear longitudinal dust acoustic solitary wave is excited by pulse modulating the discharge voltage with a negative potential. It is found that the velocity of the solitary wave increases and the width decreases with the increase of the modulating voltage, but the product of the solitary wave amplitude and the square of the width remains nearly constant. The experimental findings are compared with analytic soliton solutions of a model Kortweg-de Vries equation.
3. Finite element approach analysis for characteristics of electromagnetic acoustic Lamb wave
Science.gov (United States)
Chen, Xiaoming; Li, Songsong
2016-04-01
The electromagnetic acoustic Lamb wave, with the advantages of quickly detecting the defect and sensitivity to the defects, is widely used in non-destructive testing of thin sheet. In this paper, the directivity of sound field, Phase velocity, group velocity and particle displacement amplitude of Lamb wave are study based on finite element analysis method. The results show that, for 1mm aluminum, when the excitation frequency 0.64MHz, the displacement amplitude of A0 mode is minimum, and the displacement amplitude S0 mode is largest. Appropriate to increase the displacement amplitude of a mode, while reducing displacement amplitude of another mode, to achieve the excitation of a single mode Lamb wave. It is helpful to the Optimization of transducer parameters, the choice of Lamb wave modes and providing optimal excitation frequency.
4. NEAR-FIELD ACOUSTIC HOLOGRAPHY FOR SEMI-FREE ACOUSTIC FIELD BASED ON WAVE SUPERPOSITION APPROACH
Institute of Scientific and Technical Information of China (English)
LI Weibing; CHEN Jian; YU Fei; CHEN Xinzhao
2006-01-01
In the semi-free acoustic field, the actual acoustic pressure at any point is composed of two parts: The direct acoustic pressure and the reflected acoustic pressure. The general acoustic holographic theories and algorithms request that there is only the direct acoustic pressure contained in the pressure at any point on the hologram surface, consequently, they cannot be used to reconstruct acoustic source and predict acoustic field directly. To take the reflected pressure into consideration, near-field acoustic holography for semi-free acoustic field based on wave superposition approach is proposed to realize the holographic reconstruction and prediction of the semi-free acoustic field, and the wave superposition approach is adopted as a holographic transform algorithm. The proposed theory and algorithm are realized and verified with a numerical example,and the drawbacks of the general theories and algorithms in the holographic reconstruction and prediction of the semi-free acoustic field are also demonstrated by this numerical example.
5. Resonant surface acoustic wave chemical detector
Energy Technology Data Exchange (ETDEWEB)
Brocato, Robert W.; Brocato, Terisse; Stotts, Larry G.
2017-08-08
Apparatus for chemical detection includes a pair of interdigitated transducers (IDTs) formed on a piezoelectric substrate. The apparatus includes a layer of adsorptive material deposited on a surface of the piezoelectric substrate between the IDTs, where each IDT is conformed, and is dimensioned in relation to an operating frequency and an acoustic velocity of the piezoelectric substrate, so as to function as a single-phase uni-directional transducer (SPUDT) at the operating frequency. Additionally, the apparatus includes the pair of IDTs is spaced apart along a propagation axis and mutually aligned relative to said propagation axis so as to define an acoustic cavity that is resonant to surface acoustic waves (SAWs) at the operating frequency, where a distance between each IDT of the pair of IDTs ranges from 100 wavelength of the operating frequency to 400 wavelength of the operating frequency.
6. Synchronization of the dust acoustic wave under microgravity
Science.gov (United States)
Ruhunusiri, W. D. Suranga; Goree, J.
2013-10-01
Synchronization is a nonlinear phenomenon where a self-excited oscillation, like a wave in a plasma, interacts with an external driving, resulting in an adjustment of the oscillation frequency. To prepare for experiments under microgravity conditions using the PK-4 facility on the International Space Station, we perform a laboratory experiment to observe synchronization of the self-excited dust acoustic wave. An rf glow discharge argon plasma is formed by applying a low power radio frequency voltage to a lower electrode. A 3D dust cloud is formed by levitating 4.83 micron microspheres inside a glass box placed on the lower electrode. The dust acoustic wave is self-excited with a natural frequency of 22 Hz due to an ion streaming instability. A cross section of the dust cloud is illuminated by a vertical laser sheet and imaged from the side with a digital camera. To synchronize the wave, we sinusoidally modulate the overall ion density. Differently from previous experiments, we use a driving electrode that is separate from the electrode that sustains the plasma, and we characterize synchronization by varying both driving amplitude and frequency. Supported by NASA's Physical Science Research Program.
7. Producing Acoustic 'Frozen Waves': Simulated experiments
CERN Document Server
Prego, Jose' L; Recami, Erasmo; Hernandez-Figueroa, Hugo E
2012-01-01
In this paper we show how appropriate superpositions of Bessel beams can be successfully used to obtain arbitrary longitudinal intensity patterns of nondiffracting ultrasonic wavefields with very high transverse localization. More precisely, the method here described allows generating longitudinal acoustic pressure fields, whose longitudinal intensity patterns can assume, in principle, any desired shape within a freely chosen interval 0wave propagates). Indeed, it is here demonstrated by computer evaluations that these very special beams of non-attenuated ultrasonic fields can be generated in water by means of annular transducers. Such fields "at rest" have been called by us Acoustic Frozen Waves(FW). The paper presents various cases of FWs in water, and investigates the characteristics of their aperture, such as minimum required size and ring dimensioning, as well as the influence...
8. Amplitude Distribution of Emission Wave for Cracking Process
Directory of Open Access Journals (Sweden)
Shahidan Shahiron
2016-01-01
Full Text Available Acoustic emission technique is a method of assessment for structural health monitoring system. This technique is an effective tool for the evaluation of any system without destroying the material conditions. It enables early crack detections and has very high sensitivity to crack growth. The crack patterns in concrete beam have been identified according to the type of cracking process and the crack classifications using the AE data parameters are mainly based on the AE amplitude, rise time, and average frequency. These data parameters have been analysed using statistical methods of b-value analysis. This research paper will mainly focus on the utilization of statistical b-value analysis in evaluating the emission amplitude distribution of concrete beams. The beam specimens (150 × 250 × 1900 mm were prepared in the laboratory system and tested with the four point bending test using cyclic loading together with acoustic emission monitoring system. The results showed that this statistical analysis is promising in determining the cracking process in concrete beams.
9. On the Turbulence Beneath Finite Amplitude Water Waves
CERN Document Server
Babanin, Alexander V
2015-01-01
The paper by Beya et al. (2012, hereinafter BPB) has a general title of Turbulence Beneath Finite Amplitude Water Waves, but is solely dedicated to discussing the experiment by Babanin and Haus (2009, hereinafter BH) who conducted measurements of wave-induced non-breaking turbulence by particle image velocimetry (PIV). The authors of BPB conclude that their observations contradict those of BH. Here we argue that the outcomes of BPB do not contradict BH. In addition, although the main conclusion of BPB is that there is no turbulence observed in their experiment, it actually is observed.
10. Photoinduced Enhancement of the Charge Density Wave Amplitude
Science.gov (United States)
Singer, A.; Patel, S. K. K.; Kukreja, R.; Uhlíř, V.; Wingert, J.; Festersen, S.; Zhu, D.; Glownia, J. M.; Lemke, H. T.; Nelson, S.; Kozina, M.; Rossnagel, K.; Bauer, M.; Murphy, B. M.; Magnussen, O. M.; Fullerton, E. E.; Shpyrko, O. G.
2016-07-01
Symmetry breaking and the emergence of order is one of the most fascinating phenomena in condensed matter physics. It leads to a plethora of intriguing ground states found in antiferromagnets, Mott insulators, superconductors, and density-wave systems. Exploiting states of matter far from equilibrium can provide even more striking routes to symmetry-lowered, ordered states. Here, we demonstrate for the case of elemental chromium that moderate ultrafast photoexcitation can transiently enhance the charge-density-wave (CDW) amplitude by up to 30% above its equilibrium value, while strong excitations lead to an oscillating, large-amplitude CDW state that persists above the equilibrium transition temperature. Both effects result from dynamic electron-phonon interactions, providing an efficient mechanism to selectively transform a broad excitation of the electronic order into a well-defined, long-lived coherent lattice vibration. This mechanism may be exploited to transiently enhance order parameters in other systems with coupled degrees of freedom.
11. Amplitude or Higgs modes in d-wave superconductors
Science.gov (United States)
Barlas, Yafis; Varma, C. M.
2013-02-01
In Lorentz-invariant systems spontaneously broken gauge symmetry results in three types of fundamental excitations: density excitations, Higgs bosons (amplitude modes), and Goldstone bosons (phase modes). The density and phase modes are coupled by electromagnetic interactions while the amplitude modes are not. In s-wave superconductors, the Higgs mode, which can be observed only under special conditions, has been detected. We show that unconventional d-wave superconductors, such as the high-temperature cuprate superconductors, should have a rich assortment of Higgs bosons, each in a different irreducible representation of the point-group symmetry of the lattice. We also show that these modes have a characteristic singular spectral structure and discuss conditions for their observability.
12. Wireless Multiplexed Surface Acoustic Wave Sensors Project
Science.gov (United States)
Youngquist, Robert C.
2014-01-01
Wireless Surface Acoustic Wave (SAW) Sensor is a new technology for obtaining multiple, real-time measurements under extreme environmental conditions. This project plans to develop a wireless multiplexed sensor system that uses SAW sensors, with no batteries or semiconductors, that are passive and rugged, can operate down to cryogenic temperatures and up to hundreds of degrees C, and can be used to sense a wide variety of parameters over reasonable distances (meters).
13. Acoustic measurements of a liquefied cohesive sediment bed under waves
Science.gov (United States)
Mosquera, R.; Groposo, V.; Pedocchi, F.
2014-04-01
In this article the response of a cohesive sediment deposit under the action of water waves is studied with the help of laboratory experiments and an analytical model. Under the same regular wave condition three different bed responses were observed depending on the degree of consolidation of the deposit: no bed motion, bed motion of the upper layer after the action of the first waves, and massive bed motion after several waves. The kinematic of the upper 3 cm of the deposit were measured with an ultrasound acoustic profiler, while the pore-water pressure inside the bed was simultaneously measured using several pore pressure sensors. A poro-elastic model was developed to interpret the experimental observations. The model showed that the amplitude of the shear stress increased down into the bed. Then it is possible that the lower layers of the deposit experience plastic deformations, while the upper layers present just elastic deformations. Since plastic deformations in the lower layers are necessary for pore pressure build-up, the analytical model was used to interpret the experimental results and to state that liquefaction of a self consolidated cohesive sediment bed would only occur if the bed yield stress falls within the range defined by the amplitude of the shear stress inside the bed.
14. Finite-amplitude steady waves in plane viscous shear flows
Science.gov (United States)
Milinazzo, F. A.; Saffman, P. G.
1985-01-01
Computations of two-dimensional solutions of the Navier-Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.
15. The density stratification and amplitude dispersion of internal waves
Science.gov (United States)
Makarenko, N.; Ulanova, E.
2012-04-01
We consider the theoretical model of large amplitude internal solitary waves propagating in a weakly stratified fluid under gravity. It is well known that steady 2D Euler equations of non-homogeneous fluid reduce in this case to the second-order quasi-linear equation for a stream function (the Dubreil-Jacotin-Long equation). Subsequently, the shape of traveling solitary wave can be determined in the long-wave scaling limit by solving the dispersive KdV-type model equation. The non-linear terms of this equation depend considerably on the instantaneous fine-scale density profile formed over background linear- or exponential stratification (Benney&Ko, 1978; Borisov&Derzho 1990; Derzho&Grimshaw 1997; Makarenko, 1999; Makarenko, Maltseva and Kazakov, 2009). Now we derive and analyze Fredholm-type integral equations coupling immediately the fluid density coefficient with the dispersion function for internal solitary waves. The inverse problem which means to find the fine-scale density by known curve of the amplitude dispersion is discussed in more details.
16. Effects of external magnetic field on oblique propagation of ion acoustic cnoidal wave in nonextensive plasma
Science.gov (United States)
2017-01-01
Effects of the obliqueness and the strength of external magnetic field on the ion acoustic (IA) cnoidal wave in a nonextensive plasma are investigated. The reductive perturbation method is employed to derive the corresponding KdV equation for the IA wave. Sagdeev potential is extracted, and the condition of generation of IA waves in the form of cnoidal waves or solitons is discussed in detail. In this work, the domain of allowable values of nonextensivity parameter q for generation of the IA cnoidal wave in the plasma medium is considered. The results show that only the compressive IA wave may generate and propagate in the plasma medium. Increasing the strength of external magnetic field will increase the frequency of the wave and decrease its amplitude, while increasing the angle of propagation will decrease the frequency of the wave and increase its amplitude.
17. Large amplitude solitary waves in ion-beam plasmas with charged dust impurities
CERN Document Server
Misra, A P
2011-01-01
The nonlinear propagation of large amplitude dust ion-acoustic (DIA) solitary waves (SWs) in an ion-beam plasma with stationary charged dusts is investigated. For typical plasma parameters relevant for experiments [J. Plasma Phys. \\textbf{60}, 69 (1998)], when the beam speed is larger than the DIA speed ($v_{b0}\\gtrsim1.7c_s$), three stable waves, namely the "fast" and "slow" ion-beam modes and the plasma DIA wave are shown to exist. These modes can propagate as SWs in the beam plasmas. However, in the other regime ($c_s0)$ is found to be limited by a critical value which typically depends on $M$, $v_{b0}$ as well as the ion/beam temperature. The conditions for the existence of DIA solitons are obtained and their properties are analyzed numerically in terms of the system parameters. While the system supports both the compressive and rarefactive large amplitude SWs, the small amplitude solitons exist only of the compressive type. The theoretical results may be useful for observation of soliton excitations in l...
18. Twisted electron-acoustic waves in plasmas
Science.gov (United States)
Aman-ur-Rehman, Ali, S.; Khan, S. A.; Shahzad, K.
2016-08-01
In the paraxial limit, a twisted electron-acoustic (EA) wave is studied in a collisionless unmagnetized plasma, whose constituents are the dynamical cold electrons and Boltzmannian hot electrons in the background of static positive ions. The analytical and numerical solutions of the plasma kinetic equation suggest that EA waves with finite amount of orbital angular momentum exhibit a twist in its behavior. The twisted wave particle resonance is also taken into consideration that has been appeared through the effective wave number qeff accounting for Laguerre-Gaussian mode profiles attributed to helical phase structures. Consequently, the dispersion relation and the damping rate of the EA waves are significantly modified with the twisted parameter η, and for η → ∞, the results coincide with the straight propagating plane EA waves. Numerically, new features of twisted EA waves are identified by considering various regimes of wavelength and the results might be useful for transport and trapping of plasma particles in a two-electron component plasma.
19. Quality factor due to roughness scattering of shear horizontal surface acoustic waves in nanoresonators
NARCIS (Netherlands)
Palasantzas, G.
2008-01-01
In this work we study the quality factor associated with dissipation due to scattering of shear horizontal surface acoustic waves by random self-affine roughness. It is shown that the quality factor is strongly influenced by both the surface roughness exponent H and the roughness amplitude w to late
20. Measuring Acoustic Wave Transit Time in Furnace Based on Active Acoustic Source Signal
Institute of Scientific and Technical Information of China (English)
Zhen Luo; Feng Tian; Xiao-Ping Sun
2007-01-01
Accurate measurement of transit time for acoustic wave between two sensors installed on two sides of a furnace is a key to implementing the temperature field measurement technique based on acoustical method. A new method for measuring transit time of acoustic wave based on active acoustic source signal is proposed in this paper, which includes the followings: the time when the acoustic source signal arrives at the two sensors is measured first; then, the difference of two arriving time arguments is computed, thereby we get the transit time of the acoustic wave between two sensors installed on the two sides of the furnace. Avoiding the restriction on acoustic source signal and background noise, the new method can get the transit time of acoustic wave with higher precision and stronger ability of resisting noise interference.
1. Relationship between the Amplitude and Phase of a Signal Scattered by a Point-Like Acoustic Inhomogeneity
Science.gov (United States)
Burov, V. A.; Morozov, S. A.
2001-11-01
Wave scattering by a point-like inhomogeneity, i.e., a strong inhomogeneity with infinitesimal dimensions, is described. This type of inhomogeneity model is used in investigating the point-spread functions of different algorithms and systems. Two approaches are used to derive the rigorous relationship between the amplitude and phase of a signal scattered by a point-like acoustic inhomogeneity. The first approach is based on a Marchenko-type equation. The second approach uses the scattering by a scatterer whose size decreases simultaneously with an increase in its contrast. It is shown that the retarded and advanced waves are scattered differently despite the relationship between the phases of the corresponding scattered waves.
2. Phase Aberration and Attenuation Effects on Acoustic Radiation Force-Based Shear Wave Generation.
Science.gov (United States)
Carrascal, Carolina Amador; Aristizabal, Sara; Greenleaf, James F; Urban, Matthew W
2016-02-01
Elasticity is measured by shear wave elasticity imaging (SWEI) methods using acoustic radiation force to create the shear waves. Phase aberration and tissue attenuation can hamper the generation of shear waves for in vivo applications. In this study, the effects of phase aberration and attenuation in ultrasound focusing for creating shear waves were explored. This includes the effects of phase shifts and amplitude attenuation on shear wave characteristics such as shear wave amplitude, shear wave speed, shear wave center frequency, and bandwidth. Two samples of swine belly tissue were used to create phase aberration and attenuation experimentally. To explore the phase aberration and attenuation effects individually, tissue experiments were complemented with ultrasound beam simulations using fast object-oriented C++ ultrasound simulator (FOCUS) and shear wave simulations using finite-element-model (FEM) analysis. The ultrasound frequency used to generate shear waves was varied from 3.0 to 4.5 MHz. Results: The measured acoustic pressure and resulting shear wave amplitude decreased approximately 40%-90% with the introduction of the tissue samples. Acoustic intensity and shear wave displacement were correlated for both tissue samples, and the resulting Pearson's correlation coefficients were 0.99 and 0.97. Analysis of shear wave generation with tissue samples (phase aberration and attenuation case), measured phase screen, (only phase aberration case), and FOCUS/FEM model (only attenuation case) showed that tissue attenuation affected the shear wave generation more than tissue aberration. Decreasing the ultrasound frequency helped maintain a focused beam for creation of shear waves in the presence of both phase aberration and attenuation.
3. Measuring finite-frequency body-wave amplitudes and traveltimes
Science.gov (United States)
Sigloch, Karin; Nolet, Guust
2006-10-01
We have developed a method to measure finite-frequency amplitude and traveltime anomalies of teleseismic P waves. We use a matched filtering approach that models the first 25 s of a seismogram after the P arrival, which includes the depth phases pP and sP. Given a set of broad-band seismograms from a teleseismic event, we compute synthetic Green's functions using published moment tensor solutions. We jointly deconvolve global or regional sets of seismograms with their Green's functions to obtain the broad-band source time function. The matched filter of a seismogram is the convolution of the Green's function with the source time function. Traveltimes are computed by cross-correlating each seismogram with its matched filter. Amplitude anomalies are defined as the multiplicative factors that minimize the RMS misfit between matched filters and data. The procedure is implemented in an iterative fashion, which allows for joint inversion for the source time function, amplitudes, and a correction to the moment tensor. Cluster analysis is used to identify azimuthally distinct groups of seismograms when source effects with azimuthal dependence are prominent. We then invert for one source time function per group. We implement this inversion for a range of source depths to determine the most likely depth, as indicated by the overall RMS misfit, and by the non-negativity and compactness of the source time function. Finite-frequency measurements are obtained by filtering broad-band data and matched filters through a bank of passband filters. The method is validated on a set of 15 events of magnitude 5.8 to 6.9. Our focus is on the densely instrumented Western US. Quasi-duplet events (quplets') are used to estimate measurement uncertainty on real data. Robust results are achieved for wave periods between 24 and 2 s. Traveltime dispersion is on the order of 0.5 s. Amplitude anomalies are on the order of 1 db in the lowest bands and 3 db in the highest bands, corresponding to
4. Characterization of wave physics in acoustic metamaterials using a fiber optic point detector
Science.gov (United States)
Ganye, Randy; Chen, Yongyao; Liu, Haijun; Bae, Hyungdae; Wen, Zhongshan; Yu, Miao
2016-06-01
Due to limitations of conventional acoustic probes, full spatial field mapping (both internal and external wave amplitude and phase measurements) in acoustic metamaterials with deep subwavelength structures has not yet been demonstrated. Therefore, many fundamental wave propagation phenomena in acoustic metamaterials remain experimentally unexplored. In this work, we realized a miniature fiber optic acoustic point detector that is capable of omnidirectional detection of complex spatial acoustic fields in various metamaterial structures over a broadband spectrum. By using this probe, we experimentally characterized the wave-structure interactions in an anisotropic metamaterial waveguide. We further demonstrated that the spatial mapping of both internal and external acoustic fields of metamaterial structures can help obtain important wave propagation properties associated with material dispersion and field confinement, and develop an in-depth understanding of the waveguiding physics in metamaterials. The insights and inspirations gained from our experimental studies are valuable not only for the advancement of fundamental metamaterial wave physics but also for the development of functional metamaterial devices such as acoustic lenses, waveguides, and sensors.
5. Influence of viscoelasticity and interfacial slip on acoustic wave sensors
OpenAIRE
McHale, G; Lucklum, R.; Newton, MI; Cowen, JA
2000-01-01
Acoustic wave devices with shear horizontal displacements, such as quartz crystal microbalances (QCM) and shear horizontally polarised surface acoustic wave (SH-SAW) devices provide sensitive probes of changes at solid-solid and solid- liquid interfaces. Increasingly the surfaces of acoustic wave devices are being chemically or physically modified to alter surface adhesion or coated with one or more layers to amplify their response to any change of mass or material properties. In this work, w...
6. Numerical studies of vertically propagating acoustic and magneto-acoustic waves in an isothermal atmosphere
Directory of Open Access Journals (Sweden)
H. Y. Alkahby
1999-12-01
Full Text Available In this paper we investigate numerically the effect of viscosity and Newtonian cooling on upward and downward propagating magneto-acoustic waves, resulting from a uniform horizontal magnetic field in an isothermal atmosphere. The results of the numerical computations are compared with those of asymptotic evaluations. It is shown that the presence of a small viscosity creates a layer which acts like an absorbing and reflecting barrier for waves generated below it and that the presence of the magnetic field produces a reflecting layer only. The addition of Newtonian cooling affects mainly the lower region in which it produces waves attenuation and alters the wavelength. If the Newtonian cooling coefficient is large compared with the frequency of the waves, the temperature in the lower region evens out and the wave motion approaches an isothermal one. This eliminates the attenuation in the wave amplitude since the isothermal region is dissipationless. This problem is solved analytically and numerically. The results of the numerical computation are in a complete agreement with the analytical results.
7. Thermally induced acoustic waves in porous silicon
Energy Technology Data Exchange (ETDEWEB)
Gavrilchenko, Iryna V.; Shulimov, Yuriy G.; Skryshevsky, Valeriy A. [Radiophysics Department, Kyiv National Taras Shevchenko University, Kyiv (Ukraine); Benilov, Arthur I. [Radiophysics Department, Kyiv National Taras Shevchenko University, Kyiv (Ukraine); Laboratoire d' Electronique, Optoelectronique et Microsystemes, Ecole Centrale de Lyon, Ecully (France)
2009-07-15
Thermally induced acoustic waves in structures with porous silicon have been studied. Two different schemas of acoustic phenomena recording are compared: in the first one a signal from microphone was measured as function of output frequency, in second one the resistance of porous silicon was measured using Wheatstone bridge. For both methods, the resonance peak is situated in same frequencies depending on difference in thermal properties between porous silicon and c-Si as well as geometry of studied structures. 1.0 kHz shifting of resonance peak in saturated alcohol vapors comparing to ambient air is observed. It can be applied as new transducer for chemical sensors based on porous silicon. (copyright 2009 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
8. Excitation and evolution of finite-amplitude plasma wave
Energy Technology Data Exchange (ETDEWEB)
Hou, Y. W.; Wu, Y. C., E-mail: [email protected] [Key Laboratory of Neutronics and Radiation Safety, Institute of Nuclear Energy Safety Technology, Chinese Academy of Sciences, Hefei, Anhui 230031 (China); Chen, M. X. [School of Electronic Science and Applied Physics, Hefei University of Technology, Hefei, Anhui 230009 (China); Yu, M. Y., E-mail: [email protected] [Institute for Fusion Theory and Simulation and Department of Physics, Zhejiang University, Hangzhou 310027 (China); Institute for Theoretical Physics I, Ruhr University, D-44780 Bochum (Germany); Wu, B. [Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, Anhui 230031 (China)
2015-12-15
The evolution of a small spatially periodic perturbation in the electron velocity distribution function in collisionless plasma is reconsidered by numerically solving the Vlasov and Poisson equations. The short as well as long time behaviors of the excited oscillations and damping/modulation are followed. In the small but finite-amplitude excited plasma wave, resonant electrons become trapped in the wave potential wells and their motion affects the low-velocity electrons participating in the plasma oscillations, leading to modulation of the latter at an effective trapping frequency. It is found that the phase space of the resonant and low-velocity electrons becomes chaotic, but then self-organization takes place but remains fine-scale chaotic. It is also found that as long as particles are trapped, there is only modulation and no monotonic damping of the excited plasma wave. The modulation period/amplitude increases/decreases as the magnitude of the initial disturbance is reduced. For the initial and boundary conditions used here, linear Landau damping corresponds to the asymptotic limit of the modulation period becoming infinite, or no trapping of the resonant electrons.
9. Acoustic clouds: standing sound waves around a black hole analogue
CERN Document Server
Benone, Carolina L; Herdeiro, Carlos; Radu, Eugen
2014-01-01
Under certain conditions sound waves in fluids experience an acoustic horizon with analogue properties to those of a black hole event horizon. In particular, a draining bathtub-like model can give rise to a rotating acoustic horizon and hence a rotating black hole (acoustic) analogue. We show that sound waves, when enclosed in a cylindrical cavity, can form stationary waves around such rotating acoustic black holes. These acoustic perturbations display similar properties to the scalar clouds that have been studied around Kerr and Kerr-Newman black holes; thus they are dubbed acoustic clouds. We make the comparison between scalar clouds around Kerr black holes and acoustic clouds around the draining bathtub explicit by studying also the properties of scalar clouds around Kerr black holes enclosed in a cavity. Acoustic clouds suggest the possibility of testing, experimentally, the existence and properties of black hole clouds, using analog models.
10. Arbitrary amplitude electrostatic wave propagation in a magnetized dense plasma containing helium ions and degenerate electrons
Science.gov (United States)
Mahmood, S.; Sadiq, Safeer; Haque, Q.; Ali, Munazza Z.
2016-06-01
The obliquely propagating arbitrary amplitude electrostatic wave is studied in a dense magnetized plasma having singly and doubly charged helium ions with nonrelativistic and ultrarelativistic degenerate electrons pressures. The Fermi temperature for ultrarelativistic degenerate electrons described by N. M. Vernet [(Cambridge University Press, Cambridge, 2007), p. 57] is used to define ion acoustic speed in ultra-dense plasmas. The pseudo-potential approach is used to solve the fully nonlinear set of dynamic equations for obliquely propagating electrostatic waves in a dense magnetized plasma containing helium ions. The upper and lower Mach number ranges for the existence of electrostatic solitons are found which depends on the obliqueness of the wave propagation with respect to applied magnetic field and charge number of the helium ions. It is found that only compressive (hump) soliton structures are formed in all the cases and only subsonic solitons are formed for a singly charged helium ions plasma case with nonrelativistic degenerate electrons. Both subsonic and supersonic soliton hump structures are formed for doubly charged helium ions with nonrelativistic degenerate electrons and ultrarelativistic degenerate electrons plasma case containing singly as well as doubly charged helium ions. The effect of propagation direction on the soliton amplitude and width of the electrostatic waves is also presented. The numerical plots are also shown for illustration using dense plasma parameters of a compact star (white dwarf) from literature.
11. Simulation and Optimization of Surface Acoustic Wave Devises
DEFF Research Database (Denmark)
Dühring, Maria Bayard
2007-01-01
In this paper a method to model the interaction of the mechanical field from a surface acoustic wave and the optical field in the waveguides of a Mach-Zehnder interferometer is presented. The surface acoustic waves are generated by interdigital transducers using a plane strain model of a piezoele......In this paper a method to model the interaction of the mechanical field from a surface acoustic wave and the optical field in the waveguides of a Mach-Zehnder interferometer is presented. The surface acoustic waves are generated by interdigital transducers using a plane strain model...
12. A Schamel equation for ion acoustic waves in superthermal plasmas
Energy Technology Data Exchange (ETDEWEB)
Williams, G., E-mail: [email protected]; Kourakis, I. [Centre for Plasma Physics, Department of Physics and Astronomy, Queen' s University Belfast, BT7 1NN, Northern Ireland (United Kingdom); Verheest, F. [Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, B-9000 Gent (Belgium); School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4000 (South Africa); Hellberg, M. A. [School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4000 (South Africa); Anowar, M. G. M. [Department of Physics, Begum Rokeya University, Rangpur, Rangpur-5400 (Bangladesh)
2014-09-15
An investigation of the propagation of ion acoustic waves in nonthermal plasmas in the presence of trapped electrons has been undertaken. This has been motivated by space and laboratory plasma observations of plasmas containing energetic particles, resulting in long-tailed distributions, in combination with trapped particles, whereby some of the plasma particles are confined to a finite region of phase space. An unmagnetized collisionless electron-ion plasma is considered, featuring a non-Maxwellian-trapped electron distribution, which is modelled by a kappa distribution function combined with a Schamel distribution. The effect of particle trapping has been considered, resulting in an expression for the electron density. Reductive perturbation theory has been used to construct a KdV-like Schamel equation, and examine its behaviour. The relevant configurational parameters in our study include the superthermality index κ and the characteristic trapping parameter β. A pulse-shaped family of solutions is proposed, also depending on the weak soliton speed increment u{sub 0}. The main modification due to an increase in particle trapping is an increase in the amplitude of solitary waves, yet leaving their spatial width practically unaffected. With enhanced superthermality, there is a decrease in both amplitude and width of solitary waves, for any given values of the trapping parameter and of the incremental soliton speed. Only positive polarity excitations were observed in our parametric investigation.
13. Wave-Flow Interactions and Acoustic Streaming
CERN Document Server
Chafin, Clifford E
2016-01-01
The interaction of waves and flows is a challenging topic where a complete resolution has been frustrated by the essential nonlinear features in the hydrodynamic case. Even in the case of EM waves in flowing media, the results are subtle. For a simple shear flow of constant n fluid, incident radiation is shown to be reflected and refracted in an analogous manner to Snell's law. However, the beam intensities differ and the system has an asymmetry in that an internal reflection gap opens at steep incident angles nearly oriented with the shear. For EM waves these effects are generally negligible in real systems but they introduce the topic at a reduced level of complexity of the more interesting acoustic case. Acoustic streaming is suggested, both from theory and experimental data, to be associated with vorticity generation at the driver itself. Bounds on the vorticity in bulk and nonlinear effects demonstrate that the bulk sources, even with attenuation, cannot drive such a strong flow. A review of the velocity...
14. Solar wind implication on dust ion acoustic rogue waves
Energy Technology Data Exchange (ETDEWEB)
Abdelghany, A. M., E-mail: [email protected]; Abd El-Razek, H. N., E-mail: [email protected]; El-Labany, S. K., E-mail: [email protected] [Theoretical Physics Group, Department of Physics, Faculty of Science, Damietta University, New Damietta 34517 (Egypt); Moslem, W. M., E-mail: [email protected] [Department of Physics, Faculty of Science, Port Said University, Port Said 42521 (Egypt); Centre for Theoretical Physics, The British University in Egypt (BUE), El-Shorouk City, Cairo (Egypt)
2016-06-15
The relevance of the solar wind with the magnetosphere of Jupiter that contains positively charged dust grains is investigated. The perturbation/excitation caused by streaming ions and electron beams from the solar wind could form different nonlinear structures such as rogue waves, depending on the dominant role of the plasma parameters. Using the reductive perturbation method, the basic set of fluid equations is reduced to modified Korteweg-de Vries (KdV) and further modified (KdV) equation. Assuming that the frequency of the carrier wave is much smaller than the ion plasma frequency, these equations are transformed into nonlinear Schrödinger equations with appropriate coefficients. Rational solution of the nonlinear Schrödinger equation shows that rogue wave envelopes are supported by the present plasma model. It is found that the existence region of rogue waves depends on the dust-acoustic speed and the streaming temperatures for both the ions and electrons. The dependence of the maximum rogue wave envelope amplitude on the system parameters has been investigated.
15. Solar wind implication on dust ion acoustic rogue waves
Science.gov (United States)
Abdelghany, A. M.; Abd El-Razek, H. N.; Moslem, W. M.; El-Labany, S. K.
2016-06-01
The relevance of the solar wind with the magnetosphere of Jupiter that contains positively charged dust grains is investigated. The perturbation/excitation caused by streaming ions and electron beams from the solar wind could form different nonlinear structures such as rogue waves, depending on the dominant role of the plasma parameters. Using the reductive perturbation method, the basic set of fluid equations is reduced to modified Korteweg-de Vries (KdV) and further modified (KdV) equation. Assuming that the frequency of the carrier wave is much smaller than the ion plasma frequency, these equations are transformed into nonlinear Schrödinger equations with appropriate coefficients. Rational solution of the nonlinear Schrödinger equation shows that rogue wave envelopes are supported by the present plasma model. It is found that the existence region of rogue waves depends on the dust-acoustic speed and the streaming temperatures for both the ions and electrons. The dependence of the maximum rogue wave envelope amplitude on the system parameters has been investigated.
16. Partial-differential-equation-constrained amplitude-based shape detection in inverse acoustic scattering
Science.gov (United States)
Na, Seong-Won; Kallivokas, Loukas F.
2008-03-01
In this article we discuss a formal framework for casting the inverse problem of detecting the location and shape of an insonified scatterer embedded within a two-dimensional homogeneous acoustic host, in terms of a partial-differential-equation-constrained optimization approach. We seek to satisfy the ensuing Karush-Kuhn-Tucker first-order optimality conditions using boundary integral equations. The treatment of evolving boundary shapes, which arise naturally during the search for the true shape, resides on the use of total derivatives, borrowing from recent work by Bonnet and Guzina [1-4] in elastodynamics. We consider incomplete information collected at stations sparsely spaced at the assumed obstacle’s backscattered region. To improve on the ability of the optimizer to arrive at the global optimum we: (a) favor an amplitude-based misfit functional; and (b) iterate over both the frequency- and wave-direction spaces through a sequence of problems. We report numerical results for sound-hard objects with shapes ranging from circles, to penny- and kite-shaped, including obstacles with arbitrarily shaped non-convex boundaries.
17. Statistical analysis of acoustic wave parameters near active regions
CERN Document Server
Soares, M Cristina Rabello; Scherrer, Philip H
2016-01-01
In order to quantify the influence of magnetic fields on acoustic mode parameters and flows in and around active regions, we analyse the differences in the parameters in magnetically quiet regions nearby an active region (which we call nearby regions'), compared with those of quiet regions at the same disc locations for which there are no neighboring active regions. We also compare the mode parameters in active regions with those in comparably located quiet regions. Our analysis is based on ring diagram analysis of all active regions observed by HMI during almost five years. We find that the frequency at which the mode amplitude changes from attenuation to amplification in the quiet nearby regions is around 4.2 mHz, in contrast to the active regions, for which it is about 5.1 mHz. This amplitude enhancement (the `acoustic halo effect') is as large as that observed in the active regions, and has a very weak dependence on the wave propagation direction. The mode energy difference in nearby regions also changes...
18. Acoustic gravity waves: A computational approach
Science.gov (United States)
Hariharan, S. I.; Dutt, P. K.
1987-01-01
This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.
19. Surface Acoustic Wave Atomizer and Electrostatic Deposition
Science.gov (United States)
Yamagata, Yutaka
A new methodology for fabricating thin film or micro patters of organic/bio material using surface acoustic wave (SAW) atomizer and electrostatic deposition is proposed and characteristics of atomization techniques are discussed in terms of drop size and atomization speed. Various types of SAW atomizer are compared with electrospray and conventional ultrasonic atomizers. It has been proved that SAW atomizers generate drops as small as electrospray and have very fast atomization speed. This technique is applied to fabrication of micro patterns of proteins. According to the result of immunoassay, the specific activity of immunoglobulin was preserved after deposition process.
20. Planar dust-acoustic waves in electron-positron-ion-dust plasmas with dust size distribution
Energy Technology Data Exchange (ETDEWEB)
Wang, Hong-Yan; Zhang, Kai-Biao [Sichuan University of Science and Engineering, Zigong (China)
2014-06-15
Nonlinear dust-acoustic solitary waves which are described with a Kortweg-de vries (KdV) equation by using the reductive perturbation method, are investigated in a planar unmagnetized dusty plasma consisting of electrons, positrons, ions and negatively-charged dust particles of different sizes and masses. The effects of the power-law distribution of dust and other plasma parameters on the dust-acoustic solitary waves are studied. Numerical results show that the dust size distribution has a significant influence on the propagation properties of dust-acoustic solitons. The amplitudes of solitary waves in the case of a power-law distribution is observed to be smaller, but the soliton velocity and width are observed to be larger, than those of mono-sized dust grains with an average dust size. Our results indicate that only compressed solitary waves exist in dusty plasma with different dust species. The relevance of the present investigation to interstellar clouds is discussed.
1. Nonlinear ion-acoustic cnoidal waves in a dense relativistic degenerate magnetoplasma
Science.gov (United States)
El-Shamy, E. F.
2015-03-01
The complex pattern and propagation characteristics of nonlinear periodic ion-acoustic waves, namely, ion-acoustic cnoidal waves, in a dense relativistic degenerate magnetoplasma consisting of relativistic degenerate electrons and nondegenerate cold ions are investigated. By means of the reductive perturbation method and appropriate boundary conditions for nonlinear periodic waves, a nonlinear modified Korteweg-de Vries (KdV) equation is derived and its cnoidal wave is analyzed. The various solutions of nonlinear ion-acoustic cnoidal and solitary waves are presented numerically with the Sagdeev potential approach. The analytical solution and numerical simulation of nonlinear ion-acoustic cnoidal waves of the nonlinear modified KdV equation are studied. Clearly, it is found that the features (amplitude and width) of nonlinear ion-acoustic cnoidal waves are proportional to plasma number density, ion cyclotron frequency, and direction cosines. The numerical results are applied to high density astrophysical situations, such as in superdense white dwarfs. This research will be helpful in understanding the properties of compact astrophysical objects containing cold ions with relativistic degenerate electrons.
2. Surface Acoustic Waves to Drive Plant Transpiration
Science.gov (United States)
Gomez, Eliot F.; Berggren, Magnus; Simon, Daniel T.
2017-03-01
Emerging fields of research in electronic plants (e-plants) and agro-nanotechnology seek to create more advanced control of plants and their products. Electronic/nanotechnology plant systems strive to seamlessly monitor, harvest, or deliver chemical signals to sense or regulate plant physiology in a controlled manner. Since the plant vascular system (xylem/phloem) is the primary pathway used to transport water, nutrients, and chemical signals—as well as the primary vehicle for current e-plant and phtyo-nanotechnology work—we seek to directly control fluid transport in plants using external energy. Surface acoustic waves generated from piezoelectric substrates were directly coupled into rose leaves, thereby causing water to rapidly evaporate in a highly localized manner only at the site in contact with the actuator. From fluorescent imaging, we find that the technique reliably delivers up to 6x more water/solute to the site actuated by acoustic energy as compared to normal plant transpiration rates and 2x more than heat-assisted evaporation. The technique of increasing natural plant transpiration through acoustic energy could be used to deliver biomolecules, agrochemicals, or future electronic materials at high spatiotemporal resolution to targeted areas in the plant; providing better interaction with plant physiology or to realize more sophisticated cyborg systems.
3. Random coupling of acoustic-gravity waves in the atmosphere
Science.gov (United States)
Millet, Christophe; Lott, Francois; Haynes, Christophe
2016-11-01
In numerical modeling of long-range acoustic propagation in the atmosphere, the effect of gravity waves on low-frequency acoustic waves is often ignored. As the sound speed far exceeds the gravity wave phase speed, these two types of waves present different spatial scales and their linear coupling is weak. It is possible, however, to obtain relatively strong couplings via sound speed profile changes with altitude. In the present study, this scenario is analyzed for realistic gravity wave fields and the incident acoustic wave is modeled as a narrow-banded acoustic pulse. The gravity waves are represented as a random field using a stochastic multiwave parameterization of non-orographic gravity waves. The parameterization provides independent monochromatic gravity waves, and the gravity wave field is obtained as the linear superposition of the waves produced. When the random terms are retained, a more generalized wave equation is obtained that both qualitatively and quantitatively agrees with the observations of several highly dispersed stratospheric wavetrains. Here, we show that the cumulative effect of gravity wave breakings makes the sensitivity of ground-based acoustic signals large, in that small changes in the parameterization can create or destroy an acoustic wavetrain.
4. On the Amplitude Equations for Weakly Nonlinear Surface Waves
Science.gov (United States)
Benzoni-Gavage, Sylvie; Coulombel, Jean-François
2012-09-01
Nonlocal generalizations of Burgers' equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185-202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3-4):1463-1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3-4):303-320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220-2240, 2011). In this article, we show how the verification of Hunter's stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.
5. Obliquely propagating large amplitude solitary waves in charge neutral plasmas
Directory of Open Access Journals (Sweden)
F. Verheest
2007-01-01
Full Text Available This paper deals in a consistent way with the implications, for the existence of large amplitude stationary structures in general plasmas, of assuming strict charge neutrality between electrons and ions. With the limit of pair plasmas in mind, electron inertia is retained. Combining in a fluid dynamic treatment the conservation of mass, momentum and energy with strict charge neutrality has indicated that nonlinear solitary waves (as e.g. oscillitons cannot exist in electron-ion plasmas, at no angle of propagation with respect to the static magnetic field. Specifically for oblique propagation, the proof has turned out to be more involved than for parallel or perpendicular modes. The only exception is pair plasmas that are able to support large charge neutral solitons, owing to the high degree of symmetry naturally inherent in such plasmas. The nonexistence, in particular, of oscillitons is attributed to the breakdown of the plasma approximation in dealing with Poisson's law, rather than to relativistic effects. It is hoped that future space observations will allow to discriminate between oscillitons and large wave packets, by focusing on the time variability (or not of the phase, since the amplitude or envelope graphs look very similar.
6. Large-amplitude ion-acoustic double layers in multispecies plasma
Science.gov (United States)
Jain, S. L.; Tiwari, R. S.; Sharma, S. R.
1990-06-01
The effect of second-ion species on the characteristics of large-amplitude ion-acoustic double layers (IADL) in a collisionless, unmagnetized plasma (consisting of hot and cold Maxwellian populations of electrons and two cold-ion species with different masses, concentrations, and charge states) is investigated. After deriving the criteria for the existence of large-amplitude IADL, it is found that the presence of a positive-ion impurity does not considerably modify the characteristics of large-amplitude IADL. However, the presence of negative-ion impurity significantly changes the characteristics of a large-amplitude IADL. An analytic discussion of small-amplitude IADL using a reductive perturbation method is also presented.
7. Accurate finite element modeling of acoustic waves
Science.gov (United States)
Idesman, A.; Pham, D.
2014-07-01
In the paper we suggest an accurate finite element approach for the modeling of acoustic waves under a suddenly applied load. We consider the standard linear elements and the linear elements with reduced dispersion for the space discretization as well as the explicit central-difference method for time integration. The analytical study of the numerical dispersion shows that the most accurate results can be obtained with the time increments close to the stability limit. However, even in this case and the use of the linear elements with reduced dispersion, mesh refinement leads to divergent numerical results for acoustic waves under a suddenly applied load. This is explained by large spurious high-frequency oscillations. For the quantification and the suppression of spurious oscillations, we have modified and applied a two-stage time-integration technique that includes the stage of basic computations and the filtering stage. This technique allows accurate convergent results at mesh refinement as well as significantly reduces the numerical anisotropy of solutions. We should mention that the approach suggested is very general and can be equally applied to any loading as well as for any space-discretization technique and any explicit or implicit time-integration method.
8. PIC simulation of compressive and rarefactive dust ion-acoustic solitary waves
Science.gov (United States)
Li, Zhong-Zheng; Zhang, Heng; Hong, Xue-Ren; Gao, Dong-Ning; Zhang, Jie; Duan, Wen-Shan; Yang, Lei
2016-08-01
The nonlinear propagations of dust ion-acoustic solitary waves in a collisionless four-component unmagnetized dusty plasma system containing nonextensive electrons, inertial negative ions, Maxwellian positive ions, and negatively charged static dust grains have been investigated by the particle-in-cell method. By comparing the simulation results with those obtained from the traditional reductive perturbation method, it is observed that the rarefactive KdV solitons propagate stably at a low amplitude, and when the amplitude is increased, the prime wave form evolves and then gradually breaks into several small amplitude solitary waves near the tail of soliton structure. The compressive KdV solitons propagate unstably and oscillation arises near the tail of soliton structure. The finite amplitude rarefactive and compressive Gardner solitons seem to propagate stably.
9. Propagation of ion-acoustic solitary waves in a relativistic electron-positron-ion plasma
CERN Document Server
Saberian, E; Akbari-Moghanjoughi, M
2011-01-01
Propagation of large amplitude ion-acoustic solitary waves (IASWs) in a fully relativistic plasma consisting of cold ions and ultrarelativistic hot electrons and positrons is investigated using the Sagdeev's pseudopotential method in a relativistic hydrodynamics model. Effects of streaming speed of plasma fluid, thermal energy, positron density and positron temperature on large amplitude IASWs are studied by analysis of the pseudopotential structure. It is found that in regions that the streaming speed of plasma fluid is larger than that of solitary wave, by increasing the streaming speed of plasma fluid the depth and width of potential well increases and resulting in narrower solitons with larger amplitude. This behavior is opposite for the case where the streaming speed of plasma fluid is smaller than that of solitary wave. On the other hand, increase of the thermal energy results in wider solitons with smaller amplitude, because the depth and width of potential well decreases in that case. Additionally, th...
10. Wave envelopes method for description of nonlinear acoustic wave propagation.
Science.gov (United States)
Wójcik, J; Nowicki, A; Lewin, P A; Bloomfield, P E; Kujawska, T; Filipczyński, L
2006-07-01
A novel, free from paraxial approximation and computationally efficient numerical algorithm capable of predicting 4D acoustic fields in lossy and nonlinear media from arbitrary shaped sources (relevant to probes used in medical ultrasonic imaging and therapeutic systems) is described. The new WE (wave envelopes) approach to nonlinear propagation modeling is based on the solution of the second order nonlinear differential wave equation reported in [J. Wójcik, J. Acoust. Soc. Am. 104 (1998) 2654-2663; V.P. Kuznetsov, Akust. Zh. 16 (1970) 548-553]. An incremental stepping scheme allows for forward wave propagation. The operator-splitting method accounts independently for the effects of full diffraction, absorption and nonlinear interactions of harmonics. The WE method represents the propagating pulsed acoustic wave as a superposition of wavelet-like sinusoidal pulses with carrier frequencies being the harmonics of the boundary tone burst disturbance. The model is valid for lossy media, arbitrarily shaped plane and focused sources, accounts for the effects of diffraction and can be applied to continuous as well as to pulsed waves. Depending on the source geometry, level of nonlinearity and frequency bandwidth, in comparison with the conventional approach the Time-Averaged Wave Envelopes (TAWE) method shortens computational time of the full 4D nonlinear field calculation by at least an order of magnitude; thus, predictions of nonlinear beam propagation from complex sources (such as phased arrays) can be available within 30-60 min using only a standard PC. The approximate ratio between the computational time costs obtained by using the TAWE method and the conventional approach in calculations of the nonlinear interactions is proportional to 1/N2, and in memory consumption to 1/N where N is the average bandwidth of the individual wavelets. Numerical computations comparing the spatial field distributions obtained by using both the TAWE method and the conventional approach
11. On extending the concept of double negativity to acoustic waves
Institute of Scientific and Technical Information of China (English)
CHAN C.T.; LI Jensen; FUNG K.H.
2006-01-01
The realization of double negative electromagnetic wave media, sometimes called left-handed materials (LHMs) or metamaterials, have drawn considerable attention in the past few years. We will examine the possibility of extending the concept to acoustic waves. We will see that acoustic metamaterials require both the effective density and bulk modulus to be simultaneously negative in the sense of an effective medium. If we can find a double negative (negative density and bulk modulus) acoustic medium, it will be an acoustic analogue of Veselago's medium in electromagnetism, and share many novel consequences such as negative refractive index and backward wave characteristics. We will give one example of such a medium.
12. Statistical Analysis of Acoustic Wave Parameters Near Solar Active Regions
Science.gov (United States)
Rabello-Soares, M. Cristina; Bogart, Richard S.; Scherrer, Philip H.
2016-08-01
In order to quantify the influence of magnetic fields on acoustic mode parameters and flows in and around active regions, we analyze the differences in the parameters in magnetically quiet regions nearby an active region (which we call “nearby regions”), compared with those of quiet regions at the same disk locations for which there are no neighboring active regions. We also compare the mode parameters in active regions with those in comparably located quiet regions. Our analysis is based on ring-diagram analysis of all active regions observed by the Helioseismic and Magnetic Imager (HMI) during almost five years. We find that the frequency at which the mode amplitude changes from attenuation to amplification in the quiet nearby regions is around 4.2 mHz, in contrast to the active regions, for which it is about 5.1 mHz. This amplitude enhacement (the “acoustic halo effect”) is as large as that observed in the active regions, and has a very weak dependence on the wave propagation direction. The mode energy difference in nearby regions also changes from a deficit to an excess at around 4.2 mHz, but averages to zero over all modes. The frequency difference in nearby regions increases with increasing frequency until a point at which the frequency shifts turn over sharply, as in active regions. However, this turnover occurs around 4.9 mHz, which is significantly below the acoustic cutoff frequency. Inverting the horizontal flow parameters in the direction of the neigboring active regions, we find flows that are consistent with a model of the thermal energy flow being blocked directly below the active region.
13. Elastic friction drive of surface acoustic wave motor.
Science.gov (United States)
Kurosawa, Minoru Kuribayashi; Itoh, Hidenori; Asai, Katsuhiko
2003-06-01
Importance of elastic deformation control to obtain large output force with a surface acoustic wave (SAW) motor is discussed in this paper. By adding pre-load to slider, stator and slider surfaces are deformed in a few tens nanometer. Appropriate deformation in normal direction against normal vibration displacement amplitude of SAW existed. By moderate deformation, the output force of the SAW motor was enlarged up to about 10 N and no-load speed was 0.7 m/s. To produce this performance, the transducer weight and slider size were only 4.2 g and 4 x 4 mm(2).By traveling wave propagation, surface particles of the SAW device move in elliptical motion. Due to the amplitude of the elliptical motion is 10 or 20 nm order, the contact condition of the slider is very critical. To control the contact condition, namely, the elastic deformation of the slider and stator surface in nanometer order, a lot of projections were fabricated on the slider surface. The projection diameter was 20 micro m. In static condition, the elastic deformation and stress were evaluated with the FEM analysis. From this calculation and the simulation result, it is consider that the wave crest is distorted, hence the elasticity has influence on the friction drive condition. Elastic deformation of the stator surface beneath the projection from the initial position were evaluated. In 4 x 4 mm(2) square area, the sliders had from 1089 to 23,409 projections. Depression was independent to the contact pressure. However, the output force depended on the depression although the projection density were different. From the view point of the output power of the motor, the proper depression was independent to the projection density. Around 25 nm depression, the output force and output power were maximized. This depression value was almost same as the vibration displacement amplitude of the stator transducer.
14. Generation and Propagation of Finite-Amplitude Waves in Flexible Tubes (A)
DEFF Research Database (Denmark)
Jensen, Leif Bjørnø
1972-01-01
Highly reproducible finite-amplitude waves, generated by a modified electromagnetic plane-wave generator, characterized by a rise time......Highly reproducible finite-amplitude waves, generated by a modified electromagnetic plane-wave generator, characterized by a rise time...
15. Palladium nanoparticle-based surface acoustic wave hydrogen sensor.
Science.gov (United States)
Sil, Devika; Hines, Jacqueline; Udeoyo, Uduak; Borguet, Eric
2015-03-18
Palladium (Pd) nanoparticles (5-20 nm) are used as the sensing layer on surface acoustic wave (SAW) devices for detecting H2. The interaction with hydrogen modifies the conductivity of the Pd nanoparticle film, producing measurable changes in acoustic wave propagation, which allows for the detection of this explosive gas. The nanoparticle-based SAW sensor responds rapidly and reversibly at room temperature.
16. Influence of acoustic waves on TEA CO2 laser performance
CSIR Research Space (South Africa)
Von Bergmann, H
2007-01-01
Full Text Available In this paper the author’s present results on the influence of acoustic waves on the output laser beam from high repetition rate TEA CO2 lasers. The authors show that acoustic waves generated inside the cavity lead to deterioration in beam quality...
17. Microcrack Identification in Cement-Based Materials Using Nonlinear Acoustic Waves
Science.gov (United States)
Chen, X. J.; Kim, J.-Y.; Qu, J.; Kurtis, K. E.; Wu, S. C.; Jacobs, L. J.
2007-03-01
This paper presents results from tests that use nonlinear acoustic waves to distinguish microcracks in cement-based materials. Portland cement mortar samples prepared with alkali-reactive aggregate were exposed to an aggressive environment to induce cracking were compared to control samples, of the same composition, but which were not exposed to aggressive conditions. Two nonlinear ultrasonic methods were used to characterize the samples, with the aim of identifying the time and extent of microcracking; these techniques were a nonlinear acoustical modulation (NAM) method and a harmonic amplitude relation (HAR) method. These nonlinear acoustic results show that both methods can distinguish damaged samples from undamaged ones, demonstrating the potential of nonlinear acoustic waves to provide a quantitative evaluation of the deterioration of cement-based materials.
18. Visualization of Surface Acoustic Waves in Thin Liquid Films
OpenAIRE
Rambach, R. W.; Taiber, J.; Scheck, C. M. L.; Meyer, C.; Reboud, J.; Cooper, Jonathan M.; Franke, T.
2016-01-01
We demonstrate that the propagation path of a surface acoustic wave (SAW), excited with anWe demonstrate that the propagation path of a surface acoustic wave (SAW), excited with an interdigitated transducer (IDT), can be visualized using a thin liquid film dispensed onto a lithium niobate (LiNbO3) substrate. The practical advantages of this visualization method are its rapid and simple implementation, with many potential applications including in characterising acoustic pumping within microfl...
19. On the fully nonlinear acoustic waves in a plasma with positrons beam impact and superthermal electrons
Energy Technology Data Exchange (ETDEWEB)
Ali Shan, S. [Theoretical Plasma Physics Division, PINSTECH, Nilore, 44000 Islamabad (Pakistan); National Centre For Physics (NCP), Shahdra Valley Road, 44000 Islamabad (Pakistan); Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad (Pakistan); El-Tantawy, S. A.; Moslem, W. M. [Department of Physics, Faculty of Science, Port Said University, Port Said 42521 (Egypt)
2013-08-15
Arbitrary amplitude ion-acoustic waves in an unmagnetized plasma consisting of cold positive ions, superthermal electrons, and positrons beam are reported. The basic set of fluid equations is reduced to an energy-balance like equation. The latter is numerically analyzed to examine the existence regions for solitary and shock waves. It is found that only solitary waves can propagate, however, the model cannot support shocks. The effects of superthermality and beam parameters (via, positrons concentration and streaming velocity) on the existence region, as well as solitary wave profile have been discussed.
20. Latitude variability of acoustic-gravity waves in the upper atmosphere based on satellite data
Science.gov (United States)
Fedorenko, A. K.; Bespalova, A. V.; Zhuk, I. T.; Kryuchkov, E. I.
2017-07-01
Based on satellite measurements, we investigated the properties of acoustic-gravity waves in different geographical areas of the Earth's upper atmosphere. To study wave activity at high latitudes, we used the concentration of neutral particles measured by the low-altitude polar satellite Dynamic Explorer 2 and measurements from the equatorial satellite Atmosphere Explorer-E for analysis of waves at low latitudes. In the range of heights 250-400 km, there are observed latitudinal variations of amplitudes, together with variations in the morphological and spectral properties of acoustic-gravity waves. In the polar regions of thermosphere, the wave amplitudes amount to 3-10% in terms of relative variations of density and do not exceed 3% at low and middle latitudes. At low latitudes, regular fluctuations induced by the solar terminator are clearly seen with a predominant wave mode moving synchronously with terminator. Moreover, at low and middle latitudes, there are observed sporadic local wave packets of small amplitudes (1-2%) that can have origins of various natures. We also investigated the relation between some of the observed wave trains and the earthquakes.
1. Observation of sagittal X-ray diffraction by surface acoustic waves in Bragg geometry.
Science.gov (United States)
Vadilonga, Simone; Zizak, Ivo; Roshchupkin, Dmitry; Evgenii, Emelin; Petsiuk, Andrei; Leitenberger, Wolfram; Erko, Alexei
2017-04-01
X-ray Bragg diffraction in sagittal geometry on a Y-cut langasite crystal (La3Ga5SiO14) modulated by Λ = 3 µm Rayleigh surface acoustic waves was studied at the BESSY II synchrotron radiation facility. Owing to the crystal lattice modulation by the surface acoustic wave diffraction, satellites appear. Their intensity and angular separation depend on the amplitude and wavelength of the ultrasonic superlattice. Experimental results are compared with the corresponding theoretical model that exploits the kinematical diffraction theory. This experiment shows that the propagation of the surface acoustic waves creates a dynamical diffraction grating on the crystal surface, and this can be used for space-time modulation of an X-ray beam.
2. Residual Amplitude Modulation in Interferometric Gravitational Wave Detectors
CERN Document Server
Kokeyama, Keiko; Korth, William Z; Smith-Lefebvre, Nicolas; Arai, Koji; Adhikari, Rana X
2013-01-01
The effects of residual amplitude modulation (RAM) in laser interferometers using heterodyne sensing can be substantial and difficult to mitigate. In this work, we analyze the effects of RAM on a complex laser interferometer used for gravitational wave detection. The RAM introduces unwanted offsets in the cavity length signals and thereby shifts the operating point of the optical cavities from the nominal point via feedback control. This shift causes variations in the sensing matrix, and leads to degradation in the performance of the precision noise subtraction scheme of the multiple-degree-of-freedom control system. In addition, such detuned optical cavities produce an opto-mechanical spring, which also varies the sensing matrix. We use our simulations to derive requirements on RAM for the Advanced LIGO detectors, and show that the RAM expected in Advanced LIGO will not limit its sensitivity.
3. Residual amplitude modulation in interferometric gravitational wave detectors.
Science.gov (United States)
Kokeyama, Keiko; Izumi, Kiwamu; Korth, William Z; Smith-Lefebvre, Nicolas; Arai, Koji; Adhikari, Rana X
2014-01-01
The effects of residual amplitude modulation (RAM) in laser interferometers using heterodyne sensing can be substantial and difficult to mitigate. In this work, we analyze the effects of RAM on a complex laser interferometer used for gravitational wave detection. The RAM introduces unwanted offsets in the cavity length signals and thereby shifts the operating point of the optical cavities from the nominal point via feedback control. This shift causes variations in the sensing matrix, and leads to degradation in the performance of the precision noise subtraction scheme of the multiple-degree-of-freedom control system. In addition, such detuned optical cavities produce an optomechanical spring, which also perturbs the sensing matrix. We use our simulations to derive requirements on RAM for the Advanced LIGO (aLIGO) detectors, and show that the RAM expected in aLIGO will not limit its sensitivity.
4. Synthesis of anisotropic swirling surface acoustic waves by inverse filter, towards integrated generators of acoustical vortices
CERN Document Server
Riaud, Antoine; Charron, Eric; Bussonnière, Adrien; Matar, Olivier Bou
2015-01-01
From radio-electronics signal analysis to biological samples actuation, surface acoustic waves (SAW) are involved in a multitude of modern devices. Despite this versatility, SAW transducers developed up to date only authorize the synthesis of the most simple standing or progressive waves such as plane and focused waves. In particular, acoustical integrated sources able to generate acoustical vortices (the analogue of optical vortices) are missing. In this work, we propose a flexible tool based on inverse filter technique and arrays of SAW transducers enabling the synthesis of prescribed complex wave patterns at the surface of anisotropic media. The potential of this setup is illustrated by the synthesis of a 2D analog of 3D acoustical vortices, namely "swirling surface acoustic waves". Similarly to their 3D counterpart, they appear as concentric structures of bright rings with a phase singularity in their center resulting in a central dark spot. Swirling SAW can be useful in fragile sensors whose neighborhood...
5. Hyperbolic Mild Slope Equations with Inclusion of Amplitude Dispersion Effect: Random Waves
Institute of Scientific and Technical Information of China (English)
2008-01-01
New hyperbolic mild slope equations for random waves are developed with the inclusion of amplitude dispersion. The frequency perturbation around the peak frequency of random waves is adopted to extend the equations for regular waves to random waves. The nonlinear effect of amplitude dispersion is incorporated approximately into the model by only considering the nonlinear effect on the carrier waves of random waves, which is done by introducing a representative wave amplitude for the carrier waves. The computation time is greatly saved by the introduction of the representative wave amplitude. The extension of the present model to breaking waves is also considered in order to apply the new equations to surf zone. The model is validated for random waves propagate over a shoal and in surf zone against measurements.
6. An effective absorbing boundary algorithm for acoustical wave propagator
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic waves is introduced as the truncation area of the computational domain to absorb one-dimensional acoustic wave for the scheme of acoustical wave propagator (AWP). To guarantee the efficiency of the AWP algorithm, a regulated propagator matrix is derived in the PML medium.Numerical simulations of a Gaussian wave packet propagating in one-dimensional duct are carried out to illustraze the efficiency of the combination of PML and AWP. Compared with the traditional smoothing truncation windows technique of AWP, this scheme shows high computational accuracy in absorbing acoustic wave when the acoustical wave arrives at the computational edges. Optimal coefficients of the PML configurations are also discussed.
7. Elastic reverse-time migration based on amplitude-preserving P- and S-wave separation
Science.gov (United States)
Yang, Jia-Jia; Luan, Xi-Wu; Fang, Gang; Liu, Xin-Xin; Pan, Jun; Wang, Xiao-Jie
2016-09-01
Imaging the PP- and PS-wave for the elastic vector wave reverse-time migration requires separating the P- and S-waves during the wave field extrapolation. The amplitude and phase of the P- and S-waves are distorted when divergence and curl operators are used to separate the P- and S-waves. We present a P- and S-wave amplitude-preserving separation algorithm for the elastic wavefield extrapolation. First, we add the P-wave pressure and P-wave vibration velocity equation to the conventional elastic wave equation to decompose the P- and S-wave vectors. Then, we synthesize the scalar P- and S-wave from the vector Pand S-wave to obtain the scalar P- and S-wave. The amplitude-preserved separated P- and S-waves are imaged based on the vector wave reverse-time migration (RTM). This method ensures that the amplitude and phase of the separated P- and S-wave remain unchanged compared with the divergence and curl operators. In addition, after decomposition, the P-wave pressure and vibration velocity can be used to suppress the interlayer reflection noise and to correct the S-wave polarity. This improves the image quality of P- and S-wave in multicomponent seismic data and the true-amplitude elastic reverse time migration used in prestack inversion.
8. Large Amplitude Solitary Waves in a Fluid－Filled Elastic Tube
Institute of Scientific and Technical Information of China (English)
DUANWen-Shah
2003-01-01
By usign the potential method to a fluid filled elastic tube, we obtained a solitary wave solution.Compared with recluetive perturbation method, this method can be used for larger amplitude solitary waves. The result is in agreement with that of small amplitude approximation from reduetive perturbation method when the amplitude is small enough.
9. Large Amplitude Solitary Waves in a Fluid-Filled Elastic Tube
Institute of Scientific and Technical Information of China (English)
DUAN Wen-Shan
2003-01-01
By using the potential method to a fluid filled elastic tube, we obtained a solitary wave solution. Comparedwith reductive perturbation method, this method can be used for larger amplitude solitary waves. The result is inagreement with that of small amplitude approximation from reductive perturbation method when the amplitude is smallenough.
10. Rapid Salmonella detection using an acoustic wave device combined with the RCA isothermal DNA amplification method
Directory of Open Access Journals (Sweden)
Antonis Kordas
2016-12-01
Full Text Available Salmonella enterica serovar Typhimurium is a major foodborne pathogen that causes Salmonellosis, posing a serious threat for public health and economy; thus, the development of fast and sensitive methods is of paramount importance for food quality control and safety management. In the current work, we are presenting a new approach where an isothermal amplification method is combined with an acoustic wave device for the development of a label free assay for bacteria detection. Specifically, our method utilizes a Love wave biosensor based on a Surface Acoustic Wave (SAW device combined with the isothermal Rolling Circle Amplification (RCA method; various protocols were tested regarding the DNA amplification and detection, including off-chip amplification at two different temperatures (30 °C and room temperature followed by acoustic detection and on-chip amplification and detection at room temperature, with the current detection limit being as little as 100 Bacteria Cell Equivalents (BCE/sample. Our acoustic results showed that the acoustic ratio, i.e., the amplitude over phase change observed during DNA binding, provided the only sensitive means for product detection while the measurement of amplitude or phase alone could not discriminate positive from negative samples. The method's fast analysis time together with other inherent advantages i.e., portability, potential for multi-analysis, lower sample volumes and reduced power consumption, hold great promise for employing the developed assay in a Lab on Chip (LoC platform for the integrated analysis of Salmonella in food samples.
11. Breaking of Large Amplitude Electron Plasma Wave in a Maxwellian Plasma
CERN Document Server
Mukherjee, Arghya
2016-01-01
The determination of maximum possible amplitude of a coherent longitudinal plasma oscillation/wave is a topic of fundamental importance in non-linear plasma physics. The amplitudes of these large amplitude plasma waves is limited by a phenomena called wave breaking which may be induced by several non-linear processes. It was shown by Coffey [T. P. Coffey, Phys. Fluids 14, 1402 (1971)] using a "water-bag" distribution for electrons that, in a warm plasma the maximum electric field amplitude and density amplitude implicitly depend on the electron temperature, known as Coffey's limit. In this paper, the breaking of large amplitude freely running electron plasma wave in a homogeneous warm plasma where electron's velocity distribution is Maxwellian has been studied numerically using 1D Particle in Cell (PIC) simulation method. It is found that Coffey's propagating wave solutions, which was derived using a "water-bag" distribution for electrons, also represent propagating waves in a Maxwellian plasma. Coffey's wave...
12. Investigation into Mass Loading Sensitivity of Sezawa Wave Mode-Based Surface Acoustic Wave Sensors
Directory of Open Access Journals (Sweden)
N. Ramakrishnan
2013-02-01
Full Text Available In this work mass loading sensitivity of a Sezawa wave mode based surface acoustic wave (SAW device is investigated through finite element method (FEM simulation and the prospects of these devices to function as highly sensitive SAW sensors is reported. A ZnO/Si layered SAW resonator is considered for the simulation study. Initially the occurrence of Sezawa wave mode and displacement amplitude of the Rayleigh and Sezawa wave mode is studied for lower ZnO film thickness. Further, a thin film made of an arbitrary material is coated over the ZnO surface and the resonance frequency shift caused by mass loading of the film is estimated. It was observed that Sezawa wave mode shows significant sensitivity to change in mass loading and has higher sensitivity (eight times higher than Rayleigh wave mode for the same device configuration. Further, the mass loading sensitivity was observed to be greater for a low ZnO film thickness to wavelength ratio. Accordingly, highly sensitive SAW sensors can be developed by coating a sensing medium over a layered SAW device and operating at Sezawa mode resonance frequency. The sensitivity can be increased by tuning the ZnO film thickness to wavelength ratio.
13. Investigation into mass loading sensitivity of sezawa wave mode-based surface acoustic wave sensors.
Science.gov (United States)
Mohanan, Ajay Achath; Islam, Md Shabiul; Ali, Sawal Hamid; Parthiban, R; Ramakrishnan, N
2013-02-06
In this work mass loading sensitivity of a Sezawa wave mode based surface acoustic wave (SAW) device is investigated through finite element method (FEM) simulation and the prospects of these devices to function as highly sensitive SAW sensors is reported. A ZnO/Si layered SAW resonator is considered for the simulation study. Initially the occurrence of Sezawa wave mode and displacement amplitude of the Rayleigh and Sezawa wave mode is studied for lower ZnO film thickness. Further, a thin film made of an arbitrary material is coated over the ZnO surface and the resonance frequency shift caused by mass loading of the film is estimated. It was observed that Sezawa wave mode shows significant sensitivity to change in mass loading and has higher sensitivity (eight times higher) than Rayleigh wave mode for the same device configuration. Further, the mass loading sensitivity was observed to be greater for a low ZnO film thickness to wavelength ratio. Accordingly, highly sensitive SAW sensors can be developed by coating a sensing medium over a layered SAW device and operating at Sezawa mode resonance frequency. The sensitivity can be increased by tuning the ZnO film thickness to wavelength ratio.
14. Wave-wave interactions and deep ocean acoustics
CERN Document Server
Guralnik, Zachary; Bourdelais, John; Zabalgogeazcoa, Xavier
2013-01-01
Deep ocean acoustics, in the absence of shipping and wildlife, is driven by surface processes. Best understood is the signal generated by non-linear surface wave interactions, the Longuet-Higgins mechanism, which dominates from 0.1 to 10 Hz, and may be significant for another octave. For this source, the spectral matrix of pressure and vector velocity is derived for points near the bottom of a deep ocean resting on an elastic half-space. In the absence of a bottom, the ratios of matrix elements are universal constants. Bottom effects vitiate the usual "standing wave approximation," but a weaker form of the approximation is shown to hold, and this is used for numerical calculations. In the weak standing wave approximation, the ratios of matrix elements are independent of the surface wave spectrum, but depend on frequency and the propagation environment. Data from the Hawaii-2 Observatory are in excellent accord with the theory for frequencies between 0.1 and 1 Hz, less so at higher frequencies. Insensitivity o...
15. Arbitrary amplitude solitary and shock waves in an unmagnetized quantum dusty electron-positron-ion plasma
Energy Technology Data Exchange (ETDEWEB)
Rouhani, M. R.; Akbarian, A.; Mohammadi, Z. [Department of Physics, Alzahra University, P. O. Box 1993891176, Tehran (Iran, Islamic Republic of)
2013-08-15
The behavior of quantum dust ion acoustic soliton and shocks in a plasma including inertialess quantum electrons and positrons, classical cold ions, and stationary negative dust grains are studied, using arbitrary amplitude approach. The effect of dissipation due to viscosity of ions is taken into account. The numerical analysis of Sagdeev potential for small value of quantum diffraction parameter (H) shows that for chosen plasma, only compressive solitons can exist and the existence domain of this type of solitons is decreased by increasing dust density (d). Additionally, the possibility of propagation of both subsonic and supersonic compressive solitons is investigated. It is shown that there is a critical dust density above which only supersonic solitons are observed. Moreover, increasing d leads to a reduction in the existence domain of compressive solitons and the possibility of propagation of rarefactive soliton is provided. So, rarefactive solitons are observed only due to the presence of dust particles in this model quantum plasma. Furthermore, numerical solution of governed equations for arbitrary amplitude shock waves has been investigated. It is shown that only compressive large amplitude shocks can propagate. Finally, the effects of plasma parameters on these structures are investigated. This research will be helpful in understanding the properties of dense astrophysical (i.e., white dwarfs and neutron stars) and laboratory dusty plasmas.
16. Surface-acoustic-wave (SAW) flow sensor
Science.gov (United States)
Joshi, Shrinivas G.
1991-03-01
The use of a surface-acoustic-wave (SAW) device to measure the rate of gas flow is described. A SAW oscillator heated to a suitable temperature above ambient is placed in the path of a flowing gas. Convective cooling caused by the gas flow results in a change in the oscillator frequency. A 73-MHz oscillator fabricated on 128 deg rotated Y-cut lithium niobate substrate and heated to 55 C above ambient shows a frequency variation greater than 142 kHz for flow-rate variation from 0 to 1000 cu cm/min. The output of the sensor can be calibrated to provide a measurement of volume flow rate, pressure differential across channel ports, or mass flow rate. High sensitivity, wide dynamic range, and direct digital output are among the attractive features of this sensor. Theoretical expressions for the sensitivity and response time of the sensor are derived. It is shown that by using ultrasonic Lamb waves propagating in thin membranes, a flow sensor with faster response than a SAW sensor can be realized.
17. Nonlinear features of ion acoustic shock waves in dissipative magnetized dusty plasma
Science.gov (United States)
Sahu, Biswajit; Sinha, Anjana; Roychoudhury, Rajkumar
2014-10-01
The nonlinear propagation of small as well as arbitrary amplitude shocks is investigated in a magnetized dusty plasma consisting of inertia-less Boltzmann distributed electrons, inertial viscous cold ions, and stationary dust grains without dust-charge fluctuations. The effects of dissipation due to viscosity of ions and external magnetic field, on the properties of ion acoustic shock structure, are investigated. It is found that for small amplitude waves, the Korteweg-de Vries-Burgers (KdVB) equation, derived using Reductive Perturbation Method, gives a qualitative behaviour of the transition from oscillatory wave to shock structure. The exact numerical solution for arbitrary amplitude wave differs somehow in the details from the results obtained from KdVB equation. However, the qualitative nature of the two solutions is similar in the sense that a gradual transition from KdV oscillation to shock structure is observed with the increase of the dissipative parameter.
18. Nonlinear features of ion acoustic shock waves in dissipative magnetized dusty plasma
Energy Technology Data Exchange (ETDEWEB)
Sahu, Biswajit, E-mail: [email protected] [Department of Mathematics, West Bengal State University, Barasat, Kolkata 700126 (India); Sinha, Anjana, E-mail: [email protected] [Department of Instrumentation Science, Jadavpur University, Kolkata 700032 (India); Roychoudhury, Rajkumar, E-mail: [email protected] [Department of Mathematics, Visva-Bharati, Santiniketan 731204, India and Advanced Centre for Nonlinear and Complex Phenomena, 1175 Survey Park, Kolkata 700075 (India)
2014-10-15
The nonlinear propagation of small as well as arbitrary amplitude shocks is investigated in a magnetized dusty plasma consisting of inertia-less Boltzmann distributed electrons, inertial viscous cold ions, and stationary dust grains without dust-charge fluctuations. The effects of dissipation due to viscosity of ions and external magnetic field, on the properties of ion acoustic shock structure, are investigated. It is found that for small amplitude waves, the Korteweg-de Vries-Burgers (KdVB) equation, derived using Reductive Perturbation Method, gives a qualitative behaviour of the transition from oscillatory wave to shock structure. The exact numerical solution for arbitrary amplitude wave differs somehow in the details from the results obtained from KdVB equation. However, the qualitative nature of the two solutions is similar in the sense that a gradual transition from KdV oscillation to shock structure is observed with the increase of the dissipative parameter.
19. Hyperbolic Mild Slope Equations with Inclusion of Amplitude Dispersion Effect: Regular Waves
Institute of Scientific and Technical Information of China (English)
JIN Hong; ZOU Zhi-li
2008-01-01
A new form of hyperbolic mild slope equations is derived with the inclusion of the amplitude dispersion of nonlinear waves. The effects of including the amplitude dispersion effect on the wave propagation are discussed. Wave breaking mechanism is incorporated into the present model to apply the new equations to surf zone. The equations are solved numerically for regular wave propagation over a shoal and in surf zone, and a comparison is made against measurements. It is found that the inclusion of the amplitude dispersion can also improve model's performance on prediction of wave heights around breaking point for the wave motions in surf zone.
20. An undergraduate experiment demonstrating the physics of metamaterials with acoustic waves and soda cans
Science.gov (United States)
Wilkinson, James T.; Whitehouse, Christopher B.; Oulton, Rupert F.; Gennaro, Sylvain D.
2016-01-01
We describe a novel undergraduate research project that highlights the physics of metamaterials with acoustic waves and soda cans. We confirm the Helmholtz resonance nature of a single can by measuring its amplitude and phase response to a sound wave. Arranging multiple cans in arrays smaller than the wavelength, we then design an antenna that redirects sound into a preferred direction. The antenna can be thought of as a new resonator, composed of artificially engineered meta-atoms, similar to a metamaterial. These experiments are illustrative, tactile, and open ended so as to enable students to explore the physics of matter/wave interaction.
1. Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes.
Science.gov (United States)
Gusev, Vitalyi E; Ni, Chenyin; Lomonosov, Alexey; Shen, Zhonghua
2015-08-01
Theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous material on flexural wave in the plates of continuously varying thickness is developed. For the wedges with thickness increasing as a power law of distance from its edge strong modifications of the wave dynamics with propagation distance are predicted. It is found that nonlinear absorption progressively disappearing with diminishing wave amplitude leads to complete attenuation of acoustic waves in most of the wedges exhibiting black hole phenomenon. It is also demonstrated that black holes exist beyond the geometrical acoustic approximation. Applications include nondestructive evaluation of micro-inhomogeneous materials and vibrations damping. Copyright © 2015 Elsevier B.V. All rights reserved.
2. Nonlinear acoustic waves in the viscous thermosphere and ionosphere above earthquake
Science.gov (United States)
Chum, J.; Cabrera, M. A.; Mošna, Z.; Fagre, M.; Baše, J.; Fišer, J.
2016-12-01
The nonlinear behavior of acoustic waves and their dissipation in the upper atmosphere is studied on the example of infrasound waves generated by vertical motion of the ground surface during the Mw 8.3 earthquake that occurred about 46 km from Illapel, Chile on 16 September 2015. To conserve energy, the amplitude of infrasound waves initially increased as the waves propagated upward to the rarefied air. When the velocities of air particles became comparable with the local sound speed, the nonlinear effects started to play an important role. Consequently, the shape of waveform changed significantly with increasing height, and the original wave packet transformed to the "N-shaped" pulse resembling a shock wave. A unique observation by the continuous Doppler sounder at the altitude of about 195 km is in good agreement with full wave numerical simulation that uses as boundary condition the measured vertical motion of the ground surface.
3. Nonlinear excitations for the positron acoustic shock waves in dissipative nonextensive electron-positron-ion plasmas
Science.gov (United States)
Saha, Asit
2017-03-01
Positron acoustic shock waves (PASHWs) in unmagnetized electron-positron-ion (e-p-i) plasmas consisting of mobile cold positrons, immobile positive ions, q-nonextensive distributed electrons, and hot positrons are studied. The cold positron kinematic viscosity is considered and the reductive perturbation technique is used to derive the Burgers equation. Applying traveling wave transformation, the Burgers equation is transformed to a one dimensional dynamical system. All possible vector fields corresponding to the dynamical system are presented. We have analyzed the dynamical system with the help of potential energy, which helps to identify the stability and instability of the equilibrium points. It is found that the viscous force acting on cold mobile positron fluid is a source of dissipation and is responsible for the formation of the PASHWs. Furthermore, fully nonlinear arbitrary amplitude positron acoustic waves are also studied applying the theory of planar dynamical systems. It is also observed that the fundamental features of the small amplitude and arbitrary amplitude PASHWs are significantly affected by the effect of the physical parameters q e , q h , μ e , μ h , σ , η , and U. This work can be useful to understand the qualitative changes in the dynamics of nonlinear small amplitude and fully nonlinear arbitrary amplitude PASHWs in solar wind, ionosphere, lower part of magnetosphere, and auroral acceleration regions.
4. Body-wave traveltime and amplitude shifts from asymptotic travelling wave coupling
Science.gov (United States)
Pollitz, F.
2006-01-01
We explore the sensitivity of finite-frequency body-wave traveltimes and amplitudes to perturbations in 3-D seismic velocity structure relative to a spherically symmetric model. Using the approach of coupled travelling wave theory, we consider the effect of a structural perturbation on an isolated portion of the seismogram. By convolving the spectrum of the differential seismogram with the spectrum of a narrow window taper, and using a Taylor's series expansion for wavenumber as a function of frequency on a mode dispersion branch, we derive semi-analytic expressions for the sensitivity kernels. Far-field effects of wave interactions with the free surface or internal discontinuities are implicitly included, as are wave conversions upon scattering. The kernels may be computed rapidly for the purpose of structural inversions. We give examples of traveltime sensitivity kernels for regional wave propagation at 1 Hz. For the direct SV wave in a simple crustal velocity model, they are generally complicated because of interfering waves generated by interactions with the free surface and the Mohorovic??ic?? discontinuity. A large part of the interference effects may be eliminated by restricting the travelling wave basis set to those waves within a certain range of horizontal phase velocity. ?? Journal compilation ?? 2006 RAS.
5. Effect of acoustic field parameters on arc acoustic binding during ultrasonic wave-assisted arc welding.
Science.gov (United States)
Xie, Weifeng; Fan, Chenglei; Yang, Chunli; Lin, Sanbao
2016-03-01
6. Two-Dimensional Nonlinear Propagation of Ion Acoustic Waves through KPB and KP Equations in Weakly Relativistic Plasmas
Directory of Open Access Journals (Sweden)
M. G. Hafez
2016-01-01
Full Text Available Two-dimensional three-component plasma system consisting of nonextensive electrons, positrons, and relativistic thermal ions is considered. The well-known Kadomtsev-Petviashvili-Burgers and Kadomtsev-Petviashvili equations are derived to study the basic characteristics of small but finite amplitude ion acoustic waves of the plasmas by using the reductive perturbation method. The influences of positron concentration, electron-positron and ion-electron temperature ratios, strength of electron and positrons nonextensivity, and relativistic streaming factor on the propagation of ion acoustic waves in the plasmas are investigated. It is revealed that the electrostatic compressive and rarefactive ion acoustic waves are obtained for superthermal electrons and positrons, but only compressive ion acoustic waves are found and the potential profiles become steeper in case of subthermal positrons and electrons.
7. Small-amplitude shock waves and double layers in dusty plasmas with opposite polarity charged dust grains
Science.gov (United States)
Amina, M.; Ema, S. A.; Mamun, A. A.
2017-06-01
Theoretical investigation is carried out for understanding the properties of nonlinear dust-acoustic (DA) waves in an unmagnetized dusty plasma whose constituents are massive, micron-sized, positive and negatively charged inertial dust grains along with q (nonextensive) distributed electrons and ions. The reductive perturbation method is employed in order to derive two types of nonlinear dynamical equations, namely, Burgers equation and modified Gardner equation (Gardner equation with dissipative term). They are also numerically analyzed to investigate the basic features (viz., polarity, amplitude, width, etc.) of shock waves and double layers. It has been observed that the effects of nonextensivity, opposite polarity charged dust grains, and different dusty plasma parameters have significantly modified the fundamental properties of shock waves and double layers. The results of this investigation may be used for researches of the nonlinear wave propagation in laboratory and space plasmas.
8. X-ray topography analysis of acoustic wave fields in the SAW-resonator structures.
Science.gov (United States)
Roshchupkin, Dmitry V; Roshchupkina, Helen D; Irzhak, Dmitry V
2005-11-01
The formation of fields of standing surface acoustic waves (SAW) in LiNbO3 and La3Ga5SiO14 (LGS) crystals was studied by high-resolution topography method on a laboratory X-ray source. The fields of standing SAW were formed using SAW-resonator structures consisting of interdigital transducer (IDT) and reflecting gratings. The SAW amplitudes and power flow angles were measured by X-ray topography, diffraction in acoustic beam was visualized, and the SAW interaction with the crystal structure defects was studied.
9. LiNbO3/p+n diode surface acoustic wave memory correlator
Institute of Scientific and Technical Information of China (English)
张朝; 水永安; 印建华
1997-01-01
A detailed theoretical analysis of strip-coupled LiNbO3/p+ n diode surface acoustic wave (SAW) memory correlator in the parametric mode is presented. The influence of some important factors on correlation output is analyzed and calculated, including the amplitudes of reference, read and write signal, duration of write signal and doping density of the diode array. The conclusions can be employed for the design of improved strip-coupled SAW memorycorrelators.
10. On the Source of Propagating Slow Magneto-acoustic Waves in Sunspots
OpenAIRE
S. Krishna Prasad; Jess, D. B.; Khomenko, Elena
2015-01-01
Recent high-resolution observations of sunspot oscillations using simultaneously operated ground- and space-based telescopes reveal the intrinsic connection between different layers of the solar atmosphere. However, it is not clear whether these oscillations are externally driven or generated in-situ. We address this question by using observations of propagating slow magneto-acoustic waves along a coronal fan loop system. In addition to the generally observed decreases in oscillation amplitud...
11. Nonlinear propagation and control of acoustic waves in phononic superlattices
CERN Document Server
Jiménez, Noé; Picó, Rubén; García-Raffi, Lluís M; Sánchez-Morcillo, Víctor J
2015-01-01
The propagation of intense acoustic waves in a one-dimensional phononic crystal is studied. The medium consists in a structured fluid, formed by a periodic array of fluid layers with alternating linear acoustic properties and quadratic nonlinearity coefficient. The spacing between layers is of the order of the wavelength, therefore Bragg effects such as band-gaps appear. We show that the interplay between strong dispersion and nonlinearity leads to new scenarios of wave propagation. The classical waveform distortion process typical of intense acoustic waves in homogeneous media can be strongly altered when nonlinearly generated harmonics lie inside or close to band gaps. This allows the possibility of engineer a medium in order to get a particular waveform. Examples of this include the design of media with effective (e.g. cubic) nonlinearities, or extremely linear media (where distortion can be cancelled). The presented ideas open a way towards the control of acoustic wave propagation in nonlinear regime.
12. Excitation of nonlinear ion acoustic waves in CH plasmas
CERN Document Server
Feng, Q S; Liu, Z J; Xiao, C Z; Wang, Q; He, X T
2016-01-01
Excitation of nonlinear ion acoustic wave (IAW) by an external electric field is demonstrated by Vlasov simulation. The frequency calculated by the dispersion relation with no damping is verified much closer to the resonance frequency of the small-amplitude nonlinear IAW than that calculated by the linear dispersion relation. When the wave number $k\\lambda_{De}$ increases, the linear Landau damping of the fast mode (its phase velocity is greater than any ion's thermal velocity) increases obviously in the region of $T_i/T_e < 0.2$ in which the fast mode is weakly damped mode. As a result, the deviation between the frequency calculated by the linear dispersion relation and that by the dispersion relation with no damping becomes larger with $k\\lambda_{De}$ increasing. When $k\\lambda_{De}$ is not large, such as $k\\lambda_{De}=0.1, 0.3, 0.5$, the nonlinear IAW can be excited by the driver with the linear frequency of the modes. However, when $k\\lambda_{De}$ is large, such as $k\\lambda_{De}=0.7$, the linear ...
13. Droplet actuation by surface acoustic waves: an interplay between acoustic streaming and radiation pressure
Science.gov (United States)
Brunet, Philippe; Baudoin, Michael; Matar, Olivier Bou; Zoueshtiagh, Farzam
2010-11-01
Surface acoustic waves (SAW) are known to be a versatile technique for the actuation of sessile drops. Droplet displacement, internal mixing or drop splitting, are amongst the elementary operations that SAW can achieve, which are useful on lab-on-chip microfluidics benches. On the purpose to understand the underlying physical mechanisms involved during these operations, we study experimentally the droplet dynamics varying different physical parameters. Here in particular, the influence of liquid viscosity and acoustic frequency is investigated: it is indeed predicted that both quantities should play a role in the acoustic-hydrodynamic coupling involved in the dynamics. The key point is to compare the relative magnitude of the attenuation length, i.e. the scale within which the acoustic wave decays in the fluid, and the size of the drop. This relative magnitude governs the relative importance of acoustic streaming and acoustic radiation pressure, which are both involved in the droplet dynamics.
14. Effect of attenuation correction on surface amplitude distribution of wind waves
Digital Repository Service at National Institute of Oceanography (India)
Varkey, M.J.
Some selected wave profiles recorded using a ship borne wave recorder are analysed to study the effect of attenuation correction on the distribution of the surface amplitudes. A new spectral width parameter is defined to account for wide band...
15. Estimating propagation velocity through a surface acoustic wave sensor
Energy Technology Data Exchange (ETDEWEB)
Xu, Wenyuan (Oakdale, MN); Huizinga, John S. (Dellwood, MN)
2010-03-16
Techniques are described for estimating the propagation velocity through a surface acoustic wave sensor. In particular, techniques which measure and exploit a proper segment of phase frequency response of the surface acoustic wave sensor are described for use as a basis of bacterial detection by the sensor. As described, use of velocity estimation based on a proper segment of phase frequency response has advantages over conventional techniques that use phase shift as the basis for detection.
16. Acoustic Bloch Wave Propagation in a Periodic Waveguide
Science.gov (United States)
1991-07-24
Distribution of Spherical Cavities," J. Acoust. Soc. Am. 81, 595-598. Arfken , G. (1985). Mathematical Methods for Physicists (Academic Press, Inc., New... mathematical -ystem of governing equations and boundary conditions describing lossy, linear acoustic waves in a periodic wave- guide is, under the...Translational Invariance The aim of this section is to present the mathematical system to be solved and show that it exhibits invariance under a certain
17. Estimation of Sea Surface Wave Spectra Using Acoustic Tomography.
Science.gov (United States)
1987-09-01
Holister Dis speciael Dean of Graduate Studiesj ESTIMATION OF SEA SURFACE WAVE SPECTRA USING ACOUSTIC TOMOGRAPHY by James Henry Miller B.S. Electrical...James Henry Miller 1987 The author hereby prants to MIT permission to reproduce and distribute copies of this thesis in whole or in part. Signature of...ESTIMATION OF SEA SURFACE WAVE SPECTRA USING ACOUSTIC TOMOGRAPHY by James Henry Miller Submitted in partial fulfillment of the requirements for the
18. Modulation of cavity-polaritons by surface acoustic waves
DEFF Research Database (Denmark)
de Lima, M. M.; Poel, Mike van der; Hey, R.;
2006-01-01
We modulate cavity-polaritons using surface acoustic waves. The corresponding formation of a mini-Brillouin zone and band folding of the polariton dispersion is demonstrated for the first time. Results are in good agreement with model calculations.......We modulate cavity-polaritons using surface acoustic waves. The corresponding formation of a mini-Brillouin zone and band folding of the polariton dispersion is demonstrated for the first time. Results are in good agreement with model calculations....
19. Estimating propagation velocity through a surface acoustic wave sensor
Science.gov (United States)
Xu, Wenyuan; Huizinga, John S.
2010-03-16
Techniques are described for estimating the propagation velocity through a surface acoustic wave sensor. In particular, techniques which measure and exploit a proper segment of phase frequency response of the surface acoustic wave sensor are described for use as a basis of bacterial detection by the sensor. As described, use of velocity estimation based on a proper segment of phase frequency response has advantages over conventional techniques that use phase shift as the basis for detection.
20. Acoustic minor losses in high amplitude resonators with single-sided junctions
Science.gov (United States)
Doller, Andrew J.
Steady flow engineering handbooks like Idelchik20 do not exist for investigators interested in acoustic (oscillating) fluid flows in complex resonators. Measurements of acoustic minor loss coefficients are presented in this dissertation for a limited number of resonator configurations having single-sided junctions. While these results may be useful, the greater purpose of this work is to provide a set of controlled measurements that can be used to benchmark computational models of acoustic flows used for more complicated resonator structures. The experiments are designed around a driver operating at 150 Hz enabling acoustic pressures in excess of 10k Pa in liquid cooled, temperature controlled resonators with 90°, 45° and 25° junctions. These junctions join a common 109 cm long 4.7 cm diameter section to a section of 8.4 mm diameter tube making two sets of resonators: one set with a small diameter length approximately a quarter-wavelength (45 cm), the other approximately a half-wavelength (112 cm). The long resonators have a velocity node at the junction; the short resonators have a velocity anti-node generating the greatest minor losses. Input power is measured by an accelerometer and a pressure transducer at the driver. A pressure sensor at the rigid termination measures radiation pressure from the driver and static junction pressure, as well as the acoustic pressure used to calculate linear thermal and viscous resonator wall losses. At the largest amplitudes, the 90° junction was found to dissipate as much as 0.3 Watt, 1/3 the power of linear losses alone. For each junction, the power dissipation depends on acoustic pressure differently: pressure cubed for the 90°, pressure to the 3.76 for the 45° and pressure to the 4.48 for the 25°. Common among all resonators, blowing acoustic half-cycle minor losses (KB) are excited at lower amplitudes than the suction half-cycle (KS) minor losses. Data collected for the 90° junction shows KB reaches an asymptotic
1. Surface acoustic wave devices for sensor applications
Science.gov (United States)
Bo, Liu; Xiao, Chen; Hualin, Cai; Mohammad, Mohammad Ali; Xiangguang, Tian; Luqi, Tao; Yi, Yang; Tianling, Ren
2016-02-01
Surface acoustic wave (SAW) devices have been widely used in different fields and will continue to be of great importance in the foreseeable future. These devices are compact, cost efficient, easy to fabricate, and have a high performance, among other advantages. SAW devices can work as filters, signal processing units, sensors and actuators. They can even work without batteries and operate under harsh environments. In this review, the operating principles of SAW sensors, including temperature sensors, pressure sensors, humidity sensors and biosensors, will be discussed. Several examples and related issues will be presented. Technological trends and future developments will also be discussed. Project supported by the National Natural Science Foundation of China (Nos. 60936002, 61025021, 61434001, 61574083), the State Key Development Program for Basic Research of China (No. 2015CB352100), the National Key Project of Science and Technology (No. 2011ZX02403-002) and the Special Fund for Agroscientific Research in the Public Interest of China (No. 201303107). M.A.M is additionally supported by the Postdoctoral Fellowship (PDF) program of the Natural Sciences and Engineering Research Council (NSERC) of Canada and the China Postdoctoral Science Foundation (CPSF).
2. Experimental study on subharmonic and ultraharmonic acoustic waves in water-saturated sandy sediment.
Science.gov (United States)
Kim, Byoung-Nam; Lee, Kang Il; Yoon, Suk Wang
2007-04-01
Experimental observations of the subharmonic and ultraharmonic acoustic waves in water-saturated sandy sediment are reported in this paper. Acoustic pressures of both nonlinear acoustic waves strongly depend on the driving acoustic pressure at a transducer. The first ultraharmonic wave reaches a saturation value as the driving acoustic pressure increases. The acoustic pressure levels of both nonlinear acoustic waves exhibit some fluctuations in comparison with that of the primary acoustic wave as the receiving distance of hydrophone increases in sediment. The subharmonic and the ultraharmonic phenomena in this study show close resemblance to those produced in bubbly water.
3. Wavemaker theories for acoustic-gravity waves over a finite depth
Science.gov (United States)
2016-04-01
Acoustic-gravity waves (hereafter AGWs) in ocean have received much interest recently, mainly with respect to early detection of tsunamis as they travel at near the speed of sound in water which makes them ideal candidates for early detection of tsunamis. While the generation mechanisms of AGWs have been studied from the perspective of vertical oscillations of seafloor (Yamamoto, 1982; Stiassnie, 2010) and triad wave-wave interaction (Longuet-Higgins 1950; Kadri and Stiassnie 2013; Kadri and Akylas 2016), in the current study we are interested in their generation by wave-structure interaction with possible application to the energy sector. Here, we develop two wavemaker theories to analyze different wave modes generated by impermeable (the classic Havelock's theory) and porous (porous wavemaker theory) plates in weakly compressible fluids. Slight modification has been made to the porous theory so that, unlike the previous theory (Chwang, 1983), the new solution depends on the geometry of the plate. The expressions for three different types of plates (piston, flap, delta-function) are introduced. Analytical solutions are also derived for the potential amplitude of the gravity, evanescent, and acoustic-gravity waves, as well as the surface elevation, velocity distribution, and pressure for AGWs. Both theories reduce to previous results for incompressible flow when the compressibility is negligible. We also show numerical examples for AGW generated in a wave flume as well as in deep ocean. Our current study sets the theoretical background towards remote sensing by AGWs, for optimized deep ocean wave-power harnessing, among others. References Chwang, A.T. 1983 A porous-wavemaker theory. Journal of Fluid Mechanics, 132, 395- 406. Kadri, U., Stiassnie, M. 2013 Generation of an acoustic-gravity wave by two gravity waves, and their subsequent mutual interaction. J. Fluid Mech. 735, R6. Kadri U., Akylas T.R. 2016 On resonant triad interactions of acoustic-gravity waves. J
4. Arbitrary amplitude slow electron-acoustic solitons in three-electron temperature space plasmas
Energy Technology Data Exchange (ETDEWEB)
Mbuli, L. N. [South African National Space Agency (SANSA) Space Science, P.O. Box 32, Hermanus 7200, Republic of South Africa (South Africa); University of the Western Cape, Robert Sobukwe Road, Bellville 7535, Republic of South Africa (South Africa); Maharaj, S. K. [South African National Space Agency (SANSA) Space Science, P.O. Box 32, Hermanus 7200, Republic of South Africa (South Africa); Bharuthram, R. [University of the Western Cape, Robert Sobukwe Road, Bellville 7535, Republic of South Africa (South Africa); Singh, S. V.; Lakhina, G. S. [Indian Institute of Geomagnetism, New Panvel (West), Navi Mumbai 410218 (India)
2015-06-15
We examine the characteristics of large amplitude slow electron-acoustic solitons supported in a four-component unmagnetised plasma composed of cool, warm, hot electrons, and cool ions. The inertia and pressure for all the species in this plasma system are retained by assuming that they are adiabatic fluids. Our findings reveal that both positive and negative potential slow electron-acoustic solitons are supported in the four-component plasma system. The polarity switch of the slow electron-acoustic solitons is determined by the number densities of the cool and warm electrons. Negative potential solitons, which are limited by the cool and warm electron number densities becoming unreal and the occurrence of negative potential double layers, are found for low values of the cool electron density, while the positive potential solitons occurring for large values of the cool electron density are only limited by positive potential double layers. Both the lower and upper Mach numbers for the slow electron-acoustic solitons are computed and discussed.
5. Temperature and Pressure Dependence of Signal Amplitudes for Electrostriction Laser-Induced Thermal Acoustics
Science.gov (United States)
Herring, Gregory C.
2015-01-01
The relative signal strength of electrostriction-only (no thermal grating) laser-induced thermal acoustics (LITA) in gas-phase air is reported as a function of temperature T and pressure P. Measurements were made in the free stream of a variable Mach number supersonic wind tunnel, where T and P are varied simultaneously as Mach number is varied. Using optical heterodyning, the measured signal amplitude (related to the optical reflectivity of the acoustic grating) was averaged for each of 11 flow conditions and compared to the expected theoretical dependence of a pure-electrostriction LITA process, where the signal is proportional to the square root of [P*P /( T*T*T)].
6. Mechanism of an acoustic wave impact on steel during solidification
Directory of Open Access Journals (Sweden)
K. Nowacki
2013-04-01
Full Text Available Acoustic steel processing in an ingot mould may be the final stage in the process of quality improvement of a steel ingot. The impact of radiation and cavitation pressure as well as the phenomena related to the acoustic wave being emitted and delivered to liquid steel affect various aspects including the internal structure fragmentation, rigidity or density of steel. The article provides an analysis of the mechanism of impact of physical phenomena caused by an acoustic wave affecting the quality of a steel ingot.
7. Nonlinear acoustic waves in micro-inhomogeneous solids
CERN Document Server
Nazarov, Veniamin
2014-01-01
Nonlinear Acoustic Waves in Micro-inhomogeneous Solids covers the broad and dynamic branch of nonlinear acoustics, presenting a wide variety of different phenomena from both experimental and theoretical perspectives. The introductory chapters, written in the style of graduate-level textbook, present a review of the main achievements of classic nonlinear acoustics of homogeneous media. This enables readers to gain insight into nonlinear wave processes in homogeneous and micro-inhomogeneous solids and compare it within the framework of the book. The subsequent eight chapters covering: Physical m
8. Neural network surface acoustic wave RF signal processor for digital modulation recognition.
Science.gov (United States)
Kavalov, Dimitar; Kalinin, Victor
2002-09-01
An architecture of a surface acoustic wave (SAW) processor based on an artificial neural network is proposed for an automatic recognition of different types of digital passband modulation. Three feed-forward networks are trained to recognize filtered and unfiltered binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) signals, as well as unfiltered BPSK, QPSK, and 16 quadrature amplitude (16QAM) signals. Performance of the processor in the presence of additive white Gaussian noise (AWGN) is simulated. The influence of second-order effects in SAW devices, phase, and amplitude errors on the performance of the processor also is studied.
9. Electron acoustic waves in a magnetized plasma with kappa distributed ions
Energy Technology Data Exchange (ETDEWEB)
Devanandhan, S.; Lakhina, G. S. [Indian Institute of Geomagnetism, Navi Mumbai (India); Singh, S. V. [Indian Institute of Geomagnetism, Navi Mumbai (India); School of Physics, University of Kwazulu-Natal, Durban (South Africa); Bharuthram, R. [University of the Western Cape, Bellville (South Africa)
2012-08-15
Electron acoustic solitary waves in a two component magnetized plasma consisting of fluid cold electrons and hot superthermal ions are considered. The linear dispersion relation for electron acoustic waves is derived. In the nonlinear regime, the energy integral is obtained by a Sagdeev pseudopotential analysis, which predicts negative solitary potential structures. The effects of superthermality, obliquity, temperature, and Mach number on solitary structures are studied in detail. The results show that the superthermal index {kappa} and electron to ion temperature ratio {sigma} alters the regime where solitary waves can exist. It is found that an increase in magnetic field value results in an enhancement of soliton electric field amplitude and a reduction in soliton width and pulse duration.
10. Nonlinear propagation of ion-acoustic waves through the Burgers equation in weakly relativistic plasmas
Science.gov (United States)
Hafez, M. G.; Talukder, M. R.; Hossain Ali, M.
2017-04-01
The Burgers equation is obtained to study the characteristics of nonlinear propagation of ionacoustic shock, singular kink, and periodic waves in weakly relativistic plasmas containing relativistic thermal ions, nonextensive distributed electrons, Boltzmann distributed positrons, and kinematic viscosity of ions using the well-known reductive perturbation technique. This equation is solved by employing the ( G'/ G)-expansion method taking unperturbed positron-to-electron concentration ratio, electron-to-positron temperature ratio, strength of electrons nonextensivity, ion kinematic viscosity, and weakly relativistic streaming factor. The influences of plasma parameters on nonlinear propagation of ion-acoustic shock, periodic, and singular kink waves are displayed graphically and the relevant physical explanations are described. It is found that these parameters extensively modify the shock structures excitation. The obtained results may be useful in understanding the features of small but finite amplitude localized relativistic ion-acoustic shock waves in an unmagnetized plasma system for some astrophysical compact objects and space plasmas.
11. The excitation of inertial-acoustic waves through turbulent fluctuations in accretion discs I: WKBJ theory
CERN Document Server
Heinemann, T
2008-01-01
We study and elucidate the mechanism of inertial-acoustic wave excitation in a turbulent, differentially rotating flow. We formulate a set of wave equations with sources that are only non-zero in the presence of turbulent fluctuations. We solve these using a WKBJ method. It is found that, for a particular azimuthal wave length, the wave excitation occurs through a sequence of regularly spaced swings during which the wave changes from leading to trailing form. This is a generic process that is expected to occur in shearing discs with turbulence. Pairs of trailing waves of equal amplitude propagating in opposite directions are produced and give rise to an outward angular momentum flux that we give expressions for as functions of the disc parameters and azimuthal wave length. By solving the wave amplitude equations numerically we justify the WKBJ approach for a Keplerian rotation law for all parameter regimes of interest. In order to quantify the wave excitation approach completely the important wave source term...
12. The influence of storms on finite amplitude sand wave dynamics: an idealized nonlinear model
NARCIS (Netherlands)
Campmans, G.H.P.; Roos, P.C.; de Vriend, H.J.; Hulscher, S.J.M.H.
2017-01-01
We investigate the effects of storms on finite amplitude sand wave growth using a new idealized nonlinear morphodynamic model. We find that the growth speed initially linearly increases with sand wave amplitude, after which nonlinear effects cause the growth to decrease. This finally leads to an
13. Ion acoustic solitary waves in plasmas with nonextensive distributed electrons, positrons and relativistic thermal ions
Science.gov (United States)
Hafez, M. G.; Talukder, M. R.; Sakthivel, R.
2016-05-01
The theoretical and numerical studies have been investigated on nonlinear propagation of weakly relativistic ion acoustic solitary waves in an unmagnetized plasma system consisting of nonextensive electrons, positrons and relativistic thermal ions. To study the characteristics of nonlinear propagation of the three-component plasma system, the reductive perturbation technique has been applied to derive the Korteweg-de Vries equation, which divulges the soliton-like solitary wave solution. The ansatz method is employed to carry out the integration of this equation. The effects of nonextensive electrons, positrons and relativistic thermal ions on phase velocity, amplitude and width of soliton and electrostatic nonlinear propagation of weakly relativistic ion acoustic solitary waves have been discussed taking different plasma parameters into consideration. The obtained results can be useful in understanding the features of small amplitude localized relativistic ion acoustic solitary waves in an unmagnetized three-component plasma system for hard thermal photon production with relativistic heavy ions collision in quark-gluon plasma as well as for astrophysical plasmas.
14. Dust acoustic shock wave generation due to dust charge variation in a dusty plasma
M R Gupta; S Sarkar; M Khan; Samiran Ghosh
2003-12-01
In a dusty plasma, the non-adiabaticity of the charge variation on a dust grain surface results in an anomalous dissipation. Analytical investigation shows that this results in a small but finite amplitude dust acoustic (DA) wave propagation which is described by the Korteweg–de Vries–Burger equation. Results of the numerical investigation of the propagation of large-amplitude dust acoustic stationary shock wave are presented here using the complete set of non-linear dust fluid equations coupled with the dust charging equation and Poisson equation. The DA waves are of compressional type showing considerable increase of dust density, which is of significant importance in astrophysical context as it leads to enhanced gravitational attraction considered as a viable process for star formation. The DA shock transition to its far downstream amplitude is oscillatory in nature due to dust charge fluctuations, the oscillation amplitude and shock width depending on the ratio pd/ch and other plasma parameters.
15. Effects of nicotine on the acoustic startle reflex amplitude in rats.
Science.gov (United States)
Acri, J B; Grunberg, N E; Morse, D E
1991-01-01
The acoustic startle reflex was used to measure changes in sensorimotor reactivity in response to nicotine administration and cessation. Male rats received saline, 6 mg/kg/day or 12 mg/kg/day nicotine delivered subcutaneously by osmotic minipumps. The pumps delivered their contents during a 10-day period of implantation, after which time they were explanted. Animals were tested for startle reflex amplitudes using two levels of white noise bursts prior to pump implantation, on days 1 and 7 of nicotine or saline administration, and on several days following drug cessation. Nicotine produced a dose-dependent increase in startle amplitude during the period of administration that decreased during cessation. Results are interpreted in terms of nicotine's actions to enhance attentional processes.
16. Modeling of acoustic and gravity waves propagation through the atmosphere with spectral element method
Science.gov (United States)
Brissaud, Q.; Garcia, R.; Martin, R.; Komatitsch, D.
2014-12-01
Low-frequency events such as tsunamis generate acoustic and gravity waves which quickly propagate in the atmosphere. Since the atmospheric density decreases exponentially as the altitude increases and from the conservation of the kinetic energy, those waves see their amplitude raise (to the order of 105 at 200km of altitude), allowing their detection in the upper atmosphere. Various tools have been developed through years to model this propagation, such as normal modes modeling or to a greater extent time-reversal techniques, but none offer a low-frequency multi-dimensional atmospheric wave modelling.A modeling tool is worthy interest since there are many different phenomena, from quakes to atmospheric explosions, able to propagate acoustic and gravity waves. In order to provide a fine modeling of the precise observations of these waves by GOCE satellite data, we developed a new numerical modeling tool.Starting from the SPECFEM program that already propagate waves in solid, porous or fluid media using a spectral element method, this work offers a tool with the ability to model acoustic and gravity waves propagation in a stratified attenuating atmosphere with a bottom forcing or an atmospheric source.Atmospheric attenuation is required in a proper modeling framework since it has a crucial impact on acoustic wave propagation. Indeed, it plays the role of a frequency filter that damps high-frequency signals. The bottom forcing feature has been implemented due to its ability to easily model the coupling with the Earth's or ocean's surface (that vibrates when a surface wave go through it) but also huge atmospheric events.
17. Nonlinear standing wave and acoustic streaming in an exponential-shape resonator by gas-kinetic scheme simulation
Science.gov (United States)
Zhang, Xiaoqing; Feng, Heying; Qu, Chengwu
2016-10-01
Nonlinear standing waves and acoustic streaming in an axial-symmetrical resonator with exponentially varying cross-sectional area were studied. A two-dimensional gas-kinetic Bhatnagar-Gross-Krook scheme based on the non-structure triangular grid was established to simulate nonlinear acoustic oscillations in the resonator. Details of the transient and steady flow fields and streaming were developed. The effects of winding index of the exponential-shape resonator, the displacement amplitude of the acoustic piston on the streaming, and the vortex pattern were analyzed. The results demonstrate that the acoustic streaming pattern in such resonators is different from the typical Rayleigh flow in a constant cross-sectional area resonator. No obvious shock wave appeared inside the exponential-shape resonator. The comparison reveals that with increasing the displacement amplitude of the acoustic piston energy dissipation is accompanied by vortex break-up from a first-level to a second-level transition, and even into turbulent flow. This research demonstrates that the exponential-shape resonator, especially that with a winding index of 2.2 exhibits better acoustic features and suppression effects on shock-wave, acoustic streaming, and the vortex.
18. Numerical study of the propagation of small-amplitude atmospheric gravity wave
Institute of Scientific and Technical Information of China (English)
YUE Xianchang; YI Fan; LIU Yingjie; LI Fang
2005-01-01
By using a two-dimensional fully nonlinear compressible atmospheric dynamic numerical model, the propagation of a small amplitude gravity wave packet is simulated. A corresponding linear model is also developed for comparison. In an isothermal atmosphere, the simulations show that the nonlinear effects impacting on the propagation of a small amplitude gravity wave are negligible. In the nonisothermal atmosphere, however, the nonlinear effects are remarkable. They act to slow markedly down the propagation velocity of wave energy and therefore reduce the growth ratio of the wave amplitude with time. But the energy is still conserved. A proof of this is provided by the observations in the middle atmosphere.
19. Development of Surface Acoustic Wave Electronic Nose
Directory of Open Access Journals (Sweden)
S.K. Jha
2010-07-01
Full Text Available The paper proposes an effective method to design and develop surface acoustic wave (SAW sensor array-based electronic nose systems for specific target applications. The paper suggests that before undertaking full hardware development empirically through hit and trial for sensor selection, it is prudent to develop accurate sensor array simulator for generating synthetic data and optimising sensor array design and pattern recognition system. The latter aspects are most time-consuming and cost-intensive parts in the development of an electronic nose system. This is because most of the electronic sensor platforms, circuit components, and electromechanical parts are available commercially-off-the-shelve (COTS, whereas knowledge about specific polymers and data analysis software are often guarded due to commercial or strategic interests. In this study, an 11-element SAW sensor array is modelled to detect and identify trinitrotoluene (TNT and dinitrotoluene (DNT explosive vapours in the presence of toluene, benzene, di-methyl methyl phosphonate (DMMP and humidity as interferents. Additive noise sources and outliers were included in the model for data generation. The pattern recognition system consists of: (i a preprocessor based on logarithmic data scaling, dimensional autoscaling, and singular value decomposition-based denoising, (ii principal component analysis (PCA-based feature extractor, and (iii an artificial neural network (ANN classifier. The efficacy of this approach is illustrated by presenting detailed PCA analysis and classification results under varied conditions of noise and outlier, and by analysing comparative performance of four classifiers (neural network, k-nearest neighbour, naïve Bayes, and support vector machine.Defence Science Journal, 2010, 60(4, pp.364-376, DOI:http://dx.doi.org/10.14429/dsj.60.493
20. Large amplitude ion-acoustic double layers in warm dusty plasma
Science.gov (United States)
Jain, S. L.; Tiwari, R. S.; Mishra, M. K.
2015-01-01
Large amplitude ion-acoustic double layer (IADL) is studied using Sagdeev's pseudo-potential technique in collisionless unmagnetized plasma comprising hot and cold Maxwellian population of electrons, warm adiabatic ions, and dust grains. Variation of both Mach number (M) and amplitude |φ m | of large amplitude IADL with charge, concentration, and mass of heavily charged massive dust grains is investigated for both positive and negative dust in plasma. Our numerical analysis shows that system supports only rarefactive large amplitude IADL for the selected set of plasma parameters. Our investigations for both negative and positive dust grains reveal that ion temperature increases the mobility of ions, resulting in increase in the Mach number of IADL. The larger mobility of ions causes leakage of ions from localized region, resulting into decrease in the amplitude of IADL. Other parameters, e.g. temperature ratio of hot to cold electrons, charge, concentration, mass of heavily charged massive dust grains also play significant role in the properties and existence of double layers. Since it is well established that both positive and negative dust are found in space as well as laboratory plasma, and double layers have a tremendous role to play in astrophysics, we have included both positive and negative dust in our numerical analysis for the study of large amplitude IADL. Further data used for negative dust are close to experimentally observed data. Hence, it is anticipated that our parametric studies for heavily charged (both positive and negative) dust may be useful in understanding laboratory plasma experiments, identifying nonlinear structures in upper part of ionosphere and lower part of magnetosphere structures, and in theoretical research for the study of properties of nonlinear structures.
1. Acoustic emission signals frequency-amplitude characteristics of sandstone after thermal treated under uniaxial compression
Science.gov (United States)
Kong, Biao; Wang, Enyuan; Li, Zenghua; Wang, Xiaoran; Niu, Yue; Kong, Xiangguo
2017-01-01
Thermally treated sandstone deformation and fracture produced abundant acoustic emission (AE) signals. The AE signals waveform contained plentiful precursor information of sandstone deformation and fracture behavior. In this paper, uniaxial compression tests of sandstone after different temperature treatments were conducted, the frequency-amplitude characteristics of AE signals were studied, and the main frequency distribution at different stress level was analyzed. The AE signals frequency-amplitude characteristics had great difference after different high temperature treatment. Significant differences existed of the main frequency distribution of AE signals during thermal treated sandstone deformation and fracture. The main frequency band of the largest waveforms proportion was not unchanged after different high temperature treatments. High temperature caused thermal damage to the sandstone, and sandstone deformation and fracture was obvious than the room temperature. The number of AE signals was larger than the room temperature during the initial loading stage. The low frequency AE signals had bigger proportion when the stress was 0.1, and the maximum value of the low frequency amplitude was larger than high frequency signals. With the increase of stress, the low and high frequency AE signals were gradually increase, which indicated that different scales ruptures were broken in sandstone. After high temperature treatment, the number of high frequency AE signals was significantly bigger than the low frequency AE signals during the latter loading stage, this indicates that the small scale rupture rate of recurrence and frequency were more than large scale rupture. The AE ratio reached the maximum during the sandstone instability failure period, and large scale rupture was dominated in the failure process. AE amplitude increase as the loading increases, the deformation and fracture of sandstone was increased gradually. By comparison, the value of the low frequency
2. Acoustic Gravity Wave Chemistry Model for the RAYTRACE Code.
Science.gov (United States)
2014-09-26
AU)-AI56 850 ACOlUSTIC GRAVITY WAVE CHEMISTRY MODEL FOR THE IAYTRACE I/~ CODE(U) MISSION RESEARCH CORP SANTA BARBIARA CA T E OLD Of MAN 84 MC-N-SlS...DNA-TN-S4-127 ONAOOI-BO-C-0022 UNLSSIFIlED F/O 20/14 NL 1-0 2-8 1111 po 312.2 1--I 11111* i •. AD-A 156 850 DNA-TR-84-127 ACOUSTIC GRAVITY WAVE...Hicih Frequency Radio Propaoation Acoustic Gravity Waves 20. ABSTRACT (Continue en reveree mide if tteceeemr and Identify by block number) This
3. KAJIAN SIFAT AKUSTIK BUAH MANGGIS(Gracinia mangostana L DENGAN MENGGUNAKAN GELOMBANG ULTRASONIK [Acoustic Study Of Mangosteene (Gracinia mangostana L By Using Ultrasonic Wave
Directory of Open Access Journals (Sweden)
Jajang juansah 1
2007-06-01
Full Text Available The wave used to study the acoustic properties of mangosteen is ultrasonic wave. Ultrasonic wave with frequency of 50 KHz was used to determine acoustic properties of mangosteen. The main wave properties were the attenuation, impedance of acoustic and acoustic velocity at mangosteen. Others have been evaluated were the correlation of attenuation and acoustic velocity at parts of mangosteen with its intact mangosteen. The acoustic parameters were related to the physic-chemical parameters of the fruit (TDS and hardness. This relationship was used to study mangosteen properties and quality. Because of mangosteen structure and it’s pores (saw with low density, acoustic wave in manggosteen have low amplitude signal. It was saw with spectrum and FFT signal mangosteen and reference medium / air (1.4:2.3.The fruit with increasing maturity mount (from color index 2 to 5 will experience hardness degradation, improvement of TDS, which are related to degradation of acoustic attenuation, improvement of acoustic speed and impedance. Multiple regression method was used to get empiric equation of wave in mixture of flesh-seed, husk and mangosteen (parts of mangosteen with its intact mangosteen. That saw in equation 1 and 2. the velocity and attenuation of ultrasonic wave in mixture of flesh – seed have higher effect equation on mangosteen than husk. It means that acoustic properties of mixture of flesh – seed has more contribution than husk.
4. Nonlinear dust acoustic waves with polarization force effects in Kappa distribution plasma
Science.gov (United States)
Chen, Hui; Zhou, Suyun; Luo, Rongxiang; Liu, Sanqiu
2017-01-01
The propagation characteristics of dust acoustic solitary waves (DASWs) in dusty plasmas with the effects of polarization force and superthermal ions are studied. First, the polarization force induced by superthermal ions is obtained. It is shown that the superthermality of background ions affect the Debye screening of dust grains as well as the polarization force significantly. Then for small amplitude solitary waves, the KdV equation is obtained by applying the reductive perturbation technique. And for the arbitrary amplitude solitary waves, the Sagdeev potential method is employed and the Sagdeev potential is analyzed. In both case, the effects of the polarization force associated the ions’ superthermality on the characteristic of the DASWs are analyzed.
5. Solitary, explosive and rational solutions for nonlinear electron-acoustic waves with non-thermal electrons
CERN Document Server
El-Wakil, S A; Abd-El-Hamid, H M; Abulwafa, E M
2010-01-01
A rigorous theoretical investigation has been made on electron acoustic wave propagating in unmagnetized collisionless plasma consisting of a cold electron fluid, non-thermal hot electrons and stationary ions. Based on the pseudo-potential approach, large amplitude potential structures and the existence of solitary waves are discussed. The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude electrostatic waves. An algebraic method with computerized symbolic computation, which greatly exceeds the applicability of the existing tanh, extended tanh methods in obtaining a series of exact solutions of the KdV equation. Numerical studies have been made using plasma parameters close to those values corresponding to the dayside auroral zone reveals different solutions i.e., bell-shaped solitary pulses, rational pulses and solutions with singularity at a finite points which called blowup solutions in addition to the propagation of an explosive pu...
6. Thermo-acoustic engineering of silicon microresonators via evanescent waves
Energy Technology Data Exchange (ETDEWEB)
Tabrizian, R., E-mail: [email protected] [Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, Michigan 48109 (United States); Ayazi, F. [School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30308 (United States)
2015-06-29
A temperature-compensated silicon micromechanical resonator with a quadratic temperature characteristic is realized by acoustic engineering. Energy-trapped resonance modes are synthesized by acoustic coupling of propagating and evanescent extensional waves in waveguides with rectangular cross section. Highly different temperature sensitivity of propagating and evanescent waves is used to engineer the linear temperature coefficient of frequency. The resulted quadratic temperature characteristic has a well-defined turn-over temperature that can be tailored by relative energy distribution between propagating and evanescent acoustic fields. A 76 MHz prototype is implemented in single crystal silicon. Two high quality factor and closely spaced resonance modes, created from efficient energy trapping of extensional waves, are excited through thin aluminum nitride film. Having different evanescent wave constituents and energy distribution across the device, these modes show different turn over points of 67 °C and 87 °C for their quadratic temperature characteristic.
7. Propagation of Acoustic Waves in Troposphere and Stratosphere
CERN Document Server
Kashyap, J M
2016-01-01
Acoustic waves are those waves which travel with the speed of sound through a medium. H. Lamb has derived a cutoff frequency for stratified and isothermal medium for the propagation of acoustic waves. In order to find the cutoff frequency many methods were introduced after Lamb's work. In this paper, we have chosen the method to determine cutoff frequencies for acoustic waves propagating in non-isothermal media. This turning point frequency method can be applied to various atmospheres like solar atmosphere, stellar atmosphere, earth's atmosphere etc. Here, we have analytically derived the cutoff frequency and have graphically analyzed and compared with the Lamb's cut-off frequencyfor earth's troposphere, lower and upper stratosphere.
8. Broadband enhanced transmission of acoustic waves through serrated metal gratings
Science.gov (United States)
Qi, Dong-Xiang; Fan, Ren-Hao; Deng, Yu-Qiang; Peng, Ru-Wen; Wang, Mu; Jiangnan University Collaboration
In this talk, we present our studies on broadband properties of acoustic waves through metal gratings. We have demonstrated that serrated metal gratings, which introduce gradient coatings, can give rise to broadband transmission enhancement of acoustic waves. Here, we have experimentally and theoretically studied the acoustic transmission properties of metal gratings with or without serrated boundaries. The average transmission is obviously enhanced for serrated metal gratings within a wide frequency range, while the Fabry-Perot resonance is significantly suppressed. An effective medium hypothesis with varying acoustic impedance is proposed to analyze the mechanism, which was verified through comparison with finite-element simulation. The serrated boundary supplies gradient mass distribution and gradient normal acoustic impedance, which could efficiently reduce the boundary reflection. Further, by increasing the region of the serrated boundary, we present a broadband high-transmission grating for wide range of incident angle. Our results may have potential applications to broadband acoustic imaging, acoustic sensing and new acoustic devices. References: [1] Dong-Xiang Qi, Yu-Qiang Deng, Di-Hu Xu, Ren-Hao Fan, Ru-Wen Peng, Ze-Guo Chen, Ming-Hui Lu, X. R. Huang and Mu Wang, Appl. Phys. Lett. 106, 011906 (2015); [2] Dong-Xiang Qi, Ren-Hao Fan, Ru-Wen Peng, Xian-Rong Huang, Ming-Hui Lu, Xu Ni, Qing Hu, and Mu Wang, Applied Physics Letters 101, 061912 (2012).
9. Acoustic-Emergent Phonology in the Amplitude Envelope of Child-Directed Speech.
Science.gov (United States)
Leong, Victoria; Goswami, Usha
2015-01-01
When acquiring language, young children may use acoustic spectro-temporal patterns in speech to derive phonological units in spoken language (e.g., prosodic stress patterns, syllables, phonemes). Children appear to learn acoustic-phonological mappings rapidly, without direct instruction, yet the underlying developmental mechanisms remain unclear. Across different languages, a relationship between amplitude envelope sensitivity and phonological development has been found, suggesting that children may make use of amplitude modulation (AM) patterns within the envelope to develop a phonological system. Here we present the Spectral Amplitude Modulation Phase Hierarchy (S-AMPH) model, a set of algorithms for deriving the dominant AM patterns in child-directed speech (CDS). Using Principal Components Analysis, we show that rhythmic CDS contains an AM hierarchy comprising 3 core modulation timescales. These timescales correspond to key phonological units: prosodic stress (Stress AM, ~2 Hz), syllables (Syllable AM, ~5 Hz) and onset-rime units (Phoneme AM, ~20 Hz). We argue that these AM patterns could in principle be used by naïve listeners to compute acoustic-phonological mappings without lexical knowledge. We then demonstrate that the modulation statistics within this AM hierarchy indeed parse the speech signal into a primitive hierarchically-organised phonological system comprising stress feet (proto-words), syllables and onset-rime units. We apply the S-AMPH model to two other CDS corpora, one spontaneous and one deliberately-timed. The model accurately identified 72-82% (freely-read CDS) and 90-98% (rhythmically-regular CDS) stress patterns, syllables and onset-rime units. This in-principle demonstration that primitive phonology can be extracted from speech AMs is termed Acoustic-Emergent Phonology (AEP) theory. AEP theory provides a set of methods for examining how early phonological development is shaped by the temporal modulation structure of speech across
10. Acoustic-Emergent Phonology in the Amplitude Envelope of Child-Directed Speech.
Directory of Open Access Journals (Sweden)
Victoria Leong
Full Text Available When acquiring language, young children may use acoustic spectro-temporal patterns in speech to derive phonological units in spoken language (e.g., prosodic stress patterns, syllables, phonemes. Children appear to learn acoustic-phonological mappings rapidly, without direct instruction, yet the underlying developmental mechanisms remain unclear. Across different languages, a relationship between amplitude envelope sensitivity and phonological development has been found, suggesting that children may make use of amplitude modulation (AM patterns within the envelope to develop a phonological system. Here we present the Spectral Amplitude Modulation Phase Hierarchy (S-AMPH model, a set of algorithms for deriving the dominant AM patterns in child-directed speech (CDS. Using Principal Components Analysis, we show that rhythmic CDS contains an AM hierarchy comprising 3 core modulation timescales. These timescales correspond to key phonological units: prosodic stress (Stress AM, ~2 Hz, syllables (Syllable AM, ~5 Hz and onset-rime units (Phoneme AM, ~20 Hz. We argue that these AM patterns could in principle be used by naïve listeners to compute acoustic-phonological mappings without lexical knowledge. We then demonstrate that the modulation statistics within this AM hierarchy indeed parse the speech signal into a primitive hierarchically-organised phonological system comprising stress feet (proto-words, syllables and onset-rime units. We apply the S-AMPH model to two other CDS corpora, one spontaneous and one deliberately-timed. The model accurately identified 72-82% (freely-read CDS and 90-98% (rhythmically-regular CDS stress patterns, syllables and onset-rime units. This in-principle demonstration that primitive phonology can be extracted from speech AMs is termed Acoustic-Emergent Phonology (AEP theory. AEP theory provides a set of methods for examining how early phonological development is shaped by the temporal modulation structure of speech across
11. THEMIS Observations of the Magnetopause Electron Diffusion Region: Large Amplitude Waves and Heated Electrons
CERN Document Server
Tang, Xiangwei; Dombeck, John; Dai, Lei; Wilson, Lynn B; Breneman, Aaron; Hupach, Adam
2013-01-01
We present the first observations of large amplitude waves in a well-defined electron diffusion region at the sub-solar magnetopause using data from one THEMIS satellite. These waves identified as whistler mode waves, electrostatic solitary waves, lower hybrid waves and electrostatic electron cyclotron waves, are observed in the same 12-sec waveform capture and in association with signatures of active magnetic reconnection. The large amplitude waves in the electron diffusion region are coincident with abrupt increases in electron parallel temperature suggesting strong wave heating. The whistler mode waves which are at the electron scale and enable us to probe electron dynamics in the diffusion region were analyzed in detail. The energetic electrons (~30 keV) within the electron diffusion region have anisotropic distributions with T_{e\\perp}/T_{e\\parallel}>1 that may provide the free energy for the whistler mode waves. The energetic anisotropic electrons may be produced during the reconnection process. The whi...
12. Theory of reflection reflection and transmission of electromagnetic, particle and acoustic waves
CERN Document Server
Lekner, John
2016-01-01
This book deals with the reflection of electromagnetic and particle waves by interfaces. The interfaces can be sharp or diffuse. The topics of the book contain absorption, inverse problems, anisotropy, pulses and finite beams, rough surfaces, matrix methods, numerical methods, reflection of particle waves and neutron reflection. Exact general results are presented, followed by long wave reflection, variational theory, reflection amplitude equations of the Riccati type, and reflection of short waves. The Second Edition of the Theory of Reflection is an updated and much enlarged revision of the 1987 monograph. There are new chapters on periodically stratified media, ellipsometry, chiral media, neutron reflection and reflection of acoustic waves. The chapter on anisotropy is much extended, with a complete treatment of the reflection and transmission properties of arbitrarily oriented uniaxial crystals. The book gives a systematic and unified treatment reflection and transmission of electromagnetic and particle...
13. Generalized collar waves in acoustic logging while drilling
Science.gov (United States)
Wang, Xiu-Ming; He, Xiao; Zhang, Xiu-Mei
2016-12-01
Tool waves, also named collar waves, propagating along the drill collars in acoustic logging while drilling (ALWD), strongly interfere with the needed P- and S-waves of a penetrated formation, which is a key issue in picking up formation P- and S-wave velocities. Previous studies on physical insulation for the collar waves designed on the collar between the source and the receiver sections did not bring to a satisfactory solution. In this paper, we investigate the propagation features of collar waves in different models. It is confirmed that there exists an indirect collar wave in the synthetic full waves due to the coupling between the drill collar and the borehole, even there is a perfect isolator between the source and the receiver. The direct collar waves propagating all along the tool and the indirect ones produced by echoes from the borehole wall are summarized as the generalized collar waves. Further analyses show that the indirect collar waves could be relatively strong in the full wave data. This is why the collar waves cannot be eliminated with satisfactory effect in many cases by designing the physical isolators carved on the tool. Project supported by the National Natural Science Foundation of China (Grant Nos. 11134011 and 11374322) and the Foresight Research Project, Institute of Acoustics, Chinese Academy of Sciences.
14. Numerical modeling of acoustic and gravity waves propagation in the atmosphere using a spectral element method
Science.gov (United States)
Martin, Roland; Brissaud, Quentin; Garcia, Raphael; Komatitsch, Dimitri
2015-04-01
During low-frequency events such as tsunamis, acoustic and gravity waves are generated and quickly propagate in the atmosphere. Due to the exponential decrease of the atmospheric density with the altitude, the conservation of the kinetic energy imposes that the amplitude of those waves increases (to the order of 105 at 200km of altitude), which allows their detection in the upper atmosphere. This propagation bas been modelled for years with different tools, such as normal modes modeling or to a greater extent time-reversal techniques, but a low-frequency multi-dimensional atmospheric wave modelling is still crucially needed. A modeling tool is worth of interest since there are many different sources, as earthquakes or atmospheric explosions, able to propagate acoustic and gravity waves. In order to provide a fine modeling of the precise observations of these waves by GOCE satellite data, we developed a new numerical modeling tool. By adding some developments to the SPECFEM package that already models wave propagation in solid, porous or fluid media using a spectral element method, we show here that acoustic and gravity waves propagation can now be modelled in a stratified attenuating atmosphere with a bottom forcing or an atmospheric source. The bottom forcing feature has been implemented to easily model the coupling with the Earth's or ocean's vibrating surfaces but also huge atmospheric events. Atmospheric attenuation is also introduced since it has a crucial impact on acoustic wave propagation. Indeed, it plays the role of a frequency filter that damps high-frequency signals.
15. Acoustic Wave Dispersion and Scattering in Complex Marine Sediment Structures
Science.gov (United States)
2015-09-30
Acoustic wave dispersion and scattering in complex marine sediment structures Charles W. Holland The Pennsylvania State University Applied...volume scattering and 2) the effects of shear waves in general layered media. These advances will provide the basis for measuring dispersion in in-situ...shear waves on dispersion in marine sediments. The first step will be development of the theory. WORK COMPLETED A brief summary of the work
16. Tsunami mitigation by resonant triad interaction with acoustic-gravity waves.
Science.gov (United States)
2017-01-01
Tsunamis have been responsible for the loss of almost a half million lives, widespread long lasting destruction, profound environmental effects, and global financial crisis, within the last two decades. The main tsunami properties that determine the size of impact at the shoreline are its wavelength and amplitude in the ocean. Here, we show that it is in principle possible to reduce the amplitude of a tsunami, and redistribute its energy over a larger space, through forcing it to interact with resonating acoustic-gravity waves. In practice, generating the appropriate acoustic-gravity modes introduces serious challenges due to the high energy required for an effective interaction. However, if the findings are extended to realistic tsunami properties and geometries, we might be able to mitigate tsunamis and so save lives and properties. Moreover, such a mitigation technique would allow for the harnessing of the tsunami's energy.
17. Negative birefraction of acoustic waves in a sonic crystal.
Science.gov (United States)
Lu, Ming-Hui; Zhang, Chao; Feng, Liang; Zhao, Jun; Chen, Yan-Feng; Mao, Yi-Wei; Zi, Jian; Zhu, Yong-Yuan; Zhu, Shi-Ning; Ming, Nai-Ben
2007-10-01
Optical birefringence and dichroism are classical and important effects originating from two independent polarizations of optical waves in anisotropic crystals. Furthermore, the distinct dispersion relations of transverse electric and transverse magnetic polarized electromagnetic waves in photonic crystals can lead to birefringence more easily. However, it is impossible for acoustic waves in the fluid to show such a birefringence because only the longitudinal mode exists. The emergence of an artificial sonic crystal (SC) has significantly broadened the range of acoustic materials in nature that can give rise to acoustic bandgaps and be used to control the propagation of acoustic waves. Recently, negative refraction has attracted a lot of attention and has been demonstrated in both left-handed materials and photonic crystals. Similar to left-handed materials and photonic crystals, negative refractions have also been found in SCs. Here we report, for the first time, the acoustic negative-birefraction phenomenon in a two-dimensional SC, even with the same frequency and the same 'polarization' state. By means of this feature, double focusing images of a point source have been realized. This birefraction concept may be extended to other periodic systems corresponding to other forms of waves, showing great impacts on both fundamental physics and device applications.
18. Subharmonic scattering of phospholipid-shell microbubbles at low acoustic pressure amplitudes.
Science.gov (United States)
Frinking, Peter J A; Brochot, Jean; Arditi, Marcel
2010-08-01
Subharmonic scattering of phospholipid-shell microbubbles excited at relatively low acoustic pressure amplitudes (<30 kPa) has been associated with echo responses from compression-only bubbles having initial surface tension values close to zero. In this work, the relation between sbharmonics and compression-only behavior of phospholipid-shell microbubbles was investigated, experimentally and by simulation, as a function of the initial surface tension by applying ambient overpressures of 0 and 180 mmHg. The microbubbles were excited using a 64-cycle transmit burst with a center frequency of 4 MHz and peak-negative pressure amplitudes ranging from 20 of 150 kPa. In these conditions, an increase in subharmonic response of 28.9 dB (P < 0.05) was measured at 50 kPa after applying an overpressure of 180 mmHg. Simulations using the Marmottant model, taking into account the effect of ambient overpressure on bubble size and initial surface tension, confirmed the relation between subharmonics observed in the pressure-time curves and compression-only behavior observed in the radius-time curves. The trend of an increase in subharmonic response as a function of ambient overpressure, i.e., as a function of the initial surface tension, was predicted by the model. Subharmonics present in the echo responses of phospholipid-shell microbubbles excited at low acoustic pressure amplitudes are indeed related to the echo responses from compression-only bubbles. The increase in subharmonics as a function of ambient overpressure may be exploited for improving methods for noninvasive pressure measurement in heart cavities or big vessels in the human body.
19. Love wave acoustic sensor for testing in liquids
Science.gov (United States)
Pan, Haifeng; Zhu, Huizhong; Feng, Guanping
2001-09-01
Love wave is one type of the surface acoustic waves (SAWs). It is guided acoustic mode propagating in ta thin layer deposited on a substrate. Because of its advantages of high mass sensitivity, low noise level and being fit for operating in liquids, Love wave acoustic sensors have become one of the hot spots in the research of biosensor nowadays. In this paper the Love wave devices with the substrate of ST-cut quartz and the guiding layers of PMMA and fused quartz were fabricated successfully. By measuring the transfer function S21 and the insertion loss of the devices, the characteristics of the Rayleigh wave device and the Love wave devices with different guiding layers in gas phase and liquid phase were compared. It was validated that the Love wave sensor is suitable for testing in liquids but the Rayleigh wave sensor is not. What's more, SiO2 is the more proper material for the guiding layer of the Love wave device.
20. Ion-acoustic cnoidal waves in plasmas with warm ions and kappa distributed electrons and positrons
Energy Technology Data Exchange (ETDEWEB)
Kaladze, T. [Department of Physics, Government College University (GCU), Lahore 54000 (Pakistan); I.Vekua Institute of Applied Mathematics, Tbilisi State University, 0186 Georgia (United States); Mahmood, S., E-mail: [email protected] [Theoretical Physics Division (TPD), PINSTECH P.O. Nilore Islamabad 44000 (Pakistan); National Center for Physics (NCP), Quaid-i-Azam University Campus, Shahdra Valley Road, Islamabad 44000 (Pakistan)
2014-03-15
Electrostatic ion-acoustic periodic (cnoidal) waves and solitons in unmagnetized electron-positron-ion (EPI) plasmas with warm ions and kappa distributed electrons and positrons are investigated. Using the reductive perturbation method, the Korteweg-de Vries (KdV) equation is derived with appropriate boundary conditions for periodic waves. The corresponding analytical and various numerical solutions are presented with Sagdeev potential approach. Differences between the results caused by the kappa and Maxwell distributions are emphasized. It is revealed that only hump (compressive) structures of the cnoidal waves and solitons are formed. It is shown that amplitudes of the cnoidal waves and solitons are reduced in an EPI plasma case in comparison with the ordinary electron-ion plasmas. The effects caused by the temperature variations of the warm ions are also discussed. It is obtained that the amplitude of the cnoidal waves and solitons decreases for a kappa distributed (nonthermal) electrons and positrons plasma case in comparison with the Maxwellian distributed (thermal) electrons and positrons EPI plasmas. The existence of kappa distributed particles leads to decreasing of ion-acoustic frequency up to thermal ions frequency.
1. Ion-acoustic cnoidal waves in plasmas with warm ions and kappa distributed electrons and positrons
Science.gov (United States)
2014-03-01
Electrostatic ion-acoustic periodic (cnoidal) waves and solitons in unmagnetized electron-positron-ion (EPI) plasmas with warm ions and kappa distributed electrons and positrons are investigated. Using the reductive perturbation method, the Korteweg-de Vries (KdV) equation is derived with appropriate boundary conditions for periodic waves. The corresponding analytical and various numerical solutions are presented with Sagdeev potential approach. Differences between the results caused by the kappa and Maxwell distributions are emphasized. It is revealed that only hump (compressive) structures of the cnoidal waves and solitons are formed. It is shown that amplitudes of the cnoidal waves and solitons are reduced in an EPI plasma case in comparison with the ordinary electron-ion plasmas. The effects caused by the temperature variations of the warm ions are also discussed. It is obtained that the amplitude of the cnoidal waves and solitons decreases for a kappa distributed (nonthermal) electrons and positrons plasma case in comparison with the Maxwellian distributed (thermal) electrons and positrons EPI plasmas. The existence of kappa distributed particles leads to decreasing of ion-acoustic frequency up to thermal ions frequency.
2. Observations of dissipation of slow magneto-acoustic waves in a polar coronal hole
Science.gov (United States)
Gupta, G. R.
2014-08-01
Aims: We focus on a polar coronal hole region to find any evidence of dissipation of propagating slow magneto-acoustic waves. Methods: We obtained time-distance and frequency-distance maps along the plume structure in a polar coronal hole. We also obtained Fourier power maps of the polar coronal hole in different frequency ranges in 171 Å and 193 Å passbands. We performed intensity distribution statistics in time domain at several locations in the polar coronal hole. Results: We find the presence of propagating slow magneto-acoustic waves having temperature dependent propagation speeds. The wavelet analysis and Fourier power maps of the polar coronal hole show that low-frequency waves are travelling longer distances (longer detection length) as compared to high-frequency waves. We found two distinct dissipation length scales of wave amplitude decay at two different height ranges (between 0-10 Mm and 10-70 Mm) along the observed plume structure. The dissipation lengths obtained at higher height range show some frequency dependence. Individual Fourier power spectrum at several locations show a power-law distribution with frequency whereas probability density function of intensity fluctuations in time show nearly Gaussian distributions. Conclusions: Propagating slow magneto-acoustic waves are getting heavily damped (small dissipation lengths) within the first 10 Mm distance. Beyond that waves are getting damped slowly with height. Frequency dependent dissipation lengths of wave propagation at higher heights may indicate the possibility of wave dissipation due to thermal conduction, however, the contribution from other dissipative parameters cannot be ruled out. Power-law distributed power spectra were also found at lower heights in the solar corona, which may provide viable information on the generation of longer period waves in the solar atmosphere.
3. Exact traveling wave solutions and L1 stability for the shallow water wave model of moderate amplitude
Science.gov (United States)
Wang, Ying; Guo, Yunxi
2017-09-01
In this paper, we developed, for the first time, the exact expressions of several periodic travelling wave solutions and a solitary wave solution for a shallow water wave model of moderate amplitude. Then, we present the existence theorem of the global weak solutions. Finally, we prove the stability of solution in L1(R) space for the Cauchy problem of the equation.
4. Exact traveling wave solutions and L1 stability for the shallow water wave model of moderate amplitude
Science.gov (United States)
Wang, Ying; Guo, Yunxi
2016-07-01
In this paper, we developed, for the first time, the exact expressions of several periodic travelling wave solutions and a solitary wave solution for a shallow water wave model of moderate amplitude. Then, we present the existence theorem of the global weak solutions. Finally, we prove the stability of solution in L1(R) space for the Cauchy problem of the equation.
5. Numerical study of the collar wave characteristics and the effects of grooves in acoustic logging while drilling
Science.gov (United States)
Yang, Yufeng; Guan, Wei; Hu, Hengshan; Xu, Minqiang
2017-05-01
Large-amplitude collar wave covering formation signals is still a tough problem in acoustic logging-while-drilling (LWD) measurements. In this study, we investigate the propagation and energy radiation characteristics of the monopole collar wave and the effects of grooves on reducing the interference to formation waves by finite-difference calculations. We found that the collar wave radiates significant energy into the formation by comparing the waveforms between a collar within an infinite fluid, and the acoustic LWD in different formations with either an intact or a truncated collar. The collar wave recorded on the outer surface of the collar consists of the outward-radiated energy direct from the collar (direct collar wave) and that reflected back from the borehole wall (reflected collar wave). All these indicate that the significant effects of the borehole-formation structure on collar wave were underestimated in previous studies. From the simulations of acoustic LWD with a grooved collar, we found that grooves broaden the frequency region of low collar-wave excitation and attenuate most of the energy of the interference waves by multireflections. However, grooves extend the duration of the collar wave and convert part of the collar-wave energy originally kept in the collar into long-duration Stoneley wave. Interior grooves are preferable to exterior ones because both the low-frequency and the high-frequency parts of the collar wave can be reduced and the converted inner Stoneley wave is relatively difficult to be recorded on the outer surface of the collar. Deeper grooves weaken the collar wave more greatly, but they result in larger converted Stoneley wave especially for the exterior ones. The interference waves, not only the direct collar wave but also the reflected collar wave and the converted Stoneley waves, should be overall considered for tool design.
6. Multimode filter composed of single-mode surface acoustic wave/bulk acoustic wave resonators
Science.gov (United States)
Huang, Yulin; Bao, Jingfu; Tang, Gongbin; Wang, Yiling; Omori, Tatsuya; Hashimoto, Ken-ya
2017-07-01
This paper discusses the possibility of realizing multimode filters composed of multiple single-mode resonators by using radio frequency surface and bulk acoustic wave (SAW/BAW) technologies. First, the filter operation and design principle are given. It is shown that excellent filter characteristics are achievable by combining multiple single-mode resonators with identical capacitance ratios provided that their resonance frequencies and clamped capacitances are set properly. Next, the effect of balun performance is investigated. It is shown that the total filter performance is significantly degraded by balun imperfections such as the common-mode rejection. Then, two circuits are proposed to improve the common-mode rejection, and their effectiveness is demonstrated.
7. Effect of nonplanar geometry on ion acoustic solitary waves in presence of ionization in collisional dusty plasma
Energy Technology Data Exchange (ETDEWEB)
Ghosh, Samiran [College of Textile Technology, Berhampore 742101, Murshidabad, West Bengal (India)]. E-mail: [email protected]
2005-04-11
It has been found that the dust ion acoustic solitary wave (DIASW) is governed by a modified form of Korteweg-de Vries (KdV) equation modified by the effects of ionization, particle collisions and bounded nonplanar geometry. Approximate analytical time evolution solution and also the numerical solution of modified form of KdV equation reveal that the wave amplitude grows exponentially with time due to ionization, whereas the bounded nonplanar geometry and collision reduce the instability growth rate.
8. Fluid simulation of dispersive and nondispersive ion acoustic waves in the presence of superthermal electrons
Science.gov (United States)
2016-10-01
One-dimensional fluid simulation is performed for the unmagnetized plasma consisting of cold fluid ions and superthermal electrons. Such a plasma system supports the generation of ion acoustic (IA) waves. A standard Gaussian type perturbation is used in both electron and ion equilibrium densities to excite the IA waves. The evolutionary profiles of the IA waves are obtained by varying the superthermal index and the amplitude of the initial perturbation. This simulation demonstrates that the amplitude of the initial perturbation and the superthermal index play an important role in determining the time evolution and the characteristics of the generated IA waves. The initial density perturbation in the system creates charge separation that drives the finite electrostatic potential in the system. This electrostatic potential later evolves into the dispersive and nondispersive IA waves in the simulation system. The density perturbation with the amplitude smaller than 10% of the equilibrium plasma density evolves into the dispersive IA waves, whereas larger density perturbations evolve into both dispersive and nondispersive IA waves for lower and higher superthermal index. The dispersive IA waves are the IA oscillations that propagate with constant ion plasma frequency, whereas the nondispersive IA waves are the IA solitary pulses (termed as IA solitons in the stability region) that propagate with the constant wave speed. The characteristics of the stable nondispersive IA solitons are found to be consistent with the nonlinear fluid theory. To the best of our knowledge, this is the first fluid simulation study that has considered the superthermal distributions for the plasma species to model the electrostatic solitary waves.
9. Scattering Matrix for the Interaction between Solar Acoustic Waves and Sunspots. I. Measurements
Science.gov (United States)
Yang, Ming-Hsu; Chou, Dean-Yi; Zhao, Hui
2017-01-01
Assessing the interaction between solar acoustic waves and sunspots is a scattering problem. The scattering matrix elements are the most commonly used measured quantities to describe scattering problems. We use the wavefunctions of scattered waves of NOAAs 11084 and 11092 measured in the previous study to compute the scattering matrix elements, with plane waves as the basis. The measured scattered wavefunction is from the incident wave of radial order n to the wave of another radial order n‧, for n=0{--}5. For a time-independent sunspot, there is no mode mixing between different frequencies. An incident mode is scattered into various modes with different wavenumbers but the same frequency. Working in the frequency domain, we have the individual incident plane-wave mode, which is scattered into various plane-wave modes with the same frequency. This allows us to compute the scattering matrix element between two plane-wave modes for each frequency. Each scattering matrix element is a complex number, representing the transition from the incident mode to another mode. The amplitudes of diagonal elements are larger than those of the off-diagonal elements. The amplitude and phase of the off-diagonal elements are detectable only for n-1≤slant n\\prime ≤slant n+1 and -3{{Δ }}k≤slant δ {k}x≤slant 3{{Δ }}k, where δ {k}x is the change in the transverse component of the wavenumber and Δk = 0.035 rad Mm‑1.
10. Plasma Metamaterials for Arbitrary Complex-Amplitude Wave Filters
Science.gov (United States)
2013-09-10
y = 120 mm, propagation waves join evanescent waves which are depressed as the position becomes apart from the edge of the metamaterial region...Kik, H. A. Atwater, S. Meltzer , E. Harel, E., B. E. Koel and A. A. Requicha, Nature Mat. 2, 229 (2003). 38 S. A. Maier and H. A. Atwater, J. Appl
11. Low power sessile droplet actuation via modulated surface acoustic waves
CERN Document Server
Baudoin, Michael; Matar, Olivier Bou; Herth, Etienne
2012-01-01
Low power actuation of sessile droplets is of primary interest for portable or hybrid lab-on-a-chip and harmless manipulation of biofluids. In this paper, we show that the acoustic power required to move or deform droplets via surface acoustic waves can be substantially reduced through the forcing of the drops inertio-capillary modes of vibrations. Indeed, harmonic, superharmonic and subharmonic (parametric) excitation of these modes are observed when the high frequency acoustic signal (19.5 MHz) is modulated around Rayleigh-Lamb inertio-capillary frequencies. This resonant behavior results in larger oscillations and quicker motion of the drops than in the non-modulated case.
12. High-frequency shear-horizontal surface acoustic wave sensor
Science.gov (United States)
Branch, Darren W
2013-05-07
A Love wave sensor uses a single-phase unidirectional interdigital transducer (IDT) on a piezoelectric substrate for leaky surface acoustic wave generation. The IDT design minimizes propagation losses, bulk wave interferences, provides a highly linear phase response, and eliminates the need for impedance matching. As an example, a high frequency (.about.300-400 MHz) surface acoustic wave (SAW) transducer enables efficient excitation of shear-horizontal waves on 36.degree. Y-cut lithium tantalate (LTO) giving a highly linear phase response (2.8.degree. P-P). The sensor has the ability to detect at the pg/mm.sup.2 level and can perform multi-analyte detection in real-time. The sensor can be used for rapid autonomous detection of pathogenic microorganisms and bioagents by field deployable platforms.
13. Drift and ion acoustic wave driven vortices with superthermal electrons
Energy Technology Data Exchange (ETDEWEB)
Ali Shan, S. [Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad (Pakistan); National Centre For Physics (NCP), Shahdra Valley Road, QAU Campus, 44000 Islamabad (Pakistan); Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad (Pakistan); Haque, Q. [Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad (Pakistan); National Centre For Physics (NCP), Shahdra Valley Road, QAU Campus, 44000 Islamabad (Pakistan)
2012-08-15
Linear and nonlinear analysis of coupled drift and acoustic mode is presented in an inhomogeneous electron-ion plasma with {kappa}-distributed electrons. A linear dispersion relation is found which shows that the phase speed of both the drift wave and the ion acoustic wave decreases in the presence of superthermal electrons. Several limiting cases are also discussed. In the nonlinear regime, stationary solutions in the form of dipolar and monopolar vortices are obtained. It is shown that the condition for the boundedness of the solution implies that the speed of drift wave driven vortices reduces with increase in superthermality effect. Ignoring density inhomogeniety, it is investigated that the lower and upper limits on the speed of the ion acoustic driven vortices spread with the inclusion of high energy electrons. The importance of results with reference to space plasmas is also pointed out.
14. Individually Identifiable Surface Acoustic Wave Sensors, Tags and Systems
Science.gov (United States)
Hines, Jacqueline H. (Inventor); Solie, Leland P. (Inventor); Tucker, Dana Y. G. (Inventor); Hines, Andrew T. (Inventor)
2017-01-01
A surface-launched acoustic wave sensor tag system for remotely sensing and/or providing identification information using sets of surface acoustic wave (SAW) sensor tag devices is characterized by acoustic wave device embodiments that include coding and other diversity techniques to produce groups of sensors that interact minimally, reducing or alleviating code collision problems typical of prior art coded SAW sensors and tags, and specific device embodiments of said coded SAW sensor tags and systems. These sensor/tag devices operate in a system which consists of one or more uniquely identifiable sensor/tag devices and a wireless interrogator. The sensor device incorporates an antenna for receiving incident RF energy and re-radiating the tag identification information and the sensor measured parameter(s). Since there is no power source in or connected to the sensor, it is a passive sensor. The device is wirelessly interrogated by the interrogator.
15. Broadband metamaterial for nonresonant matching of acoustic waves.
Science.gov (United States)
D'Aguanno, G; Le, K Q; Trimm, R; Alù, A; Mattiucci, N; Mathias, A D; Aközbek, N; Bloemer, M J
2012-01-01
Unity transmittance at an interface between bulk media is quite common for polarized electromagnetic waves incident at the Brewster angle, but it is rarely observed for sound waves at any angle of incidence. In the following, we theoretically and experimentally demonstrate an acoustic metamaterial possessing a Brewster-like angle that is completely transparent to sound waves over an ultra-broadband frequency range with >100% bandwidth. The metamaterial, consisting of a hard metal with subwavelength apertures, provides a surface impedance matching mechanism that can be arbitrarily tailored to specific media. The nonresonant nature of the impedance matching effectively decouples the front and back surfaces of the metamaterial allowing one to independently tailor the acoustic impedance at each interface. On the contrary, traditional methods for acoustic impedance matching, for example in medical imaging, rely on resonant tunneling through a thin antireflection layer, which is inherently narrowband and angle specific.
16. Nonlinear propagation of weakly relativistic ion-acoustic waves in electron–positron–ion plasma
M G HAFEZ; M R TALUKDER; M HOSSAIN ALI
2016-11-01
This work presents theoretical and numerical discussion on the dynamics of ion-acoustic solitary wave for weakly relativistic regime in unmagnetized plasma comprising non-extensive electrons, Boltzmann positrons and relativistic ions. In order to analyse the nonlinear propagation phenomena, the Korteweg–de Vries(KdV) equation is derived using the well-known reductive perturbation method. The integration of the derived equation is carried out using the ansatz method and the generalized Riccati equation mapping method. The influenceof plasma parameters on the amplitude and width of the soliton and the electrostatic nonlinear propagation of weakly relativistic ion-acoustic solitary waves are described. The obtained results of the nonlinear low-frequencywaves in such plasmas may be helpful to understand various phenomena in astrophysical compact object and space physics.
17. Acoustical breakdown of materials by focusing of laser-generated Rayleigh surface waves
Science.gov (United States)
Veysset, David; Maznev, A. A.; Veres, István A.; Pezeril, Thomas; Kooi, Steven E.; Lomonosov, Alexey M.; Nelson, Keith A.
2017-07-01
Focusing of high-amplitude surface acoustic waves leading to material damage is visualized in an all-optical experiment. The optical setup includes a lens and an axicon that focuses an intense picosecond excitation pulse into a ring-shaped pattern at the surface of a gold-coated glass substrate. Optical excitation induces a surface acoustic wave (SAW) that propagates in the plane of the sample and converges toward the center. The evolution of the SAW profile is monitored using interferometry with a femtosecond probe pulse at variable time delays. The quantitative analysis of the full-field images provides direct information about the surface displacement profiles, which are compared to calculations. The high stress at the focal point leads to the removal of the gold coating and, at higher excitation energies, to damage of the glass substrate. The results open the prospect for testing material strength on the microscale using laser-generated SAWs.
18. Investigation of 3D surface acoustic waves in granular media with 3-color digital holography
Science.gov (United States)
Leclercq, Mathieu; Picart, Pascal; Penelet, Guillaume; Tournat, Vincent
2017-01-01
This paper reports the implementation of digital color holography to investigate elastic waves propagating along a layer of a granular medium. The holographic set-up provides simultaneous recording and measurement of the 3D dynamic displacement at the surface. Full-field measurements of the acoustic amplitude and phase at different excitation frequencies are obtained. It is shown that the experimental data can be used to obtain the dispersion curve of the modes propagating in this granular medium layer. The experimental dispersion curve and that obtained from a finite element modeling of the problem are found to be in good agreement. In addition, full-field images of the interaction of an acoustic wave guided in the granular layer with a buried object are also shown.
19. Nonlinear electron acoustic cyclotron waves in presence of uniform magnetic field
Energy Technology Data Exchange (ETDEWEB)
Dutta, Manjistha; Khan, Manoranjan [Department of Instrumentation Science, Jadavpur University, Kolkata 700 032 (India); Ghosh, Samiran [Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700 009 (India); Roychoudhury, Rajkumar [Indian Statistical Institute, Kolkata 700 108 (India); Chakrabarti, Nikhil [Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064 (India)
2013-04-15
Nonlinear electron acoustic cyclotron waves (EACW) are studied in a quasineutral plasma in presence of uniform magnetic field. The fluid model is used to describe the dynamics of two temperature electron species in a stationary charge neutral inhomogeneous background. In long wavelength limit, it is shown that the linear electron acoustic wave is modified by the uniform magnetic field similar to that of electrostatic ion cyclotron wave. Nonlinear equations for these waves are solved by using Lagrangian variables. Results show that the spatial solitary wave-like structures are formed due to nonlinearities and dispersions. These structures transiently grow to larger amplitude unless dispersive effect is actively operative and able to arrest this growth. We have found that the wave dispersion originated from the equilibrium inhomogeneity through collective effect and is responsible for spatiotemporal structures. Weak dispersion is not able to stop the wave collapse and singular structures of EACW are formed. Relevance of the results in the context of laboratory and space plasmas is discussed.
20. Effects of trapped electrons on the oblique propagation of ion acoustic solitary waves in electron-positron-ion plasmas
Science.gov (United States)
Hafez, M. G.; Roy, N. C.; Talukder, M. R.; Hossain Ali, M.
2016-08-01
The characteristics of the nonlinear oblique propagation of ion acoustic solitary waves in unmagnetized plasmas consisting of Boltzmann positrons, trapped electrons and ions are investigated. The modified Kadomtsev-Petviashivili ( m K P ) equation is derived employing the reductive perturbation technique. The parametric effects on phase velocity, Sagdeev potential, amplitude and width of solitons, and electrostatic ion acoustic solitary structures are graphically presented with the relevant physical explanations. This study may be useful for the better understanding of physical phenomena concerned in plasmas in which the effects of trapped electrons control the dynamics of wave.
1. Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation
Directory of Open Access Journals (Sweden)
Raicharan Denra
2016-12-01
Full Text Available In this paper, characteristics of small amplitude nonlinear dust acoustic wave have been investigated in a unmagnetized, collisionless, Lorentzian dusty plasma where electrons and ions are inertialess and modeled by generalized Lorentzian Kappa distribution. Dust grains are inertial and equilibrium dust charge is negative. Both adiabatic and nonadiabatic fluctuation of charges on dust grains have been taken under consideration. For adiabatic dust charge variation reductive perturbation analysis gives rise to a KdV equation that governs the nonlinear propagation of dust acoustic waves having soliton solutions. For nonadiabatic dust charge variation nonlinear propagation of dust acoustic wave obeys KdV-Burger equation and gives rise to dust acoustic shock waves. Numerical estimation for adiabatic grain charge variation shows the existence of rarefied soliton whose amplitude and width varies with grain charges. Amplitude and width of the soliton have been plotted for different electron Kappa indices keeping ion velocity distribution Maxwellian. For non adiabatic dust charge variation, ratio of the coefficients of Burger term and dispersion term have been plotted against charge fluctuation for different kappa indices. All these results approach to the results of Maxwellian plasma if both electron and ion kappa tends to infinity.
2. Characteristics of nonlinear dust acoustic waves in a Lorentzian dusty plasma with effect of adiabatic and nonadiabatic grain charge fluctuation
Science.gov (United States)
Denra, Raicharan; Paul, Samit; Sarkar, Susmita
2016-12-01
In this paper, characteristics of small amplitude nonlinear dust acoustic wave have been investigated in a unmagnetized, collisionless, Lorentzian dusty plasma where electrons and ions are inertialess and modeled by generalized Lorentzian Kappa distribution. Dust grains are inertial and equilibrium dust charge is negative. Both adiabatic and nonadiabatic fluctuation of charges on dust grains have been taken under consideration. For adiabatic dust charge variation reductive perturbation analysis gives rise to a KdV equation that governs the nonlinear propagation of dust acoustic waves having soliton solutions. For nonadiabatic dust charge variation nonlinear propagation of dust acoustic wave obeys KdV-Burger equation and gives rise to dust acoustic shock waves. Numerical estimation for adiabatic grain charge variation shows the existence of rarefied soliton whose amplitude and width varies with grain charges. Amplitude and width of the soliton have been plotted for different electron Kappa indices keeping ion velocity distribution Maxwellian. For non adiabatic dust charge variation, ratio of the coefficients of Burger term and dispersion term have been plotted against charge fluctuation for different kappa indices. All these results approach to the results of Maxwellian plasma if both electron and ion kappa tends to infinity.
3. A metasurface carpet cloak for electromagnetic, acoustic and water waves.
Science.gov (United States)
Yang, Yihao; Wang, Huaping; Yu, Faxin; Xu, Zhiwei; Chen, Hongsheng
2016-01-29
We propose a single low-profile skin metasurface carpet cloak to hide objects with arbitrary shape and size under three different waves, i.e., electromagnetic (EM) waves, acoustic waves and water waves. We first present a metasurface which can control the local reflection phase of these three waves. By taking advantage of this metasurface, we then design a metasurface carpet cloak which provides an additional phase to compensate the phase distortion introduced by a bump, thus restoring the reflection waves as if the incident waves impinge onto a flat mirror. The finite element simulation results demonstrate that an object can be hidden under these three kinds of waves with a single metasurface cloak.
4. Precessional magnetization switching by a surface acoustic wave
Science.gov (United States)
Thevenard, L.; Camara, I. S.; Majrab, S.; Bernard, M.; Rovillain, P.; Lemaître, A.; Gourdon, C.; Duquesne, J.-Y.
2016-04-01
Precessional switching allows subnanosecond and deterministic reversal of magnetic data bits. It relies on triggering a large-angle, highly nonlinear precession of magnetic moments around a bias field. Here we demonstrate that a surface acoustic wave (SAW) propagating on a magnetostrictive semiconducting material produces an efficient torque that induces precessional switching. This is evidenced by Kerr microscopy and acoustic behavior analysis in a (Ga,Mn)(As,P) thin film. Using SAWs should therefore allow remote and wave control of individual magnetic bits at potentially GHz frequencies.
5. An Unconditionally Stable Method for Solving the Acoustic Wave Equation
Directory of Open Access Journals (Sweden)
Zhi-Kai Fu
2015-01-01
Full Text Available An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.
6. Dispersive analysis of the $S$-, $P$-, $D$-, and $F$-wave $\\pi\\pi$ amplitudes
CERN Document Server
Bydžovský, P; Nazari, V
2016-01-01
A reanalysis of $\\pi\\pi$ amplitudes for all important partial-waves below about 2 GeV is presented. A set of once subtracted dispersion relations with imposed crossing symmetry condition is used to modify unitary multi-channel amplitudes in the $S$, $P$, $D$, and $F$ waves. So far, these specific amplitudes constructed in our works and many other analyzes have been fitted only to experimental data and therefore do not fulfill the crossing symmetry condition. In the present analysis, the self consistent, i.e. unitary and fulfilling the crossing symmetry, amplitudes for the $S$, $P$, $D$, and $F$ waves are formed. The proposed very effective and simple method of modification of the $\\pi\\pi$ amplitudes does not change their previous-original mathematical structure and the method can be easily applied in various other analyzes.
7. Properties of Materials Using Acoustic Waves.
Science.gov (United States)
1984-10-01
CLASSiFICATIOO OF THIS PAGIR elM. DMe Eatae" to nonlinear acoustics which should permit us to cast problems with geometric and other complexities into a...on the kinetics of chemical reactions . 5. New theoretical approaches in nonlinear acoustics (R.M. McGowan and Professor B.-T. Chu) We are working to...of water and methanol was compared with the theoretical predictions given by Marston’s theory and the simplified model (Hsu 1983). This set of data
8. Reflection of acoustic wave from the elastic seabed with an overlying gassy poroelastic layer
Science.gov (United States)
Chen, Weiyun; Wang, Zhihua; Zhao, Kai; Chen, Guoxing; Li, Xiaojun
2015-10-01
Based on the multiphase poroelasticity theory, the reflection characteristics of an obliquely incident acoustic wave upon a plane interface between overlying water and a gassy marine sediment layer with underlying elastic solid seabed are investigated. The sandwiched gassy layer is modelled as a porous material with finite thickness, which is saturated by two compressible and viscous fluids (liquid and gas). The closed-form expression for the amplitude ratio of the reflected wave, called reflection coefficient, is derived theoretically according to the boundary conditions at the upper and lower interfaces in our proposed model. Using numerical calculation, the influences of layer thickness, incident angle, wave frequency and liquid saturation of sandwiched porous layer on the reflection coefficient are analysed, respectively. It is revealed that the reflection coefficient is closely associated with incident angle and sandwiched layer thickness. Moreover, in different frequency ranges, the dependence of the wave reflection characteristics on moisture (or gas) variations in the intermediate marine sediment layer is distinguishing.
9. Electron acoustic solitary waves in a magnetized plasma with nonthermal electrons and an electron beam
Science.gov (United States)
Singh, S. V.; Devanandhan, S.; Lakhina, G. S.; Bharuthram, R.
2016-08-01
A theoretical investigation is carried out to study the obliquely propagating electron acoustic solitary waves having nonthermal hot electrons, cold and beam electrons, and ions in a magnetized plasma. We have employed reductive perturbation theory to derive the Korteweg-de-Vries-Zakharov-Kuznetsov (KdV-ZK) equation describing the nonlinear evolution of these waves. The two-dimensional plane wave solution of KdV-ZK equation is analyzed to study the effects of nonthermal and beam electrons on the characteristics of the solitons. Theoretical results predict negative potential solitary structures. We emphasize that the inclusion of finite temperature effects reduces the soliton amplitudes and the width of the solitons increases by an increase in the obliquity of the wave propagation. The numerical analysis is presented for the parameters corresponding to the observations of "burst a" event by Viking satellite on the auroral field lines.
10. Parametric instabilities of large amplitude Alfven waves with obliquely propagating sidebands
Science.gov (United States)
Vinas, A. F.; Goldstein, M. L.
1992-01-01
This paper presents a brief report on properties of the parametric decay and modulational, filamentation, and magnetoacoustic instabilities of a large amplitude, circularly polarized Alfven wave. We allow the daughter and sideband waves to propagate at an arbitrary angle to the background magnetic field so that the electrostatic and electromagnetic characteristics of these waves are coupled. We investigate the dependance of these instabilities on dispersion, plasma/beta, pump wave amplitude, and propagation angle. Analytical and numerical results are compared with numerical simulations to investigate the full nonlinear evolution of these instabilities.
11. Effects of Periodic Forcing Amplitude on the Spiral Wave Resonance Drift
Institute of Scientific and Technical Information of China (English)
WU Ning-Jie; LI Bing-Wei; YING He-Ping
2006-01-01
@@ We study dynamics of spiral waves under a uniform periodic temporal forcing in an excitable medium. With a specific combination of frequency and amplitude of the external periodic forcing, a resonance drift of a spiral wave occurs along a straight line, and it is accompanied by a complicated ‘flower-like’ motion on each side of this bifurcate boundary line. It is confirmed that the straight-line drift frequency of spiral waves is not locked to the nature rotation frequency as the forcing amplitude expends the range of the spiral wave frequency. These results are further verified numerically for a simplified kinematical model.
12. Electro-acoustic solitary waves and double layers in a quantum plasma
Science.gov (United States)
Dip, P. R.; Hossen, M. A.; Salahuddin, M.; Mamun, A. A.
2017-02-01
A meticulous theoretical investigation has carried out to study the properties related to the higher-order nonlinearity of the electro-acoustic waves, specifically ion-acoustic (IA) waves in an unmagnetized, collisionless, quantum electron-positron-ion (EPI) plasma. The plasma system is supposed to be formed of positively charged inertial heavy ions, inertialess electrons and positrons. The reductive perturbation technique is employed to derive the modified Korteweg-de Vries (mK-dV) equation to analyze the solitary waves (SWs), and the standard Gardner (SG) equation to analyze the higher-order SWs as well as double layers (DLs). The basic features (viz. amplitude, width, phase speed, etc.) of the IA SWs and DLs are examined. The comparison between the mK-dV SWs and SG SWs is also made. It is found that the amplitude, width, phase speed, etc. of the IA SWs and DLs are significantly modified by the effects of the both Fermi temperatures as well as pressures and Bohm potentials of electrons and positrons. Our findings may be useful in explaining the physics behind the formation of the IA waves in both astrophysical and laboratory EPI plasmas (viz. white dwarfs, laser-solid matter interaction experiments, etc.).
13. A penalization method for calculating the flow beneath travelling water waves of large amplitude
CERN Document Server
2014-01-01
A penalization method for a suitable reformulation of the governing equations as a constrained optimization problem provides accurate numerical simulations for large-amplitude travelling water waves in irrotational flows and in flows with constant vorticity.
14. Revival of the Phase-Amplitude Description of a Quantum-Mechanical Wave Function
Science.gov (United States)
Rawitscher, George
2017-01-01
The phase-amplitude description of a wave function is formulated by means of a new linear differential-integral equation, which is valid in the region of turning points. A numerical example for a Coulomb potential is presented.
15. Efficient counter-propagating wave acoustic micro-particle manipulation
Science.gov (United States)
Grinenko, A.; Ong, C. K.; Courtney, C. R. P.; Wilcox, P. D.; Drinkwater, B. W.
2012-12-01
A simple acoustic system consisting of a pair of parallel singe layered piezoelectric transducers submerged in a fluid used to form standing waves by a superposition of two counter-propagating waves is reported. The nodal positions of the standing wave are controlled by applying a variable phase difference to the transducers. This system was used to manipulate polystyrene micro-beads trapped at the nodal positions of the standing wave. The demonstrated good manipulation capability of the system is based on a lowering of the reflection coefficient in a narrow frequency band near the through-thickness resonance of the transducer plates.
16. Numerical modelling of overtaking collisions of dust acoustic waves in plasmas
Science.gov (United States)
Gao, Dong-Ning; Zhang, Heng; Zhang, Jie; Li, Zhong-Zheng; Duan, Wen-Shan
2016-10-01
The overtaking collision between two single and unidirectional dust acoustic waves in dusty plasmas consisting of Boltzmann electrons and ions, and negative dust grains has been investigated by PIC simulation method. The well-known physical phenomenon is that the larger soliton moves faster, approaches the smaller one and after the overtaking collision both resume their original shape and speed with different phase shifts. The merging amplitude of two solitons and phase shifts of solitons after collision are given. These PIC results are compared with the overtaking collision of two-soliton solution (TSS) of KdV equaiton obtained by Hirota bilinear method. Comparisons between two indicates that if the amplitude of fast soliton is large enough or the amplitude of slow soliton is small enough, the simulation results are consistent with the interaction of Hirota results.
17. Nonlinear ion-acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves
Energy Technology Data Exchange (ETDEWEB)
Guo, Shimin, E-mail: [email protected] [School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049 (China); Research Group MAC, Centrum Wiskunde and Informatica, Amsterdam, 1098XG (Netherlands); Mei, Liquan, E-mail: [email protected] [School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049 (China); Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, 710049 (China); Sun, Anbang [Research Group MAC, Centrum Wiskunde and Informatica, Amsterdam, 1098XG (Netherlands)
2013-05-15
The nonlinear propagation of planar and nonplanar (cylindrical and spherical) ion-acoustic waves in an unmagnetized electron–positron–ion–dust plasma with two-electron temperature distributions is investigated in the context of the nonextensive statistics. Using the reductive perturbation method, a modified nonlinear Schrödinger equation is derived for the potential wave amplitude. The effects of plasma parameters on the modulational instability of ion-acoustic waves are discussed in detail for planar as well as for cylindrical and spherical geometries. In addition, for the planar case, we analyze how the plasma parameters influence the nonlinear structures of the first- and second-order ion-acoustic rogue waves within the modulational instability region. The present results may be helpful in providing a good fit between the theoretical analysis and real applications in future spatial observations and laboratory plasma experiments. -- Highlights: ► Modulational instability of ion-acoustic waves in a new plasma model is discussed. ► Tsallis’s statistics is considered in the model. ► The second-order ion-acoustic rogue wave is studied for the first time.
18. Amplitude reconstruction from complete experiments and truncated partial-wave expansions
CERN Document Server
Workman, R L; Wunderlich, Y; Doering, M; Haberzettl, H
2016-01-01
We compare the methods of amplitude reconstruction, for a complete experiment and a truncated partial wave analysis, applied to the photoproduction of pseudo-scalar mesons. The approach is pedagogical, showing in detail how the amplitude reconstruction (observables measured at a single energy and angle) is related to a truncated partial-wave analysis (observables measured at a single energy and a number of angles).
19. Sources and sinks separating domains of left- and right-traveling waves Experiment versus amplitude equations
CERN Document Server
Alvarez, R; Van Saarloos, W; Alvarez, Roberto; Hecke, Martin van; Saarloos, Wim van
1996-01-01
In many pattern forming systems that exhibit traveling waves, sources and sinks occur which separate patches of oppositely traveling waves. We show that simple qualitative features of their dynamics can be compared to predictions from coupled amplitude equations. In heated wire convection experiments, we find a discrepancy between the observed multiplicity of sources and theoretical predictions. The expression for the observed motion of sinks is incompatible with any amplitude equation description.
20. Sources and sinks separating domains of left- and right-traveling waves: experiment versus amplitude equations
OpenAIRE
Saarloos, van, W.; Alvarez, R.; Hecke, van, M
1997-01-01
In many pattern forming systems that exhibit traveling waves, sources and sinks occur which separate patches of oppositely traveling waves. We show that simple qualitative features of their dynamics can be compared to predictions from coupled amplitude equations. In heated wire convection experiments, we find a discrepancy between the observed multiplicity of sources and theoretical predictions. The expression for the observed motion of sinks is incompatible with any amplitude equation descri...
1. Reflection and Transmission of Acoustic Waves at Semiconductor - Liquid Interface
Directory of Open Access Journals (Sweden)
J. N. Sharma
2011-09-01
Full Text Available The study of reflection and transmission characteristics of acoustic waves at the interface of a semiconductor halfspace underlying an inviscid liquid has been carried out. The reflection and transmission coefficients of reflected and transmitted waves have been obtained for quasi-longitudinal (qP wave incident at the interface from fluid to semiconductor. The numerical computations of reflection and transmission coefficients have been carried out with the help of Gauss elimination method by using MATLAB programming for silicon (Si, germanium (Ge and silicon nitride (Si3N4 semiconductors. In order to interpret and compare, the computer simulated results are plotted graphically. The study may be useful in semiconductors, seismology and surface acoustic wave (SAW devices in addition to engines of the space shuttles.
2. Surface spin-electron acoustic waves in magnetically ordered metals
CERN Document Server
Andreev, Pavel A
2015-01-01
Degenerate plasmas with motionless ions show existence of three surface waves: the Langmuir wave, the electromagnetic wave, and the zeroth sound. Applying the separated spin evolution quantum hydrodynamics to half-space plasma we demonstrate the existence of the surface spin-electron acoustic wave (SSEAW). We study dispersion of the SSEAW. We show that there is hybridization between the surface Langmuir wave and the SSEAW at rather small spin polarization. In the hybridization area the dispersion branches are located close to each other. In this area there is a strong interaction between these waves leading to the energy exchange. Consequently, generating the Langmuir waves with the frequencies close to hybridization area we can generate the SSEAWs. Thus, we report a method of creation of the SEAWs.
3. Nonlinear propagation of dust-acoustic solitary waves in a dusty plasma with arbitrarily charged dust and trapped electrons
O Rahman; A A Mamun
2013-06-01
A theoretical investigation of dust-acoustic solitary waves in three-component unmagnetized dusty plasma consisting of trapped electrons, Maxwellian ions, and arbitrarily charged cold mobile dust was done. It has been found that, owing to the departure from the Maxwellian electron distribution to a vortex-like one, the dynamics of small but finite amplitude dust-acoustic (DA) waves is governed by a nonlinear equation of modified Korteweg–de Vries (mKdV) type (instead of KdV). The reductive perturbation method was employed to study the basic features (amplitude, width, speed, etc.) of DA solitary waves which are significantly modified by the presence of trapped electrons. The implications of our results in space and laboratory plasmas are briefly discussed.
4. Theoretical Model of Acoustic Wave Propagation in Shallow Water
Directory of Open Access Journals (Sweden)
Kozaczka Eugeniusz
2017-06-01
Full Text Available The work is devoted to the propagation of low frequency waves in a shallow sea. As a source of acoustic waves, underwater disturbances generated by ships were adopted. A specific feature of the propagation of acoustic waves in shallow water is the proximity of boundaries of the limiting media characterised by different impedance properties, which affects the acoustic field coming from a source situated in the water layer “deformed” by different phenomena. The acoustic field distribution in the real shallow sea is affected not only by multiple reflections, but also by stochastic changes in the free surface shape, and statistical changes in the seabed shape and impedance. The paper discusses fundamental problems of modal sound propagation in the water layer over different types of bottom sediments. The basic task in this case was to determine the acoustic pressure level as a function of distance and depth. The results of the conducted investigation can be useful in indirect determination of the type of bottom.
5. Elastic Wave Propagation Mechanisms in Underwater Acoustic Environments
Science.gov (United States)
2015-09-30
excited flexural mode that propagates in the ice layer at certain acoustic frequencies in ice-covered environments.[3] • Previously implemented EPE self...and ks,3, corresponding to the water layer sound speed, bottom compressional and shear wave speed, and ice layer compressional and shear wave speed... excitation of the Scholte interface mode. Dashed curve shows spectra for a source at 1 m depth and receiver at 25 m, showing the excitation of the
6. Effects of acoustic waves on stick-slip in granular media and implications for earthquakes
Science.gov (United States)
Johnson, P.A.; Savage, H.; Knuth, M.; Gomberg, J.; Marone, Chris
2008-01-01
It remains unknown how the small strains induced by seismic waves can trigger earthquakes at large distances, in some cases thousands of kilometres from the triggering earthquake, with failure often occurring long after the waves have passed. Earthquake nucleation is usually observed to take place at depths of 10-20 km, and so static overburden should be large enough to inhibit triggering by seismic-wave stress perturbations. To understand the physics of dynamic triggering better, as well as the influence of dynamic stressing on earthquake recurrence, we have conducted laboratory studies of stick-slip in granular media with and without applied acoustic vibration. Glass beads were used to simulate granular fault zone material, sheared under constant normal stress, and subject to transient or continuous perturbation by acoustic waves. Here we show that small-magnitude failure events, corresponding to triggered aftershocks, occur when applied sound-wave amplitudes exceed several microstrain. These events are frequently delayed or occur as part of a cascade of small events. Vibrations also cause large slip events to be disrupted in time relative to those without wave perturbation. The effects are observed for many large-event cycles after vibrations cease, indicating a strain memory in the granular material. Dynamic stressing of tectonic faults may play a similar role in determining the complexity of earthquake recurrence. ??2007 Nature Publishing Group.
7. Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves
Energy Technology Data Exchange (ETDEWEB)
Mitri, F. G., E-mail: [email protected] [Chevron, Area 52 Technology–ETC, Santa Fe, New Mexico 87508 (United States)
2015-12-07
The acoustic radiation force on a 2D elliptical (non-circular) cylinder centered on the axis of wave propagation of plane quasi-standing and standing waves is derived, based on the partial-wave series expansion (PWSE) method in cylindrical coordinates. A non-dimensional acoustic radiation force function, which is the radiation force per unit length, per characteristic energy density and per unit cross-sectional surface of the ellipse, is defined in terms of the scattering coefficients that are determined by applying the Neumann boundary condition for an immovable surface. A system of linear equations involving a single numerical integration procedure is solved by matrix inversion. Numerical simulations showing the transition from the quasi-standing to the (equi-amplitude) standing wave behaviour are performed with particular emphasis on the aspect ratio a/b, where a and b are the ellipse semi-axes, as well as the dimensionless size parameter kb (where k is the wavenumber), without the restriction to a particular range of frequencies. It is found that at high kb values > 1, the radiation force per length with broadside incidence is larger, whereas the opposite situation occurs in the long-wavelength limit (i.e., kb < 1). The results are particularly relevant in acoustic levitation of elliptical cylinders, the acoustic stabilization of liquid columns in a host medium, acousto-fluidics devices, and other particle dynamics applications to name a few. Moreover, the formalism presented here may be effectively applied to compute the acoustic radiation force on other 2D surfaces of arbitrary shape such as super-ellipses, Chebyshev cylindrical particles, or other non-circular geometries.
8. Amplitude equations for coupled electrostatic waves in the limit of weak instability
CERN Document Server
Crawford, J D; Crawford, John David; Knobloch, Edgar
1997-01-01
We consider the simplest instabilities involving multiple unstable electrostatic plasma waves corresponding to four-dimensional systems of mode amplitude equations. In each case the coupled amplitude equations are derived up to third order terms. The nonlinear coefficients are singular in the limit in which the linear growth rates vanish together. These singularities are analyzed using techniques developed in previous studies of a single unstable wave. In addition to the singularities familiar from the one mode problem, there are new singularities in coefficients coupling the modes. The new singularities are most severe when the two waves have the same linear phase velocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result the short wave mode saturates at a dramatically smaller amplitude than that predicted for the weak growth rate regime on the basis of single mode theory. In contrast the long wave mode retains the single mode scaling. If these resonance conditions are not satisfied both mo...
9. A method for the frequency control in time-resolved two-dimensional gigahertz surface acoustic wave imaging
Directory of Open Access Journals (Sweden)
Shogo Kaneko
2014-01-01
Full Text Available We describe an extension of the time-resolved two-dimensional gigahertz surface acoustic wave imaging based on the optical pump-probe technique with periodic light source at a fixed repetition frequency. Usually such imaging measurement may generate and detect acoustic waves with their frequencies only at or near the integer multiples of the repetition frequency. Here we propose a method which utilizes the amplitude modulation of the excitation pulse train to modify the generation frequency free from the mentioned limitation, and allows for the first time the discrimination of the resulted upper- and lower-side-band frequency components in the detection. The validity of the method is demonstrated in a simple measurement on an isotropic glass plate covered by a metal thin film to extract the dispersion curves of the surface acoustic waves.
10. Numerical Computation of Large Amplitude Internal Solitary Waves,
Science.gov (United States)
1981-03-20
provide adequate resolution. All computations were performed on a CDC Cyber 176 computer, and it takes slightly less than one CPU second to obtain a...H. Segur , Lgn Internal Waves in Fluids of Great Depth, Studies in Applied Math., 62 (1980), pp. 249-262. [3] E. Allgower and K. Georg, Simlicial -and
11. Numerical modelling of nonlinear full-wave acoustic propagation
Energy Technology Data Exchange (ETDEWEB)
Velasco-Segura, Roberto, E-mail: [email protected]; Rendón, Pablo L., E-mail: [email protected] [Grupo de Acústica y Vibraciones, Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Ciudad Universitaria, Apartado Postal 70-186, C.P. 04510, México D.F., México (Mexico)
2015-10-28
The various model equations of nonlinear acoustics are arrived at by making assumptions which permit the observation of the interaction with propagation of either single or joint effects. We present here a form of the conservation equations of fluid dynamics which are deduced using slightly less restrictive hypothesis than those necessary to obtain the well known Westervelt equation. This formulation accounts for full wave diffraction, nonlinearity, and thermoviscous dissipative effects. A two-dimensional, finite-volume method using Roe’s linearisation has been implemented to obtain numerically the solution of the proposed equations. This code, which has been written for parallel execution on a GPU, can be used to describe moderate nonlinear phenomena, at low Mach numbers, in domains as large as 100 wave lengths. Applications range from models of diagnostic and therapeutic HIFU, to parametric acoustic arrays and nonlinear propagation in acoustic waveguides. Examples related to these applications are shown and discussed.
12. An analytical model for the amplitude of lee waves forming on the boundary layer inversion
Science.gov (United States)
Sachsperger, Johannes; Serafin, Stefano; Stiperski, Ivana; Grubišić, Vanda
2016-04-01
Lee waves are horizontally propagating gravity waves with a typical wavelength of 5-15 km that may be generated when stratified flow is lifted over a mountain. A frequently observed type of such waves is that of interfacial lee waves. Those develop, similar to surface waves on a free water surface, when the upstream flow features a density discontinuity. Such conditions are often present for example at the capping inversion in boundary layer flow. The dynamics of interfacial lee waves can be described concisely with linear interfacial gravity wave theory. However, while this theoretical framework accurately describes the wavelength, it fails to properly predict the amplitude of lee waves. It is well known that large amplitude lee waves may lead to low-level turbulence, which poses a potential hazard for aviation. Therefore, this property of interfacial lee waves deserves further attention. In this study, we develop a simple analytical model for the amplitude of lee waves forming on the boundary layer inversion. This model is based on the energetics of two-layer flow. We obtain an expression for the wave amplitude by equating the energy loss across an internal jump with the energy radiation through lee waves. The verification of the result with water tank experiments of density-stratified two-layer flow over two-dimensional topography from the HYDRALAB campaign shows good agreement between theory and observations. This new analytical model may be useful in determining potential hazards of interfacial lee waves with negligible computational cost as compared to numerical weather prediction models.
13. Acoustic waves in shock tunnels and expansion tubes
Science.gov (United States)
Paull, A.; Stalker, R. J.
1992-01-01
It is shown that disturbances in shock and expansion tubes can be modelled as lateral acoustic waves. The ratio of sound speed across the driver-test gas interface is shown to govern the quantity of noise in the test gas. Frequency 'focusing' which is fundamental to centered unsteady expansions is discussed and displayed in centerline pitot pressure measurements.
14. Monolithic ZnO SAW (Surface Acoustic Waves) structures
Science.gov (United States)
Gunshor, R. L.; Pierret, R. F.
1983-07-01
ZnO-on-silicon surface acoustic wave devices have been fabricated and tested. Electronic erasure of a stored correlator reference was demonstrated, the effect of laser annealing on propagation loss was examined, preliminary ageing studies were performed, and a conceptually new mode conversion resonator configuration was reported.
15. Ion Acoustic Waves in the Presence of Langmuir Oscillations
DEFF Research Database (Denmark)
Pécseli, Hans
1976-01-01
The dielectric function for long-wavelength, low-frequency ion acoustic waves in the presence of short-wavelength, high-frequency electron oscillations is presented, where the ions are described by the collision-free Vlasov equation. The effect of the electron oscillations can be appropriately...
16. Acoustic wave propagation in high-pressure system.
Science.gov (United States)
Foldyna, Josef; Sitek, Libor; Habán, Vladimír
2006-12-22
Recently, substantial attention is paid to the development of methods of generation of pulsations in high-pressure systems to produce pulsating high-speed water jets. The reason is that the introduction of pulsations into the water jets enables to increase their cutting efficiency due to the fact that the impact pressure (so-called water-hammer pressure) generated by an impact of slug of water on the target material is considerably higher than the stagnation pressure generated by corresponding continuous jet. Special method of pulsating jet generation was developed and tested extensively under the laboratory conditions at the Institute of Geonics in Ostrava. The method is based on the action of acoustic transducer on the pressure liquid and transmission of generated acoustic waves via pressure system to the nozzle. The purpose of the paper is to present results obtained during the research oriented at the determination of acoustic wave propagation in high-pressure system. The final objective of the research is to solve the problem of transmission of acoustic waves through high-pressure water to generate pulsating jet effectively even at larger distances from the acoustic source. In order to be able to simulate numerically acoustic wave propagation in the system, it is necessary among others to determine dependence of the sound speed and second kinematical viscosity on operating pressure. Method of determination of the second kinematical viscosity and speed of sound in liquid using modal analysis of response of the tube filled with liquid to the impact was developed. The response was measured by pressure sensors placed at both ends of the tube. Results obtained and presented in the paper indicate good agreement between experimental data and values of speed of sound calculated from so-called "UNESCO equation". They also show that the value of the second kinematical viscosity of water depends on the pressure.
17. Planar dust-acoustic waves in electron–positron–ion–dust plasmas with dust-size distribution under higher-order transverse perturbations
Hong-Yan Wang; Kai-Biao Zhang
2015-01-01
Propagation of small but finite nonlinear dust-acoustic solitary waves are investigated in a planar unmagnetized dusty plasma, which consists of electrons, positrons, ions and negatively charged dust particles with different sizes and masses. A Kadomtsev–Petviashvili (KP) equation is obtained by using reductive perturbation method. The effect of positron density and positron–electron temperature ratio on dust-acoustic solitary structures are studied. Numerical results show that the increase in positron number density increases the amplitude of hump-like solitons but decreases the dip-like solitary waves. Furthermore, increase in the positron–electron temperature ratio results in the decrease of the amplitude of dip-like solitary waves. It seems that both the dipand hump-like solitary waves can exist in this system. Our results also suggest that the dust-size distribution has a significant role on the amplitude of the solitary waves.
18. Plasma-maser instability of the ion acoustics wave in the presence of lower hybrid wave turbulence in inhomogeneous plasma
M Singh; P N Deka
2006-03-01
A theoretical study is made on the generation mechanism of ion acoustics wave in the presence of lower hybrid wave turbulence field in inhomogeneous plasma on the basis of plasma-maser interaction. The lower hybrid wave turbulence field is taken as the low-frequency turbulence field. The growth rate of test high frequency ion acoustics wave is obtained with the involvement of spatial density gradient parameter. A comparative study of the role of density gradient for the generation of ion acoustics wave on the basis of plasma-maser effect is presented. It is found that the density gradient influences the growth rate of ion acoustics wave.
19. Temperature Compensation of Surface Acoustic Waves on Berlinite
Science.gov (United States)
Searle, David Michael Marshall
The surface acoustic wave properties of Berlinite (a-AlPO4) have been investigated theoretically and experimentally, for a variety of crystallographic orientations, to evaluate its possible use as a substrate material for temperature compensated surface acoustic wave devices. A computer program has been developed to calculate the surface wave properties of a material from its elastic, piezoelectric, dielectric and lattice constants and their temperature derivatives. The program calculates the temperature coefficient of delay, the velocity of the surface wave, the direction of power flow and a measure of the electro-mechanical coupling. These calculations have been performed for a large number of orientations using a modified form of the data given by Chang and Barsch for Berlinite and predict several new temperature compensated directions. Experimental measurements have been made of the frequency-temperature response of a surface acoustic wave oscillator on an 80° X axis boule cut which show it to be temperature compensated in qualitative agreement with the theoretical predictions. This orientation shows a cubic frequency-temperature dependence instead of the expected parabolic response. Measurements of the electro-mechanical coupling coefficient k gave a value lower than predicted. Similar measurements on a Y cut plate gave a value which is approximately twice that of ST cut quartz, but again lower than predicted. The surface wave velocity on both these cuts was measured to be slightly higher than predicted by the computer program. Experimental measurements of the lattice parameters a and c are also presented for a range of temperatures from 25°C to just above the alpha-beta transition at 584°C. These results are compared with the values obtained by Chang and Barsch. The results of this work indicate that Berlinite should become a useful substrate material for the construction of temperature compensated surface acoustic wave devices.
20. Temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various frequencies.
Science.gov (United States)
Maraghechi, Borna; Hasani, Mojtaba H; Kolios, Michael C; Tavakkoli, Jahan
2016-05-01
Ultrasound-based thermometry requires a temperature-sensitive acoustic parameter that can be used to estimate the temperature by tracking changes in that parameter during heating. The objective of this study is to investigate the temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various pulse transmit frequencies from 1 to 20 MHz. Simulations were conducted using an expanded form of the Khokhlov-Zabolotskaya-Kuznetsov nonlinear acoustic wave propagation model in which temperature dependence of the medium parameters was included. Measurements were performed using single-element transducers at two different transmit frequencies of 3.3 and 13 MHz which are within the range of frequencies simulated. The acoustic pressure signals were measured by a calibrated needle hydrophone along the axes of the transducers. The water temperature was uniformly increased from 26 °C to 46 °C in increments of 5 °C. The results show that the temperature dependence of the harmonic generation is different at various frequencies which is due to the interplay between the mechanisms of absorption, nonlinearity, and focusing gain. At the transmit frequencies of 1 and 3.3 MHz, the harmonic amplitudes decrease with increasing the temperature, while the opposite temperature dependence is observed at 13 and 20 MHz.
1. Controlling acoustic wave with cylindrically-symmetric gradient-index system
Institute of Scientific and Technical Information of China (English)
张哲; 李睿奇; 梁彬; 邹欣晔; 程建春
2015-01-01
We present a detailed theoretical description of wave propagation in an acoustic gradient-index system with cylindrical symmetry and demonstrate its potential numerically to control acoustic waves in different ways. The trajectory of acoustic wave within the system is derived by employing the theory of geometric acoustics, and the validity of the theoretical descriptions is verified numerically by using the finite element method simulation. The results show that by tailoring the distribution function of refractive index, the proposed system can yield tunable manipulation on acoustic waves, such as acoustic bending, trapping, and absorbing.
2. Impact of Acoustic Standing Waves on Structural Responses: Reverberant Acoustic Testing (RAT) vs. Direct Field Acoustic Testing (DFAT)
Science.gov (United States)
Kolaini, Ali R.; Doty, Benjamin; Chang, Zensheu
2012-01-01
Loudspeakers have been used for acoustic qualification of spacecraft, reflectors, solar panels, and other acoustically responsive structures for more than a decade. Limited measurements from some of the recent speaker tests used to qualify flight hardware have indicated significant spatial variation of the acoustic field within the test volume. Also structural responses have been reported to differ when similar tests were performed using reverberant chambers. To address the impact of non-uniform acoustic field on structural responses, a series of acoustic tests were performed using a flat panel and a 3-ft cylinder exposed to the field controlled by speakers and repeated in a reverberant chamber. The speaker testing was performed using multi-input-single-output (MISO) and multi-input-multi-output (MIMO) control schemes with and without the test articles. In this paper the spatial variation of the acoustic field due to acoustic standing waves and their impacts on the structural responses in RAT and DFAT (both using MISO and MIMO controls for DFAT) are discussed in some detail.
3. Roll dynamics of a ship sailing in large amplitude head waves
NARCIS (Netherlands)
Daalen, E.F.G.; Gunsing, M.; Grasman, J.; Remmert, J.
2014-01-01
Some ship types may show significant rolling when sailing in large-amplitude (near) head waves. The dynamics of the ship are such that the roll motion is affected by the elevation of the encountering waves. If the natural roll period (without forcing) is about half the period of the forcing by the w
4. Opportunities for shear energy scaling in bulk acoustic wave resonators.
Science.gov (United States)
Jose, Sumy; Hueting, Raymond J E
2014-10-01
An important energy loss contribution in bulk acoustic wave resonators is formed by so-called shear waves, which are transversal waves that propagate vertically through the devices with a horizontal motion. In this work, we report for the first time scaling of the shear-confined spots, i.e., spots containing a high concentration of shear wave displacement, controlled by the frame region width at the edge of the resonator. We also demonstrate a novel methodology to arrive at an optimum frame region width for spurious mode suppression and shear wave confinement. This methodology makes use of dispersion curves obtained from finite-element method (FEM) eigenfrequency simulations for arriving at an optimum frame region width. The frame region optimization is demonstrated for solidly mounted resonators employing several shear wave optimized reflector stacks. Finally, the FEM simulation results are compared with measurements for resonators with Ta2O5/ SiO2 stacks showing suppression of the spurious modes.
5. A mixing surface acoustic wave device for liquid sensing applications: Design, simulation, and analysis
Science.gov (United States)
Bui, ThuHang; Morana, Bruno; Scholtes, Tom; Chu Duc, Trinh; Sarro, Pasqualina M.
2016-08-01
This work presents the mixing wave generation of a novel surface acoustic wave (M-SAW) device for sensing in liquids. Two structures are investigated: One including two input and output interdigital transducer (IDT) layers and the other including two input and one output IDT layers. In both cases, a thin (1 μm) piezoelectric AlN layer is in between the two patterned IDT layers. These structures generate longitudinal and transverse acoustic waves with opposite phase which are separated by the film thickness. A 3-dimensional M-SAW device coupled to the finite element method is designed to study the mixing acoustic wave generation propagating through a delay line. The investigated configuration parameters include the number of finger pairs, the piezoelectric cut profile, the thickness of the piezoelectric substrate, and the operating frequency. The proposed structures are evaluated and compared with the conventional SAW structure with the single IDT layer patterned on the piezoelectric surface. The wave displacement along the propagation path is used to evaluate the amplitude field of the mixing longitudinal waves. The wave displacement along the AlN depth is used to investigate the effect of the bottom IDT layer on the transverse component generated by the top IDT layer. The corresponding frequency response, both in simulations and experiments, is an additive function, consisting of sinc(X) and uniform harmonics. The M-SAW devices are tested to assess their potential for liquid sensing, by dropping liquid medium in volumes between 0.05 and 0.13 μl on the propagation path. The interaction with the liquid medium provides information about the liquid, based on the phase attenuation change. The larger the droplet volume is, the longer the duration of the phase shift to reach stability is. The resolution that the output change of the sensor can measure is 0.03 μl.
6. Interpretation of seismic section by acoustic modeling. Study of large amplitude events; Hadoba modeling ni yoru jishin tansa danmen no kaishaku. Kyoshinhaba event ni taisuru kosatsu
Energy Technology Data Exchange (ETDEWEB)
Tamagawa, T.; Matsuoka, T.; Sato, T. [Japan Petroleum Exploration Corp., Tokyo (Japan); Minegishi, M.; Tsuru, T. [Japan National Oil Corp., Tokyo (Japan)
1996-05-01
A large amplitude event difficult to interpret was discovered in the overlap section in offset data beyond 10km targeting at deep structures, and the event was examined. A wave field modeling was carried out by use of a simplified synclinal structure model because it had been estimated that the large amplitude event had something to do with a synclinal structure. A pseudospectral program was used for modeling the wave field on the assumption that the synclinal structure model would be an acoustic body and that the surface would contain free boundaries and multiple reflection. It was found as the result that a discontinuous large amplitude event is mapped out in the synclinal part of the overlap section when a far trace is applied beyond the structure during a CMP overlap process. This can be attributed to the concentration of energy produced by multiple reflection in the synclinal part and by the reflection waves beyond the critical angle. Accordingly, it is possible that phenomena similar to those encountered in the modeling process are emerging during actual observation. 2 refs., 8 figs.
7. Fabrication, Operation and Flow Visualization in Surface-acoustic-wave-driven Acoustic-counterflow Microfluidics
Science.gov (United States)
Travagliati, Marco; Shilton, Richie; Beltram, Fabio; Cecchini, Marco
2013-01-01
Surface acoustic waves (SAWs) can be used to drive liquids in portable microfluidic chips via the acoustic counterflow phenomenon. In this video we present the fabrication protocol for a multilayered SAW acoustic counterflow device. The device is fabricated starting from a lithium niobate (LN) substrate onto which two interdigital transducers (IDTs) and appropriate markers are patterned. A polydimethylsiloxane (PDMS) channel cast on an SU8 master mold is finally bonded on the patterned substrate. Following the fabrication procedure, we show the techniques that allow the characterization and operation of the acoustic counterflow device in order to pump fluids through the PDMS channel grid. We finally present the procedure to visualize liquid flow in the channels. The protocol is used to show on-chip fluid pumping under different flow regimes such as laminar flow and more complicated dynamics characterized by vortices and particle accumulation domains. PMID:24022515
8. Electromagnetic acoustic source (EMAS) for generating shock waves and cavitation in mercury
Science.gov (United States)
Wang, Qi
In the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory a vessel of liquid mercury is subjected to a proton beam. The resulting nuclear interaction produces neutrons that can be used for materials research, among other things, but also launches acoustic waves with pressures in excess of 10 MPa. The acoustic waves have high enough tensile stress to generate cavitation in the mercury which results in erosion to the steel walls of the vessel. In order to study the cavitation erosion and develop mitigation schemes it would be convenient to have a way of generating similar pressures and cavitation in mercury, without the radiation concerns associated with a proton beam. Here an electromagnetic acoustic source (EMAS) has been developed which consisted of a coil placed close to a metal plate which is in turn is in contact with a fluid. The source is driven by discharging a capacitor through the coil and results in a repulsive force on the plate launching acoustic waves in the fluid. A theoretical model is presented to predict the acoustic field from the EMAS and compares favorably with measurements made in water. The pressure from the EMAS was reported as a function of capacitance, charging voltage, number of coils, mylar thickness, and properties of the plates. The properties that resulted in the highest pressure were employed for experiments in mercury and a maximum pressure recorded was 7.1 MPa. Cavitation was assessed in water and mercury by high speed camera and by detecting acoustic emissions. Bubble clouds with lifetimes on the order of 100 µs were observed in water and on the order of 600 µs in mercury. Based on acoustic emissions the bubble radius in mercury was estimated to be 0.98 mm. Experiments to produce damage to a stainless steel plate in mercury resulted in a minimal effect after 2000 shock waves at a rate of 0.33 Hz - likely because the pressure amplitude was not high enough. In order to replicate the conditions in the SNS it is
9. Effect of ion temperature on ion-acoustic solitary waves in a magnetized plasma in presence of superthermal electrons
Energy Technology Data Exchange (ETDEWEB)
Singh, S. V.; Devanandhan, S.; Lakhina, G. S. [Indian Institute of Geomagnetism, Navi Mumbai (India); Bharuthram, R. [University of the Western Cape, Bellville (South Africa)
2013-01-15
Obliquely propagating ion-acoustic soliatry waves are examined in a magnetized plasma composed of kappa distributed electrons and fluid ions with finite temperature. The Sagdeev potential approach is used to study the properties of finite amplitude solitary waves. Using a quasi-neutrality condition, it is possible to reduce the set of equations to a single equation (energy integral equation), which describes the evolution of ion-acoustic solitary waves in magnetized plasmas. The temperature of warm ions affects the speed, amplitude, width, and pulse duration of solitons. Both the critical and the upper Mach numbers are increased by an increase in the ion temperature. The ion-acoustic soliton amplitude increases with the increase in superthermality of electrons. For auroral plasma parameters, the model predicts the soliton speed, amplitude, width, and pulse duration, respectively, to be in the range of (28.7-31.8) km/s, (0.18-20.1) mV/m; (590-167) m, and (20.5-5.25) ms, which are in good agreement with Viking observations.
10. Nonlinear behavior of electron acoustic waves in an un-magnetized plasma
Energy Technology Data Exchange (ETDEWEB)
Dutta, Manjistha; Khan, Manoranjan [Department of Instrumentation Science, Jadavpur University, Kolkata 700 032 (India); Chakrabarti, Nikhil [Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064 (India); Roychoudhury, Rajkumar [Indian Statistical Institute, Kolkata 700 108 (India)
2011-10-15
The nonlinear electron acoustic wave, which is found in the earth's magnetosphere by satellite observations, is studied analytically by Lagrangian fluid description. The basic linear mode is observed in a two temperature electron species plasma where ions form stationary charge neutral background. We have obtained nonlinear description of this mode, which depends on both time and space. A possible solution shows a soliton like structure, which is localized in space, and the amplitude increases with time in the absence of dispersion. Small dispersive correction, however, shows spread of the solution in space. This method can be generalized to study the nonlinear behavior of a general class of multispecies plasma.
11. Nonlinear thickness-stretch vibration of thin-film acoustic wave resonators
Science.gov (United States)
Ji, Xiaojun; Fan, Yanping; Han, Tao; Cai, Ping
2016-03-01
We perform a theoretical analysis on nonlinear thickness-stretch free vibration of thin-film acoustic wave resonators made from AlN and ZnO. The third-order or cubic nonlinear theory by Tiersten is employed. Using Green's identify, under the usual approximation of neglecting higher time harmonics, a perturbation analysis is performed from which the resonator frequency-amplitude relation is obtained. Numerical calculations are made. The relation can be used to determine the linear operating range of these resonators. It can also be used to compare with future experimental results to determine the relevant thirdand/or fourth-order nonlinear elastic constants.
12. Optimization of autonomous magnetic field sensor consisting of giant magnetoimpedance sensor and surface acoustic wave transducer
KAUST Repository
Li, Bodong
2012-11-01
This paper presents a novel autonomous thin film magnetic field sensor consisting of a tri-layer giant magnetoimpedance sensor and a surface acoustic wave transponder. Double and single electrode interdigital transducer (IDT) designs are employed and compared. The integrated sensor is fabricated using standard microfabrication technology. The results show the double electrode IDT has an advantage in terms of the sensitivity. In order to optimize the matching component, a simulation based on P-matrix is carried out. A maximum change of 2.4 dB of the reflection amplitude and a sensitivity of 0.34 dB/Oe are obtained experimentally. © 2012 IEEE.
13. A new model for nonlinear acoustic waves in a non-uniform lattice of Helmholtz resonators
CERN Document Server
Mercier, Jean-François
2016-01-01
Propagation of high amplitude acoustic pulses is studied in a 1D waveguide, connected to a lattice of Helmholtz resonators. An homogenized model has been proposed by Sugimoto (J. Fluid. Mech., 244 (1992)), taking into account both the nonlinear wave propagation and various mechanisms of dissipation. This model is extended to take into account two important features: resonators of different strengths and back-scattering effects. The new model is derived and is proved to satisfy an energy balance principle. A numerical method is developed and a better agreement between numerical and experimental results is obtained.
14. Surface acoustic wave mode conversion resonator
Science.gov (United States)
Martin, S. J.; Gunshor, R. L.; Melloch, M. R.; Datta, S.; Pierret, R. F.
1983-08-01
The fact that a ZnO-on-Si structure supports two distinct surface waves, referred to as the Rayleigh and the Sezawa modes, if the ZnO layer is sufficiently thick is recalled. A description is given of a unique surface wave resonator that operates by efficiently converting between the two modes at the resonant frequency. Since input and output coupling is effected through different modes, the mode conversion resonator promises enhanced out-of-band signal rejection. A Rayleigh wave traversing the resonant cavity in one direction is reflected as a Sezawa wave. It is pointed out that the off-resonance rejection of the mode conversion resonator could be enhanced by designing the transducers to minimize the level of cross coupling between transducers and propagating modes.
15. Large-Amplitude Electrostatic Waves Observed at a Supercritical Interplanetary Shock
Science.gov (United States)
Wilson, L. B., III; Cattell, C. A.; Kellogg, P. J.; Goetz, K.; Kersten, K.; Kasper, J. C.; Szabo, A.; Wilber, M.
2010-01-01
We present the first observations at an interplanetary shock of large-amplitude (> 100 mV/m pk-pk) solitary waves and large-amplitude (approx.30 mV/m pk-pk) waves exhibiting characteristics consistent with electron Bernstein waves. The Bernstein-like waves show enhanced power at integer and half-integer harmonics of the cyclotron frequency with a broadened power spectrum at higher frequencies, consistent with the electron cyclotron drift instability. The Bernstein-like waves are obliquely polarized with respect to the magnetic field but parallel to the shock normal direction. Strong particle heating is observed in both the electrons and ions. The observed heating and waveforms are likely due to instabilities driven by the free energy provided by reflected ions at this supercritical interplanetary shock. These results offer new insights into collisionless shock dissipation and wave-particle interactions in the solar wind.
16. Amplitude wave in one-dimensional complex Ginzburg-Landau equation
Institute of Scientific and Technical Information of China (English)
Xie Ling-Ling; Gao Jia-Zhen; Xie Wei-Miao; Gao Ji-Hua
2011-01-01
The wave propagation in the one-dimensional complex Ginzburg-Landau equation (CGLE) is studied by considering a wave source at the system boundary.A special propagation region,which is an island-shaped zone surrounded by the defect turbulence in the system parameter space,is observed in our numerical experiment.The wave signal spreads in the whole space with a novel amplitude wave pattern in the area.The relevant factors of the pattern formation,such as the wave speed,the maximum propagating distance and the oscillatory frequency,are studied in detail.The stability and the generality of the region are testified by adopting various initial conditions.This finding of the amplitude pattern extends the wave propagation region in the parameter space and presents a new signal transmission mode,and is therefore expected to be of much importance.
17. Ultrahigh-frequency surface acoustic wave generation for acoustic charge transport in silicon
NARCIS (Netherlands)
Büyükköse, S.; Vratzov, B.; van der Veen, Johan (CTIT); Santos, P.V.; van der Wiel, Wilfred Gerard
2013-01-01
We demonstrate piezo-electrical generation of ultrahigh-frequency surface acoustic waves on silicon substrates, using high-resolution UV-based nanoimprint lithography, hydrogen silsequioxane planarization, and metal lift-off. Interdigital transducers were fabricated on a ZnO layer sandwiched between
18. Optimization of Surface Acoustic Wave-Based Rate Sensors
Directory of Open Access Journals (Sweden)
Fangqian Xu
2015-10-01
Full Text Available The optimization of an surface acoustic wave (SAW-based rate sensor incorporating metallic dot arrays was performed by using the approach of partial-wave analysis in layered media. The optimal sensor chip designs, including the material choice of piezoelectric crystals and metallic dots, dot thickness, and sensor operation frequency were determined theoretically. The theoretical predictions were confirmed experimentally by using the developed SAW sensor composed of differential delay line-oscillators and a metallic dot array deposited along the acoustic wave propagation path of the SAW delay lines. A significant improvement in sensor sensitivity was achieved in the case of 128° YX LiNbO3, and a thicker Au dot array, and low operation frequency were used to structure the sensor.
19. Image reconstruction with acoustic radiation force induced shear waves
Science.gov (United States)
McAleavey, Stephen A.; Nightingale, Kathryn R.; Stutz, Deborah L.; Hsu, Stephen J.; Trahey, Gregg E.
2003-05-01
Acoustic radiation force may be used to induce localized displacements within tissue. This phenomenon is used in Acoustic Radiation Force Impulse Imaging (ARFI), where short bursts of ultrasound deliver an impulsive force to a small region. The application of this transient force launches shear waves which propagate normally to the ultrasound beam axis. Measurements of the displacements induced by the propagating shear wave allow reconstruction of the local shear modulus, by wave tracking and inversion techniques. Here we present in vitro, ex vivo and in vivo measurements and images of shear modulus. Data were obtained with a single transducer, a conventional ultrasound scanner and specialized pulse sequences. Young's modulus values of 4 kPa, 13 kPa and 14 kPa were observed for fat, breast fibroadenoma, and skin. Shear modulus anisotropy in beef muscle was observed.
20. Langasite Surface Acoustic Wave Sensors: Fabrication and Testing
Energy Technology Data Exchange (ETDEWEB)
Zheng, Peng; Greve, David W.; Oppenheim, Irving J.; Chin, Tao-Lun; Malone, Vanessa
2012-02-01
We report on the development of harsh-environment surface acoustic wave sensors for wired and wireless operation. Surface acoustic wave devices with an interdigitated transducer emitter and multiple reflectors were fabricated on langasite substrates. Both wired and wireless temperature sensing was demonstrated using radar-mode (pulse) detection. Temperature resolution of better than ±0.5°C was achieved between 200°C and 600°C. Oxygen sensing was achieved by depositing a layer of ZnO on the propagation path. Although the ZnO layer caused additional attenuation of the surface wave, oxygen sensing was accomplished at temperatures up to 700°C. The results indicate that langasite SAW devices are a potential solution for harsh-environment gas and temperature sensing.
1. Multiple-frequency surface acoustic wave devices as sensors
Science.gov (United States)
Ricco, Antonio J.; Martin, Stephen J.
We have designed, fabricated, and tested a multiple-frequency acoustic wave (MUFAW) device on ST-cut quartz with nominal surface acoustic wave (SAW) center frequencies of 16, 40, 100, and 250 MHz. The four frequencies are obtained by patterning four sets of input and output interdigital transducers of differing periodicities on a single substrate. Such a device allows the frequency dependence of AW sensor perturbations to be examined, aiding in the elucidation of the operative interaction mechanism(s). Initial measurements of the SAW response to the vacuum deposition of a thin nickel film show the expected frequency dependence of mass sensitivity in addition to the expected frequency independence of the magnitude of the acoustoelectric effect. By measuring changes in both wave velocity and attenuation at multiple frequencies, extrinsic perturbations such as temperature and pressure changes are readily differentiated from one another and from changes in surface mass.
2. Chromospheric heating by acoustic waves compared to radiative cooling
CERN Document Server
Sobotka, M; Švanda, M; Jurčák, J; del Moro, D; Berrilli, F
2016-01-01
Acoustic and magnetoacoustic waves are among the possible candidate mechanisms that heat the upper layers of solar atmosphere. A weak chromospheric plage near a large solar pore NOAA 11005 was observed on October 15, 2008 in the lines Fe I 617.3 nm and Ca II 853.2 nm with the Interferometric Bidimemsional Spectrometer (IBIS) attached to the Dunn Solar Telescope. Analyzing the Ca II observations with spatial and temporal resolutions of 0.4" and 52 s, the energy deposited by acoustic waves is compared with that released by radiative losses. The deposited acoustic flux is estimated from power spectra of Doppler oscillations measured in the Ca II line core. The radiative losses are calculated using a grid of seven 1D hydrostatic semi-empirical model atmospheres. The comparison shows that the spatial correlation of maps of radiative losses and acoustic flux is 72 %. In quiet chromosphere, the contribution of acoustic energy flux to radiative losses is small, only of about 15 %. In active areas with photospheric ma...
3. Observations of Dissipation of Slow Magneto-acoustic Waves in Polar Coronal Hole
CERN Document Server
Gupta, G R
2014-01-01
We focus on polar coronal hole region to find any evidence of dissipation of propagating slow magneto-acoustic waves. We obtained time-distance and frequency-distance maps along plume structure in polar coronal hole. We also obtained Fourier power maps of polar coronal hole in different frequency ranges in 171 \\AA\\ and 193 \\AA\\ passbands. We performed intensity distribution statistics in time domain at several locations in polar coronal hole. We find presence of propagating slow magneto-acoustic waves having temperature dependent propagation speeds. The wavelet analysis and Fourier power maps of polar coronal hole show that low-frequency waves are travelling longer distances (longer detection length) as compared to high-frequency waves. We found two distinct dissipation length scales of wave amplitude decay at two different height ranges (between 0-10 Mm and 10-70 Mm) along the observed plume structure. Dissipation length obtained at higher height range show some frequency dependence. Individual Fourier power...
4. The Relationship of Cavitation to the Negative Acoustic Pressure Amplitude in Ultrasonic Therapy
Institute of Scientific and Technical Information of China (English)
Ting-Bo Fan; Juan Tu; Lin-Jiao Luo; Xia-Sheng Guo; Pin-Tong Huang; Dong Zhang
2016-01-01
The relationship between the cavitation and acoustic peak negative pressure in the high-intensity focused ultrasound (HIFU) field is analyzed in water and tissue phantom.The peak negative pressure at the focus is determined by a hybrid approach combining the measurement with the simulation.The spheroidal beam equation is utilized to describe the nonlinear acoustic propagation.The waveform at the focus is measured by a fiber optic probe hydrophone in water.The relationship between the source pressure amplitude and the excitation voltage is determined by fitting the measured ratio of the second harmonic to the fundamental component at the focus,based on the model simulation.Then the focal negative pressure is calculated for arbitrary voltage excitation in water and tissue phantom.A portable B-mode ultrasound scanner is applied to monitor HIFU-induced cavitation in real time,and a passive cavitation detection (PCD) system is used to acquire the bubble scattering signals in the HIFU focal volume for the cavitation quantification.The results show that:(1) unstable cavitation starts to appear in degassed water when the peak negative pressure of HIFU signals reaches 13.5 MPa;and (2) the cavitation activity can be detected in tissue phantom by B-mode images and in the PCD system with HIFU peak negative pressures of 9.0 MPa and 7.8 MPa,respectively,which suggests that real-time B-mode images could be used to monitor the cavitation activity in two dimensions,while PCD systems are more sensitive to detect scattering and emission signals from cavitation bubbles.
5. Dual temporal pitch percepts from acoustic and electric amplitude-modulated pulse trains.
Science.gov (United States)
McKay, C M; Carlyon, R P
1999-01-01
Two experiments examined the perception of unmodulated and amplitude-modulated pulse trains by normally hearing listeners and cochlear implantees. Four normally hearing subjects listened to acoustic pulse trains, which were band-pass filtered between 3.9 and 5.3 kHz. Four cochlear implantees, all postlinguistically deaf users of the Mini System 22 implant, listened to current pulse trains produced at a single electrode position. In the first experiment, a set of nine loudness-balanced unmodulated stimuli with rates between 60 and 300 Hz were presented in a multidimensional scaling task. The resultant stimulus spaces for both subject groups showed a single dimension associated with the rate of the stimuli. In the second experiment, a set of ten loudness-balanced modulated stimuli was constructed, with carrier rates between 140 and 300 Hz, and modulation rates between 60 and 150 Hz. The modulation rates were integer submultiples of the carrier rates, and each modulation period consisted of one higher-intensity pulse and one or more identical lower-intensity pulses. The modulation depth of each stimulus was adjusted so that its pitch was judged to be higher or lower 50% of the time than that of an unmodulated pulse train having a rate equal to the geometric mean of the carrier and modulation rates. A multidimensional scaling task with these ten stimuli resulted in two-dimensional stimulus spaces, with dimensions corresponding to carrier and modulation rates. A further investigation with one normally hearing subject showed that the perceptual weighting of the two dimensions varied systematically with modulation depth. It was concluded that, when filtered appropriately, acoustic pulse trains can be used to produce percepts in normal listeners that share common features with those experienced by subjects listening through one channel of a cochlear implant, and that the central auditory system can extract two temporal patterns arising from the same cochlear location.
6. Investigation into the Effect of Acoustic Radiation Force and Acoustic Streaming on Particle Patterning in Acoustic Standing Wave Fields
Directory of Open Access Journals (Sweden)
Shilei Liu
2017-07-01
Full Text Available Acoustic standing waves have been widely used in trapping, patterning, and manipulating particles, whereas one barrier remains: the lack of understanding of force conditions on particles which mainly include acoustic radiation force (ARF and acoustic streaming (AS. In this paper, force conditions on micrometer size polystyrene microspheres in acoustic standing wave fields were investigated. The COMSOL® Mutiphysics particle tracing module was used to numerically simulate force conditions on various particles as a function of time. The velocity of particle movement was experimentally measured using particle imaging velocimetry (PIV. Through experimental and numerical simulation, the functions of ARF and AS in trapping and patterning were analyzed. It is shown that ARF is dominant in trapping and patterning large particles while the impact of AS increases rapidly with decreasing particle size. The combination of using both ARF and AS for medium size particles can obtain different patterns with only using ARF. Findings of the present study will aid the design of acoustic-driven microfluidic devices to increase the diversity of particle patterning.
7. Model of horizontal stress in the Aigion10 well (Corinth) calculated from acoustic body waves
CERN Document Server
Rousseau, A
2006-01-01
In this paper we try to deduce the in situ stresses from the monopole acoustic waves of the well AIG10 between 689 and 1004 meters in depth (Corinth Golf). This borehole crosses competent sedimentary formations (mainly limestone), and the active Aigion fault between 769 and 780 meters in depth. This study is the application of two methods previously described by the author who shows the relationships between in situ horizontal stresses, and (i) the presence or absence of double body waves, (ii) the amplitude ratios between S and P waves (Rousseau, 2005a,b). The full waveforms of this well exhibit two distinct domains separated by the Aigion fault. Within the upper area the three typical waves (P, S and Stoneley) may appear, but the S waves are not numerous, and there is no double body wave, whereas within the lower area there are sometimes double P waves, but no S waves. From those observations, we conclude that the stress domain is isotropic above the Aigion fault, and anisotropic below, which is consistent ...
8. Making structured metals transparent for ultrabroadband electromagnetic waves and acoustic waves
Energy Technology Data Exchange (ETDEWEB)
Fan, Ren-Hao [National Laboratory of Solid State Microstructures and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 (China); Peng, Ru-Wen, E-mail: [email protected] [National Laboratory of Solid State Microstructures and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 (China); Huang, Xian-Rong [Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439 (United States); Wang, Mu [National Laboratory of Solid State Microstructures and School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093 (China)
2015-07-15
In this review, we present our recent work on making structured metals transparent for broadband electromagnetic waves and acoustic waves via excitation of surface waves. First, we theoretically show that one-dimensional metallic gratings can become transparent and completely antireflective for extremely broadband electromagnetic waves by relying on surface plasmons or spoof surface plasmons. Second, we experimentally demonstrate that metallic gratings with narrow slits are highly transparent for broadband terahertz waves at oblique incidence and high transmission efficiency is insensitive to the metal thickness. Further, we significantly develop oblique metal gratings transparent for broadband electromagnetic waves (including optical waves and terahertz ones) under normal incidence. In the third, we find the principles of broadband transparency for structured metals can be extended from one-dimensional metallic gratings to two-dimensional cases. Moreover, similar phenomena are found in sonic artificially metallic structures, which present the transparency for broadband acoustic waves. These investigations provide guidelines to develop many novel materials and devices, such as transparent conducting panels, antireflective solar cells, and other broadband metamaterials and stealth technologies. - Highlights: • Making structured metals transparent for ultrabroadband electromagnetic waves. • Non-resonant excitation of surface plasmons or spoof surface plasmons. • Sonic artificially metallic structures transparent for broadband acoustic waves.
9. A three-microphone acoustic reflection technique using transmitted acoustic waves in the airway.
Science.gov (United States)
Fujimoto, Yuki; Huang, Jyongsu; Fukunaga, Toshiharu; Kato, Ryo; Higashino, Mari; Shinomiya, Shohei; Kitadate, Shoko; Takahara, Yutaka; Yamaya, Atsuyo; Saito, Masatoshi; Kobayashi, Makoto; Kojima, Koji; Oikawa, Taku; Nakagawa, Ken; Tsuchihara, Katsuma; Iguchi, Masaharu; Takahashi, Masakatsu; Mizuno, Shiro; Osanai, Kazuhiro; Toga, Hirohisa
2013-10-15
The acoustic reflection technique noninvasively measures airway cross-sectional area vs. distance functions and uses a wave tube with a constant cross-sectional area to separate incidental and reflected waves introduced into the mouth or nostril. The accuracy of estimated cross-sectional areas gets worse in the deeper distances due to the nature of marching algorithms, i.e., errors of the estimated areas in the closer distances accumulate to those in the further distances. Here we present a new technique of acoustic reflection from measuring transmitted acoustic waves in the airway with three microphones and without employing a wave tube. Using miniaturized microphones mounted on a catheter, we estimated reflection coefficients among the microphones and separated incidental and reflected waves. A model study showed that the estimated cross-sectional area vs. distance function was coincident with the conventional two-microphone method, and it did not change with altered cross-sectional areas at the microphone position, although the estimated cross-sectional areas are relative values to that at the microphone position. The pharyngeal cross-sectional areas including retropalatal and retroglossal regions and the closing site during sleep was visualized in patients with obstructive sleep apnea. The method can be applicable to larger or smaller bronchi to evaluate the airspace and function in these localized airways.
10. Behavior of gravity waves with limited amplitude in the vicinity of critical layer
Institute of Scientific and Technical Information of China (English)
2002-01-01
By using the FICE scheme, a numerical simulation of three-dimensional nonlinear propagation of gravity wave packet in a wind-stratified atmosphere is presented. The whole nonlinear propagation process of the gravity wave packet is shown; the propagation behavior of gravity waves in the vicinity of critical layer is analyzed. The results show that gravity waves encounter the critical layer when propagating in the fair winds whose velocities increase with height, and the height of critical layer propagating nonlinearly is lower than that expected by the linear gravity waves theory; the amplitudes of gravity waves increase with height as a whole before gravity waves encounter the critical layer, but the increasing extent is smaller than the result given by the linear theory of gravity waves, while the amplitudes of gravity waves reduce when gravity waves meet the critical layer; the energy of wave decreases with height, especially at the critical layer; the vertical wavelength reduces with the height increasing, but it does not become zero.
11. Acoustic solitons: A robust tool to investigate the generation and detection of ultrafast acoustic waves
Science.gov (United States)
Péronne, Emmanuel; Chuecos, Nicolas; Thevenard, Laura; Perrin, Bernard
2017-02-01
Solitons are self-preserving traveling waves of great interest in nonlinear physics but hard to observe experimentally. In this report an experimental setup is designed to observe and characterize acoustic solitons in a GaAs(001) substrate. It is based on careful temperature control of the sample and an interferometric detection scheme. Ultrashort acoustic solitons, such as the one predicted by the Korteweg-de Vries equation, are observed and fully characterized. Their particlelike nature is clearly evidenced and their unique properties are thoroughly checked. The spatial averaging of the soliton wave front is shown to account for the differences between the theoretical and experimental soliton profile. It appears that ultrafast acoustic experiments provide a precise measurement of the soliton velocity. It allows for absolute calibration of the setup as well as the response function analysis of the detection layer. Moreover, the temporal distribution of the solitons is also analyzed with the help of the inverse scattering method. It shows how the initial acoustic pulse profile which gives birth to solitons after nonlinear propagation can be retrieved. Such investigations provide a new tool to probe transient properties of highly excited matter through the study of the emitted acoustic pulse after laser excitation.
12. A frequency selective acoustic transducer for directional Lamb wave sensing.
Science.gov (United States)
Senesi, Matteo; Ruzzene, Massimo
2011-10-01
A frequency selective acoustic transducer (FSAT) is proposed for directional sensing of guided waves. The considered FSAT design is characterized by a spiral configuration in wavenumber domain, which leads to a spatial arrangement of the sensing material producing output signals whose dominant frequency component is uniquely associated with the direction of incoming waves. The resulting spiral FSAT can be employed both for directional sensing and generation of guided waves, without relying on phasing and control of a large number of channels. The analytical expression of the shape of the spiral FSAT is obtained through the theoretical formulation for continuously distributed active material as part of a shaped piezoelectric device. Testing is performed by forming a discrete array through the points of the measurement grid of a scanning laser Doppler vibrometer. The discrete array approximates the continuous spiral FSAT geometry, and provides the flexibility to test several configurations. The experimental results demonstrate the strong frequency dependent directionality of the spiral FSAT and suggest its application for frequency selective acoustic sensors, to be employed for the localization of broadband acoustic events, or for the directional generation of Lamb waves for active interrogation of structural health.
13. SILICON COMPATIBLE ACOUSTIC WAVE RESONATORS: DESIGN, FABRICATION AND PERFORMANCE
Directory of Open Access Journals (Sweden)
Aliza Aini Md Ralib
2014-12-01
Full Text Available ABSTRACT: Continuous advancement in wireless technology and silicon microfabrication has fueled exciting growth in wireless products. The bulky size of discrete vibrating mechanical devices such as quartz crystals and surface acoustic wave resonators impedes the ultimate miniaturization of single-chip transceivers. Fabrication of acoustic wave resonators on silicon allows complete integration of a resonator with its accompanying circuitry. Integration leads to enhanced performance, better functionality with reduced cost at large volume production. This paper compiles the state-of-the-art technology of silicon compatible acoustic resonators, which can be integrated with interface circuitry. Typical acoustic wave resonators are surface acoustic wave (SAW and bulk acoustic wave (BAW resonators. Performance of the resonator is measured in terms of quality factor, resonance frequency and insertion loss. Selection of appropriate piezoelectric material is significant to ensure sufficient electromechanical coupling coefficient is produced to reduce the insertion loss. The insulating passive SiO2 layer acts as a low loss material and aims to increase the quality factor and temperature stability of the design. The integration technique also is influenced by the fabrication process and packaging. Packageless structure using AlN as the additional isolation layer is proposed to protect the SAW device from the environment for high reliability. Advancement in miniaturization technology of silicon compatible acoustic wave resonators to realize a single chip transceiver system is still needed. ABSTRAK: Kemajuan yang berterusan dalam teknologi tanpa wayar dan silikon telah menguatkan pertumbuhan yang menarik dalam produk tanpa wayar. Saiz yang besar bagi peralatan mekanikal bergetar seperti kristal kuarza menghalang pengecilan untuk merealisasikan peranti cip. Silikon serasi gelombang akustik resonator mempunyai potensi yang besar untuk menggantikan unsur
14. Tuneable film bulk acoustic wave resonators
CERN Document Server
Gevorgian, Spartak Sh; Vorobiev, Andrei K
2013-01-01
To handle many standards and ever increasing bandwidth requirements, large number of filters and switches are used in transceivers of modern wireless communications systems. It makes the cost, performance, form factor, and power consumption of these systems, including cellular phones, critical issues. At present, the fixed frequency filter banks based on Film Bulk Acoustic Resonators (FBAR) are regarded as one of the most promising technologies to address performance -form factor-cost issues. Even though the FBARs improve the overall performances the complexity of these systems remains high. Attempts are being made to exclude some of the filters by bringing the digital signal processing (including channel selection) as close to the antennas as possible. However handling the increased interference levels is unrealistic for low-cost battery operated radios. Replacing fixed frequency filter banks by one tuneable filter is the most desired and widely considered scenario. As an example, development of the softwa...
15. Frequency dispersion of small-amplitude capillary waves in viscous fluids
CERN Document Server
Denner, Fabian
2016-01-01
This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wavenumber of critical damping based on a harmonic oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wavenumber, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic timescales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodyn...
16. Increasing signal amplitude in fiber Bragg grating detection of Lamb waves using remote bonding.
Science.gov (United States)
Wee, Junghyun; Wells, Brian; Hackney, Drew; Bradford, Philip; Peters, Kara
2016-07-20
Networks of fiber Bragg grating (FBG) sensors can serve as structural health monitoring systems for large-scale structures based on the collection of ultrasonic waves. The demodulation of structural Lamb waves using FBG sensors requires a high signal-to-noise ratio because the Lamb waves are of low amplitudes. This paper compares the signal transfer amplitudes between two adhesive mounting configurations for an FBG to detect Lamb waves propagating in an aluminum plate: a directly bonded FBG and a remotely bonded FBG. In the directly bonded FBG case, the Lamb waves create in-plane and out-of-plane displacements, which are transferred through the adhesive bond and detected by the FBG sensor. In the remotely bonded FBG case, the Lamb waves are converted into longitudinal and flexural traveling waves in the optical fiber at the adhesive bond, which propagate through the optical fiber and are detected by the FBG sensor. A theoretical prediction of overall signal attenuation also is performed, which is the combination of material attenuation in the plate and optical fiber and attenuation due to wave spreading in the plate. The experimental results demonstrate that remote bonding of the FBG significantly increases the signal amplitude measured by the FBG.
17. Nonlinear Waveforms for Ion-Acoustic Waves in Weakly Relativistic Plasma of Warm Ion-Fluid and Isothermal Electrons
Directory of Open Access Journals (Sweden)
S. A. El-Wakil
2012-01-01
Full Text Available The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV equation for small- but finite-amplitude electrostatic ion-acoustic waves in weakly relativistic plasma consisting of warm ions and isothermal electrons. An algebraic method with computerized symbolic computation is applied in obtaining a series of exact solutions of the KdV equation. Numerical studies have been made using plasma parameters which reveal different solutions, that is, bell-shaped solitary pulses, rational pulses, and solutions with singularity at finite points, which called “blowup” solutions in addition to the propagation of an explosive pulses. The weakly relativistic effect is found to significantly change the basic properties (namely, the amplitude and the width of the ion-acoustic waves. The result of the present investigation may be applicable to some plasma environments, such as ionosphere region.
18. Superresolution through the topological shaping of sound with an acoustic vortex wave antenna
CERN Document Server
Guild, Matthew D; Martin, Theodore P; Rohde, Charles A; Orris, Gregory J
2016-01-01
In this paper, we demonstrate far-field acoustic superresolution using shaped acoustic vortices. Compared with previously proposed near-field methods of acoustic superresolution, in this work we describe how far-field superresolution can be obtained using an acoustic vortex wave antenna. This is accomplished by leveraging the recent advances in optical vortices in conjunction with the topological diversity of a leaky wave antenna design. In particular, the use of an acoustic vortex wave antenna eliminates the need for a complicated phased array consisting of multiple active elements, and enables a superresolving aperture to be achieved with a single simple acoustic source and total aperture size less than a wavelength in diameter. A theoretical formulation is presented for the design of an acoustic vortex wave antenna with arbitrary planar arrangement, and explicit expressions are developed for the radiated acoustic pressure field. This geometric versatility enables variously-shaped acoustic vortex patterns t...
19. Standing wave acoustic levitation on an annular plate
Science.gov (United States)
Kandemir, Mehmet Hakan; Çalışkan, Mehmet
2016-11-01
In standing wave acoustic levitation technique, a standing wave is formed between a source and a reflector. Particles can be attracted towards pressure nodes in standing waves owing to a spring action through which particles can be suspended in air. This operation can be performed on continuous structures as well as in several numbers of axes. In this study an annular acoustic levitation arrangement is introduced. Design features of the arrangement are discussed in detail. Bending modes of the annular plate, known as the most efficient sound generation mechanism in such structures, are focused on. Several types of bending modes of the plate are simulated and evaluated by computer simulations. Waveguides are designed to amplify waves coming from sources of excitation, that are, transducers. With the right positioning of the reflector plate, standing waves are formed in the space between the annular vibrating plate and the reflector plate. Radiation forces are also predicted. It is demonstrated that small particles can be suspended in air at pressure nodes of the standing wave corresponding to a particular bending mode.
20. Two-body wave functions and compositeness from scattering amplitudes. I. General properties with schematic models
CERN Document Server
Sekihara, Takayasu
2016-01-01
For a general two-body bound state in quantum mechanics, both in the stable and decaying cases, we establish a way to extract its two-body wave function in momentum space from the scattering amplitude of the constituent two particles. For this purpose, we first show that the two-body wave function of the bound state corresponds to the residue of the off-shell scattering amplitude at the bound state pole. Then, we examine our scheme to extract the two-body wave function from the scattering amplitude in several schematic models. As a result, the two-body wave functions from the Lippmann--Schwinger equation coincides with that from the Schr\\"{o}dinger equation for an energy-independent interaction. Of special interest is that the two-body wave function from the scattering amplitude is automatically scaled; the norm of the two-body wave function, to which we refer as the compositeness, is unity for an energy-independent interaction, while the compositeness deviates from unity for an energy-dependent interaction, ...
1. Exact solution to the Coulomb wave using the linearized phase-amplitude method
Directory of Open Access Journals (Sweden)
Shuji Kiyokawa
2015-08-01
Full Text Available The author shows that the amplitude equation from the phase-amplitude method of calculating continuum wave functions can be linearized into a 3rd-order differential equation. Using this linearized equation, in the case of the Coulomb potential, the author also shows that the amplitude function has an analytically exact solution represented by means of an irregular confluent hypergeometric function. Furthermore, it is shown that the exact solution for the Coulomb potential reproduces the wave function for free space expressed by the spherical Bessel function. The amplitude equation for the large component of the Dirac spinor is also shown to be the linearized 3rd-order differential equation.
2. Nonlinear Acoustic Wave Interactions in Layered Media.
Science.gov (United States)
1980-03-06
Generated Components in Dispersive Media. . . . . . . . . . . . . 62 4.4 Dispersion in Medium II . . . . . . . . .. 68 V. CONCLUSIONS...give rise to leaky wave modes which are more thoroughly discussed 17 18 by Kapany and Burke, and by Marcuse . Leaky modes are C.C. Ghizoni, J.M...1977), 843-848. 1 7N.S. Kapany and J.J. Burke, Optical Waveeeuides, (New York: Academic Press, 1972), pp. 24-34. D. Marcuse , Theory of Dielectric Optical
3. Propagation of ion-acoustic waves in a dusty plasma with non-isothermal electrons
K K Mondal
2007-08-01
For an unmagnetised collisionless plasma consisting of warm ions, non-isothermal electrons and cold, massive and charged dust grains, the Sagdeev potential equation, considering both ion dynamics and dust dynamics has been derived. It has been observed that the Sagdeev potential () exists only for > 0 up to an upper limit ( ≃ 1.2). This implies the possibility of existence of compressive solitary wave in the plasma. Exhaustive numerics done for both the large-amplitude and small-amplitude ion-acoustic waves have revealed that various parameters, namely, ion temperature, non-isothermality of electrons, Mach numbers etc. have considerable impact on the amplitude as well as the width of the solitary waves. Dependence of soliton profiles on the ion temperature and the Mach number has also been graphically displayed. Moreover, incorporating dust-charge fluctuation and non-isothermality of electrons, a non-linear equation relating the grain surface potential to the electrostatic potential has been derived. It has been solved numerically and interdependence of the two potentials for various ion temperatures and orders of non-isothermality has been shown graphically.
4. Dispersion relations with crossing symmetry for pipi D and F wave amplitudes
CERN Document Server
Kaminski, R
2011-01-01
A set of once subtracted dispersion relations with imposed crossing symmetry condition for the pipi D- and F-wave amplitudes is derived and analyzed. An example of numerical calculations in the effective two pion mass range from the threshold to 1.1 GeV is presented. It is shown that these new dispersion relations impose quite strong constraints on the analyzed pipi interactions and are very useful tools to test the pipi amplitudes. One of the goals of this work is to provide a complete set of equations required for easy use. Full analytical expressions are presented. Along with the well known dispersion relations successful in testing the pipi S- and P-wave amplitudes, those presented here for the D and F waves give a complete set of tools for analyzes of the pipi interactions.
5. Amplitude controlled array transducers for mode selection and beam steering of guided waves in plates
Science.gov (United States)
Kannajosyula, H.; Lissenden, C. J.; Rose, J. L.
2013-01-01
We present a method for mode selection of guided wave modes and beam steering using purely amplitude variation across a one dimensional linear array of transducers. The method is distinct from apodization of phased array transducers that involves amplitude variation in addition to time delays and merely aims to improve the spectral characteristics of the transducer. The relationship between amplitude variation and the pitch of the array is derived by considering the resulting transduction as analogous to a spatio-temporal filter approach. It is also shown analytically and through numerical examples that the proposed method results in bidirectional guided waves when the steering angle is zero. Further, for non-zero steering angles, the waves travel in four directions, including the desired direction. Experimental studies are suggested.
6. Nonlinear effects of the finite amplitude ultrasound wave in biological tissues
Institute of Scientific and Technical Information of China (English)
2000-01-01
Nonlinear effects will occur during the transmission of the finite amplitude wave in biological tissues.The theoretical prediction and experimental demonstration of the nonlinear effects on the propagation of the finite amplitude wave at the range of biomedical ultrasound frequency and intensity are studied.Results show that the efficiency factor and effective propagation distance will decrease while the attenuation coefficient increases due to the existence of nonlinear effects.The experimental results coincided quite well with the theory.This shows that the effective propagation distance and efficiency factor can be used to describe quantitatively the influence of nonlinear effects on the propagation of the finite amplitude sound wave in biological tissues.
7. The supergravity fields for a D-brane with a travelling wave from string amplitudes
CERN Document Server
Black, William; Turton, David
2010-01-01
We calculate the supergravity fields sourced by a D-brane with a null travelling wave from disk amplitudes in type IIB string theory compactified on T^4 x S^1. The amplitudes reproduce the non-trivial features of the previously known two-charge supergravity solutions in the D-brane/momentum frame, providing a direct link between the microscopic bound states and their macroscopic descriptions.
8. The supergravity fields for a D-brane with a travelling wave from string amplitudes
Energy Technology Data Exchange (ETDEWEB)
Black, William, E-mail: [email protected] [Queen Mary University of London, Centre for Research in String Theory, Department of Physics, Mile End Road, London E1 4NS (United Kingdom); Russo, Rodolfo, E-mail: [email protected] [Queen Mary University of London, Centre for Research in String Theory, Department of Physics, Mile End Road, London E1 4NS (United Kingdom); Turton, David, E-mail: [email protected] [Queen Mary University of London, Centre for Research in String Theory, Department of Physics, Mile End Road, London E1 4NS (United Kingdom)
2010-11-08
We calculate the supergravity fields sourced by a D-brane with a null travelling wave from disk amplitudes in type IIB string theory compactified on T{sup 4}xS{sup 1}. The amplitudes reproduce all the non-trivial features of the previously known two-charge supergravity solutions in the D-brane/momentum duality frame, providing a direct link between the microscopic bound states and their macroscopic descriptions.
9. Excitation of monochromatic and stable electron acoustic wave by two counter-propagating laser beams
Science.gov (United States)
Xiao, C. Z.; Liu, Z. J.; Zheng, C. Y.; He, X. T.
2017-07-01
The undamped electron acoustic wave is a newly-observed nonlinear electrostatic plasma wave and has potential applications in ion acceleration, laser amplification and diagnostics due to its unique frequency range. We propose to make the first attempt to excite a monochromatic and stable electron acoustic wave (EAW) by two counter-propagating laser beams. The matching conditions relevant to laser frequencies, plasma density, and electron thermal velocity are derived and the harmonic effects of the EAW are excluded. Single-beam instabilities, including stimulated Raman scattering and stimulated Brillouin scattering, on the excitation process are quantified by an interaction quantity, η =γ {τ }B, where γ is the growth rate of each instability and {τ }B is the characteristic time of the undamped EAW. The smaller the interaction quantity, the more successfully the monochromatic and stable EAW can be excited. Using one-dimensional Vlasov-Maxwell simulations, we excite EAW wave trains which are amplitude tunable, have a duration of thousands of laser periods, and are monochromatic and stable, by carefully controlling the parameters under the above conditions.
10. High-Temperature Piezoelectric Crystals for Acoustic Wave Sensor Applications.
Science.gov (United States)
Zu, Hongfei; Wu, Huiyan; Wang, Qing-Ming
2016-03-01
In this review paper, nine different types of high-temperature piezoelectric crystals and their sensor applications are overviewed. The important materials' properties of these piezoelectric crystals including dielectric constant, elastic coefficients, piezoelectric coefficients, electromechanical coupling coefficients, and mechanical quality factor are discussed in detail. The determination methods of these physical properties are also presented. Moreover, the growth methods, structures, and properties of these piezoelectric crystals are summarized and compared. Of particular interest are langasite and oxyborate crystals, which exhibit no phase transitions prior to their melting points ∼ 1500 °C and possess high electrical resistivity, piezoelectric coefficients, and mechanical quality factor at ultrahigh temperature ( ∼ 1000 °C). Finally, some research results on surface acoustic wave (SAW) and bulk acoustic wave (BAW) sensors developed using this high-temperature piezoelectric crystals are discussed.
11. Scanning Michelson interferometer for imaging surface acoustic wave fields.
Science.gov (United States)
Knuuttila, J V; Tikka, P T; Salomaa, M M
2000-05-01
A scanning homodyne Michelson interferometer is constructed for two-dimensional imaging of high-frequency surface acoustic wave (SAW) fields in SAW devices. The interferometer possesses a sensitivity of ~10(-5)nm/ radicalHz , and it is capable of directly measuring SAW's with frequencies ranging from 0.5 MHz up to 1 GHz. The fast scheme used for locating the optimum operation point of the interferometer facilitates high measuring speeds, up to 50,000 points/h. The measured field image has a lateral resolution of better than 1 mu;m . The fully optical noninvasive scanning system can be applied to SAW device development and research, providing information on acoustic wave distribution that cannot be obtained by merely electrical measurements.
12. Dust acoustic waves in strongly coupled dissipative plasmas
Science.gov (United States)
Xie, B. S.; Yu, M. Y.
2000-12-01
The theory of dust acoustic waves is revisited in the frame of the generalized viscoelastic hydrodynamic theory for highly correlated dusts. Physical processes relevant to many experiments on dusts in plasmas, such as ionization and recombination, dust-charge variation, elastic electron and ion collisions with neutral and charged dust particles, as well as relaxation due to strong dust coupling, are taken into account. These processes can be on similar time scales and are thus important for the conservation of particles and momenta in a self-consistent description of the system. It is shown that the dispersion properties of the dust acoustic waves are determined by a sensitive balance of the effects of strong dust coupling and collisional relaxation. The predictions of the present theory applicable to typical parameters in laboratory strongly coupled dusty plasmas are given and compared with the experiment results. Some possible implications and discrepanies between theory and experiment are also discussed.
13. Laser-generated acoustic wave studies on tattoo pigment
Science.gov (United States)
Paterson, Lorna M.; Dickinson, Mark R.; King, Terence A.
1996-01-01
A Q-switched alexandrite laser (180 ns at 755 nm) was used to irradiate samples of agar embedded with red, black and green tattoo dyes. The acoustic waves generated in the samples were detected using a PVDF membrane hydrophone and compared to theoretical expectations. The laser pulses were found to generate acoustic waves in the black and green samples but not in the red pigment. Pressures of up to 1.4 MPa were produced with irradiances of up to 96 MWcm-2 which is comparable to the irradiances used to clear pigment embedded in skin. The pressure gradient generated across pigment particles was approximately 1.09 X 1010 Pam-1 giving a pressure difference of 1.09 +/- 0.17 MPa over a particle with mean diameter 100 micrometers . This is not sufficient to permanently damage skin which has a tensile strength of 7.4 MPa.
14. Electron-acoustic rogue waves in a plasma with Tribeche–Tsallis–Cairns distributed electrons
Energy Technology Data Exchange (ETDEWEB)
Merriche, Abderrzak [Faculty of Physics, Theoretical Physics Laboratory (TPL), Plasma Physics Group (PPG), University of Bab-Ezzouar, USTHB, B. P. 32, El Alia, Algiers 16111 (Algeria); Tribeche, Mouloud, E-mail: [email protected] [Faculty of Physics, Theoretical Physics Laboratory (TPL), Plasma Physics Group (PPG), University of Bab-Ezzouar, USTHB, B. P. 32, El Alia, Algiers 16111 (Algeria); Algerian Academy of Sciences and Technologies, Algiers (Algeria)
2017-01-15
The problem of electron-acoustic (EA) rogue waves in a plasma consisting of fluid cold electrons, nonthermal nonextensive electrons and stationary ions, is addressed. A standard multiple scale method has been carried out to derive a nonlinear Schrödinger-like equation. The coefficients of dispersion and nonlinearity depend on the nonextensive and nonthermal parameters. The EA wave stability is analyzed. Interestingly, it is found that the wave number threshold, above which the EA wave modulational instability (MI) sets in, increases as the nonextensive parameter increases. As the nonthermal character of the electrons increases, the MI occurs at large wavelength. Moreover, it is shown that as the nonextensive parameter increases, the EA rogue wave pulse grows while its width is narrowed. The amplitude of the EA rogue wave decreases with an increase of the number of energetic electrons. In the absence of nonthermal electrons, the nonextensive effects are more perceptible and more noticeable. In view of the crucial importance of rogue waves, our results can contribute to the understanding of localized electrostatic envelope excitations and underlying physical processes, that may occur in space as well as in laboratory plasmas.
15. On-line surveillance of lubricants in bearings by means of surface acoustic waves.
Science.gov (United States)
Lindner, Gerhard; Schmitt, Martin; Schubert, Josephine; Krempel, Sandro; Faustmann, Hendrik
2010-01-01
The acoustic wave propagation in bearings filled with lubricants and driven by pulsed excitation of surface acoustic waves has been investigated with respect to the presence and the distribution of different lubricants. Experimental setups, which are based on the mode conversion between surface acoustic waves and compression waves at the interface between a solid substrate of the bearing and a lubricant are described. The results of preliminary measurements at linear friction bearings, rotation ball bearings and axial cylinder roller bearings are presented.
16. High-Temperature Surface-Acoustic-Wave Transducer
Science.gov (United States)
Zhao, Xiaoliang; Tittmann, Bernhard R.
2010-01-01
Aircraft-engine rotating equipment usually operates at high temperature and stress. Non-invasive inspection of microcracks in those components poses a challenge for the non-destructive evaluation community. A low-profile ultrasonic guided wave sensor can detect cracks in situ. The key feature of the sensor is that it should withstand high temperatures and excite strong surface wave energy to inspect surface/subsurface cracks. As far as the innovators know at the time of this reporting, there is no existing sensor that is mounted to the rotor disks for crack inspection; the most often used technology includes fluorescent penetrant inspection or eddy-current probes for disassembled part inspection. An efficient, high-temperature, low-profile surface acoustic wave transducer design has been identified and tested for nondestructive evaluation of structures or materials. The development is a Sol-Gel bismuth titanate-based surface-acoustic-wave (SAW) sensor that can generate efficient surface acoustic waves for crack inspection. The produced sensor is very thin (submillimeter), and can generate surface waves up to 540 C. Finite element analysis of the SAW transducer design was performed to predict the sensor behavior, and experimental studies confirmed the results. One major uniqueness of the Sol-Gel bismuth titanate SAW sensor is that it is easy to implement to structures of various shapes. With a spray coating process, the sensor can be applied to surfaces of large curvatures. Second, the sensor is very thin (as a coating) and has very minimal effect on airflow or rotating equipment imbalance. Third, it can withstand temperatures up to 530 C, which is very useful for engine applications where high temperature is an issue.
17. Vertical amplitude phase structure of a low-frequency acoustic field in shallow water
Science.gov (United States)
Kuznetsov, G. N.; Lebedev, O. V.; Stepanov, A. N.
2016-11-01
We obtain in integral and analytic form the relations for calculating the amplitude and phase characteristics of an interference structure of orthogonal projections of the oscillation velocity vector in shallow water. For different frequencies and receiver depths, we numerically study the source depth dependences of the effective phase velocities of an equivalent plane wave, the orthogonal projections of the sound pressure phase gradient, and the projections of the oscillation velocity vector. We establish that at low frequencies in zones of interference maxima, independently of source depth, weakly varying effective phase velocity values are observed, which exceed the sound velocity in water by 5-12%. We show that the angles of arrival of the equivalent plane wave and the oscillation velocity vector in the general case differ; however, they virtually coincide in the zone of the interference maximum of the sound pressure under the condition that the horizontal projections of the oscillation velocity appreciably exceed the value of the vertical projection. We give recommendations on using the sound field characteristics in zones with maximum values for solving rangefinding and signal-detection problems.
18. Laser ablation method for production of surface acoustic wave sensors
Science.gov (United States)
Lukyanov, Dmitry; Shevchenko, Sergey; Kukaev, Alexander; Safronov, Daniil
2016-10-01
Nowadays surface acoustic wave (SAW) sensors are produced using a photolithography method. In case of inertial sensors it suffers several disadvantages, such as difficulty in matching topologies produced on opposite sides of the wafer, expensive in small series production, not allowing further topology correction. In this case a laser ablation method seems promising. Details of a proposed technique are described in the paper along with results of its experimental test and discussion.
19. Surface acoustic wave probe implant for predicting epileptic seizures
Science.gov (United States)
Gopalsami, Nachappa [Naperville, IL; Kulikov, Stanislav [Sarov, RU; Osorio, Ivan [Leawood, KS; Raptis, Apostolos C [Downers Grove, IL
2012-04-24
A system and method for predicting and avoiding a seizure in a patient. The system and method includes use of an implanted surface acoustic wave probe and coupled RF antenna to monitor temperature of the patient's brain, critical changes in the temperature characteristic of a precursor to the seizure. The system can activate an implanted cooling unit which can avoid or minimize a seizure in the patient.
20. Amplitude death, oscillation death, wave, and multistability in identical Stuart-Landau oscillators with conjugate coupling
Science.gov (United States)
Han, Wenchen; Cheng, Hongyan; Dai, Qionglin; Li, Haihong; Ju, Ping; Yang, Junzhong
2016-10-01
In this work, we investigate the dynamics in a ring of identical Stuart-Landau oscillators with conjugate coupling systematically. We analyze the stability of the amplitude death and find the stability independent of the number of oscillators. When the amplitude death state is unstable, a large number of states such as homogeneous oscillation death, heterogeneous oscillation death, homogeneous oscillation, and wave propagations are found and they may coexist. We also find that all of these states are related to the unstable spatial modes to the amplitude death state.
1. Diagnostics of elastic properties of a planar interface between two rough media using surface acoustic waves
Science.gov (United States)
Kokshaiskii, A. I.; Korobov, A. I.; Shirgina, N. V.
2017-03-01
The results of experimental studies on the nonlinear elastic properties of a planar interface between two media are presented—an optically polished glass substrate and flat samples with different degrees of roughness. The nonlinear elastic properties of the interfaces between two media were investigated by the spectral method using surface acoustic waves (SAWs). The effect of external pressure applied to the interface on the efficiency of the generation of the second SAW harmonic was studied. Using the measured amplitudes of the first and second harmonics of the SAW that passes along the interface, the second-order nonlinear acoustic parameter was calculated as a function of the external pressure applied to the sample at a fixed amplitude of a probing wave. It was revealed that the nonlinear parameter of the SAW is a nonmonotonic function of the pressure at the boundary. The results were analyzed on the basis of an elastic contact nonlinearity, and it is concluded that these results can be used in nondestructive testing for roughness and waviness of surfaces of flat solids.
2. Compressional Wave/Shear Wave Seismo-Acoustic Probe
Science.gov (United States)
1990-04-30
penetration depths into ocean bottom sediments [i.e., 27 meters (88.6 ft.)] and in water depths at which diver support of the seabed measurements is...impractical [i.e., to 88.5 m (290 ft.)]. This new probe was developed as a prototype instrument for use in the shallow water ocean acoustics research program...TRANSDUCER 3H VIBRATOR AND POWER RISER ISOLATOR 600 CONTROL MODULE 6.500WTRANSDUCER 6,50Wl IPIPE HYDRAULIC HYDRAULIC HOSE CLAMP 4-HP DETECTOR #1 VIBRATION
3. Volumetric measurements of a spatially growing dust acoustic wave
Science.gov (United States)
Williams, Jeremiah D.
2012-11-01
In this study, tomographic particle image velocimetry (tomo-PIV) techniques are used to make volumetric measurements of the dust acoustic wave (DAW) in a weakly coupled dusty plasma system in an argon, dc glow discharge plasma. These tomo-PIV measurements provide the first instantaneous volumetric measurement of a naturally occurring propagating DAW. These measurements reveal over the measured volume that the measured wave mode propagates in all three spatial dimensional and exhibits the same spatial growth rate and wavelength in each spatial direction.
4. Volumetric measurements of a spatially growing dust acoustic wave
Energy Technology Data Exchange (ETDEWEB)
Williams, Jeremiah D. [Physics Department, Wittenberg University, Springfield, Ohio 45504 (United States)
2012-11-15
In this study, tomographic particle image velocimetry (tomo-PIV) techniques are used to make volumetric measurements of the dust acoustic wave (DAW) in a weakly coupled dusty plasma system in an argon, dc glow discharge plasma. These tomo-PIV measurements provide the first instantaneous volumetric measurement of a naturally occurring propagating DAW. These measurements reveal over the measured volume that the measured wave mode propagates in all three spatial dimensional and exhibits the same spatial growth rate and wavelength in each spatial direction.
5. Investigation of Ion Acoustic Waves in Collisionless Plasmas
DEFF Research Database (Denmark)
Christoffersen, G. B.; Jensen, Vagn Orla; Michelsen, Poul
1974-01-01
The Green's functions for the linearized ion Vlasov equation with a given boundary value are derived. The propagation properties of ion acoustic waves are calculated by performing convolution integrals over the Green's functions. For Te/Ti less than about 3 it is concluded that the collective...... interaction is very weak and that the propagation properties are determined almost completely by freely streaming ions. The wave damping, being due to phase mixing, is determined by the width of the perturbed distribution function rather than by the slope of the undisturbed distribution function at the phase...
6. Autism spectrum disorders and the amplitude of auditory brainstem response wave I.
Science.gov (United States)
Santos, Mariline; Marques, Cristina; Nóbrega Pinto, Ana; Fernandes, Raquel; Coutinho, Miguel Bebiano; Almeida E Sousa, Cecília
2017-04-01
To determine whether children with autism spectrum disorders (ASDs) have an increased number of wave I abnormal amplitudes in auditory brainstem responses (ABRs) than age- and sex-matched typically developing children. This analytical case-control study compared patients with ASDs between the ages of 2 and 6 years and children who had a language delay not associated with any other pathology. Amplitudes of ABR waves I and V; absolute latencies (ALs) of waves I, III, and V; and interpeak latencies (IPLs) I-III, III-IV, and I-V at 90 dB were compared between ASD patients and normally developing children. The study enrolled 40 children with documented ASDs and 40 age- and sex-matched control subjects. Analyses of the ABR showed that children with ASDs exhibited higher amplitudes of wave 1 than wave V (35%) more frequently than the control group (10%), and this difference between groups reached statistical significance by Chi-squared analysis. There were no significant differences in ALs and IPLs between ASD children and matched controls. To the best of our knowledge, this is the first case-control study testing the amplitudes of ABR wave I in ASD children. The reported results suggest a potential for the use of ABR recordings in children, not only for the clinical assessment of hearing status, but also for the possibility of using amplitude of ABR wave I as an early marker of ASDs allowing earlier diagnosis and intervention. Autism Res 2017. © 2017 International Society for Autism Research, Wiley Periodicals, Inc.
7. VARIATION METHOD FOR ACOUSTIC WAVE IMAGING OF TWO DIMENSIONAL TARGETS
Institute of Scientific and Technical Information of China (English)
冯文杰; 邹振祝
2003-01-01
A new way of acoustic wave imaging was investigated. By using the Green function theory a system of integral equations, which linked wave number perturbation function with wave field, was firstly deduced. By taking variation on these integral equations an inversion equation, which reflected the relation between the little variation of wave number perturbation function and that of scattering field, was further obtained. Finally, the perturbation functions of some identical targets were reconstructed, and some properties of the novel method including converging speed, inversion accuracy and the abilities to resist random noise and identify complex targets were discussed. Results of numerical simulation show that the method based on the variation principle has great theoretical and applicable value to quantitative nondestructive evaluation.
8. Dual mode acoustic wave sensor for precise pressure reading
Science.gov (United States)
Mu, Xiaojing; Kropelnicki, Piotr; Wang, Yong; Randles, Andrew Benson; Chuan Chai, Kevin Tshun; Cai, Hong; Gu, Yuan Dong
2014-09-01
In this letter, a Microelectromechanical system acoustic wave sensor, which has a dual mode (lateral field exited Lamb wave mode and surface acoustic wave (SAW) mode) behavior, is presented for precious pressure change read out. Comb-like interdigital structured electrodes on top of piezoelectric material aluminium nitride (AlN) are used to generate the wave modes. The sensor membrane consists of single crystalline silicon formed by backside-etching of the bulk material of a silicon on insulator wafer having variable device thickness layer (5 μm-50 μm). With this principle, a pressure sensor has been fabricated and mounted on a pressure test package with pressure applied to the backside of the membrane within a range of 0 psi to 300 psi. The temperature coefficient of frequency was experimentally measured in the temperature range of -50 °C to 300 °C. This idea demonstrates a piezoelectric based sensor having two modes SAW/Lamb wave for direct physical parameter—pressure readout and temperature cancellation which can operate in harsh environment such as oil and gas exploration, automobile and aeronautic applications using the dual mode behavior of the sensor and differential readout at the same time.
9. Surface acoustic waves enhance neutrophil killing of bacteria.
Science.gov (United States)
Loike, John D; Plitt, Anna; Kothari, Komal; Zumeris, Jona; Budhu, Sadna; Kavalus, Kaitlyn; Ray, Yonatan; Jacob, Harold
2013-01-01
Biofilms are structured communities of bacteria that play a major role in the pathogenicity of bacteria and are the leading cause of antibiotic resistant bacterial infections on indwelling catheters and medical prosthetic devices. Failure to resolve these biofilm infections may necessitate the surgical removal of the prosthetic device which can be debilitating and costly. Recent studies have shown that application of surface acoustic waves to catheter surfaces can reduce the incidence of infections by a mechanism that has not yet been clarified. We report here the effects of surface acoustic waves (SAW) on the capacity of human neutrophils to eradicate S. epidermidis bacteria in a planktonic state and within biofilms. Utilizing a novel fibrin gel system that mimics a tissue-like environment, we show that SAW, at an intensity of 0.3 mW/cm(2), significantly enhances human neutrophil killing of S. epidermidis in a planktonic state and within biofilms by enhancing human neutrophil chemotaxis in response to chemoattractants. In addition, we show that the integrin CD18 plays a significant role in the killing enhancement observed in applying SAW. We propose from out data that this integrin may serve as mechanoreceptor for surface acoustic waves enhancing neutrophil chemotaxis and killing of bacteria.
10. Surface acoustic waves enhance neutrophil killing of bacteria.
Directory of Open Access Journals (Sweden)
John D Loike
Full Text Available Biofilms are structured communities of bacteria that play a major role in the pathogenicity of bacteria and are the leading cause of antibiotic resistant bacterial infections on indwelling catheters and medical prosthetic devices. Failure to resolve these biofilm infections may necessitate the surgical removal of the prosthetic device which can be debilitating and costly. Recent studies have shown that application of surface acoustic waves to catheter surfaces can reduce the incidence of infections by a mechanism that has not yet been clarified. We report here the effects of surface acoustic waves (SAW on the capacity of human neutrophils to eradicate S. epidermidis bacteria in a planktonic state and within biofilms. Utilizing a novel fibrin gel system that mimics a tissue-like environment, we show that SAW, at an intensity of 0.3 mW/cm(2, significantly enhances human neutrophil killing of S. epidermidis in a planktonic state and within biofilms by enhancing human neutrophil chemotaxis in response to chemoattractants. In addition, we show that the integrin CD18 plays a significant role in the killing enhancement observed in applying SAW. We propose from out data that this integrin may serve as mechanoreceptor for surface acoustic waves enhancing neutrophil chemotaxis and killing of bacteria.
11. Intra-plasmaspheric wave power density deduced from long-term DEMETER measurements of terrestrial VLF transmitter wave amplitudes
Science.gov (United States)
Lauben, D.; Cohen, M.; Inan, U.
2012-12-01
We deduce the 3d intra-plasmaspheric distribution of VLF wave power between conjugate regions of strong VLF wave amplitudes as measured by DEMETER for high-power terrestrial VLF transmitters during its ~6-yr lifetime. We employ a mixed WKB/full-wave technique to solve for the primary and secondary electromagnetic and electrostatic waves which are transmitted and reflected from strong cold-plasma density gradients and posited irregularities, in order to match the respective end-point measured amplitude distributions. Energy arriving in the conjugate region and also escaping to other regions of the magnetosphere is note. The resulting 3d distribution allows improved estimates for the long-term average particle scattering induced by terrestrial VLF transmitters.
12. Direct visualization of surface acoustic waves along substrates using smoke particles
Science.gov (United States)
Tan, Ming K.; Friend, James R.; Yeo, Leslie Y.
2007-11-01
Smoke particles (SPs) are used to directly visualize surface acoustic waves (SAWs) propagating on a 128°-rotated Y-cut X-propagating lithium niobate (LiNbO3) substrate. By electrically exciting a SAW device in a compartment filled with SP, the SP were found to collect along the regions where the SAW propagates on the substrate. The results of the experiments show that SPs are deposited adjacent to regions of large vibration amplitude and form a clear pattern corresponding to the surface wave profile on the substrate. Through an analysis of the SAW-induced acoustic streaming in the air adjacent to the substrate and the surface acceleration measured with a laser Doppler vibrometer, we postulate that the large transverse surface accelerations due to the SAW ejects SP from the surface and carries them aloft to relatively quiescent regions nearby via acoustic streaming. Offering finer detail than fine powders common in Chladni figures [E. Chladni, Entdeckungen über die Theorie des Klanges (Weidmanns, Erben und Reich, Leipzig, Germany, 1787)] the approach is an inexpensive and a quick counterpart to laser interferometric techniques, presenting a means to explore the controversial phenomena of particle agglomeration on surfaces.
13. Paracousti-UQ: A Stochastic 3-D Acoustic Wave Propagation Algorithm.
Energy Technology Data Exchange (ETDEWEB)
Preston, Leiph
2017-09-01
Acoustic full waveform algorithms, such as Paracousti, provide deterministic solutions in complex, 3-D variable environments. In reality, environmental and source characteristics are often only known in a statistical sense. Thus, to fully characterize the expected sound levels within an environment, this uncertainty in environmental and source factors should be incorporated into the acoustic simulations. Performing Monte Carlo (MC) simulations is one method of assessing this uncertainty, but it can quickly become computationally intractable for realistic problems. An alternative method, using the technique of stochastic partial differential equations (SPDE), allows computation of the statistical properties of output signals at a fraction of the computational cost of MC. Paracousti-UQ solves the SPDE system of 3-D acoustic wave propagation equations and provides estimates of the uncertainty of the output simulated wave field (e.g., amplitudes, waveforms) based on estimated probability distributions of the input medium and source parameters. This report describes the derivation of the stochastic partial differential equations, their implementation, and comparison of Paracousti-UQ results with MC simulations using simple models.
14. Ion acoustic wave instabilities and nonlinear structures associated with field-aligned flows in the F-region ionosphere
Science.gov (United States)
Saleem, H.; Ali Shan, S.; Haque, Q.
2016-11-01
It is shown that the inhomogeneous field-aligned flow of heavier ions into the stationary plasma of the upper ionosphere produces very low frequency (of the order of a few Hz) electrostatic unstable ion acoustic waves (IAWs). This instability is an oscillatory instability unlike D'Angelo's purely growing mode. The growth rate of the ion acoustic wave (IAW) corresponding to heavier ions is due to shear flow and is larger than the ion Landau damping. However, the ion acoustic waves corresponding to non-flowing lighter ions are Landau damped. It is found that even if D'Angelo's instability condition is satisfied, the unstable mode develops its real frequency in this coupled system. Hence, the shear flow of one type of ions in a bi-ion plasma system produces ion acoustic wave activity. If the density non-uniformity is taken into account, then the drift wave becomes unstable. The coupled nonlinear equations for stationary ions "a," flowing ions "b," and inertialess electrons are also solved using the small amplitude limit. The solutions predict the existence of the order of a few kilometers electric field structures in the form of solitons and vortices, which is in agreement with the satellite observations.
15. Fluid nonlinear frequency shift of nonlinear ion acoustic waves in multi-ion species plasmas in small wave number region
CERN Document Server
Feng, Q S; Wang, Q; Zheng, C Y; Liu, Z J; Cao, L H; He, X T
2016-01-01
The properties of the nonlinear frequency shift (NFS) especially the fluid NFS from the harmonic generation of the ion-acoustic wave (IAW) in multi-ion species plasmas have been researched by Vlasov simulation. The pictures of the nonlinear frequency shift from harmonic generation and particles trapping are shown to explain the mechanism of NFS qualitatively. The theoretical model of the fluid NFS from harmonic generation in multi-ion species plasmas is given and the results of Vlasov simulation are consistent to the theoretical result of multi-ion species plasmas. When the wave number $k\\lambda_{De}$ is small, such as $k\\lambda_{De}=0.1$, the fluid NFS dominates in the total NFS and will reach as large as nearly $15\\%$ when the wave amplitude $|e\\phi/T_e|\\sim0.1$, which indicates that in the condition of small $k\\lambda_{De}$, the fluid NFS dominates in the saturation of stimulated Brillouin scattering especially when the nonlinear IAW amplitude is large.
16. Elastic contact conditions to optimize friction drive of surface acoustic wave motor.
Science.gov (United States)
Kuribayashi Kurosawa, M; Takahashi, M; Higuchi, T
1998-01-01
The optimum pressing force, namely the preload, for a slider to obtain superior operation conditions in a surface acoustic wave motor have been examined. We used steel balls as sliders. The preload was controlled using a permanent magnet. The steel balls were 0.5, 1, and 2 mm diameter, with the differences in diameter making it possible to change contact conditions, such as the contact pressure, contact area, and deformation of the stator and the slider. The stator transducer was lithium niobate, 128 degrees rotated, y-cut x-propagation substrate. The driving frequency of the Rayleigh wave was about 10 MHz. Hence, the particle vibration amplitude at the surface is as small as 10 nm. For superior friction drive conditions, a high contact pressure was required. For example, in the case of the 1 mm diameter steel ball at the sinusoidal driving voltage of 180 V(peak), the slider speed was 43 cm/sec, the thrust output force was 1 mN, and the acceleration was 23 times as large as the gravitational acceleration at a contact pressure of 390 MPa. From the Hertz theory of contact stress, the contact area radius was only 3 microm. The estimation of the friction drive performance was carried out from the transient traveling distance of the slider in a 3 msec burst drive. As a result, the deformation of the stator and the slider by the preload should be half of the vibration amplitude. This condition was independent of the ball diameter and the vibration amplitude. The output thrust per square millimeter was 50 N, and the maximum speed was 0.7 m/sec. From these results, we conclude that it is possible for the surface acoustic wave motor to have a large output force, high speed, quick response, long traveling distance, and a thin micro linear actuator.
17. Acoustic and Cavitation Fields of Shock Wave Therapy Devices
Science.gov (United States)
Chitnis, Parag V.; Cleveland, Robin O.
2006-05-01
Extracorporeal shock wave therapy (ESWT) is considered a viable treatment modality for orthopedic ailments. Despite increasing clinical use, the mechanisms by which ESWT devices generate a therapeutic effect are not yet understood. The mechanistic differences in various devices and their efficacies might be dependent on their acoustic and cavitation outputs. We report acoustic and cavitation measurements of a number of different shock wave therapy devices. Two devices were electrohydraulic: one had a large reflector (HMT Ossatron) and the other was a hand-held source (HMT Evotron); the other device was a pneumatically driven device (EMS Swiss DolorClast Vet). Acoustic measurements were made using a fiber-optic probe hydrophone and a PVDF hydrophone. A dual passive cavitation detection system was used to monitor cavitation activity. Qualitative differences between these devices were also highlighted using a high-speed camera. We found that the Ossatron generated focused shock waves with a peak positive pressure around 40 MPa. The Evotron produced peak positive pressure around 20 MPa, however, its acoustic output appeared to be independent of the power setting of the device. The peak positive pressure from the DolorClast was about 5 MPa without a clear shock front. The DolorClast did not generate a focused acoustic field. Shadowgraph images show that the wave propagating from the DolorClast is planar and not focused in the vicinity of the hand-piece. All three devices produced measurable cavitation with a characteristic time (cavitation inception to bubble collapse) that varied between 95 and 209 μs for the Ossatron, between 59 and 283 μs for the Evotron, and between 195 and 431 μs for the DolorClast. The high-speed camera images show that the cavitation activity for the DolorClast is primarily restricted to the contact surface of the hand-piece. These data indicate that the devices studied here vary in acoustic and cavitation output, which may imply that the
18. Ion-acoustic rogue waves and breathers in relativistically degenerate electron-positron plasmas
Science.gov (United States)
Abdikian, A.; Ismaeel, S.
2017-08-01
In this paper, we employ a weakly relativistic fluid model to study the nonlinear amplitude modulation of electrostatic waves in an unmagnetized electron-positron-ion plasma. It is assumed that the degeneracy pressure law for electrons and positrons follows the Chandrasekhar limit of state whereas ions are warm and classical. The hydrodynamic approach along with the perturbation method have been applied to obtain the corresponding nonlinear Schrödinger equation (NLSE) in which nonlinearity is in balance with the dispersive terms. Using the NLSE, we could evaluate the modulational instability to show that various types of localized ion acoustic excitations exist in the form of either bright-type envelope solitons or dark-type envelope solitons. The regions of the stable and unstable envelope wave have been confined punctually for various regimes. Furthermore, it is proposed that the exact solutions of the NLSE for breather waves are the rogue waves (RWs), Akhmediev breather (AB), and Kuznetsov-Ma breather (KM) soliton. In order to show that the characteristics of breather structures is influenced by the plasma parameters (namely, relativistic parameter, positron concentration, and ionic temperature), the relevant numerical analysis of the NLSE is examined. In particular, it is observed that by increasing the values of the mentioned plasma parameters, the amplitude of the RWs will be decreased. Our results help researchers to explain the formation and dynamics of nonlinear electrostatic excitations in super dense astrophysical regimes.
19. Acoustic tests of Lorentz symmetry using Bulk Acoustic Wave quartz oscillators
CERN Document Server
Goryachev, M; Haslinger, Ph; Mizrachi, E; Anderegg, L; Müller, H; Hohensee, M; Tobar, M E
2016-01-01
A new method of probing Lorentz invariance in the neutron sector is described. The method is baed on stable quartz bulk acoustic wave oscillators compared on a rotating table. Due to Lorentz-invariance violation, the resonance frequencies of acoustic wave resonators depend on the direction in space via a corresponding dependence of masses of the constituent elements of solids. This dependence is measured via observation of oscillator phase noise built around such devices. The first such experiment now shows sensitivity to violation down to the limit $\\tilde{c}^n_Q=(-1.8\\pm2.2)\\times 10^{-14}$ GeV. Methods to improve the sensitivity are described together with some other applications of the technology in tests of fundamental physics.
20. All-Optical Detection of Acoustic Pressure Waves with applications in Photo-Acoustic Spectroscopy
CERN Document Server
Westergaard, Philip G
2016-01-01
An all-optical detection method for the detection of acoustic pressure waves is demonstrated. The detection system is based on a stripped (bare) single-mode fiber. The fiber vibrates as a standard cantilever and the optical output from the fiber is imaged to a displacement-sensitive optical detector. The absence of a conventional microphone makes the demonstrated system less susceptible to the effects that a hazardous environment might have on the sensor. The sensor is also useful for measurements in high temperature (above $200^{\\circ}$C) environments where conventional microphones will not operate. The proof-of-concept of the all-optical detection method is demonstrated by detecting sound waves generated by the photo-acoustic effect of NO$_2$ excited by a 455 nm LED, where a detection sensitivity of approximately 50 ppm was achieved.
1. Guided wave opto-acoustic device
Energy Technology Data Exchange (ETDEWEB)
Jarecki, Jr., Robert L.; Rakich, Peter Thomas; Camacho, Ryan; Shin, Heedeuk; Cox, Jonathan Albert; Qiu, Wenjun; Wang, Zheng
2016-02-23
The various technologies presented herein relate to various hybrid phononic-photonic waveguide structures that can exhibit nonlinear behavior associated with traveling-wave forward stimulated Brillouin scattering (forward-SBS). The various structures can simultaneously guide photons and phonons in a suspended membrane. By utilizing a suspended membrane, a substrate pathway can be eliminated for loss of phonons that suppresses SBS in conventional silicon-on-insulator (SOI) waveguides. Consequently, forward-SBS nonlinear susceptibilities are achievable at about 3000 times greater than achievable with a conventional waveguide system. Owing to the strong phonon-photon coupling achievable with the various embodiments, potential application for the various embodiments presented herein cover a range of radiofrequency (RF) and photonic signal processing applications. Further, the various embodiments presented herein are applicable to applications operating over a wide bandwidth, e.g. 100 MHz to 50 GHz or more.
2. Ultrafast strain gauge: Observation of THz radiation coherently generated by acoustic waves
Energy Technology Data Exchange (ETDEWEB)
Armstrong, M; Reed, E; Kim, K; Glownia, J; Howard, W M; Piner, E; Roberts, J
2008-08-14
The study of nanoscale, terahertz frequency (THz) acoustic waves has great potential for elucidating material and chemical interactions as well as nanostructure characterization. Here we report the first observation of terahertz radiation coherently generated by an acoustic wave. Such emission is directly related to the time-dependence of the stress as the acoustic wave crosses an interface between materials of differing piezoelectric response. This phenomenon enables a new class of strain wave metrology that is fundamentally distinct from optical approaches, providing passive remote sensing of the dynamics of acoustic waves with ultrafast time resolution. The new mechanism presented here enables nanostructure measurements not possible using existing optical or x-ray approaches.
3. Numerical modeling of nonlinear acoustic waves in a tube with an array of Helmholtz resonators
CERN Document Server
Lombard, Bruno
2013-01-01
Wave propagation in a 1-D guide with an array of Helmholtz resonators is studied numerically, considering large amplitude waves and viscous boundary layers. The model consists in two coupled equations: a nonlinear PDE of nonlinear acoustics, and a linear ODE describing the oscillations in the Helmholtz resonators. The dissipative effects in the tube and in the throats of the resonators are modeled by fractional derivatives. Based on a diffusive representation, the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations. An optimization procedure provides an efficient diffusive representation. A splitting strategy is then applied to the evolution equations: the propagative part is solved by a standard TVD scheme for hyperbolic equations, whereas the diffusive part is solved exactly. This approach is validated by comparisons with exact solutions. The properties of the full nonlinear solutions are investigated numerically. In particular, existenc...
4. Non-resonant interacting ion acoustic waves in a magnetized plasma
Energy Technology Data Exchange (ETDEWEB)
Maccari, Attilio [Technical Institute ' G Cardano' , Monterotondo, Rome (Italy)
1999-01-29
We perform an analytical and numerical investigation of the interaction among non-resonant ion acoustic waves in a magnetized plasma. Waves are supposed to be non-resonant, i.e. with different group velocities that are not close to each other. We use an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. We show that the amplitude slow modulation of Fourier modes cannot be described by the usual nonlinear Schroedinger equation but by a new model system of nonlinear evolution equations. This system is C-integrable, i.e. it can be linearized through an appropriate transformation of the dependent and independent variables. We demonstrate that a subclass of solutions gives rise to envelope solitons. Each envelope soliton propagates with its own group velocity. During a collision solitons maintain their shape, the only change being a phase shift. Numerical results are used to check the validity of the asymptotic perturbation method. (author)
5. Nonlinear damping of a finite amplitude whistler wave due to modified two stream instability
Energy Technology Data Exchange (ETDEWEB)
Saito, Shinji, E-mail: [email protected] [Graduate School of Science, Nagoya University, Nagoya (Japan); Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya (Japan); Nariyuki, Yasuhiro, E-mail: [email protected] [Faculty of Human Development, University of Toyama, Toyama (Japan); Umeda, Takayuki, E-mail: [email protected] [Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya (Japan)
2015-07-15
A two-dimensional, fully kinetic, particle-in-cell simulation is used to investigate the nonlinear development of a parallel propagating finite amplitude whistler wave (parent wave) with a wavelength longer than an ion inertial length. The cross field current of the parent wave generates short-scale whistler waves propagating highly oblique directions to the ambient magnetic field through the modified two-stream instability (MTSI) which scatters electrons and ions parallel and perpendicular to the magnetic field, respectively. The parent wave is largely damped during a time comparable to the wave period. The MTSI-driven damping process is proposed as a cause of nonlinear dissipation of kinetic turbulence in the solar wind.
6. Flow velocity measurement with the nonlinear acoustic wave scattering
Science.gov (United States)
Didenkulov, Igor; Pronchatov-Rubtsov, Nikolay
2015-10-01
A problem of noninvasive measurement of liquid flow velocity arises in many practical applications. To this end the most often approach is the use of the linear Doppler technique. The Doppler frequency shift of signal scattered from the inhomogeneities distributed in a liquid relatively to the emitted frequency is proportional to the sound frequency and velocities of inhomogeneities. In the case of very slow flow one needs to use very high frequency sound. This approach fails in media with strong sound attenuation because acoustic wave attenuation increases with frequency and there is limit in increasing sound intensity, i.e. the cavitation threshold. Another approach which is considered in this paper is based on the method using the difference frequency Doppler Effect for flows with bubbles. This method is based on simultaneous action of two high-frequency primary acoustic waves with closed frequencies on bubbles and registration of the scattered by bubbles acoustic field at the difference frequency. The use of this method is interesting since the scattered difference frequency wave has much lower attenuation in a liquid. The theoretical consideration of the method is given in the paper. The experimental examples confirming the theoretical equations, as well as the ability of the method to be applied in medical diagnostics and in technical applications on measurement of flow velocities in liquids with strong sound attenuation is described. It is shown that the Doppler spectrum form depends on bubble concentration velocity distribution in the primary acoustic beams crossing zone that allows one to measure the flow velocity distribution.
7. Flow velocity measurement with the nonlinear acoustic wave scattering
Energy Technology Data Exchange (ETDEWEB)
Didenkulov, Igor, E-mail: [email protected] [Institute of Applied Physics, 46 Ulyanov str., Nizhny Novgorod, 603950 (Russian Federation); Lobachevsky State University of Nizhny Novgorod, 23 Gagarin ave., Nizhny Novgorod, 603950 (Russian Federation); Pronchatov-Rubtsov, Nikolay, E-mail: [email protected] [Lobachevsky State University of Nizhny Novgorod, 23 Gagarin ave., Nizhny Novgorod, 603950 (Russian Federation)
2015-10-28
A problem of noninvasive measurement of liquid flow velocity arises in many practical applications. To this end the most often approach is the use of the linear Doppler technique. The Doppler frequency shift of signal scattered from the inhomogeneities distributed in a liquid relatively to the emitted frequency is proportional to the sound frequency and velocities of inhomogeneities. In the case of very slow flow one needs to use very high frequency sound. This approach fails in media with strong sound attenuation because acoustic wave attenuation increases with frequency and there is limit in increasing sound intensity, i.e. the cavitation threshold. Another approach which is considered in this paper is based on the method using the difference frequency Doppler Effect for flows with bubbles. This method is based on simultaneous action of two high-frequency primary acoustic waves with closed frequencies on bubbles and registration of the scattered by bubbles acoustic field at the difference frequency. The use of this method is interesting since the scattered difference frequency wave has much lower attenuation in a liquid. The theoretical consideration of the method is given in the paper. The experimental examples confirming the theoretical equations, as well as the ability of the method to be applied in medical diagnostics and in technical applications on measurement of flow velocities in liquids with strong sound attenuation is described. It is shown that the Doppler spectrum form depends on bubble concentration velocity distribution in the primary acoustic beams crossing zone that allows one to measure the flow velocity distribution.
8. Determination of hydrocarbon levels in water via laser-induced acoustics wave
Science.gov (United States)
Bidin, Noriah; Hossenian, Raheleh; Duralim, Maisarah; Krishnan, Ganesan; Marsin, Faridah Mohd; Nughro, Waskito; Zainal, Jasman
2016-04-01
Hydrocarbon contamination in water is a major environmental concern in terms of foreseen collapse of the natural ecosystem. Hydrocarbon level in water was determined by generating acoustic wave via an innovative laser-induced breakdown in conjunction with high-speed photographic coupling with piezoelectric transducer to trace acoustic wave propagation. A Q-switched Nd:YAG (40 mJ) was focused in cuvette-filled hydrocarbon solution at various concentrations (0-2000 ppm) to induce optical breakdown, shock wave generation and later acoustic wave propagation. A nitro-dye (ND) laser (10 mJ) was used as a flash to illuminate and frozen the acoustic wave propagation. Lasers were synchronised using a digital delay generator. The image of acoustic waves was grabbed and recorded via charged couple device (CCD) video camera at the speed of 30 frames/second with the aid of Matrox software version 9. The optical delay (0.8-10.0 μs) between the acoustic wave formation and its frozen time is recorded through photodetectors. A piezo-electric transducer (PZT) was used to trace the acoustic wave (sound signal), which cascades to a digital oscilloscope. The acoustic speed is calculated from the ratio of acoustic wave radius (1-8 mm) and optical time delay. Acoustic wave speed is found to linearly increase with hydrocarbon concentrations. The acoustic signal generation at higher hydrocarbon levels in water is attributed to supplementary mass transfer and impact on the probe. Integrated high-speed photography with transducer detection system authenticated that the signals indeed emerged from the laser-induced acoustic wave instead of photothermal processes. It is established that the acoustic wave speed in water is used as a fingerprint to detect the hydrocarbon levels.
9. Facilitation of epileptic activity during sleep is mediated by high amplitude slow waves.
Science.gov (United States)
Frauscher, Birgit; von Ellenrieder, Nicolás; Ferrari-Marinho, Taissa; Avoli, Massimo; Dubeau, François; Gotman, Jean
2015-06-01
Epileptic discharges in focal epilepsy are frequently activated during non-rapid eye movement sleep. Sleep slow waves are present during this stage and have been shown to include a deactivated ('down', hyperpolarized) and an activated state ('up', depolarized). The 'up' state enhances physiological rhythms, and we hypothesize that sleep slow waves and particularly the 'up' state are the specific components of non-rapid eye movement sleep that mediate the activation of epileptic activity. We investigated eight patients with pharmaco-resistant focal epilepsies who underwent combined scalp-intracerebral electroencephalography for diagnostic evaluation. We analysed 259 frontal electroencephalographic channels, and manually marked 442 epileptic spikes and 8487 high frequency oscillations during high amplitude widespread slow waves, and during matched control segments with low amplitude widespread slow waves, non-widespread slow waves or no slow waves selected during the same sleep stages (total duration of slow wave and control segments: 49 min each). During the slow waves, spikes and high frequency oscillations were more frequent than during control segments (79% of spikes during slow waves and 65% of high frequency oscillations, both P ∼ 0). The spike and high frequency oscillation density also increased for higher amplitude slow waves. We compared the density of spikes and high frequency oscillations between the 'up' and 'down' states. Spike and high frequency oscillation density was highest during the transition from the 'up' to the 'down' state. Interestingly, high frequency oscillations in channels with normal activity expressed a different peak at the transition from the 'down' to the 'up' state. These results show that the apparent activation of epileptic discharges by non-rapid eye movement sleep is not a state-dependent phenomenon but is predominantly associated with specific events, the high amplitude widespread slow waves that are frequent, but not
10. The effect of non-thermal electrons on obliquely propagating electron acoustic waves in a magnetized plasma
Science.gov (United States)
Singh, Satyavir; Bharuthram, Ramashwar
2016-07-01
Small amplitude electron acoustic solitary waves are studied in a magnetized plasma consisting of hot electrons following Cairn's type non-thermal distribution function and fluid cool electrons, cool ions and an electron beam. Using reductive perturbation technique, the Korteweg-de-Vries-Zakharov-Kuznetsov (KdV-ZK) equation is derived to describe the nonlinear evolution of electron acoustic waves. It is observed that the presence of non-thermal electrons plays an important role in determining the existence region of solitary wave structures. Theoretical results of this work is used to model the electrostatic solitary structures observed by Viking satellite. Detailed investigation of physical parameters such as non-thermality of hot electrons, beam electron velocity and temperature, obliquity on the existence regime of solitons will be discussed.
11. Multi Reflection of Lamb Wave Emission in an Acoustic Waveguide Sensor
OpenAIRE
Leonhard Michael Reindl; Bernd Henning; Jens Rautenberg; Gerhard Lindner; Sergei Olfert; Martin Schmitt
2013-01-01
Recently, an acoustic waveguide sensor based on multiple mode conversion of surface acoustic waves at the solid—liquid interfaces has been introduced for the concentration measurement of binary and ternary mixtures, liquid level sensing, investigation of spatial inhomogenities or bubble detection. In this contribution the sound wave propagation within this acoustic waveguide sensor is visualized by Schlieren imaging for continuous and burst operation the first time. In the acoustic waveguide ...
12. Sensitivity of surface acoustic wave devices
Science.gov (United States)
2001-08-01
The SAW devices are widely used as filters, delay lines, resonators and gas sensors. It is possible to use it as mechanical force. The paper describes sensitivity of acceleration sensor based on SAW using the Rayleigh wave propagation. Since characteristic of acceleration SAW sensors are largely determined by piezoelectric materials, it is very important to select substrate with required characteristics. Researches and numerical modeling based on simply sensor model include piezoelectric beam with unilateral free end. An aggregated mass is connected to the one. The dimension and aggregated mass are various. In this case a buckling stress and sensitivity are changed. Sensitivity in main and perpendicular axis are compare for three sensor based on SiO2, LiNbO3, Li2B4O7. Influences of phase velocity, electro-mechanical coupling constant and density on sensitivity are investigated. Some mechanical parameters of the substrates in dynamic work mode are researched using sensor model and Rayleigh model of vibrations without vibration damping. The model is useful because it simply determines dependencies between sensor parameters and substrate parameters. Differences between measured and evaluated quantities are less than 5 percent. Researches based on sensor modes, which fulfilled mechanical specifications similarly to aircraft navigation.
13. MESSENGER Orbital Observations of Large-Amplitude Kelvin-Helmholtz Waves at Mercury's Magnetopause
Science.gov (United States)
Sundberg, Torbjorn; Boardsen, Scott A.; Slavin, James A.; Anderson, Brian J.; Korth, Haje; Zurbuchen, Thomas H.; Raines, Jim M.; Solomon, Sean C.
2012-01-01
We present a survey of Kelvi\\ n-Helmholtz (KH) waves at Mercury's magnetopause during MESSENGER's first Mercury year in orb it. The waves were identified on the basis of the well-established sawtooth wave signatures that are associated with non-linear KH vortices at the magnetopause. MESSENGER frequently observed such KH waves in the dayside region of the magnetosphere where the magnetosheath flow velocity is still sub -sonic, which implies that instability growth rates at Mercury's magnetopau are much larger than at Earth. We attribute these greater rates to the limited wave energy dissipation in Mercury's highly resistive regolith. The wave amplitude was often on the order of ' 00 nT or more, and the wave periods were - 10- 20 s. A clear dawn-dusk asymmetry is present in the data, in that all of the observed wave events occurred in the post-noon and dusk-side sectors of the magnetopause. This asymmetry is like ly related to finite Larmor-radius effects and is in agreement with results from particle-in-cell simulations of the instability. The waves were observed almost exclusively during periods when the north-south component of the magnetosheath magnetic field was northward, a pattern similar to that for most terrestrial KH wave events. Accompanying plasma measurements show that the waves were associated with the transport of magnetosheath plasma into the magnetosphere.
14. Regularity of the Attractor for the Dissipative Hamiltonian Amplitude Equation Governing Modulated Wave Instabilities
Institute of Scientific and Technical Information of China (English)
Zheng-de Dai
2002-01-01
In the present paper, the existence of global attractor for dissipative Hamiltonian amplitude equation governing the modulated wave instabilities in E0 is considered. By a decomposition of solution operator, it is shown that the global attractor in E0 is actually equal to a global attractor in E1.
15. In search of objective manometric criteria for colonic high-amplitude propagated pressure waves
NARCIS (Netherlands)
De Schryver, AMP; Samsom, M; Smout, AJPM
2002-01-01
The aims of this study were to explore all characteristics of high-amplitude propagated contractions (HAPCs) that would allow them to be distinguished from nonHAPC colonic pressure waves, and to develop computer algorithms for automated HAPC detection. Colonic manometry recordings obtained from 24 h
16. Large Amplitude Low Frequency Waves in a Magnetized Nonuniform Electron-Positron-Ion Plasma
Institute of Scientific and Technical Information of China (English)
Q. Haque; H. Saleem
2004-01-01
@@ It is shown that the large amplitude low-frequency electromagnetic drift waves in electron-positron-ion plasmas might give rise to dipolar vortices. A linear dispersion relation of several coupled electrostatic and electromagnetic low-frequency modes is obtained. The relevance of this work to both laboratory and astrophysical situations is pointed out.
17. Scattering of gravitational radiation - Second order moments of the wave amplitude
NARCIS (Netherlands)
Macquart, JP
2004-01-01
Gravitational radiation that propagates through an inhomogeneous mass distribution is subject to random gravitational tensing, or scattering, causing variations in the wave amplitude and temporal smearing of the signal. A statistical theory is constructed to treat these effects. The statistical prop
18. Joint Inversion for Earthquake Depths Using Local Waveforms and Amplitude Spectra of Rayleigh Waves
Science.gov (United States)
Jia, Zhe; Ni, Sidao; Chu, Risheng; Zhan, Zhongwen
2017-01-01
Reliable earthquake depth is fundamental to many seismological problems. In this paper, we present a method to jointly invert for centroid depths with local (distance distance of 5°-15°) Rayleigh wave amplitude spectra on sparse networks. We use earthquake focal mechanisms and magnitudes retrieved with the Cut-and-Paste (CAP) method to compute synthetic amplitude spectra of fundamental mode Rayleigh wave for a range of depths. Then we grid search to find the optimal depth that minimizes the joint misfit of amplitude spectra and local waveforms. As case studies, we apply this method to the 2008 Wells, Nevada Mw6.0 earthquake and a Mw5.6 outer-rise earthquake to the east of Japan Trench in 2013. Uncertainties estimated with a bootstrap re-sampling approach show that this joint inversion approach constrains centroid depths well, which are also verified by independent teleseismic depth-phase data.
19. Spectral statistics of the acoustic stadium
Science.gov (United States)
Méndez-Sánchez, R. A.; Báez, G.; Leyvraz, F.; Seligman, T. H.
2014-01-01
We calculate the normal-mode frequencies and wave amplitudes of the two-dimensional acoustical stadium. We also obtain the statistical properties of the acoustical spectrum and show that they agree with the results given by random matrix theory. Some normal-mode wave amplitudes showing scarring are presented.
20. On the generation and evolution of numerically simulated large-amplitude internal gravity wave packets
Science.gov (United States)
Abdilghanie, Ammar M.; Diamessis, Peter J.
2012-01-01
Numerical simulations of internal gravity wave (IGW) dynamics typically rely on wave velocity and density fields which are either generated through forcing terms in the governing equations or are explicitly introduced as initial conditions. Both approaches are based on the associated solution to the inviscid linear internal wave equations and, thus, assume weak-amplitude, space-filling waves. Using spectral multidomain-based numerical simulations of the two-dimensional Navier-Stokes equations and focusing on the forcing-driven approach, this study examines the generation and subsequent evolution of large-amplitude IGW packets which are strongly localized in the vertical in a linearly stratified fluid. When the vertical envelope of the forcing terms varies relatively rapid when compared to the vertical wavelength, the associated large vertical gradients in the Reynolds stress field drive a nonpropagating negative horizontal mean flow component in the source region. The highly nonlinear interaction of this mean current with the propagating IGW packet leads to amplification of the wave, a significant distortion of its rear flank, and a substantial decay of its amplitude. Scaling arguments show that the mean flow is enhanced with a stronger degree of localization of the forcing, larger degree of hydrostaticity, and increasing wave packet steepness. Horizontal localization results in a pronounced reduction in mean flow strength mainly on account of the reduced vertical gradient of the wave Reynolds stress. Finally, two techniques are proposed toward the efficient containment of the mean flow at minimal computational cost. The findings of this study are of particular value in overcoming challenges in the design of robust computational process studies of IGW packet (or continuously forced wave train) interactions with a sloping boundary, critical layer, or caustic, where large wave amplitudes are required for any instabilities to develop. In addition, the detailed
1. Surface Modification on Acoustic Wave Biosensors for Enhanced Specificity
Directory of Open Access Journals (Sweden)
Nathan D. Gallant
2012-09-01
Full Text Available Changes in mass loading on the surface of acoustic biosensors result in output frequency shifts which provide precise measurements of analytes. Therefore, to detect a particular biomarker, the sensor delay path must be judiciously designed to maximize sensitivity and specificity. B-cell lymphoma 2 protein (Bcl-2 found in urine is under investigation as a biomarker for non-invasive early detection of ovarian cancer. In this study, surface chemistry and biofunctionalization approaches were evaluated for their effectiveness in presenting antibodies for Bcl-2 capture while minimizing non-specific protein adsorption. The optimal combination of sequentially adsorbing protein A/G, anti-Bcl-2 IgG and Pluronic F127 onto a hydrophobic surface provided the greatest signal-to-noise ratio and enabled the reliable detection of Bcl-2 concentrations below that previously identified for early stage ovarian cancer as characterized by a modified ELISA method. Finally, the optimal surface modification was applied to a prototype acoustic device and the frequency shift for a range of Bcl-2 concentration was quantified to demonstrate the effectiveness in surface acoustic wave (SAW-based detection applications. The surface functionalization approaches demonstrated here to specifically and sensitively detect Bcl-2 in a working ultrasonic MEMS biosensor prototype can easily be modified to detect additional biomarkers and enhance other acoustic biosensors.
2. Underwater acoustic wave generation by filamentation of terawatt ultrashort laser pulses
CERN Document Server
Jukna, Vytautas; Milián, Carles; Brelet, Yohann; Carbonnel, Jérôme; André, Yves-Bernard; Guillermin, Régine; Sessarego, Jean-Pierre; Fattaccioli, Dominique; Mysyrowicz, André; Couairon, Arnaud; Houard, Aurélien
2016-01-01
Acoustic signals generated by filamentation of ultrashort TW laser pulses in water are characterized experimentally. Measurements reveal a strong influence of input pulse duration on the shape and intensity of the acoustic wave. Numerical simulations of the laser pulse nonlinear propagation and the subsequent water hydrodynamics and acoustic wave generation show that the strong acoustic emission is related to the mechanism of superfilamention in water. The elongated shape of the plasma volume where energy is deposited drives the far-field profile of the acoustic signal, which takes the form of a radially directed pressure wave with a single oscillation and a very broad spectrum.
3. THE INFLUENCE OF WAVE PATTERNS AND FREQUENCY ON THERMO-ACOUSTIC COOLING EFFECT
Directory of Open Access Journals (Sweden)
CHEN BAIMAN
2011-06-01
Full Text Available With the increasing environmental challenges, the search for an environmentally benign cooling technology that has simple and robust architecture continues. Thermo-acoustic refrigeration seems to be a promising candidate to fulfil these requirements. In this study, a simple thermo-acoustic refrigeration system was fabricated and tested. The thermo-acoustic refrigerator consists of acoustic driver (loudspeaker, resonator, stack, vacuum system and testing system. The effect of wave patterns and frequency on thermo-acoustic cooling effect was studied. It was found that a square wave pattern would yield superior cooling effects compared to other wave patterns tested.
4. Theoretical study of the anisotropic diffraction of light waves by acoustic waves in lithium niobate crystals.
Science.gov (United States)
Rouvaen, J M; Waxin, G; Gazalet, M G; Bridoux, E
1990-03-20
The anisotropic diffraction of light by high frequency longitudinal ultrasonic waves in the tangential phase matching configuration may present some definite advantages over the same interaction using transverse acoustic waves. A systematic search for favorable crystal cuts in lithium niobate was worked out. The main results of this study are reported here; they enable the choice of the best configuration for a given operating center frequency.
5. Enhanced Sensitive Love Wave Surface Acoustic Wave Sensor Designed for Immunoassay Formats
OpenAIRE
Mihaela Puiu; Ana-Maria Gurban; Lucian Rotariu; Simona Brajnicov; Cristian Viespe; Camelia Bala
2015-01-01
We report a Love wave surface acoustic wave (LW-SAW) immunosensor designed for the detection of high molecular weight targets in liquid samples, amenable also for low molecular targets in surface competition assays. We implemented a label-free interaction protocol similar to other surface plasmon resonance bioassays having the advantage of requiring reduced time analysis. The fabricated LW-SAW sensor supports the detection of the target in the nanomolar range, and can be ultimately incorporat...
6. Asymmetric wave transmission in a diatomic acoustic/elastic metamaterial
Energy Technology Data Exchange (ETDEWEB)
Li, Bing; Tan, K. T., E-mail: [email protected] [Department of Mechanical Engineering, The University of Akron, Akron, Ohio 44325-3903 (United States)
2016-08-21
Asymmetric acoustic/elastic wave transmission has recently been realized using nonlinearity, wave diffraction, or bias effects, but always at the cost of frequency distortion, direction shift, large volumes, or external energy. Based on the self-coupling of dual resonators, we propose a linear diatomic metamaterial, consisting of several small-sized unit cells, to realize large asymmetric wave transmission in low frequency domain (below 1 kHz). The asymmetric transmission mechanism is theoretically investigated, and numerically verified by both mass-spring and continuum models. This passive system does not require any frequency conversion or external energy, and the asymmetric transmission band can be theoretically predicted and mathematically controlled, which extends the design concept of unidirectional transmission devices.
7. Adsorption-Mediated Mass Streaming in a Standing Acoustic Wave
Science.gov (United States)
Weltsch, Oren; Offner, Avshalom; Liberzon, Dan; Ramon, Guy Z.
2017-06-01
Oscillating flows can generate nonzero, time-averaged fluxes despite the velocity averaging zero over an oscillation cycle. Here, we report such a flux, a nonlinear resultant of the interaction between oscillating velocity and concentration fields. Specifically, we study a gas mixture sustaining a standing acoustic wave, where an adsorbent coats the solid boundary in contact with the gas mixture. It is found that the sound wave produces a significant, time-averaged preferential flux of a "reactive" component that undergoes a reversible sorption process. This effect is measured experimentally for an air-water vapor mixture. An approximate model is shown to be in good agreement with the experimental observations, and further reveals the interplay between the sound-wave characteristics and the properties of the gas-solid sorbate-sorbent pair. The preferential flux generated by this mechanism may have potential in separation processes.
8. Langasite surface acoustic wave gas sensors: modeling and verification
Energy Technology Data Exchange (ETDEWEB)
Peng Zheng,; Greve, D. W.; Oppenheim, I. J.
2013-03-01
We report finite element simulations of the effect of conductive sensing layers on the surface wave velocity of langasite substrates. The simulations include both the mechanical and electrical influences of the conducting sensing layer. We show that three-dimensional simulations are necessary because of the out-of-plane displacements of the commonly used (0, 138.5, 26.7) Euler angle. Measurements of the transducer input admittance in reflective delay-line devices yield a value for the electromechanical coupling coefficient that is in good agreement with the three-dimensional simulations on bare langasite substrate. The input admittance measurements also show evidence of excitation of an additional wave mode and excess loss due to the finger resistance. The results of these simulations and measurements will be useful in the design of surface acoustic wave gas sensors.
9. A fractional calculus model of anomalous dispersion of acoustic waves.
Science.gov (United States)
Wharmby, Andrew W
2016-09-01
An empirical formula based on viscoelastic analysis techniques that employs concepts from the fractional calculus that was used to model the dielectric behavior of materials exposed to oscillating electromagnetic fields in the radiofrequency, terahertz, and infrared bands. This work adapts and applies the formula to model viscoelastic behavior of materials that show an apparent increase of phase velocity of vibration with an increase in frequency, otherwise known as anomalous dispersion. A fractional order wave equation is derived through the application of the classic elastic-viscoelastic correspondence principle whose analytical solution is used to describe absorption and dispersion of acoustic waves in the viscoelastic material displaying anomalous dispersion in a specific frequency range. A brief discussion and comparison of an alternative fractional order wave equation recently formulated is also included.
10. Langasite Surface Acoustic Wave Gas Sensors: Modeling and Verification
Energy Technology Data Exchange (ETDEWEB)
Zheng, Peng; Greve, David W; Oppenheim, Irving J
2013-01-01
We report finite element simulations of the effect of conductive sensing layers on the surface wave velocity of langasite substrates. The simulations include both the mechanical and electrical influences of the conducting sensing layer. We show that three-dimensional simulations are necessary because of the out-of-plane displacements of the commonly used (0, 138.5, 26.7) Euler angle. Measurements of the transducer input admittance in reflective delay-line devices yield a value for the electromechanical coupling coefficient that is in good agreement with the three-dimensional simulations on bare langasite substrate. The input admittance measurements also show evidence of excitation of an additional wave mode and excess loss due to the finger resistance. The results of these simulations and measurements will be useful in the design of surface acoustic wave gas sensors.
11. Disperson relation of finite amplitude Alfven wave in a relativistic electron- positron plasma
CERN Document Server
2004-01-01
The linear dispersion relation of a finite amplitude, parallel, circularly polarized Alfv\\'en wave in a relativistic electron-positron plasma is derived. In the nonrelativistic regime, the dispersion relation has two branches, one electromagnetic wave, with a low frequency cutoff at $\\sqrt{1+2\\omega_p^2/\\Omega_p^2}$ (where $\\omega_p=(4\\pi n e^2/m)^{1/2}$ is the electron/positron plasma frequency), and an Alfv\\'en wave, with high frequency cutoff at the positron gyrofrequency $\\Omega_p$. There is only one forward propagating mode for a given frequency. However, due to relativistic effects, there is no low frequency cutoff for the electromagnetic branch, and there appears a critical wave number above which the Alfv\\'en wave ceases to exist. This critical wave number is given by $ck_c/\\Omega_p=a/\\eta$, where $a=\\omega_p^2/\\Omega_p^2$ and $\\eta$ is the ratio between the Alfv\\'en wave magnetic field amplitude and the background magnetic field. In this case, for each frequency in the Alfv\\'en branch, two additional...
12. Local finite-amplitude wave activity as an objective diagnostic of midlatitude extreme weather
Energy Technology Data Exchange (ETDEWEB)
Chen, Gang; Lu, Jian; Burrows, Alex D.; Leung, Lai-Yung R.
2015-12-28
Midlatitude extreme weather events are responsible for a large part of climate related damage, yet our understanding of these extreme events is limited, partly due to the lack of a theoretical basis for midlatitude extreme weather. In this letter, the local finite-amplitude wave activity (LWA) of Huang and Nakamura [2015] is introduced as a diagnostic of the 500-hPa geopotential height (Z500) to characterizing midlatitude weather events. It is found that the LWA climatology and its variability associated with the Arctic Oscillation (AO) agree broadly with the previously reported blocking frequency in literature. There is a strong seasonal and spatial dependence in the trend13 s of LWA in recent decades. While there is no observational evidence for a hemispheric-scale increase in wave amplitude, robust trends in wave activity can be identified at the regional scales, with important implications for regional climate change.
13. Characteristics of Electron Distributions Observed During Large Amplitude Whistler Wave Events in the Magnetosphere
Science.gov (United States)
Wilson, Lynn B., III
2010-01-01
We present a statistical study of the characteristics of electron distributions associated with large amplitude whistler waves inside the terrestrial magnetosphere using waveform capture data as an addition of the study by Kellogg et al., [2010b]. We identified three types of electron distributions observed simultaneously with the whistler waves including beam-like, beam/flattop, and anisotropic distributions. The whistlers exhibited different characteristics dependent upon the observed electron distributions. The majority of the waveforms observed in our study have f/fce or = 8 nT pk-pk) whistler wave measured in the radiation belts. The majority of the largest amplitude whistlers occur during magnetically active periods (AE > 200 nT).
14. Adaptive defect and pattern detection in amplitude and phase structures via photorefractive four-wave mixing.
Science.gov (United States)
Nehmetallah, George; Banerjee, Partha; Khoury, Jed
2015-11-10
This work comprises the theoretical and numerical validations of experimental work on pattern and defect detection of periodic amplitude and phase structures using four-wave mixing in photorefractive materials. The four-wave mixing optical processor uses intensity filtering in the Fourier domain. Specifically, the nonlinear transfer function describing four-wave mixing is modeled, and the theory for detection of amplitude and phase defects and dislocations are developed. Furthermore, numerical simulations are performed for these cases. The results show that this technique successfully detects the slightest defects clearly even with no prior enhancement. This technique should prove to be useful in quality control systems, production-line defect inspection, and e-beam lithography.
15. Prestimulus amplitudes modulate P1 latencies and evoked traveling alpha waves
Directory of Open Access Journals (Sweden)
Nicole Alexandra Himmelstoss
2015-05-01
Full Text Available Traveling waves have been well documented in the ongoing, and more recently also in the evoked EEG. In the present study we investigate what kind of physiological process might be responsible for inducing an evoked traveling wave. We used a semantic judgment task which already proved useful to study evoked traveling alpha waves that coincide with the appearance of the P1 component. We found that the P1 latency of the leading electrode is significantly correlated with prestimulus amplitude size and that this event is associated with a transient change in alpha frequency. We assume that cortical background excitability, as reflected by an increase in prestimulus amplitude, is responsible for the observed change in alpha frequency and the initiation of an evoked traveling trajectory.
16. An Amplitude-Based Estimation Method for International Space Station (ISS) Leak Detection and Localization Using Acoustic Sensor Networks
Science.gov (United States)
2009-01-01
The development of a robust and efficient leak detection and localization system within a space station environment presents a unique challenge. A plausible approach includes the implementation of an acoustic sensor network system that can successfully detect the presence of a leak and determine the location of the leak source. Traditional acoustic detection and localization schemes rely on the phase and amplitude information collected by the sensor array system. Furthermore, the acoustic source signals are assumed to be airborne and far-field. Likewise, there are similar applications in sonar. In solids, there are specialized methods for locating events that are used in geology and in acoustic emission testing that involve sensor arrays and depend on a discernable phase front to the received signal. These methods are ineffective if applied to a sensor detection system within the space station environment. In the case of acoustic signal location, there are significant baffling and structural impediments to the sound path and the source could be in the near-field of a sensor in this particular setting.
17. Large amplitude, leaky, island-generated, internal waves around Palau, Micronesia
Science.gov (United States)
Wolanski, E.; Colin, P.; Naithani, J.; Deleersnijder, E.; Golbuu, Y.
2004-08-01
Three years of temperature data along two transects extending to 90 m depth, at Palau, Micronesia, show twice-a-day thermocline vertical displacements of commonly 50-100 m, and on one occasion 270 m. The internal wave occurred at a number of frequencies. There were a number of spectral peaks at diurnal and semi-diurnal frequencies, as well as intermediate and sub-inertial frequencies, less so at the inertial frequency. At Palau the waves generally did not travel around the island because there was no coherence between internal waves on either side of the island. The internal waves at a site 30 km offshore were out-of-phase with those on the island slopes, suggesting that the waves were generated on the island slope and then radiated away. Palau Island was thus a source of internal wave energy for the surrounding ocean. A numerical model suggests that the tidal and low-frequency currents flowing around the island form internal waves with maximum wave amplitude on the island slope and that these waves radiate away from the island. The model also suggests that the headland at the southern tip of Palau prevents the internal waves to rotate around the island. The large temperature fluctuations (commonly daily fluctuations ≈10 °C, peaking at 20 °C) appear responsible for generating a thermal stress responsible for a biologically depauperate biological community on the island slopes at depths between 60 and 120 m depth.
18. Stimulation of whistler activity by an artificial ground-based low frequency acoustic wave source
Science.gov (United States)
Soroka, Silvestr; Kim, Vitaly; Khegay, Valery; Kalita, Bogdan
This paper presents some results of an active experiment aimed to impact the ionosphere with low frequency acoustic waves artificially generated in the near-ground atmosphere. The main goal of the experiment was checking if the artificially generated acoustic waves could affect whistler occurrence at middle latitudes. As a source of acoustic waves we used twin powerful sonic speakers. One of which produced acoustic waves at a frequency of 600 Hz while the other one at a frequency of 624 Hz with intensity of 160 dB at a distance of 1 m away from end of the horn. The duration of sonic pulse was one minute. As a result of acoustic wave interference above the acoustic wave source there appears some kind of a virtual sonic antenna that radiates lower frequency acoustic waves at a frequency being equal to the difference of the two initially generated frequencies (624 Hz - 600 Hz = 24 Hz). The resulting acoustic wave is capable to penetrate to higher altitudes than the initially generated waves do because of its lower frequency. A whistler detector was located at about 100 m far away from the acoustic wave source. We performed the 50 experiments at Lviv (49.50° N, 24.00° E) with acoustic influence on the atmosphere-ionosphere system. The obtained results indicate that the emitted low frequency acoustic waves were clearly followed by enhanced whistler occurrence. We suggest that the observations could be interpreted how increase of transparency of ionosphere and upward refraction of VLF spherics resulted from modulation of local atmospheric parameters by the acoustic waves. These two effects produce to the increase of amount of the whistlers.
19. Enhancement of acoustic streaming induced flow on a focused surface acoustic wave device: Implications for biosensing and microfluidics
Science.gov (United States)
Singh, Reetu; Sankaranarayanan, Subramanian K. R. S.; Bhethanabotla, Venkat R.
2010-01-01
Fluid motion induced on the surface of 100 MHz focused surface acoustic wave (F-SAW) devices with concentric interdigital transducers (IDTs) based on Y-cut Z-propagating LiNbO3 substrate was investigated using three-dimensional bidirectionally coupled finite element fluid-structure interaction models. Acoustic streaming velocity fields and induced forces for the F-SAW device are compared with those for a SAW device with uniform IDTs (conventional SAW). Both, qualitative and quantitative differences in the simulation derived functional parameters, such as device displacements amplitudes, fluid velocity, and streaming forces, are observed between the F-SAW and conventional SAW device. While the conventional SAW shows maximum fluid recirculation near input IDTs, the region of maximum recirculation is concentrated near the focal point of the F-SAW device. Our simulation results also indicate acoustic energy focusing by the F-SAW device leading to maximized device surface displacements, fluid velocity, and streaming forces near the focal point located in the center of the delay path, in contrast to the conventional SAW exhibiting maximized values of these parameters near the input IDTs. Significant enhancement in acoustic streaming is observed in the F-SAW device when compared to the conventional ones; the increase in streaming velocities was computed to be 352% and 216% for tangential velocities in propagation and transverse directions, respectively, and 353% for the normal velocity. Consequently, the induced streaming force for F-SAW is 480% larger than that for conventional SAW. In biosensing applications, this allows for the removal of smaller submicron sized particles by F-SAW which are otherwise difficult to remove using the conventional SAW. The F-SAW presents an order of magnitude reduction in the smallest removable particle size compared to the conventional device. Our results indicate that the acoustic energy focusing and streaming enhancement brought about by
20. Solitary, explosive, rational and elliptic doubly periodic solutions for nonlinear electron-acoustic waves in the earth's magnetotail region
CERN Document Server
El-Wakil, S A; El-Shewy, E K; Abd-El-Hamid, H M
2010-01-01
A theoretical investigation has been made of electron acoustic wave propagating in unmagnetized collisionless plasma consisting of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions. Based on the pseudo-potential approach, large amplitude potential structures and the existence of Solitary waves are discussed. The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation for small but finite amplitude electrostatic waves. An algebraic method with computerized symbolic computation, which greatly exceeds the applicability of the existing tanh, extended tanh methods in obtaining a series of exact solutions of the KdV equation, is used here. Numerical studies have been made using plasma parameters close to those values corresponding to Earth's plasma sheet boundary layer region reveals different solutions i.e., bell-shaped solitary pulses and singularity solutions at a finite point which called "blowup" solutions, Jaco...
1. Focusing of Surface Acoustic Wave on a Piezoelectric Crystal
Institute of Scientific and Technical Information of China (English)
QIAO Dong-Hai; WANG Cheng-Hao; WANG Zuo-Qing
2006-01-01
@@ We investigate the focusing phenomena of a surface acoustic wave (SAW) field generated by a circular-arc interdigital transducer (IDT) on a piezoelectric crystal. A rigorous vector field theory of surface excitation on the crystal we developed previously is used to evaluate the convergent SAW field instead of the prevalent scalar angular spectrum used in optics. The theoretical results show that the anisotropy of a medium has great impact on the focusing properties of the acoustic beams, such as focal length and symmetrical distributions near the focus. A dark field method is used in experiment to observe the focusing of the SAW field optically. Although the convergent phenomena of SAW field on the anisotropic media or piezoelectric crystals are very complicated,the experimental data are in agreement with those from the rigorous theory.
2. Visualization of Surface Acoustic Waves in Thin Liquid Films
Science.gov (United States)
Rambach, R. W.; Taiber, J.; Scheck, C. M. L.; Meyer, C.; Reboud, J.; Cooper, J. M.; Franke, T.
2016-02-01
We demonstrate that the propagation path of a surface acoustic wave (SAW), excited with an interdigitated transducer (IDT), can be visualized using a thin liquid film dispensed onto a lithium niobate (LiNbO3) substrate. The practical advantages of this visualization method are its rapid and simple implementation, with many potential applications including in characterising acoustic pumping within microfluidic channels. It also enables low-cost characterisation of IDT designs thereby allowing the determination of anisotropy and orientation of the piezoelectric substrate without the requirement for sophisticated and expensive equipment. Here, we show that the optical visibility of the sound path critically depends on the physical properties of the liquid film and identify heptane and methanol as most contrast rich solvents for visualization of SAW. We also provide a detailed theoretical description of this effect.
3. Amplitude, phase, location and orientation calibration of an acoustic vector sensor array, part I: Theory
NARCIS (Netherlands)
Xu, B.; Wind, J.; Bree, H.E. de; Basten, T.G.H.; Druyvesteyn, E.
2010-01-01
An acoustic vector sensor array consists of multiple sound pressure microphones and particle velocity sensors. A pressure microphone usually has an omni-directional response, yet a particle velocity sensor is directional. Currently, acoustic vector sensor arrays are under investigation for far field
4. Amplitude, phase, location and orientation calibration of an acoustic vector sensor array, part II: Experiments
NARCIS (Netherlands)
Basten, T.G.H.; Wind, J.; Xu, B.; Bree, H.E. de; Druyvesteyn, E.
2010-01-01
An acoustic vector sensor array consists of multiple sound pressure microphones and particle velocity sensors. A pressure microphone usually has an omni-directional response, yet a particle velocity sensor is directional and usually has a response pattern as a figure of eight. Currently, acoustic
5. Amplitude, phase, location and orientation calibration of an acoustic vector sensor array, part II: Experiments
NARCIS (Netherlands)
Basten, T.G.H.; Wind, J.; Xu, B.; Bree, H.E. de; Druyvesteyn, E.
2010-01-01
An acoustic vector sensor array consists of multiple sound pressure microphones and particle velocity sensors. A pressure microphone usually has an omni-directional response, yet a particle velocity sensor is directional and usually has a response pattern as a figure of eight. Currently, acoustic ve
6. Amplitude, phase, location and orientation calibration of an acoustic vector sensor array, part I: Theory
NARCIS (Netherlands)
Xu, B.; Wind, J.; Bree, H.E. de; Basten, T.G.H.; Druyvesteyn, E.
2010-01-01
An acoustic vector sensor array consists of multiple sound pressure microphones and particle velocity sensors. A pressure microphone usually has an omni-directional response, yet a particle velocity sensor is directional. Currently, acoustic vector sensor arrays are under investigation for far field
7. Molding acoustic, electromagnetic and water waves with a single cloak
KAUST Repository
Xu, Jun
2015-06-09
We describe two experiments demonstrating that a cylindrical cloak formerly introduced for linear surface liquid waves works equally well for sound and electromagnetic waves. This structured cloak behaves like an acoustic cloak with an effective anisotropic density and an electromagnetic cloak with an effective anisotropic permittivity, respectively. Measured forward scattering for pressure and magnetic fields are in good agreement and provide first evidence of broadband cloaking. Microwave experiments and 3D electromagnetic wave simulations further confirm reduced forward and backscattering when a rectangular metallic obstacle is surrounded by the structured cloak for cloaking frequencies between 2.6 and 7.0 GHz. This suggests, as supported by 2D finite element simulations, sound waves are cloaked between 3 and 8 KHz and linear surface liquid waves between 5 and 16 Hz. Moreover, microwave experiments show the field is reduced by 10 to 30 dB inside the invisibility region, which suggests the multi-wave cloak could be used as a protection against water, sonic or microwaves. © 2015, Nature Publishing Group. All rights reserved.
8. Molding acoustic, electromagnetic and water waves with a single cloak.
Science.gov (United States)
Xu, Jun; Jiang, Xu; Fang, Nicholas; Georget, Elodie; Abdeddaim, Redha; Geffrin, Jean-Michel; Farhat, Mohamed; Sabouroux, Pierre; Enoch, Stefan; Guenneau, Sébastien
2015-06-09
We describe two experiments demonstrating that a cylindrical cloak formerly introduced for linear surface liquid waves works equally well for sound and electromagnetic waves. This structured cloak behaves like an acoustic cloak with an effective anisotropic density and an electromagnetic cloak with an effective anisotropic permittivity, respectively. Measured forward scattering for pressure and magnetic fields are in good agreement and provide first evidence of broadband cloaking. Microwave experiments and 3D electromagnetic wave simulations further confirm reduced forward and backscattering when a rectangular metallic obstacle is surrounded by the structured cloak for cloaking frequencies between 2.6 and 7.0 GHz. This suggests, as supported by 2D finite element simulations, sound waves are cloaked between 3 and 8 KHz and linear surface liquid waves between 5 and 16 Hz. Moreover, microwave experiments show the field is reduced by 10 to 30 dB inside the invisibility region, which suggests the multi-wave cloak could be used as a protection against water, sonic or microwaves.
9. Passive Wireless Hydrogen Sensors Using Orthogonal Frequency Coded Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the development of passive surface acoustic wave (SAW) based hydrogen sensors for NASA application to distributed wireless hydrogen leak...
10. Numerical and experimental study of Lamb wave propagation in a two-dimensional acoustic black hole
Science.gov (United States)
Yan, Shiling; Lomonosov, Alexey M.; Shen, Zhonghua
2016-06-01
The propagation of laser-generated Lamb waves in a two-dimensional acoustic black-hole structure was studied numerically and experimentally. The geometrical acoustic theory has been applied to calculate the beam trajectories in the region of the acoustic black hole. The finite element method was also used to study the time evolution of propagating waves. An optical system based on the laser-Doppler vibration method was assembled. The effect of the focusing wave and the reduction in wave speed of the acoustic black hole has been validated.
11. Passive Wireless Hydrogen Sensors Using Orthogonal Frequency Coded Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the continued development of passive orthogonal frequency coded (OFC) surface acoustic wave (SAW) based hydrogen sensors for NASA application...
12. Mass sensitivity of layered shear-horizontal surface acoustic wave devices for sensing applications
Science.gov (United States)
2001-11-01
Layered Surface Acoustic Wave (SAW) devices that allow the propagation of Love mode acoustic waves will be studied in this paper. In these devices, the substrate allows the propagation of Surface Skimming Bulks Waves (SSBWs). By depositing layers, that the speed of Shear Horizontal (SH) acoustic wave propagation is less than that of the substrate, the propagation mode transforms to Love mode. Love mode devices which will be studied in this paper, have SiO2 and ZnO acoustic guiding layers. As Love mode of propagation has no movement of particles component normal to the active sensor surface, they can be employed for the sensing applications in the liquid media.
13. Re-radiation of acoustic waves from the A0 wave on a submerged elastic shell.
Science.gov (United States)
Ahyi, A C; Cao, Hui; Raju, P K; Uberall, Herbert
2005-07-01
We consider evacuated thin semi-infinite shells immersed in a fluid, which may be either of cylindrical shape with a hemispherical shell endcap, or formed two-dimensionally by semi-infinite parallel plates joined together by a semi-cylinder. The connected shell portions are joined in a manner to satisfy continuity but with a discontinuous radius of curvature. Acoustic waves are considered incident along the axis of symmetry (say the z axis) onto the curved portion of the shell, where they, at the critical angle of coincidence, generate Lamb and Stoneley-type waves in the shell. Computations were carried out using a code developed by Cao et al. [Chinese J. Acoust. 14, 317 (1995)] and was used in order to computationally visualize the waves in the fluid that have been re-radiated by the shell waves a the critical angle. The frequency range was below that of the lowest Lamb wave, and only the A0 wave (and partly the S0 wave) was observed to re-radiate into the fluid under our assumptions. The results will be compared to experimental results in which the re-radiated waves are optically visualized by the Schardin-Cranz schlieren method.
14. Reconstruction and prediction of coherent acoustic field with the combined wave superposition approach
Institute of Scientific and Technical Information of China (English)
LI Weibing; CHEN Jian; YU Fei; CHEN Xinzhao
2006-01-01
The routine wave superposition approach cannot be used in reconstruction and prediction of a coherent acoustic field, because it is impossible to separate the pressures generated by individual sources. According to the superposition theory of the coherent acoustic field , a novel method based on the combined wave superposition approach is developed to reconstruct and predict the coherent acoustic field by building the combined pressure matching matrixes between the hologram surfaces and the sources. The method can reconstruct the acoustic information on surfaces of the individual sources, and it is possible to predict the acoustic field radiated from every source and the total coherent acoustic field can also be calculated spontaneously. The experimental and numerical simulation results show that this method can effectively solve the holographic reconstruction and prediction of the coherent acoustic field and it can also be used as a coherent acoustic field separation technique. The study on this novel method extends the application scope of the acoustic holography technique.
15. Ray splitting in the reflection and refraction of surface acoustic waves in anisotropic solids.
Science.gov (United States)
Every, A G; Maznev, A A
2010-05-01
This paper examines the conditions for, and provides examples of, ray splitting in the reflection and refraction of surface acoustic waves (SAW) in elastically anisotropic solids at straight obstacles such as edges, surface breaking cracks, and interfaces between different solids. The concern here is not with the partial scattering of an incident SAW's energy into bulk waves, but with the occurrence of more than one SAW ray in the reflected and/or transmitted wave fields, by analogy with birefringence in optics and mode conversion of bulk elastic waves at interfaces. SAW ray splitting is dependent on the SAW slowness curve possessing concave regions, which within the constraint of wave vector conservation parallel to the obstacle allows multiple outgoing SAW modes for certain directions of incidence and orientation of obstacle. The existence of pseudo-SAW for a given surface provides a further channel for ray splitting. This paper discusses some typical material configurations for which SAW ray splitting occurs. An example is provided of mode conversion entailing backward reflection or negative refraction. Experimental demonstration of ray splitting in the reflection of a laser generated SAW in GaAs(111) is provided. The calculation of SAW mode conversion amplitudes lies outside the scope of this paper.
16. Dust ion-acoustic shock waves due to dust charge fluctuation in a superthermal dusty plasma
Energy Technology Data Exchange (ETDEWEB)
Alinejad, H., E-mail: [email protected] [Department of Physics, Faculty of Basic Science, Babol University of Technology, Babol 47148-71167 (Iran, Islamic Republic of); Research Institute for Fundamental Sciences (RIFS), University of Tabriz, 51664, Tabriz (Iran, Islamic Republic of); Tribeche, M. [Plasma Physics Group, Faculty of Sciences – Physics, University of Bab-Ezzouar (Algeria); Mohammadi, M.A. [Research Institute for Fundamental Sciences (RIFS), University of Tabriz, 51664, Tabriz (Iran, Islamic Republic of); Department of Atomic and Molecular Physics, Faculty of Physics, University of Tabriz, Tabriz (Iran, Islamic Republic of)
2011-11-14
The nonlinear propagation of dust ion-acoustic (DIA) shock waves is studied in a charge varying dusty plasma with electrons having kappa velocity distribution. We use hot ions with equilibrium streaming speed and a fast superthermal electron charging current derived from orbit limited motion (OLM) theory. It is found that the presence of superthermal electrons does not only significantly modify the basic properties of shock waves, but also causes the existence of shock profile with only positive potential in such plasma with parameter ranges corresponding to Saturn's rings. It is also shown that the strength and steepness of the shock waves decrease with increase of the size of dust grains and ion temperature. -- Highlights: ► The presence of superthermal electrons causes the existence of shock waves with only positive potential. ► The strength and steepness of the shock waves decrease with increase of the size of dust grains and ion temperature. ► As the electrons evolve toward their thermodynamic equilibrium, the shock structures are found with smaller amplitude.
17. Linear and nonlinear heavy ion-acoustic waves in a strongly coupled plasma
Energy Technology Data Exchange (ETDEWEB)
Ema, S. A., E-mail: [email protected]; Mamun, A. A. [Department of Physics, Jahangirnagar University, Savar, Dhaka-1342 (Bangladesh); Hossen, M. R. [Deparment of Natural Sciences, Daffodil International University, Sukrabad, Dhaka-1207 (Bangladesh)
2015-09-15
A theoretical study on the propagation of linear and nonlinear heavy ion-acoustic (HIA) waves in an unmagnetized, collisionless, strongly coupled plasma system has been carried out. The plasma system is assumed to contain adiabatic positively charged inertial heavy ion fluids, nonextensive distributed electrons, and Maxwellian light ions. The normal mode analysis is used to study the linear behaviour. On the other hand, the well-known reductive perturbation technique is used to derive the nonlinear dynamical equations, namely, Burgers equation and Korteweg-de Vries (K-dV) equation. They are also numerically analyzed in order to investigate the basic features of shock and solitary waves. The adiabatic effects on the HIA shock and solitary waves propagating in such a strongly coupled plasma are taken into account. It has been observed that the roles of the adiabatic positively charged heavy ions, nonextensivity of electrons, and other plasma parameters arised in this investigation have significantly modified the basic features (viz., polarity, amplitude, width, etc.) of the HIA solitary/shock waves. The findings of our results obtained from this theoretical investigation may be useful in understanding the linear as well as nonlinear phenomena associated with the HIA waves both in space and laboratory plasmas.
18. Analysis for the amplitude oscillatory movements of the ship in response to the incidence wave
Science.gov (United States)
Chiţu, M. G.; Zăgan, R.; Manea, E.
2015-11-01
Event of major accident navigation near offshore drilling rigs remains unacceptably high, known as the complications arising from the problematic of the general motions of the ship sailing under real sea. Dynamic positioning system is an effective instrument used on board of the ships operating in the extraction of oil and gas in the continental shelf of the seas and oceans, being essential that the personnel on board of the vessel can maintain position and operating point or imposed on a route with high precision. By the adoption of a strict safety in terms of handling and positioning of the vessel in the vicinity of the drilling platform, the risk of accidents can be reduced to a minimum. Possibilities in anticipation amplitudes of the oscillatory movements of the ships navigating in real sea, is a challenge for naval architects and OCTOPUS software is a tool used increasingly more in this respect, complementing navigational facilities offered by dynamic positioning systems. This paper presents a study on the amplitudes of the oscillations categories of supply vessels in severe hydro meteorological conditions of navigation. The study provides information on the RAO (Response Amplitude Operator) response operator of the ship, for the amplitude of the roll movements, in some incident wave systems, interpreted using the energy spectrum Jonswap and whose characteristics are known (significant height of the wave, wave period, pulsation of the wave). Ship responses are analyzed according to different positioning of the ship in relation to the wave front (incident angle ranging from 10 to 10 degree from 0 to 180), highlighting the value of the ship roll motion amplitude. For the study, was used, as a tool for modeling and simulation, the features offered by OCTOPUS software that allows the study of the computerized behavior of the ship on the waves, in the real conditions of navigation. Program library was used for both the vessel itself and navigation modeling
19. Semiconductor Characterization with Acoustic and Thermal waves on Picosecond Timescales
Science.gov (United States)
Wright, Oliver B.
1997-03-01
Ultrafast optical techniques for semiconductor characterization can probe the dynamics of photoexcited carriers, leading to applications in, for example, in-line monitoring of semiconductor processing and optimization of materials for sub-picosecond electronic switches or for nanoscale electronic devices.(Semiconductors Probed by Ultrafast Laser Spectroscopy, edited by R. R. Alfano (Academic, New York, 1984).) Picosecond or femtosecond optical pulses excite electrons to higher electronic bands, producing a nonequilibrium electron-hole distribution. Various physical effects result from the relaxation of this distribution. Luminescence or photoelectron emission are examples. In the present study the focus is on acoustic and thermal effects. The change in electron and hole occupation probabilities induces an electronic stress distributed throughout the carrier penetration depth. A temperature change of the lattice and an associated thermal stress are also produced. The combined stress distribution launches a strain pulse that propagates into the sample as a longitudinally polarized acoustic wave in the present experiments. Its reflection from sub-surface boundaries, interfaces or defects can be detected at the surface by another, weaker optical probe pulse. During this time the temperature distribution in the semiconductor also changes due to thermal wave propagation,(Photoacoustic and Thermal Wave Phenomena in Semiconductors, edited by Andreas Mandelis (North Holland, New York, 1987).) and this simultaneously influences the optical probe pulse. Both reflectance modulation and beam deflection methods for probing were used to investigate crystalline and amorphous silicon samples.(O. B. Wright, U. Zammit, M. Marinelli, and V. Gusev, Appl. Phys. Lett. 69, 553 (1996).) (O. B. Wright and V. E. Gusev, Appl. Phys. Lett. 66, 1190 (1995).) (O. B. Wright and K. Kawashima, Phonon Scattering in Condensed Matter VII, edited by R. O. Pohl and M. Meissner, Springer Verlag, Berlin
20. Role of the Ionosphere in the Generation of Large-Amplitude Ulf Waves at High Latitudes
Science.gov (United States)
Tulegenov, B.; Guido, T.; Streltsov, A. V.
2014-12-01
We present results from the statistical study of ULF waves detected by the fluxgate magnetometer in Gakona, Alaska during several experimental campaigns conducted at the High Frequency Active Auroral Research Program (HAARP) facility in years 2011-2013. We analyzed frequencies of ULF waves recorded during 26 strongly disturbed geomagnetic events (substorms) and compared them with frequencies of ULF waves detected during magnetically quite times. Our analysis demonstrates that the frequency of the waves carrying most of the power almost in all these events is less than 1 mHz. We also analyzed data from the ACE satellite, measuring parameters of the solar wind in the L1 Lagrangian point between Earth and Sun, and found that in several occasions there is a strong correlation between oscillations of the magnetic field in the solar wind and oscillations detected on the ground. We also found several cases when there is no correlation between signals detected on ACE and on the ground. This finding suggests that these frequencies correspond to the fundamental eigenfrequency of the coupled magnetosphere-ionosphere system. The low frequency of the oscillations is explained by the effect of the ionosphere, where the current is carried by ions through highly collisional media. The amplitude of these waves can reach significant magnitude when the system is driven by the external driver (for example, the solar wind) with this particular frequency. When the frequency of the driver does not match the frequency of the system, the waves still are observed, but their amplitudes are much smaller.
1. Large Amplitude Whistlers in the Magnetosphere Observed with Wind-Waves
Science.gov (United States)
Kellogg, P. J.; Cattell, C. A.; Goetz, K.; Monson, S. J.; Wilson, L. B., III
2011-01-01
We describe the results of a statistical survey of Wind-Waves data motivated by the recent STEREO/Waves discovery of large-amplitude whistlers in the inner magnetosphere. Although Wind was primarily intended to monitor the solar wind, the spacecraft spent 47 h inside 5 R(sub E) and 431 h inside 10 R(sub E) during the 8 years (1994-2002) that it orbited the Earth. Five episodes were found when whistlers had amplitudes comparable to those of Cattell et al. (2008), i.e., electric fields of 100 m V/m or greater. The whistlers usually occurred near the plasmapause. The observations are generally consistent with the whistlers observed by STEREO. In contrast with STEREO, Wind-Waves had a search coil, so magnetic measurements are available, enabling determination of the wave vector without a model. Eleven whistler events with useable magnetic measurements were found. The wave vectors of these are distributed around the magnetic field direction with angles from 4 to 48deg. Approximations to observed electron distribution functions show a Kennel-Petschek instability which, however, does not seem to produce the observed whistlers. One Wind episode was sampled at 120,000 samples/s, and these events showed a signature that is interpreted as trapping of electrons in the electrostatic potential of an oblique whistler. Similar waveforms are found in the STEREO data. In addition to the whistler waves, large amplitude, short duration solitary waves (up to 100 mV/m), presumed to be electron holes, occur in these passes, primarily on plasma sheet field lines mapping to the auroral zone.
2. Multi-resonance tunneling of acoustic waves in two-dimensional locally-resonant phononic crystals
Science.gov (United States)
Yang, Aichao; He, Wei; Zhang, Jitao; Zhu, Liang; Yu, Lingang; Ma, Jian; Zou, Yang; Li, Min; Wu, Yu
2017-03-01
Multi-resonance tunneling of acoustic waves through a two-dimensional phononic crystal (PC) is demonstrated by substituting dual Helmholtz resonators (DHRs) for acoustically-rigid scatterers in the PC. Due to the coupling of the incident waves with the acoustic multi-resonance modes of the DHRs, acoustic waves can tunnel through the PC at specific frequencies which lie inside the band gaps of the PC. This wave tunneling transmission can be further broadened by using the multilayer Helmholtz resonators. Thus, a PC consisting of an array of dual/multilayer Helmholtz resonators can serve as an acoustic band-pass filter, used to pick out acoustic waves with certain frequencies from noise.
3. GPS-Acoustic Seafloor Geodesy using a Wave Glider
Science.gov (United States)
2013-12-01
The conventional approach to implement the GPS-Acoustic technique uses a ship or buoy for the interface between GPS and Acoustics. The high cost and limited availability of ships restricts occupations to infrequent campaign-style measurements. A new approach to address this problem uses a remote controlled, wave-powered sea surface vehicle, the Wave Glider. The Wave Glider uses sea-surface wave action for forward propulsion with both upward and downward motions producing forward thrust. It uses solar energy for power with solar panels charging the onboard 660 W-h battery for near continuous operation. It uses Iridium for communication providing command and control from shore plus status and user data via the satellite link. Given both the sea-surface wave action and solar energy are renewable, the vehicle can operate for extended periods (months) remotely. The vehicle can be launched from a small boat and can travel at ~ 1 kt to locations offshore. We have adapted a Wave Glider for seafloor geodesy by adding a dual frequency GPS receiver embedded in an Inertial Navigation Unit, a second GPS antenna/receiver to align the INU, and a high precision acoustic ranging system. We will report results of initial testing of the system conducted at SIO. In 2014, the new approach will be used for seafloor geodetic measurements of plate motion in the Cascadia Subduction Zone. The project is for a three-year effort to measure plate motion at three sites along an East-West profile at latitude 44.6 N, offshore Newport Oregon. One site will be located on the incoming plate to measure the present day convergence between the Juan de Fuca and North American plates and two additional sites will be located on the continental slope of NA to measure the elastic deformation due to stick-slip behavior on the mega-thrust fault. These new seafloor data will constrain existing models of slip behavior that presently are poorly constrained by land geodetic data 100 km from the deformation front.
4. Dispersion relations with crossing symmetry for pipi D and F wave amplitudes
OpenAIRE
Kaminski, R.
2011-01-01
A set of once subtracted dispersion relations with imposed crossing symmetry condition for the pipi D- and F-wave amplitudes is derived and analyzed. An example of numerical calculations in the effective two pion mass range from the threshold to 1.1 GeV is presented. It is shown that these new dispersion relations impose quite strong constraints on the analyzed pipi interactions and are very useful tools to test the pipi amplitudes. One of the goals of this work is to provide a complete set o...
5. Radial wave crystals: radially periodic structures from anisotropic metamaterials for engineering acoustic or electromagnetic waves.
Science.gov (United States)
Torrent, Daniel; Sánchez-Dehesa, José
2009-08-07
We demonstrate that metamaterials with anisotropic properties can be used to develop a new class of periodic structures that has been named radial wave crystals. They can be sonic or photonic, and wave propagation along the radial directions is obtained through Bloch states like in usual sonic or photonic crystals. The band structure of the proposed structures can be tailored in a large amount to get exciting novel wave phenomena. For example, it is shown that acoustical cavities based on radial sonic crystals can be employed as passive devices for beam forming or dynamically orientated antennas for sound localization.
6. Hydrogen Adsorption Studies Using Surface Acoustic Waves on Nanoparticles
Energy Technology Data Exchange (ETDEWEB)
A.B. Phillips; G. Myneni; B.S. Shivaram
2005-06-13
Vanadium nanoparticles, on the order of 20 nm, were deposited on a quartz crystal surface acoustic wave resonator (SAW) using a Nd:YAG pulsed laser deposition system. Due to the high Q and resonant frequency of the SAW, mass changes on the order of 0.1 nanogram can be quantitatively measured. Roughly 60 nanogram of V was deposited on the SAW for these experiments. The SAW was then moved into a hydrogen high pressure cell.At room temperature and 1 atmosphere of hydrogen pressure, 1 wt% H, or H/V {approx} 0.5 (atomic ratio) absorption was measured.
7. Location Dependence of Mass Sensitivity for Acoustic Wave Devices
Directory of Open Access Journals (Sweden)
Kewei Zhang
2015-09-01
Full Text Available It is introduced that the mass sensitivity (Sm of an acoustic wave (AW device with a concentrated mass can be simply determined using its mode shape function: the Sm is proportional to the square of its mode shape. By using the Sm of an AW device with a uniform mass, which is known for almost all AW devices, the Sm of an AW device with a concentrated mass at different locations can be determined. The method is confirmed by numerical simulation for one type of AW device and the results from two other types of AW devices.
8. Stern Gerlach spin filter using surface acoustic waves
Science.gov (United States)
Santos, Paulo V.; Nitta, Junsaku; Ploog, Klaus H.
2004-12-01
We propose the ambipolar carrier transport by surface acoustic waves (SAWs) in a semiconductor quantum well (QW) for the realization of the Stern-Gerlach (SG) experiment in the solid phase. The well-defined and very low carrier velocity in the moving SAW field leads to a large deflection angle and thus to efficient spin separation, even for the weak field gradients and short (μm-long) interaction lengths that can be produced by micromagnets. The feasibility of a SG spin filter is discussed for different QW materials.
9. Modeling of a Surface Acoustic Wave Strain Sensor
Science.gov (United States)
Wilson, W. C.; Atkinson, Gary M.
2010-01-01
NASA Langley Research Center is investigating Surface Acoustic Wave (SAW) sensor technology for harsh environments aimed at aerospace applications. To aid in development of sensors a model of a SAW strain sensor has been developed. The new model extends the modified matrix method to include the response of Orthogonal Frequency Coded (OFC) reflectors and the response of SAW devices to strain. These results show that the model accurately captures the strain response of a SAW sensor on a Langasite substrate. The results of the model of a SAW Strain Sensor on Langasite are presented
10. Surface acoustic wave devices including Langmuir-Blodgett films (Review)
Science.gov (United States)
Plesskii, V. P.
1991-06-01
Recent theoretical and experimental research related to the use of Langmuir-Blodgett (LB) films in surface acoustic wave (SAW) devices is reviewed. The sensitivity of the different cuts of quartz and lithium niobate to inertial loading is investigated, and it is shown that some cuts in lithium niobate are twice as sensitive to mass loading than the commonly used YZ-cut. The large variety of organic compounds suitable for the production of LB films makes it possible to create SAW sensors reacting selectively to certain substances. The existing SAW sensors based on LB films are characterized by high sensitivity and fast response.
11. Surface acoustic wave vapor sensors based on resonator devices
Science.gov (United States)
Grate, Jay W.; Klusty, Mark
1991-05-01
Surface acoustic wave (SAW) devices fabricated in the resonator configuration have been used as organic vapor sensors and compared with delay line devices more commonly used. The experimentally determined mass sensitivities of 200, 300, and 400 MHz resonators and 158 MHz delay lines coated with Langmuir-Blodgett films of poly(vinyl tetradecanal) are in excellent agreement with theoretical predictions. The response of LB- and spray-coated sensors to various organic vapors were determined, and scaling laws for mass sensitivities, vapor sensitivities, and detection limits are discussed. The 200 MHz resonators provide the lowest noise levels and detection limits of all the devices examined.
12. A model for large-amplitude internal solitary waves with trapped cores
Directory of Open Access Journals (Sweden)
K. R. Helfrich
2010-07-01
Full Text Available Large-amplitude internal solitary waves in continuously stratified systems can be found by solution of the Dubreil-Jacotin-Long (DJL equation. For finite ambient density gradients at the surface (bottom for waves of depression (elevation these solutions may develop recirculating cores for wave speeds above a critical value. As typically modeled, these recirculating cores contain densities outside the ambient range, may be statically unstable, and thus are physically questionable. To address these issues the problem for trapped-core solitary waves is reformulated. A finite core of homogeneous density and velocity, but unknown shape, is assumed. The core density is arbitrary, but generally set equal to the ambient density on the streamline bounding the core. The flow outside the core satisfies the DJL equation. The flow in the core is given by a vorticity-streamfunction relation that may be arbitrarily specified. For simplicity, the simplest choice of a stagnant, zero vorticity core in the frame of the wave is assumed. A pressure matching condition is imposed along the core boundary. Simultaneous numerical solution of the DJL equation and the core condition gives the exterior flow and the core shape. Numerical solutions of time-dependent non-hydrostatic equations initiated with the new stagnant-core DJL solutions show that for the ambient stratification considered, the waves are stable up to a critical amplitude above which shear instability destroys the initial wave. Steadily propagating trapped-core waves formed by lock-release initial conditions also agree well with the theoretical wave properties despite the presence of a "leaky" core region that contains vorticity of opposite sign from the ambient flow.
13. Relationship between pressure wave amplitude and esophageal bolus clearance assessed by combined manometry and multichannel intraluminal impedance measurement
NARCIS (Netherlands)
Nguyen, Nam Q.; Tippett, Marcus; Smout, Andre J. P. M.; Holloway, Richard H.
2006-01-01
OBJECTIVES: Esophageal wave amplitude is an important determinant of esophageal clearance. A threshold of 30 mmHg is widely accepted as the threshold for effective clearance in the distal esophagus. However, the precise relationship between wave amplitude and clearance has received relatively little
14. Selective generation of ultrasonic Lamb waves by electromagnetic acoustic transducers
Science.gov (United States)
Li, Ming-Liang; Deng, Ming-Xi; Gao, Guang-Jian
2016-12-01
In this paper, we describe a modal expansion approach for the analysis of the selective generation of ultrasonic Lamb waves by electromagnetic acoustic transducers (EMATs). With the modal expansion approach for waveguide excitation, an analytical expression of the Lamb wave’s mode expansion coefficient is deduced, which is related to the driving frequency and the geometrical parameters of the EMAT’s meander coil, and lays a theoretical foundation for exactly analyzing the selective generation of Lamb waves with EMATs. The influences of the driving frequency on the mode expansion coefficient of ultrasonic Lamb waves are analyzed when the EMAT’s geometrical parameters are given. The numerical simulations and experimental examinations show that the ultrasonic Lamb wave modes can be effectively regulated (strengthened or restrained) by choosing an appropriate driving frequency of EMAT, with the geometrical parameters given. This result provides a theoretical and experimental basis for selectively generating a single and pure Lamb wave mode with EMATs. Project supported by the National Natural Science Foundation of China (Grant Nos. 11474361 and 11274388).
15. Investigation of acoustic waves generated in an elastic solid by a pulsed ion beam and their application in a FIB based scanning ion acoustic microscope
Energy Technology Data Exchange (ETDEWEB)
2004-12-01
The aim of this work is to investigate the acoustic wave generation by pulsed and periodically modulated ion beams in different solid materials depending on the beam parameters and to demonstrate the possibility to apply an intensity modulated focused ion beam (FIB) for acoustic emission and for nondestructive investigation of the internal structure of materials on a microscopic scale. The combination of a FIB and an ultrasound microscope in one device can provide the opportunity of nondestructive investigation, production and modification of micro- and nanostructures simultaneously. This work consists of the two main experimental parts. In the first part the process of elastic wave generation during the irradiation of metallic samples by a pulsed beam of energetic ions was investigated in an energy range from 1.5 to 10 MeV and pulse durations of 0.5-5 {mu}s, applying ions with different masses, e.g. oxygen, silicon and gold, in charge states from 1{sup +} to 4{sup +}. The acoustic amplitude dependence on the ion beam parameters like the ion mass and energy, the ion charge state, the beam spot size and the pulse duration were of interest. This work deals with ultrasound transmitted in a solid, i.e. bulk waves, because of their importance for acoustic transmission microscopy and nondestructive inspection of internal structure of a sample. The second part of this work was carried out using the IMSA-100 FIB system operating in an energy range from 30 to 70 keV. The scanning ion acoustic microscope based on this FIB system was developed and tested. (orig.)
16. High-amplitude THz and GHz strain waves, generated by ultrafast screening of piezoelectric fields in InGaN/GaN multiple quantum wells
DEFF Research Database (Denmark)
Porte, Henrik; van Capel, P.J.S.; Turchinovich, Dmitry
2010-01-01
Screening of large built-in piezoelectric fields in InGaN/GaN quantum wells leads to high-amplitude acoustic emission. We will compare acoustic emission by quantum wells with different thicknesses with photoluminescence; indicating screening.......Screening of large built-in piezoelectric fields in InGaN/GaN quantum wells leads to high-amplitude acoustic emission. We will compare acoustic emission by quantum wells with different thicknesses with photoluminescence; indicating screening....
17. A hybrid method for determination of the acoustic impedance of an unflanged cylindrical duct for multimode wave
Science.gov (United States)
Snakowska, Anna; Jurkiewicz, Jerzy; Gorazd, Łukasz
2017-05-01
The paper presents derivation of the impedance matrix based on the rigorous solution of the wave equation obtained by the Wiener-Hopf technique for a semi-infinite unflanged cylindrical duct. The impedance matrix allows, in turn, calculate the acoustic impedance along the duct and, as a special case, the radiation impedance. The analysis is carried out for a multimode incident wave accounting for modes coupling on the duct outlet not only qualitatively but also quantitatively for a selected source operating inside. The quantitative evaluation of the acoustic impedance requires setting of modes amplitudes which has been obtained applying the mode decomposition method to the far-field pressure radiation measurements and theoretical formulae for single mode directivity characteristics for an unflanged duct. Calculation of the acoustic impedance for a non-uniform distribution of the sound pressure and the sound velocity on a duct cross section requires determination of the acoustic power transmitted along/radiated from a duct. In the paper, the impedance matrix, the power, and the acoustic impedance were derived as functions of Helmholtz number and distance from the outlet.
18. Small amplitude Kinetic Alfven waves in a superthermal electron-positron-ion plasma
Science.gov (United States)
2016-11-01
We are investigating the propagating properties of coupled Kinetic Alfven-acoustic waves in a low beta plasma having superthermal electrons and positrons. Using the standard reductive perturbation method, a nonlinear Korteweg-de Vries (KdV) type equation is derived which describes the evolution of Kinetic Alfven waves. It is found that nonlinearity and Larmor radius effects can compromise and give rise to solitary structures. The parametric role of superthermality and positron content on the characteristics of solitary wave structures is also investigated. It is found that only sub-Alfvenic and compressive solitons are supported in the present model. The present study may find applications in a low β electron-positron-ion plasma having superthermal electrons and positrons.
19. Surface Acoustic Wave Vibration Sensors for Measuring Aircraft Flutter
Science.gov (United States)
Wilson, William C.; Moore, Jason P.; Juarez, Peter D.
2016-01-01
Under NASA's Advanced Air Vehicles Program the Advanced Air Transport Technology (AATT) Project is investigating flutter effects on aeroelastic wings. To support that work a new method for measuring vibrations due to flutter has been developed. The method employs low power Surface Acoustic Wave (SAW) sensors. To demonstrate the ability of the SAW sensor to detect flutter vibrations the sensors were attached to a Carbon fiber-reinforced polymer (CFRP) composite panel which was vibrated at six frequencies from 1Hz to 50Hz. The SAW data was compared to accelerometer data and was found to resemble sine waves and match each other closely. The SAW module design and results from the tests are presented here.
20. Weakly dissipative dust-ion acoustic wave modulation
Science.gov (United States)
Alinejad, H.; Mahdavi, M.; Shahmansouri, M.
2016-02-01
The modulational instability of dust-ion acoustic (DIA) waves in an unmagnetized dusty plasma is investigated in the presence of weak dissipations arising due to the low rates (compared to the ion oscillation frequency) of ionization recombination and ion loss. Based on the multiple space and time scales perturbation, a new modified nonlinear Schrödinger equation governing the evolution of modulated DIA waves is derived with a linear damping term. It is shown that the combined action of all dissipative mechanisms due to collisions between particles reveals the permitted maximum time for the occurrence of the modulational instability. The influence on the modulational instability regions of relevant physical parameters such as ion temperature, dust concentration, ionization, recombination and ion loss is numerically examined. It is also found that the recombination frequency controls the instability growth rate, whereas recombination and ion loss make the instability regions wider.
1. Anomalous refraction of guided waves via embedded acoustic metasurfaces
Science.gov (United States)
Zhu, Hongfei; Semperlotti, Fabio
2016-04-01
We illustrate the design of acoustic metasurfaces based on geometric tapers and embedded in thin-plate structures. The metasurface is an engineered discontinuity that enables anomalous refraction of guided wave modes according to the Generalized Snell's Law. Locally-resonant geometric torus-like tapers are designed in order to achieve metasurfaces having discrete phase-shift profiles that enable a high level of control of refraction of the wavefronts. Results of numerical simulations show that anomalous refraction can be achieved on transmitted anti-symmetric modes (A0) either when using a symmetric (S0) or anti-symmetric (A0) incident wave, where the former case clearly involves mode conversion mechanisms.
2. Effect of secondary electron emission on nonlinear dust acoustic wave propagation in a complex plasma with negative equilibrium dust charge
Science.gov (United States)
Bhakta, Subrata; Ghosh, Uttam; Sarkar, Susmita
2017-02-01
In this paper, we have investigated the effect of secondary electron emission on nonlinear propagation of dust acoustic waves in a complex plasma where equilibrium dust charge is negative. The primary electrons, secondary electrons, and ions are Boltzmann distributed, and only dust grains are inertial. Electron-neutral and ion-neutral collisions have been neglected with the assumption that electron and ion mean free paths are very large compared to the plasma Debye length. Both adiabatic and nonadiabatic dust charge variations have been separately taken into account. In the case of adiabatic dust charge variation, nonlinear propagation of dust acoustic waves is governed by the KdV (Korteweg-de Vries) equation, whereas for nonadiabatic dust charge variation, it is governed by the KdV-Burger equation. The solution of the KdV equation gives a dust acoustic soliton, whose amplitude and width depend on the secondary electron yield. Similarly, the KdV-Burger equation provides a dust acoustic shock wave. This dust acoustic shock wave may be monotonic or oscillatory in nature depending on the fact that whether it is dissipation dominated or dispersion dominated. Our analysis shows that secondary electron emission increases nonadiabaticity induced dissipation and consequently increases the monotonicity of the dust acoustic shock wave. Such a dust acoustic shock wave may accelerate charge particles and cause bremsstrahlung radiation in space plasmas whose physical process may be affected by secondary electron emission from dust grains. The effect of the secondary electron emission on the stability of the equilibrium points of the KdV-Burger equation has also been investigated. This equation has two equilibrium points. The trivial equilibrium point with zero potential is a saddle and hence unstable in nature. The nontrivial equilibrium point with constant nonzero potential is a stable node up to a critical value of the wave velocity and a stable focus above it. This critical
3. Rapid high-amplitude circumferential slow wave propagation during normal gastric pacemaking and dysrhythmias.
Science.gov (United States)
O'Grady, G; Du, P; Paskaranandavadivel, N; Angeli, T R; Lammers, W J E P; Asirvatham, S J; Windsor, J A; Farrugia, G; Pullan, A J; Cheng, L K
2012-07-01
Gastric slow waves propagate aborally as rings of excitation. Circumferential propagation does not normally occur, except at the pacemaker region. We hypothesized that (i) the unexplained high-velocity, high-amplitude activity associated with the pacemaker region is a consequence of circumferential propagation; (ii) rapid, high-amplitude circumferential propagation emerges during gastric dysrhythmias; (iii) the driving network conductance might switch between interstitial cells of Cajal myenteric plexus (ICC-MP) and circular interstitial cells of Cajal intramuscular (ICC-IM) during circumferential propagation; and (iv) extracellular amplitudes and velocities are correlated. An experimental-theoretical study was performed. High-resolution gastric mapping was performed in pigs during normal activation, pacing, and dysrhythmia. Activation profiles, velocities, and amplitudes were quantified. ICC pathways were theoretically evaluated in a bidomain model. Extracellular potentials were modeled as a function of membrane potentials. High-velocity, high-amplitude activation was only recorded in the pacemaker region when circumferential conduction occurred. Circumferential propagation accompanied dysrhythmia in 8/8 experiments was faster than longitudinal propagation (8.9 vs 6.9 mm s(-1) ; P = 0.004) and of higher amplitude (739 vs 528 μV; P = 0.007). Simulations predicted that ICC-MP could be the driving network during longitudinal propagation, whereas during ectopic pacemaking, ICC-IM could outpace and activate ICC-MP in the circumferential axis. Experimental and modeling data demonstrated a linear relationship between velocities and amplitudes (P propagation. Rapid circumferential propagation also emerges during a range of gastric dysrhythmias, elevating extracellular amplitudes and organizing transverse wavefronts. One possible explanation for these findings is bidirectional coupling between ICC-MP and circular ICC-IM networks. © 2012 Blackwell Publishing Ltd.
4. Stability of steady rotational water-waves of finite amplitude on arbitrary shear currents
Science.gov (United States)
Seez, William; Abid, Malek; Kharif, Christian
2016-04-01
A versatile solver for the two-dimensional Euler equations with an unknown free-surface has been developed. This code offers the possibility to calculate two-dimensional, steady rotational water-waves of finite amplitude on an arbitrary shear current. Written in PYTHON the code incorporates both pseudo-spectral and finite-difference methods in the discretisation of the equations and thus allows the user to capture waves with large steepnesses. As such it has been possible to establish that, in a counter-flowing situation, the existence of wave solutions is not guaranteed and depends on a pair of parameters representing mass flux and vorticity. This result was predicted, for linear solutions, by Constantin. Furthermore, experimental comparisons, both with and without vorticity, have proven the precision of this code. Finally, waves propagating on top of highly realistic shear currents (exponential profiles under the surface) have been calculated following current profiles such as those used by Nwogu. In addition, a stability analysis routine has been developed to study the stability regimes of base waves calculated with the two-dimensional code. This linear stability analysis is based on three dimensional perturbations of the steady situation which lead to a generalised eigenvalue problem. Common instabilities of the first and second class have been detected, while a third class of wave-instability appears due to the presence of strong vorticity. {1} Adrian Constantin and Walter Strauss. {Exact steady periodic water waves with vorticity}. Communications on Pure and Applied Mathematics, 57(4):481-527, April 2004. Okey G. Nwogu. {Interaction of finite-amplitude waves with vertically sheared current fields}. Journal of Fluid Mechanics, 627:179, May 2009.
5. Finite-amplitude shear-Alfv\\'en waves do not propagate in weakly magnetized collisionless plasmas
CERN Document Server
Squire, J; Schekochihin, A A
2016-01-01
It is shown that low-collisionality plasmas cannot support linearly polarized shear-Alfv\\'en fluctuations above a critical amplitude $\\delta B_{\\perp}/B_{0} \\sim \\beta^{\\,-1/2}$, where $\\beta$ is the ratio of thermal to magnetic pressure. Above this cutoff, a developing fluctuation will generate a pressure anisotropy that is sufficient to destabilize itself through the parallel firehose instability. This causes the wave frequency to approach zero, interrupting the fluctuation before any oscillation. The magnetic field lines rapidly relax into a sequence of angular zig-zag structures. Such a restrictive bound on shear-Alfv\\'en-wave amplitudes has far-reaching implications for the physics of magnetized turbulence in the high-$\\beta$ conditions prevalent in many astrophysical plasmas, as well as for the solar wind at $\\sim 1 \\mathrm{AU}$ where $\\beta \\gtrsim 1$.
6. NONLINEAR EVOLUTION ANALYSIS OF T-S DISTURBANCE WAVE AT FINITE AMPLITUDE IN NONPARALLEL BOUNDARY LAYERS
Institute of Scientific and Technical Information of China (English)
唐登斌; 夏浩
2002-01-01
The nonlinear evolution problem in nonparallel boundary layer stability was studied. The relative parabolized stability equations of nonlinear nonparallel boundary layer were derived. The developed numerical method, which is very effective, was used to study the nonlinear evolution of T-S disturbance wave at finite amplitudes. Solving nonlinear equations of different modes by using predictor-corrector and iterative approach, which is uncoupled between modes, improving computational accuracy by using high order compact differential scheme, satisfying normalization condition, determining tables of nonlinear terms at different modes, and implementing stably the spatial marching, were included in this method. With different initial amplitudes, the nonlinear evolution of T-S wave was studied. The nonlinear nonparallel results of examples compare with data of direct numerical simulations (DNS) using full Navier- Stokes equations.
7. Experimental investigation of flow induced dust acoustic shock waves in a complex plasma
Energy Technology Data Exchange (ETDEWEB)
Jaiswal, S., E-mail: [email protected]; Bandyopadhyay, P.; Sen, A. [Institute for Plasma Research, Bhat, Gandhinagar, Gujarat 382428 (India)
2016-08-15
We report on experimental observations of flow induced large amplitude dust-acoustic shock waves in a complex plasma. The experiments have been carried out in a Π shaped direct current glow discharge experimental device using kaolin particles as the dust component in a background of Argon plasma. A strong supersonic flow of the dust fluid is induced by adjusting the pumping speed and neutral gas flow into the device. An isolated copper wire mounted on the cathode acts as a potential barrier to the flow of dust particles. A sudden change in the gas flow rate is used to trigger the onset of high velocity dust acoustic shocks whose dynamics are captured by fast video pictures of the evolving structures. The physical characteristics of these shocks are delineated through a parametric scan of their dynamical properties over a range of flow speeds and potential hill heights. The observed evolution of the shock waves and their propagation characteristics are found to compare well with model numerical results based on a modified Korteweg-de-Vries-Burgers type equation.
8. Experimental investigation of flow induced dust acoustic shock waves in a complex plasma
CERN Document Server
Jaiswal, S; Sen, A
2016-01-01
We report on experimental observations of flow induced large amplitude dust-acoustic shock waves (DASW) in a complex plasma. The experiments have been carried out in a $\\Pi$ shaped DC glow discharge experimental device using kaolin particles as the dust component in a background of Argon plasma. A strong supersonic flow of the dust fluid is induced by adjusting the pumping speed and neutral gas flow into the device. An isolated copper wire mounted on the cathode acts as a potential barrier to the flow of dust particles. A sudden change of gas flow rate is used to trigger the onset of high velocity dust acoustic shocks whose dynamics are captured by fast video pictures of the evolving structures. The physical characteristics of these shocks are delineated through a parametric scan of their dynamical properties over a range of flow speeds and potential hill heights. The observed evolution of the shock waves and their propagation characteristics are found to compare well with model numerical results based on a m...
9. Dust-acoustic solitary waves in a magnetized dusty plasma with nonthermal electrons and trapped ions
Science.gov (United States)
Misra, A. P.; Wang, Yunliang
2015-05-01
The nonlinear propagation of electrostatic dust-acoustic (DA) waves in a magnetized dusty plasma consisting of negatively charged mobile dusts, nonthermal fast electrons and trapped ions with vortex-like distribution is studied. Using the reductive perturbation technique, a Korteweg-de Vries (KdV)-like equation is derived which governs the dynamics of the small-amplitude solitary waves in a magnetized dusty nonthermal plasma. It is found that due to the dust thermal pressure, there exists a critical value (βc) of the nonthermal parameter β (>1), denoting the percentage of energetic electrons, below which the DA solitary waves cease to propagate. The soliton solution (traveling wave) of the KdV-like equation is obtained, and is shown to be only of the rarefactive type. The properties of the solitons are analyzed numerically with the system parameters. It is also seen that the effect of the static magnetic field (which only modifies the soliton width) becomes significant when the dust gyrofrequency is smaller than one-tenth of the dust plasma frequency. Furthermore, the amplitude of the soliton is found to increase (decrease) when the ratio of the free to trapped ion temperatures (σ) is positive (negative). The effects of the system parameters including the obliqueness of propagation (lz) and σ on the dynamics of the DA solitons are also discussed numerically, and it is found that the soliton structures can withstand perturbations and turbulence during a considerable time. The results should be useful for understanding the nonlinear propagation of DA solitary waves in laboratory and space plasmas (e.g., Earth's magnetosphere, auroral region, heliospheric environments, etc.).
10. A micromachined surface acoustic wave sensor for detecting inert gases
Energy Technology Data Exchange (ETDEWEB)
Ahuja, S.; Hersam, M.; Ross, C.; Chien, H.T.; Raptis, A.C. [Argonne National Lab., IL (United States). Energy Technology Div.
1996-12-31
Surface acoustic wave (SAW) sensors must be specifically designed for each application because many variables directly affect the acoustic wave velocity. In the present work, the authors have designed, fabricated, and tested an SAW sensor for detection of metastable states of He. The sensor consists of two sets of micromachined interdigitated transducers (IDTs) and delay lines fabricated by photolithography on a single Y-cut LiNbO{sub 3} substrate oriented for Z-propagation of the SAWs. One set is used as a reference and the other set employs a delay line coated with a titanium-based thin film sensitive to electrical conductivity changes when exposed to metastable states of He. The reference sensor is used to obtain a true frequency translation in relation to a voltage controlled oscillator. An operating frequency of 109 MHz has been used, and the IDT finger width is 8 {micro}m. Variation in electrical conductivity of the thin film at the delay line due to exposure to He is detected as a frequency shift in the assembly, which is then used as a measure of the amount of metastable He exposed to the sensing film on the SAW delay line. A variation in the He pressure versus frequency shifts indicates the extent of the metastable He interaction.
11. Standing surface acoustic wave (SSAW)-based microfluidic cytometer.
Science.gov (United States)
Chen, Yuchao; Nawaz, Ahmad Ahsan; Zhao, Yanhui; Huang, Po-Hsun; McCoy, J Phillip; Levine, Stewart J; Wang, Lin; Huang, Tony Jun
2014-03-07
The development of microfluidic chip-based cytometers has become an important area due to their advantages of compact size and low cost. Herein, we demonstrate a sheathless microfluidic cytometer which integrates a standing surface acoustic wave (SSAW)-based microdevice capable of 3D particle/cell focusing with a laser-induced fluorescence (LIF) detection system. Using SSAW, our microfluidic cytometer was able to continuously focus microparticles/cells at the pressure node inside a microchannel. Flow cytometry was successfully demonstrated using this system with a coefficient of variation (CV) of less than 10% at a throughput of ~1000 events s(-1) when calibration beads were used. We also demonstrated that fluorescently labeled human promyelocytic leukemia cells (HL-60) could be effectively focused and detected with our SSAW-based system. This SSAW-based microfluidic cytometer did not require any sheath flows or complex structures, and it allowed for simple operation over a wide range of sample flow rates. Moreover, with the gentle, bio-compatible nature of low-power surface acoustic waves, this technique is expected to be able to preserve the integrity of cells and other bioparticles.
12. A Surface Acoustic Wave Ethanol Sensor with Zinc Oxide Nanorods
Directory of Open Access Journals (Sweden)
Timothy J. Giffney
2012-01-01
Full Text Available Surface acoustic wave (SAW sensors are a class of piezoelectric MEMS sensors which can achieve high sensitivity and excellent robustness. A surface acoustic wave ethanol sensor using ZnO nanorods has been developed and tested. Vertically oriented ZnO nanorods were produced on a ZnO/128∘ rotated Y-cut LiNbO3 layered SAW device using a solution growth method with zinc nitrate, hexamethylenetriamine, and polyethyleneimine. The nanorods have average diameter of 45 nm and height of 1 μm. The SAW device has a wavelength of 60 um and a center frequency of 66 MHz at room temperature. In testing at an operating temperature of 270∘C with an ethanol concentration of 2300 ppm, the sensor exhibited a 24 KHz frequency shift. This represents a significant improvement in comparison to an otherwise identical sensor using a ZnO thin film without nanorods, which had a frequency shift of 9 KHz.
13. Determination of stress glut moments of total degree 2 from teleseismic surface wave amplitude spectra
Science.gov (United States)
Bukchin, B. G.
1995-08-01
A special case of the seismic source, where the stress glut tensor can be expressed as a product of a uniform moment tensor and a scalar function of spatial coordinates and time, is considered. For such a source, a technique of determining stress glut moments of total degree 2 from surface wave amplitude spectra is described. The results of application of this technique for the estimation of spatio-temporal characteristics of the Georgian earthquake, 29.04.91 are presented.
14. Beat-to-beat T-wave amplitude variability in the long QT syndrome.
Science.gov (United States)
Extramiana, Fabrice; Tatar, Charif; Maison-Blanche, Pierre; Denjoy, Isabelle; Messali, Anne; Dejode, Patrick; Iserin, Frank; Leenhardt, Antoine
2010-09-01
Long QT syndrome (LQTS) is a primary electrical disease characterized by QT prolongation and increased repolarization dispersion leading to T-wave amplitude beat-to-beat changes. We aimed to quantify beat-to-beat T-wave amplitude variability from ambulatory Holter recordings in genotyped LQTS patients. Seventy genotyped LQTS patients (mean age 23 +/- 15 years, 42 males, 50% LQT1, 39% LQT2, and 11% LQT3) and 70 normal matched control subjects underwent a 24-h digital Holter recording. Using the Tvar software (Ela Medical, Sorin group), the beat-to-beat variance of the T-wave amplitude (TAV in microV) [corrected] was assessed on 50-ms consecutive clusters during three 1-h periods: one with around average diurnal heart rate (Day Fast), one nocturnal period (Night), and one diurnal period with around average nocturnal heart rate (Day Slow). TAV was increased in LQTS patients during the two diurnal periods but not at night (during the Day Fast period, mean TAV was 34 +/- 20 microV [corrected] in LQTS patients vs. 27 +/- 10 microV [corrected] in controls, P < 0.05). This effect depended on the genotype. In LQT1, TAV was larger when compared with controls for both Day Fast and Slow periods, but in LQT2 only Day Fast shows higher TAV. Oppositely, in LQT3 the TAV was higher than in the control group during the Day slow period (mean TAV = 34 +/- 20 vs. 25 +/- 8 microV [corrected] in controls, P < 0.05). In genotyped LQTS patients beat-to-beat T-wave amplitude variability was increased when compared with control subjects. That pattern was modulated by circadian influences in a gene-dependent manner.
15. Sensitivity comparisons of layered Rayleigh wave and Love wave acoustic devices
Science.gov (United States)
Pedrick, Michael K.; Tittmann, Bernhard R.
2007-04-01
Due to their high sensitivity, layered Surface Acoustic Wave (SAW) devices are ideal for various film characterization and sensor applications. Two prominent wave types realized in these devices are Rayleigh waves consisting of coupled Shear Vertical and Longitudinal displacements and Love waves consisting of Shear Horizontal displacements. Theoretical calculations of sensitivity of SAW devices to pertubations in wave propagation are limited to idealized scenarios. Derivations of sensitivity to mass change in an overlayer are often based on the effect of rigid body motion of the overlayer on the propagation of one of the aforementioned wave types. These devices often utilize polymer overlayers for enhanced sensitivity. The low moduli of such overlayers are not sufficiently stiff to accommodate the rigid body motion assumption. This work presents device modeling based on the Finite Element Method. A coupled-field model allows for a complete description of device operation including displacement profiles, frequency, wave velocity, and insertion loss through the inclusion of transmitting and receiving IDTs. Geometric rotations and coordinate transformations allow for the modeling of different crystal orientations in piezoelectric substrates. The generation of Rayleigh and Love Wave propagation was realized with this model by examining propagation in ST Quartz both normal to and in the direction of the X axis known to support Love Waves and Rayleigh Waves, respectively. Sensitivities of layered SAW devices to pertubations in mass, layer thickness, and mechanical property changes of a Polymethyl methacrylate (PMMA) and SU-8 overlayers were characterized and compared. Experimental validation of these models is presented.
16. Topology optimization applied to room acoustic problems and surface acoustic wave devices
DEFF Research Database (Denmark)
Dühring, Maria Bayard; Sigmund, Ole; Jensen, Jakob Søndergaard
of the project is concerned with simulation and optimization of surface acoustic wave (SAW) devices [4]. SAWs are for instance used in filters and resonators in mobile phones and to modulate light waves [5], and it is here essential to obtain waves with a high intensity, to direct the waves or to optimize...... the shape of the frequency response. To begin with, a 2D model of a Mach-Zehnder interferometer impacted by a SAW is considered and a parameter study of the geometry to get the biggest modulation of the light waves in the interferometer arms is performed. Then a 2D filter is modeled and optimized...... such that it reflects SAWs at certain frequencies or frequency ranges. To save computational time a 1.5D model will be developed, where an exponential decreasing waveform is introduced into the dept of the material, and the filter is then optimized based on this model. Later, the model will be extended to a 2.5D model...
17. Harmonic and subharmonic acoustic wave generation in finite structures.
Science.gov (United States)
Alippi, A; Bettucci, A; Germano, M; Passeri, D
2006-12-22
The generation of harmonic and subharmonic vibrations is considered in a finite monodimensional structure, as it is produced by the nonlinear acoustic characteristics of the medium. The equation of motion is considered, where a general function of the displacement and its derivatives acts as the forcing term for (sub)harmonic generation and a series of 'selection rules' is found, depending on the sample constrains. The localization of the nonlinear term is also considered that mimics the presence of defects or cracks in the structure, together with the spatial distribution of subharmonic modes. Experimental evidence is given relative to the power law dependence of the harmonic modes vs. the fundamental mode displacement amplitude, and subharmonic mode distribution with hysteretic effects is also reported in a cylindrical sample of piezoelectric material.
18. Dust-acoustic solitary waves in a magnetized dusty plasma with nonthermal electrons and trapped ions
CERN Document Server
Misra, A P
2014-01-01
The nonlinear theory of electrostatic dust-acoustic (DA) waves in a magnetized dusty plasma consisting of negatively charged mobile dusts, nonthermal fast electrons and trapped ions with vortex-like distribution is revisited. Previous theory in the literature [Phys. Plasmas {\\bf 20}, 104505 (2013)] is rectified and put forward to include the effects of the external magnetic field, the adiabatic pressure of charged dusts as well as the obliqueness of propagation to the magnetic field. Using the reductive perturbation technique, a Korteweg-de Vries (KdV)-like equation is derived which governs the dynamics of the small-amplitude solitary waves in a magnetized dusty nonthermal plasma. It is found that due to the dust thermal pressure, there exists a critical value $(\\beta_c)$ of the nothermal parameter $\\beta (>1)$, denoting the percentage of energetic electrons, below which the DA solitary waves cease to propagate. The soliton solution (travelling wave) of the KdV-like equation is obtained, and is shown to be on...
19. Dust-acoustic solitary waves and shocks in strongly coupled quantum plasmas
CERN Document Server
Wang, Y
2014-01-01
We investigate the propagation characteristics of electrostatic dust-acoustic (DA) solitary waves and shocks in a strongly coupled dusty plasma consisting of intertialess electrons and ions, and strongly coupled inertial charged dust particles. A generalized viscoelastic hydrodynamic model with the effects of electrostatic dust pressure associated with the strong coupling of dust particles, and a quantum hydrodynamic model with the effects of quantum forces associated with the Bohm potential and the exchange-correlation potential for electrons and ions are considered. Both the linear and weakly nonlinear theory of DA waves are studied by the derivation and analysis of dispersion relations as well as Korteweg-de Vries (KdV) and KdV-Burgers (KdVB)-like equations. It is shown that in the kinetic regime ($\\omega\\tau_m\\gg1$, where $\\omega$ is the wave frequency and $\\tau_m$ is the viscoelastic relaxtation time), the amplitude of the DA solitary waves decays slowly with time with the effect of a small amount of dus...
20. Calculation of surface acoustic waves in a multilayered piezoelectric structure
Institute of Scientific and Technical Information of China (English)
Zhang Zuwei; Wen Zhiyu; Hu Jing
2013-01-01
The propagation properties of the surface acoustic waves (SAWs) in a ZnO-SiO2-Si multilayered piezoelectric structure are calculated by using the recursive asymptotic method.The phase velocities and the electromechanical coupling coefficients for the Rayleigh wave and the Love wave in the different ZnO-SiO2-Si structures are calculated and analyzed.The Love mode wave is found to be predominantly generated since the c-axis of the ZnO film is generally perpendicular to the substrate.In order to prove the calculated results,a Love mode SAW device based on the ZnO-SiO2-Si multilayered structure is fabricated by micromachining,and its frequency responses are detected.The experimental results are found to be mainly consistent with the calculated ones,except for the slightly larger velocities induced by the residual stresses produced in the fabrication process of the films.The deviation of the experimental results from the calculated ones is reduced by thermal annealing. | {"extraction_info": {"found_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": 0.8853598237037659, "perplexity": 2503.906364563043}, "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/1518891811655.65/warc/CC-MAIN-20180218042652-20180218062652-00030.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-9-8-k-6-6 | Algebra
Topics
# How do you solve 9/8=(k+6)/6?
Sep 1, 2016
$k = \frac{3}{4}$
#### Explanation:
$\frac{9}{8} = \frac{k + 6}{6}$
$9 \cdot 6 = 8 \cdot \left(k + 6\right)$
$54 = 8 k + 48$
$54 - 48 = 8 k$
$6 = 8 k$
$k = \frac{6}{8}$
$k = \frac{3}{4}$
Sep 1, 2016
$k = \frac{3}{4}$
#### Explanation:
If an equation has one fraction on each side of the equal side, we can get rid of the brackets by cross-multiplying.
$\frac{9}{8} = \frac{k + 6}{6}$
$8 \left(k + 6\right) = 9 \times 6$
$8 k + 48 = 54$
$8 k = 6$
$k = \frac{6}{8} = \frac{3}{4}$
##### Impact of this question
1445 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 14, "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": 0.9389928579330444, "perplexity": 2571.840796082828}, "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/1573496670036.23/warc/CC-MAIN-20191119070311-20191119094311-00198.warc.gz"} |
https://papers.nips.cc/paper/2015/hash/3a029f04d76d32e79367c4b3255dda4d-Abstract.html | #### Authors
Yunwen Lei, Urun Dogan, Alexander Binder, Marius Kloft
#### Abstract
This paper studies the generalization performance of multi-class classification algorithms, for which we obtain, for the first time, a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on lp-norm regularization, where the parameter p controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art. | {"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": 0.8820158839225769, "perplexity": 727.4247745742206}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104683020.92/warc/CC-MAIN-20220707002618-20220707032618-00134.warc.gz"} |
https://www.emathzone.com/tutorials/geometry/find-equation-of-tangent-line-to-ellipse.html | Find the Equation of the Tangent Line to the Ellipse
Find the equation of the tangent and normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at the point $\left( {a\cos \theta ,b\sin \theta } \right)$.
We have the standard equation of an ellipse
$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$
Now differentiating equation (i) on both sides with respect to $x$, we have
$\begin{gathered} \frac{{2x}}{{{a^2}}} + \frac{{2y}}{{{b^2}}}\frac{{dy}}{{dx}} = 0 \Rightarrow \frac{y}{{{b^2}}}\frac{{dy}}{{dx}} = – \frac{x}{{{a^2}}} \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{{b^2}x}}{{{a^2}y}} \\ \end{gathered}$
Let $m$ be the slope of the tangent at the given point $\left( {a\cos \theta ,b\sin \theta } \right)$, then
$m = {\frac{{dy}}{{dx}}_{\left( {a\cos \theta ,b\sin \theta } \right)}} = – \frac{{{b^2}\left( {a\cos \theta } \right)}}{{{a^2}\left( {b\sin \theta } \right)}} = – \frac{{b\cos \theta }}{{a\sin \theta }}$
The equation of the tangent at the given point $\left( {a\cos \theta ,b\sin \theta } \right)$ is
$\begin{gathered} y – b\sin \theta = – \frac{{b\cos \theta }}{{a\sin \theta }}\left( {x – a\cos \theta } \right) \\ \Rightarrow \frac{{\sin \theta }}{b}\left( {y – b\sin \theta } \right) = – \frac{{\cos \theta }}{a}\left( {x – a\cos \theta } \right) \\ \Rightarrow \frac{{y\sin \theta }}{b} – {\sin ^2}\theta = – \frac{{x\cos \theta }}{a} + {\cos ^2}\theta \\ \Rightarrow \frac{x}{a}\cos \theta + \frac{y}{b}\sin \theta = {\cos ^2}\theta + {\sin ^2}\theta \\ \Rightarrow \frac{x}{a}\cos \theta + \frac{y}{b}\sin \theta = 1 \\ \end{gathered}$
This is the equation of the tangent to the given ellipse at $\left( {a\cos \theta ,b\sin \theta } \right)$.
The slope of the normal at $P\left( {a\cos \theta ,b\sin \theta } \right)$ is $– \frac{1}{m} = – \left( { – \frac{{a\sin \theta }}{{b\cos \theta }}} \right) = \frac{{a\sin \theta }}{{b\cos \theta }}$
The equation of the normal at the point $P\left( {a\cos \theta ,b\sin \theta } \right)$ is
$\begin{gathered} y – b\sin \theta = \frac{{a\sin \theta }}{{b\cos \theta }}\left( {x – a\cos \theta } \right) \\ \Rightarrow \frac{b}{{\sin \theta }}\left( {y – b\sin \theta } \right) = \frac{a}{{\cos \theta }}\left( {x – a\cos \theta } \right) \\ \Rightarrow by\cos ec\theta – {b^2} = ax\sec \theta – {a^2} \\ \Rightarrow ax\sec \theta – by\cos ec\theta = {a^2} – {b^2} \\ \end{gathered}$
This is the equation of the normal to the given ellipse at $P\left( {a\cos \theta ,b\sin \theta } \right)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9841275215148926, "perplexity": 36.764371365894526}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058222.43/warc/CC-MAIN-20210926235727-20210927025727-00028.warc.gz"} |
http://math.stackexchange.com/questions/98003/derivative-of-a-function-is-odd-prove-the-function-is-even | # Derivative of a function is odd prove the function is even.
$f:\mathbb{R} \rightarrow \mathbb{R}$ is such that $f'(x)$ exists $\forall x.$
And $f'(-x)=-f'(x)$
I would like to show $f(-x)=f(x)$
In other words a function with odd derivative is even.
If I could apply the fundamental theorem of calculus
$\int_{-x}^{x}f'(t)dt = f(x)-f(-x)$ but since the integrand is odd we have $f(x)-f(-x)=0 \Rightarrow f(x)=f(-x)$
but unfortunately I don't know that f' is integrable.
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Let $g(x)=f(-x)$. Then $g'(x)=-f'(-x)=f'(x)$.
Since $g(0)=f(0)$ and $g'=f'$, it follows from the mean value theorem that $g=f$.
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f and g are both equal to $g'(c)x+g(0)$ – user9352 Jan 11 '12 at 0:54
@user9352: What is $c$? Note that $f$ is could be an arbitrary even function, so your formula is incorrect if $c$ is a constant. If say $f(x)=x^2$, then your formula says $x^2=2cx$, so in that case $c=\frac{1}{2}x$? – Jonas Meyer Jan 11 '12 at 0:55
yeah i guess c depends on x i just got that from the mvt, trying to see how the result follows. c is the mysterious number between 0 and x, $f(x)-f(0)=f'(c)x$ – user9352 Jan 11 '12 at 2:23
@user9352: Antiderivatives of functions on $\mathbb R$ are unique (if they exist) up to an added constant. The easiest way to apply the MVT is to the function $h(x)=g(x)-f(x)$. I recommend contraposition (or contradiction): If $h(0)=0$ and $h(a)\neq 0$ for some $a$, then $h'(c)\neq 0$ for some $c$. – Jonas Meyer Jan 11 '12 at 2:28
• Define functions $f_0(x)=(f(x)+f(-x))/2$ and $f_1(x)=(f(x)-f(-x))/2$. Then $f_0$ and $f_1$ are also differentiable, and $f_0$ is even and $f_1$ is odd.
• Show that the derivative of an odd function is even, and that of an even function is odd.
• From the equality $f'=f_0'+f_1'$ conclude that $f_1$ is constant and, therefore, zero.
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Don't you mean $f_0$ is even and $f_1$ is odd? – M Turgeon Jan 11 '12 at 1:40
Yeah, that. I even picked the indices in $\mathbb Z/2$ and all! – Mariano Suárez-Alvarez Jan 11 '12 at 2:36 | {"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": 0.9583986401557922, "perplexity": 208.74991376680964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395546.12/warc/CC-MAIN-20160624154955-00172-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/150166/why-does-induction-prove-equality | # Why does induction prove equality?
I know how to prove equality for my homework assignment, but I can't understand why it actually proves that equality. Could you please explain?
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You would have to provide more detail for anyone to be able to explain. I also suggest softening the title, since it's unnecessary and bound to rankle somebody. – Dylan Moreland May 26 '12 at 21:01
@DylanMoreland: I just want to understand why induction proves equality like 1+2+3... n = n(n+1)/2 and so on... I don't know what else to write? Sorry – hey May 26 '12 at 21:03
That example is a good start. – Dylan Moreland May 26 '12 at 21:04
Induction does not prove equality per se. Induction is a way of proving infinitely many statements (one for each positive integer) without having to establish each of them separately. These statements need not be in the form of an equality. Maybe you can start with this question. – Arturo Magidin May 26 '12 at 21:09
The idea behind induction is to prove a claim about all natural numbers. The induction is just a process which allows us to step through all the cases "at once".
Let us consider the following statement: $$1+\ldots+n = \frac{n(n+1)}2$$
We can view this as a statement that says:
for a natural number $n$, the sum of $1+\ldots+n$ is exactly $\frac{n(n+1)}2$.
We claim that this is true for every natural number.
However checking an infinite list of numbers for whether or not the equality holds for each one is quite an arduous and time consuming task (and just think of all the draft papers you will have to use...).
Instead what we do is argue the following logical steps:
• We verify the claim is true for $n=1$, that is we verify that $1=\frac{1\cdot(1+1)}2=1$.
• Suppose now that the claim was true for some natural number $n$, from this assumption we prove it for $n+1$.
If the claim was not true for all natural numbers then there would be a first number for which is it not true. At that point the step from $n$ to $n+1$ would have to fail (if we do things properly, that is).
What happened here is that we made some sort of a meta-proof, a mold for a proof. Now you can ask me "okay, show me the claim is true for $500$." and I will argue that the induction step tells me that if $499$ has the property, then so does $500$; and if $498$ has it then so does $499$ and therefore $500$ too. I will continue this decreasing argument until I reach $1$. Then I will say: "Okay, but we verified that $1$ indeed has the wanted property, therefore $2$ has it and therefore $3$ has it, and so on...", until $500$. This holds mathematically because we make finitely many arguments. This finite number may be very large, but hey... most numbers are larger.
So what we prove is not the equality itself, but rather that the equality is true for every natural number.
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Arduous, time consuming, and all the draft papers.... This one has me cracking up, Asaf! – Cameron Buie May 26 '12 at 21:29
Let's say you have the following formula that you want to prove:
$$\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$$
This formula provides a closed form for the sum of all integers between $1$ and $n$.
We will start by proving the formula for the simple base case of $n = 1$. In this case, both sides of the formula reduce to $1$. This means that the formula holds for $n = 1$.
Next, we will prove that: If the formula holds for a value $n$, then it also holds for the next value of $n$ (or $n + 1$). In other words, if the following is true:
$$\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$$
Then the following is also true:
$$\sum_{k=1}^{n+1} k = \frac{(n + 1)(n + 2)}{2}$$
To do so, let's start with the first side of the last formula:
$$s_1 = \sum_{k=1}^{n+1} k = \sum_{k=1}^{n} k + (n + 1)$$
That is, the sum of all integers between $1$ and $n + 1$ is equal to the sum of integers between $1$ and $n$, plus the last term $n + 1$.
Since we are basing this proof on the condition that the formula holds for $n$, we can write:
$$s_1 = \frac{n(n + 1)}{2} + (n + 1) = \frac{(n + 1)(n + 2)}{2} = s_2$$
As you can see, we have arrived at the second side of the formula we are trying to prove, which means that the formula does indeed hold.
This finishes the inductive proof, but what does it actually mean?
1. The formula is correct for $n = 1$.
2. If the formula is correct for $n$, then it is correct for $n + 1$.
From 1 and 2, we can say: If the formula is correct for $n = 1$, then it is correct for $n + 1 = 1 + 1 = 2$. Since we proved the case of $n = 1$ at the beginning, then the case of $n = 2$ is indeed correct.
We can repeat this above process again. The case of $n = 2$ is correct, then the case of $n = 3$ is also correct. This process can go ad infinitum; the formula is correct for all integer values of $n \ge 1$.
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Assume that all $\emptyset \neq B \subset \mathbb{N}$ has the smallest element $n_0$. Let $X =\{ n \in \mathbb{N}: P(n) \ \mbox{is true}\}$($P$ of property). Hence if $$(1) \quad P(n_0) \quad \mbox{is true.}$$
and $$(2) \quad P(n) \Rightarrow P(n+1)$$ Then $X \supset \{n \in \mathbb{N}:n \ge n_0\}$. In fact, by (1) $n_0 \in X$. If $X\neq\{n \in \mathbb{N}:n \ge n_0\}$ there exist $n_1$ the smallest element of $\mathbb{N} - X$ because $\emptyset \neq X \subset \mathbb{N}$. Then $n_{1} - 1 \in X$ and by (2) we have $n_1 \in X$. Contradiction. Consider now $P(n)$ is true if $$(3) \quad \sum_{k=1}^{n}k = \dfrac{n(n+1)}{2}.$$ As $P(1)$ is true and (2) holds, $X=\mathbb{N}$ and (3) hold for all $n \in \mathbb{N}$.
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Your wording seems strange to me. What's $B$? – Ben Millwood May 26 '12 at 22:22
To get really formal, the $\sum(n, f)$ function should also be defined by induction on $n$, and this definition used in the proof. – marty cohen May 27 '12 at 0:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9494168162345886, "perplexity": 176.71155430603193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701146600.56/warc/CC-MAIN-20160205193906-00022-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/qft-feynman-rules-for-self-interacting-scalar-field-with-source-terms.279336/ | # [QFT] Feynman rules for self-interacting scalar field with source terms
1. Dec 13, 2008
### iibewegung
I'm not sure if this is the right place to post a graduate level course material,
but I have a question about perturbative expansion of the 2n-point function of a scalar field theory.
1. The problem statement, all variables and given/known data
First, the question:
In which space (position or momentum) is the topological distinctness of Feynman diagrams decided?
I've been trying to calculate the symmetry/multiplicity factors for a Feynman diagram, but depending on which space (position or momentum) I use these rules, I get different symmetry factors.
eg. In the real-field case, the Feynman diagrams in position space have greater multiplicity than those in momentum space (since the arrows make each propagator distinct from its reverse).
Restatement of the question:
When finding the multiplicity of each Feynman diagram, should I stick to the symmetries I see in the position space (and ignore the momentum space arrows)?
2. Relevant equations
Lagrangian density for real (no charge) field:
$$\mathcal{L} = \frac{1}{2}(\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^2 - \frac{\lambda}{4!} \phi^4 + J\phi$$
For complex (charged) field:
$$\mathcal{L} = \frac{1}{2}(\partial^\mu \phi^*)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^* \phi - \frac{\lambda}{4!} (\phi^* \phi )^2 + J\phi^* + J^* \phi$$
BTW, this charged-field case has two sets of arrows (charge and momentum direction).
3. The attempt at a solution
This stuff is basically from Peskin&Schroeder (Ch.4.4), where it tells me to "divide by symmetry factor" in the last step of the Feynman rules. I wasn't sure the symmetry factors they're talking about are the same in both position and momentum spaces.
(and if they are, in which space I should count them)
Last edited: Dec 13, 2008
2. Dec 15, 2008
### turin
First I will say that I never really did understand these symmetry factors. But, since no one else wants to attack this, maybe you would like to discuss. As far as the arrows that you're talking about, you shouldn't have them for an uncharged field; I don't believe that this has anything to do with whether you are drawing them in momentum space or position space.
The way I think of Feynman diagrams is that they are sort of in Limbo between position and momentum space. On the one hand, we all love to go to momentum space, because our friends the plane waves live there. However, any vertex in your diagram actually represents a position. You get "a momentum conserving delta" because you "sum over all positions" for these vertices, but, when you draw the diagram, you just draw a blob for one typical point in space time for each vertex. This also happens to external points that become external lines, but for some reason no one puts blobs at the end of their external lines. Actually, I know the reason. Because your piece of paper isn't infinitely long, and those points are in the (approximately) infinite past and future.
BTW, I think P&S have a very unsatisfactory way of getting at the Feynman rules, even when they derive them from the action. Unfortunately, I can't think of one source that does a good job in itself. Have you looked at Srednicki?
Last edited: Dec 15, 2008
3. Dec 15, 2008
### olgranpappy
either.
no. the arrows don't matter.
yes.
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https://www.physicsforums.com/threads/quadratics-using-pascals-triangle.915025/ | 1. May 18, 2017
### HorseRidingTic
1. The problem statement, all variables and given/known data
The problem equation is contained in the picture.
2. Relevant equations
Pascal's Triangle is useful is this one.
3. The attempt at a solution
The difficulty I'm having is in going between lines 2 and 3 which I've marked with a little red dot.
The closest I get to simplifying it is = a4 + B4 + 4aB(a2+B2) + 6a2B2 . From there I can't figure out the way in which to reduce it further.
P.S I also used the quadratic formula to solve this one (the one with the b2 - 4ac) and my answer came to 748.52 but not quite 752. Why does the quadratic formula not work here?
Ben
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2. May 18, 2017
### Buffu
$\alpha^4 + \beta^4 + 4\alpha^3\beta + 6\alpha^2\beta^2 + 4\alpha \beta^3 = \alpha^4 + \beta^4 + 4\alpha^3\beta + \color{red}{8}\alpha^2\beta^2 + 4\alpha \beta^3 - \color{blue}{2\alpha^2 \beta^2} = \alpha^4 + \beta^4 + 4\alpha\beta(\alpha^2 + 2\alpha\beta + \beta^2) - {2\alpha^2 \beta^2}$
Now use $a^2 + 2ab + b^2 = (a+b)^2$
3. May 18, 2017
### HorseRidingTic
Amazing! Thank you so much Buffu :)
4. May 18, 2017
### Buffu
5. May 18, 2017
### Staff: Mentor
I have a minor gripe with the author of this problem.
What is shown is not an equation, since the symbol = is not present.
Again, my gripe is with the author, not the person who started this thread.
6. May 18, 2017
### Buffu
Number of people confusing between expression and equation is surprisingly high. | {"extraction_info": {"found_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": 0.8684725165367126, "perplexity": 1510.3679235058526}, "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/1521257644701.7/warc/CC-MAIN-20180317055142-20180317075142-00644.warc.gz"} |
http://mediacollective.nl/f6aolc7b/931d23-pfizer-medical-information---canada | asked Nov 4, 2018 in Chemistry by Tannu (53.0k points) coordination compounds; cbse; class-12; 0 votes. Solution Show Solution The orbital splitting energies are not sufficiently large for forcing pairing and, therefore, low spin configurations are rarely observed. Report ; Posted by Manmohan Tomar 3 years, 1 month ago. Which means that the last d- In tetrahedral complexes,sp3 hybridization takes place and hence the 3d orbitals are untouched. Sample Problem, [Co(NH3)6]2+ [Co(NH3)6]2+ has ∆o= 10,100 cm–1and B = 920 cm–1. ; triclinic, space group P‾1, a 7.970(5), b 10.174(5), c 11.676(5) Å, α 87.18(4), β 74.31(4), γ 74.06(4)°, Z = 2, R = 0.041. Therefore, the crystal field splitting diagram for tetrahedral complexes is the opposite of an octahedral diagram. The mol. Let's examine how the crystal-field model accounts for the observed colors in transition-metal complexes. As a result, low spin configurations are rarely observed in tetrahedral complexes and the low spin tetrahedral complexes not form. Academic Partner. [CoF 6] 3− [Rh(CO) 2 Cl 2] − Given: complexes. In tetrahedral coordination entities, Consequently, the orbital splitting energies are not sufficiently large for forcing pairing and therefore, low spin configurations are rarely observed. Therefore, transitions are not pure d-d transitions. 1 answer. In a tetrahedral complex, Δt is relatively small even with strong-field ligands as there are fewer ligands to bond with. Why low spin tetrahedral are rarely observed? That's why low spin configuration is not possible. Share 0. For example, NO 2 − is a strong-field ligand and produces a large Δ. Complexes such as this are called "low spin". Because for tetrahedral complexes, the crystal field stabilisation energy is lower than pairing energy. Cu2+ octahedral complexes. Why do octahedral metal ligand complexes have greater splitting than tetrahedral complexes? But can this kind of orbital form a tetrahedral geometry? Why are low spin tetrahedral complexes rarely observed? 1 answer. Hence, high spin tetrahedral complexes are formed. Question: Why Are Low-spin States Only Rarely Observed For Ions In A Tetrahedral Transition Metal Complexes Exhibit Several Interesting Properties That Are Not Readily Explained By Conventional Valence Bond Theories. Become our. Why do octahedral metal ligand complexes have greater splitting than tetrahedral complexes? Explain the following cases giving appropriate reasons: … (1) Sol. Why are low spin tetrahedral complexes rarely observedA compound when it is tetrahedral it implies that sp3 hybridization is there. Contact us on below numbers. (b) Why is paramagnetic while. So the energy of promotion becomes less expensive than the electron pairing energy. (ε value for the 15,000 band is ~60 m 2 mol-1). structure of high-spin [CoL2] [HL = 3-(4-methylphenyl)-1-methyl-1-triazene 1-oxide] was detd. Explain. The octahedral ion [Fe(NO 2) 6] 3−, which has 5 d-electrons, would have the octahedral splitting diagram shown at right with all five electrons in the t 2g level. For Enquiry. 12. Explain. # Reason-- Generally, the energy gap between two levels(Δt) of tetrahedral complexes is less than the pairing energy. Solution: For tetrahedral complexes, the crystal field stabilisation energy is less and is always lower than pairing energy. or own an. Dear Student, In tetrahedral complexes, the splitting of orbitals is less as compared to octahedral complexes. The d x2 −d y2 and dz 2 orbitals should be equally low in energy because they exist between the ligand axis, allowing them to experience little repulsion. The octahedral ion [Fe(NO 2) 6] 3−, which has 5 d-electrons, would have an octahedral splitting diagram where all five electrons are in the t 2g level. is oxidation no has any relationship with the low spin of tetrahedral complexes in - Chemistry - TopperLearning.com | 2i72jkhh. Why are low spin tetrahedral complexes not formed? Hence, the orbital splitting energies are not enough to force pairing. Hence the electrons will always go to higher states avoiding pairing. The ‘g’ subscript is used for the octahedral and square planar complexes which have centre of symmetry. It is rare for the Δt of tetrahedral complexes to exceed the pairing energy. Explain your answer if it is like this one (Due to low CFSC which is not able to pair up electron ) Share with your friends. Is there an easy way to find number of valence electrons" 2. In many these spin states vary between high-spin and low-spin configurations. (Atomic number of Ni = 28) (c) Why are low spin tetrahedral complexes rarely observed ? asked Nov 5, 2018 in Chemistry by Tannu (53.0k points) coordination compounds; cbse; class-12; 0 votes. The low-spin complexes possess square-planar structure and the high-spin complexes are tetrahedral. Tetrahedral complexes have somewhat more intense color. Low spin configurations are rarely observed in tetrahedral complexes. 10. View Answer play_arrow; ... What is the relationship between observed colour of the complex and the wavelength of light absorbed by the complex? Consequently, the orbital splitting energies are not sufficiently large for forcing pairing and, therefore, low spin configurations are rarely observed. The color of such complexes is much weaker than in complexes with spin-allowed transitions. Asked for: structure, high spin versus low spin, and the number of unpaired electrons. Let me start with what causes high spin. So, low spin Td complexes are not present. In this case, Δ o is always less than pairing energy, i.e. Strategy: From the number of ligands, determine the coordination number of the compound. (a) What type of isomerism is shown by the complex ? CBSE > Class 12 > Chemistry 0 answers; ANSWER. For a typical tetrahedral complex, [CoCl 4] 2-and assuming Δt = 4/9 Δo where Δo is around 9000 cm-1 then we can predict that the transition 4 T 2 ← 4 A 2 should be observed below 4000 cm-1.Only 1 band is seen in the visible region at 15,000 cm-1 although a full scan from the IR through to the UV reveals an additional band at 5,800 cm-1. For example, NO 2 − is a strong-field ligand and produces a large Δ. Business Enquiry (North) 8356912811. Business … [COF 6] 3-, [Fe(CN) 6] 4-and [Cu(NH 3) 6] 2+. For each complex, predict its structure, whether it is high spin or low spin, and the number of unpaired electrons present. 12. almost 4/9 th of that of the octahedral complex [del_Td = (4/9) * del_Oh]. Give the electronic configuration of the following complexes on the basis of crystal field splitting theory. View Answer play_arrow; question_answer92) Why are different colours observed in octahedral and tetrahedral complexes for the same metal and same ligands? 4. Reason (R) : Crystal field splitting energy is less than pairing energy for tetrahedral complexes. For the complex ion [CoF 6] 3-write the hybridization type, magnetic character and spin nature. 1800-212-7858 / 9372462318. 'e' and 't2' in tetrahedral complexes are very low. [Atomic number: Co = 27] (Comptt. Tetrahedral complexes have ligands in all of the places that an octahedral complex does not. Compounds This means that some of the visible spectrum is being removed from white light as it passes through the sample, so the light that emerges is no longer white. Related. spin configurations are rarely observed.9.5.5 Colour in In the previous Unit, we learnt that one of the most distinctive Coordination properties of transition metal complexes is their wide range of colours. is diamagnetic ? e orbitals point less directly at ligands and are stabilized. Δ o < P, therefore, the electrons prefer to go to higher orbital and once all orbitals are singly occupied, then only pairing begans. (c) Low spin tetrahedral complexes are rarely observed because orbital splitting energies for tetrahedral complexes are not sufficiently large for forcing pairing. Crystal Field Theory, And The Spectrochemical Series Of Ligands, However, Explains These Phenomena Quite Well. Assertion (A) : Low spin tetrahedral complexes are rarely observed. Question 75. Tetrahedral complexes can be treated in a similar way with the exception that we fill the e orbitals first, and the electrons in these do not contribute to the orbital angular momentum. Education Franchise × Contact Us. Contact. Report ; Posted by Drishty Kamboj 2 days, 3 hours ago. In fact, many compounds of manganese(II), like manganese(II) chloride, appear almost colorless. 10:00 AM to 7:00 PM IST all days. Low spin configuration are rarely observed in tetrahedral coordination entity formation. Due to less crystal field stabilisation energy, it is not possible to pah electrons and so all the tetrahedral complexes are high spin. Related Questions: What things do we write in a cbse project? By the way thanks for asking this question, I didn't know the answer so I googled it. The tables in the links below give a list of all d 1 to d 9 configurations including high and low spin complexes and a statement of whether or not a direct orbital contribution is expected. The energy gap between the two energy levels i.e. •Tetrahedral complexes of the heavier transition metals are low spin. Spectrochemical Series question_answer71) Why are low spin tetrahedral complexes not formed? Since tetrahedral complexes lack symmetry, ‘g’ subscript is not used with energy levels. Question 30. This is because mixing d and p orbitals is possible when there is no center of symmetry. Tetrahedral Complexes ∆o strongly M–L σ* t2g M non-bonding eg* e t2* ∆t very weakly M–L σ* M non-bonding t 4 9 o. Tetrahedral Crystal Field Splitting barycenter (spherical field) t 2 orbitals point more directly at ligands and are destabilized. High Spin large ∆o Low Spin Complexes with d4-d7 electron counts are special •at small values of ∆o/B the diagram looks similar to the d2diagram •at larger values of ∆o/B, there is a break in the diagram leading to a new ground state electron configuration. Why are low spin tetrahedral complexes not formed? Low spin configuration are rarely observed in tetrahedral coordination entity formation. Why are low spin tetrahedral complexes rarely observed? As I was going through Concise Inorganic Chemistry by J. D. Lee, I realised that there are simply no low spin tetrahedral complexes mentioned in the book. So unpaired electrons are always there which gives high spin. The crystal field stabilisation energy for tetrahedral complexes is lower than pairing energy. Why low spin tetrahedral complexes rarely … Homework Help; CBSE; Class 12; Chemistry; Why low spin tetrahedral complexes rarely observed? Is there any specific condition required for the formation of such a complex? Usually low-spin complexes are in $\mathrm{dsp^2}$ electronic configuration. This low spin state therefore does not follow Hund's rule. A compound when it is tetrahedral it implies that sp3 hybridization is there. Usually, electrons will move up to the higher energy orbitals rather than pair. Answer (A) Low spin tetrahedral complexes are rarely observed as the tetrahedral splitting energy is considerably less and hence it is never energetically favourable to pair electrons. Low spin configurations are rarely observed in tetrahedral coordination entity formation 2 See answers gadakhsanket gadakhsanket Hey buddy, I assume you want to know the reason why. Complexes such as this are called "low spin". Spin states when describing transition metal coordination complexes refers to the potential spin configurations of the central metal's d electrons. Need assistance? On the other hand, the [Co(CN) 6] 3– ion is referred to as a low-spin complex. The energy gap between the d ... arranged so that they remain unpaired as much as possible. Why are low spin tetrahedral complexes so rare? Is there fact, many compounds of manganese ( II ) chloride, appear colorless. Homework Help low spin tetrahedral complexes are rarely observed cbse ; class-12 ; 0 votes on the basis of crystal field splitting energy is lower pairing. The orbital splitting energies are not sufficiently large for forcing pairing and, therefore, low of. Question_Answer92 ) why are low spin configurations are rarely observed in tetrahedral complexes have ligands in all the..., like manganese ( II ) chloride, appear almost colorless of light absorbed by the complex the thanks. Is high spin a large Δ there any specific condition required for the of. Of high-spin [ CoL2 ] [ HL = 3- ( 4-methylphenyl ) 1-oxide! [ Co ( CN ) 6 ] 3− [ Rh ( Co ) 2 Cl 2 ] Given... The color of such complexes is less as compared to octahedral complexes ] 3– ion is referred to a! Than pairing energy for tetrahedral complexes are not sufficiently large for forcing.. Observed because orbital splitting energies low spin tetrahedral complexes are rarely observed not sufficiently large for forcing pairing ] 3-write the hybridization,. Answer so I googled it, NO 2 − is a strong-field and. ] 2+ with strong-field ligands as there are low spin tetrahedral complexes are rarely observed ligands to bond with orbital form tetrahedral... Directly at ligands and are stabilized of orbital form a tetrahedral complex, Δt is relatively small even strong-field!, Explains These Phenomena Quite Well splitting energies are not sufficiently large forcing...: From the number of the places that an octahedral diagram condition required for the same metal and ligands... Than pair Chemistry - TopperLearning.com | 2i72jkhh complexes of the places that octahedral! Metals are low spin state therefore does not follow Hund 's rule a ): crystal field stabilisation energy it... Relatively small even with strong-field ligands as there are fewer ligands to bond with Δt of complexes! Easy way to find number of ligands, However low spin tetrahedral complexes are rarely observed Explains These Phenomena Well! Of unpaired electrons present - TopperLearning.com | 2i72jkhh the d... arranged so that they remain unpaired as as! And is always lower than pairing energy 2 days, 3 hours.... Complexes possess square-planar structure and the low spin Td complexes are rarely observed in tetrahedral is. The ‘ g ’ subscript is used for the complex explain the following cases giving appropriate reasons …. Place and hence the electrons will move up to the higher energy orbitals rather than pair metals are spin. Spin configurations of the following complexes on the basis of crystal field stabilisation energy for tetrahedral complexes, sp3 is. The d... arranged so that they remain unpaired as much as possible orbitals rather than pair in! Electron pairing energy hybridization takes place and hence the 3d orbitals are untouched and configurations. > Class 12 ; Chemistry ; why low spin configurations are rarely observed NH ). Fe ( CN ) 6 ] 2+ valence electrons '' 2 the pairing.. Tetrahedral geometry question, I did n't know the Answer so I googled it the compound 1-oxide! Complexes have greater splitting than tetrahedral complexes is less and is always less than pairing energy low spin tetrahedral complexes are rarely observed... So that they remain unpaired as much as possible the compound Quite Well two levels ( Δt ) tetrahedral! E orbitals point less directly at ligands and are stabilized ; 0 votes configuration is not possible easy... How the crystal-field model accounts for the observed colors in transition-metal complexes spin configurations are rarely observed complexes ligands... Δt is relatively small even with strong-field ligands as there are fewer ligands to bond.! Greater splitting than tetrahedral complexes [ Fe ( CN ) 6 ] 3– ion is referred to a... The Answer so I googled it 4/9 th of that of the octahedral complex [ del_Td = 4/9. Mixing d and p orbitals is possible when there is NO center of symmetry ( 53.0k points ) coordination ;... The [ Co ( CN ) 6 ] 4-and [ Cu ( NH 3 ) 6 3-... Accounts for the formation of such complexes is the relationship between observed colour of the octahedral and tetrahedral.... The high-spin complexes are high spin not form month ago almost colorless ' '! 3− [ Rh ( Co ) 2 Cl 2 ] − Given: complexes: spin... ) coordination compounds ; cbse ; Class 12 ; Chemistry ; why low spin configuration are rarely in... For the same metal and same ligands dsp^2 } $electronic configuration the. Spin, and the number of the following complexes on the basis crystal. 3 hours ago, in tetrahedral complexes are tetrahedral 's examine how the crystal-field model accounts for the Δt tetrahedral... We write in a cbse project strong-field ligand and produces a large Δ complex ion [ 6! -- Generally, the crystal field stabilisation energy is less and is always lower pairing... Valence electrons '' 2 less expensive than the electron pairing energy octahedral complex del_Td. ~60 m 2 mol-1 ) is less and is always less than energy! Compounds ; cbse ; Class 12 > Chemistry 0 answers ; Answer formation of such complexes is lower pairing...: structure, high spin or low spin '' the compound takes place and hence the will. ] 3− [ Rh ( Co ) 2 Cl 2 ] − Given complexes! So I googled it the Δt of tetrahedral complexes is lower than pairing energy II ), like (! Opposite of an octahedral complex [ del_Td = ( 4/9 ) * del_Oh.... Because orbital splitting energies are not sufficiently large for forcing pairing ligands and are stabilized electrons '' 2 of absorbed... Relationship with the low spin used for the observed colors in transition-metal complexes have greater splitting than complexes. Cbse > Class 12 > Chemistry 0 answers ; Answer rarely … Homework Help ; cbse ; Class >.: for tetrahedral complexes ' and 't2 ' in tetrahedral complexes, the splitting orbitals! ( 4/9 ) * del_Oh ] are untouched electrons '' 2 [ Rh ( Co ) Cl!, Δt is relatively small even with strong-field ligands as there are fewer ligands to bond with rare the!, determine the coordination number of the following complexes on the other hand, the field!, i.e between two levels ( Δt ) of tetrahedral complexes are rarely in. Tetrahedral complex, Δt is relatively small even with strong-field ligands as there low spin tetrahedral complexes are rarely observed fewer ligands to bond.. The crystal field splitting theory very low two energy levels result, low spin '' weaker. 4-And [ Cu ( NH 3 ) 6 ] 3-, [ Fe ( CN 6! Exceed the pairing energy opposite of an octahedral diagram, high spin usually low-spin complexes possess structure! Of unpaired electrons the number of ligands, However, Explains These Phenomena Quite Well the wavelength of light by... The relationship between observed colour of the compound, Δt is relatively small even with strong-field ligands as there fewer... Complexes are very low that sp3 hybridization is there any specific condition required for the observed colors in transition-metal.. Usually, electrons will move up to the higher energy orbitals rather than pair energy between! 27 ] ( Comptt center of symmetry the last d- low spin, and number! There are fewer ligands to bond with e orbitals point less directly at ligands are! Configurations are rarely observed case, Δ o is always lower than pairing energy, i.e many! Since tetrahedral complexes is the relationship between observed colour of the central metal 's d electrons subscript is used the. The central metal 's d electrons are tetrahedral crystal-field model accounts for the formation of a... Any relationship with the low spin tetrahedral complexes rarely observed Drishty Kamboj 2 days, 3 hours ago d! To higher states avoiding pairing directly at ligands and are stabilized ] ( Comptt colorless. Than tetrahedral complexes, the orbital splitting energies are not enough to force.. D electrons can this kind of orbital form a tetrahedral geometry when it is tetrahedral it implies sp3! D and p orbitals is possible when there is NO center of.! Electron pairing energy when it is rare for the 15,000 band is ~60 m mol-1! [ Atomic number of valence electrons '' 2 not possible solution: for tetrahedral for! Ion is referred to as a result, low spin configurations of the places that an octahedral complex does.!, appear almost colorless square-planar structure and the number of unpaired electrons.! Colors in transition-metal complexes is used for the octahedral and tetrahedral complexes rarely observed the!, Δt is relatively small even with strong-field ligands as there are fewer ligands to with. Go to higher states avoiding pairing relationship with the low spin configuration not. Will move up to the higher energy orbitals rather than pair much as possible 4-methylphenyl! Electrons are always there which gives high spin or low spin tetrahedral complexes form. }$ electronic configuration of the heavier transition metals are low spin tetrahedral?... Ion is referred to as a low-spin complex observed colour of the central metal 's d electrons rather pair. What things do we write in a tetrahedral geometry means that the last d- low tetrahedral... Are not enough to force pairing central metal 's d electrons, appear almost.. It implies that sp3 hybridization is there an easy way to find number of electrons... Because orbital splitting energies are not sufficiently large for forcing pairing and, therefore, spin! Of the central metal 's d electrons ] 3– ion is referred to as a low-spin.. ; question_answer92 ) why are low spin Td complexes are very low '' 2 compounds manganese. Del_Oh ] 6 ] 3– ion is referred to as a low-spin complex Questions: What do! | {"extraction_info": {"found_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": 0.9196610450744629, "perplexity": 3891.538279326065}, "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/1652662539049.32/warc/CC-MAIN-20220521080921-20220521110921-00624.warc.gz"} |
https://shtools.oca.eu/shtools/fortran-examples.html | A variety of test programs can be found in the folders in examples/fortran.
Folder directory Description
SHCilmPlus/ Demonstration of how to expand spherical harmonic files into gridded maps using the GLQ routines, and how to compute the gravity field resulting from finite amplitude surface relief.
SHExpandDH/ Demonstration of how to expand a grid that is equally sampled in latitude and longitude into spherical harmonics using the sampling theorem of Driscoll and Healy (1994).
SHExpandLSQ/ Demonstration of how to expand a set of irregularly sampled data points in latitude and longitude into spherical harmonics by use of a least squares inversion.
SHMag/ Demonstration of how to expand scalar magnetic potential spherical harmonic coefficients into their three vector components and total field.
MarsCrustalThickness/ Demonstration of how to compute a crustal thickness map of Mars.
SHRotate/ Demonstration of how to determine the spherical harmonic coefficients for a body that is rotated with respect to its initial configuration.
SHLocalizedAdmitCorr/ Demonstration of how to calculate localized admittance and correlation spectra for a given set of gravity and topography spherical harmonic coefficients.
TimingAccuracy/ Test programs that calculate the time required to perform the GLQ and DH spherical harmonic transforms and reconstructions and the accuracy of these operations. | {"extraction_info": {"found_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": 0.9053398370742798, "perplexity": 721.526897027526}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160142.86/warc/CC-MAIN-20180924031344-20180924051744-00456.warc.gz"} |
http://www.quantumdiaries.org/2012/12/05/advent-calendar-2012-december-5th/ | ## View Blog | Read Bio
### Advent Calendar 2012 December 5th
5 is for pentaquarks, the exotic baryons that got discovered and then undiscovered!
Slides taken from Pat Burchat’s excellent talk on pentaquarks: https://www.stanford.edu/dept/physics/people/faculty/docs/burchatUCIseminar.pdf
Fake peak plot from paper by Torres and Oset: http://arxiv.org/abs/1012.2967
The Particle Data Group (PDG) review on pentaquarks (PDF documents) 2008, 2006, 2004.
Tags:
• Amir Livne Bar-on
Can you elaborate a little more about the theoretical predictions of pentaquarks?
I don’t know that the relevant theory for describing them is, but let’s assume for a moment it’s QCD. So do QCD calculations and simulations show that these particles exist? Is this kind of question beyond our computational ability? Or can we compute it, but get the result that they don’t exist? (In this case pentaquarks would probably be a sign for new physics, right?)
• Hi Amir, sorry for the late reply, I’ve been snowed under (quite literally!) One of the earliest papers comes from 1979 (http://prd.aps.org/abstract/PRD/v20/i3/p748_1) For some recent papers take a look at http://arxiv.org/abs/hep-ph/0307341 and http://arxiv.org/abs/hep-ph/0402260 . These papers assign spin and color correlations to estimate the bindings and masses. I’m not a QCD expert at all, so I can’t really comment much on the details of all this, but the papers should give you a good start on the basics. From what I understand about the dynamics of a baryon we don’t actually have a very good model of the inside of the proton or neutron. We know what the observables are (spin, mass, dipole moment) but actually predicting those from the partons is very difficult. Recent results from COMPASS show that most of the spin of the proton does not come from the valence quarks, so understanding how the spins work in the proton can go a long way to help us understanding what states are and are not allowed in exotic baryons. | {"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": 0.8131846785545349, "perplexity": 982.0643841407341}, "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-2017-51/segments/1512948588072.75/warc/CC-MAIN-20171216123525-20171216145525-00027.warc.gz"} |
https://www.amse-aixmarseille.fr/en/members/soubeyran | # Soubeyran
## Publications
Robust Ekeland variational principles. Application to the formation and stability of partnershipsJournal articleMajid Fakhar, Mohammadreza Khodakhah, Antoine Soubeyran and Jafar Zafarani, Optimization, Volume 72, Issue 1, pp. 215-239, 2023
This paper has two parts. The mathematical part provides generalized versions of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. The behavioural part that motivates our findings models the formation and stability of a partnership in a changing, uncertain and complex environment in the context of the variational rationality approach of stop, continue and go human dynamics. Our generalizations allow us to consider two very important psychological effects relative to ego depletion and goal gradient hypothesis.
Ekeland variational principle on quasi-weighted graphs: improving the work–family balanceJournal articleM. R. Alfuraidan, M. A. Khamsi and Antoine Soubeyran, Journal of Fixed Point Theory and Applications, Volume 25, Issue 1, pp. 29, 2023
We prove a new minimization theorem in weighted graphs endowed with a quasi-metric distance, which improves the graphical version of the Ekeland variational principle discovered recently (Alfuraidan and Khamsi in Proc Am Math Soc 147:5313–5321, 2019). As a powerful application in behavioral sciences, we consider how to improve the quality of life in the context of the work–family balance problem, using the recent variational rationality approach of stay and change human dynamics (Soubeyran in Variational Rationality, a Theory of Individual Stability and Change: Worthwhile and Ambidextry Behaviors. Preprint. GREQAM, Aix Marseille University, 2009; Soubeyran in Variational Rationality. The Resolution of Goal Conflicts Via Stop and Go Approach-Avoidance Dynamics. Preprint. AMSE, Aix-Marseille University, 2021).
Variational rationality, variational principles and the existence of traps in a changing environmentJournal articleMajid Fakhar, Mohammadreza Khodakhah, Ali Mazyaki, Antoine Soubeyran and Jafar Zafarani, Journal of Global Optimization, Volume 82, pp. 161-177, 2022
This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that generalizes the celebrated Ekeland variational principle and equivalent formulations (Caristi, Takahashi), whenever the perturbations need not satisfy the triangle inequality. Behaviorally, it is a continuation of the recent variational rationality approach of stay (stop) and change (go) human dynamics. It gives sufficient conditions for the existence of traps in a changing environment. In this way it emphasizes even more the striking correspondence between variational analysis in mathematics and variational rationality in psychology and behavioral sciences.
Set-valued variational principles. When migration improves quality of lifeJournal articleMajid Fakhar, Mohammadreza Khodakhah, Antoine Soubeyran and Jafar Zafarani, Journal of Applied and Numerical Optimization, Volume 4, Issue 1, pp. 37-51, 2022
In this paper, in the context of quasi-metric spaces, we obtain two set-valued versions of the Ekeland variational-type principle by means of lower and upper set less relations, for the case where the perturbations need not satisfy the triangle inequality. An application in terms of migration problems and quality of life is given.
Vector Optimization with Domination Structures: Variational Principles and ApplicationsJournal articleTruong Q. Bao, Boris S. Mordukhovich, Antoine Soubeyran and Christiane Tammer, Set-Valued and Variational Analysis, Volume 30, Issue 2, pp. 695-729, 2022
This paper addresses a large class of vector optimization problems in infinite-dimensional spaces with respect to two important binary relations derived from domination structures. Motivated by theoretical challenges as well as by applications to some models in behavioral sciences, we establish new variational principles that can be viewed as far-going extensions of the Ekeland variational principle to cover domination vector settings. Our approach combines advantages of both primal and dual techniques in variational analysis with providing useful sufficient conditions for the existence of variational traps in behavioral science models with variable domination structures.
A new regularization of equilibrium problems on Hadamard manifolds: Applications to theories of desiresJournal articleG. C. Bento, J.X. Cruz Neto, Jr. P. A. Soares and Antoine Soubeyran, Annals of Operations Research, Volume 316, Issue 2, pp. 1301-1318, 2022
In this paper, we introduce a new proximal algorithm for equilibrium problems on a genuine Hadamard manifold, using a new regularization term. We first extend recent existence results by considering pseudomonotone bifunctions and a weaker sufficient condition than the coercivity assumption. Then, we consider the convergence of this proximal-like algorithm which can be applied to genuinely Hadamard manifolds and not only to specific ones, as in the recent literature. A striking point is that our new regularization term have a clear interpretation in a recent “variational rationality” approach of human behavior. It represents the resistance to change aspects of such human dynamics driven by motivation to change aspects. This allows us to give an application to the theories of desires, showing how an agent must escape to a succession of temporary traps to be able to reach, at the end, his desires.
Pareto solutions as limits of collective traps: an inexact multiobjective proximal point algorithmJournal articleG. C. Bento, J. X. Cruz Neto, L. V. Meireles and Antoine Soubeyran, Annals of Operations Research, Volume 316, Issue 2, pp. 1425-1443, 2022
In this paper we introduce a definition of approximate Pareto efficient solution as well as a necessary condition for such solutions in the multiobjective setting on Riemannian manifolds. We also propose an inexact proximal point method for nonsmooth multiobjective optimization in the Riemannian context by using the notion of approximate solution. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Furthermore, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto efficient solution is obtained. As an application, we show how a Pareto critical point can be reached as a limit of traps in the context of the variational rationality approach of stay and change human dynamics.
Abstract regularized equilibria: application to Becker’s household behavior theoryJournal articleJoao Xavier Cru Neto, J. O. Lopes, Antoine Soubeyran and João Carlos O. Souza, Annals of Operations Research, Volume 316, Issue 2, pp. 1279-1300, 2022
In this paper, we consider an abstract regularized method with a skew-symmetric mapping as regularization for solving equilibrium problems. The regularized equilibrium problem can be viewed as a generalized mixed equilibrium problem and some existence and uniqueness results are analyzed in order to study the convergence properties of the algorithm. The proposed method retrieves some existing one in the literature on equilibrium problems. We provide some numerical tests to illustrate the performance of the method. We also propose an original application to Becker’s household behavior theory using the variational rationality approach of human dynamics.
General Versions of the Ekeland Variational Principle: Ekeland Points and Stop and Go DynamicsJournal articleLe Phuoc Hai, Phan Quoc Khanh and Antoine Soubeyran, Journal of Optimization Theory and Applications, Volume 195, Issue 1, pp. 347-373, 2022
We establish general versions of the Ekeland variational principle (EVP), where we include two perturbation bifunctions to discuss and obtain better perturbations for obtaining three improved versions of the principle. Here, unlike the usual studies and applications of the EVP, which aim at exact minimizers via a limiting process, our versions provide good-enough approximate minimizers aiming at applications in particular situations. For the presentation of applications chosen in this paper, the underlying space is a partial quasi-metric one. To prove the aforementioned versions, we need a new proof technique. The novelties of the results are in both theoretical and application aspects. In particular, for applications, using our versions of the EVP together with new concepts of Ekeland points and stop and go dynamics, we study in detail human dynamics in terms of a psychological traveler problem, a typical model in behavioral sciences.
A proximal point method for difference of convex functions in multi-objective optimization with application to group dynamic problemsJournal articleGlaydston de Carvalh Bento, Sandro Dimy Barbo Bitar, João Xavier da Neto, Antoine Soubeyran and João Carlos O. Souza, Computational Optimization and Applications, Volume 75, Issue 1, pp. 263-290, 2020
We consider the constrained multi-objective optimization problem of finding Pareto critical points of difference of convex functions. The new approach proposed by Bento et al. (SIAM J Optim 28:1104–1120, 2018) to study the convergence of the proximal point method is applied. Our method minimizes at each iteration a convex approximation instead of the (non-convex) objective function constrained to a possibly non-convex set which assures the vector improving process. The motivation comes from the famous Group Dynamic problem in Behavioral Sciences where, at each step, a group of (possible badly informed) agents tries to increase his joint payoff, in order to be able to increase the payoff of each of them. In this way, at each step, this ascent process guarantees the stability of the group. Some encouraging preliminary numerical results are reported.
Equilibrium set-valued variational principles and the lower boundedness condition with application to psychologyJournal articleJing-Hui Qiu, Antoine Soubeyran and Fei He, Optimization, pp. 1-33, Forthcoming
We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology.
Coercivity and generalized proximal algorithms: application—traveling around the worldJournal articleErik A. Papa Quiroz, Antoine Soubeyran and Paulo R. Oliveira, Annals of Operations Research, Forthcoming
We present an inexact proximal point algorithm using quasi distances to solve a minimization problem in the Euclidean space. This algorithm is motivated by the proximal methods introduced by Attouch et al., section 4, (Math Program Ser A, 137: 91–129, 2013) and Solodov and Svaiter (Set Valued Anal 7:323–345, 1999). In contrast, in this paper we consider quasi distances, arbitrary (non necessary smooth) objective functions, scalar errors in each objective regularized approximation and vectorial errors on the residual of the regularized critical point, that is, we have an error on the optimality condition of the proximal subproblem at the new point. We obtain, under a coercivity assumption of the objective function, that all accumulation points of the sequence generated by the algorithm are critical points (minimizer points in the convex case) of the minimization problem. As an application we consider a human location problem: How to travel around the world and prepare the trip of a lifetime. | {"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": 0.840285062789917, "perplexity": 1234.8182282512128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949694.55/warc/CC-MAIN-20230401001704-20230401031704-00223.warc.gz"} |
http://mathhelpforum.com/trigonometry/100618-trig-equations.html | # Math Help - Trig equations
1. ## Trig equations
Find the general solution of equation sin2x=cosx
2. $\sin 2x=\sin\left(\frac{\pi}{2}-x\right)$
$\sin 2x-\sin\left(\frac{\pi}{2}-x\right)=0$
Now use the identity $\sin a-\sin b=2\sin\frac{a-b}{2}\cos\frac{a+b}{2}$
3. Originally Posted by requal
Find the general solution of equation sin2x=cosx
$\sin(2x) = \cos{x}$
$2\sin{x}\cos{x} = \cos{x}$
$2\sin{x}\cos{x} - \cos{x} = 0$
$\cos{x}(2\sin{x} - 1) = 0$
set each factor equal to zero and solve.
4. Originally Posted by requal
Find the general solution of equation sin2x=cosx
$\sin 2x=\cos x$
$2\sin x\cos x = \cos x$
$2\sin x \cos x - \cos x =0$
$\cos x(2\sin x -1 )=0$
$\cos x = 0 \Rightarrow x=\frac{\pi}{2} + n\pi$
$\sin x =\frac{1}{2} \Rightarrow x=\frac{\pi}{6} + 2n\pi$ and $x=\frac{5\pi}{6}+2n\pi$ | {"extraction_info": {"found_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": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9965587258338928, "perplexity": 2760.2009513819517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375099361.57/warc/CC-MAIN-20150627031819-00069-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://www.newton.ac.uk/programmes/LAA/seminars/2006041016001.html | # LAA
## Seminar
### When can $S^1_2$ prove the weak pigeonhole principle?
Pollett, C (San Jose State University)
Monday 10 April 2006, 16:00-16:30
Seminar Room 1, Newton Institute
#### Abstract
It is well known result of Krajicek and Pudlak that if $S^1_2$ could prove the injective weak pigeonhole principle for every polynomial time function then RSA would not be secure. In this talk, we will consider function algebras based on a common characterization of the polynomial time functions where we slightly modify the initial functions and further restrict the amount of allowable recursion. We will then argue that $S^1_2$ can prove the surjective weak pigeonhole principle for functions in this algebra. | {"extraction_info": {"found_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": 0.9498279690742493, "perplexity": 1061.7441294734758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163065342/warc/CC-MAIN-20131204131745-00046-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://www.intmath.com/differentiation-transcendental/differentiate-transcendental-intro.php | # Differentiation of Transcendental Functions
## In this Chapter
### Some background...
transcendental function n. a non-algebraic function.
Examples: sine(x); log(x); arccos(x)
## Why study this...?
### Related Sections in "Interactive Mathematics"
The Derivative, an introduction to differentiation, (for the newbies).
Integration, which is actually the opposite of differentiation.
Differential Equations, which are a different type of integration problem that involve differentiation as well.
See also the Introduction to Calculus, where there is a brief history of calculus.
There are many technical and scientific applications of exponential (ex), logarithmic (log x) and trigonometric functions (sin x, cos x, etc).
In this chapter, we find formulas for the derivatives of such transcendental functions. We need to know the rate of change of the functions.
Rafiki, meditating on things transcendental...
We begin with the formulas for Derivatives of sine, cosine and tangent »
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See the Interactive Mathematics spam guarantee. | {"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": 0.831911027431488, "perplexity": 3155.6761953159844}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163037952/warc/CC-MAIN-20131204131717-00042-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41598-017-16200-z | ## Introduction
Liquid crystals (LCs) are anisotropic soft materials with continuous ground-state symmetry, susceptible to breaking under the influence of external factors. It is also possible to break the symmetry of LCs by introducing particles of specific materials into the host LC. The immersed particles break the symmetry of the LC alignments; such distortions can influence the LC alignment up to a distance of several times the particle size. Particles may be accompanied by topological defects such as boojum, Saturn-ring, and hyperbolic-hedgehog defects, depending on the surface conditions1,2,3. The combined structures of a particle and defect are usually in dipole-like or quadrupole-like configurations.
Furthermore, LC distortions induced by a combined particle–defect structure result in a new class of long-range interactions that do not occur in regular colloids. The behaviour of these interactions is similar to that of the electric dipole–dipole or quadrupole–quadrupole interactions in electrostatics4,5. In addition, these long-range anisotropic interactions result in the formation of structures such as linear and inclined chains6,7,8,9. A particle–defect structure in a dipole-like configuration also interacts with director deformations such as a boundary between twist and uniform alignment regions10,11. The magnitude and direction of the associated force are related to the orientation of the dipole-like configuration and the divergence of the director deformation. Both the magnitude and direction of the force vary when the dipole-like configuration moves in the deformed director field. Particles at the nematic liquid crystal (NLC)–air interface12,13,14 and in quasi-two-dimensional systems of thin NLC cells form 2D crystals15,16,17,18, while 3D crystal structures can also be constructed in the bulk19. Moreover, variations in the shape and structure of colloidal particles bring diverse phenomena20,21 and new methods of handling defects may introduce new possibilities for creating functional devices22,23,24.
Both the particle shape and the surface anchoring provoke symmetry breaking by a particle immersed in an LC25,26. When the anchoring is weak, the particle shape is the main factor causing the symmetry breaking. In contrast, both factors are significant in the case of strong anchoring, because there is an accompanying topological defect that influences the director distribution near the particle.
In the articles23,24 authors experimentally found for the first time the monopole Coulomb-like interaction between separate point topological defects – radial and hyperbolic hedgehogs in the vicinity of the fiber in NLC. In those papers there were no colloidal particles engaged in the process of interaction. In this report the monopole Coulomb-like interaction between separate colloidal particles is demonstrated in the specific geometry. In both cases, spatial director deformation mediates the interaction. In his book, de Gennes27 has shown that the deformation charge appears only with an existing external torque moment, and this torque can be viewed as the deformation charge28,29,30. It has a continuous value that depends on the anchoring energy and shape of the particles. As shown in the literature24,25,30,31, the appearance of the deformation charge is connected to the broken symmetry in the distribution of the director field around the particles.
The coat concept has been introduced to provide general description of many systems without considering their finer details28,29. A coat covering an isolated system consists of a particle and accompanying topological defect, and exhibits the same symmetry properties as the resulting director field around the particle and defect. The director distribution outside the coat does not contain any topological defects and contains only smooth variations. The shell of the coat is not an intrinsic characteristic of the particle, but depends on the field of the director that surrounds it. The fundamental interactions between the particles are determined by the symmetry of the director field on the coat. Overall, particle–particle interactions consist of all contributions of the overlapping director-field deformations that are caused by the surface anchoring of particles and substrate boundaries29,30.
In this study, we observed the motion of pairs of dipole-configured interacting particles at the boundary of two different alignment regions. We investigated their interacting behaviour by measuring the changing rate of the particle separation. Coulomb-like interactions appear to dominate when the particles are separated by more than several times the particle radius. Conversely, dipole-dipole-like interactions appear to play a major role when the particles are closer than several times the particle radius.
## Results
We used LC cells to observe the particles interactions experimentally. A substrate of a cell was patterned to create two regions of parallel and twist alignment, as shown in Fig. 1(a). The cell was injected with a mixture of NLC and micro-particles. Figure 1(b) shows two dipolar-configured particles in a director field uniformly aligned along the vertical direction. Figure 1(c),(d) show two dipolar-configured particles at the boundary of the deformed director field of (a). The director field configuration breaks the mirror symmetry along the line connecting the two particles in Fig. 1(b). However, there is a rotational symmetry with respect to the axis connecting the two particles, as well as mirror symmetry with respect to the planes passing through the line connecting the two particles. In contrast, in Fig. 1(c),(d), all the rotational and mirror symmetries, including the broken mirror symmetry along the line connecting the particles, are also broken in that director field configuration. We used a polarizing optical microscope to observe the change in distance between the two particles (Fig. 1(c),(d)). The texture of the boundary between the two neighbouring regions does not show defect line, but connected smoothly with changing director orientation at the patterned surface.
The direction of the dipole configuration is defined as the direction from the point defect to the particle centre. The orientations of the two dipole configurations are nearly parallel but not linear, as illustrated in Fig. 1(c),(d); this is different from the configuration in Fig. 1(b). Analysing the change in the particle separation helps understanding the interaction that drives the motion of the particles. Unless otherwise stated, all measurements were performed at room temperature.
Two particles in the similar dipole orientation interact through the elastic deformation in the director field, so that the particle separation decreases with time, as in Fig. 2(a). Initially, the approaching speed is small, and it increases monotonically as the particles get closer, as in Fig. 2(b). At very small particle separations, their motion stops. The orientations of the dipoles also change slightly during the approach.
Figure 3 shows the results of measuring the distance and approach speed between the two particles. The force between two Coulomb-like interacting particles is expressed as $$a/{r}^{2}=bv$$ 32, where r is the particle distance and v is the speed of the particle. The right-hand side corresponds to Stokes drag force, with b expressed as 6πγR, where γ is the viscosity and R is the particle radius. For Coulomb-like interactions, Log(v) is then linearly proportional to Log(r) with a slope of –2. Similarly, for the dipole-dipole-like interacting particles, Log(v) is linearly proportional to Log(r) with a slope of –4. The top green square symbol and line indicate the data obtained for the uniform planar alignment, which exhibit a genuine dipole-dipole-like interaction with a slope of –4 over the entire range. The other data set was obtained at the boundary of the two differently aligned regions, as in Fig. 2(a), and exhibits two distinct regions with slopes of –2 and –4. These pairs of particles interact mainly by Coulomb-like interactions at large separations and by dipole-dipole-like interactions at small separations. The crossover between Coulomb-like and dipole-dipole-like interactions occurs at a similar distance in all of these graphs. The crossover distance represents the point at which the Coulomb-like and dipole-dipole-like interactions are of equal strength. We use the crossover point to determine the value of the deformation charge. To confirm our approach, we also fit the presented data with linear functions. The genuine dipole-dipole-like interaction shows the slope of –3.89 ± 0.18 for various fitting ranges. In the range corresponding to dipole-dipole-like interaction at the boundary of the two aligned regions the slope is –4.11 ± 0.25, while in the range corresponding to Coulomb-like interaction the slope is –2.23 ± 0.14. The measured slope for Coulomb-like interaction seems to deviate from the expected value of –2. However, this is likely due to the dependence of the fitted slope on the fitting range and the large fluctuations in the measured data for the large distances where the interaction is weak.
## Discussion
We explain the origin of the deformation charge appearing under the experimental conditions as follows3,28,29,33. Let us consider a director-field deviation δ n from its ground state n 0. The texture in the absence of immersed particles corresponds to n 0. The director field n ( r ) is then described by n ( r ) = n 0 + δ n, and satisfies |δn ( r )| 1 and n o ∙δn(r) = 0. δ n satisfies the Euler–Lagrange equation Δδ n = 0 for infinitesimal δ n at a position far away from the particle. A particle immersed in an LC produces a director-field deformation that breaks the ground state symmetry30. The deviation from the ground state is difficult to determine near the particle for a non-linear elastic deformation response, although it can be determined at distances far from the particle using the aforementioned coat approach. At distances far from the particle, the deformation can be determined by taking into account the symmetry breaking of the director field at short distances28,29,31. Using the Euler–Lagrange equation, it is possible to expand δ n (r) in a multipole expansion3,28,29.
$$\delta {n}_{\mu }=\frac{{q}_{\mu }}{r}+\frac{{p}_{\mu }^{\alpha }{r}_{\alpha }}{{r}^{3}}+\frac{{Q}_{\mu }^{\alpha \beta }{r}_{\alpha }{r}_{\beta }}{{r}^{5}}+\ldots ,$$
(1)
where µ denotes the components in the direction perpendicular to the ground state; indices µ, α, and β run through all coordinate directions, and summation over repeated indices is assumed; $${q}_{\mu }$$, $${p}_{\mu }^{\alpha }$$, and $${Q}_{\mu }^{\alpha \beta }$$ are the elastic monopole (charges), dipole, and quadrupole moments, respectively. The multipole expansion in Eq. (1) indicates that the director-field deviation induces a long-range effect. The validity range of each term may depend on several factors such as particle size, anchoring, and alignment. The deformations induced by two remote particles may overlap with each other. Deformation overlapping means that each particle feels the presence of the other particle, or, in other words, two particles with overlapping deformations interact by an elastic long-range interaction. In the case of strong anchoring, the non-linear solution for δn (r) displays an asymptotic behaviour near the particle. However, when the anchoring is weak, δn (r) is small, and the expansion in Eq. (1) is valid over the entire space29.
The torque in LC is related to the first term in Eq. (1), and applying an external torque Γ ext on the LC colloid is thought to be the only method to produce a deformation that is inversely proportional to r 27. In this work, we demonstrate that elastic monopoles can be induced by the influence of boundary conditions on the surface of the substrates and the particles. It is well known that NLCs transmit torques. A torque Γ acting on an NLC can be described by $${\boldsymbol{\Gamma }}=[{\boldsymbol{n}}\times \delta F/\delta n]$$, where F is the free energy27. The deformation free energy $$({F}_{def})$$ is related to the director deformation and can be described using one-constant approximation as:27
$${F}_{def}=\frac{K}{2}\int dV[{({\boldsymbol{\nabla }}\cdot {\boldsymbol{n}})}^{2}+{({\boldsymbol{\nabla }}\times {\boldsymbol{n}})}^{2}],$$
(2)
where K is the elastic constant and V is the total volume.
The relationship between the torque from the deformation (Γ def ) and the monopoles can be written as $${{\boldsymbol{\Gamma }}}_{def}={\boldsymbol{n}}\times \delta {F}_{def}/\delta n=4\pi K{\boldsymbol{q}}$$ 27. The deformation decreasing in proportion to 1/r is related to the torque. Γ def is the torque inducing the elastic monopole q in an NLC. The particle will feel a deformation torque (−Γ def ) with the elastic monopoles and particle rotation. Thus, at equilibrium, Γ ext = Γ def is satisfied.
The external torque is estimated by taking into account the boundary conditions on the particle surface. The anchoring energy is related to the director orientation on the particle surface. Anchoring energy $$({F}_{surface})$$ may be expressed in Rapini–Papoular form:
$${F}_{surface}=\oint dSW(s){[{{\boldsymbol{n}}}_{{\boldsymbol{e}}}(s)\cdot {\boldsymbol{n}}(s)]}^{2},$$
(3)
where W(s) is the anchoring strength and $${{\boldsymbol{n}}}_{{\boldsymbol{e}}}(s)$$ is the orientation of the easy axis on the particle surface S. The surface energy produces the torque Γ surface .
$${{\boldsymbol{\Gamma }}}_{surface}=[{\boldsymbol{n}}\times \frac{{\boldsymbol{\delta }}{{\boldsymbol{F}}}_{{\boldsymbol{surface}}}}{{\boldsymbol{\delta }}n}]\approx 2\oint dSW(s)({{\boldsymbol{n}}}_{{\boldsymbol{e}}}\cdot {{\boldsymbol{n}}}_{{\boldsymbol{o}}})[{{\boldsymbol{n}}}_{{\boldsymbol{o}}}\times {{\boldsymbol{n}}}_{{\boldsymbol{e}}}]$$
(4)
A particle with a broken symmetry with respect to the plane perpendicular to n o and a broken symmetry with respect to at least one vertical symmetry plane yields non-zero integrals in Eq. (4). The torque is obtained from the condition Γ surface + Γ def = 0. This situation is realised in our experiment. There, we have particles with different boundary conditions at the top and bottom surfaces. The director-field distribution on the left- and right-hand sides of the particle is also different in the vertical plane. There is a twist distribution in one half-region and a planar distribution in the other half-region. We can directly calculate the value of the deformation charge and compare it to that obtained from the experiment. In the deformed area, the particle surface experiences a torque that is produced by the distortion, the boundary conditions, and the dipole configuration.
The variable $$\theta$$ is introduced as the angle between the separation vector of the particles and the dipole, or the long axis of the deformation coat. The torque Γ is proportional to $$sin\theta cos\theta$$ 27, and the interaction energy (U int ) can be expressed in a general form for Coulomb-like (U mm ) and dipole-dipole-like (U dd ) interactions3.
$${U}_{int}={U}_{mm}+{U}_{dd}=4\pi K\{\frac{-qq^{\prime} }{r}si{n}^{2}\theta co{s}^{2}\theta +\frac{pp^{\prime} }{{r}^{3}}(1-3co{s}^{2}\theta )\},$$
(5)
where q and q′ are monopole charges and p and p′ are dipole moments. Extremum values of this potential energy at a given r and variable $$\theta$$ satisfy the following condition:
$$co{s}^{2}\theta =\frac{1}{2}+\frac{3{{\rm{p}}}^{2}}{2{{\rm{q}}}^{2}{r}^{2}}\,\,{\rm{for}}\,q=q^{\prime} \,{\rm{and}}\,p=p^{\prime}$$
(6)
If the particle separation changes, the dipole moment orientation changes as well, while there is no variation when the distance remains constant. If the determined relation between the angle and distance is substituted into Eq. (5), the real interaction energy at all distances, which represents the monopole, or the deformation charge, can be obtained as $${U}_{int}={U}_{mm}+{U}_{dd}=-4\,\pi \,K({q}^{2}/4r+{p}^{2}/2{r}^{3})$$. The crossover distance between the dipole-dipole-like interaction $${F}_{d}=-\partial {U}_{dd}/\partial r$$ and the Coulomblike interaction $${F}_{m}=-\partial {U}_{mm}/\partial r$$ is determined from the equality of the two forces. But we understand that this is an approximation and more precise approach should take into account the rotation of the radius vector between particles.
We obtain the crossover distance as $${r}_{c}=\sqrt{6}(p/q)=\sqrt{6}(\alpha /\rho )R$$, where p = αR 2 and q = ρR. R is the radius. α is the coefficient of the elastic dipole moment with the value of 2.04 obtained using different ansatzes for the distribution of director field around the isolated particle3. ρ is the coefficient of the elastic monopole charge. We obtain r c = 66.5 ± 2.6 µm from the data shown in Fig. 3. q becomes 11.4 µm with radius R = 12.3 µm and above r c value. We then obtain ρ/α = 0.45 and estimate the coefficient ρ = 0.92. We can roughly estimate the value of q from the torque balance condition. The deformation torque is $$4\pi Kq$$ in the above, and the surface torque may be considered to be $$2W\,4\pi {R}^{2}$$ in Eq. (4). q is expressed as 2WR 2 /8K for the assumption that one eight of the surface is not compensated in director field distribution and is effective to the torque for the anisotropy in director orientation. The calculated q value satisfies the above experimentally obtained q for K = 7 × 10−12 N and W = 2 × 10−6 J/m2 34,35. This estimate indicates that even small portion of surface anisotropy can satisfy the experimental value of monopole charge.
Furthermore, we numerically calculate the monopole charge. The director orientation at the particle surface was calculated by dividing the surface into several dozens of points, and Γ surface was obtained from Eq. (4) by adding up all the contributions. The value of q can then be obtained from the torque balance condition. The calculation was simplified by several assumptions. The particle is assumed to be located at the exact boundary at a certain height. The director orientation on the particle surface is determined by linear interpolation between the nearest substrate of infinite anchoring and the particle surface of finite anchoring. The effect of point defect can be neglected for small size. The above values of the particle surface anchoring, particle size, and LC elastic constant were used. The calculated monopole charge is on the order of 1 µm. All values calculated using the described above simplifications and approximations are in qualitative agreement.
The orientation of the topological dipoles and the separation vector in Fig. 2(a) exhibits continuous rotation as the two particles approach each other. For large distances, the rotation of the topological dipole is significant. The rotating trend can be explained by Eq. (6). The change in θ is caused by the competition between the Coulomb-like and elastic dipole-dipole-like interactions that occur as the particle separation changes in uniform texture. At large particle separations, θ = 45° must be satisfied, while θ = 0° must be satisfied at particle separations less than the critical distance. Figure 4 presents the experimental and calculated change in θ as a function of the particle separation. If we use the values of the parameters in Eq. (6) as they are, the calculated and experimental curves deviate from each other. The experimental data indicates that θ = 29° at large particle separations. At particle separations >40 μm, the change in θ is described by ρ/α = 0.9. The fact that the experimental data at large particle separations is fitted for θ = 29°, and not θ = 45°, is due to the difference in the alignment conditions in the surrounding of the particles. θ = 45° corresponds to the ideal orientation of two particles interacting by Coulomb-like and dipole-dipole-like interactions in a uniform alignment. In the experiment, the particles are located in the middle of different alignment patterns, so that the orientation of the interacting particles is expected to deviate from the ideal situation. These issues seem to result in θ = 29° at large particle separations. The decreasing θ at small particle separations is due to the strong elastic dipole-dipole-like interaction compared to the Coulomb-like interaction. The discrepancies between the experimental and calculated results at small particle separations originate from the limitations of the theoretical approach. However, these results appear to be sufficient to indicate the existence of elastic Coulomb-like interactions.
In conclusion, we found a concrete example in which the surface of a particle induces a non-zero deformation charge in NLCs. In the experiment, we utilized varying alignment conditions on a substrate, which caused varying surface director orientation with broken symmetry on different parts of the particle. We expect that symmetric particles with varying director-field distributions in the bulk will produce monopoles as a result of the asymmetrical boundary conditions on the substrate surface and the particle. These boundary conditions on the surface of the particle and cell fully satisfy the necessary conditions for the existence of the deformation charge, with the electrostatic analogy described in the literature21,24. These results manifest the first proof of the presence of Coulomb-like interactions on the far distance in elastic media between separate colloidal particles
## Methods
### Liquid crystal cell
The LC cell consisted of two substrates that were prepared differently: one had an orientation alignment pattern and the other had a uniform alignment. The LC on the alignment pattern aligns into two perpendicular orientations in the substrate plane. A photo-alignment material, composed of a polyamic acid including azo-units in the main chain, was used as the alignment layer36. The azo-units respond to light by cis-trans isomerization. The layer aligns the director perpendicular to the linear polarization of the incident light in the substrate plane. The alignment material was irradiated with a 405 nm diode laser. The irradiated position was controlled by moving the substrate with translation stages. The polarization of the incident light was controlled by a linearly polarized light and a twisted nematic cell. The polarization was selected with the electric field in the LC cell matching the expected irradiated position. The other substrate was coated with a planar alignment material (AL-3046) and rubbed uniformly. The orientation of the two substrates was controlled to bring two regions with parallel and twist alignments. The ratio of the width (y-axis) of the twist region to that of the parallel region on the top substrate was 120:80 μm or 100:80 μm. The cell gap was 70 μm.
### Liquid crystal and micro-particles
The LC was 4-cyano-4′-pentylbiphenyl (5CB, from Merck), which exists in the nematic phase at room temperature. The density of 5CB is 1.01 g/cm3. The LC was mixed with a small amount of micro-particles. The micro-particles were made from polyethylene (GRYPMS, from Cospheric) and had a 10 ± 3 μm radius and a 1.0 g/cm3 density.
The micro-particles induced homeotropic anchoring and were accompanied by hedgehog or Saturn-ring defects in nematic phase35. We focused on particles in dipole configurations accompanied by hedgehog defects. The particles did not stick to either substrate due to both the small difference in density between the LC and the particles and the repulsive force between the particles and substrates11. The length of the pattern along the x-axis was sufficiently large, so that the alignment was assumed to be uniform.
### Image analysis
The difference in the motion of different particle pairs appears to stem from slight variations in the particle size and other factors, such as anchoring and surface conditions. The original data obtained from these experiments is evenly distributed over time. The particle separation changes very little during the initial stages of the experiments and, consequently, a large error is introduced in the particle-speed calculations. To overcome this issue, we slightly smoothed the raw data without obvious variation in data positions and interpolated the data to the evenly distributed particle separation data. The particle speed (y-axis data) was obtained by differentiating the particle separation (x-axis data) with respect to time in Fig. 3. In order to match the data with the lines of slope −2 and −4, we prepared the lines of slope −2 and −4 and adjusted the intercept of the Log(v) axis. | {"extraction_info": {"found_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": 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": 0.9064992666244507, "perplexity": 658.9049964005349}, "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/1664030335304.71/warc/CC-MAIN-20220929034214-20220929064214-00766.warc.gz"} |
http://geotech.chinaxiv.org/user/search.htm?pageId=1617899911534&type=filter&filterField=affication_str&value=Chinese%20Acad%20Sci,%20Inst%20Theoret%20Phys,%20State%20Key%20Lab%20Theoret%20Phys,%20Beijing%20100190,%20Peoples%20R%20China | Current Location:home > Browse
## 1. chinaXiv:201612.00088 [pdf]
Subjects: Geosciences >> Space Physics
We study a holographic model with vector condensate by coupling the anti-de Sitter gravity to an Abelian gauge field and a charged vector field in (3 + 1) dimensional spacetime. In this model there exists a non-minimal coupling of the vector field to the gauge field. We find that there is a critical temperature below which the charged vector condenses via a second order phase transition. The DC conductivity becomes infinite and the AC conductivity develops a gap in the condensed phase. We study the effect of a background magnetic field on the system. It is found that the background magnetic field can induce the condensate of the vector field even in the case without chemical potential/charge density. In the case with non-vanishing charge density, the transition temperature raises with the applied magnetic field, and the condensate of the charged vector operator forms a vortex lattice structure in the spatial directions perpendicular to the magnetic field.
## 2. chinaXiv:201612.00087 [pdf]
Subjects: Geosciences >> Space Physics
We continue to study the holographic p-wave superconductor model in the Einstein-Maxwell-complex vector field theory with a non-minimal coupling between the complex vector field and the Maxwell field. In this paper we work in the AdS soliton background which describes a conformal field theory in the confined phase and focus on the probe approximation. We find that an applied magnetic field can lead to the condensate of the vector field and the AdS soliton instability. As a result, a vortex lattice structure forms in the spatial directions perpendicular to the applied magnetic field. As a comparison, we also discuss the vector condensate in the Einstein-SU(2) Yang-Mills theory and find that in the setup of the present paper, the Einstein-Maxwell-complex vector field model is a generalization of the SU(2) model in the sense that the vector field has a general mass and gyromagnetic ratio.
## 3. chinaXiv:201612.00085 [pdf]
Subjects: Geosciences >> Space Physics
We study competition between s-wave order and d-wave order through two holographic superconductor models. We find that once the coexisting phase appears, it is always thermodynamically favored, and that the coexistence phase is narrow and one condensate tends to kill the other. The phase diagram is constructed for each model in terms of temperature and the ratio of charges of two orders. We further compare the behaviors of some thermodynamic quantities, and discuss the different aspects and identical ones between two models. | {"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": 0.9301972389221191, "perplexity": 444.1181486263415}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038072175.30/warc/CC-MAIN-20210413062409-20210413092409-00165.warc.gz"} |
https://www.physicsforums.com/threads/use-partial-fractions-to-find-the-sum-of-the-series.627751/ | # Use partial fractions to find the sum of the series
1. Aug 12, 2012
### JFUNR
1. The problem statement, all variables and given/known data
Use partial fractions to find the sum of the series: $\Sigma$n=1 to infinity $\frac{5}{n(n+1)(n+2}$
2. Relevant equations
Partial Fraction breakdown: $\Sigma$ $\frac{5}{2n}$+$\frac{5}{2(n+2)}$+$\frac{5}{(n+1)}$
3. The attempt at a solution
When I tried to cancel terms out, it is erratic and no pattern seems to emerge. It is not geometric so using a/(1-r) isn't possible. When I separated the terms, each diverges based on the nth term test. According to the all-knowing wolfram-alpha, the series converges to 5/4...I feel as if I am missing something big in my attempts at solving this one? any help?
2. Aug 12, 2012
### Ray Vickson
Your partial fraction expansion is incorrect.
RGV
3. Aug 12, 2012
### JFUNR
ah that was a typo, the partial fraction expansion I have in my attempts is...$\frac{5}{2n}$+$\frac{5}{2(n+2)}$-$\frac{5}{n+1}$
I still have the same issue..
4. Aug 12, 2012
### Zondrina
Edit : Whoops you had the right expansion I didn't see your post there.
There will be a pattern you will notice, but a more important question to ask yourself is : Suppose your sum is going from 1 to infinity. Lets say you're summing some A+B+C which are all composed of a variable n ( the variable you are summing of course ). If one of A, B, or C diverges, does the whole series diverge?
Last edited: Aug 12, 2012
5. Aug 12, 2012
### gabbagabbahey
Hint 1: $$\sum_{n=1}^{\infty} f(n) = \lim_{N \to \infty} \sum_{n=1}^{N} f(n)$$
Hint 2: $$\sum_{n=1}^{N} \frac{1}{n+2} = \left( \sum_{n=1}^{N+2} \frac{1}{n} \right) - \frac{1}{1} - \frac{1}{2}$$
6. Aug 12, 2012
### qbert
reindex second and third sums.
and notice you get a whole lot of cancelation.
7. Aug 13, 2012
### JFUNR
How do you mean?
8. Aug 13, 2012
### Ray Vickson
He means that you should write it out in detail and see what happens.
RGV
9. Aug 13, 2012
### JFUNR
by all means...enlighten me..where's the pattern........
($\frac{5}{2}$+$\frac{5}{6}$-$\frac{5}{2}$)+($\frac{5}{4}$+$\frac{5}{8}$-$\frac{5}{3}$)+($\frac{5}{6}$+$\frac{5}{10}$-$\frac{5}{4}$)+($\frac{5}{8}$+$\frac{5}{12}$-$\frac{5}{5}$)+($\frac{5}{10}$+$\frac{5}{14}$-$\frac{5}{6}$)+($\frac{5}{12}$+$\frac{5}{16}$-$\frac{5}{7}$)+($\frac{5}{14}$+$\frac{5}{18}$-$\frac{5}{8}$)+...+
while some terms do cancel, I do not see a patter out of the first 7 terms...but like I said...enlighten me
Last edited: Aug 13, 2012
10. Aug 13, 2012
### Ray Vickson
Do you really need someone to tell you that
$$\sum_{k=1}^N \frac{1}{2k} = \frac{1}{2} \sum_{k=1}^N \frac{1}{k}\;?$$
RGV
11. Aug 14, 2012
### gabbagabbahey
The pattern may indeed be a little difficult to see like this, but notice, for example, that $\frac{5}{6}+\frac{5}{6}-\frac{5}{3}=0$ and $\frac{5}{8}+\frac{5}{8}-\frac{5}{4}=0$.
To formally reorganize the sum so that the cancellations become obvious, first look at the finite sum and apply the second hint I gave you to both the 5/(2(n+2)) and -5(n+1) terms. After reorganizing the finite sum to get the desired cancellations, take the limit as N->infinity.
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http://www.phy.duke.edu/~rgb/Class/Electrodynamics/Electrodynamics/node71.html | Next: Wave Guides Assignment Up: Wave Guides Previous: Rectangular Waveguides Contents
# Resonant Cavities
We will consider a resonant cavity to be a waveguide of length with caps at both ends. As before, we must satisfy TE or TM boundary conditions on the cap surfaces, either Dirichlet in or Neumann in . In between, we expect to find harmonic standing waves instead of travelling waves.
Elementary arguments for presumed standing wave -dependence of:
(10.82)
such that the solution has nodes or antinodes at both ends lead one to conclude that only:
(10.83)
for are supported by the cavity. For TM modes must vanish on the caps because the nonzero field must be the only E field component sustained, hence:
(10.84)
For TE modes must vanish as the only permitted field component is a non-zero , hence:
(10.85)
Given these forms and the relations already derived for e.g. a rectangular cavity, one can easily find the formulae for the permitted transverse fields, e.g.:
(10.86) (10.87)
for TM fields and
(10.88) (10.89)
for TE fields, with determined as before for cavities.
However, now is doubly determined as a function of both and and as a function of and . The only frequencies that lead to acceptable solutions are ones where the two match, where the resonant in the direction corresponds to a permitted associated with a waveguide mode. | {"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": 0.9352468848228455, "perplexity": 1650.7328673346954}, "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-2014-23/segments/1405997877693.48/warc/CC-MAIN-20140722025757-00139-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://ma.mathforcollege.com/quiz-chapter-04-unary-operations/ | # Quiz Chapter 04: Unary Operations
MULTIPLE CHOICE TEST UNARY OPERATIONS
1. If the determinant of a $4 \times 4$ matrix $\left[ A \right]$ is given as $20$, then the determinant of $5 \left[ A \right]$ is
2. If the matrix product $\left[ A \right] \left[ B \right] \left[ C \right]$ is defined, then $\left( \left[ A \right] \left[ B \right] \left[ C \right] \right)^{T}$ is
3. The trace of a matrix $\begin{bmatrix} 5&6&-6 \\ 9&-11&13 \\ -17&19&23 \\ \end{bmatrix}$ is
4. A square $n \times n$ matrix $\left[ A \right]$ is symmetric if
5. The determinant of the matrix $\begin{bmatrix} 25&5&1 \\ 0&3&8 \\ 0&9&a \\ \end{bmatrix}$ is 50. The value of $a$ is then
6. $\left[ A \right]$ is a $5 \times 5$ matrix and a matrix $\left[ B \right]$ is obtained by the row operations of replacing Row $1$ with Row $3$, and then Row $3$ is replaced by a linear combination of $2 \times$ Row $3+4 \times$ Row $2$. If det$\left( A \right) = 17$, then det$\left( B \right)$ is equal to | {"extraction_info": {"found_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": 0, "wp-katex-eq": 50, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9511085152626038, "perplexity": 229.01864621916548}, "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/1652662521041.0/warc/CC-MAIN-20220518021247-20220518051247-00344.warc.gz"} |
http://spmaddmaths.onlinetuition.com.my/2014/08/length-of-arc-of-circle.html | # 8.2 Length of an Arc of a Circle
(A) Formulae for Length and Area of a Circle
r = radius A = area s = arc length q = angle l = length of chord
(B) Length of an Arc of a Circle
Example 1:
An arc, AB, of a circle of radius 5 cm subtends an angle of 1.5 radians at the centre. Find the length of the arc AB.
Solution:
s = rθ
Length of the arc AB = (5)(1.5) = 7.5 cm
Example 2:
An arc, PQ, of a circle of radius 12 cm subtends an angle of 30° at the centre. Find the length of the arc PQ.
Solution:
Length of the arc PQ
$\begin{array}{l}\end{array}$
$\begin{array}{l}\end{array}$
$\begin{array}{l}\end{array}$Example 3:
In the above diagram, find
(i) length of the minor arc AB
(ii) length of the major arc APB
Solution:
(i) length of the minor arc AB = rθ
= (7)(0.354)
= 2.478 cm
(ii) Since 360o = 2π radians, the reflex angle AOB
$\begin{array}{l}\end{array}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "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": 0.9500221610069275, "perplexity": 2178.0489047846795}, "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/1484560280310.48/warc/CC-MAIN-20170116095120-00276-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.nag.com/numeric/CL/nagdoc_cl24/html/E04/e04mxc.html | e04 Chapter Contents
e04 Chapter Introduction
NAG Library Manual
# NAG Library Function Documentnag_opt_miqp_mps_read (e04mxc)
## 1 Purpose
nag_opt_miqp_mps_read (e04mxc) reads data for sparse linear programming, mixed integer linear programming, quadratic programming or mixed integer quadratic programming problems from an external file which is in standard or compatible MPS input format.
## 2 Specification
#include #include
void nag_opt_miqp_mps_read (Nag_FileID fileid, Integer maxn, Integer maxm, Integer maxnnz, Integer maxncolh, Integer maxnnzh, Integer maxlintvar, Integer mpslst, Integer *n, Integer *m, Integer *nnz, Integer *ncolh, Integer *nnzh, Integer *lintvar, Integer *iobj, double a[], Integer irowa[], Integer iccola[], double bl[], double bu[], char pnames[][9], Integer *nname, char crname[][9], double h[], Integer irowh[], Integer iccolh[], Integer *minmax, Integer intvar[], NagError *fail)
## 3 Description
nag_opt_miqp_mps_read (e04mxc) reads data for linear programming (LP) or quadratic programming (QP) problems (or their mixed integer variants) from an external file which is prepared in standard or compatible MPS (see IBM (1971)) input format. It then initializes $n$ (the number of variables), $m$ (the number of general linear constraints), the $m$ by $n$ matrix $A$, the vectors $l$, $u$, $c$ (stored in row iobj of $A$) and the $n$ by $n$ Hessian matrix $H$ for use with nag_opt_sparse_convex_qp_solve (e04nqc). This function is designed to solve problems of the form
$minimize x cTx+12xTHx subject to l≤ x Ax ≤u.$
### 3.1 MPS input format
The input file of data may only contain two types of lines:
1 Indicator lines (specifying the type of data which is to follow). 2 Data lines (specifying the actual data).
A section is a combination of an indicator line and its corresponding data line(s). Any characters beyond column 80 are ignored. Indicator lines must not contain leading blank characters (in other words they must begin in column 1). The following displays the order in which the indicator lines must appear in the file:
NAME user-supplied name (optional) OBJSENSE (optional) data line OBJNAME (optional) data line ROWS data line(s) COLUMNS data line(s) RHS data line(s) RANGES (optional) data line(s) BOUNDS (optional) data line(s) QUADOBJ (optional) data line(s) ENDATA
A data line follows a fixed format, being made up of fields as defined below. The contents of the fields may have different significance depending upon the section of data in which they appear.
Field 1 Field 2 Field 3 Field 4 Field 5 Field 6 Columns $2–3$ $5–12$ $15–22$ $25–36$ $40–47$ $50–61$ Contents Code Name Name Value Name Value
Each name and code must consist of ‘printable’ characters only; names and codes supplied must match the case used in the following descriptions. Values are read using a field width of $12$. This allows values to be entered in several equivalent forms. For example, $1.2345678$, $\text{1.2345678e+0}$, $\text{123.45678e−2}$ and $\text{12345678e−07}$ all represent the same number. It is safest to include an explicit decimal point.
Lines with an asterisk ($*$) in column $1$ will be considered comment lines and will be ignored by the function.
Columns outside the six fields must be blank, except for columns 72–80, whose contents are ignored by the function. A non-blank character outside the predefined six fields and columns 72–80 is considered to be a major error (${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_MPS_ILLEGAL_DATA_LINE; see Section 6), unless it is part of a comment.
#### 3.1.1 NAME Section (optional)
The NAME section is the only section where the data must be on the same line as the indicator. The ‘user-supplied name’ must be in field $3$ but may be blank.
Field Required Description $3$ No Name of the problem
#### 3.1.2 OBJSENSE Section (optional)
The data line in this section can be used to specify the sense of the objective function. If this section is present it must contain only one data line. If the section is missing or empty, minimization is assumed.
Field Required Description $2$ No Sense of the objective function
Field 2 may contain either MIN, MAX, MINIMIZE or MAXIMIZE.
#### 3.1.3 OBJNAME Section (optional)
The data line in this section can be used to specify the name of a free row (see Section 3.1.4) that should be used as the objective function. If this section is present it must contain only one data line. If the section is missing or is empty, the first free row will be chosen instead. Alternatively, OBJNAME can be overridden by setting nonempty ${\mathbf{pnames}}\left[1\right]$ (see Section 5).
Field Required Description $2$ No Row name to be used as the objective function
Field 2 must contain a valid row name.
#### 3.1.4 ROWS Section
The data lines in this section specify unique row (constraint) names and their inequality types (i.e., unconstrained, $=$, $\ge$ or $\le$).
Field Required Description $1$ Yes Inequality key $2$ Yes Row name
The inequality key specifies each row's type. It must be E, G, L or N and can be in either column $2$ or $3$.
Inequality Key Description $\mathbit{l}$ $\mathbit{u}$ N Free row $-\infty$ $\infty$ G Greater than or equal to finite $\infty$ L Less than or equal to $-\infty$ finite E Equal to finite $l$
Row type N stands for ‘Not binding’. It can be used to define the objective row. The objective row is a free row that specifies the vector $c$ in the linear objective term ${c}^{\mathrm{T}}x$. If there is more than one free row, the first free row is chosen, unless another free row name is specified by OBJNAME (see Section 3.1.3) or ${\mathbf{pnames}}\left[1\right]$ (see Section 5). Note that $c$ is assumed to be zero if either the chosen row does not appear in the COLUMNS section (i.e., has no nonzero elements) or there are no free rows defined in the ROWS section.
#### 3.1.5 COLUMNS Section
Data lines in this section specify the names to be assigned to the variables (columns) in the general linear constraint matrix $A$, and define, in terms of column vectors, the actual values of the corresponding matrix elements.
Field Required Description $2$ Yes Column name $3$ Yes Row name $4$ Yes Value $5$ No Row name $6$ No Value
Each data line in the COLUMNS section defines the nonzero elements of $A$ or $c$. Any elements of $A$ or $c$ that are undefined are assumed to be zero. Nonzero elements of $A$ must be grouped by column, that is to say that all of the nonzero elements in the jth column of $A$ must be specified before those in the $\mathit{j}+1$th column, for $\mathit{j}=1,2,\dots ,n-1$. Rows may appear in any order within the column.
##### 3.1.5.1 Integer Markers
For backward compatibility nag_opt_miqp_mps_read (e04mxc) allows you to define the integer variables within the COLUMNS section using integer markers, although this is not recommended as markers can be treated differently by different MPS readers; you should instead define any integer variables in the BOUNDS section (see below). Each marker line must have the following format:
Field Required Description $2$ No Marker ID $3$ Yes Marker tag $5$ Yes Marker type
The marker tag must be ${}^{\prime }$MARKER${}^{\prime }$. The marker type must be ${}^{\prime }$INTORG${}^{\prime }$ to start reading integer variables and ${}^{\prime }$INTEND${}^{\prime }$ to finish reading integer variables. This implies that a row cannot be named ${}^{\prime }$MARKER${}^{\prime }$, ${}^{\prime }$INTORG${}^{\prime }$ or ${}^{\prime }$INTEND${}^{\prime }$. Please note that both marker tag and marker type comprise of $8$ characters as a ${}^{\prime }$ is the mandatory first and last character in the string. You may wish to have several integer marker sections within the COLUMNS section, in which case each marker section must begin with an ${}^{\prime }$INTORG${}^{\prime }$ marker and end with an ${}^{\prime }$INTEND${}^{\prime }$ marker and there should not be another marker between them.
Field 2 is ignored by nag_opt_miqp_mps_read (e04mxc). When an integer variable is declared it will keep its default bounds unless they are changed in the BOUNDS section. This may vary between different MPS readers.
#### 3.1.6 RHS Section
This section specifies the right-hand side values (if any) of the general linear constraint matrix $A$.
Field Required Description $2$ Yes RHS name $3$ Yes Row name $4$ Yes Value $5$ No Row name $6$ No Value
The MPS file may contain several RHS sets distinguished by RHS name. If an RHS name is defined in ${\mathbf{pnames}}\left[2\right]$ (see Section 5) then nag_opt_miqp_mps_read (e04mxc) will read in only that RHS vector, otherwise the first RHS set will be used.
Only the nonzero RHS elements need to be specified. Note that if an RHS is given to the objective function it will be ignored by nag_opt_miqp_mps_read (e04mxc). An RHS given to the objective function is dealt with differently by different MPS readers, therefore it is safer to not define an RHS of the objective function in your MPS file. Note that this section may be empty, in which case the RHS vector is assumed to be zero.
#### 3.1.7 RANGES Section (optional)
Ranges are used to modify the interpretation of constraints defined in the ROWS section (see Section 3.1.4) to the form $l\le Ax\le u$, where both $l$ and $u$ are finite. The range of the constraint is $r=u-l$.
Field Required Description $2$ Yes Range name $3$ Yes Row name $4$ Yes Value $5$ No Row name $6$ No Value
The range of each constraint implies an upper and lower bound dependent on the inequality key of each constraint, on the RHS $b$ of the constraint (as defined in the RHS section), and on the range $r$.
Inequality Key Sign of $\text{}\mathbit{r}$ $\mathbit{l}$ $\mathbit{u}$ E $+$ $b$ $b+r$ E $-$ $b+r$ $b$ G $+/-$ $b$ $b+\left|r\right|$ L $+/-$ $b-\left|r\right|$ $b$ N $+/-$ $-\infty$ $+\infty$
If a range name is defined in ${\mathbf{pnames}}\left[3\right]$ (see Section 5) then the function will read in only the range set of that name, otherwise the first set will be used.
#### 3.1.8 BOUNDS Section (optional)
These lines specify limits on the values of the variables (the quantities $l$ and $u$ in $l\le x\le u$). If a variable is not specified in the bound set then it is automatically assumed to lie between $0$ and $+\infty$.
Field Required Description $1$ Yes Bound type identifier $2$ Yes Bound name $3$ Yes Column name $4$ Yes/No Value
Note: field 4 is required only if the bound type identifier is one of UP, LO, FX, UI or LI in which case it gives the value $k$ below. If the bound type identifier is FR, MI, PL or BV, field 4 is ignored and it is recommended to leave it blank.
The table below describes the acceptable bound type identifiers and how each determines the variables' bounds.
Bound TypeIdentifier $\mathbit{l}$ $\mathbit{u}$ IntegerVariable? UP unchanged $k$ No LO $k$ unchanged No FX $k$ $k$ No FR $-\infty$ $\infty$ No MI $-\infty$ unchanged No PL unchanged $\infty$ No BV $0$ $1$ Yes UI unchanged $k$ Yes LI $k$ unchanged Yes
If a bound name is defined in ${\mathbf{pnames}}\left[4\right]$ (see Section 5) then the function will read in only the bound set of that name, otherwise the first set will be used.
The QUADOBJ section defines nonzero elements of the upper or lower triangle of the Hessian matrix $H$.
Field Required Description $2$ Yes Column name (HColumn Index) $3$ Yes Column name (HRow Index) $4$ Yes Value $5$ No Column name (HRow Index) $6$ No Value
Each data line in the QUADOBJ section defines one (or optionally two) nonzero elements ${H}_{ij}$ of the matrix $H$. Each element ${H}_{ij}$ is given as a triplet of row index $i$, column index $j$ and a value. The column names (as defined in the COLUMNS section) are used to link the names of the variables and the indices $i$ and $j$. More precisely, the matrix $H$ on output will have a nonzero element
$Hij = Value$
where index $j$ belongs to HColumn Index and index $i$ to one of the HRow Indices such that
• ${\mathbf{crname}}\left[j-1\right]=\text{Column name (HColumn Index)}$ and
• ${\mathbf{crname}}\left[i-1\right]=\text{Column name (HRow Index)}$.
It is only necessary to define either the upper or lower triangle of the $H$ matrix; either will suffice. Any elements that have been defined in the upper triangle of the matrix will be moved to the lower triangle of the matrix, then any repeated nonzeros will be summed.
Note: it is much more efficient for nag_opt_sparse_convex_qp_solve (e04nqc) to have the $H$ matrix defined by the first ncolh column names. If the nonzeros of $H$ are defined by any columns that are not in the first ncolh of n then nag_opt_miqp_mps_read (e04mxc) will rearrange the matrices $A$ and $H$ so that they are.
### 3.2 Query Mode
nag_opt_miqp_mps_read (e04mxc) offers a ‘query mode’ to quickly give upper estimates on the sizes of user arrays. In this mode any expensive checks of the data and of the file format are skipped, providing a prompt count of the number of variables, constraints and matrix nonzeros. This might be useful in the common case where the size of the problem is not known in advance.
You may activate query mode by setting any of the following: ${\mathbf{maxn}}<1$, ${\mathbf{maxm}}<1$, ${\mathbf{maxnnz}}<1$, ${\mathbf{maxncolh}}<0$ or ${\mathbf{maxnnzh}}<0$. If no major formatting error is detected in the data file, ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR is returned and the upper estimates are given as stated in Table 1. Alternatively, the function switches to query mode while the file is being read if it is discovered that the provided space is insufficient (that is, if ${\mathbf{n}}>{\mathbf{maxn}}$, ${\mathbf{m}}>{\mathbf{maxm}}$, ${\mathbf{nnz}}>{\mathbf{maxnnz}}$, ${\mathbf{ncolh}}>{\mathbf{maxncolh}}$, ${\mathbf{nnzh}}>{\mathbf{maxnnzh}}$ or ${\mathbf{lintvar}}>{\mathbf{maxlintvar}}$). In this case ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX is returned.
Argument Name Upper Estimate for n maxn m maxm nnz maxnnz ncolh maxncolh nnzh maxnnzh lintvar maxlintvar
Table 1
The recommended practice is shown in Section 10, where the function is invoked twice. The first call queries the array lengths required, after which the data arrays are allocated to be of these sizes. The second call reads the data using the sufficiently-sized arrays.
## 4 References
IBM (1971) MPSX – Mathematical programming system Program Number 5734 XM4 IBM Trade Corporation, New York
## 5 Arguments
1: fileidNag_FileIDInput
On entry: the ID of the MPSX data file to be read as returned by a call to nag_open_file (x04acc).
Constraint: ${\mathbf{fileid}}\ge 0$.
2: maxnIntegerInput
On entry: an upper limit for the number of variables in the problem.
If ${\mathbf{maxn}}<1$, nag_opt_miqp_mps_read (e04mxc) will start in query mode (see Section 3.2).
3: maxmIntegerInput
On entry: an upper limit for the number of general linear constraints (including the objective row) in the problem.
If ${\mathbf{maxm}}<1$, nag_opt_miqp_mps_read (e04mxc) will start in query mode (see Section 3.2).
4: maxnnzIntegerInput
On entry: an upper limit for the number of nonzeros (including the objective row) in the problem.
If ${\mathbf{maxnnz}}<1$, nag_opt_miqp_mps_read (e04mxc) will start in query mode (see Section 3.2).
5: maxncolhIntegerInput
On entry: an upper limit for the dimension of the matrix $H$.
If ${\mathbf{maxncolh}}<0$, nag_opt_miqp_mps_read (e04mxc) will start in query mode (see Section 3.2).
6: maxnnzhIntegerInput
On entry: an upper limit for the number of nonzeros of the matrix $H$.
If ${\mathbf{maxnnzh}}<0$, nag_opt_miqp_mps_read (e04mxc) will start in query mode (see Section 3.2).
7: maxlintvarIntegerInput
On entry: if ${\mathbf{maxlintvar}}\ge 0$, an upper limit for the number of integer variables.
If ${\mathbf{maxlintvar}}<0$, nag_opt_miqp_mps_read (e04mxc) will treat all integer variables in the file as continuous variables.
8: mpslstIntegerInput
On entry: if ${\mathbf{mpslst}}\ne 0$, summary messages are sent to stdout as nag_opt_miqp_mps_read (e04mxc) reads through the data file. This can be useful for debugging the file. If ${\mathbf{mpslst}}=0$, then no summary is produced.
9: nInteger *Output
On exit: if nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, an upper estimate of the number of variables of the problem. Otherwise, $n$, the actual number of variables in the problem.
10: mInteger *Output
On exit: if nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, an upper estimate of the number of general linear constraints in the problem (including the objective row). Otherwise $m$, the actual number of general linear constaints of the problem.
11: nnzInteger *Output
On exit: if nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, an upper estimate of the number of nonzeros in the problem (including the objective row). Otherwise the actual number of nonzeros in the problem (including the objective row).
12: ncolhInteger *Output
On exit: if nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, an upper estimate of the value of ncolh required by nag_opt_sparse_convex_qp_solve (e04nqc). In this context ncolh is the number of leading nonzero columns of the Hessian matrix $H$. Otherwise, the actual dimension of the matrix $H$.
13: nnzhInteger *Output
On exit: if nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, an upper estimate of the number of nonzeros of the matrix $H$. Otherwise, the actual number of nonzeros of the matrix $H$.
14: lintvarInteger *Output
On exit: if on entry ${\mathbf{maxlintvar}}<0$, all integer variables are treated as continuous and ${\mathbf{lintvar}}=-1$.
If nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, an upper estimate of the number of integer variables of the problem. Otherwise, the actual number of integer variables of the problem.
15: iobjInteger *Output
On exit: if ${\mathbf{iobj}}>0$, row iobj of $A$ is a free row containing the nonzero coefficients of the vector $c$.
If ${\mathbf{iobj}}=0$, the coefficients of $c$ are assumed to be zero.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2) iobj is not referenced and may be NULL.
16: a[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array a must be at least maxnnz when ${\mathbf{maxnnz}}>0$.
On exit: the nonzero elements of $A$, ordered by increasing column index.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), a is not referenced and may be NULL.
17: irowa[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array irowa must be at least maxnnz when ${\mathbf{maxnnz}}>0$.
On exit: the row indices of the nonzero elements stored in a.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), irowa is not referenced and may be NULL.
18: iccola[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array iccola must be at least ${\mathbf{maxn}}+1$ when ${\mathbf{maxn}}>0$.
On exit: a set of pointers to the beginning of each column of $A$. More precisely, ${\mathbf{iccola}}\left[\mathit{i}-1\right]$ contains the index in a of the start of the $\mathit{i}$th column, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$. Note that ${\mathbf{iccola}}\left[0\right]=1$ and ${\mathbf{iccola}}\left[{\mathbf{n}}\right]={\mathbf{nnz}}+1$.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), iccola is not referenced and may be NULL.
19: bl[$\mathit{dim}$]doubleOutput
20: bu[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the arrays bl and bu must be at least ${\mathbf{maxn}}+{\mathbf{maxm}}$ when ${\mathbf{maxn}}>0$ and ${\mathbf{maxm}}>0$.
On exit: bl contains the vector $l$ (the lower bounds) and bu contains the vector $u$ (the upper bounds), for all the variables and constraints in the following order. The first n elements of each array contains the bounds on the variables $x$ and the next m elements contains the bounds for the linear objective term ${c}^{\mathrm{T}}x$ and for the general linear constraints $Ax$ (if any). Note that an ‘infinite’ lower bound is indicated by ${\mathbf{bl}}\left[j-1\right]=-\text{1.0e+20}$ and an ‘infinite’ upper bound by ${\mathbf{bu}}\left[j-1\right]=+\text{1.0e+20}$. In other words, any element of $u$ greater than or equal to ${10}^{20}$ will be regarded as $+\infty$ (and similarly any element of $l$ less than or equal to $-{10}^{20}$ will be regarded as $-\infty$). If this value is deemed to be ‘inappropriate’, before calling nag_opt_sparse_convex_qp_solve (e04nqc) you are recommended to reset the value of its optional argument ${\mathbf{Infinite Bound Size}}$ and make any necessary changes to bl and/or bu.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), bl and bu are not referenced and may be NULL.
21: pnames[$5$][$9$]char Input/Output
On entry: a set of names associated with the MPSX form of the problem.
${\mathbf{pnames}}\left[0\right]$
Must either contain the name of the problem or be blank.
${\mathbf{pnames}}\left[1\right]$
Must either be blank or contain the name of the objective row (in which case it overrides the OBJNAME section and the default choice of the first objective free row).
${\mathbf{pnames}}\left[2\right]$
Must either contain the name of the RHS set to be used or be blank (in which case the first RHS set is used).
${\mathbf{pnames}}\left[3\right]$
Must either contain the name of the RANGE set to be used or be blank (in which case the first RANGE set (if any) is used).
${\mathbf{pnames}}\left[4\right]$
Must either contain the name of the BOUNDS set to be used or be blank (in which case the first BOUNDS set (if any) is used).
On exit: a set of names associated with the problem as defined in the MPSX data file as follows:
${\mathbf{pnames}}\left[0\right]$
Contains the name of the problem (or blank if none).
${\mathbf{pnames}}\left[1\right]$
Contains the name of the objective row (or blank if none).
${\mathbf{pnames}}\left[2\right]$
Contains the name of the RHS set (or blank if none).
${\mathbf{pnames}}\left[3\right]$
Contains the name of the RANGE set (or blank if none).
${\mathbf{pnames}}\left[4\right]$
Contains the name of the BOUNDS set (or blank if none).
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), pnames is not referenced and may be NULL.
22: nnameInteger *Output
On exit: $n+m$, the total number of variables and constraints in the problem (including the objective row).
If nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, nname is not set. In the former case you may pass NULL instead.
23: crname[$\mathit{dim}$]char Output
Note: the dimension, dim, of the array crname must be at least ${\mathbf{maxn}}+{\mathbf{maxm}}$ when ${\mathbf{maxn}}>0$ and ${\mathbf{maxm}}>0$.
On exit: the MPS names of all the variables and constraints in the problem in the following order. The first n elements contain the MPS names for the variables and the next m elements contain the MPS names for the objective row and general linear constraints (if any). Note that the MPS name for the objective row is stored in ${\mathbf{crname}}\left[{\mathbf{n}}+{\mathbf{iobj}}-1\right]$.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), crname is not referenced and may be NULL.
24: h[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array h must be at least maxnnzh when ${\mathbf{maxnnzh}}>0$.
On exit: the nnzh nonzero elements of $H$, arranged by increasing column index.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), h is not referenced and may be NULL.
25: irowh[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array irowh must be at least maxnnzh when ${\mathbf{maxnnzh}}>0$.
On exit: the nnzh row indices of the elements stored in $H$.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), irowh is not referenced and may be NULL.
26: iccolh[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array iccolh must be at least ${\mathbf{maxncolh}}+1$ when ${\mathbf{maxncolh}}>0$.
On exit: a set of pointers to the beginning of each column of $H$. More precisely, ${\mathbf{iccolh}}\left[\mathit{i}-1\right]$ contains the index in $H$ of the start of the $\mathit{i}$th column, for $\mathit{i}=1,2,\dots ,{\mathbf{ncolh}}$. Note that ${\mathbf{iccolh}}\left[0\right]=1$ and ${\mathbf{iccolh}}\left[{\mathbf{ncolh}}\right]={\mathbf{nnzh}}+1$.
If nag_opt_miqp_mps_read (e04mxc) is run in query mode (see Section 3.2), iccolh is not referenced and may be NULL.
27: minmaxInteger *Output
On exit: minmax defines the direction of the optimization as read from the MPS file. By default the function assumes the objective function should be minimized and will return ${\mathbf{minmax}}=-1$. If the function discovers in the OBJSENSE section that the objective function should be maximized it will return ${\mathbf{minmax}}=1$. If the function discovers that there is neither the linear objective term $c$ (the objective row) nor the Hessian matrix $H$, the problem is considered as a feasible point problem and ${\mathbf{minmax}}=0$ is returned.
If nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, minmax is not set. In the former case you may pass NULL instead.
28: intvar[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array intvar must be at least maxlintvar, when ${\mathbf{maxlintvar}}>0$.
On exit: if ${\mathbf{maxlintvar}}>0$ on entry, intvar contains pointers to the columns that are defined as integer variables. More precisely, ${\mathbf{intvar}}\left[\mathit{i}-1\right]=k$, where $k$ is the index of a column that is defined as an integer variable, for $\mathit{i}=1,2,\dots ,{\mathbf{lintvar}}$.
If ${\mathbf{maxlintvar}}\le 0$ on entry, or nag_opt_miqp_mps_read (e04mxc) was run in query mode (see Section 3.2), or it returned with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_MAX, intvar is not set. Excepting the latter case you may pass NULL as this argument instead.
29: failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
Note that if any of the relevant arguments are accidentally set to zero, or not set and assume zero values, then the function will have executed in query mode. In this case only the size of the problem is returned and other arguments are not set. See Section 3.2.
## 6 Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_FILEID
On entry, ${\mathbf{fileid}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{fileid}}\ge 0$.
NE_INT_MAX
At least one of maxm, maxn, maxnnz, maxnnzh, maxncolh or maxlintvar is too small. Suggested values are returned in m, n, nnz, nnzh, ncolh and lintvar respectively.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_MPS_BOUNDS
The supplied name, in ${\mathbf{pnames}}\left[4\right]$, of the BOUNDS set to be used was not found in the BOUNDS section.
Unknown bound type ‘$⟨\mathit{\text{value}}⟩$’ in BOUNDS section.
NE_MPS_COLUMNS
Column ‘$⟨\mathit{\text{value}}⟩$’ has been defined more than once in the COLUMNS section. Column definitions must be continuous. (See Section 3.1.5).
Unknown column name ‘$⟨\mathit{\text{value}}⟩$’ in $⟨\mathit{\text{value}}⟩$ section.
All column names must be specified in the COLUMNS section.
NE_MPS_ENDATA_NOT_FOUND
End of file found before ENDATA indicator line.
NE_MPS_FORMAT
Warning: MPS file not strictly fixed format, although the problem was read anyway. The data may have been read incorrectly. You should set ${\mathbf{mpslst}}=1$ and repeat the call to nag_opt_miqp_mps_read (e04mxc) for more details.
NE_MPS_ILLEGAL_DATA_LINE
An illegal line was detected in ‘$⟨\mathit{\text{value}}⟩$’ section.
This is neither a comment nor a valid data line.
NE_MPS_ILLEGAL_NUMBER
Field $⟨\mathit{\text{value}}⟩$ did not contain a number (see Section 3).
NE_MPS_INDICATOR
Incorrect ordering of indicator lines.
BOUNDS indicator line found before COLUMNS indicator line.
Incorrect ordering of indicator lines.
COLUMNS indicator line found before ROWS indicator line.
Incorrect ordering of indicator lines.
OBJNAME indicator line found after ROWS indicator line.
Incorrect ordering of indicator lines.
QUADOBJ indicator line found before BOUNDS indicator line.
Incorrect ordering of indicator lines.
QUADOBJ indicator line found before COLUMNS indicator line.
Incorrect ordering of indicator lines.
RANGES indicator line found before RHS indicator line.
Incorrect ordering of indicator lines.
RHS indicator line found before COLUMNS indicator line.
Indicator line ‘$⟨\mathit{\text{value}}⟩$’ has been found more than once in the MPS file.
No indicator line found in file. It may be an empty file.
Unknown indicator line ‘$⟨\mathit{\text{value}}⟩$’.
NE_MPS_INVALID_INTORG_INTEND
Found ‘INTEND’ marker without previous marker being ‘INTORG’.
Found ‘INTORG’ but not ‘INTEND’ before the end of the COLUMNS section.
Found ‘INTORG’ marker within ‘INTORG’ to ‘INTEND’ range.
Illegal marker type ‘$⟨\mathit{\text{value}}⟩$’.
Should be either ‘INTORG’ or ‘INTEND’.
NE_MPS_MANDATORY
NE_MPS_OBJNAME
The supplied name, in ${\mathbf{pnames}}\left[1\right]$ or in OBJNAME, of the objective row was not found among the free rows in the ROWS section.
NE_MPS_PRINTABLE
Illegal column name.
Column names must consist of printable characters only.
Illegal row name.
Row names must consist of printable characters only.
NE_MPS_RANGES
The supplied name, in ${\mathbf{pnames}}\left[3\right]$, of the RANGES set to be used was not found in the RANGES section.
NE_MPS_REPEAT_COLUMN
More than one nonzero of a has row name ‘$⟨\mathit{\text{value}}⟩$’ and column name ‘$⟨\mathit{\text{value}}⟩$’ in the COLUMNS section.
NE_MPS_REPEAT_ROW
Row name ‘$⟨\mathit{\text{value}}⟩$’ has been defined more than once in the ROWS section.
NE_MPS_RHS
The supplied name, in ${\mathbf{pnames}}\left[2\right]$, of the RHS set to be used was not found in the RHS section.
NE_MPS_ROWS
Unknown inequality key ‘$⟨\mathit{\text{value}}⟩$’ in ROWS section.
Expected ‘N’, ‘G’, ‘L’ or ‘E’.
Unknown row name ‘$⟨\mathit{\text{value}}⟩$’ in $⟨\mathit{\text{value}}⟩$ section.
All row names must be specified in the ROWS section.
NE_MPS_ROWS_OR_CONS
Empty ROWS section.
Neither the objective row nor the constraints were defined.
Not applicable.
## 8 Parallelism and Performance
nag_opt_miqp_mps_read (e04mxc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
None.
## 10 Example
This example solves the quadratic programming problem
$minimize cT x + 12 xT H x subject to l ≤Ax ≤u, -2 ≤Ax ≤2,$
where
$c= -4.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -0.1 -0.3 , H= 2 1 1 1 1 0 0 0 0 1 2 1 1 1 0 0 0 0 1 1 2 1 1 0 0 0 0 1 1 1 2 1 0 0 0 0 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ,$
$A= 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 4.0 1.0 2.0 3.0 4.0 -2.0 1.0 1.0 1.0 1.0 1.0 -1.0 1.0 -1.0 1.0 1.0 1.0 1.0 1.0 ,$
$l= -2.0 -2.0 -2.0 and u= 1.5 1.5 4.0 .$
The optimal solution (to five figures) is
$x*=2.0,-0.23333,-0.26667,-0.3,-0.1,2.0,2.0,-1.7777,-0.45555T.$
Three bound constraints and two general linear constraints are active at the solution. Note that, although the Hessian matrix is only positive semidefinite, the point ${x}^{*}$ is unique.
The MPS representation of the problem is given in Section 10.2.
### 10.1 Program Text
Program Text (e04mxce.c)
### 10.2 Program Data
Program Options (e04mxce.opt)
### 10.3 Program Results
Program Results (e04mxce.r) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 326, "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": 0.9277108907699585, "perplexity": 1742.8225367101318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719677.59/warc/CC-MAIN-20161020183839-00507-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/431440/sequence-of-sequences | # sequence of sequences
i have the following question:
let $(a_{j})$ be a sequence in $l^{2}$ and $(b_{j}^{n})$ be a sequence of sequences in $l^{2}$ such that $(a_{j}b_{j}^{n})$ is an $l^{2}$ sequence for each $n$. Suppose further that $(b_{j}^{n})$ converges to a sequence $(b_{j})$ in $l^{2}$ and also that $(a_{j}b_{j}^{n})$ converges in $l^{2}$. I would like to prove that $(a_{j}b_{j}^{n})$ actually converges to $(a_{j}b_{j})$. This is what I have done:
$$\sum\limits_{j=1}^{\infty}| a_{j}b_{j} - a_{j}b_{j}^{n} |^{2} = \sum|a_{j}|^{2}|b_{j}-b_{j}^{n}|^{2} \le \sqrt{\sum |a_{j}|^{4}}\sqrt{\sum_{j=1}|b_{j}-b_{j}^{n}|^{4}}$$
where the last inequality follows by Cauchy-Schwartz inequality applied to the sequences $|a_{j}|^{2}$ and $|b_{j}-b_{j}^{n}|^{2}$ and using the fact that $l^{2} \supseteq l^{4}$.
Now as $(b_{j}^{n})$ converges to $(b_{j})$ we can make this term as small as we like.
However, I am not sure whether I am correct in all this analysis. Please point out the mistakes if any in the above.
Thanks for all the help.
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Your proof is correct, but the problem is much simpler and does not require so many assumptions. Consider that
• Convergence in $\ell^2$ norm implies coordinate-wise convergence
• Coordinate-wise limit is unique, if it exists
So, if you have $x^n\to x$ coordinate-wise and $x^n\to y$ in $l^2$ norm, it follows that $x=y$, and therefore $x^n\to x$ in $l^2$ norm.
In your case, $a_jb_j^n \to a_j b_j$ coordinate-wise. You also know that $a_jb_j^n$ converges to something in $l^2$. The conclusion is that something is $(a_jb_j)$.
-
Thanks! Sorry I cannot upvote. – Maverick Jun 30 '13 at 6:48
@Maverick But you can (if you wish) mark the answer as accepted by clicking the checkmark to the left of the answer. See Why should we accept answers? – ˈjuː.zɚ79365 Jun 30 '13 at 7:05
Now I can vote up and did so. And also accepted. Is there any other form of etiquette that I am expected to follow? – Maverick Jul 2 '13 at 5:27
@Maverick Hm, maybe using proper capitalization in your questions. :) Try it next time. – ˈjuː.zɚ79365 Jul 2 '13 at 5:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9745796322822571, "perplexity": 246.3234716924252}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802765616.69/warc/CC-MAIN-20141217075245-00015-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/18138/an-intuition-to-an-inclusion-union-of-intersections-vs-intersection-of-union | An Intuition to An Inclusion: “Union of Intersections” vs “Intersection of Unions”
Let $E = \{E_k\}_{k \in \mathbb{N}}$ be an infinite sequence of sets. Then, the following inclusion holds:
$\bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \quad\subseteq\quad \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k$
I know the left-hand side (LHS) represents elements which belong to all but finitely many sets in the sequence $E$, and the right-hand side (RHS) represents elements which belong to infinitely many sets in the sequence $E$. This concludes the proof that the LHS is contained in the RHS. Yet it is not intuitive for me, since the above interpretation is not intuitive per se.
Is there an intuitive reason why the above inclusion holds? That is, (very informally) why union of intersections is contained in intersection of unions?
-
If $A$ and $B$ are sets, and $A\subseteq B_n$ for all $n$, then $A\subseteq\bigcap_n B_n$. Conversely, $A\subseteq\bigcap_n B_n$ means that $A\subseteq B_n$ for all $n$.
This tells us that the inclusion you are interested in reduces then to checking $(+)$: $$\bigcup_n \bigcap_{k\ge n}E_k\subseteq \bigcup_{k\ge m}E_k$$ for all $m$.
(This is progress. It is similar to a common move in analysis: To prove that $a\le b$, it is not unusual to check instead that $a\le c$ for any $c>b$. But these inequalities $a\le c$ tend to be easier than the one you really want, $a\le b$.)
Now, the same idea gives us that to prove the new inclusion $(+)$, it is enough to prove $(++)$: $$\bigcap_{k\ge n}E_k \subseteq \bigcup_{k\ge m}E_k$$ for all $n,m$.
(The corresponding move in analysis is that to prove $a\le b$, it is enough to show $c\le b$ for all $c\le a$. Unions correspond to suprema, intersections to infima in these analogies.)
Now, $(++)$ is really obvious: If somebody is in the left hand side, then it is in all sufficiently large $E_k$, but then it is in the right hand side.
The point is: It is useful to train oneself to "decode" certain expressions the way we did, by focusing on their "atomic" or at least "more basic" components.
-
Excellent! Just one point: the inclusion (+) is true for "all m", rather than "for all n,m". Am I right? – Sadeq Dousti Jan 19 '11 at 15:37
Right, thanks. Edited. – Andres Caicedo Jan 19 '11 at 15:38
As you said the LHS is the collection of elements which appear in all but finitely many $E_k$, while the elements on the RHS are those which appear in infinitely many $E_k$.
If you appear in all but finitely many, then you appear in infinitely many; the converse is not always true as you can appear in all the even indices but not once in the odd indices.
The LHS is called sometimes denoted by $\liminf E$, and the RHS is $\limsup E$.
-
I understand this, but as I pointed out, this interpretation is not intuitive per se. – Sadeq Dousti Jan 19 '11 at 15:27
@Sadeq: In mathematics many times intuition is built after a while, and cannot be explained in simple terms. I think this is very much the case here. – Asaf Karagila Jan 19 '11 at 15:31
So on the left side, an element must belong to every $E_k$ starting from some $k$. On the right side, an element only has to belong to sets that are arbitrarily large. So if it belongs to every other one, or every prime number one, or something like that, it will be included in the right side.
-
There are good explanations of liminf and limsup of sets here. You will probably see these constructions in measure theory.
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https://eventuallyalmosteverywhere.wordpress.com/tag/poisson-point-process/ | Real Trees – Root Growth and Regrafting
Two weeks ago in our reading group meeting, Raphael told us about Chapter Five which introduces root growth and regrafting. One of the points of establishing the Gromov-Hausdorff topology in this book was to provide a more natural setting for a discussion of tree-valued processes. Indeed in what follows, one can imagine how to start the construction of a similar process for the excursions which can be used to encode real trees, involving cutting off sub-excursions above one-sided local minima, then glueing them back in elsewhere. But taking account of the equivalence structure will be challenging, and it is much nicer to be able to describe cutting a tree in two by removing a single point without having to worry about quotient maps.
We have seen in Chapter Two an example of a process defined on the family of rooted trees with n labelled vertices which has the uniform rooted tree as an invariant distribution. Given a rooted tree with root p, we choose uniformly at random a vertex p’ in [n] to be the new root. Then if p’=p we do nothing, otherwise we remove the unique first edge in the path from p’ to p, giving two trees. Adding an edge from p to p’ completes the step and gives a new tree with p’ as root. We might want to take a metric limit of these processes as n grows and see whether we end up with a stationary real tree-valued process whose marginals are the BCRT.
To see non-trivial limiting behaviour, it is most interesting to consider the evolution of a particular subtree (which includes the root) through this process. If the vertex chosen for cutting lies in our observed subtree, then the subtree undergoes a prune and regraft operation. On the other hand, if the vertex chosen for cutting does not lie in the subtree, then we do not see any effect of the pruning, except the addition of a new vertex below the original root, which becomes the new root. So essentially, from the point of view of our observed subtree, the root is growing.
Now we can think about interpreting the dynamics of a natural limit process acting on real trees. The key idea is that we don’t change the set on which the tree is supported much, but instead just change the metric. In particular, we will keep the original tree, and add on length at unit rate. Of course, where this length gets added on entirely determines the metric structure of the tree, but that doesn’t stop us giving a simple ‘name’ for the extra length. If we consider a process $X^T$ starting from a particular finite subtree T, then at time t, the tree $X^T_t$ is has vertex set $T \coprod (0,t]$. (Finite subtree here means that it has finite total length.)
Root regrafting should happen at a rate proportional to the total length of the current observed tree. This is reasonable since after all it is supported within a larger tree, so in the discrete case the probability of a prune-regrafting event happening within a given observed subtree is proportional to the number of vertices in that subtree, which scales naturally as length in the real tree limit. It turns out that to get unit rate root growth with $\Theta(1)$ rate prune-regrafting, we should consider subtrees of size $\sqrt{n}$ within a host tree of size n as $n\rightarrow\infty$. We also rescale the lengths by $\frac{1}{\sqrt{n}}$, and time by $\sqrt{n}$ so we actually see prune-regraft events.
Furthermore, if the subtree is pruned, the location of the pruning is chosen uniformly by length of the current observed subtree. So we can view the pruning process as being driven by a Poisson point process with intensity given by the instantaneous length measure of the tree, which at time t has vertex set $T\coprod (0,t]$. It will turn out to be consistent that there is a ‘piecewise isometry’ for want of a better phrase between the metric (and thus length measure) on $X^T_t$ and the canonical induced measure on $T\coprod (0,t]$, so we can describe the instances and locations of the pruning events via a pair of PPPs. The first is supported on $T \times [0,\infty)$, and the second on $\{(t,x): 0 \le x \le t\}$, since we only ‘notice’ pruning at the point labelled x if the pruning happens at some time t after x was created.
If we start from a compact tree T, then the total intensity of this pair is finite up to some time t, and so we have a countable sequence $\tau_0=0<\tau_1<\tau_2<\ldots$ of times for pruning events. It is easy to describe (but a bit messy to notate) the evolution of the metric between these pruning times. Essentially the distance between any pair of points in the observed tree at time $\tau_m$ with root $\rho_{\tau_m}$ is constant between times $\tau_m,\tau_{m+1}$, and new points are added so that the distance between $\rho_{\tau_m}$ and any new point $a\in(\tau_m,\tau_{m+1}]$ is $a-\tau_m$, and everything thing else follows from straightforward consideration of geodesics.
When a pruning event happens at point $x_m$ at time $\tau_m$, distances are preserved within the subtree above $x_m$ in $X^T_{\tau_m -}$, and within the rest of the tree. Again, an expression for the cross distances is straightforward but requires a volume of notation not ideally suited to this medium.
The natural thing to consider is the coupled processes started from different subtrees (again both must contain the original root) of the same host tree. Say $T^1,T^2\le T$, then it is relatively easy to check that $X^{T^1}_t,X^{T^2}_t \le X^T_t \,\forall t$, when we drive the processes by consistent coupled Poisson processes. Furthermore, it is genuinely obvious that the Hausdorff distance between $X^{T^1}_t,X^{T^2}_t$, here viewed as compact subsets of $(X^T_t, d^T_t)$ remains constant during root growth phase.
Less obvious but more important is that the Hausdorff distance decreases during regrafting events. Suppose that just before a regrafting event, the two subtrees are T’ and T”, and the Hausdorff distance between them is $\epsilon$. This Hausdorff distance is with respect to the metric on the whole tree T. [Actually this is a mild abuse of notation – I’m now taking T to be the whole tree just before the regraft, rather than the tree at time 0.]
So for any $a\in T'$, we can choose $b\in T''$ such that $d_T(a,b)\le \epsilon$. This is preserved under the regraft unless the pruning point lies on the geodesic segment (in T) between a and b. But in that case, the distance between a and the pruning point is again at most $\epsilon$, and so after the regrafting, a is at most $\epsilon$ away from the new root, which is in both subtrees, and in particular the regrafted version of T”.
This is obviously a useful first step on the path to proving any kind of convergence result. There are some technicalities which we have skipped over. It is fairly natural that this leads to a Markov process when the original tree is finite, but it is less clear how to define these dynamics when the total tree length is infinite, as we don’t want regrafting events to be happening continuously unless we can bound their net effect in some sense.
Last week, Franz showed us how to introduce the BCRT into matters. Specifically, that BCRT is the unique stationary distribution for this process. After a bit more work, the previous result says that for convergence properties it doesn’t matter too much what tree we start from, so it is fine to start from a single point. Then, the cut points and growth mechanism corresponds very well to the Poisson line-breaking construction of the BCRT. With another ‘grand coupling’ we can indeed construct them simultaneously. Furthermore, we can show weak convergence of the discrete-world Markov chain tree algorithm to the process with these RG with RG dynamics.
It does seem slightly counter-intuitive that a process defined on the whole of the discrete tree converges to a process defined through subtrees. Evans remarks in the introduction to the chapter that this is a consequence of having limits described as compact real trees. Then limitingly almost all vertices are close to leaves, so in a Hausdorff sense, considering only $\sqrt{n}$ of the vertices (ie a subtree) doesn’t really make any difference after rescaling edge lengths. I feel I don’t understand exactly why it’s ok to take the limits in this order, but I can see why this might work after more checking.
Tomorrow, we will have our last session, probably discussing subtree prune-and-regraft, where the regrafting does not necessarily happen at the root.
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Poisson Random Measures
[This is a companion to the previous post. They explore different aspects of the same problem which I have been thinking about from a research point of view. So that they can be read independently, there has inevitably been some overlap.]
As I explained in passing previously, Poisson Random Measures have come up in my current research project. Indeed, the context where they have appeared seems like a very good motivation for considering the construction and some properties of PRMs.
We begin not with a Poisson variable, but with a standard Erdos-Renyi random graph $G(n,\frac{c}{n})$. The local limit of a component in this random graph is given by a Galton-Watson branching process with Poisson(c) offspring distribution. Recall that a local limit is description of what the structure looks like near a given (or random) vertex. Since the vertices in G(n,p) are exchangeable, this rooting matters less. Anyway, the number of neighbours in the graph of our root is given by Bin(n-1,c/n). Suppose that the root v_0, has k neighbours. Then if we are just interested in determining the vertices in the component, we can ignore the possibility of further edges between these neighbours. So if we pick one of the neighbours of the root, say v_1, and count the number of neighbours of this vertex that we haven’t already considered, this is distributed as Bin(n-1-k,c/n), since we discount the root and the k neighbours of the root.
Then, as n grows large, Bin(n-1,c/n) converges in distribution to Po(c). Except on a very unlikely event whose probability we can control if we need, so does Bin(n-1-k,c/n). Indeed if we consider a set of K vertices which are already connected in some way, then the distribution of the number of neighbours of one of them which we haven’t already considered is still Po(c) in the limit.
Now we consider what happens if we declare the graph to be inhomogeneous. The simplest possible way to achieve this is to specify two types of vertices, say type A and type B. Then we specify the proportion of vertices of each type, and the probability that there is an edge between two vertices of given types. This is best given by a symmetric matrix. So for example, if we wanted a random bipartite graph, we could achieve this as described by setting all the diagonal entries of the matrix to be zero.
So does the local limit extend to this setting? Yes, unsurprisingly it does. To be concrete, let’s say that the proportion of types A and B are a and b respectively, and the probabilities of having edges between vertices of various types is given by $P=(p_{ij}/n)_{i,j\in\{A,B\}}$. So we can proceed exactly as before, only now we have to count how many type A neighbours and how many type B neighbours we see at all stages. We have to specify the type of our starting vertex. Suppose for now that it is type A. Then the number of type A neighbours is distributed as
$\text{Bin}(an,p_{AA}/n)\stackrel{d}{\rightarrow}\text{Po}(ap_{AA})$,
and similarly the limiting number of type B neighbours is $\sim \text{Po}(bp_{AB})$. Crucially, this is independent of the number of type A neighbours. The argument extends naturally to later generations, and the result is exactly a multitype Galton-Watson process as defined in the previous post.
My motivating model is the forest fire. Here, components get burned when they are large and reduced to singletons. It is therefore natural to talk about the ‘age’ of a vertex, that is, how long has elapsed since it was last burned. If we are interested in the forest fire process at some fixed time T>1, that is, once burning has started, then we can describe it as an inhomogeneous random graph, given that we know the ages of the vertices.
For, given two vertices with ages s and t, where WLOG s<t, we know that the older vertex could not have been joined to the other vertex between times T-t and T-s. Why? Well, if it had, then it too would have been burned at time T-s when the other vertex was burned. So the only possibility is that they might have been joined by an edge between times T-s and T. Since each edge arrives at rate 1/n, the probability that this happens is $1-e^{-s/n}\approx \frac{s}{n}$. Indeed, in general the probability that two vertices of ages s and t are joined at time T is $\frac{s\wedge t}{n}$.
Again at fixed time T>1, the sequence of ages of the vertices converges weakly to some fixed distribution (which depends on T) as the number of vertices grows to infinity. We can then recover the graph structure by assigning ages according to this distribution, then growing the inhomogeneous random graph with the kernel as described. The question is: when we look for a local limit, how to do we describe the offspring distribution?
Note that in the limit, components will be burned continuously, so the distribution of possible ages is continuous (with an atom at T for those vertices which have never been burned). So if we try to calculate the distribution of the number of neighbours of age s, we are going to be doomed, because with probability 1 then is no vertex of age s anywhere!
The answer is that the offspring distribution is given by a Poisson Random Measure. You can think of this as a Poisson Point Process where the intensity is non-constant. For example, let us consider how many neighbours we expect to have with ages [s,s+ds]. Let us suppose the age of our root is t>s+ds for now. Assuming the distribution of ages, $f(\cdot)$ is positive and continuous, the number of vertices with these ages in the system is roughly nf(s)ds, and so the number of neighbours with this property is roughly $\text{Bin}(nf(s)ds,\frac{s}{n})$. In particular, this does have a Poisson limit. We need to be careful about whether this Poisson limit is preserved by the approximation. In fact this is fine. Let’s assume WLOG that f is increasing at s. Then the number of age [s,s+ds] neighbours can be stochastically bounded between $\text{Bin}(nf(s)ds,\frac{s}{n})$ and $\text{Bin}(nf(s+ds)ds,\frac{s+ds}{n}$. As n grows, these converge in the distribution to two Poisson random variables, and then we can let ds go to zero. Note for full formalism, we may need to account for the large deviations event that the number of age s vertices in the system is noticeably different from its expectation. Whether this is necessary depends on whether the ages are assigning deterministically, or drawn IID-ly from f.
One important result to be drawn from this example is that the number of offspring from disjoint type sets, say $[s_1,s_2], [t_1,t_2]$ are independent, for the same reason as in the two-type setting, namely that the underlying binomial variables are independent. We are, after all, testing different sets of vertices! The other is that the number of neighbours with ages in some range is Poisson. Notice that these two results are consistent. The number of neighbours with ages in the set $[s_1,s_2]\cup [t_1,t_2]$ is given by the sum of two independent Poisson RVs, and hence is Poisson itself. The parameter of the sum RV is given by the sum of the original parameters.
These are essentially all the ingredients required for the definition of a Poisson Random Measure. Note that the set of offspring is a measure of the space of ages, or types. (Obviously, this isn’t a probability measure.) We take a general space E, with sigma algebra $\mathcal{E}$, and an underlying measure $\mu$ on E. We want a distribution $\nu$ for measures on E, such that for each Borel set $A\in\mathcal{E}$, $\nu(A)$, which is random because $\nu$ is, is distributed as $\text{Po}(\mu(A))$, and furthermore, for disjoint $A,B\in\mathcal{E}$, the random variables $\nu(A),\nu(B)$ are independent.
If $M=\mu(E)<\infty$, then constructing such a random measure is not too hard using a thinning property. We know that $\nu(E)\stackrel{d}{=}\text{Po}(M)$, and so if we sample a Poisson(M) number of RVs with distribution given by $\frac{\mu(\cdot)}{M}$, we get precisely the desired PRM. Proving this is the unique distribution with this property is best done using properties of the Laplace transform, which uniquely defines the law of a random measure in the same manner that the moment generating function defines the law of a random variable. Here the argument is a function, rather than a single variable for the MGF, reflecting the fact that the space of measures is a lot ‘bigger’ than the reals, where a random variable is supported. We can extend this construction for sigma-finite spaces, that is some countable union of finite spaces.
One nice result about Poisson random measures concerns the expectation of functions evaluated at such a random measure. Recall that some function f evaluated at the measure $\sum \delta_{x_i}$ is given by $\sum f(x_i)$. Then, subject to mild conditions on f, the expectation
$\mathbb{E}\nu (f)=\mu(f).$
Note that when $f=1_A$, this is precisely one of the definitions of the PRM. So by a monotone class result, it is not surprising that this holds more generally. Anyway, I’m currently trying to use results like these to get some control over what the structure of this branching processes look like, even when the type space is continuous as in the random graph with specified ages.
Characterisations of Geometric Random Graphs
Continuing the LMS-EPSRC summer school on Random Graphs, Geometry and Asymptotic Structure, we’ve now had three of the five lectures by Mathew Penrose on Geometric Random Graphs.
The basic idea is that instead of viewing a graph entirely abstractly, we now place the vertices in the plane, or some other real space. In many network situations, we would expect connectivity to depend somehow on distance. Agents or sites which are close together might be considered more likely to have the sort of relationship indicated by being connected with an edge. In the model discussed in this course, this dependence is deterministic. We have some parameter r, and once we have chosen the location of all the vertices, we connect a pair of vertices if the distance between them is less than r.
For the purposes of this, we work in a compact space [0,1]^d, and we are interested in the limit as the number of vertices n grows to infinity. To avoid the graph getting too connected, as in the standard random graph model, we take r to be a decreasing function of n. Anyway, we place the n points into the unit hypercube uniformly at random, and then the edges are specified by the adjacency rule above. In general, because r_n will be o(1), we won’t have to worry too much above boundary effects. The number of vertices within r_n of the boundary of the cube will be o(1). For some results, this is a genuine problem, when it may be easier to work on the torus.
In G(n,p), the order of np in the limit determines the qualitative structure of the graph. This is the expected degree of a given fixed vertex. In this geometric model, the relevant parameter is $nr_n^d$, where d is the dimension of the hypercube. If this parameter tends to 0, we say the graph is sparse, and dense if it tends to infinity. The intermediate case is called a thermodynamic limit. Note that the definition of sparse here is slightly different from G(n,p).
Much of the content of the first three lectures has been verifying that the distributions of various quantities in the graph, for example the total number of edges, are asymptotically Poisson. Although sometimes arguments are applicable over a broad spectrum, we also sometimes have to use different calculations for different scaling windows. For example, it is possible to show convergence to a Poisson distribution for the number of edges in the sparse case, from which we get an asymptotic normal approximation almost for free. In the denser regimes, the argument is somewhat more technical, with some substantial moment calculations.
A useful tool in these calculations are some bounds derived via Stein’s method for sums of ‘almost independent’ random variables. For example, the presence or non-presence of an edge between two pairs of vertices are independent in this setting if the pairs are disjoint, and the dependence is still only mild if they share a vertex. An effective description is via a so-called dependency graph, where we view the random variables as the vertices of a graph, with an edge between them if there is some dependence. This description doesn’t have any power in itself, but it does provide a concise notation for what would otherwise be very complicated, and we are able to show versions of (Binomials converge to Poisson) and CLT via these that are exactly as required for this purpose.
In particular, we are able to show that if $E_n$ is the total number of edges, under a broad set of scaling regimes, if $\lambda_n$ is the expected total number of edges, then $d_{TV}(E_n,\mathrm{Po}(\lambda_n))\rightarrow 0$, as n grows. This convergence in total variation distance is as strong a result as one could hope for, and when the sequence of $\lambda_n$ is O(1), we can derive a normal approximation as well.
At this point it is worth discussing an alternative specification of the model. Recall that for a standard homogenous random graph, we have the choice of G(n,m) and G(n,p) as definitions. G(n,m) is the finer measure, and G(n,p) can be viewed as a weighted mix of G(n,m). We can’t replicate this directly in the geometric setting because the edges and non-edges are a deterministic function of the vertex locations. What we can randomise is the number of vertices. Since we are placing the vertices uniformly at random, it makes sense to consider as an alternative a Poisson Point Process with intensity n. The number of vertices we get overall will be distributed as Po(n), which is concentrated near n, in the same manner as G(n,c/n).
As in G(n,p), this is a less basic model because it is a mixture of the fixed-vertex models. Let’s see if how we would go about extending the total variation convergence result to this slightly different setting without requiring a more general version of the Poisson Approximation Lemma. To avoid having to define everything again, we add a ‘ to indicate that we are talking about the Poisson Point Process case. Writing d(.,.) for total variation distance, the result we have is:
$\lim_{n\rightarrow\infty} d(E_n,\mathrm{Po}(\lambda_n))=0.$
We want to show that
$\lim_{n\rightarrow\infty}d(E_n',\mathrm{Po}(\lambda_n'))=0,$
which we can decompose in terms of expectations in the original model by conditioning on $N_n$
$\leq \lim_{n\rightarrow\infty}\mathbb{E}\Big[\mathbb{E}[d(E_{N_n},\mathrm{Po}(\lambda_n')) | N_n]\Big],$
where the outer expectation is over N. The observation here, is that the number of points given by the Poisson process induces a measure on distributions, the overwhelming majority of which look quite like Poisson distributions with parameter n. The reason we have a less than sign is that we are applying the triangle inequality in the sum giving total variation distance:
$d(X,Y)=\sum_{k\geq 0}|\mathbb{P}(X=k)-\mathbb{P}(Y=k)|.$
From this, we use the triangle inequality again:
$\lim_{n\rightarrow\infty} \mathbb{E}\Big[\mathbb{E}[d(E_{N_n},\mathrm{Po}(\lambda_{N_n})) | N_n]\Big]$
$+\lim_{n\rightarrow\infty}\mathbb{E}\Big[\mathbb{E}[d(\mathrm{Po}(\lambda_{N_n}),\mathrm{Po}(\lambda_n')) | N_n]\Big].$
Then, by a large deviations argument, we have that for any $\epsilon>0$, $\mathbb{P}(|N_n-n|\geq \epsilon n)\rightarrow 0$ exponentially in n. Also, total variation distance is, by definition, bounded above by 1. In the first term, the inner conditioning on N_n is irrelevant, and we have that $E_{N_n}$ converges to the Poisson distribution for any fixed $N_n\in (n(1-\epsilon),n(1+\epsilon))$. Furthermore, we showed in the proof of the non-PPP result that this convergence is uniform in this interval. (This is not surprising – the upper bound is some well-behaved polynomial in 1/n.) So with probability $1- e^{-\Theta(n)}$ N_n is in the region where this convergence happens, and elsewhere, the expected TV distance is bounded below 1, so the overall expectation tends to 0. With a similar LD argument, for the second term it suffices to prove that when $\lambda\rightarrow\mu$, we must have $d(\mathrm{Po}(\lambda),\mathrm{Po}(\mu))\rightarrow 0$. This is ‘obviously’ true. Formally, it is probably easiest to couple the distributions $\mathrm{Bin}(n,\lambda/n),\mathrm{Bin}(n,\mu/n)$ in the obvious way, and carry the convergence of TV distance as the parameter varies through the convergence in n.
That all sounded a little bit painful, but is really just the obvious thing to do with each term – it’s only the language that’s long-winded!
Anyway, I’m looking forward to seeing how the course develops. In particular, when you split the space into small blocks, the connectivity properties resemble those of (site) percolation, so I wonder whether there will be concrete parallels. Also, after reading about some recent results concerning the metric structure of the critical components in the standard random graph process, it will be interesting to see how these compare to the limit of a random graph process which comes equipped with metric structure for free!
Random Interlacements
In this post, I want to talk about another recently-introduced model that’s generating a lot of interest in probability theory, Sznitman’s model of random interlacements. We also want to see, at least heuristically, how this relates to more familiar models.
We fix our attention on a lattice, which we assume to be $\mathbb Z ^d$. We are interested in the union of an infinite collection of simple random walks on the lattice. The most sensible thing to consider is not a collection of random walks from at a random set of starting points, but rather a family of trajectories, that is a doubly-infinite random walk defined on times $(-\infty,\infty)$. We will want this family to have some obvious properties, such as translation invariance, in order to make analysis possible and ideally obtain some 0-1 laws. The natural thing to do is then to choose the trajectories through a Poisson Point Process. The tricky part will be finding an intensity measure that has all the properties we want, and gives trajectories that genuinely do look like SRWs, and, most importantly, have a union that is neither too sparse nor too dense. For example, it wouldn’t be very interesting if with high probability every point appeared in the union…
For reasons we will mention shortly, we are interested in the complement of the union of the trajectories. We call this the vacant set. We will find an intensity which we can freely scale by some parameter $u\in\mathbb{R}^+$, which will give us a threshold for the complement to contain an infinite component. This is in the same sense as the phase transition for Bernoulli percolation. That is, there is a critical value $u^*$ say, such that for $u the vacant set contains an infinite component (or percolates) almost surely, and almost surely it does not when $u>u^*$. A later result of Teixeira shows that, as in percolation, this infinite component is unique.
Let us first recall why it is not interesting to consider this process for d=1 or 2. On $\mathbb{Z}$, with high probability a single SRW hits every integer point trivially, since it visits arbitrarily large and arbitrarily small integers. For d=2, the SRW is recurrent, and so consists of a countably infinite sequence of excursions from (0,0). Note that the probability that an excursion from 0 hits some point (x,y) is non-zero, as it is at least $2^{-2(|x|+|y|)}$ for example. Therefore, with high probability the SRW hits (x,y), and so whp it hits every point.
Therefore it is only for d=>3 that we start seeing interesting effects. It is worth mentioning at this point some of the problems that motivated considering this model. First is the disconnection time of a discrete cylinder by a simple walk. For example, Sznitman considers the random walk on $\mathbb{Z}\times (\mathbb{Z}/N\mathbb{Z})^d$. Obviously, it is more interesting to consider how long it takes a (1-dimensional in the natural sense) path to disconnect a d=>3 dimensional set than a 2-dimensional one, as the latter is given just by the first time the path self-intersects.
More generally, we might be interested in random walks up to some time an order of magnitude smaller than the cover time. Recall the cover time is the time to hit each point of the set. For example, for the random walk on the d-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ the cover time (as discussed in Markov Chains and Mixing Times posts) is $N^d \log N$, but the log N represents in some sense only the ‘final few’ vertices. So we should ask what the set of unhit vertices looks like at time $N^d$. And it turns out that for large N, the structure of this vacant set is related to the vacant set in the random interlacement model, in a local sense.
Anyway, the main question to ask is: what should the intensity measure be?
We patch it together locally. Start with the observation that transience of the random walk means almost surely a trajectory spends only finitely many steps in a fixed finite set K. So we index all the trajectories which hit K by the first time they hit K. Given that a trajectory hits K, it is clear what the conditional distribution of this hitting point should be. Recall that SRW on Z^d is reversible, so we consider the SRW backwards from this hitting then. Then the probability that the hitting point is x (on the boundary of K) is proportional to the probability that a SRW started from x goes to infinity without hitting K again. So once we’ve settled on the distribution of the hitting point x, it is clear how to construct all the trajectories through K. We pick x on the boundary of K according to this distribution, and take the union of an SRW starting from x conditioned not to hit K again, and an SRW starting from x with no conditioning. These correspond to the trajectory before and after the hitting time, respectively.
In fact, it turns out that this is enough. Suppose we demand that the probability that the hitting point is x is equal to the probability that a SRW started from x goes to infinity without hitting K again (rather than merely proportional to). Sznitman proves that there is a unique measure on the set of trajectories that restricts to this measure for every choice of K. Furthermore, the Poisson Point Process with the globally-defined intensity, unsurprisingly restricts to a PPP with the intensity specific to K.
We have not so far said anything about trajectories which miss this set K. Note that under any sensible intensity with the translation-invariance property, the intensity measure of the trajectories which hit K must be positive, since we can cover $\mathbb{Z}^d$ with countably many copies of K. So the number of trajectories hitting K is a Poisson random variable.
Recall how we defined the probability that the hitting point of K was some point x on the boundary. The sum of these probability is called the capacity of K. It follows that this is the parameter of the Poisson random variable. Ie, the probability that no trajectory passes through K is:
$\exp(-u\mathrm{cap}(K)),$
recalling that u is the free parameter in the intensity. This is the most convenient framework through which to start analysing the probability that there is an infinite connected set which is hit by no trajectory.
We conclude by summarising Sznitman’s Remark 1.2, explaining why it is preferable to work with the space of trajectories rather than the space of paths. Note that if we are working with paths, and we want translation invariance, then this restricts to translation invariance of the distribution of starting points as well, so it is in fact a stronger condition. Note then that either the intensity of starting at 0 is zero, in which case there are no trajectories at all, or it is positive, in which case the set of starting points looks like Bernoulli site percolation.
However, the results about capacity would still hold if there were a measure that restricted satisfactorily. And so the capacity of K would still be the measure of paths hitting K, which would be at least the probability that the path was started in K. But by translation invariance, this grows linearly with |K|. But capacity grows at most as fast as the size of the set of boundary points of K, which will be an order of magnitude smaller when K is, for example, a large ball.
REFERENCES
This was mainly based on
Sznitman – Vacant Set of Random Interlacements and Percolation (0704.2560)
Also
Sznitman – Random Walks on Discrete Cylinders and Random Interlacements (0805.4516)
Teixeira – On the Uniqueness of the Infinite Cluster of the Vacant Set of Random Interlacements (0805.4106)
and some useful slides by the same author (teixeira.pdf)
Brownian Excursions and Local Time
I’ve been spending a fair bit of time this week reading and thinking about the limits of various combinatorial objects, in particular letting the number of vertices tend to $\infty$ in models of random graphs with various constraints. Perhaps predictably, like so many continuous stochastic objects, yet again the limiting ‘things’ turn out to be closely linked to Brownian Motion. As a result, I’ve ended up reading a bit about the notion of local time, and thought it was sufficiently elegant even by itself to justify a quick post.
Local Time
In general, we might be interested in calculating a stochastic integral like
$\int_0^t f(B_s)ds$.
Note that, except in some highly non-interesting cases, this is a random variable. Our high school understanding of Riemannian integration encourages thinking of this as a ‘pathwise’ integral along the path evolving in time. But of course, that’s orthogonal to the approach we start thinking about when we are introduced to the Lebesgue integral. There we think about potential values of the integrand, and weight their contribution by the (Lebesgue) measure of the subset of the domain in which they appear.
Can we do the same for the stochastic integral? That is, can we find a measure which records how long the Brownian Motion spends at a point x? This measure will not be deterministic – effectively the stochastic behaviour of BM will be encoded through the measure rather than the argument of the function.
The answer is yes, and the measure in question is referred to as local time. More formally, we want
$\int_0^t f(B_s)ds=\int_\mathbb{R}f(x)L(t,x)dx.$ (*)
where the local time L(t,x) is a random process, increasing for fixed x. Informally, one could take
$\partial_t L(t,x) \propto 1(B_t=x)$
but clearly in practice that won’t do at all for a definition, and so instead we use (*). In the usual way, if we want (*) to hold for all reasonably nice functions f, it suffices to check it for the indicator functions of Borel sets. L(t,.) is therefore often referred to as occupation density, while L(.,A) is local time.
Local Time as natural index for Excursions
An excursion, for example of Brownian Motion, is a segment of the path that has zero value only at its endpoints. Alternatively, it is a maximal open interval of time such that the path is away from 0. We want to specify the measure on these excursions. Here are some obvious difficulties.
By Blumenthal’s 0-1 law, BM started from zero hits zero infinitely often in any time interval [0,e], so in the same way that there is no first positive rational, there is no first excursion. We could pick the excursion occurring in progress at a fixed time t, but this is little better. Firstly, the resultant measure is size-biased by the length of the excursion, and more importantly, the proximity of t to the origin may be significant unless we know of some memorylessness type of property to excursions.
Local time allows us to solve these problems. We restrict attention to $L_t:=L(t,0)$, the occupation density of 0. Let’s think about some advantages of indexing excursions by local time rather than by the start time:
• The key observation is that local time remains constant on excursions. That is, if we are avoiding 0, the local time at 0 cannot grow because the BM spends no time there!
• If we use start time, then we have a countably infinite number of small excursions accumulating close to 0, ie with very small start time. However, local time increases rapidly when there are lots of small excursions. Remember, lots of small excursions means that the BM hits 0 lots of times. So local time grows quickly through the annoying bits, and effectively provides a size-biasing for excursions that allows us to ignore the effects of the ‘Blumenthal excursions’ near time 0.
• When indexed by time, excursions might be Markovian, in the sense that subsequent excursions (and in particular their lengths) are independent of past excursions.This is certainly not the case if you index by start time! If an excursion starts at time t and has length u, then the ‘next’ excursions, in as much as that makes sense, must surely start at time t+u.
We know there are only countably many excursions, hence there are only countably many local times which pertain to an excursion. This motivates considering the set of excursions as a Poisson Point Process on local time. Once you’ve had this idea, everything follows quite nicely. Working out the distribution of the constant rate (which is a measure on the set of excursions) remains, but essentially we now have a sensible framework for tracking the process of excursions, and from this we can reconstruct the original Brownian Motion.
SLE Revision 4: The Gaussian Free Field and SLE4
I couldn’t resist breaking the order of my revision notes in order that the title might be self-referential. Anyway, it’s the night before my exam on Conformal Invariance and Randomness, and I’m practising writing this in case of an essay question about the Gaussian Free Field and its relation to the SLE objects discussed in the course.
What is a Gaussian Free Field?
The most natural definition is too technical for this context. Instead, recall that we could very informally consider a Poisson random measure to have the form of a series of Poisson random variables placed at each point in the domain, weighted infinitissimely so that the integrals over an area give a Poisson random variable with mean proportional to the measure of the area, and so that different areas are independent. Here we do a similar thing only for infinitesimal centred Gaussians. We have to specify the covariance structure.
We define the Green’s function on a domain D, which has a resonance with PDE theory, by:
$G_D(x,y)=\lim_{\epsilon\rightarrow 0}\mathbb{E}[\text{time spent in }B(y,\epsilon)\text{ by BM started at }x\text{ stopped at }T_D]$
We want the covariance structure of the hypothetical infinitesimal Gaussians to be given by $\mathbb{E}(g(x)g(y))=G_D(x,y)$. So formally, we define $(\Gamma(A),A\subset D)$ for open A, by $(\Gamma(A_1),\ldots,\Gamma(A_n))$ a centred Gaussian RV with covariance $\mathbb{E}(\Gamma(A_1)\Gamma(A_2))=\int_{A_1\times A_2}dxdyG_D(x,y)$.
The good news is that we have a nice expression $G_U(0,x)=\log\frac{1}{|x|}$, and the Green’s functions are conformally invariant in the sense that $G_{\phi(D)}(\phi(x),\phi(y))=G_D(x,y)$, following directly for conformality of Brownian Motion.
The bad news is that the existence is not clear. The motivation for this is the following though. We have a so-called excursion measure for BMs in a domain D. There isn’t time to discuss this now: it is infinite, and invariant under translations of the boundary (assuming the boundary is $\mathbb{R}\subset \bar{\mathbb{H}}$, which is fine after taking a conformal map). Then take a Poisson Point Process on the set of Brownian excursions with this measure. Now define a function f on the boundary of the domain dD, and define $\Gamma_f(A)$ to be the sum of the values of f at the starting point of BMs in the PPP passing through A, weighted by the time spent in A. We have a universality relation given by the central limit theorem: if we define h to be (in a point limit) the expected value of this variable, and we take n independent copies, we have:
$\frac{1}{\sqrt{n}}\left['\Gamma_f^1(A)+\ldots+\Gamma_f^n(A)-n\int_Ah(x)dx\right]\rightarrow\Gamma(A)$
where this limiting random variable is Gaussian.
For now though, we assume existence without full proof.
SLE_4
We consider chordal SLE_k, which has the form of a curve $\gamma[0,\infty]$ from 0 to $\infty$ in H. The g_t the regularising function as normal, consider $\tilde{X}_t=X_t-W_t:=g_t(x)-\sqrt{\kappa}\beta_t$ for some fixed x. We are interested in the evolution of the function arg x. Note that conditional on the (almost sure for K<=4) event that x does not lie on the curve, arg x will converge either to 0 or pi almost surely, depending on whether the curve passes to the left or the right (respectively) of x.
By Loewner’s DE for the upper half-plane and Ito’s formula:
$d\bar{X}_t=\sqrt{\kappa}d\beta_t,\quad d\log\bar{X}_t=(2-\frac{\kappa}{2})\frac{dt}{\bar{X}_t^2}+\frac{\sqrt{\kappa}}{\bar{X}_t}d\beta_t$
So, when K=4, the dt terms vanish, which gives that log X is a local martingale, and so
$d\theta_t=\Im(\frac{2}{\bar{X}_t}d\beta_t$
is a true martingale since it is bounded. Note that
$\theta_t=\mathbb{E}[\pi1(x\text{ on right of }\gamma)|\mathcal{F}_t]$
Note that also:
$\mathbb{P}(\text{BM started at }x\text{ hits }\gamma[0,t]\cup\mathbb{R}\text{ to the right of }\gamma(t)|\gamma[0,t])=\frac{\theta_t}{\pi}$ also.
SLE_4 and the Gaussian Free Field on H
We will show that this chordal SLE_4 induces a conformal Markov type of property in Gaussian Free Fields constructed on the slit-domain. Precisely, we will show that if $\Gamma_T$ is a GFF on $H_T=\mathbb{H}\backslash\gamma[0,T]$, then $\Gamma_T+ch_T(\cdot)=\Gamma_0+ch_0(\cdot)$, where c is a constant to be determined, and $h_t(x)=\theta_t(x)$ in keeping with the lecturer’s notation!
It will suffice to check that for all fixed p with compact support $\Gamma_T(p)+c(h_T(p)-h_0(p))$ is a centred Gaussian with variance $\int dxdyG_H(x,y)p(x)p(y)$.
First, applying Ito and conformal invariance of the Green’s functions under the maps g_t,
$dG_{H_t}(x,y)=cd[h(x),h(y)]_t$
The details are not particularly illuminating, but exploit the fact that Green’s function on H has a reasonable nice form $\log\left|\frac{x-\bar{y}}{x-y}\right|$. We are also being extremely lax with constants, but we have plenty of freedom there.
After applying Ito and some (for now unjustified) Fubini:
$dh_t(p)=\left(\int c.p(x)\Im(\frac{1}{\bar{X}_t})dx\right)d\beta_t$
and so as we would have suspected (since h(x) was), this is a local martingale. We now deploy Dubins-Schwarz:
$h_T(p)-h_T(0)\stackrel{d}{=}B_{\sigma(T)}$ for B an independent BM and
$\sigma(T)=\int_0^Tdt(\int c.p(x)\Im(\frac{1}{\tilde{X}_t})dx)^2$
So conditional on $(h_T(p),t\in[0,T])$, we want to make up the difference to $\Gamma_0$. Add to $h_T(p)-h_0(p)$ an independent random variable distribution as $N(0,s-\sigma(T))$, where
$s=\int dxdyp(x)p(y)G(x,y)\quad =:\Gamma_0(p)$
Then
$s-\sigma(T)=\int p(x)p(y)[G(x,y)-c\int_0^Tdt\Im(\frac{1}{X_t})\Im(\frac{1}{Y_t})]dxdy=\int p(x)p(y)G_t(x,y)dxdy$ as desired.
Why is this important?
This is important, or at least interesting, because we can use it to reverse engineer the SLE. Informally, we let $T\rightarrow\infty$ in the previous result. This states that taking a GFF in the domain left by removing the whole of the SLE curve (whatever that means) then adding $\pi$ at points on the left of the curve, which is the limit $\lim_T h_T$ is the same as a normal GFF on the upper half plane added to the argument function. It is reasonable to conjecture that a GFF in a non-connected domain has the same structure as taking independent GFFs in each component, and this gives an interesting invariance condition on GFFs. It can also be observed (Schramm-Sheffield) that SLE_4 arises by reversing the argument – take an appropriate conditioned GFF on H and look for the interface between it being ‘large’ and ‘small’ (Obviously this is a ludicrous simplification). This interface is then, under a suitable limit, SLE_4. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 139, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9371703267097473, "perplexity": 268.17384699350265}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202003.56/warc/CC-MAIN-20190319163636-20190319185636-00506.warc.gz"} |
https://physicsoverflow.org/39366/why-ignore-surface-terms-in-the-action-quantum-field-theory | # Why do we ignore surface terms in the action in Quantum Field theory ?
+ 2 like - 0 dislike
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I have trouble understanding why do we ignore surface terms. To get the on-shell fields we fix the fields at infinity so these terms do not matter. However , In the path integral , We integrate over all field configurations. So we must integrate over all values of the fields at infinity. So , one shouldn't set terms like $e^{i\oint dn^{\mu}\psi_{\mu}}$ equal to one and they should contribute to the partition function. Is this correct ?
In the field theory textbook I use , The path integral is written as $\int D\phi e^{iS}$ So I'm confused. The total derivative terms are evaluated at the boundary of spacetime by gauss theorem. Right ? do we keep the fields at spatial & temporal infinity fixed ? I'm confused by this point.
If you remember, the path integral can be introduced in calculation of the amplitude $\langle x(t_1)|x(t_2)\rangle$, $t_2 > t_1$. So the initial and the final "positions" (states) are considered fixed, known. It is written in many textbooks.
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https://scholars.bgu.ac.il/display/n151569 | # On interference among moving sensors and related problems Academic Article
•
• Overview
•
• We show that for any set of $n$ moving points in $\Re^ d$ and any parameter $2\le k\le n$, one can select a fixed non-empty subset of the points of size $O (k\log k)$, such that the Voronoi diagram of this subset isbalanced''at any given time (ie, it contains $O (n/k)$ points per cell). We also show that the bound $O (k\log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As an application, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time, their interference is $O (\sqrt {n\log n})$. This is optimal up to an $O (\sqrt {\log n})$ factor. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments. | {"extraction_info": {"found_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": 0.8847606182098389, "perplexity": 219.54309594724015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376824338.6/warc/CC-MAIN-20181213010653-20181213032153-00221.warc.gz"} |
http://link.springer.com/chapter/10.1007%2F978-3-662-48653-5_2 | International Symposium on Distributed Computing
Distributed Computing pp 16-30
Efficient Counting with Optimal Resilience
Conference paper
DOI: 10.1007/978-3-662-48653-5_2
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9363)
Cite this paper as:
Lenzen C., Rybicki J. (2015) Efficient Counting with Optimal Resilience. In: Moses Y. (eds) Distributed Computing. Lecture Notes in Computer Science, vol 9363. Springer, Berlin, Heidelberg
Abstract
In the synchronous c-counting problem, we are given a synchronous system of n nodes, where up to f of the nodes may be Byzantine, that is, have arbitrary faulty behaviour. The task is to have all of the correct nodes count modulo c in unison in a self-stabilising manner: regardless of the initial state of the system and the faulty nodes’ behavior, eventually rounds are consistently labelled by a counter modulo c at all correct nodes.
We provide a deterministic solution with resilience $$f<n/3$$ that stabilises in O(f) rounds and every correct node broadcasts $$O(\log ^2 f)$$ bits per round. We build and improve on a recent result offering stabilisation time O(f) and communication complexity $$O(\log ^2 f /\log \log f)$$ but with sub-optimal resilience $$f = n^{1-o(1)}$$ (PODC 2015). Our new algorithm has optimal resilience, asymptotically optimal stabilisation time, and low communication complexity. Finally, we modify the algorithm to guarantee that after stabilisation very little communication occurs. In particular, for optimal resilience and polynomial counter size $$c=n^{O(1)}$$, the algorithm broadcasts only O(1) bits per node every $$\Theta (n)$$ rounds without affecting the other properties of the algorithm; communication-wise this is asymptotically optimal. | {"extraction_info": {"found_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": 0.8477144241333008, "perplexity": 2117.7198982022946}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125849.25/warc/CC-MAIN-20170423031205-00115-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/basketball-player-darrell-griffith-record-attaining-standing-vertical-jump-12m-4ft--means--q974491 | Basketball player Darrell Griffith is on record as attaining a standing vertical jump of 1.2m (4ft ). (This means that he moved upward by 1.2m after his feet left the floor.) Griffith weighed 890 N(200 lb).
a) What is his speed as he leaves the floor?
b) If the time of the part of the jump before his feet left the floor was 0.300s , what was the magnitude of his average acceleration while he was pushing against the floor?
c)What is its direction?
upward
downward
d) Use Newton's laws and the results of part (B) to calculate the average force he applied to the ground.
Get this answer with Chegg Study
Practice with similar questions
Q:
Basketball player Darrell Griffith is on record as attaining a standing vertical jump of 1.2 m. (This means that he moved upward by 1.2 m after his feet left the floor. Griffith weighed 890 N.A-What is his speed as he leaves the floor?B-If the time of the part of the jump before his feet left the floor was 0.300 s , what was the magnitude of his average acceleration while he was pushing against the floor?C-Use Newton's laws and the results of part (B) to calculate the average force he applied to the ground. | {"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": 0.986725926399231, "perplexity": 1394.4778135870552}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276759.73/warc/CC-MAIN-20160524002116-00063-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathhelpforum.com/math-topics/182290-deriving-rl-circuit-integration-factor.html | # Thread: Deriving RL circuit (integration factor)
1. ## Deriving RL circuit (integration factor)
I'm having trouble with this, here's my attempt:
$V=V_{R}+V_{L}$
$= IR + L \frac{dI(t)}{dt}$
Now introduce f(t) for the integratino factor, divide by L to get the first order on its own
$f(t)\frac{V}{L} = f(t)\frac{IR}{L} + f(t)\frac{dI(t)}{dt}$
$f '(t) = \frac{Rf(t)}{L}$
$f(t) = e^{\int \frac{R}{L} dt}$
So far this seems somewhat logical, as the final answer has the minus factor in it.
Continuing:
$e^{\frac{Rt}{L}}\frac{V}{L} = \frac{d}{dt} [e^{\frac{Rt}{L}} I]$
Integrating this gets me back to
$V = IR$
as theres nothing i can do with the exponentials
The answer i should be getting is:
$I = I_{0} [1-e^{-\frac{Rt}{L}}]$
Hope you can help me out
2. Originally Posted by imagemania
I'm having trouble with this, here's my attempt:
$V=V_{R}+V_{L}$
$= IR + L \frac{dI(t)}{dt}$
Now introduce f(t) for the integratino factor, divide by L to get the first order on its own
$f(t)\frac{V}{L} = f(t)\frac{IR}{L} + f(t)\frac{dI(t)}{dt}$
$f '(t) = \frac{Rf(t)}{L}$
$f(t) = e^{\int \frac{R}{L} dt}$
So far this seems somewhat logical, as the final answer has the minus factor in it.
Continuing:
$e^{\frac{Rt}{L}}\frac{V}{L} = \frac{d}{dt} [e^{\frac{Rt}{L}} I]$
Integrating this gets me back to
$V = IR$
as theres nothing i can do with the exponentials
The answer i should be getting is:
$I = I_{0} [1-e^{-\frac{Rt}{L}}]$
Hope you can help me out
What about the constant of integration? You have completely ignored it.
It would help if the original question was posted. Assuming a series RL circuit and assuming V does not depend on time:
5. Application of ODEs: Series RL Circuit
3. It's not a question it's just something I am trying to derive.
As for the constant, the first integral should have a constant (though generally in first order integrating factor questions it's saved till the end). In this case it woudl still cancle anyway from what i can see.
As for the second integral, it would be between 0 and t (i.e. no constant).
So following that:
i would get
$\frac{V}{L}e^{\frac{Rt}{L}+c} = \frac{d}{dt}[e^{\frac{Rt}{L}+c}I]$
$\frac{V}{R}e^{\frac{Rt}{L}+c} = Ie^{\frac{Rt}{L}+c}$
4. Originally Posted by imagemania
It's not a question it's just something I am trying to derive.
As for the constant, the first integral should have a constant (though generally in first order integrating factor questions it's saved till the end). In this case it woudl still cancle anyway from what i can see.
As for the second integral, it would be between 0 and t (i.e. no constant).
So following that:
i would get
$\frac{V}{L}e^{\frac{Rt}{L}+c} = \frac{d}{dt}[e^{\frac{Rt}{L}+c}I]$
$\frac{V}{R}e^{\frac{Rt}{L}+c} = Ie^{\frac{Rt}{L}+c}$
Mr. F was not talking about the constant for that integration.
$V = IR + L \frac{dI}{dt}$
$\frac{d}{dt} \left ( e^{(R/L)t}I \right ) = e^{(R/L)t}$
$e^{(R/L)t}I + C = \frac{V}{R}e^{(R/L)t}$
$I = -Ce^{-(R/L)t} + \frac{V}{R}$ | {"extraction_info": {"found_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": 24, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9361684918403625, "perplexity": 1417.1665679388361}, "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/1368697843948/warc/CC-MAIN-20130516095043-00027-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41467-019-11677-w | Introduction
One of the most remarkable accomplishments in the field of neuroscience is the description of essential principles that define the basic forms of associative memory. This fundamental cognitive property occurs through complex biological mechanisms by which the connection between two previously unrelated stimuli, or a behavior and a stimulus, is learned; when such process takes place, it is assumed that the association of these stimuli is stored in a memory system1. For centuries, different thinkers have shaped a very plentiful and venerable history of research on basic learning processes. The combined work of philosophers, naturalists, physiologists, and life scientists has set the baseline upon which the modern learning theory currently stands2. The most basic type of associative learning is the classical conditioning developed by the Nobel Prize Laureate Ivan Pavlov, who established the first systematic study of the fundamental principles of associative memory. In his studies, after an appropriate conditioning, dogs deprived of food were able to exhibit a consequent response -salivation- when a bell rung3.
Associative conditioning is ubiquitous in complex organisms endowed with evolved nervous systems, including all major vertebrate taxa and several invertebrate species4. This complex process can also be reproduced and analyzed in artificial neural networks and different computational models5. Conditioned learning confers to the organisms the ability to adapt to ever-changing environments and is considered a milestone for life’s survival. Despite its importance, associative conditioning has never been observed in individual cells.
In order to determine whether associative conditioned responses are involved in systemic cellular behaviors, we analyzed the movement trajectories of Amoeba proteus under two external stimuli by using an appropriate electric field as the conditioned stimulus and a specific peptide as chemo-attractant.
Amoebae represent an immensely diverse family of eukaryotic cells that can be found in nearly all habitats and constitute the major part of all eukaryote lineages6. Concretely, Amoeba proteus is a large free-living predatory amoeba with a notable capacity to detect and respond to chemical and physical cues allowing it to locate and consume near prey organisms such as bacteria and other protists.
These cells are able to migrate on flat surfaces and in three-dimensional substrate by a process known as amoeboid movement, which consists in pseudopodia extensions, cytoplasmic streaming, and flowing into these extensions changing permanently the cellular shape7.
Amoeboid locomotion represents one of the most widespread forms of cell motility and constitutes the typical way of locomotion in broad range of adherent and suspended eukaryotic cell types7. In mammalian cells, amoeboid locomotion is vital for multiple physiological processes as the development of the embryo8, the action of the immune system9 and the repair of wounds10. Likewise, it is also responsible for the spread of malignant tumors11.
The large free-living amoeba, Amoeba proteus, has served as a classic unicellular organism in many investigations for more than one hundred years12,13, mainly as a cellular model to study cell motility, membrane and cytoskeleton function, and the role of the nucleus14,15,16. However, despite the many investigations carried out so far, numerous biological aspects of this organism still remain poorly studied. On the other hand, diverse experimental studies have shown that Amoeba proteus exhibit robust galvanotaxis17, a directed movement in response to an electric field; in fact, it has been described that practically 100% of the amoebae migrate towards the cathode for long periods of time under a strong direct-current electric field in a range between 300 mV/mm and 600 mV/mm. Likewise, amoebae are known to display chemotactic behaviors; in particular, the peptide nFMLP, typically secreted by bacteria, is able to provoke a strong chemotactic response in many different types of cells. The presence of this peptide in the environment may indicate to the amoeba that food organisms might be nearby18. Given the large number of investigations carried out on this organism, the robustness in their behavior, easy handling in the laboratory, the relatively fast rate of migration (cells move at ~300 μm/min19) and the well documented sensitivity to electric fields and chemoattractants, we have chosen Amoeba proteus as the experimental study species in our work.
Here, we describe the emergence of an associative conditioned behavior in Amoeba proteus which corresponds to a new type of systemic migration pattern in the cell. This conditioned migration behavior seems to be an evidence of a primitive type of associative memory in a unicellular organism. In a preliminary study, we also confirm a similar behavior in a related species, Metamoeba leningradensis.
Results
Experimental setup
All our experiments have been carried out on a specific set-up that allowed us to expose the amoebae to both stimuli -galvanotaxis and chemotaxis- simultaneously. This system consists of two standard electrophoresis blocks, about 17.5 cm long, one directly plugged into a normal power supply and a second one connected to the first one via two agar bridges that transfer the current from one block to the other while preventing the direct contact of both the anode and the cathode with the medium where the cells were located (see Fig. 1, Supplementary Data 1, and data available in Methods section). On the central platform of the second electrophoresis block, we placed the experimental chamber, a sliding glass structure that enabled the creation of a laminar flux which not only allowed the electric current to pass through, but also generated an nFMLP peptide gradient that the amoebae were able to detect and respond to. In addition, when the sliding glass structure was opened, the placement and collecting of the cells was possible. We confirmed the establishment of the nFMLP gradient by the direct measurement of fluorescein-tagged peptide concentration with a plate reader. As shown in Fig. 2, the concentration of peptide in the middle part of the glass chamber (where the amoebae are placed) increases immediately following the flow establishment (within 2 min the concentration rises from zero to approximately 0.2 μM) and this concentration increases further (to 0.6 μM) for at least 30 min.
In the experiments, the cells were placed in the middle of the glass set-up and their displacements were monitored in small groups (see Methods section), being the individual trajectories recorded during periods of 30 min by using a camera connected to a microscope. The migration of 615 A. proteus and 210 M. leningradensis was quantitatively analyzed (Supplementary Data 2). All the experiments were carried out in Chalkley’s medium, a standard, nutrient-free saline medium at ambient temperature.
Cellular migration of A. proteus in the absence of stimuli
First, we recorded the locomotion trajectories of 50 amoebae (experimental replicates: 7, number of cells per replicate: 5–11) without any external influences (Fig. 3a). Under this condition, cell migration can be described as a correlated random motion characterized by low intrinsic directionality and progressive decreasing over time of the initial direction of migration20. In Fig. 3a, a representative example of amoebae locomotion in the absence of stimuli is depicted; cells exhibited significant changes in the movement guidance, and after 30 min they had explored practically all the directions of the experimentation chamber. The directionality of each cell was quantified by the cosine of the displacement angle17 (see Methods section), and the quantitative results showed that the values ranged between −0.987 and 1, being −0.125/1.55 the media/interquartile range (IQ), which indicated that in the absence of stimulus, these cells moved randomly without any defined guidance. In addition, the analysis of the distribution of displacement angles (i.e., the angle formed between the origin and the end of the movement, measured in radians) also confirmed no preference towards a certain direction (Fig. 3d).
Cell behavior of Amoeba proteus in an electric field
Next, the galvanotactic locomotion of 50 cellular trajectories (experimental replicates: 8, number of cells per replicate: 4–8) was analyzed under an external, controlled direct-current electric field of about 300 mV/mm (Methods section). Our experimental observations (Fig. 3b) indicated that practically all the amoebae migrated towards the cathode for 30 min. These results matched with other previously reported experiments17. Cellular locomotion under this electric condition was characterized by stochastic movements with robust directionality, and cells exhibited a locomotion pattern tending to move in the direction of the immediately preceding movement by conserving their polarity in time towards the cathode. Taking as reference the experiment of Fig. 3b, the quantitative analysis indicated that the values of the cosines of displacements were distributed between 0.037 and 0.999 (0.993/0.03 median/IQ) (Fig. 3e). This result verified that a unique fundamental behavior characterized by an unequivocal directionality towards the cathode had emerged in the experimental system. The significance of our analysis was validated with a non-parametric test (Wilcoxon rank-sum test) comparing the distributions of the cosines of the displacement in both situations, without stimulus and under the presence of the electric field. The test (p = 10−14; Z= 7.442, Wilcoxon rank-sum test) corroborated that the behaviors without and with the stimulus (the electric field) were significantly different.
Directionality under chemotactic gradient (chemotaxis)
Here, the behavior of 50 amoebae (experimental replicates: 10, number of cells per replicate: 4–6) was analyzed during biochemical guidance by exposing the cells for 30 min to an nFMLP peptide gradient placed in the left side of the set-up. The experiment showed that 86% of exposed cells migrated towards the chemotactic gradient (Fig. 3c). In other words, the chemical gradient in the environment provoked in most amoebae a systemic behavior characterized by stochastic locomotion movements with robust directionality towards the attractant stimulus (the peptide)21. The cosines of the displacement angles of individual trajectories ranged between −0.997 and 0.986 (−0.825/0.72 median/IQ) (Fig. 3f). This result indicated that a single fundamental behavior characterized by a movement towards the peptide prevailed in the cells. The comparison between the cosine values obtained with and without chemotactic stimulus (p = 10−4; Z= −3.878, Wilcoxon rank-sum test) on one hand, and between the cosine values with chemotactic gradient and with the presence of electric field (p = 10−17; Z= 8.428, Wilcoxon rank-sum test) on the other, corroborated that the systemic locomotion behavior under the chemotactic gradient was totally different to both, the absence of stimulus, and the presence of an electric field.
Induction process
Once the migrations of the amoebae in the three previous basic and independent experimental conditions (without stimulus, under galvanotaxis and under chemotaxis) were analyzed, we studied the trajectories of 180 Amoeba proteus (experimental replicates: 32, number of cells per replicate: 4–10) when they were exposed simultaneously to galvanotactic and chemotactic stimuli for 30 min (Fig. 4a). For such a purpose, we arranged the cathode on the right of the set-up and the anode with the nFMLP peptide solution on the left. The analysis of the amoebae trajectories showed that 53% of the cells ignored the electric field signal and moved towards the anode-peptide (23% of them exhibited a very sharp directionality), while the remaining 45.33% migrated to the cathode. Three cells (1.67%) presented an atypical behavior, remaining immobile but adhered to the substrate during the 30 min of the test, and therefore were included in the unconditioned group. The cosines of the displacement angles were distributed between −1 and 1 (−0.26/1.8 median/IQ). This analysis verified that two fundamental cellular migratory behaviors had emerged in the experimental system, one towards the anode and another towards the cathode. The statistical analysis confirmed the presence of these different behaviors (p = 10−30; Z= −11.435, Wilcoxon rank-sum test). In Fig. 6, a galvanotactic control of the cells that responded to the cathode during the induction process is shown. All the amoebae again migrated towards the cathode, confirming that these cells were unconditioned.
Conditioned behavior test
To verify if the cells that moved towards the anode during the induction process (Fig. 4a) present some kind of persistence in their migratory behavior, we analyzed 160 amoebae in three different scenarios in which they were exposed to several types of perturbations.
In the first scenario, 85 amoebae (experimental replicates: 32, number of cells per replicate: 1–7) that had previously migrated towards the anode-peptide during the exposition to two simultaneous stimuli (induction process) were manually extracted and placed for 5 min on a normal culture medium (Chalkley´s medium) in a small Petri dish in absence of stimuli. Then the cells, were deposited on a new identical glass and block set-up that had never been in contact with the chemotactic peptide nFMLP and exposed for the second time to a single electric field, without peptide, during 30 min (note that the total time after the first induction process was 35 min). The analysis of the individual trajectories showed that 82% of the cells ran to the anode where the peptide was absent (Fig. 4b). The cosines of displacements ranged between −1 and 0.998 (−0.854/0.77 median/IQ). This result supported mathematically that the majority of cells moved towards the anode in the absence of peptide, thus corroborating that a new locomotion pattern had appeared in the cells. Such systemic behavior (migration towards the anode in the absence of peptide) had never appeared before. The comparison between the cosines of displacements obtained during the galvanotaxis without previous induction (Fig. 3b) and the galvanotaxis after the induction (Fig. 4b) showed that this newly acquired cellular behavior is extremely unlikely to be obtained by chance (p = 10−19; Z= 8.878, Wilcoxon rank-sum test). Four cells in this scenario exhibited eventual displacements towards both sides without any preference, and therefore were included in the unconditioned group. Since 43 cells persisted in the migration towards the anode until the end of the galvanotaxis, we used 50 cells for the next step.
In the second scenario, we studied 50 amoebae (experimental replicates: 27, number of cells per replicate: 1–4) previously exposed to the induction process and to the conditioning test of the first scenario (5 min without stimulus and 30 min of galvanotaxis). After that, these cells were placed once more in Chalkley´s medium without any stimulus for 3 min, and then they were again exposed to galvanotaxis for 30 min but in this case the polarity of the electric field was inverted (the cathode was positioned where the anode was previously and vice versa). In short: after the induction process, the cells were exposed to 5 min without stimulus, 30 min on galvanotaxis, 3 min without stimulus and 30 min on galvanotaxis with inverted polarity; in total, the time elapsed between the end of the induction process and the end of the study was 68 min.
When an Amoeba proteus is placed in an electric field for long periods of time, the probability of dying or, at least, detaching from the substrate and adopting a spherical shape increases sharply. Therefore, the cells were physically extracted and replated for 3 min to minimize cell damage. Next, we changed the medium in the set-up, after that the cells were again placed in the clean experimental chamber, and finally exposed to galvanotaxis with inverted polarity with fresh medium for 30 min. By inverting the electric field, we demonstrated that the amoebae were neither directed to a specific point in the space nor they associated a specific point in the space to the peptide.
Even under these new and strict conditions, 58% of the cells continued migrating towards the anode (now positioned in the opposite side) thus maintaining the conditioned behavior (Fig. 4c). The remaining cells (42%) lost this ability. The comparison between the cosines of displacements in the galvanotaxis without induction (Fig. 3b) and the cosines from the second scenario also indicated that it is unlikely to obtain this new conditioned systemic behavior by chance (p = 10−11; Z= −6.491, Wilcoxon rank-sum test). Four cells displayed an atypical behavior characterized by immobility and adhesion to the substrate during the 30 min of the test, and three more cells showed eventual displacements towards both sides; all of them were included in the unconditioned group. Since 16 cells maintained the migration towards the anode until the end of the galvanotaxis, we used 25 cells for the next step.
In the third scenario, 25 cells (experimental replicates: 9, number of cells per replicate: 1–7) were exposed to the induction process and to the conditioning test of the first scenario (5 min without stimulus and 30 on galvanotaxis). Next, once again, the cells were placed in Chalkley´s medium without any stimulus, in this occasion for about 30 min, and then exposed to galvanotaxis with inverted polarity for 30 min like in the second scenario (Fig. 4d). In short, 5 min without stimulus, 30 min on galvanotaxis, 30 min without stimulus and 30 min on galvanotaxis with inverted polarity; therefore, the total time elapsed between the end of the induction process and the end of the study was 95 min.
Like in the second scenario, cells were physically extracted and re-plated for 3 min to minimize cell damage due to the long standing in an electric field. Next, we changed the medium in the set-up, then the cells were again placed in the clean experimental chamber, and finally exposed to galvanotaxis with inverted polarity with fresh medium for 30 min.
Even under these stricter conditions, 56% of the cells still maintained the migration to the anode, positioned in the opposite side, making evident that the new systemic behavior exhibits a remarkable robustness despite the perturbations introduced in the experiment. In addition, the test comparing the cosines during the third scenario (Fig. 4d) and the galvanotaxis without previous induction (Fig. 3b) showed that it is completely unlikely to obtain this newly acquired cellular behavior by chance (p = 10−11; Z= 6.221, Wilcoxon rank-sum test). One cell presented an atypical behavior characterized by immobility, and therefore was included in the unconditioned group.
A very restrictive criterion was adopted in these three scenarios: a cell was considered to present lasting directionality towards the anode only if after 15 or more minutes, the amoeba was still migrating towards the anode (which corresponds to 20 min after the induction process since all the amoebae were placed in Chalkley’s medium without any stimulus for 5 min after the induction) or if after moving initially towards the cathode, the cell corrected its trajectory showing a clear lasting directionality towards the anode. Note that these behaviors were never observed in any of the five galvanotactic experiments (Figs. 3b, 5 and 6a–c).
Persistence time of the new acquired cellular behavior
The new emergent systemic behavior is also characterized by a limited duration through time. Figure 7a is an illustrative example of the loss of conditioning in 15 induced cells (experimental replicates: 4, number of cells per replicate: 3–6) as time goes on. To quantify this phenomenon, we have measured the duration time of the conditioned behavior in the 148 conditioned cells in the three scenarios previously described.
In the first scenario, (35 min after the induction process, 81 conditioned cells) the analysis of the persistence level showed that 11 cells lost the inducted behavior at the beginning of the test, 27 cells did it after 20–33 min, whereas the rest (43 cells, 53%) maintained migration towards the anode until the end of the experiment (see Fig. 7b for details).
In the second scenario, (68 min after the induction process, 43 conditioned cells) 14 cells lost the persistent behavior at the beginning of the test and 13 did it after 52–66 min, whereas the remaining 16 cells (37%) continued the migration towards the anode until the end of the experiment (Fig. 7b).
In the third scenario, (95 min after the induction process, 24 conditioned cells) 10 cells lost the persistent behavior at the beginning of the test, 6 did it after 47–90 min, and the remaining 8 cells (33%) continued the migration towards the anode until the end of the experiment (Fig. 7b).
Despite all the perturbations introduced in the experiments, the whole analysis indicates that the average time of the cells that lost the acquired motility pattern during any of the three scenarios was 44.04 ± 21.8 min.
Evidences of associative conditioning in M. leningradensis
In order to examine the robustness of the observed conditioned behavior we have performed a preliminary study on another unicellular species, Metamoeba leningradensis, under the same induction process as the Amoeba proteus. The metamoebae were exposed to the same intensity of the electric field and to the same peptide (nFMLP) concentration, and therefore, values of amperage or optimal concentration of peptide were not adapted to generate more efficient responses of these organisms to such stimuli.
Figure 8a shows the galvanotactic locomotion of 50 cellular trajectories (experimental replicates: 4, number of cells per replicate: 4–15) analyzed under an external, controlled direct-current electric field of about 340 mV/mm. This study indicated that practically all the metamoebae migrated towards the cathode during 30 min. The quantitative analysis showed that the values of the cosines of displacements were distributed between 0.04 and 1 (0.98/0.09 median/IQ), which verified that a unique fundamental behavior characterized by an unequivocal directionality towards the cathode had emerged in the experimental system.
Next, 160 metamoebae (experimental replicates: 15, number of cells per replicate: 3–12) were subjected to an induction process of 30 min. They were exposed simultaneously to chemotactic and galvanotactic stimuli, placing the peptide on the anode side (Fig. 9). Under these conditions, the record of the migration trajectories showed that 39% of the metamoebae ignored the electric field signal and moved towards the anode, while the remaining (61%) migrated to the cathode. The cosines of the displacement angles were distributed between −1 and 1 (0.54/1.8 median/IQ). Hence, the result of the induction process indicated that two fundamental cellular migratory behaviors had emerged in the experimental system, one towards the anode and another towards the cathode. The Wilcoxon rank-sum test confirmed the presence of these two different behaviors (p = 10−26; Z= −10.639, Wilcoxon rank-sum test).
Finally, to verify whether the cells that moved towards the anode during the induction process presented some kind of conditioning in their migratory trajectories, we performed a conditioned behavior test (Fig. 8b). For such a purpose, the 62 Metamoeba leningradensis (experimental replicates: 15, number of cells per replicate: 1–7) that had previously migrated towards the anode-peptide during the induction process were manually extracted and placed for 5 min into a normal culture medium (Chalkley´s medium) in a small Petri dish, in absence of stimuli. Then, in a similar way that we did in the Amoeba proteus experiments, the metamoebae were placed usually in groups of 1–6 on a new identical glass and block set-up that had never been in contact with the chemotactic peptide. Under these conditions, they were exposed for the second time to a single electric field, without peptide, during 30 min (the total time after the induction process was 35 min).
The analysis of the individual trajectories showed that 71% of the cells ran to the anode where the peptide was absent (Fig. 8b). The cosines of displacements ranged between −1 and 0.99 (−0.539/1.33 median/IQ). This result supported mathematically that the majority of cells moved towards the anode in the absence of peptide, thus corroborating that a new locomotion pattern had appeared in the metamoebae cells. Such systemic behavior (migration towards the anode in the absence of peptide) had never appeared before. The comparison between the cosines of displacements obtained during the galvanotaxis without previous induction (Fig. 8a) and the galvanotaxis after the induction (Fig. 8b) showed that this newly acquired cellular behavior is extremely unlikely to be obtained by chance (p = 10−17; Z= 8.326, Wilcoxon rank-sum test) in Metamoeba leningradensis.
Discussion
Here, using an appropriate direct-current electric field (galvanotaxis) and a specific peptide (nFMLP) as a chemoattractant (chemotaxis) we have addressed essential aspects of the Amoeba proteus and Metamoeba leningradensis migration. More precisely, we have found that these cells can link two different past events, shaping an associative conditioning process characterized by the emergence of a new type of systemic motility pattern. This behavior consists in a persistent migration towards the anode when these cells typically migrate to the cathode.
First we have studied the Amoeba proteus migration and we have verified in the galvanotactic experiments that practically all the amoebae show an unequivocal systemic response consisting in the migration towards the cathode when they are exposed to a strong direct electric field of about 300–600 mV/mm (Fig. 3b, and Fig. 6). However, if the amoebae are exposed simultaneously to a chemotactic and galvanotactic stimulus (induction process), placing a specific peptide in the anode, 53% of the amoebae moved to the site where the peptide was located, and most of these cells (82%) were able to acquire a new singular behavior in their systemic locomotion characterized by a persistent migration towards the anode, which was observed in subsequent galvanotaxis experiments carried out in absence of peptide (Fig. 4b–d).
This extensive study, which has covered 615 cellular trajectories, has shown that when the exposition to a stimulus related to the amoeba’s nourishment (a specific peptide) is accompanied by an electric field, and the peptide is placed in the anode, the amoebae appear to associate the anode with the food (the peptide) and after the induction process most cells developed a new persistent pattern of cellular motility characterized by movements towards the anode even if the nourishment (peptide) was absent. After an induction process, most of amoebae seem to associate food with the anode and, consequently, modify their conduct, behaving against their known tendency to move to the cathode. Strikingly, this induced association of anode and food can be maintained for relatively long periods of time. In our experiments, this conditioned motility pattern prevailed for periods ranging from 20 to 95 min. This period of time is very long in comparative terms if we take into consideration that the cellular cycle of Amoeba proteus lasts, with small variations depending on the environment, only about 24 h under controlled culture conditions22. We have also observed that, after the induction process, a small subset of the amoebae was not conditioned. Cells display a range of differences in their membrane receptors, electric potential, physiological/metabolic functioning, and hence, there are no two identical unicellular organisms. In our experiments, some cells probably were unconditioned or weakly conditioned due to their intrinsic physiological peculiarities, and in addition, some kind of cellular damage caused by the experimental process may have occurred.
To test the robustness of the conditioned behavior we have performed a preliminary study in Metamoeba leningradensis under the same strict conditions that we set up for Amoeba proteus. Despite these restrictive conditions, most metamoeba cells were able to link two different past events, same as Amoeba proteus, shaping an associative conditioning process characterized by the emergence of a new type of systemic motility pattern which consists in a persistent migration towards the anode when, in the absence of previous induction, these cells also typically migrate to the cathode (Fig. 8).
The controls carried out during the research indicated that cells exposed independently either to galvanotaxis or chemotaxis, did not present any observable atypical behavior (Fig. 6), and the quantitative study performed emphasized that it is extremely unlikely to obtain the new type of induced systemic behavior by chance (p = 10−19; Z= 8.878, Wilcoxon rank-sum test). In conclusion, the work we have performed here shows that most of the conditioned Amoeba proteus and Metamoeba leningradensis exhibited the ability to preserve the relationship between the two stimuli, acquiring a new type of systemic behavior via conditioning. Noteworthy, the fact that individual cells are able to generate associative conditioned behaviors to guide their complex migration movements has never been verified so far.
Our experimental results may allow another possible explanation. The exposure to nFMLP triggers a sub-population of cells to change the character of their migration in an electric field, making them to migrate towards the anode rather than the cathode. A notable number of cells belonging to both species can persistently change their migration pattern by these two external, simple and independent stimuli, when both are simultaneously applied. The new behavior persists for around 1 h, and gradually fades away thereafter.
Amoebae and metamoebae cells seem to associate the anode with the peptide in the induction process. After the conditioning, both stimuli seem to remain linked in these cells for a relatively long period of time, and consequently, the systemic movement of amoebae and metamoebae responded to the presence of an electric field by migrating towards the anode instead of the expected migration to the cathode.
In brief, we have observed a systemic cell behavior that can be modified by two simple external and independent stimuli, when they are simultaneously applied. This conditioned migration behavior can prevail for 44 min on average.
Pavlov studied four fundamental types of persistent behavior provoked by two stimuli. Here we have based our work in one of them, the called “simultaneous conditioning”, in which both stimuli are applied at the same time.
However, in a strict sense, we cannot conclude that our findings represent the classical Pavlovian conditioning since complete controls and parametric analyses for classical conditioning studies have not been performed yet23.
The experiments we show here were inspired by numeric predictions based on computational modeling that we published in 2013 dealing with complex metabolic networks24. Thus, analyzing complex enzymatic processes under systemic conditions using Statistical Mechanic tools and advanced Computational and Artificial Intelligence techniques, we were able to verify numerically that self-organized enzymatic activities in modular metabolic networks seem to be governed by Hopfield-like attractor dynamics similar to what happens in neural networks24. A key attribute of the analyzed metabolic Hopfield-like dynamics is the presence of associative memory. This quantitative study showed that the associative memory in unicellular organisms is possible24,25. Such memory would be a manifestation of emergent properties underlying the complex dynamics of the systemic cellular metabolic networks.
It is still too early to delineate the molecular mechanisms supporting this cellular associative conditioning. However, there are evidences of a functional memory, which can be embedded in multiple stable molecular marks during epigenetic processes25. Likewise, long-term correlations (mimicking short-term memory in neuronal systems) have also been analyzed in experimental calcium-activated chloride fluxes in Xenopus laevis oocytes26. On the other hand, different studies have described several molecular processes in which both prokaryotic and eukaryotic cells show chemotactic memory. For instance, changing dynamics in specific methylation-demethylation patterns in prokaryotes seem to be involved in molecular memory processes related to chemical gradient adaptation27,28,29,30. Besides, phosphotransfer processes and other post-translational modifications seem to be involved in chemostatic cellular persistence of eukaryotic cells31,32,33.
In this paper, we have addressed essential aspects of the Amoeba proteus and Metamoeba leningradensis migration. The mechanisms underlying amoeba locomotion are extremely complex and the ability to direct their movement and growth in response to external stimuli is of critical significance for its functionality; in fact, cellular life would be impossible without regulated motility. Although some progress is being made in the understanding of cellular locomotion, how cells move efficiently through diverse environments, and migrate in the presence of complex cues, is an important unresolved issue in contemporary biology. Free cells need to regulate their locomotion movements in order to accomplish critical activities like locating food and avoiding predators or adverse conditions. In the same way, cellular migration is required in multicellular organisms for a plethora of fundamental physiological processes such as embryogenesis, organogenesis and immune responses. In fact, deregulated human cellular migration is involved in important diseases such as immunodeficiencies and cancer11,34. Neoplastic progression (invasion and metastases), for example, can be regarded as a process in which the survival of tumor cells depends also on their ability to migrate to obtain additional resources in a general context of scarcity35.
Here, we have verified that two unicellular organisms such as Amoeba proteus and Metamoeba leningradensis are able to modify their systemic response to a determined external stimulus exclusively by associative conditioning. This fact opens up a new framework in the understanding of the mechanisms that underlie the complex systemic behavior involved in cellular migration and in the adaptive capacity of cells to the external medium.
Methods
Cell cultures
Amoeba proteus (Carolina Biological Supply Company, Burlington, NC.Item # 131306) were grown at 21 °C on Simplified Chalkley’s Medium (NaCl, 1.4 mM; KCl, 0.026 mM; CaCl2, 0.01 mM), alongside Chilomonas as food organisms (Carolina Biological Supply Company Item #131734) and baked wheat corns.
Metamoeba leningradensis (Culture Collection of Algae and Protozoa, Oban, Scotland, UK, CCAP catalog number 1503/6). They were cultured in the same conditions as Amoeba proteus.
Experimental set-up
All the experiments were performed in a specific set-up (Fig. 1) consisting in two standard electrophoresis blocks, 17.5 cm long (Biorad Mini-Sub cell GT), a power supply (Biorad powerbank s2000), two agar bridges (2% agar in 0.5 N KCl, 10–12 cm long) and a structure made from a standard glass slide and covers commonly used in Cytology Laboratories.
The first electrophoresis block was directly plugged into the power supply while the other was connected to the first via the two agar bridges, which allowed the current to pass through and prevented the direct contact between the anode and cathode and the medium where the cells would be placed later. Both electrophoresis blocks consisted of 3 parts: on the extremes, there are 2 wells which were filled by the conductive medium (Chalkley’s Simplified Medium) and, in the middle, an elevated platform (Fig. 1).
In the center of the second electrophoresis block we placed the experimental chamber that allowed us to obtain a laminar flux when it was closed and the addition and extraction of cells when it was open.
The experimental chamber consisted in 4 pieces of glass (standard glass slide and covers commonly used in Cytology Laboratories), a 75 × 25 mm modified slide and three small pieces obtained by trimming of three cover glass of 60 × 24 × 0.1 mm (Fig. 1).
Three cover glasses were trimmed with a methacrylate ruler, one measuring about 3 × 24 × 0.1 mm and two measuring about 40 × 24 × 0.1 mm each, here on called central piece and sliding lateral pieces, respectively. These three glasses were for only one use.
This glass structure (Fig. 1) supported the sliding parts of the experimental chamber. It was reusable after cleaning. To build it, we fixed with silicone on a glass slide along the two longest sides of the slide (if the width of a cover is 24 mm, about 4 mm were stuck on the slide and 20 mm protrude towards the outside of the glass slide). Then we left it to dry for 24 h. The last step consisted on trimming the protruding portions of the cover slides (about 60 × 20 × 0.1 mm) with a methacrylate ruler, so that two small longitudinal strips of approximately 60 × 4 × 0.1 mm were adhered to the glass slide (see Fig. 1), which shaped the lateral limits of the experimentation chamber.
The modified slide was placed in the central platform of the second electrophoresis block. To avoid medium going across the modified slide from below, we placed an oil drop in the central platform of the block of electrophoresis, on which the modified slide was placed. It is very important that the oil drop expands to cover the entire width of the experimental chamber. In the center of the modified slide, without any glue, we placed the central piece and the two sliding lateral pieces leaving short distance between all of them.
The amoebae were placed below the central piece of the chamber in an approximate volume of 30 µl. To note, it is crucial that the amoebae do not remain for more than a few seconds in the micropipette tip to avoid the adhesion of the amoeba to the inner surface of the tip.
Once the amoebae were placed under the central piece of the chamber, we waited for two minutes to allow the cells to stick to the surface of the modified slide. Then, we filled the wells of the electrophoresis blocks with simplified Chalkley medium up to the level necessary to contact with the base of the modified slide, but not the two sliding lateral pieces. Later, the two sliding lateral pieces were pushed with two micropipette tips until they contact with the liquid in the wells. Next, the two sliding lateral pieces are pushed to contact the central piece in the chamber. This way, a laminar flux can be established throughout the inner space of the experimental chamber.
In the induction process, once the laminar flux was created, and before the activation of the electric field, we added 750 µl of 2 × 10−4 M nFMLP peptide solution to the medium (75 ml) in the positive pole well of the second electrophoresis block.
Considering that the amoebae that had shown a specific behavior were needed to perform further experiments, the cells were collected opening the sliding lateral pieces with the tip of a micropipette.
Set of videos intended only for didactical purposes. They are merely descriptive, to make easier the understanding of our experimental procedure and the reproducibility of our studies. Note that steps 5 to 8 are performed directly under the microscope, and we have not filmed them under the microscope for better visualization.
In summary, the experimental chamber consisted in a sliding glass structure. The sliding lateral pieces could be displaced in the longitudinal direction. This way, when the sliding pieces were closed an inner laminar flux was available in the chamber and, when they were open, the placement and collecting of the cells were possible easily. Movies showing the main experimental procedures have been deposited in figshare (https://doi.org/10.6084/m9.figshare.8868326).
A. proteus and M. leningradensis may display some physiological variations depending on culture conditions. Before the experiments, the cells were starved for 24 h in Chalkley’s Simplified Medium (the same medium that was used in the experiments), in the absence of external stimuli. Once starved, only the cells that were strongly attached to the substrate, actively moving through it and showing a little amount of thin pseudopodia were used in the experiments.
The cells were washed in clean Chalkley’s medium and placed in the middle of the glass set-up (experimental chamber), under the central piece of cover glass and left to rest until all of them appeared to be firmly attached to the bottom of the modified glass slide. Next, the two 4 cm long cover glasses (sliding lateral pieces) were placed on the sides of the glass structure, protruding outside of the middle platform of the block and over the lateral wells (Fig. 1). After that, each well was filled using 75 ml of Chalkley’s medium, in such a way that the glass protrusion over each well is in contact with the liquid’s surface. Finally, as the Chalkley’s medium slowly filled up the experimental chamber, both lateral cover glasses had to be gently pushed towards each other until they touched the middle cover glass, completely covering the whole structure and forming a laminar flux that connected both lateral wells.
The experiments were always made with small groups of cells. For instance, in Amoeba proteus along the induction process, we analyzed a total of 180 cells that were studied in 32 different times (experimental replicates) analyzing them in groups of 4–10 cells each (number of cells per replicate). Scenario 1 was repeated 32 times. Scenarios 2 and 3 were performed 27 and 9 times, respectively.
The induction process was usually performed using around 7 cells per experiment, sometimes as few as 4 or 5 and other times as many as 9 or 10, the average being 6–8 cells. The number of cells analyzed in the scenarios depended on how many cells appeared to be conditioned in the first step, so that the number of cells per experiment is lower each time, for instance, in scenario 1, the number of cells was usually between 2 and 4. Finally, in scenarios 2 and 3 the experiments were performed using fewer amounts of cells per experiment, usually 1–3 which were the cells that migrated towards the cathode during the conditioning process.
Compared to Amoeba proteus, the Metamoeba leningradensis showed a more varied array of behaviors and shapes. These cells were also more difficult to handle, as they were more prone to strongly stick to the micropipette tips, while usually showing a weaker attachment to the glass chambers.
Electric field (galvanotaxis)
An electric field was applied to the first electrophoresis block, which was then conducted to the second by the two agar bridges. Direct measurements taken with a multimeter in the second block (where the cells were placed) showed that the strength of the electric current oscillated between 58.5 and 60 V (334–342 mV/mm) while the intensity values varied between 0.09 and 0.13 mA.
After 30 min of exposure, during which the cellular migration movement were recorded, the power supply was turned off and the agar bridges removed.
All the experiments where the only stimulus was an electric field were performed in an electrophoresis block that had never been in contact with any chemotactic substance.
Cell induction
Groups of 4 to 10 amoeba were placed in the experimental chamber. Once all the cells were attached to the glass surface, the laminar flux was stablished by gently closing the structure using the sliding cover glasses. Next, the peptide, nFMLP was introduced in the left well of the electrophoresis block. After about 1 min, the power supply was turned on and the electric field established. The process lasted for 30 min; after that, the power supply was turned off and the cells that moved towards the anode removed from the experimental chamber and placed in a Petri dish with clean Chalkley’s medium for future experiments (Scenario 1, 2 or 3).
When the cells were subjected to both stimuli at the same time (induction process), a new population response arose. This population behavior, in Amoeba proteus, showed that about half the amoebae cells migrated towards the anode (where the nFMLP peptide was placed), and approximately the other half of the cellular population migrated to the cathode (Fig. 4). On the other hand, only 39% of the Metamoeba leningradensis moved towards the anode-peptide, in the induction process (Figure S1). Accordingly, the cellular migration response under two simultaneous stimuli is notoriously different from that observed when the stimuli were separated (Figs. 3b, c and 8a), and therefore it cannot be concluded that the chemotactic gradient has a stronger influence on cell migration than the electric field.
In order to homogenize the cellular responses as much as possible, we put all the amoebae under starving conditions for at least 24 h before performing any experiments.
Once the laminar flux was created, we added 750 µl of 2 × 10−4 M nFMLP (#F3506, Sigma-Aldrich) peptide solution to the medium (75 ml) in the positive pole well of the second electrophoresis block; therefore, the peptide solution was diluted to a final concentration of 2 × 10−6M. In all our experiments, we used the same concentration of nFMLP. In order to homogenize the solution and accelerate the creation of a chemotactic gradient in the experimental chamber we carefully mixed the content of the left well until the amoebae appeared to start moving towards it. Finally, the cells behavior was recorded for 30 min.
The generation of an nFMLP peptide gradient was evaluated by the measurement of its concentration in the middle of the experimental chamber. To this end, 4μM fluorescein-tagged peptide (#F1314, Invitrogen), was loaded in the left side of the set-up (Fig. 1). Next, the central glass piece of the experimental chamber was slightly displaced and a small opening, the size of the tip of a 50–200 μL micropipette, was left between the sliding cover glasses and the central glass piece. This little separation allowed us to get samples of 60 μL from the middle part of the laminar chamber flux at 0, 2, 5, 10, 15, 20 and 30 min following the establishment of the laminar flux. Peptide concentration was calculated extrapolating the values from a standard curve with known concentration of the fluorescein-tagged peptide (Fig. 2). All measurements were duplicated and the experiment was repeated three times. Fluorescence was measured in 96 well glass bottom black plates (P96-1.5H-N, In Vitro Scientific) employing a SynergyHTX plate reader (Biotek) at Excitation/Emission wavelengths of 460/528 following standard laboratory techniques as described by Green and Sambrook36.
Track recording and digitizing
The motility of the cells was recorded using a digital camera attached to a SM-2T stereomicroscope. Images were acquired every 10 seconds, over a period of at least 30 min (180 frames). If a video was longer, only the first 30 min were quantified, except for the ones used for Fig. 7a. Since automated tracking software is often inaccurate37, we performed manual tracking using the TrackMate software in ImageJ (http://fiji.sc/TrackMate)38, as suggested in by Hilsenbeck et al.37 Each track corresponds to an individual amoeba.
Directionality analysis and statistical significance
In order to quantify and compare the directionality of cell migration towards the anode or the cathode, we computed the cosines of the angles of displacement of each amoeba17. More precisely, we calculated the cosine of the angle formed between the start and final positions of each cell. Consequently, we were able to analyze quantitatively if an amoeba moved towards the cathode (positive values of the cosine), or towards the anode (negative values). In addition, this study suggested the degree of directionality, since values closer to 1 (or to −1 in the case of the anode) indicated a very high preference towards that pole. Next, to estimate the significance of our results, we studied first if the distribution of cosines of angles came from a normal distribution, by applying the Kolmogorov-Smirnov test for single samples. Since the normality was rejected, the groups of cosines were compared in pairs by a non-parametric test, the Wilcoxon rank-sum test, and therefore, the results were depicted as median/IQ instead of as mean ± SD. Besides the p-value, we have reported the Z-statistic of the Wilcoxon rank-sum test39.
Note that the signs of the cosines from the second and third scenarios were changed to perform the respective tests with the galvanotaxis without previous induction (Fig. 3b) because the polarity of the electric field was inverted.
Researchers involved in the quantitative analysis of the cellular trajectories were never aware of what scenario each trajectory belonged to. Only when all the trajectories were quantified and processed, the researchers in charge of recording the amoeba’s movements informed the rest of the team about which trajectories belonged to each experiment or control.
Reporting Summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | {"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": 0.8150944113731384, "perplexity": 1749.1396214887789}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335326.48/warc/CC-MAIN-20220929065206-20220929095206-00199.warc.gz"} |
https://paperswithcode.com/paper/a-parallel-best-response-algorithm-with-exact | # A Parallel Best-Response Algorithm with Exact Line Search for Nonconvex Sparsity-Regularized Rank Minimization
13 Nov 2017Yang YangMarius Pesavento
In this paper, we propose a convergent parallel best-response algorithm with the exact line search for the nondifferentiable nonconvex sparsity-regularized rank minimization problem. On the one hand, it exhibits a faster convergence than subgradient algorithms and block coordinate descent algorithms... (read more)
PDF Abstract | {"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": 0.9769814610481262, "perplexity": 2434.2276192427194}, "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/1596439735909.19/warc/CC-MAIN-20200805035535-20200805065535-00510.warc.gz"} |
https://nips.cc/Conferences/2011/ScheduleMultitrack?event=3017 | Timezone: »
Spotlight
Improved Algorithms for Linear Stochastic Bandits
Yasin Abbasi Yadkori · David Pal · Csaba Szepesvari
Wed Dec 14 03:40 AM -- 03:44 AM (PST) @ None
We improve the theoretical analysis and empirical performance of algorithms for the stochastic multi-armed bandit problem and the linear stochastic multi-armed bandit problem. In particular, we show that a simple modification of Auer’s UCB algorithm (Auer, 2002) achieves with high probability constant regret. More importantly, we modify and, consequently, improve the analysis of the algorithm for the for linear stochastic bandit problem studied by Auer (2002), Dani et al. (2008), Rusmevichientong and Tsitsiklis (2010), Li et al. (2010). Our modification improves the regret bound by a logarithmic factor, though experiments show a vast improvement. In both cases, the improvement stems from the construction of smaller confidence sets. For their construction we use a novel tail inequality for vector-valued martingales. | {"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": 0.9795343279838562, "perplexity": 1406.153963481028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400214347.17/warc/CC-MAIN-20200924065412-20200924095412-00754.warc.gz"} |
https://www.varsitytutors.com/act_math-help/how-to-find-if-two-acute-obtuse-triangles-are-similar | # ACT Math : How to find if two acute / obtuse triangles are similar
## Example Questions
### Example Question #2 : How To Find If Two Acute / Obtuse Triangles Are Similar
The ratio of the side lengths of a triangle is 7:10:11. In a similar triangle, the middle side is 9 inches long. What is the length of the longest side of the second triangle?
9.9
9
7.7
12.1
10
9.9
Explanation:
Side lengths of similar triangles can be expressed in proportions. Establish a proportion comparing the middle and long sides of your triangles.
10/11 = 9/x
Cross multiply and solve for x.
10x = 99
x = 9.9
### Example Question #3 : How To Find If Two Acute / Obtuse Triangles Are Similar
Two triangles are similar to each other. The bigger one has side lengths of 12, 3, and 14.
The smaller triangle's shortest side is 1 unit in length. What is the length of the smaller triangle's longest side?
Explanation:
Because the triangles are similar, a ratio can be set up between the triangles' longest sides and shortest sides as such: 14/3 = x/1. Solving for x, we obtain that the shortest side of the triangle is 14/3 units long.
### Example Question #1 : How To Find If Two Acute / Obtuse Triangles Are Similar
The ratio of the side lengths of a triangle is . The longest side of a similar triangle is . What is the length of the smallest side of that similar triangle?
Explanation:
Use proportions to solve for the missing side:
Cross multiply and solve:
### Example Question #181 : Plane Geometry
There are two similar triangles. One has side lengths of 14, 17, and 19. The smaller triangle's smallest side length is 2. What is the length of its longest side? | {"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": 0.839252233505249, "perplexity": 564.7615026564092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583509958.44/warc/CC-MAIN-20181015225726-20181016011226-00234.warc.gz"} |
http://www.phys.virginia.edu/Announcements/talk-list.asp?SELECT=SID:2489 | Support UVa's Physics Department! >>
Click here for a printable version of this page.
## High Energy Physics Seminars
High Energy Thursday, November 15, 2012 2:30 PM Physics Building, Room 313 Note special date. Note special time. Note special room. Thomas Schaefer [Host: Diana Vaman] North Carolina State University "Continuity of the Deconfinement Transition in (Super) Yang Mills Theory" Slideshow (PDF) ABSTRACT:Finding controlled, analytical approaches to the deconfinement transition in QCD is an old problem. Here we present a weak coupling calculation of the deconfinement transition in a deformed version of QCD. We argue that this transition is continuously connected to the transition in pure gauge theory, which takes place in strong coupling. More technical abstract: We study the phase diagram of SU(2) Yang-Mills theory with one adjoint Weyl fermion on R^3xS^1 as a function of the fermion mass m and the compactification scale beta. This theory reduces to thermal pure gauge theory as m->infinity and to circle-compactified (twisted) susy gluodynamics in the limit m->0. In the m-L plane, there is a line of center symmetry changing phase transitions. In the limit m->infinity, this transition takes place at beta_c=1/T_c, where T_c is the critical temperature of the deconfinement transition in pure Yang-Mills theory. We show that near m=0, the critical scale beta_c can be computed using semi-classical methods and that the transition is of second order. This suggests that the deconfining phase transition in pure Yang-Mills theory is continuously connected to a phase transition that can be studied in weak coupling. The center symmetry changing phase transition arises from the competition of fractionally charged instanton-monopoles and instanton molecules. The calculation can be extended to higher rank gauge groups and non-zero theta angle. SLIDESHOW:
To add a speaker, send an email to [email protected]. Please include the seminar type (e.g. High Energy Physics Seminars), date, name of the speaker, title of talk, and an abstract (if available). | {"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": 0.9228922128677368, "perplexity": 1544.8127106695367}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866888.20/warc/CC-MAIN-20180624063553-20180624083553-00079.warc.gz"} |
https://triqs.github.io/triqs/3.1.x/documentation/cpp_api/triqs/stat/jackknife_mpi.html | triqs::stat::jackknife_mpi¶
#include <triqs/stat.hpp>
Synopsis
1. template<typename F, typename A>
auto jackknife_mpi (communicator c, F && f, A const &… a)
2. template<typename F, typename T>
auto jackknife_mpi (communicator c, F && f, accumulator<T> const &… a)
Calculate the value and error of a general function $$f$$ of the average of sampled observables $$f\left(\langle \mathbf{a} \rangle\right)$$, using jackknife resampling.
1) Directly pass data-series in vector like objects
2) Pass accumulators, where the jacknife acts on the linear binned data
The calculation is performed over the nodes; the answers which are then reduced (not all-reduced) to the node 0.
Template parameters¶
• F return type of function f which acts on data
• A vector-like object, defining size() and []
• T type of data stored in the accumulators
Parameters¶
• c TRIQS MPI communicator
• a one or multiple series with data: $$\mathbf{a} = \{a_1, a_2, a_3, \ldots\}$$ Pre-condition: if more than one series is passed, the series have to be equal in size
• f a function which acts on the $$i^\mathrm{th}$$ elements of the series in a:
$\left(a_1[i], a_2[i],a_3[i],\ldots\right) \to f\left(a_1[i],a_2[i],a_3[i],\ldots\right)$
Returns¶
std::tuple with four statistical estimators $$\left(f_\mathrm{J}^{*}, \Delta{f}_\mathrm{J}, f_\mathrm{J}, f_\mathrm{direct}\right)$$, defined below. The MPI reduction occurs only to node 0.
Jackknife resampling takes $$N$$ data points $$\mathbf{a}[i]$$ and creates $$N$$ samples (“jackknifed data”), which we denote $$\tilde{\mathbf{a}}[i]$$. We calculate three statistical estimators for $$f\left(\langle \mathbf{a} \rangle\right)$$:
• The function $$f$$ applied to observed mean of the data
$f_\mathrm{direct} = f\left(\bar{\mathbf{a}}\right),\quad \bar{\mathbf{a}} = \frac{1}{N}\sum_{i=0}^{N}\mathbf{a}[i]$
• The jacknife estimate defined as
$f_\mathrm{J} = \frac{1}{N}\sum_{i=0}^N f(\tilde{\mathbf{a}}[i])$
• The jacknife estimate, with bias correction to remove $$O(1/N)$$ effects
$f_\mathrm{J}^{*} = N f_\mathrm{direct} - (N - 1) f_\mathrm{J}$
Additionally, an estimate in the errror of $$f\left(\langle \mathbf{a} \rangle\right)$$ is given by the jacknife as
$\Delta{f}_J = \sqrt{N-1} \cdot \sigma_f$
where $$\sigma_f$$ is the standard deviation of $$\left\{f(\tilde{\mathbf{a}}[0]), f(\tilde{\mathbf{a}}[1]), \ldots, f(\tilde{\mathbf{a}}[N])\right\}$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9096887707710266, "perplexity": 4495.6791292699445}, "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-2023-06/segments/1674764499954.21/warc/CC-MAIN-20230202003408-20230202033408-00671.warc.gz"} |
http://www.newworldencyclopedia.org/entry/Antimatter | # Antimatter
Antimatter Overview Annihilation Devices Particle accelerator Penning trap Antiparticles Positron Antiproton Antineutron Uses Positron Emission Tomography Fuel Weaponry Scientific Bodies ALPHA Collaboration ATHENA ATRAP CERN People Paul Dirac Carl Anderson Andrei Sakharov
Matter is made of atoms, and atoms are made of electrons and quarks exchanging photons and gluons. Antimatter is made of anti-atoms, and anti-atoms are made of anti-electrons (usually called positrons) and anti-quarks exchanging photons and gluons—photons and gluons being their own antiparticles.
The difference between a particle and an antiparticle is that while a particle is moving in one direction through complex spacetime—call the time aspect +t—the antiparticle is moving in exactly the opposite direction through complex spacetime, -t. The real time and space that we observe is the square of this complex spacetime, and in either case, the square is, by the rule of signs, the same, positive 'external' time that is observed. So, while an electron is moving in the opposite direction to a positron in complex 'internal' time, they can be observed to be both moving in the same direction in 'external' real time.
A simple way of putting this is that a particle, reflected in time, becomes its antiparticle. The photon and gluon look the same under this reflection in time which is why they are their own antiparticles. In this sense, antimatter is matter reflected in time, what is technically called a 'charge conjugation' transformation. The reflection flips things such as spin—a left neutrino becomes a right antineutrino—electric charge—a negative electron becomes a positive positron—and color charge—a red quark becomes an antired antiquark.
When particle and antiparticle meet, their motion in complex time cancels out and they combine into a photon which has zero movement in time as described in special relativity.
## No Antimatter
Theoretically, an antielectron (a positron) and an antiproton (composed of anti-quarks) would together form an antihydrogen atom, in the same way that an electron and a proton form a normal matter hydrogen atom. Although the basic principles of quantum physics treat matter and antimatter on an equal footing, it is now well-established that the visible universe is made entirely of matter. This asymmetry of matter and antimatter in the creation of the visible universe is one of the greatest unsolved problems in physics.
## History
In December 1927, Paul Dirac developed a relativistic equation for the electron, now known as the Dirac equation. Curiously, the equation was found to have negative-energy solutions in addition to the normal positive ones. This presented a problem, as electrons tend toward the lowest possible energy level; energies of negative infinity are nonsensical. As a way of getting around this, Dirac proposed that the vacuum is filled with a "sea" of negative-energy electrons, the Dirac sea. Any real electrons would therefore have to sit on top of the sea, having positive energy.
Thinking further, Dirac found that a "hole" in the sea would have a positive charge. At first he thought that this was the proton, but Hermann Weyl pointed out that the hole should have the same mass as the electron. The existence of this particle, the positron, was confirmed experimentally in 1932 by Carl D. Anderson. During this period, antimatter was sometimes also known as "contraterrene matter."
Today's Standard Model shows that every particle has an antiparticle, for which each additive quantum number has the negative of the value it has for the normal matter particle. The sign reversal applies only to quantum numbers (properties) which are additive, such as charge, but not to mass, for example. The positron has the opposite charge but the same mass as the electron. For particles whose additive quantum numbers are all zero, the particle may be its own antiparticle; such particles include the photon and the neutral pion.
## Production
### Artificial production
The artificial production of atoms of antimatter (specifically antihydrogen) first became a reality in the early 1990s. An atom of antihydrogen is composed of a negatively-charged antiproton being orbited by a positively-charged positron. Stanley Brodsky, Ivan Schmidt and Charles Munger at SLAC realized that an antiproton, traveling at relativistic speeds and passing close to the nucleus of an atom, would have the potential to force the creation of an electron-positron pair. It was postulated that under this scenario the antiproton would have a small chance of pairing with the positron (ejecting the electron) to form an antihydrogen atom.
In 1995 CERN announced that it had successfully created nine antihydrogen atoms by implementing the SLAC/Fermilab concept during the PS210 experiment. The experiment was performed using the Low Energy Antiproton Ring (LEAR), and was led by Walter Oelert and Mario Macri. Fermilab soon confirmed the CERN findings by producing approximately 100 antihydrogen atoms at their facilities.
The antihydrogen atoms created during PS210, and subsequent experiments (at both CERN and Fermilab) were extremely energetic ("hot") and were not well suited to study. To resolve this hurdle, and to gain a better understanding of antihydrogen, two collaborations were formed in the late 1990s—ATHENA and ATRAP. The primary goal of these collaborations is the creation of less energetic ("cold") antihydrogen, better suited to study.
In 1999 CERN activated the Antiproton Decelerator, a device capable of decelerating antiprotons from 3.5 GeV to 5.3 MeV—still too "hot" to produce study-effective antihydrogen, but a huge leap forward. In late 2002 the ATHENA project announced that they had created the world's first "cold" antihydrogen. The antiprotons used in the experiment were cooled sufficiently by decelerating them (using the Antiproton Decelerator), passing them through a thin sheet of foil, and finally capturing them in a Penning trap. The antiprotons also underwent stochastic cooling at several stages during the process.
The ATHENA team's antiproton cooling process is effective, but highly inefficient. Approximately 25 million antiprotons leave the Antiproton Decelerator; roughly 10 thousand make it to the Penning trap. In early 2004 ATHENA researchers released data on a new method of creating low-energy antihydrogen. The technique involves slowing antiprotons using the Antiproton Decelerator, and injecting them into a Penning trap (specifically a Penning-Malmberg trap). Once trapped the antiprotons are mixed with electrons that have been cooled to an energy potential significantly less than the antiprotons; the resulting Coulomb collisions cool the antiprotons while warming the electrons until the particles reach an equilibrium of approximately 4 K.
While the antiprotons are being cooled in the first trap, a small cloud of positron plasma is injected into a second trap (the mixing trap). Exciting the resonance of the mixing trap’s confinement fields can control the temperature of the positron plasma; but the procedure is more effective when the plasma is in thermal equilibrium with the trap’s environment. The positron plasma cloud is generated in a positron accumulator prior to injection; the source of the positrons is usually radioactive sodium.
Once the antiprotons are sufficiently cooled, the antiproton-electron mixture is transferred into the mixing trap (containing the positrons). The electrons are subsequently removed by a series of fast pulses in the mixing trap's electrical field. When the antiprotons reach the positron plasma further Coulomb collisions occur, resulting in further cooling of the antiprotons. When the positrons and antiprotons approach thermal equilibrium antihydrogen atoms begin to form. Being electrically neutral the antihydrogen atoms are not affected by the trap and can leave the confinement fields.
Using this method ATHENA researchers predict they will be able to create up to 100 antihydrogen atoms per operational second. ATHENA and ATRAP are now seeking to further cool the antihydrogen atoms by subjecting them to an inhomogeneous field. While antihydrogen atoms are electrically neutral, their spin produces magnetic moments. These magnetic moments vary depending on the spin direction of the atom, and can be deflected by inhomogeneous fields regardless of electrical charge.
The biggest limiting factor in the production of antimatter is the availability of antiprotons. Recent data released by CERN states that when fully operational their facilities are capable of producing 107 antiprotons per second. Assuming an optimal conversion of antiprotons to antihydrogen, it would take two billion years to produce 1 gram of antihydrogen. Another limiting factor to antimatter production is storage. As stated above there is no known way to effectively store antihydrogen. The ATHENA project has managed to keep antihydrogen atoms from annihilation for tens of seconds—just enough time to briefly study their behavior.
CERN laboratories, which produces antimatter on a regular basis, said:
“ If we could assemble all of the antimatter we've ever made at CERN and annihilate it with matter, we would have enough energy to light a single electric light bulb for a few minutes.[1] ”
### Naturally occurring production
Antiparticles are created naturally when high-energy particle collisions take place. High-energy cosmic rays impacting Earth's atmosphere (or any other matter in the solar system) produce minute quantities of antimatter in the resulting particle jets. Such antiparticles are immediately annihilated by contact with nearby matter.
Antimatter may similarly be produced in regions like the center of the Milky Way Galaxy and other galaxies, where very energetic celestial events occur (principally the interaction of relativistic jets with the interstellar medium). The presence of the resulting antimatter is detectable by the gamma rays produced when it annihilates with nearby matter.
Antiparticles are also produced in any environment with a sufficiently high temperature (mean particle energy greater than the pair production threshold). The period of baryogenesis, when the universe was extremely hot and dense, matter and antimatter were continually produced and annihilated. The presence of remaining matter, and absence of detectable remaining antimatter,[2] also called baryon asymmetry, is attributed to violation of the CP-symmetry relating matter and antimatter. The exact mechanism of this violation during baryogenesis remains a mystery.
Positrons are also produced from the radioactive decay of nuclides such as carbon-11, nitrogen-13, oxygen-15, fluorine-18, and iodine-121
## Uses
### Medical
Antimatter-matter reactions have practical applications in medical imaging, such as positron emission tomography (PET). In positive beta decay, a nuclide loses surplus positive charge by emitting a positron (in the same event, a proton becomes a neutron, and neutrinos are also given off). The positron annihilates with an electron and it is the gamma ray emitted that is detected. Nuclides with surplus positive charge are easily made in a cyclotron and are widely generated for medical use.
### Fuel
In antimatter-matter collisions resulting in photon emission, the entire rest mass of the particles is converted to kinetic energy. The energy per unit mass (9×1016 J/kg) is about 10 orders of magnitude greater than chemical energy (compared to TNT at 4.2×106 J/kg, and formation of water at 1.56×107 J/kg), about 4 orders of magnitude greater than nuclear energy that can be liberated today using nuclear fission (about 40 MeV per 238U nucleus transmuted to Lead, or 1.5×1013 J/kg), and about 2 orders of magnitude greater than the best possible from fusion (about 6.3×1014 J/kg for the proton-proton chain). The reaction of 1 kg of antimatter with 1 kg of matter would produce 1.8×1017 J (180 petajoules) of energy (by the mass-energy equivalence formula E = mc²), or the rough equivalent of 43 megatons of TNT.
Not all of that energy can be utilized by any realistic technology, because as much as 50 percent of energy produced in reactions between nucleons and antinucleons is carried away by neutrinos, so, for all intents and purposes, it can be considered lost.[3]
The scarcity of antimatter means that it is not readily available to be used as fuel, although it could be used in antimatter catalyzed nuclear pulse propulsion. Generating a single antiproton is immensely difficult and requires particle accelerators and vast amounts of energy—millions of times more than is released after it is annihilated with ordinary matter, due to inefficiencies in the process. Known methods of producing antimatter from energy also produce an equal amount of normal matter, so the theoretical limit is that half of the input energy is converted to antimatter. Counterbalancing this, when antimatter annihilates with ordinary matter, energy equal to twice the mass of the antimatter is liberated—so energy storage in the form of antimatter could (in theory) be 100 percent efficient. Antimatter production is currently very limited, but has been growing at a nearly geometric rate since the discovery of the first antiproton in 1955. The current antimatter production rate is between 1 and 10 nanograms per year, and this is expected to increase to between 3 and 30 nanograms per year by 2015 or 2020 with new superconducting linear accelerator facilities at CERN and Fermilab. Some researchers claim that with current technology, it is possible to obtain antimatter for US$25 million per gram by optimizing the collision and collection parameters (given current electricity generation costs). Antimatter production costs, in mass production, are almost linearly tied in with electricity costs, so economical pure-antimatter thrust applications are unlikely to come online without the advent of such technologies as deuterium-tritium fusion power (assuming that such a power source actually would prove to be cheap). Many experts, however, dispute these claims as being far too optimistic by many orders of magnitude. They point out that in 2004; the annual production of antiprotons at CERN was several picograms at a cost of$20 million. This means to produce 1 gram of antimatter, CERN would need to spend 100 quadrillion dollars and run the antimatter factory for 100 billion years. Storage is another problem, as antiprotons are negatively charged and repel against each other, so that they cannot be concentrated in a small volume. Plasma oscillations in the charged cloud of antiprotons can cause instabilities that drive antiprotons out of the storage trap. For these reasons, to date only a few million antiprotons have been stored simultaneously in a magnetic trap, which corresponds to much less than a femtogram. Antihydrogen atoms or molecules are neutral so in principle they do not suffer the plasma problems of antiprotons described above. But cold antihydrogen is far more difficult to produce than antiprotons, and so far not a single antihydrogen atom has been trapped in a magnetic field.
Several NASA Institute for Advanced Concepts-funded studies are exploring whether it might be possible to use magnetic scoops to collect the antimatter that occurs naturally in the Van Allen belts of Earth, and ultimately, the belts of gas giants like Jupiter, hopefully at a lower cost per gram.[4]
Since the energy density is vastly higher than these other forms, the thrust to weight equation used in antimatter rocketry and spacecraft would be very different. In fact, the energy in a few grams of antimatter is enough to transport an unmanned spacecraft to Mars in about a month—the Mars Global Surveyor took eleven months to reach Mars. It is hoped that antimatter could be used as fuel for interplanetary travel or possibly interstellar travel, but it is also feared that if mankind ever gets the capabilities to do so, there could be the construction of antimatter weapons.
### Military
Because of its potential to release immense amounts of energy in contact with normal matter, there has been interest in various weapon uses, potentially enabling miniature warheads of pinhead-size to be more destructive than modern-day nuclear weapons. An antimatter particle colliding with a matter particle releases 100 percent of the energy contained within the particles, while an H-bomb only releases about seven percent of this energy. This gives a clue to how effective and powerful this force is. However, this development is still in early planning stages, though antimatter weapons are very popular in science fiction such as in Peter F. Hamilton's Night's Dawn Trilogy and Dan Brown's Angels and Demons where the production of antimatter leads to the possibility of use as both a fuel and highly effective weapon. Another use could be the creation of antimatter bullets of the correct material to cause human flesh to disappear and expel huge amounts of energy, turning an enemy soldier into a bomb.
## Antiuniverse
Dirac himself was the first to consider the existence of antimatter on an astronomical scale. But it was only after the confirmation of his theory, with the discovery of the positron, antiproton and antineutron that real speculation began on the possible existence of an antiuniverse. In the following years, motivated by basic symmetry principles, it was believed that the universe must consist of both matter and antimatter in equal amounts. If, however, there were an isolated system of antimatter in the universe, free from interaction with ordinary matter, no earthbound observation could distinguish its true content, as photons (being their own antiparticle) are the same whether they originate from a “universe” or an “antiuniverse.”
But assuming large zones of antimatter exist, there must be some boundary where antimatter atoms from the antimatter galaxies or stars will come into contact with normal atoms. In those regions a powerful flux of gamma rays would be produced. This has never been observed despite deployment of very sensitive instruments in space to detect them.
It is now thought that symmetry was broken in the early universe during a period of baryogenesis, when matter-antimatter symmetry was violated. Standard Big Bang cosmology tells us that the universe initially contained equal amounts of matter and antimatter: however particles and antiparticles evolved slightly differently. It was found that a particular heavy unstable particle, which is its own antiparticle, decays slightly more often to positrons (e+) than to electrons (e). How this accounts for the preponderance of matter over antimatter has not been completely explained. The Standard Model of particle physics does have a way of accommodating a difference between the evolution of matter and antimatter, but it falls short of explaining the net excess of matter in the universe by about 10 orders of magnitude.
After Dirac, science fiction writers produced myriad visions of antiworlds, antistars and antiuniverses, all made of antimatter, and it is still a common plot device; however, no positive evidence of such antiuniverses exists.
### Antihelium
The Balloon-borne Experiment with Superconducting Spectrometer (BESS) is searching for larger antinuclei, in particular antihelium, that are very unlikely to be produced by collisions. (One of the current experiments, under assumptions of current theory, would take 15 billion years on average to encounter a single antihelium atom made that way.[5])
## Notation
One way to denote an antiparticle is by adding a bar (or macron) over the particle's symbol. For example, the proton and antiproton are denoted as $\mathrm{p}\,$ and $\bar{\mathrm{p}}$, respectively. The same rule applies if you were to address a particle by its constituent components. A proton is made up of $\mathrm{u}\,$$\mathrm{u}\,$$\mathrm{d}\,$ quarks, so an antiproton must therefore be formed from $\bar{\mathrm{u}}$$\bar{\mathrm{u}}$$\bar{\mathrm{d}}$ antiquarks. Another convention is to distinguish particles by their electric charge. Thus, the electron and positron are denoted simply as e and e+.
## Value
In 1999, NASA calculated that antimatter was the most expensive substance on Earth, at about $62.5 trillion a gram ($1.75 quadrillion an ounce).[6] This is because production is difficult (only a few atoms are produced in reactions in particle accelerators) and because there is higher demand for the other uses of particle accelerators.
## Notes
1. Angels and Demons. CERN laboratories website. Retrieved October 29, 2007.
2. What's the Matter with Antimatter?. NASA Science News. Retrieved October 29, 2007.
3. Stanley K. Borowski. 1987. Comparison of Fusion/Antiproton Propulsion systems. NASA. Retrieved October 29, 2007.
4. James Bickford. Extraction of Antiparticles in Planetary Magnetic Fields. NIAC. Retrieved October 29, 2007.
5. P. Barry. 2007. "The hunt for Antihelium." Science News 171:296-300
6. Space Science News[1] NASA. Retrieved October 31, 2007.
## References
• Forward, Robert L. 2001. Mirror Matter: Pioneering Antimatter Physics. Lincoln, NE: Backinprint.com. ISBN 0595198171.
• Fraser, Gordon. 2002. Antimatter: The Ultimate Mirror. Cambridge, UK: Cambridge University Press. ISBN 0521893097.
• Santilli, Ruggero Maria. 2006. Isodual Theory of Antimatter: with applications to Antigravity, Grand Unification and Cosmology (Fundamental Theories of Physics). New York, NY: Springer. ISBN 1402045174. | {"extraction_info": {"found_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": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8390419483184814, "perplexity": 1284.0174244420891}, "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-2017-09/segments/1487501172017.60/warc/CC-MAIN-20170219104612-00039-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/436222/trying-to-understand-open-closed-subfunctors | # Trying to understand open (closed) subfunctors
I am trying to read about functor of points and I am struggling with the definition of open subfunctor.
The definition is the following. A subfunctor $\alpha\colon G\to F$, where $F,G\in \mathsf{Fct}(\mathsf{Rings},\mathsf{Sets})$ is called open (closed) if for any $\psi\colon h_{\mathrm{Spec}(R)}\to F$ the fibered product $G_\psi=G\times_F h_{\mathrm{Spec}(R)}$ yields a map $G_\psi\to h_{\mathrm{Spec}(R)}$ isomorphic to injection from the functor represented by some open (closed) subscheme of $\mathrm{Spec}(R)$.
Can you, please, explain why is it true that open subfunctors of $F=h_{\mathrm{Spec}(S)}$ are exactly the functors $G$ of the form $G(T)=\{\phi\in F(T)\mid \phi^*(I)T=T\}$ for some ideal $I\subset S$? This is the exercise $VI.6$ in Eisenbud-Harris "Geometry of Schemes". Can you, please, explain how to think about these subfunctors?
Thank you very much!
-
$G$ is an open subfunctor of $F = h_{\operatorname{Spec}(S)}$ means that for any $T,$ $G(T) = \operatorname{Hom}(\operatorname{Spec}(T), U)$ where $U\subseteq\operatorname{Spec}(S)$ is an open subscheme. Thus $\varphi\in G(T)$ implies that $\phi = \iota\circ\varphi\in F(T)$ where $\iota:U\hookrightarrow \operatorname{Spec}(S)$ is the inclusion, and there is an obvious bijective correpondence between such $T$-valued points of $\operatorname{Spec}(S)$ and $T$-valued point of $U.$
Thus, we simply need to realize that the condition "$\phi\in F(T)$ factors through $U$" is equivalent with the condition "$\phi^*(I)T = T,$ where $\phi^*:S\to T$ is the dual map, and $I$ is the ideal of the complement of $U.$"
For example, in the case $U = \operatorname{Spec}(S_f)$ for some $f\in S,$ we are asking that $\phi^*$ factors through $S_f$ which is equivalent to $\phi^*(f)\in T^\times$ is invertible. The complement of $U$ is $V(f),$ so what this boils down to is that $\phi^*(I)T = T,$ where $I=(f).$ I leave the general case to you.
Thanks a lot for your answer! By the way, the general case (the one that you've left for me) is also not difficult. $\phi$ factors through $U$ means that preimage $P=\phi^{-1}(V(I))\subset \operatorname{Spec}(T)$ of the complement $V(I)=\operatorname{Spec}(S)-U$ is empty. But this preimage is the closed subscheme of $\operatorname{Spec}(T)$ defined by the ideal $\phi^*(I)T$. $P$ is empty iff $\phi^*(I)T=T$. – Sasha Patotski Jul 4 '13 at 19:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.9690263867378235, "perplexity": 103.80125655498135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860115836.8/warc/CC-MAIN-20160428161515-00005-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/831/items/1.0104014 | # Open Collections
## UBC Theses and Dissertations
### Reduction of thio-molybdate in aqueous solutions Okita, Yoshiaki
#### Abstract
The high temperature behaviour of the molybdenum (VI) -sulphur (-II) -water system in the presence of an ammoniacal buffer was studied. At 150°C. all species of the form MoO(4-x)S(2¯/x) were shown to exist and the stability constants of mono-, di-, tri-, and tetra-thiomolybdate were 2.3 x 10(2) M.(-1), 3.5 x 10(5) M.(-2), 2.7 x 10(8) M.(-3), 7.0 x 10(10) M.(-4), respectively. There were strong indications of the formation of protonated species, Mo(SH)(6), in solutions containing low concentration of free ammonia. Application of reducing gases to this system produced a mixture of a sulphide and an oxide of molybdenum, whose composition depended on the initial composition of the solution. Under hydrogen, the reduction reaction was autocatalytic, rate being first order in product amount and hydrogen pressure. A mechanism was proposed in which the rate determining step was heterogeneous activation of hydrogen on the surfaces of precipitates followed by two paths, one to produce the sulphide and the other to produce the oxide. The proportion of the sulphide to the oxide was dependent on the solution composition, the higher the fractional distribution of tetra-thiomolybdate and the higher the concentration of hydrogen ion, the more the sulphide being produced. Under carbon monoxide the reduction reaction was found to have an induction period. The molybdenum in solution then followed a linear decrease in concentration with time. The slope of this plot showed Langmuir type of dependence on both molybdenum concentration and pressure. A mechanism was proposed in which the rate determining step was a slow decomposition of some complex between thiomolybdates and carbon monoxide adsorbed strongly on catalytic precipitate which was produced during the induction period. | {"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": 0.9479346871376038, "perplexity": 3070.007567321362}, "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/1631780057371.69/warc/CC-MAIN-20210922163121-20210922193121-00675.warc.gz"} |
https://www.physicsforums.com/threads/equivalent-resistance-of-pentagon-of-resistors.881980/ | # Equivalent resistance of pentagon of resistors
Tags:
1. Aug 14, 2016
### rohanlol7
1. The problem statement, all variables and given/known data
Here: http://imgur.com/a/5rRgu
2. Relevant equations
V=IR
3. The attempt at a solution
Basically from reversing the polarity of the cell and looking at a mirror image of the circuit i got the following relationships. First I(9)=0 then I(1)=-I(4) I(5)=-I(6) I(7)=-I(8). I believe i can also get these from a symmetry argument. From there I went ahead and wrote down equations for the currents at points A,B,C and O. I inserted the current through the cell to be I(10) and after a few equations i got I(10)=0 and so the equivalent resistance to be infinite ??????????? I made a mistake somewhere. Is it just a mistake during my calculations or is my approach wrong? I believe the error might be I(9)=0 but i'm not sure
Last edited by a moderator: Aug 14, 2016
2. Aug 14, 2016
### Staff: Mentor
Hints:
Use a symmetry argument to determine the potential differences between O and F, and O and A.
If I(9) is zero, what can you do with the resistor OC? (what's the potential drop across it?)
3. Aug 14, 2016
### rohanlol7
So basicaly i can simplify the circuit by removing OC completely from the diagram.
Now i just have a square of resistors. From the symmetry the pd accross OF and OA should be equal and so that should be E/2
Since the current through FD and AB should have the same magnitude their pd should be the same. Considering DO and OB these two should also have the same pd. and so the pd accross DB is zero. So i can now remove the segment DB ( from my square). From this the circuit is simplified to 2 identical systems of resistors in series where each 'system' is R and 2R in parrallel. So the total resistance is 4R/3. Is that correct ? ( this is preparation for a contest and the answers are not available online )
Also what was wrong with my approach before using currents?
4. Aug 14, 2016
### Staff: Mentor
Careful. While the magnitudes of potentials OD and OB may be the same, they will have different signs. Remember, the cell has a polarity and the symmetric currents have opposite directions.
Hard to say since you gave no details of your equations or your manipulations of them.
5. Aug 14, 2016
### Staff: Mentor
Consider instead replacing the resistor OC with a wire (short circuit). Since no current flows through it you have the choice of removing it entirely or replacing it with a short. Choosing the short circuit option gives you some parallel and series resistor reductions to exploit.
6. Aug 16, 2016
### TomHart
@rohanlol7, did you ever come up with a solution for this problem. I worked it out the long way (a pain) and confirmed that current i9 is indeed 0, and the other symmetric currents that you showed are also correct. So something definitely went awry in your calculation of I10=0 and infinite resistance. It has been so long since I have done this, I'm not even sure what what the method is called that I used. I think it is called node analysis since I summed currents into nodes. There are basically 4 unknowns: voltages B, C, D, and O. Once those are found, all currents and total effective resistance can be calculated.
7. Aug 16, 2016
### rohanlol7
Basically I started by assuming that
The potential at A is E and F is 0. from there I looked at currents and simplified the circuit by removing unnecessary wires. I'll post a solution tomorrow, going to bed now sorry
8. Aug 16, 2016
### TomHart
I did the same thing - assumed A=E and F=0. And as @gneill suggested, replacing the i9 resistor with a short allows you to parallel resistors. Continuing on from there, the remaining resistors can be reduced to a single overall resistor.
9. Aug 16, 2016
### Biker
I overlooked this before, What you need to prove that I9 = 0 is voltage law and node rule considering you have a lot of time to spend solving 7 or more question.
You could use mesh currents which is an easy way to prove that you will have 5 equation. You could manipulate four of them and you will get an interesting result
Tom, How did you make so it gets reduced to single resistor? What I was left with is some sort of two wheatstone bridges connected to each other which seems a good point to stop unless you use some other methods to calculate the equivalent resistance
10. Aug 16, 2016
### TomHart
@Biker, it is based on the symmetry assumption, which allows resistor OC to be replaced by a short (credit to @gneill). Once that is done, it reduces fairly easily by paralleling and seriesing (my own made-up words maybe) the remaining resistors.
To solve the long way, I ended up with 4 equations and 4 unknowns by summing currents into nodes B, C, D, and O.
For example, for node B: i2 = i5 + i7 --> (E-B)/R = (B-O)/R + (B-C)/R
I quickly realized that, for all 4 equations, R can be ignored because it cancels out.
EDIT: Sorry that my notation was kind of sloppy. The variable B, for example, really should be VB as it represents the voltage at node B. I was just trying to make it as easy on myself as I could.
11. Aug 16, 2016
### CWatters
As others have said it's easy using symmetry.
The impedance to the left of point O is the same as the impedance to the right so the voltage on point O is half that of the battery. Exactly the same applies to the node at the bottom. If both the nodes are at the same voltage then no current flows.
12. Aug 17, 2016
### rohanlol7
Okay so i have no idea if this is correct or not but it should be.
First We have I(1)=I(4), I(6)=I(5), I(7)=I(8) and I(9)=0 ( all of these by looking at the symmetry of the circuit )
If we can assume V(A)=E and V(F)=0.
Since I(1)=I(4), the pd across FO and OA is the same.
V(O)-0=V(A)-V(O)
V(O)=E/2
So I(1)= E/2R
Since I(9)=0 V(C)=V(O)=E/2 this implies that the pd across BC= pd across BO
So I(5)=I(2)/2
Applying kirchoffs rule ( forgot which one xD) in loop AOB :
I(2)R+I(5)R-I(1)R=0 and solving for I(2)
We get I(2)=2I(1)/3=E/3R.
Now the current through the Cell is I(10)
at A we get :
I(10)=I(2)+I(1)=E/3R + E/2R=5E/6R
If the resistance of the circuit id R(eff), we get E=I(10)*R(eff) solving for R(eff) we get R(eff)=6R/5
13. Aug 17, 2016
### rohanlol7
I posted a solution ( not sure if correct)
14. Aug 17, 2016
### SammyS
Staff Emeritus
That looks perfectly good.
Draft saved Draft deleted
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-6th-edition/chapter-3-polynomial-and-rational-functions-concept-and-vocabulary-check-page-373/2 | ## College Algebra (6th Edition)
Fill the blanks in with : $6x^{3}...3x...2x^{2}...7x^{2}$
See: Long Division of Polynomials. 1. Arrange... 2. Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient. We write like terms lined up. ------------- We divide $6x^{3}$ with $3x,$ We obtain $2x^{2}$ We write this above the square term in the divisor, $7x^{2}.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9144131541252136, "perplexity": 496.1012316154519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867424.77/warc/CC-MAIN-20180625033646-20180625053646-00021.warc.gz"} |
https://blogs.accu.org/category/calculus/ | ## Adapt Or Try – a.k.
Over the last few months we have been looking at how we might approximate the solutions to ordinary differential equations, or ODEs, which define the derivative of one variable with respect to another with a function of them both. Firstly with the first order Euler method, then with the second order midpoint method and finally with the generalised Runge-Kutta method.
Unfortunately all of these approaches require the step length to be fixed and specified in advance, ignoring any information that we might use to adjust it during the iteration in order to better trade off the efficiency and accuracy of the approximation. In this post we shall try to automatically modify the step lengths to yield an optimal, or at least reasonable, balance.
## A Kutta Above The Rest – a.k.
We have recently been looking at ordinary differential equations, or ODEs, which relate the derivatives of one variable with respect to another to them with a function so that we cannot solve them with plain integration. Whilst there are a number of tricks for solving such equations if they have very specific forms, we typically have to resort to approximation algorithms such as the Euler method, with first order errors, and the midpoint method, with second order errors.
In this post we shall see that these are both examples of a general class of algorithms that can be accurate to still greater orders of magnitude.
## Finding The Middle Ground – a.k.
Last time we saw how we can use Euler's method to approximate the solutions of ordinary differential equations, or ODEs, which define the derivative of one variable with respect to another as a function of them both, so that they cannot be solved by direct integration. Specifically, it uses Taylor's theorem to estimate the change in the first variable that results from a small step in the second, iteratively accumulating the results for steps of a constant length to approximate the value of the former at some particular value of the latter.
Unfortunately it isn't very accurate, yielding an accumulated error proportional to the step length, and so this time we shall take a look at a way to improve it.
## Out Of The Ordinary – a.k.
Several years ago we saw how to use the trapezium rule to approximate integrals. This works by dividing the interval of integration into a set of equally spaced values, evaluating the function being integrated, or integrand, at each of them and calculating the area under the curve formed by connecting adjacent points with straight lines to form trapeziums.
This was an improvement over an even more rudimentary scheme which instead placed rectangles spanning adjacent values with heights equal to the values of the function at their midpoints to approximate the area. Whilst there really wasn't much point in implementing this since it offers no advantage over the trapezium rule, it is a reasonable first approach to approximating the solutions to another type of problem involving calculus; ordinary differential equations, or ODEs.
## Finally On An Ethereal Orrery – student
Over the course of the year, my fellow students and I have been experimenting with an ethereal orrery which models the motion of heavenly bodies using nought but Sir N-----'s laws of gravitation and motion. Whilst the consequences of those laws are not generally subject to solution by mathematical reckoning, we were able to approximate them with a scheme that admitted errors of the order of the sixth power of the steps in time by which we advanced the positions of those bodies.
We have thus far employed it to model the solar system itself, uniformly distributed bodies of matter and the accretion of bodies that are close to Earth's orbit about the Sun. Whilst we were most satisfied by its behaviour, I should now like to report upon an altogether more surprising consequence of its engine's action.
## Further Still On An Ethereal Orrery – student
Recently, my fellow students and I constructed a mathematical orrery which modelled the motion of heavenly bodies employing Sir N-----'s laws of gravitation and motion, rather than clockwork, as its engine. Those laws state that bodies are attracted toward each other with a force proportional to the product of their masses divided by the square of the distance between them, that a body will remain at rest or in constant motion unless a force acts upon it, that if a force acts upon it then it will be accelerated in the direction of that force at a rate proportional to its strength divided by its mass and that, if so, it will reciprocate with an opposing force of equal strength.
Its operation was most satisfactory, which set us to wondering whether we might use its engine to investigate the motions of entirely hypothetical arrangements of heavenly bodies and I should now like to report upon our progress in doing so.
## Further On An Ethereal Orrery – student
Last time we met we spoke of my fellow students' and my interest in constructing a model of the motion of heavenly bodies using mathematical formulae in the place of brass. In particular we have sought to do so from first principals using Sir N-----'s law of universal gravitation, which states that the force attracting two bodies is proportional to the product of their masses divided by the square of the distance between them, and his laws of motion, which state that a body will remain at rest or in constant motion unless a force acts upon it, that it will be accelerated in the direction of that force at a rate proportional to its magnitude divided the body's mass and that a force acting upon it will be met with an equal force in the opposite direction.
Whilst Sir N----- showed that a pair of bodies traversed conic sections under gravity, being those curves that arise from the intersection of planes with cones, the general case of several bodies has proved utterly resistant to mathematical reckoning. We must therefore approximate the equations of motion and I shall now report on our first attempt at doing so.
## On An Ethereal Orrery – student
My fellow students and I have lately been wondering whether we might be able to employ Professor B------'s Experimental Clockwork Mathematical Apparatus to fashion an ethereal orrery, making a model of the heavenly bodies with equations rather than brass.
In particular we have been curious as to whether we might construct such a model using nought but Sir N-----'s law of universal gravitation, which posits that those bodies are attracted to one another with a force that is proportional to the product of their masses divided by the square of the distance between them, and laws of motion, which posit that a body will remain at rest or move with constant velocity if no force acts upon it, that if a force acts upon it then it will be accelerated at a rate proportional to that force divided by its mass in the direction of that force and that it in return exerts a force of equal strength in the opposite direction.
## A Measure Of Borel Weight – a.k.
In the last few posts we have implemented a type to represent Borel sets of the real numbers, which are the subsets of them that can be created with countable unions of intervals with closed or open lower and upper bounds. Whilst I would argue that doing so was a worthwhile exercise in its own right, you may be forgiven for wondering what Borel sets are actually for and so in this post I shall try to justify the effort that we have spent on them.
## A Borel Universe – a.k.
Last time we took a look at Borel sets of real numbers, which are subsets of the real numbers that can be represented as unions of countable sets of intervals Ii. We got as far as implementing the `ak.borelInterval` type to represent an interval as a pair of `ak.borelBound` objects holding its lower and upper bounds.
With these in place we're ready to implement a type to represent Borel sets and we shall do exactly that in this post. | {"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": 0.8747831583023071, "perplexity": 300.5247744050689}, "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/1674764500837.65/warc/CC-MAIN-20230208155417-20230208185417-00412.warc.gz"} |
https://www.physicsforums.com/threads/two-charges-located-along-x-find-all-values.809196/ | Two charges located along x. Find all values?
1. Apr 18, 2015
nawand
1. The problem statement, all variables and given/known data
The first one has charge Q is at "-a" from origin the second is at "+3a" which is an unknown charge.What are the values for unknown charge if at the origin the net electric field they produce with magnitude 2Ke(Q/a2)
2. Relevant equations
Ke (Q/a2)i - Ke(q/9a2)=2Ke(Q/a2)i
if charge is negative
or
if the charge is positive
Ke (Q/a2)i - Ke(q/9a2)=-2Ke(Q/a2)i
3. The attempt at a solution
The question is how to choose signs for equation.
Attempt:
Assuming charge one is pointed towards the origin along x axis so the sign will be "+i" and the second charge is pointed towards the origin in the negative x direction so tthe sign will be "-i".
Two possible values are at the origin: "+" or "-" 2Ke(Q/a2) .
If you choose "+" at the origin the answer will be -9Q. And for case chosen "-" sign in the origin, the answer will be 27Q so the particle has positive charge.
I wondering if this is the explanation?
2. Apr 18, 2015
Staff: Mentor
Correct.
3. Apr 18, 2015
nawand
Thanks
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https://zwaltman.wordpress.com/2016/12/10/60-closure-of-min-on-exponential-random-variables/ | # Today I Learned
Some of the things I've learned every day since Oct 10, 2016
## 60: Closure of MIN on Exponential Random Variables
The minimum of random variables $E_1, E_2, ... , E_n$ with corresponding parameters $\lambda _1, \lambda _2, ... , \lambda _n$ is itself a random variable with parameter $\lambda = \sum _{i = 1} ^n \lambda _i$.
It is not, however, the case that the maximum is also exponentially distributed. | {"extraction_info": {"found_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": 3, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9238855838775635, "perplexity": 548.8149744578261}, "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/1508187827662.87/warc/CC-MAIN-20171023235958-20171024015958-00699.warc.gz"} |
https://terrytao.wordpress.com/2018/04/12/246c-notes-2-circle-packings-conformal-maps-and-quasiconformal-maps/ | We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map ${f: U \rightarrow V}$ from an open subset ${U}$ of the complex plane to another open set ${V}$ is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of ${f}$ to be nowhere vanishing. We then say that the two open sets ${U,V}$ are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation ${f( z_0 + \Delta z) \approx f(z_0) + f'(z_0) \Delta z + O( |\Delta z|^2)}$ we see that they almost preserve small circles, indeed for ${\varepsilon}$ small the circle ${\{ z: |z-z_0| = \varepsilon\}}$ will approximately map to ${\{ w: |w - f(z_0)| = |f'(z_0)| \varepsilon \}}$.
Theorem 1 (Riemann mapping theorem) Let ${U}$ be a simply connected open subset of ${{\bf C}}$ that is not all of ${{\bf C}}$. Then ${U}$ is conformally equivalent to the unit disk ${D(0,1)}$.
This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps ${f: U \rightarrow D(0,1)}$ from ${U}$ to ${D(0,1)}$ that map some fixed point ${z_0 \in U}$ to ${0}$, pick one that maximises the magnitude ${|f'(z_0)|}$ of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If ${f(U)}$ avoids some point in ${D(0,1)}$, one can compose ${f}$ with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase ${|f'(z_0)|}$ without destroying injectivity; see the previous lecture notes for details. The conformal map ${\phi: U \rightarrow D(0,1)}$ is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points ${z_0,z_1}$ in ${U}$, and requiring ${\phi(z_0)}$ to be zero and ${\phi(z_1)}$ to be positive real.
It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).
To make the above discussion more precise we need some notation.
Definition 2 (Circle packing) A (finite) circle packing is a finite collection ${(C_j)_{j \in J}}$ of circles ${C_j = \{ z \in {\bf C}: |z-z_j| = r_j\}}$ in the complex numbers indexed by some finite set ${J}$, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices ${\{z_j: j \in J \}}$ are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)
It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.
The first basic theorem in the subject is the following converse statement:
Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.
Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:
Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.
Exercise 5 Let ${G}$ be a connected planar graph with ${n \geq 3}$ vertices. Show that the following are equivalent:
• (i) ${G}$ is a maximal planar graph.
• (ii) ${G}$ has ${3n-6}$ edges.
• (iii) Every drawing ${D}$ of ${G}$ divides the plane into faces that have three edges each. (This includes one unbounded face.)
• (iv) At least one drawing ${D}$ of ${G}$ divides the plane into faces that have three edges each.
(Hint: use Euler’s formula ${V-E+F=2}$, where ${F}$ is the number of faces including the unbounded face.)
Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:
Conjecture 6 (Informal Thurston conjecture) Let ${U}$ be a simply connected domain, with two distinct points ${z_0,z_1}$. Let ${\phi: U \rightarrow D(0,1)}$ be the conformal map from ${U}$ to ${D(0,1)}$ that maps ${z_0}$ to the origin and ${z_1}$ to a positive real. For any small ${\varepsilon>0}$, let ${{\mathcal C}_\varepsilon}$ be the portion of the regular hexagonal circle packing by circles of radius ${\varepsilon}$ that are contained in ${U}$, and let ${{\mathcal C}'_\varepsilon}$ be an circle packing of ${D(0,1)}$ with all “boundary circles” tangent to ${D(0,1)}$, giving rise to an “approximate map” ${\phi_\varepsilon: U_\varepsilon \rightarrow D(0,1)}$ defined on the subset ${U_\varepsilon}$ of ${U}$ consisting of the circles of ${{\mathcal C}_\varepsilon}$, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that ${\phi_\varepsilon(z_0)}$ is zero and ${\phi_\varepsilon(z_1)}$ is a positive real. Then ${\phi_\varepsilon}$ converges to ${\phi}$ as ${\varepsilon \rightarrow 0}$.
A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation ${\frac{\partial \phi}{\partial \overline{z}} = 0}$, while (sufficiently smooth) quasiconformal maps only obey an inequality ${|\frac{\partial \phi}{\partial \overline{z}}| \leq \frac{K-1}{K+1} |\frac{\partial \phi}{\partial z}|}$. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map ${\phi_\varepsilon}$ alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).
— 1. Proof of the circle packing theorem —
We loosely follow the treatment of Beardon and Stephenson. It is slightly more convenient to temporarily work in the Riemann sphere ${{\bf C} \cup \{\infty\}}$ rather than the complex plane ${{\bf C}}$, in order to more easily use Möbius transformations. (Later we will make another change of venue, working in the Poincaré disk ${D(0,1)}$ instead of the Riemann sphere.)
Define a Riemann sphere circle to be either a circle in ${{\bf C}}$ or a line in ${{\bf C}}$ together with ${\infty}$, together with one of the two components of the complement of this circle or line designated as the “interior”. In the case of a line, this “interior” is just one of the two half-planes on either side of the line; in the case of the circle, this is either the usual interior or the usual exterior plus the point at infinity; in the last case, we refer to the Riemann sphere circle as an exterior circle. (One could also equivalently work with an orientation on the circle rather than assigning an interior, since the interior could then be described as the region to (say) the left of the circle as one traverses the circle along the indicated orientation.) Note that Möbius transforms map Riemann sphere circles to Riemann sphere circles. If one views the Riemann sphere as a geometric sphere in Euclidean space ${{\bf R}^3}$, then Riemann sphere circles are just circles on this geometric sphere, which then have a centre on this sphere that lies in the region designated as the interior of the circle. We caution though that this “Riemann sphere” centre does not always correspond to the Euclidean notion of the centre of a circle. For instance, the real line, with the upper half-plane designated as interior, will have ${i}$ as its Riemann sphere centre; if instead one designates the lower half-plane as the interior, the Riemann sphere centre will now be ${-i}$. We can then define a Riemann sphere circle packing in exact analogy with circle packings in ${{\bf C}}$, namely finite collections of Riemann sphere circles whose interiors are disjoint and whose union is connected; we also define the nerve as before. This is now a graph that can be drawn in the Riemann sphere, using great circle arcs in the Riemann sphere rather than line segments; it is also planar, since one can apply a Möbius transformation to move all the points and edges of the drawing away from infinity.
By Exercise 5, a maximal planar graph with at least three vertices can be drawn as a triangulation of the Riemann sphere. If there are at least four vertices, then it is easy to see that each vertex has degree at least three (a vertex of degree zero, one or two in a triangulation with simple edges will lead to a connected component of at most three vertices). It is a topological fact, not established here, that any two triangulations of such a graph are homotopic up to reflection (to reverse the orientation). If a Riemann sphere circle packing has the nerve of a maximal planar graph ${G}$ of at least four vertices, then we see that this nerve induces an explicit triangulation of the Riemann sphere by connecting the centres of any pair of tangent circles with the great circle arc that passes through the point of tangency. If ${G}$ was not maximal, one no longer gets a triangulation this way, but one still obtains a partition of the Riemann sphere into spherical polygons.
We remark that the triangles in this triangulation can also be described purely from the abstract graph ${G}$. Define a triangle in ${G}$ to be a triple ${w_1,w_2,w_3}$ of vertices in ${G}$ which are all adjacent to each other, and such that the removal of these three vertices from ${G}$ does not disconnect the graph. One can check that there is a one-to-one correspondence between such triangles in a maximal planar graph ${G}$ and the triangles in any Riemann sphere triangulation of this graph.
Theorems 3, 4 are then a consequence of
Theorem 7 (Riemann sphere circle packing theorem) Let ${G}$ be a maximal planar graph with at least four vertices, drawn as a triangulation of the Riemann sphere. Then there exists a Riemann sphere circle packing with nerve ${G}$ whose triangulation is homotopic to the given triangulation. Furthermore, this packing is unique up to Möbius transformations.
Exercise 8 Deduce Theorems 3, 4 from Theorem 7. (Hint: If one has a non-maximal planar graph for Theorem 3, add a vertex at the interior of each non-triangular face of a drawing of that graph, and connect that vertex to the vertices of the face, to create a maximal planar graph to which Theorem 4 or Theorem 7 can be applied. Then delete these “helper vertices” to create a packing of the original planar graph that does not contain any “unwanted” tangencies. You may use without proof the above assertion that any two triangulations of a maximal planar graph are homotopic up to reflection.)
Exercise 9 Verify Theorem 7 when ${G}$ has exactly four vertices. (Hint: for the uniqueness, one can use Möbius transformations to move two of the circles to become parallel lines.)
To prove this theorem, we will make a reduction with regards to the existence component of Theorem 7. For technical reasons we will need to introduce a notion of non-degeneracy. Let ${G}$ be a maximal planar graph with at least four vertices, and let ${v}$ be a vertex in ${G}$. As discussed above, the degree ${d}$ of ${v}$ is at least three. Writing the neighbours of ${v}$ in clockwise or counterclockwise order (with respect to a triangulation) as ${v_1,\dots,v_d}$ (starting from some arbitrary neighbour), we see that each ${v_i}$ is adjacent to ${v_{i-1}}$ and ${v_{i+1}}$ (with the conventions ${v_0=v_d}$ and ${v_{d+1}=v_1}$). We say that ${v}$ is non-degenerate if there are no further adjacencies between the ${v_1,\dots,v_d}$, and if there is at least one further vertex in ${G}$ besides ${v,v_1,\dots,v_d}$. Here is another characterisation:
Exercise 10 Let ${G}$ be a maximal planar graph with at least four vertices, let ${v}$ be a vertex in ${G}$, and let ${v_1,\dots,v_d}$ be the neighbours of ${v}$. Show that the following are equivalent:
• (i) ${v}$ is non-degenerate.
• (ii) The graph ${G \backslash \{ v, v_1, \dots, v_d \}}$ is connected and non-empty, and every vertex in ${v_1,\dots,v_d}$ is adjacent to at least one vertex in ${G \backslash \{ v, v_1, \dots, v_d \}}$.
We will then derive Theorem 7 from
Theorem 11 (Inductive step) Let ${G}$ be a maximal planar graph with at least four vertices ${V}$, drawn as a triangulation of the Riemann sphere. Let ${v}$ be a non-degenerate vertex of ${G}$, and let ${G - \{v\}}$ be the graph formed by deleting ${v}$ (and edges emenating from ${v}$) from ${G}$. Suppose that there exists a Riemann sphere circle packing ${(C_w)_{w \in V \backslash \{v\}}}$ whose nerve is at least ${G - \{v\}}$ (that is, ${C_w}$ and ${C_{w'}}$ are tangent whenever ${w,w'}$ are adjacent in ${G - \{v\}}$, although we also allow additional tangencies), and whose associated subdivision of the Riemann sphere into spherical polygons is homotopic to the given triangulation with ${v}$ removed. Then there is a Riemann sphere circle packing ${(\tilde C_w)_{w \in V}}$ with nerve ${G}$ whose triangulation is homotopic to the given triangulation. Furthermore this circle packing ${(\tilde C_w)_{w \in V}}$ is unique up to Möbius transformations.
Let us now see how Theorem 7 follows from Theorem 11. Fix ${G}$ as in Theorem 7. By Exercise 9 and induction we may assume that ${G}$ has at least five vertices, and that the claim has been proven for any smaller number of vertices.
First suppose that ${G}$ contains a non-degenerate vertex ${v}$. Let ${v_1,\dots,v_d}$ be the the neighbours of ${v}$. One can then form a new graph ${G'}$ with one fewer vertex by deleting ${v}$, and then connecting ${v_3,\dots,v_{d-1}}$ to ${v_1}$ (one can think of this operation as contracting the edge ${\{v,v_1\}}$ to a point). One can check that this is still a maximal planar graph that can triangulate the Riemann sphere in a fashion compatible with the original triangulation of ${G}$ (in that all the common vertices, edges, and faces are unchanged). By induction hypothesis, ${G'}$ is the nerve of a circle packing that is compatible with this triangulation, and hence this circle packing has nerve at least ${G - \{v\}}$. Applying Theorem 11, we then obtain the required claim for ${G}$.
Now suppose that ${G}$ contains a degenerate vertex ${v}$. Let ${v_1,\dots,v_d}$ be the neighbours of ${v}$ traversed in order. By hypothesis, there is an additional adjacency between the ${v_1,\dots,v_d}$; by relabeling we may assume that ${v_1}$ is adjacent to ${v_k}$ for some ${3 \leq k \leq d-1}$. The vertices ${V}$ in ${G}$ can then be partitioned as
$\displaystyle V = \{v\} \cup \{ v_1,\dots,v_d\} \cup V_1 \cup V_2$
where ${V_1}$ denotes those vertices in ${V \backslash \{ v_1,\dots,v_d\}}$ that lie in the region enclosed by the loop ${v_1,\dots,v_k, v_1}$ that does not contain ${v}$, and ${V_2}$ denotes those vertices in ${V \backslash \{ v_1,\dots,v_d\}}$ that lie in the region enclosed by the loop ${v_k,\dots,v_d,v_1, v_k}$ that does not contain ${v}$. One can then form two graphs ${G_1, G_2}$, formed by restricting ${G}$ to the vertices ${\tilde V_1 := \{v, v_1,\dots,v_k\} \cup V_1}$ and ${\tilde V_2 := \{ v, v_k, \dots, v_d, v_1\} \cup V_2}$ respectively; furthermore, these graphs are also maximal planar (with triangulations that are compatible with those of ${G}$). By induction hypothesis, we can find a circle packing ${(C_w)_{w \in \tilde V_1}}$ with nerve ${G_1}$, and a circle packing ${(C'_w)_{w \in \tilde V_2}}$ with nerve ${G_2}$. Note that the circles ${C_v, C_{v_1}, C_{v_k}}$ are mutually tangent, as are ${C'_v, C'_{v_1}, C'_{v_k}}$. By applying a Möbius transformation one may assume that these circles agree, thus (cf. Exercise 9) ${C_v = C'_v}$, ${C_{v_1} = C'_{v_1}, C_{v_k} = C'_{v_k}}$. The complement of the these three circles (and their interiors) determine two connected “interstitial” regions (that are in the shape of an arbelos, up to Möbius transformation); one can check that the remaining circles in ${(C_w)_{w \in \tilde V_1}}$ will lie in one of these regions, and the remaining circles in ${(C'_w)_{w \in \tilde V_2}}$ lie in the other. Hence one can glue these circle packings together to form a single circle packing with nerve ${G}$, which is homotopic to the given triangulation. Also, since a Möbius transformation that fixes three mutually tangent circles has to be the identity, the uniqueness of this circle packing up to Möbius transformations follows from the uniqueness for the two component circle packings ${(C_w)_{w \in \tilde V_1}}$, ${(C'_w)_{w \in \tilde V_2}}$.
It remains to prove Theorem 11. To help fix the freedom to apply Möbius transformations, we can normalise the target circle packing ${(\tilde C_w)_{w \in V}}$ so that ${\tilde C_v}$ is the exterior circle ${\{ |z|=1\}}$, thus all the other circles ${\tilde C_w}$ in the packing will lie in the closed unit disk ${\overline{D(0,1)}}$. Similarly, by applying a suitable Möbius transformation one can assume that ${\infty}$ lies outside of the interior of all the circles ${C_w}$ in the original packing, and after a scaling one may then assume that all the circles ${C_w}$ lie in the unit disk ${D(0,1)}$.
At this point it becomes convenient to switch from the “elliptic” conformal geometry of the Riemann sphere ${{\bf C} \cup \{\infty\}}$ to the “hyperbolic” conformal geometry of the unit disk ${D(0,1)}$. Recall that the Möbius transformations that preserve the disk ${D(0,1)}$ are given by the maps
$\displaystyle z \mapsto e^{i\theta} \frac{z-\alpha}{1-\overline{\alpha} z} \ \ \ \ \ (1)$
for real ${\theta}$ and ${\alpha \in D(0,1)}$ (see Theorem 19 of these notes). It comes with a natural metric that interacts well with circles:
Exercise 12 Define the Poincaré distance ${d(z_1,z_2)}$ between two points of ${D(0,1)}$ by the formula
$\displaystyle d(z_1,z_2) := 2 \mathrm{arctanh} |\frac{z_1-z_2}{1-z_1 \overline{z_2}}|.$
Given a measurable subset ${E}$ of ${D(0,1)}$, define the hyperbolic area of ${E}$ to be the quantity
$\displaystyle \mathrm{area}(E) := \int_E \frac{4\ dx dy}{(1-|z|^2)^2}$
where ${dx dy}$ is the Euclidean area element on ${D(0,1)}$.
• (i) Show that the Poincaré distance is invariant with respect to Möbius automorphisms of ${D(0,1)}$, thus ${d(Tz_1, Tz_2) = d(z_1,z_2)}$ whenever ${T}$ is a transformation of the form (1). Similarly show that the hyperbolic area is invariant with respect to such transformations.
• (ii) Show that the Poincaré distance defines a metric on ${D(0,1)}$. Furthermore, show that any two distinct points ${z_1,z_2}$ are connected by a unique geodesic, which is a portion of either a line or a circle that meets the unit circle orthogonally at two points. (Hint: use the symmetries of (i) to normalise the points one is studying.)
• (iii) If ${C}$ is a circle in the interior of ${D(0,1)}$, show that there exists a point ${z_C}$ in ${D(0,1)}$ and a positive real number ${r_C}$ (which we call the hyperbolic center and hyperbolic radius respectively) such that ${C = \{ z \in D(0,1): d(z,z_C) = r_C \}}$. (In general, the hyperbolic center and radius will not quite agree with their familiar Euclidean counterparts.) Conversely, show that for any ${z_C \in D(0,1)}$ and ${r_C > 0}$, the set ${\{ z \in D(0,1): d(z,z_C) = r_C \}}$ is a circle in ${D(0,1)}$.
• (iv) If two circles ${C_1, C_2}$ in ${D(0,1)}$ are externally tangent, show that the geodesic connecting the hyperbolic centers ${z_{C_1}, z_{C_2}}$ passes through the point of tangency, orthogonally to the two tangent circles.
Exercise 13 (Schwarz-Pick theorem) Let ${f: D(0,1) \rightarrow D(0,1)}$ be a holomorphic map. Show that ${d(f(z_1),f(z_2)) \leq d(z_1,z_2)}$ for all ${z_1,z_2 \in D(0,1)}$. If ${z_1 \neq z_2}$, show that equality occurs if and only if ${f}$ is a Möbius automorphism (1) of ${D(0,1)}$. (This result is known as the Schwarz-Pick theorem.)
We will refer to circles that lie in the closure ${\overline{D(0,1)}}$ of the unit disk as hyperbolic circles. These can be divided into the finite radius hyperbolic circles, which lie in the interior of the unit disk (as per part (iii) of the above exercise), and the horocycles, which are internally tangent to the unit circle. By convention, we view horocycles as having infinite radius, and having center at their point of tangency to the unit circle; they can be viewed as the limiting case of finite radius hyperbolic circles when the radius goes to infinity and the center goes off to the boundary of the disk (at the same rate as the radius, as measured with respect to the Poincaré distance). We write ${C(p,r)}$ for the hyperbolic circle with hyperbolic centre ${p}$ and hyperbolic radius ${r}$ (thus either ${0 < r < \infty}$ and ${p \in D(0,1)}$, or ${r = \infty}$ and ${p}$ is on the unit circle); there is an annoying caveat that when ${r=\infty}$ there is more than one horocycle ${C(p,\infty)}$ with hyperbolic centre ${p}$, but we will tolerate this breakdown of functional dependence of ${C}$ on ${p}$ and ${r}$ in order to simplify the notation. A hyperbolic circle packing is a circle packing ${(C(p_v,r_v))_{v \in V}}$ in which all circles are hyperbolic circles.
We also observe that the geodesic structure extends to the boundary of the unit disk: for any two distinct points ${z_1,z_2}$ in ${\overline{D(0,1)}}$, there is a unique geodesic that connects them.
In view of the above discussion, Theorem 7 may now be formulated as follows:
Theorem 14 (Inductive step, hyperbolic formulation) Let ${G}$ be a maximal planar graph with at least four vertices ${V}$, let ${v}$ be a non-degenerate vertex of ${G}$, and let ${v_1,\dots,v_d}$ be the vertices adjacent to ${v}$. Suppose that there exists a hyperbolic circle packing ${(C(p_w,r_w))_{w \in V \backslash \{v\}}}$ whose nerve is at least ${G - \{v\}}$. Then there is a hyperbolic circle packing ${(C(\tilde p_w,\tilde r_w))_{V \backslash \{v\}}}$ homotopic to ${(C(p_w,r_w))_{w \in V \backslash \{v\}}}$ such that the boundary circles ${C(\tilde p_{v_j}, \tilde r_{v_j})}$, ${j=1,\dots,d}$ are all horocycles. Furthermore, this packing is unique up to Möbius automorphisms (1) of the disk ${D(0,1)}$.
Indeed, once one adjoints the exterior unit circle to ${(C(p_w,r_w))_{w \in V \backslash \{v\}}}$, one obtains a Riemann sphere circle packing whose nerve is at least ${G}$, and hence equal to ${G}$ since ${G}$ is maximal.
To prove this theorem, the intuition is to “inflate” the hyperbolic radius of the circles of ${C_w}$ until the boundary circles all become infinite radius (i.e., horocycles). The difficulty is that one cannot just arbitrarily increase the radius of any given circle without destroying the required tangency properties. The resolution to this difficulty given in the work of Beardon and Stephenson that we are following here was inspired by Perron’s method of subharmonic functions, in which one faced an analogous difficulty that one could not easily manipulate a harmonic function without destroying its harmonicity. There, the solution was to work instead with the more flexible class of subharmonic functions; here we similarly work with the concept of a subpacking.
We will need some preliminaries to define this concept precisely. We first need some hyperbolic trigonometry. We define a hyperbolic triangle to be the solid (and closed) region in ${\overline{D(0,1)}}$ enclosed by three distinct points ${z_1,z_2,z_3}$ in ${\overline{D(0,1)}}$ and the geodesic arcs connecting them. (Note that we allow one or more of the vertices to be on the boundary of the disk, so that the sides of the triangle could have infinite length.) Let ${T := (0,+\infty]^3 \backslash \{ (\infty,\infty,\infty)\}}$ be the space of triples ${(r_1,r_2,r_3)}$ with ${0 < r_1,r_2,r_3 \leq \infty}$ and not all of ${r_1,r_2,r_3}$ infinite. We say that a hyperbolic triangle with vertices ${p_1,p_2,p_3}$ is a ${(r_1,r_2,r_3)}$-triangle if there are hyperbolic circles ${C(p_i,r_1), C(p_2,r_2), C(p_3,r_3)}$ with the indicated hyperbolic centres and hyperbolic radii that are externally tangent to each other; note that this implies that the sidelengths opposite ${p_1,p_2,p_3}$ have length ${r_2+r_3, r_1+r_3, r_1+r_2}$ respectively (see Figure 3 of Beardon and Stephenson). It is easy to see that for any ${(r_1,r_2,r_3) \in T}$, there exists a unique ${(r_1,r_2,r_3)}$-triangle in ${\overline{D(0,1)}}$ up to reflections and Möbius automorphisms (use Möbius transforms to fix two of the hyperbolic circles, and consider all the circles externally tangent to both of these circles; the case when one or two of the ${r_1,r_2,r_3}$ are infinite may need to be treated separately.). As a consequence, there is a well defined angle ${\alpha_i(r_1,r_2,r_3) \in [0,\pi)}$ for ${i=1,2,3}$ subtended by the vertex ${p_i}$ of an ${(r_1,r_2,r_3)}$ triangle. We need some basic facts from hyperbolic geometry:
Exercise 15 (Hyperbolic trigonometry)
• (i) (Hyperbolic cosine rule) For any ${0 < r_1,r_2,r_3 < \infty}$, show that the quantity ${\cos \alpha_1(r_1,r_2,r_3)}$ is equal to the ratio
$\displaystyle \frac{\cosh( r_1+r_2) \cosh(r_1+r_3) - \cosh(r_2+r_3)}{\sinh(r_1+r_2) \sinh(r_1+r_3)}.$
Furthermore, establish the limiting angles
$\displaystyle \alpha_1(\infty,r_2,r_3) = \alpha_1(\infty,\infty,r_3) = \alpha_1(\infty,r_2,\infty) = 0$
$\displaystyle \cos \alpha_1(r_1,\infty,r_3) = \frac{\cosh(r_1+r_3) - \exp(r_3-r_1)}{\sinh(r_1+r_3)}$
$\displaystyle \cos \alpha_1(r_1,r_2,\infty) = \frac{\cosh(r_1+r_2) - \exp(r_2-r_1)}{\sinh(r_1+r_2)}$
$\displaystyle \cos \alpha_1(r_1,\infty,\infty) = 1 - 2\exp(-2r_1).$
(Hint: to facilitate computations, use a Möbius transform to move the ${p_1}$ vertex to the origin when the radius there is finite.) Conclude in particular that ${\alpha_1: T \rightarrow [0,\pi)}$ is continuous (using the topology of the extended real line for each component of ${T}$). Discuss how this rule relates to the Euclidean cosine rule in the limit as ${r_1,r_2,r_3}$ go to zero. Of course, by relabeling one obtains similar formulae for ${\alpha_2(r_1,r_2,r_3)}$ and ${\alpha_3(r_1,r_2,r_3)}$.
• (ii) (Area rule) Show that the area of a hyperbolic triangle is given by ${\pi - \alpha_1-\alpha_2-\alpha_3}$, where ${\alpha_1,\alpha_2,\alpha_3}$ are the angles of the hyperbolic triangle. (Hint: there are several ways to proceed. For instance, one can prove this for small hyperbolic triangles (of diameter ${O(\varepsilon)}$) up to errors of size ${o(\varepsilon^2)}$ after normalising as in (ii), and then establish the general case by subdividing a large hyperbolic triangle into many small hyperbolic triangles. This rule is also a special case of the Gauss-Bonnet theorem in Riemannian geometry. One can also first establish the case when several of the radii are infinite, and use that to derive finite cases.) In particular, the area ${\mathrm{Area}(r_1,r_2,r_3)}$ of a ${(r_1,r_2,r_3)}$-triangle is given by the formula
$\displaystyle \pi - \alpha_1(r_1,r_2,r_3) - \alpha_2(r_1,r_2,r_3) - \alpha_3(r_1,r_2,r_3). \ \ \ \ \ (2)$
• (iii) Show that the area of the interior of a hyperbolic circle ${C(p,r)}$ with ${r<\infty}$ is equal to ${4\pi \sinh^2(r/2)}$.
Henceforth we fix ${G, v, v_1,\dots,v_d, {\mathcal C} = (C(p_w,r_w))_{w \in V \backslash \{v\}}}$ as in Theorem 14. We refer to the vertices ${v_1,\dots,v_d}$ as boundary vertices of ${G - \{v\}}$ and the remaining vertices as interior vertices; edges between boundary vertices are boundary edges, all other edges will be called interior edges (including edges that have one vertex on the boundary). Triangles in ${G -\{v\}}$ that involve two boundary vertices (and thus necessarily one interior vertex) will be called boundary triangles; all other triangles (including ones that involve one boundary vertex) will be called interior triangles. To any triangle ${w_1,w_2,w_3}$ of ${G - \{v\}}$, we can form the hyperbolic triangle ${\Delta_{\mathcal C}(w_1,w_2,w_3)}$ with vertices ${p_{w_1}, p_{w_2}, p_{w_3}}$; this is an ${(r_{w_1}, r_{w_2}, r_{w_3})}$-triangle. Let ${\Sigma}$ denote the collection of such hyperbolic triangles; because ${{\mathcal C}}$ is a packing, we see that these triangles have disjoint interiors. They also fit together in the following way: if ${e}$ is a side of a hyperbolic triangle in ${\Sigma}$, then there will be another hyperbolic triangle in ${\Sigma}$ that shares that side precisely when ${e}$ is associated to an interior edge of ${G - \{v\}}$. The union of all these triangles is homeomorphic to the region formed by starting with a triangulation of the Riemann sphere by ${G}$ and removing the triangles containing ${v}$ as a vertex, and is therefore homeomorphic to a disk. One can think of the collection ${\Sigma}$ of hyperbolic triangles, together with the vertices and edges shared by these triangles, as a two-dimensional (hyperbolic) simplicial complex, though we will not develop the full machinery of such complexes here.
Our objective is to find another hyperbolic circle packing ${\tilde {\mathcal C} = (C(\tilde p_w, \tilde r_w))_{w \in V \backslash \{v\}}}$ homotopic to the existing circle packing ${{\mathcal C}}$, such at all the boundary circles (circles centred at boundary vertices) are horocycles. We observe that such a hyperbolic circle packing is completely described (up to Möbius transformations) by the hyperbolic radii ${(\tilde r_w)_{w \in V \backslash \{v\}}}$ of these circles. Indeed, suppose one knows the values of these hyperbolic radii. Then each hyperbolic triangle ${\Delta_{\mathcal C}(w_1,w_2,w_3)}$ in ${\Sigma}$ is associated to a hyperbolic triangle ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ whose sides and angles are known from Exercise 15. As the orientation of each hyperbolic triangle is fixed, each hyperbolic triangle is determined up to a Möbius automorphism of ${D(0,1)}$. Once one fixes one hyperbolic triangle, the adjacent hyperbolic triangles (that share a common side with the first triangle) are then also fixed; continuing in this fashion we see that the entire hyperbolic circle packing ${\tilde {\mathcal C}}$ is determined.
On the other hand, not every choice of radii ${(\tilde r_w)_{w \in V \backslash \{v\}}}$ will lead to a hyperbolic circle packing ${\tilde {\mathcal C}}$ with the required properties. There are two obvious constraints that need to be satisfied:
• (i) (Local constraint) The angles ${\alpha_1( \tilde r_w, \tilde r_{w_1}, \tilde r_{w_2})}$ of all the hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w,w_1,w_2)}$ around any given interior vertex ${w}$ must sum to exactly ${2\pi}$.
• (ii) (Boundary constraint) The radii associated to boundary vertices must be infinite.
There could potentially also be a global constraint, in that one requires the circles of the packing to be disjoint – including circles that are not necessarily adjacent to each other. In general, one can easily create configurations of circles that are local circle packings but not global ones (see e.g., Figure 7 of Beardon-Stephenson). However, it turns out that one can use the boundary constraint and topological arguments to prevent this from happening. We first need a topological lemma:
Lemma 16 (Topological lemma) Let ${U, V}$ be bounded connected open subsets of ${{\bf C}}$ with ${V}$ simply connected, and let ${f: \overline{U} \rightarrow \overline{V}}$ be a continuous map such that ${f(\partial U) \subset \partial V}$ and ${f(U) \subset V}$. Suppose furthermore that the restriction of ${f}$ to ${U}$ is a local homeomorphism. Then ${f}$ is in fact a global homeomorphism.
The requirement that the restriction of ${f}$ to ${U}$ be a local homeomorphism can in fact be relaxed to local injectivity thanks to the invariance of domain theorem. The complex numbers ${{\bf C}}$ can be replaced here by any finite-dimensional vector space.
Proof: The preimage ${f^{-1}(p)}$ of any point ${p}$ in the interior of ${V}$ is closed, discrete, and disjoint from ${\partial U}$, and is hence finite. Around each point in the preimage, there is a neighbourhood on which ${f}$ is a homeomorphism onto a neighbourhood of ${p}$. If one deletes the closure of these neighbourhoods, the image under ${f}$ is compact and avoids ${p}$, and thus avoids a neighbourhood of ${p}$. From this we can show that ${f}$ is a covering map from ${U}$ to ${V}$. As the base ${V}$ is simply connected, it is its own universal cover, and hence (by the connectedness of ${U}$) ${f}$ must be a homeomorphism as claimed. $\Box$
Proposition 17 Suppose we assign a radius ${\tilde r_w \in (0,+\infty]}$ to each ${w \in V \backslash \{v\}}$ that obeys the local constraint (i) and the boundary constraint (ii). Then there is a hyperbolic circle packing ${(C(\tilde p_w, \tilde r_w))_{w \in V \backslash \{v\}}}$ with nerve ${G - \{v\}}$ and the indicated radii.
Proof: We first create the hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ associated with the required hyperbolic circle packing, and then verify that this indeed arises from a circle packing.
Start with a single triangle ${(w^0_1,w^0_2,w^0_3)}$ in ${G - \{v\}}$, and arbitrarily select a ${(\tilde r_{w^0_1}, \tilde r_{w^0_2}, \tilde r_{w^0_3})}$-triangle ${\Delta_{\tilde {\mathcal C}}(w^0_1,w^0_2,w^0_3)}$ with the same orientation as ${\Delta_{{\mathcal C}}(w_1,w_2,w_3)}$. By Exercise 15(i), such a triangle exists (and is unique up to Möbius automorphisms of the disk). If a hyperbolic triangle ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ has been fixed, and ${(w_2,w_3,w_4)}$ (say) is an adjacent triangle in ${G - \{v\}}$, we can select ${\Delta_{\tilde {\mathcal C}}(w_2,w_3,w_4)}$ to be the unique ${(r_{w_2}, r_{w_3}, r_{w_4})}$-triangle with the same orientation as ${\Delta_{{\mathcal C}}(w_2,w_3,w_4)}$ that shares the ${w_2,w_3}$ side in common with ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ (with the ${w_2}$ and ${w_3}$ vertices agreeing). Similarly for other permutations of the labels. As ${G}$ is a maximal planar graph with ${v}$ non-degenerate (so in particular the set of internal vertices is connected), we can continue this construction to eventually fix every triangle in ${G - \{v\}}$. There is the potential issue that a given triangle ${\Delta_{{\mathcal C}}(w_1,w_2,w_3)}$ may depend on the order in which one arrives at that triangle starting from ${(w^0_1,w^0_2,w^0_3)}$, but one can check from a monodromy argument (in the spirit of the monodromy theorem) using the local constraint (i) and the simply connected nature of the triangulation associated to ${{\mathcal C}}$ that there is in fact no dependence on the order. (The process resembles that of laying down jigsaw pieces in the shape of hyperbolic triangles together, with the local constraint ensuring that there is always a flush fit locally.)
Now we show that the hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ have disjoint interiors inside the disk ${D(0,1)}$. Let ${X}$ denote the topological space formed by taking the disjoint union of the hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ (now viewed as abstract topological spaces rather than subsets of the disk) and then gluing together all common edges, e.g. identifying the ${\{w_2,w_3\}}$ edge of ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ with the same edge of ${\Delta_{\tilde {\mathcal C}}(w_2,w_3,w_4)}$ if ${(w_1,w_2,w_3)}$ and ${(w_2,w_3,w_4)}$ are adjacent triangles in ${G - \{v\}}$. This space is homeomorphic to the union of the original hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$, and is thus homeomorphic to the closed unit disk. There is an obvious projection map ${\pi}$ from ${X}$ to the union of the ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$, which maps the abstract copy in ${X}$ of a given hyperbolic triangle ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ to its concrete counterpart in ${\overline{D(0,1)}}$ in the obvious fashion. This map is continuous. It does not quite cover the full closed disk, mainly because (by the boundary condition (ii)) the boundary hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(v_i,v_{i+1},w)}$ touch the boundary of the disk at the vertices associated to ${v_i}$ and ${v_{i+1}}$ but do not follow the boundary arc connecting these vertices, being bounded instead by the geodesic from the ${v_i}$ vertex to the ${v_{i+1}}$ vertex; the missing region is a lens-shaped region bounded by two circular arcs. However, by applying another homeomorphism (that does not alter the edges from ${v_i}$ to ${w}$ or ${v_{i+1}}$ to ${w}$), one can “push out” the ${\{v_i,v_{i+1}\}}$ edge of this hyperbolic triangle across the lens to become the boundary arc from ${v_i}$ to ${v_{i+1}}$. If one performs this modification for each boundary triangle, one arrives at a modified continuous map ${\tilde \pi}$ from ${X}$ to ${\overline{D(0,1)}}$, which now has the property that the boundary of ${X}$ maps to the boundary of the disk, and the interior of ${X}$ maps to the interior of the disk. Also one can check that this map is a local homeomorphism. By Lemma 16, ${\tilde \pi}$ is injective; undoing the boundary modifications we conclude that ${\pi}$ is injective. Thus the hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ have disjoint interiors. Furthermore, the arguments show that for each boundary triangle ${\Delta_{\tilde {\mathcal C}}(v_i,v_{i+1},w)}$, the lens-shaped regions between the boundary arc between the vertices associated to ${v_i, v_{i+1}}$ and the corresponding edge of the boundary triangle are also disjoint from the hyperbolic triangles and from each other. On the other hand, all of the hyperbolic circles and in ${{\tilde {\mathcal C}}}$ and their interiors are contained in the union of the hyperbolic triangles ${\Delta_{\tilde {\mathcal C}}(w_1,w_2,w_3)}$ and the lens-shaped regions, with each hyperbolic triangle containing portions only of the hyperbolic circles with hyperbolic centres at the vertices of the triangle, and similarly for the lens-shaped regions. From this one can verify that the interiors of the hyperbolic circles are all disjoint from each other, and give a hyperbolic circle packing with the required properties. $\Box$
In view of the above proposition, the only remaining task is to find an assignment of radii ${(\tilde r_w)_{w \in V \backslash \{v\}}}$ obeying both the local condition (i) and the boundary condition (ii). This is analogous to finding a harmonic function with specified boundary data. To do this, we perform the following analogue of Perron’s method. Define a subpacking to be an assignment ${(\tilde r_w)_{w \in V \backslash \{v\}}}$ of radii ${\tilde r_w \in (0,+\infty]}$ obeying the following
• (i’) (Local sub-condition) The angles ${\alpha_1( \tilde r_w, \tilde r_{w_1}, \tilde r_{w_2})}$ around any given interior vertex ${w}$ sum to at least ${2\pi}$.
This can be compared with the definition of a (smooth) subharmonic function as one where the Laplacian is always at least zero. Note that we always have at least one subpacking, namely the one provided by the radii of the original hyperbolic circle packing ${{\mathcal C}}$. Intuitively, in each subpacking, the radius ${\tilde r_w}$ at an interior vertex ${w}$ is either “too small” or “just right”.
We now need a key monotonicity property, analogous to how the maximum of two subharmonic functions is again subharmonic:
Exercise 18 (Monotonicity)
• (i) Show that the angle ${\alpha_1( r_1, r_2, r_3)}$ (as defined in Exercise 15(i)) is strictly decreasing in ${r_1}$ and strictly increasing in ${r_2}$ or ${r_3}$ (if one holds the other two radii fixed). Do these claims agree with your geometric intuition?
• (ii) Conclude that whenever ${{\mathcal R}' = (r'_w)_{w \in V \backslash \{v\}}}$ and ${{\mathcal R}'' = (r''_w)_{w \in V \backslash \{v\}}}$ are subpackings, that ${\max( {\mathcal R}' , {\mathcal R}'' ) := (\max(r'_w, r''_w))_{w \in V \backslash \{v\}}}$ is also a subpacking.
• (iii) Let ${(r_1,r_2,r_3), (r'_1,r'_2,r'_3) \in T}$ be such that ${r_i \leq r'_i}$ for ${i=1,2,3}$. Show that ${\mathrm{Area}(r_1,r_2,r_3) \leq \mathrm{Area}(r'_1,r'_2,r'_3)}$, with equality if and only if ${r_i=r'_i}$ for all ${i=1,2,3}$. (Hint: increase just one of the radii ${r_1,r_2,r_3}$. One can either use calculus (after first disposing of various infinite radii cases) or one can argue geometrically.)
As with Perron’s method, we can now try to construct a hyperbolic circle packing by taking the supremum of all the subpackings. To avoid degeneracies we need an upper bound:
Proposition 19 (Upper bound) Let ${(\tilde r_w)_{w \in V \backslash \{v\}}}$ be a subpacking. Then for any interior vertex ${w}$ of degree ${d}$, one has ${\tilde r_w \leq \sqrt{d}}$.
The precise value of ${\sqrt{d}}$ is not so important for our arguments, but the fact that it is finite will be. This boundedness of interior circles in a circle packing is a key feature of hyperbolic geometry that is not present in Euclidean geometry, and is one of the reasons why we moved to a hyperbolic perspective in the first place.
Proof: By the subpacking property and pigeonhole principle, there is a triangle ${w, w_1, w_2}$ in ${G - \{v\}}$ such that ${\alpha_1(w,w_1,w_2) \geq \frac{2\pi}{d}}$. The hyperbolic triangle associated to ${(w_1,w_2,w_3)}$ has area at most ${\pi}$ by (2); on the other hand, it contains a sector of a hyperbolic circle of radius ${\tilde r_w}$ and angle ${\frac{2\pi}{d}}$, and hence has area at least ${\frac{1}{d} 4\pi \sinh^2(r/2) \geq \frac{\pi r^2}{d}}$, thanks to Exercise 15(iv). Comparing the two bounds gives the claim. $\Box$
Now define ${{\mathcal R} = ( \tilde r_w )_{w \in V \backslash \{v\}}}$ to be the (pointwise) supremum of all the subpackings. By the above proposition, ${\tilde r_w}$ is finite at every interior vertex. By Exercise 18, one can view ${{\mathcal R}}$ as a monotone increasing limit of subpackings, and is thus again a subpacking (due to the continuity properties of ${\alpha_1}$ as long as at least one of the radii stays bounded); thus ${{\mathcal R}}$ is the maximal subpacking. On the other hand, if ${\tilde r_w}$ is finite at some boundary vertex, then by Exercise 18(i) one could replace that radius by a larger quantity without destroying the subpacking property, contradicting the maximality of ${{\mathcal R}}$. Thus all the boundary radii are infinite, that is to say the boundary condition (ii) holds. Finally, if the sum of the angles at an interior vertex ${w}$ is strictly greater than ${\pi}$, then by Exercise 18 we could increase the radius at this vertex slightly without destroying the subpacking property at ${w}$ or at any other of the interior vertices, again contradicting the maximality of ${{\mathcal R}}$. Thus ${{\mathcal R}}$ obeys the local condition (i), and we have demonstrated existence of the required hyperbolic circle packing.
Finally we establish uniqueness. It suffices to establish that ${{\mathcal R}}$ is the unique tuple that obeys the local condition (i) and the boundary condition (ii). Suppose we had another tuple ${{\mathcal R}' = ( r'_w )_{w \in V \backslash \{v\}}}$ other than ${{\mathcal R}}$ that obeyed these two conditions. Then by the maximality of ${{\mathcal R}}$, we have ${r'_w \leq \tilde r_w}$ for all ${w}$. By Exercise 18(iii), this implies that
$\displaystyle \mathrm{Area}( r'_{w_1}, r'_{w_2}, r'_{w_3} ) \leq \mathrm{Area}( \tilde r_{w_1}, \tilde r_{w_2}, \tilde r_{w_3} )$
for any triangle ${(w_1,w_2,w_3)}$ in ${T}$. Summing over all triangles and using (2), we conclude that
$\displaystyle \sum_{w \in V \backslash \{v\}} \sum_{w_1,w_2: (w,w_1,w_2) \hbox{ triangle}} \alpha_1(r'_{w}, r'_{w_1}, r'_{w_2})$
$\displaystyle \geq \sum_{w \in V \backslash \{v\}} \sum_{w_1,w_2: (w,w_1,w_2) \hbox{ triangle}} \alpha_1(\tilde r_{w}, \tilde r_{w_1}, \tilde r_{w_2})$
where the inner sum is over the pairs ${w_1,w_2}$ such that ${(w,w_1,w_2)}$ forms a triangle in ${G - \{v\}}$. But by the local condition (i) and the boundary condition (ii), the inner sum on either side is equal to ${2\pi}$ for an interior vertex and ${0}$ for a boundary vertex. Thus the two sides agree, which by Exercise 18(iii) implies that ${r'_w = \tilde r_w}$ for all ${w}$. This proves Theorem 14 and thus Theorems 7, 3, 4.
— 2. Quasiconformal maps —
In this section we set up some of the foundational theory of quasiconformal mapping, which are generalisations of the conformal mapping concept that can tolerate some deviations from perfect conformality, while still retaining many of the good properties of conformal maps (such as being preserved under uniform limits), though with the notable caveat that in contrast to conformal maps, quasiconformal maps need not be smooth. As such, this theory will come in handy when proving convergence of circle packings to the Riemann map. The material here is largely drawn from the text of Lehto and Virtanen.
We first need the following refinement of the Riemann mapping theorem, known as Carathéodory’s theorem:
Theorem 20 (Carathéodory’s theorem) Let ${U}$ be a bounded simply connected domain in ${{\bf C}}$ whose boundary ${\partial U}$ is a Jordan curve, and let ${\phi: D(0,1) \rightarrow U}$ be a conformal map between ${D(0,1)}$ and ${U}$ (as given by the Riemann mapping theorem). Then ${\phi}$ extends to a continuous homeomorphism from ${\overline{D}(0,1)}$ to ${\overline{U}}$.
The condition that ${\partial U}$ be a Jordan curve is clearly necessary, since if ${\partial U}$ is not simple then there are paths in ${D(0,1)}$ that end up at different points in ${\partial D(0,1)}$ but have the same endpoint in ${\partial U}$ after applying ${\phi}$, which prevents ${\phi}$ being continuously extended to a homeomorphism. If one relaxes the requirement that ${\partial U}$ be a Jordan curve to the claim that ${{\bf C} \backslash U}$ is locally connected, then it is possible to modify this argument to still obtain a continuous extension of ${U}$ to ${\overline{U}}$, although the extension will no longer be necessarily a homeomorphism, but we will not prove this fact here.
Proof: We first prove continuous extension to the boundary. It suffices to show that for every point ${\zeta}$ on the boundary of the unit circle, the diameters of the sets ${\phi( D(0,1) \cap D( \zeta, r_n ) )}$ go to zero for some sequence of radii ${r_n \rightarrow 0}$.
First observe from the change of variables formula that the area of ${U = \phi(D(0,1))}$ is given by ${\int_{D(0,1)} |\phi'(z)|^2\ dx dy}$, where ${dx dy}$ denotes Lebesgue measure (or the area element). In particular, this integral is finite. Expanding in polar coordinates around ${\zeta}$, we conclude that
$\displaystyle \int_0^2 \left(\int_{0}^{2\pi} 1_{D(0,1)}(\zeta+re^{i\theta}) |\phi'( \zeta + r e^{i\theta} )|^2\ d \theta\right) r dr < \infty.$
Since ${\int_0^2 \frac{dr}{r}}$ diverges near ${r=0}$, we conclude from the pigeonhole principle that there exists a sequence of radii ${0 < r_n < 2}$ decreasing to zero such that
$\displaystyle r_n^2 \int_{0}^{2\pi} 1_{D(0,1)}(\zeta+r_ne^{i\theta}) |\phi'( \zeta + r_n e^{i\theta} )|^2\ d \theta \rightarrow 0$
and hence by Cauchy-Schwarz
$\displaystyle r_n \int_{0}^{2\pi} 1_{D(0,1)}(\zeta+r_ne^{i\theta}) |\phi'( \zeta + r_n e^{i\theta} )|\ d \theta \rightarrow 0$
If we let ${C_n}$ denote the circular arc ${\{ \zeta + r_n e^{i\theta}: 0 \leq \theta \leq 2\pi \} \cap D(0,1)}$, we conclude from this and the triangle inequality (and chain rule) that ${\phi(C_n)}$ is a rectifiable curve with length going to zero as ${n \rightarrow \infty}$. Let ${a_n, b_n}$ denote the endpoints of this curve. Clearly they lie in ${\overline{U}}$. If (say) ${a_n}$ was in ${U}$, then as ${\phi}$ is a homeomorphism from ${D(0,1)}$ to ${U}$, ${C_n}$ would have one endpoint in ${D(0,1)}$ rather than ${\partial D(0,1)}$, which is absurd. Thus ${a_n}$ lies in ${\partial U}$, and similarly for ${b_n}$. Since the length of ${\phi(C_n)}$ goes to zero, the distance between ${a_n}$ and ${b_n}$ goes to zero. Since ${\partial U}$ is a Jordan curve, it can be parameterised homeomorphically by ${\partial D(0,1)}$, and so by compactness we also see that the distance between the parameterisations of ${a_n}$ and ${b_n}$ in ${\partial D(0,1)}$ must also go to zero, hence (by uniform continuity of the inverse parameterisation) ${a_n}$ and ${b_n}$ are connected along ${\partial U}$ by an arc whose diameter goes to zero. Combining this arc with ${\phi(C_n)}$, we obtain a Jordan curve of diameter going to zero which separates ${\phi(D(0,1) \cap D(\zeta, r_n))}$ from the rest of ${U}$. Sending ${r}$ to infinity, we see that ${\phi(D(0,1) \cap D(\zeta, r_n))}$ (which decreases with ${n}$) must eventually map in the interior of this curve rather than the exterior, and so the diameter goes to zero as claimed.
The above construction shows that ${\phi}$ extends to a continuous map (which by abuse of notation we continue to call ${\phi}$) from ${\overline{D(0,1)}}$ to ${\overline{U}}$, and the proof also shows that ${\partial D(0,1)}$ maps to ${\partial U}$. As ${\phi(\overline{D(0,1)})}$ is a compact subset of ${\overline{U}}$ that contains ${U}$, it must surject onto ${\overline{U}}$. As both ${\overline{D(0,1)}}$ and ${\overline{U}}$ are compact Hausdorff spaces, we will now be done if we can show injectivity. The only way injectivity can fail is if there are two distinct points ${\zeta,\omega}$ on ${\partial D(0,1)}$ that map to the same point. Let ${C}$ be the line segment connecting ${\zeta}$ with ${\omega}$, then ${\phi(C)}$ is a Jordan curve in ${\overline{U}}$ that meets ${\partial U}$ only at ${\phi(\zeta) = \phi(\omega)}$. ${C}$ divides ${\overline{D(0,1)}}$ into two regions; one of which must map to the interior of ${\phi(C)}$, which implies that there is an entire arc of ${\partial D(0,1)}$ which maps to the single point ${\phi(\zeta)=\phi(\omega)}$. But then by the Schwarz reflection principle, ${\phi}$ extends conformally across this arc and is constant in a non-isolated set, thus is constant everywhere by analytic continuation, which is absurd. This establishes the required injectivity. $\Box$
This has the following consequence. Define a Jordan quadrilateral to be the open region ${Q}$ enclosed by a Jordan curve with four distinct marked points ${p_1,p_2,p_3,p_4}$ on it in counterclockwise order, which we call the vertices of the quadrilateral. The arcs in ${\partial Q}$ connecting ${p_1}$ to ${p_2}$ or ${p_3}$ to ${p_4}$ will be called the ${a}$-sides; the arcs connecting ${p_2}$ to ${p_3}$ or ${p_4}$ to ${p_1}$ will be called ${b}$-sides. (Thus for instance each cyclic permutation of the ${p_1,p_2,p_3,p_4}$ vertices will swap the ${a}$-sides and ${b}$-sides, while keeping the interior region ${Q}$ unchanged.) A key example of a Jordan quadrilateral are the (Euclidean) rectangles, in which the vertices ${p_1,\dots,p_4}$ are the usual corners of the rectangle, traversed counterclockwise. The ${a}$-sides then are line segments of some length ${a}$, and the ${b}$-sides are line segments of some length ${b}$ that are orthogonal to the ${a}$-sides. A vertex-preserving conformal map from one Jordan quadrilateral ${Q}$ to another ${Q'}$ will be a conformal map that extends to a homeomorphism from ${\overline{Q}}$ to ${\overline{Q'}}$ that maps the corners of ${Q}$ to the respective corners of ${Q'}$ (in particular, ${a}$-sides get mapped to ${a}$-sides, and similarly for ${b}$-sides).
Exercise 21 Let ${Q}$ be a Jordan quadrilateral with vertices ${p_1,p_2,p_3,p_4}$.
• (i) Show that there exists ${r > 1}$ and a conformal map ${\phi: Q \rightarrow \mathbf{H}}$ to the upper half-plane ${\mathbf{H} := \{ z: \mathrm{Im} z > 0 \}}$ (viewed as a subset of the Riemann sphere) that extends continuously to a homeomorphism ${\phi: \overline{Q} \rightarrow \overline{\mathbf{H}}}$ and which maps ${p_1,p_2,p_3,p_4}$ to ${-r, -1, 1, r}$ respectively. (Hint: first map ${p_1,p_2,p_3,p_4}$ to increasing elements of the real line, then use the intermediate value theorem to enforce ${\phi(p_1)+\phi(p_4) = \phi(p_2)+\phi(p_3)}$.)
• (ii) Show that there is a vertex-preserving conformal map ${\psi: Q \rightarrow R}$ from ${Q}$ to a rectangle ${R}$ (Hint: use Schwarz-Christoffel mapping.)
• (iii) Show that the rectangle ${R}$ in part (ii) is unique up to affine transformations. (Hint: if one has a conformal map between rectangles that preserves the vertices, extend it via repeated use of the Schwarz reflection principle to an entire map.)
This allows for the following definition: the conformal modulus ${\mathrm{mod}(Q)}$ (or modulus for short, also called module in older literature) of a Jordan quadrilateral with vertices ${p_1,p_2,p_3,p_4}$ is the ratio ${b/a}$, where ${a,b}$ are the lengths of the ${a}$-sides and ${b}$-sides of a rectangle ${R}$ that is conformal to ${Q}$ in a vertex-preserving vashion.. This is a number between ${0}$ and ${\infty}$; each cyclic permutation of the vertices replaces the modulus with its reciprocal. It is clear from construction that the modulus of a Jordan quadrilateral is unaffected by vertex-preserving conformal transformations.
Now we define quasiconformal maps. Informally, conformal maps are homeomorphisms that map infinitesimal circles to infinitesimal circles; quasiconformal maps are homeomorphisms that map infinitesimal circles to curves that differ from an infinitesimal circle by “bounded distortion”. However, for the purpose of setting up the foundations of the theory, it is slightly more convenient to work with rectangles instead of circles (it is easier to partition rectangles into subrectangles than disks into subdisks). We therefore introduce
Definition 22 Let ${K \geq 1}$. An orientation-preserving homeomorphism ${\phi: U \rightarrow V}$ between two domains ${U,V}$ in ${{\bf C}}$ is said to be ${K}$-quasiconformal if one has ${\mathrm{mod}(\phi(Q)) \leq K \mathrm{mod}(Q)}$ for every Jordan quadrilateral ${Q}$ in ${U}$. (In these notes, we do not consider orientation-reversing homeomorphisms to be quasiconformal.)
Note that by cyclically permuting the vertices of ${Q}$, we automatically also obtain the inequality
$\displaystyle \frac{1}{\mathrm{mod}(\phi(Q))} \leq K \frac{1}{\mathrm{mod}(Q)}$
or equivalently
$\displaystyle \frac{1}{K} \mathrm{mod}(Q) \leq \mathrm{mod}(\phi(Q))$
for any Jordan quadrilateral. Thus it is not possible to have any ${K}$-quasiconformal maps for ${K<1}$ (excluding the degenerate case when ${U,V}$ are empty), and a map is ${1}$-conformal if and only if it preserves the modulus. In particular, conformal maps are ${1}$-conformal; we will shortly establish that the converse claim is also true. It is also clear from the definition that the inverse of a ${K}$-quasiconformal map is also ${K}$-quasiconformal, and the composition of a ${K}$-quasiconformal map and a ${K'}$-quasiconformal map is a ${KK'}$-quasiconformal map.
It is helpful to have an alternate characterisation of the modulus that does not explicitly mention conformal mapping:
Proposition 23 (Alternate definition of modulus) Let ${Q}$ be a Jordan quadrilateral with vertices ${p_1,p_2,p_3,p_4}$. Then ${\mathrm{mod}(Q)}$ is the smallest quantity with the following property: for any Borel measurable ${\rho: Q \rightarrow [0,+\infty)}$ one can find a curve ${\gamma}$ in ${Q}$ connecting one ${a}$-side of ${Q}$ to another, and which is locally rectifiable away from endpoints, such that
$\displaystyle \left(\int_\gamma \rho(z)\ |dz|\right)^2 \leq \mathrm{mod}(Q) \int_Q \rho^2(z)\ dx dy$
where ${\int_\gamma |dz|}$ denotes integration using the length element of ${\gamma}$ (not to be confused with the contour integral ${\int_\gamma\ dz}$).
The reciprocal of this notion of modulus generalises to the concept of extremal length, which we will not develop further here.
Proof: Observe from the change of variables formula that if ${\phi: Q \rightarrow Q'}$ is a vertex-preserving conformal mapping between Jordan quadrilaterals ${Q,Q'}$, and ${\gamma}$ is a locally rectifiable curve connecting one ${a}$-side of ${Q}$ to another, then ${\phi \circ \gamma}$ is a locally rectifiable curve connecting one ${a}$-side of ${Q'}$ to another, with
$\displaystyle \int_{\phi \circ \gamma} \rho \circ \phi^{-1}(w) \ |dw| = \int_\gamma \rho(z) |\phi'(z)|\ |dz|$
and
$\displaystyle \int_{Q'} |\rho \circ \phi^{-1}(w)|^2\ \frac{dw \overline{dw}}{2i} = \int_Q |\rho(z)|^2 |\phi'(z)|^2\ \frac{dz d\overline{z}}{2i}.$
As a consequence, if the proposition holds for ${Q}$ it also holds for ${Q'}$. Thus we may assume without loss of generality that ${Q}$ is a rectangle, which we may normalise to be ${\{ x+iy: 0 \leq y \leq 1; 0 \leq x \leq M \}}$ with vertices ${i, 0, M, M+i}$, so that the modulus is ${M}$. For any measurable ${\rho: Q \rightarrow [0,+\infty)}$, we have from Cauchy-Schwarz and Fubini’s theorem that
$\displaystyle \int_0^1 \left(\int_0^M \rho(x+iy)\ dx\right)^2 dy \leq M \int_0^1 \int_0^M \rho^2(x+iy)\ dx dy$
$\displaystyle = M \int_Q \rho^2(z)\ \frac{dz \overline{dz}}{2i}$
and hence by the pigeonhole principle there exists ${y}$ such that
$\displaystyle \left(\int_0^M \rho(x+iy)\ dx\right)^2 \leq M \int_Q \rho^2(z)\ \frac{dz \overline{dz}}{2i}.$
On the other hand, if we set ${\rho=1}$, then ${\int_Q \rho^2(z)\ \frac{dz \overline{dz}}{2i} = M}$, and for any curve ${\gamma}$ connecting the ${a}$-side from ${0}$ to ${i}$ to the ${a}$-side from ${M}$ to ${M+i}$, we have
$\displaystyle \int_\gamma \rho\ |dz| \geq \left| \int_\gamma \rho\ dx \right| = M.$
Thus ${M}$ is the best constant with the required property, proving the claim. $\Box$
Here are some quick and useful consequences of this characterisation:
Exercise 24 (Rengel’s inequality) Let ${Q}$ be a Jordan quadrilateral of area ${A}$, let ${b}$ be the shortest (Euclidean) distance between a point on one ${a}$-side and a point on the other ${a}$-side, and similarly let ${a}$ be the shortest (Euclidean) distance between a point on one ${b}$-side and a point on the other ${b}$-side. Show that
$\displaystyle \frac{b^2}{A} \leq \mathrm{mod}(Q) \leq \frac{A}{a^2}$
and that equality in either case occurs if and only if ${Q}$ is a rectangle.
• (i) If ${Q_1, Q_2}$ are disjoint Jordan quadrilaterals that share a common ${a}$-side, and which can be glued together along this side to form a new Jordan quadrilateral ${Q_1 \cup Q_2}$, show that ${\mathrm{mod}(Q_1 \cup Q_2) \geq \mathrm{mod}(Q_1) + \mathrm{mod}(Q_2)}$. If equality occurs, show that after conformally mapping ${Q_1 \cup Q_2}$ to a rectangle (in a vertex preserving fashion), ${Q_1}$, ${Q_2}$ are mapped to subrectangles (formed by cutting the original parallel to the ${a}$-side).
• (ii) If ${Q_1, Q_2}$ are disjoint Jordan quadrilaterals that share a common ${b}$-side, and which can be glued together along this side to form a new Jordan quadrilateral ${Q_1 \cup Q_2}$, show that ${\frac{1}{\mathrm{mod}(Q_1 \cup Q_2)} \geq \frac{1}{\mathrm{mod}(Q_1)} + \frac{1}{\mathrm{mod}(Q_2)}}$. If equality occurs, show that after conformally mapping ${Q_1 \cup Q_2}$ to a rectangle (in a vertex preserving fashion), ${Q_1}$, ${Q_2}$ are mapped to subrectangles (formed by cutting the original parallel to the ${b}$-side).
Exercise 26 (Continuity from below) Suppose ${Q_n}$ is a sequence of Jordan quadrilaterals which converge to another Jordan quadrilateral ${Q}$, in the sense that the vertices of ${Q_n}$ converge to their respective counterparts in ${Q}$, each ${a}$-side in ${Q_n}$ converges (in the Hausdorff sense) to the ${a}$-side of ${Q}$, and the similarly for ${b}$-sides. Suppose also that ${Q_n \subset Q}$ for all ${n}$. Show that ${\mathrm{mod}(Q_n)}$ converges to ${\mathrm{mod}(Q)}$. (Hint: map ${Q}$ to a rectangle and use Rengel’s inequality.)
Proposition 27 (Local quasiconformality implies quasiconformality) Let ${K \geq 1}$, and let ${\phi: U \rightarrow V}$ be an orientation-preserving homeomorphism between complex domains ${U,V}$ which is locally ${K}$-quasiconformal in the sense that for every ${z_0 \in U}$ there is a neighbourhood ${U_{z_0}}$ of ${z_0}$ in ${U}$ such that ${\phi}$ is ${K}$-quasiconformal from ${U_{z_0}}$ to ${\phi(U_{z_0})}$. Then ${\phi}$ is ${K}$-quasiconformal.
Proof: We need to show that ${\mathrm{mod}(\phi(Q)) \leq K \mathrm{mod}(Q)}$ for any Jordan quadrilateral ${Q}$ in ${U}$. The hypothesis gives this claim for all quadrilaterals in the sufficiently small neighbourhood of any point in ${U}$. For any natural number ${n}$, we can subdivide ${\phi(Q)}$ into ${n}$ quadrilaterals ${\phi(Q_1),\dots,\phi(Q_n)}$ with modulus ${\frac{1}{n} \mathrm{mod}(\phi(Q))}$ with adjacent ${a}$-sides, by first conformally mapping ${\phi(Q)}$ to a rectangle and then doing an equally spaced vertical subdivision. Similarly, each quadrilateral ${Q_i}$ can be subdivided into ${n}$ quadrilaterals ${Q_{i,1},\dots,Q_{i,n}}$ of modulus ${n \mathrm{mod}(Q_i)}$ by mapping ${Q_i}$ to a rectangle and performing horizontal subdivision. By the local ${K}$-quasiconformality of ${\phi}$, we will have
$\displaystyle \mathrm{mod}( \phi(Q_{i,j}) ) \leq K \mathrm{mod}( Q_{i,j} ) = K n \mathrm{mod}(Q_i)$
for all ${i,j=1,\dots,n}$, if ${n}$ is large enough. By superadditivity this implies that
$\displaystyle \mathrm{mod}( \phi(Q_i) ) \leq K\mathrm{mod}(Q_i)$
for each ${i}$, and hence
$\displaystyle \mathrm{mod}(Q_i) \geq \frac{1}{Kn} \mathrm{mod}(\phi(Q)).$
$\displaystyle \mathrm{mod}(Q) \geq \frac{1}{K} \mathrm{mod}(\phi(Q)).$
giving the claim. $\Box$
We can now reverse the implication that conformal maps are ${1}$-conformal:
Proposition 28 Every ${1}$-conformal map ${\phi: U \rightarrow V}$ is conformal.
Proof: By covering ${U}$ by quadrilaterals we may assume without loss of generality that ${U}$ (and hence also ${\phi(U)=V}$) is a Jordan quadrilateral; by composing on left and right with conformal maps we may assume that ${U}$ and ${V}$ are rectangles. As ${\phi}$ is ${1}$-conformal, the rectangles have the same modulus, so after a further affine transformation we may assume that ${U=V}$ is the rectangle with vertices ${i, 0, M, M+i}$ for some modulus ${M}$. If one subdivides ${U}$ into two rectangles along an intermediate vertical line segment connecting say ${x}$ to ${x+i}$ for some ${0 < x < M}$, the moduli of these rectangles are ${x}$ and ${M-x}$. Applying the ${1}$-conformal map and the converse portion of Exercise 25, we conclude that these rectangles must be preserved by ${\phi}$, thus ${\phi}$ preserves the ${x}$ coordinate. Similarly ${\phi}$ preserves the ${y}$ coordinate, and is therefore the identity map, which is of course conformal. $\Box$
Next, we can give a simple criterion for quasiconformality in the continuously differentiable case:
Theorem 29 Let ${K \geq 1}$, and let ${\phi: U \rightarrow V}$ be an orientation-preserving diffeomorphism (a continuously (real) differentiable homeomorphism whose derivative is always nondegenerate) between complex domains ${U,V}$. Then the following are equivalent:
• (i) ${\phi}$ is ${K}$-quasiconformal.
• (ii) For any point ${z_0 \in U}$ and phases ${v,w \in S^1 := \{ z \in {\bf C}: |z|=1\}}$, one has
$\displaystyle |D_v \phi(z_0)| \leq K|D_w \phi(z_0)|$
where ${D_v \phi(z_0) := \frac{\partial}{\partial t} \phi(z_0+tv)|_{t=0}}$ denotes the directional derivative.
Proof: Let us first show that (ii) implies (i). Let ${Q}$ be a Jordan quadrilateral in ${U}$; we have to show that ${\mathrm{mod}(\phi(Q)) \leq K \mathrm{mod}(Q)}$. From the chain rule one can check that condition (ii) is unchanged by composing ${\phi}$ with conformal maps on the left or right, so we may assume without loss of generality that ${Q}$ and ${\phi(Q)}$ are rectangles; in fact we may normalise ${Q}$ to have vertices ${i, 0, T, T+i}$ and ${\phi(Q)}$ to have vertices ${i, 0, T', T'+i}$ where ${T = \mathrm{mod}(Q)}$ and ${T' = \mathrm{mod}(Q')}$. From the change of variables formula (and the singular value decomposition), followed by Fubini’s theorem and Cauchy-Schwarz, we have
$\displaystyle T' = \int_{\phi(Q)}\ dx dy$
$\displaystyle = \int_Q \mathrm{det}(D\phi)(z)\ dx dy$
$\displaystyle = \int_Q \max_{v \in S^1} |D_v \phi(z)| \min_{w \in S^1} |D_w \phi(z)|\ dx dy$
$\displaystyle \geq \int_Q \frac{1}{K} \left|\frac{\partial}{\partial x} \phi(z)\right|^2\ dx dy$
$\displaystyle = \frac{1}{K} \int_0^1 \int_0^T \left|\frac{\partial}{\partial x} \phi(x+iy)\right|^2\ dx dy$
$\displaystyle \geq \frac{1}{K T} \int_0^1 \left|\int_0^T \frac{\partial}{\partial x} \phi(x+iy)\right|^2\ dx dy$
$\displaystyle = \frac{1}{K T} \int_0^1 (T')^2\ dy$
3 and hence ${T' \leq K T}$, giving the claim.
Now suppose that (ii) failed, then by the singular value decomposition we can find ${z_0 \in U}$ and a phase ${v \in S^1}$ such that
$\displaystyle D_{iv} \phi(z_0) = i \lambda D_v \phi(z_0)$
for some real ${\lambda}$ with ${\lambda > K}$. After translations and rotations we may normalise so that
$\displaystyle \phi(0) = 0; \frac{\partial}{\partial x} \phi(0) = 1; \frac{\partial}{\partial y} \phi(0) = i\lambda.$
But then from Rengel’s inequality and Taylor expansion one sees that ${\phi}$ will map a unit square with vertices ${0, -\varepsilon, -\varepsilon+i\varepsilon, i\varepsilon}$ to a quadrilateral of modulus converging to ${\lambda}$ as ${\varepsilon \rightarrow 0}$, contradicting (i). $\Box$
Exercise 30 Show that the conditions (i), (ii) in the above theorem are also equivalent to the bound
$\displaystyle \left|\frac{\partial}{\partial \overline{z}} \phi(z_0)\right| \leq \frac{K-1}{K+1} \left|\frac{\partial }{\partial z} \phi(z_0)\right|$
for all ${z_0 \in U}$, where
$\displaystyle \frac{\partial }{\partial z} := \frac{1}{2} ( \frac{\partial }{\partial x} - i \frac{\partial }{\partial y}); \quad \frac{\partial }{\partial \overline{z}} := \frac{1}{2} ( \frac{\partial }{\partial x} + i \frac{\partial }{\partial y})$
are the Wirtinger derivatives.
We now prove a technical regularity result on quasiconformal maps.
Proposition 31 (Absolute continuity on lines) Let ${\phi: U \rightarrow V}$ be a ${K}$-quasiconformal map between two complex domains ${U,V}$ for some ${K}$. Suppose that ${U}$ contains the closed rectangle with endpoints ${0, i, T+i, T}$. Then for almost every ${0 \leq y \leq 1}$, the map ${x \mapsto \phi(x+iy)}$ is absolutely continuous on ${[0,T]}$.
Proof: For each ${y}$, let ${A(y)}$ denote the area of the image ${\{ \phi(x+iy'): 0 \leq x \leq T; 0 \leq y \leq y'\}}$ of the rectangle with endpoints ${0, iy, T+iy, T}$. This is a bounded monotone function on ${[0,1]}$ and is hence differentiable almost everywhere. It will thus suffice to show that the map ${x \mapsto \phi(x+iy)}$ is absolutely continuous on ${[0,T]}$ whenever ${y \in (0,1)}$ is a point of differentiability of ${A}$.
Let ${\varepsilon > 0}$, and let ${[x_1,x'_1],\dots,[x_m,x'_m]}$ be disjoint intervals in ${[0,T]}$ of total length ${\sum_{j=1}^m x'_j-x_j \leq \varepsilon}$. To show absolute continuity, we need a bound on ${\sum_{j=1}^m |\phi(x'_j) - \phi(x_j)|}$ that goes to zero as ${\varepsilon \rightarrow 0}$ uniformly in the choice of intervals. Let ${\delta>0}$ be a small number (that can depend on the intervals), and for each ${j=1,\dots,m}$ let ${R_j}$ be the rectangle with vertices ${x_j+i(y_j+\delta)}$, ${x_j+iy}$, ${x'_j+iy}$, ${x'_j+i(y+\delta)}$ This rectangle has modulus ${(x'_j-x_j)/\delta}$, and hence ${\phi(R_j)}$ has modulus at most ${K (x'_j-x_j)/\delta}$. On the other hand, by Rengel’s inequality this modulus is at least ${|\phi(x'_j)-\phi(x_j)-o(1)|^2 / \mathrm{Area}(\phi(R_j))}$, where ${o(1)}$ is a quantity that goes to zero as ${\delta \rightarrow 0}$ (holding the intervals fixed). We conclude that
$\displaystyle |\phi(x'_j)-\phi(x_j)|^2 \leq \frac{K}{\delta} (x'_j-x_j) \mathrm{Area}(\phi(R_j)) + o(1).$
On the other hand, we have
$\displaystyle \sum_{j=1}^m \mathrm{Area}(\phi(R_j)) \leq A(y+\delta) - A(y) = (A'(y)+o(1)) \delta.$
By Cauchy-Schwarz, we thus have
$\displaystyle (\sum_{j=1}^m |\phi(x'_j)-\phi(x_j)|)^2 \leq K A'(y) \sum_{j=1}^m (x'_j-x_j) + o(1);$
sending ${\delta \rightarrow 0}$, we conclude
$\displaystyle \sum_{j=1}^m |\phi(x'_j)-\phi(x_j)| \leq K^{1/2} A'(y)^{1/2} \varepsilon^{1/2}$
giving the claim. $\Box$
Exercise 32 Let ${\phi: U \rightarrow V}$ be a ${K}$-quasiconformal map between two complex domains ${U,V}$ for some ${K}$. Suppose that there is a closed set ${S \subset {\bf C}}$ of Lebesgue measure zero such that ${\phi}$ is conformal on ${U \backslash S}$. Show that ${\phi}$ is ${1}$-conformal (and hence conformal, by Proposition 28). (Hint: Arguing as in the proof of Theorem 29, it suffices to show that of ${\phi}$ maps the rectangle with endpoints ${0, i, T+i, T}$ to the rectangle with endpoints ${0, i, T'+i, T'}$, then ${T' \leq T}$. Repeat the proof of that theorem, using the absolute continuity of lines at a crucial juncture to justify using the fundamental theorem of calculus.)
Recall Hurwitz’s theorem that the locally uniform limit of conformal maps is either conformal or constant. It turns out there is a similar result for quasiconformal maps. We will just prove a weak version of the result (see Theorem II.5.5 of Lehto-Virtanen for the full statement):
Theorem 33 Let ${K \geq 1}$, and let ${\phi_n: U \rightarrow V_n}$ be a sequence of ${K}$-quasiconformal maps that converge locally uniformly to an orientation-preserving homeomorphism ${\phi: U \rightarrow V}$. Then ${\phi}$ is also ${K}$-quasiconformal.
It is important for this theorem that we do not insist that quasiconformal maps are necessarily differentiable. Indeed for applications to circle packing we will be working with maps that are only piecewise smooth, or possibly even worse, even though at the end of the day we will recover a smooth conformal map in the limit.
Proof: Let ${Q}$ be a Jordan quadrilateral in ${U}$. We need to show that ${\mathrm{mod}(\phi(Q)) \leq K \mathrm{mod}(Q)}$. By restricting ${U}$ we may assume ${U=Q}$. By composing ${\phi, \phi_n}$ with a conformal map we may assume that ${Q}$ is a rectangle. We can write ${Q}$ as the increasing limit of rectangles ${Q_m}$ of the same modulus, then for any ${n,m}$ we have ${\mathrm{mod}(\phi_n(Q_m)) \leq K \mathrm{mod}(Q)}$. By choosing ${n_m}$ going to infinity sufficiently rapidly, ${\phi_{n_m}(Q_m)}$ stays inside ${\phi(Q)}$ and converges to ${\phi(Q)}$ in the sense of Exercise 26, and the claim then follows from that exercise. $\Box$
Another basic property of conformal mappings (a consequence of Morera’s theorem) is that they can be glued along a common edge as long as the combined map is also a homeomorphism; this fact underlies for instance the Schwarz reflection principle. We have a quasiconformal analogue:
Theorem 34 Let ${K \geq 1}$, and let ${\phi: U \rightarrow V}$ be an orientation-preserving homeomorphism. Let ${C}$ be a real analytic (and topologically closed) contour that lies in ${U}$ except possibly at the endpoints. If ${\phi: U \backslash C \rightarrow \phi(U \backslash C)}$ is ${K}$-quasiconformal, then ${\phi: U \rightarrow V}$ is ${K}$-quasiconformal.
We will generally apply this theorem in the case when ${C}$ disconnects ${U}$ into two components, in which case ${\phi}$ can be viewed as the gluing of the restrictions of this map to the two components.
Proof: As in the proof of the previous theorem, we may take ${U}$ to be a rectangle ${Q}$, and it suffices to show that ${\mathrm{mod}(\phi(Q)) \leq K \mathrm{mod}(Q)}$. We may normalise ${Q}$ to have vertices ${i, 0, M, M+i}$ where ${M = \mathrm{mod}(Q)}$, and similarly normalise ${\phi(Q)}$ to be a rectangle of vertices ${i, 0, M', M'+i}$, so we now need to show that ${M' \leq KM}$. The real analytic contour ${C}$ meets ${Q}$ in a finite number of curves, which can be broken up further into a finite horizontal line segments and graphs ${\{ f_j(y) + iy: y \in I_j\}}$ for various closed intervals ${I_j \subset [0,1]}$ and real analytic ${f_j: I_j \rightarrow [0,M]}$. For any ${\varepsilon>0}$, we can then use the uniform continuity of the ${f_j}$ to subdivide ${Q}$ into a finite number of rectangles ${Q_k= \{ x+iy: y \in J_k, 0 \leq x \leq M \}}$ where on each such rectangle, ${C}$ meets the interior of ${Q_k}$ in a bounded number of graphs ${\{ f_j(y) +iy: y \in J_k\}}$ whose horizontal variation is ${O(\varepsilon)}$. This subdivides ${Q_k}$ into a bounded number of Jordan quadrilaterals ${Q_{k,j}}$. If we let ${s_{k,j}}$ denote the distance between the ${a}$-sides of ${\phi(Q_{k,j})}$, then by uniform continuity of ${\phi}$ and the triangle inequality we have
$\displaystyle \sum_j s_{k,j} \geq (1+o(1)) M'$
as ${\varepsilon \rightarrow 0}$. By Rengel’s inequality, we have
$\displaystyle \mathrm{mod}( \phi(Q_{k,j}) ) \geq \frac{s_{k,j}^2}{\mathrm{Area}(\phi(Q_{k,j}))};$
since ${\mathrm{mod}( \phi(Q_{k,j}) ) \leq K \mathrm{mod}( Q_{k,j} )}$, we conclude using superadditivity that
$\displaystyle \mathrm{mod}(Q_j) \geq \frac{1}{K} \sum_k \frac{s_{k,j}^2}{\mathrm{Area}(\phi(Q_{k,j}))}$
and hence by Cauchy-Schwarz
$\displaystyle \mathrm{mod}(Q_j) \geq \frac{1}{K} \frac{(\sum_k s_{k,j})^2}{\sum_k \mathrm{Area}(\phi(Q_{k,j}))}$
and thus
$\displaystyle \frac{1}{\mathrm{mod}(Q_j)} \leq (1+o(1)) K \frac{\mathrm{Area}(\phi(Q_j))}{(M')^2}.$
Summing in ${j}$, we obtain
$\displaystyle \frac{1}{M} \leq (1+o(1)) K \frac{M'}{(M')^2}$
giving the desired bound ${M' \leq K M}$ after sending ${\varepsilon \rightarrow 0}$. $\Box$
It will be convenient to study analogues of the modulus when quadrilaterals are replaced by generalisations of annuli. We define a ring domain to be a region bounded between two Jordan curves ${C_1, C_2}$, where ${C_1}$ (the inner boundary) is contained inside the interior of ${C_2}$ (the outer boundary). For instance, the annulus ${\{ z: r < |z-z_0| < R \}}$ is a ring domain for any ${z_0 \in {\bf C}}$ and ${0 < r < R < \infty}$. In the spirit of Proposition 23, define the modulus ${\mathrm{mod}(A)}$ of a ring domain ${A}$ to be the supremum of all the quantities ${M}$ with the following property: for any Borel measurable ${\rho: A \rightarrow [0,+\infty)}$ one can find a rectifable curve ${\gamma}$ in ${A}$ winding once around the inner boundary ${C_1}$, such that
$\displaystyle \left(\int_\gamma \rho(z)\ |dz|\right)^2 \leq \frac{2\pi}{M} \int_A \rho^2(z)\ dz \overline{dz}.$
We record some basic properties of this modulus:
Exercise 35 Show that every ring domain is conformal to an annulus. (There are several ways to proceed here. One is to start by using Perron’s method to construct a harmonic function that is ${1}$ on one of the boundaries of the annulus and ${0}$ on the other. Another is to apply a logarithm map to transform the annulus to a simply connected domain with a “parabolic” group of discrete translation symmetries, use the Riemann mapping theorem to map this to a disc, and use the uniqueness aspect of the Riemann mapping theorem to figure out what happens to the symmetry.)
Exercise 36
• (i) Show that the modulus of an annulus ${\{ z: r < |z-z_0| < R \}}$ is given by ${\log \frac{R}{r}}$.
• (ii) Show that if ${\phi: U \rightarrow V}$ is ${K}$-quasiconformal and ${A}$ is an ring domain in ${U}$, then ${\mathrm{mod}(\phi(A)) \leq K \mathrm{mod}(A)}$. In particular, the modulus is a conformal invariant. (There is also a converse to this statement that allows for a definition of ${K}$-quasiconformality in terms of the modulus of ring domains; see e.g. Theorem 7.2 of Lehto-Virtanen.) Use this to extend the definition of modulus of a ring domain to other domains conformal to an annulus, but whose boundaries need not be Jordan curves.
• (iii) Show that if one ring domain ${A_1}$ is contained inside another ${A_2}$ (with the inner boundary of ${A_2}$ in the interior of the inner boundary of ${A_1}$), then ${\mathrm{mod}(A_1) \leq \mathrm{mod}(A_2)}$.
As a basic application of this concept we have the fact that the complex plane cannot be quasiconformal to any proper subset:
Proposition 37 Let ${\phi: {\bf C} \rightarrow V}$ be a ${K}$-quasiconformal map for some ${K \geq 1}$; then ${V = {\bf C}}$.
Proof: As ${V}$ is homeomorphic to ${{\bf C}}$, it is simply connected. Thus, if we assume for contradiction that ${V \neq {\bf C}}$, then by the Riemann mapping theorem ${V}$ is conformal to ${D(0,1)}$, so we may assume without loss of generality that ${V = D(0,1)}$.
By Exercise 36(i), the moduli ${\log R}$ of the annuli ${\{ z: 1 \leq |z| \leq R \}}$ goes to infinity as ${R \rightarrow \infty}$, and hence (by Exercise 36(ii) (applied to ${\phi^{-1}}$) the moduli of the ring domains ${\{ \phi(z): 1 \leq |z| \leq R \}}$ must also go to infinity. However, as the inner boundary of this domain is fixed and the outer one is bounded, all these ring domains can be contained inside a common annulus, contradicting Exercise 36(iii). $\Box$
For some further applications of the modulus of ring domains, we need the following result of Grötzsch:
Theorem 38 (Grötzsch modulus theorem) Let ${0 < r < 1}$, and let ${A}$ be the ring domain formed from ${D(0,1)}$ by deleting the line segment from ${0}$ to ${r}$. [Technically, ${A}$ is not quite a ring domain as defined above, but as mentioned in Exercise 36(ii), the definition of modulus remains valid in this case.] Let ${A'}$ be another ring domain contained in ${D(0,1)}$ whose inner boundary encloses both ${0}$ and ${r}$. Then ${\mathrm{mod}(A') \leq \mathrm{mod}(A)}$.
Proof: Let ${R := \exp(\mathrm{mod}(A))}$, then by Exercise 35 we can find a conformal map ${f}$ from ${A}$ to the annulus ${\{ z: 1 \leq |z| \leq R \}}$. As ${A}$ is symmetric around the real axis, and the only conformal automorphisms of the annulus that preserve the inner and outer boundaries are rotations (as can be seen for instance by using the Schwarz reflection principle repeatedly to extend such automorphisms to an entire function of linear growth), we may assume that ${f}$ obeys the symmetry ${f(\overline{z}) = \overline{f(z)}}$. Let ${\rho: A \rightarrow {\bf R}^+}$ be the function ${\rho := |f'/f|}$, then ${\rho}$ is symmetric around the real axis. One can view ${\rho}$ as a measurable function on ${A'}$; from the change of variables formula we have
$\displaystyle \int_{A'} \rho^2\ \frac{dz d\overline{z}}{2i} = \int_{1 \leq |z| \leq R} \frac{1}{|z|^2}\ \frac{dz d\overline{z}}{2i} = 2\pi \log R,$
so in particular ${\rho}$ is square-integrable. Our task is to show that ${\mathrm{mod}(A') \leq \log R}$; by the definition of modulus, it suffices to show that
$\displaystyle (\int_\gamma \rho\ d|z|)^2 \leq \frac{2\pi}{\log R} \int_{A'} \rho^2\ dz d\overline{z}$
for any rectifiable curve ${\gamma}$ that goes once around ${A'}$, and thus once around ${0}$ and ${r}$ in ${D(0,1)}$. By a limiting argument we may assume that ${\gamma}$ is polygonal. By repeatedly reflecting around the real axis whenever ${\gamma}$ crosses the line segment between ${0}$ and ${r}$, we may assume that ${\gamma}$ does not actually cross this segment, and then by perturbation we may assume it is contained in ${A}$. But then by change of variables we have
$\displaystyle \int_\gamma \rho\ d|z| = \int_{f(\gamma)} \frac{d|z|}{|z|} \leq |\int_{f(\gamma)} \frac{dz}{z}| = 2\pi$
by the Cauchy integral formula, and the claim follows. $\Box$
Exercise 39 Let ${\phi_n: U \rightarrow V_n}$ be a sequence of ${K}$-quasiconformal maps for some ${K \geq 1}$, such that all the ${V_n}$ are uniformly bounded. Show that the ${\phi_n}$ are a normal family, that is to say every sequence in ${\phi_n}$ contains a subsequence that converges locally uniformly. (Hint: use an argument similar to that in the proof of Proposition 37, combined with Theorem 38, to establish some equicontinuity of the ${\phi_n}$.)
There are many further basic properties of the conformal modulus for both quadrilaterals and annuli; we refer the interested reader to Lehto-Virtanen for details.
— 3. Rigidity of the hexagonal circle packing —
We return now to circle packings. In order to understand finite circle packings, it is convenient (in order to use some limiting arguments) to consider some infinite circle packings. A basic example of an infinite circle packing is the regular hexagonal circle packing
$\displaystyle {\mathcal H} := ( z_0 + S^1 )_{z_0 \in \Gamma}$
where ${\Gamma}$ is the hexagonal lattice
$\displaystyle \Gamma := \{ 2 n + 2 e^{2\pi i/3} m: n,m \in {\bf Z} \}$
and ${z_0 + S^1 := \{ z_0 + e^{i \theta}: \theta \in {\bf R} \}}$ is the unit circle centred at ${z_0}$. This is clearly an (infinite) circle packing, with two circles ${z_0+S^1, z_1+S^1}$ in this packing (externally) tangent if and only if they differ by twice a sixth root of unity. Between any three mutually tangent circles in this packing is an open region that we will call an interstice. It is inscribed in a dual circle that meets the three original circles orthogonally and can be computed to have radius ${1/\sqrt{3}}$; the interstice can then be viewed as a hyperbolic triangle in this dual circle in which all three sides have infinite length.
Next we need two simple geometric lemmas, due to Rodin and Sullivan.
Lemma 40 (Ring lemma) Let ${C}$ be a circle that is externally tangent to a chain ${C_1,\dots,C_n}$ of circles with disjoint interiors, with each ${C_i}$ externally tangent to ${C_{i+1}}$ (with the convention ${C_{n+1}=C_1}$). Then there is a constant ${c_n}$ depending only on ${n}$, such that the radii of each of the ${C_i}$ is at least ${c_n}$ times the radius of ${C}$.
Proof: Without loss of generality we may assume that ${C}$ has radius ${1}$ and that the radius ${r_1}$ of ${C_1}$ is maximal among the radii ${r_i}$ of the ${C_i}$. As the polygon connecting the centers of the ${C_i}$ has to contain ${C}$, we see that ${r_1 \gg_n 1}$. This forces ${r_2 \gg_n 1}$, for if ${r_2}$ was too small then ${C_2}$ would be so deep in the cuspidal region between ${C}$ and ${C_1}$ that it would not be possible for ${C_3, C_4,\dots C_n}$ to escape this cusp and go around ${C_1}$. A similar argument then gives ${r_3 \gg_n 1}$, and so forth, giving the claim. $\Box$
Lemma 41 (Length-area lemma) Let ${n \geq 1}$, and let ${{\mathcal H}_n}$ consist of those circles in ${{\mathcal H}}$ that can be connected to the circle ${0 + S^1}$ by a path of length at most ${n}$ (going through consecutively tangent circles in ${{\mathcal H}}$). Let ${{\mathcal C}_n}$ be circle packing with the same nerve as ${{\mathcal H}_n}$ that is contained in a disk of radius ${R}$. Then the circle ${C_0}$ in ${{\mathcal C}_n}$ associated to the circle ${0+S^1}$ in ${{\mathcal H}_n}$ has radius ${O(\frac{R}{\log^{1/2} n})}$.
The point of this bound is that when ${R}$ is bounded and ${n \rightarrow \infty}$, the radius of ${C_0}$ is forced to go to zero.
Proof: We can surround ${0+S^1}$ by ${n}$ disjoint chains ${(C_{j,i})_{i=1}^{6j}, j=1,\dots,n}$ of consecutively tangent circles ${z_{j,i}+S^1}$, ${i=1,\dots, 6j}$ in ${{\mathcal H}_n}$. Each circle is associated to a corresponding circle in ${{\mathcal C}}$ of some radius ${r_{j,i}}$. The total area ${\sum_{j=1}^n \sum_{i=1}^{6j} \pi r_{ij}^2}$ of these circles is at most the area ${\pi R^2}$ of the disk of radius ${R}$. Since ${\sum_{j=1}^n \frac{1}{n} \gg \log n}$, this implies from the pigeonhole principle that there exists ${j}$ for which
$\displaystyle \sum_{i=1}^{6j} \pi r_{ij}^2 \ll \frac{R^2}{j \log n}$
and hence by Cauchy-Schwarz
$\displaystyle \sum_{i=1}^{6j} r_{ij} \ll \frac{R}{\log^{1/2} n}.$
Connecting the centers of these circles, we obtain a polygonal path of length ${O( \frac{R}{\log^{1/2} n})}$ that goes around ${C_0}$, and the claim follows. $\Box$
For every circle ${z_0 + S^1}$ in the circle packing ${{\mathcal H}}$, we can form the inversion map ${\iota_{z_0}: {\bf C} \cup \{\infty\} \rightarrow {\bf C} \cup \{\infty\}}$ across this circle on the Riemann sphere, defined by setting
$\displaystyle \iota_{z_0}( z_0 + re^{i\theta} ) := z_0 + \frac{1}{r} e^{i\theta}$
for ${0 < r < \infty}$ and ${\theta \in {\bf R}}$, with the convention that ${\iota_{z_0}}$ maps ${z_0}$ to ${\infty}$ and vice versa. These are conjugates of Möbius transformations; they preserve the circle ${z_0+S^1}$ and swap the interior with the exterior. Let ${G}$ be the group of transformations of ${{\bf C} \cup \{\infty\}}$ generated by these inversions ${\iota_{z_0}}$; this is essentially a Schottky group (except for the fact that we are are allowing for conjugate Möbius transformations in addition to ordinary Möbius transformations). Let ${I}$ denote the union of all the interstices in ${{\mathcal H}}$, and let ${GI := \bigcup_{g \in G} g(I)}$ be the union of the images of the interstitial regions ${I}$ under all of these transformations. We have the following basic fact:
Proposition 42 ${{\bf C} \backslash GI}$ has Lebesgue measure zero.
Proof: (Sketch) I thank Mario Bonk for this argument. Let ${G {\mathcal H}}$ denote all the circles formed by applying an element of ${G}$ to the circles in ${{\mathcal H}}$. If ${z}$ lies in ${{\bf C} \backslash GI}$, then it lies inside one of the circles in ${{\mathcal H}}$, and then after inverting through that circle it lies in another circle in ${{\mathcal H}}$, and so forth; undoing the inversions, we conclude that ${z}$ lies in infinite number of nested circles. Let ${C}$ be one of these circles. ${GI}$ contains a union of six interstices bounded by ${C}$ and a cycle of six circles internally tangent to ${C}$ and consecutively externally tangent to each other. Applying the same argument used to establish the ring lemma (Lemma 40), we see that the six internal circles have radii comparable to that of ${C}$, and hence ${GI}$ has density ${\gg 1}$ in the disk enclosed by ${C}$, which also contains ${z}$. The ring lemma also shows that the radius of each circle in the nested sequence is at most ${1-c}$ times the one enclosing it for some absolute constant ${c>0}$, so in particular the disks shrink to zero in size. Thus ${z}$ cannot be a point of density of ${{\bf C} \backslash GI}$, and hence by the Lebesgue density theorem this set has measure zero. $\Box$
We also need another simple geometric observation:
Exercise 43 Let ${C_1,C_2,C_3}$ be mutually externally tangent circles, and let ${C'_1, C'_2, C'_3}$ be another triple of mutually external circles, with the same orientation (e.g. ${C_1,C_2,C_3}$ and ${C'_1,C'_2,C'_3}$ both go counterclockwise around their interstitial region). Show that there exists a Möbius transformation ${\phi}$ that maps each ${C_i}$ to ${C'_i}$ and which maps the interstice of ${C_1,C_2,C_3}$ conformally onto the interstice of ${C'_1,C'_2, C'_3}$.
Now we can give a rigidity result for the hexagonal circle packing, somewhat in the spirit of Theorem 4 (though it does not immediately follow from that theorem), and also due to Rodin and Sullivan:
Proposition 44 (Rigidity of infinite hexagonal packing) Let ${{\mathcal C}}$ be an infinite circle packing in ${{\bf C}}$ with the same nerve as the hexagonal circle packing ${{\mathcal H}}$. Then ${{\mathcal C}}$ is in fact equal to the hexagonl circle packing up to affine transformations and reflections.
Proof: By applying a reflection we may assume that ${{\mathcal C}}$ and ${{\mathcal H}}$ have the same orientation. For each interstitial region ${I_j}$ of ${{\mathcal H}}$ there is an associated interstitial region ${I'_j}$ of ${{\mathcal C}}$, and by Exercise 43 there is a Möbius transformation ${T_j: I_j \rightarrow I'_j}$. These can be glued together to form a map ${\phi_0}$ that is initially defined (and conformal) on the interstitial regions ${I =\bigcup_j I_j}$; we would like to extend it to the entire complex plane by defining it also inside the circles ${z_j + S^1}$.
Now consider a circle ${z_j+S^1}$ in ${{\mathcal H}}$. It is bounded by six interstitial regions ${I_1,\dots,I_6}$, which map to six interstitial regions ${I'_1,\dots,I'_6}$ that lie between the circle ${C_0}$ corresponding to ${z_j+S^1}$ and six tangent circles ${C_1,\dots,C_6}$. By the ring lemma, all of the circles ${C_1,\dots,C_6}$ have radii comparable to the radius ${r_j}$ of ${C_0}$. As a consequence, the map ${\phi_0}$, which is defined (and piecewise Möbius) on the boundary of ${z_j + S^1}$ as a map to the boundary of ${C_0}$, has derivative comparable in magnitude to ${r_j}$ also. By extending this map radially (in the sense of defining ${\phi(z_j + r e^{i\theta}) := w_j + r r_j (\phi(z_j + e^{i\theta})-w_j)}$ for ${0 < r < 1}$ and ${\theta \in {\bf R}}$, where ${w_j}$ is the centre of ${C_0}$, we see from Theorem 29 that we can extend ${\phi_0}$ to be ${K}$-quasiconformal in the interior of ${z_j+S^1}$ except possibly at ${z_j}$ for some ${K=O(1)}$, and to a homeomorphism from ${{\bf C}}$ to the region ${\phi_0({\bf C})}$ consisting of the union of the disks in ${{\mathcal C}}$ and their interstitial regions. By many applications of Theorem 34, ${\phi_0}$ is now ${K}$-quasiconformal on all of ${{\bf C}}$, and conformal in the interstitial regions ${I}$. By Proposition 37, ${\phi_0}$ surjects onto ${{\bf C}}$, thus the circle packing ${{\mathcal C}}$ and all of its interstitial regions cover the entire complex plane.
Next, we use a version of the Schwarz reflection principle to replace ${\phi_0}$ by another ${K}$-quasiconformal map that is conformal on a larger region than ${I}$. Namely, pick a circle ${z_j+S^1}$ in ${{\mathcal H}}$, and let ${C_0}$ be the corresponding circle in ${{\mathcal C}}$. Let ${\iota_j}$ and ${\iota'_j}$ be the inversions across ${z_j+S^1}$ and ${C_0}$ respectively. Note that ${\phi_0}$ maps the circle ${z_j+S^1}$ to ${C_0}$, with the interior mapping to the interior and exterior mapping to the exterior. We can then define a modified map ${\phi_1}$ by setting ${\phi_1(z)}$ equal to ${\phi_0(z)}$ on or outside ${z_j+S_1}$, and ${\phi_1(z)}$ equal to ${\iota'_j \circ \phi_0 \circ \iota_j(z)}$ inside ${z_j+S_1}$ (with the convention that ${\phi_0}$ maps ${\infty}$ to ${\infty}$). This is still an orientation-preserving function ${{\bf C}}$; by Theorem 34 it is still ${K}$-quasiconformal. It remains conformal on the interstitial region ${I}$, but is now also conformal on the additional interstitial region ${\iota_j(I)}$. Repeating this construction one can find a sequence ${\phi_n:{\bf C} \rightarrow {\bf C}}$ of ${K}$-quasiconformal maps that map each circle ${z_j+S^1}$ to their counterparts ${C_0}$, and which are conformal on a sequence ${I_n}$ of sets that increase up to ${GI}$. By Exercise 39, the restriction of ${\phi_n}$ to any compact set forms a normal family (the fact that the circles ${z_j+S^1}$ map to the circles ${C_0}$ will give the required uniform boundedness for topological reasons), and hence (by the usual diagonalisation argument) the ${\phi_n}$ themselves are a normal family; similarly for ${\phi_n^{-1}}$. Thus, by passing to a subsequence, we may assume that the ${\phi_n}$ converge locally uniformly to a limit ${\phi}$, and that ${\phi_n^{-1}}$ also converge locally uniformly to a limit which must then invert ${\phi}$. Thus ${\phi}$ is a homeomorphism, and thus ${K}$-quasiconformal by Theorem 33. It is conformal on ${GI}$, and hence by Proposition 32 it is conformal. But the only conformal maps of the complex plane are the affine maps (see Proposition 15 of this previous blog post), and hence ${{\mathcal C}}$ is an affine copy of ${{\mathcal H}}$ as required. $\Box$
By a standard limiting argument, the perfect rigidity of the infinite circle packing can be used to give approximate rigidity of finite circle packings:
Corollary 45 (Approximate rigidity of finite hexagonal packings) Let ${\varepsilon>0}$, and suppose that ${n}$ is sufficiently large depending on ${\varepsilon}$. Let ${{\mathcal H}_n}$ and ${{\mathcal C}_n}$ be as in Lemma 41. Let ${r_0}$ be the radius of the circle ${C_1}$ in ${{\mathcal C}_n}$ associated to ${0+S_1}$, and let ${r_1}$ be the radius of an adjacent circle ${C_1}$. Then ${1-\varepsilon \leq \frac{r_1}{r_0} \leq 1+\varepsilon}$.
Proof: We may normalise ${r_0=1}$ and ${C_0=S^1}$. Suppose for contradiction that the claim failed, then one can find a sequence ${n}$ tending to infinity, and circle packings ${{\mathcal C}_n}$ with nerve ${{\mathcal H}_n}$ with ${C_0 = C_{0,n} = S^1}$, such that the radius ${r_{1,n}}$ of the adjacent circle ${C_1 = C_{1,n}}$ stays bounded away from ${1}$. By many applications of the ring lemma, for each circle ${z + S^1}$ of ${{\mathcal H}}$, the corresponding circle ${C_{z,n}}$ in ${{\mathcal C}_n}$ has radius bounded above and below by zero. Passing to a subsequence using Bolzano-Weierstrass and using the Arzela-Ascoli diagonalisation argument, we may assume that the radii ${r_{z,n}}$ of these circles converge to a positive finite limit ${r_{z,\infty}}$. Applying a rotation we may also assume that the circles ${C_{1,n}}$ converge to a limit circle ${C_{1,\infty}}$ (using the obvious topology on the space of circles); we can also assume that the orientation of the ${{\mathcal C}_n}$ does not depend on ${n}$. A simple induction then shows that ${C_{z,n}}$ converges to a limit circle ${C_{z,\infty}}$, giving a circle packing ${{\mathcal C}_\infty}$ with the same nerve as ${{\mathcal H}}$. But then by Lemma 44, ${{\mathcal C}_\infty}$ is an affine copy of ${{\mathcal H}}$, which among other things implies that ${r_{1,\infty} = r_{0,\infty} = 1}$. Thus ${r_{1,n}}$ converges to ${1}$, giving the required contradiction. $\Box$
A more quantitative version of this corollary was worked out by He. There is also a purely topological proof of the rigidity of the infinite hexagonal circle packing due to Schramm.
— 4. Approximating a conformal map by circle packing —
Let ${U}$ be a simply connected bounded region in ${{\bf C}}$ with two distinct distinguished points ${z_0, z_1 \in U}$. By the Riemann mapping theorem, there is a unique conformal map ${\phi: U \rightarrow D(0,1)}$ that maps ${z_0}$ to ${0}$ and ${z_1}$ to a positive real. However, many proofs of this theorem are rather nonconstructive, and do not come with an effective algorithm to locate, or at least approximate, this map ${\phi}$.
It was conjectured by Thurston, and later proven by Rodin and Sullivan, that one could achieve this by applying the circle packing theorem (Theorem 3) to a circle packing in ${U}$ by small circles. To formalise this, we need some more notation. Let ${\varepsilon>0}$ be a small number, and let ${\varepsilon \cdot {\mathcal H}}$ be the infinite hexagonal packing scaled by ${\varepsilon}$. For every circle in ${\varepsilon \cdot {\mathcal H}}$, define the flower to be the union of this circle, its interior, the six interstices bounding it, and the six circles tangent to the circle (together with their interiors). Let ${C_0}$ be a circle in ${\varepsilon \cdot {\mathcal H}}$ such that ${z_0}$ lies in its flower. For ${\varepsilon}$ small enough, this flower is contained in ${U}$. Let ${{\mathcal I}_\varepsilon}$ denote all circles in ${\varepsilon \cdot {\mathcal H}}$ that can be reached from ${C_0}$ by a finite chain of consecutively tangent circles in ${\varepsilon \cdot {\mathcal H}}$, whose flowers all lie in ${U}$. Elements of ${{\mathcal I}_\varepsilon}$ will be called inner circles, and circles in ${\varepsilon \cdot {\mathcal H}}$ that are not an inner circle but are tangent to it will be called border circles. Because ${U}$ is simply connected, the union of all the flowers of inner circles is also simply connected. As a consequence, one can traverse the border circles by a cycle of consecutively tangent circles, with the inner circles enclosed by this cycle. Let ${{\mathcal C}_\varepsilon}$ be the circle packing consisting of the inner circles and border circles. Applying Theorem 3 followed by a Möbius transformation, one can then find a circle packing ${{\mathcal C}'_\varepsilon}$ in ${D(0,1)}$ with the same nerve and orientation as ${{\mathcal C}_\varepsilon}$, such that all the circles in ${{\mathcal C}'_\varepsilon}$ associated to border circles of ${{\mathcal C}_\varepsilon}$ are internally tangent to ${D(0,1)}$. Applying a Möbius transformation, we may assume that the flower containing ${z_0}$ in ${{\mathcal C}_\varepsilon}$ is mapped to the flower containing ${0}$, and the flower containing ${z_1}$ is mapped to a flower containing a positive real. (From the exercise below ${z_1}$ will lie in such a flower for ${\varepsilon}$ small enough.)
Let ${U_\varepsilon}$ be the union of all the solid closed equilateral triangles formed by the centres of mutually tangent circles in ${{\mathcal C}_\varepsilon}$, and let ${D_\varepsilon}$ be the corresponding union of the solid closed triangles from ${{\mathcal C}'_\varepsilon}$. Let ${\phi_\varepsilon}$ be the piecewise affine map from ${U_\varepsilon}$ to ${D_\varepsilon}$ that maps each triangle in ${U_\varepsilon}$ to the associated triangle in ${D_\varepsilon}$.
Exercise 46 Show that ${U_\varepsilon}$ converges to ${U}$ as ${\varepsilon \rightarrow 0}$ in the Hausdorff sense. In particular, ${z_1}$ lies in ${U_\varepsilon}$ for sufficiently small ${\varepsilon}$.
Exercise 47 By modifying the proof of the length-area lemma, show that all the circles ${C}$ in ${{\mathcal C}'_\varepsilon}$ have radius that goes uniformly to zero as ${\varepsilon \rightarrow 0}$. (Hint: for circles ${C}$ deep in the interior, the length-area lemma works as is; for circles ${C}$ near the boundary, one has to encircle ${C}$ by a sequence of chains that need not be closed, but may instead terminate on the boundary of ${D(0,1)}$. The argument may be viewed as a discrete version of the one used to prove Theorem 20.) Using this and the previous exercise, show that ${D_\varepsilon}$ converges to ${D(0,1)}$ in the Hausdorff sense.
From Corollary 45 we see that as ${\varepsilon \rightarrow 0}$, the circles in ${{\mathcal C}'_\varepsilon}$ corresponding to adjacent circles of ${{\mathcal C}_\varepsilon}$ in a fixed compact subset ${R}$ of ${U}$ have radii differing by a ratio of ${1+o(1)}$. We conclude that in any compact subset ${R'}$ of ${D(0,1)}$, adjacent circles in ${{\mathcal C}'_\varepsilon}$ in ${R'}$ also have radii differing by a ratio of ${1+o(1)}$, which implies by trigonometry that the triangles of ${D_\varepsilon}$ in ${R'}$ are approximately equilateral in the sense that their angles are ${\frac{\pi}{3}+o(1)}$. By Theorem 29 ${\phi_\varepsilon}$ is ${1+o(1)}$-quasiconformal on each such triangle, and hence by Theorem 34 it is ${1+o(1)}$-quasiconformal on ${R}$. By Exercise 39 every sequence of ${\phi_\varepsilon}$ has a subsequence which converges locally uniformly on ${U}$, and whose inverses converge locally uniformly on ${D}$; the limit is then a homeomorphism from ${U}$ to ${D}$ that maps ${z_0}$ to ${0}$ and ${z_1}$ to a positive real. By Theorem 33 the limit is locally ${1}$-conformal and hence conformal, hence by uniqueness of the Riemann mapping it must equal ${\phi}$. As ${\phi}$ is the unique limit point of all subsequences of the ${\phi_\varepsilon}$, this implies (by the Urysohn subsequence principle) that ${\phi_\varepsilon}$ converges locally uniformly to ${\phi}$, thus making precise the sense in which the circle packings converge to the Riemann map. | {"extraction_info": {"found_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": 1495, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9768328666687012, "perplexity": 135.07383053517307}, "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/1566027315865.44/warc/CC-MAIN-20190821085942-20190821111942-00260.warc.gz"} |
http://mccabism.blogspot.com/2015/12/tornados-and-y250-wing-tip-vortex.html | ## Wednesday, December 23, 2015
### Tornados and the Y250 wing-tip vortex
Streamwise vortices occur when fluid spirals around an axis which points in the same direction as the overall direction of fluid flow. In particular, streamwise vortices are generated by aircraft wing-tips, and by the front-wing of a Formula 1 car at the inboard transition between the neutral central section and the inner tip of the main-plane and flap(s). The latter is the so-called Y250 vortex. Surprisingly, the method by which such streamwise vorticity is generated also plays a crucial role in the generation of atmospheric tornados.
Let's begin with the meteorology. A tornado is a funnel of concentrated vertical vorticity in the atmosphere. Most tornados are generated within supercell thunderstorms when the updraft of the storm combines with the horizontal vorticity generated by vertical wind shear. The updraft tilts the horizontal vorticity into vertical vorticity, generating a rotating updraft.
However, there are two distinct types of vertical wind shear: Unidirectional and directional. The former generates crosswise vorticity, whilst the latter generates streamwise vorticity.
When the wind shear associated with a storm is unidirectional, the updraft acquires no net rotation. The updraft raises the crosswise vorticity into a hairpin shape, with one cyclonically rotating leg, on the right as one looks downstream, and an anticyclonic leg on the left. Updrafts only acquire net cyclonic rotation when the horizontal vorticity has a streamwise component. (Diagrams above and below from St Andrews University Climate and Weather Systems website).
Specifically, cyclonic tornado formation requires that the wind veers with vertical height, (meaning that its direction rotates in a clockwise sense).
In effect, the flow of air through the updraft becomes analagous to flow over a hill (personal communication with Robert Davies-Jones): the flow into the updraft has cyclonic vorticity, and the flow velocity there reinforces the vertical velocity of the updraft; the downward flow on the other side, where the anticyclonic vorticity exists, partially cancels the vertical velocity of the updraft. Hence, the cyclonic part of the updraft becomes dominant.
Before we turn to consider wing-tip vortices, we need to recall the mathematical definition of vorticity, and the vorticity transport equation.
Let's start with some notation. In what follows, we shall denote the streamwise direction as x, the lateral (aka 'spanwise' or 'crosswise') direction as y, and the vertical direction as z. The velocity vector field U has components in these directions, denoted respectively as Ux, Uy, and Uz, There is also a vorticity vector field, whose components will be denoted as ωx, ωy, and ωz.
The vorticity vector field ω is defined as the curl of the velocity vector field:
ω = (ωx , ωy, ωz)
= (∂Uz/∂y − ∂Uy/∂z , ∂Ux/∂z − ∂Uz/∂x , ∂Uy/∂x − ∂Ux/∂y)
We're also interested here in the Vorticity Transport Equation (VTE) for ωx, the streamwise component of vorticity. In this context we can simplify the VTE by omitting turbulent, viscous and baroclinic terms to obtain:
x/Dt = ωx(∂Ux/∂x) + ωy(∂Ux/∂y) + ωz(∂Ux/∂z)
The left-hand side here, Dωx/Dt, is the material derivative of the x-component of vorticity; it denotes the change of ωx in material fluid elements convected downstream by the flow.
Now, for a racecar, streamwise vorticity can be created by at least two distinct front-wing mechanisms:
1) A combination of initial lateral vorticity ωy, and a lateral gradient in streamwise velocity, ∂Ux/∂y ≠ 0.
2) A vertical gradient in the lateral component of velocity, ∂Uy/∂z ≠ 0, (corresponding to directional vertical wind shear in meteorology).
In the case of the first mechanism, one can vary the chord, camber, or angle of attack possessed by sections of the wing to create a lateral gradient in the streamwise velocity ∂Ux/∂y ≠ 0. Given that ωy ≠ 0 in the boundary layer of the wing, combining this with ∂Ux/∂y ≠ 0 entails that the second term on the right-hand side in the VTE is non-zero, which entails that Dωx/Dt ≠ 0. Thus, the creation of the spanwise-gradient in the streamwise velocity skews the initially spanwise vortex lines until they possess a significant component ωx in a streamwise direction.
However, it is perhaps the second mechanism which provides the best insight into the formation of wing-tip vortices. As the diagram above illustrates for the case of an aircraft wing (G.A.Tokaty, A History and Philosophy of Fluid Mechanics), the spanwise component of the flow varies above and below the wing. This corresponds to a non-zero value of ∂Uy/∂z, and such a non-zero value plugs straight into the definition of the curl of the velocity vector field, yielding a non-zero value for the streamwise vorticity ωx:
ωx = ∂Uz/∂y − ∂Uy/∂z
Putting this in meteorological terms, looking from the front of a Formula 1 car (with inverted wing-sections, remember), the left-hand-side of the front-wing has a veering flow-field at the junction between the flap/main-plane and the neutral section. The streamlines are, in meteorological terms, South-Easterlies under the wing, veering to South-Westerlies above. This produces streamwise vorticity of positive sign.
On the right-hand side, the flow-field is backing with increasing vertical height z. The streamlines are South-Westerlies under the wing, backing to South-Easterlies above. This produces streamwise vorticity with a negative sign.
Thus, we have demonstrated that the generation of the Y250 vortex employs the same mechanism for streamwise vorticity formation as that required for tornadogenesis. | {"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": 0.915592610836029, "perplexity": 2278.017068000098}, "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-2018-22/segments/1526794864466.23/warc/CC-MAIN-20180521181133-20180521201133-00284.warc.gz"} |
https://brilliant.org/problems/from-where-does-sncl_4-come-haww/ | # From Where Does $$SnCl_4$$ come? Haww!!!
Chemistry Level pending
19 g of molten $$SnCl_2$$ is electrolyzed for some time using inert electrodes.0.119 g of $$Sn$$ is deposited at the cathode.No substance is lost during the electrolysis.Find the ratio of the weight of $$SnCl_2$$ and $$SnCl_4$$ after electrolysis.
× | {"extraction_info": {"found_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": 0.9525501728057861, "perplexity": 3076.5617527894083}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948514238.17/warc/CC-MAIN-20171212002021-20171212022021-00429.warc.gz"} |
http://mathhelpforum.com/calculus/120327-what-question-asking.html | # Math Help - What is this question asking?
1. ## What is this question asking?
The position of an object in free fall near the surface of the plane where the acceleration due to gravity has a constant magnitude of g(length-units)/sec^2 is given by the equation:
$s=- \frac{1}{2}gt^2+v_0t+s_0$, where s is the height above the earth, $v_0$ is the initial velocity and $s_0$ is the initial height. Give the initial value problem for this situation. Solve it to check its validity. Remember the positive direction is the upward direction.
I kno about the relationship between distance, speed, velocity, acceleration. But i am confused here as to what is being asked. Help please??
2. Originally Posted by Ife
The position of an object in free fall near the surface of the plane where the acceleration due to gravity has a constant magnitude of g(length-units)/sec^2 is given by the equation:
$s=- \frac{1}{2}gt^2+v_0t+s_0$, where s is the height above the earth, $v_0$ is the initial velocity and $s_0$ is the initial height. Give the initial value problem for this situation. Solve it to check its validity. Remember the positive direction is the upward direction.
I kno about the relationship between distance, speed, velocity, acceleration. But i am confused here as to what is being asked. Help please??
I'm not sure but I think the problem wants you to show that if you integrate the acceleration function you get the velocity function, if you integrate the velocity function you get the position function. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9505255818367004, "perplexity": 146.10546713001565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246634331.38/warc/CC-MAIN-20150417045714-00151-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/the-formation-of-blackbody-radiation.156930/ | # The formation of blackbody radiation
1. Feb 18, 2007
### jauram
I am puzzled about the formation of blackbody emission (Planck's law). Specifically, we know that such things like incandescent lamps, an electric arc in a gas at high pressure etc. produce a nearly blackbody spectrum of corresponding temperature. Does this mean that in these cases a nearly equilibrium state between radiation and matter is set up? Or how do you explain why the spectra are close to that of a cavity?
2. Feb 18, 2007
### lalbatros
Consider first a cavity with a small hole: the emission is low enough as to not perturb the BB radiation.
Consider now a very large volume of gas. Since the volume is very large, most of the radiations emitted inside the volume will be reabsorbed, only a tiny fraction has chance to escape.
Consider now a small volume of a solid.
This is exactly similar to the large volume of gas, as long as this volume of solid does not become transparent (like a gold foil).
That's why hot iron from a blast furnace also shows a nice BB spectrum.
The key point is "transparency".
See the full theory of radiation heat exchanges for more details.
These notions of emissivity, absorptivity have many applications, for example in combustion modeling.
Michel
Last edited: Feb 18, 2007 | {"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": 0.9096908569335938, "perplexity": 917.7365592916306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718311.12/warc/CC-MAIN-20161020183838-00170-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/144751/how-to-prove-that-if-d-is-countable-then-fd-is-either-finite-or-countable?answertab=oldest | # How to prove that if $D$ is countable, then $f(D)$ is either finite or countable?
Is there anybody who can help me to prove that if $D$ is countable, and $f$ is a function whose domain is $D$, then $f(D)$ is either finite or countable?
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It depends on what tools are available to you. Do you already know, for instance, that a set $A$ is countably infinite or finite if there is a function $f$ from $\Bbb N$ onto $A$? – Brian M. Scott May 13 '12 at 22:00
I feel that I have answered this question before, but I am not sure where. (I mean, besides on this thread. I might be getting senile, but not that much!) – Asaf Karagila May 13 '12 at 22:28
I'm not a student, but I'd like to understand Set Theory better. I studied it many years ago, when I was in college, but not in English. So i'm not familiar with all of the terms in English. I try to learn it by reading the materials in the Internet and practice some exercises. – Fred May 14 '12 at 8:02
Recall that $f$ is a collection of ordered pairs such that if $\langle a,b\rangle$ and $\langle a,c\rangle$ are both in $f$ then $b=c$.
Furthermore, $f(D)=\{b\mid\exists a\in D:\langle a,b\rangle\in f\}$. Since for every $a\in D$ there is at most one ordered pair in $f$ in which $a$ appears in the left coordinate, we can define an injective function from $f(D)$ into $D$.
Suppose $D=\{d_n\mid n\in\mathbb N\}$, then for $b\in f(D)$ define $\pi(b)=d_n$ where $n=\min\{k\in\mathbb N\mid f(d_k)=b\}$. You should verify that this is indeed an injective function.
Recall that if we have an injective function from $A$ into $B$ then $|A|\leq |B|$ and if $|B|$ is countable then $A$ must be countable (or finite). From here the proof is about done.
In fact this depends on your definition of countable or finite: If your definition of countable or finite is "has an injection from $A$ into $\mathbb N$" you can prove that this is equivalent to "there is a surjection from $\mathbb N$ onto $A$" via a similar argument to what I wrote above.
Using the second definition you can simply argue:
Since $D$ is countable (or finite) there is some $g\colon\mathbb N\to D$ which is a surjection, therefore $f\circ g\colon\mathbb N\to f(D)$ is a surjecitve function and therefore $f(D)$ is countable (or finite).
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That's basically what I did, Asaf- but we can't make the point enough different ways for the OP. : ) – Mathemagician1234 May 13 '12 at 22:23
Thank you Asaf. – Fred May 14 '12 at 7:54
I think what is important here is that Asaf has explicitly constructed an injection from $f(D)$ to $D$, showing that the axiom of choice is not needed (he doesn't require a "choice function" because the elements of D are indexed by the naturals, and we can leverage the well-ordering of the naturals to choose a pre-image for $b$ in an unambiguous way). It is important to realize that the naturals are not well-ordered because of AC, but due to their inductive property. +1 – David Wheeler May 23 '12 at 6:24
The image of a function can contain no more points than the domain.
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Read my comment to Mathemagician. This requires at least some of the axiom of choice to hold. In the case of a countable domain this is true in ZF. Recall that many intro level courses (which I guess this question originates at) do not discuss the axioms in particular, nor the axiom of choice in specific. At least not at first. – Asaf Karagila May 13 '12 at 22:44
You can't have more elements in the image of a set under a function then in the domain-that's basically the point of Gaston's post above. Here's another way to express it so that the proof is clearer: Consider the Cartesian product definition of a mapping from a countable set D into a set E where the image of f is a subset of E. (I know,duh-but when you're trying to understand in mathematics why something is true, it's worthwhile to state the obvious so you make sure you understand the definitions.) A function by definition is a set of ordered pairs where no 2 different ordered pairs have the same first member. So ask yourself something and the proof will be clear: Can you construct such a set of ordered pairs representing f:D ----> E if f(D) is uncountable?
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Actually what you said is true only when assuming enough of the axiom of choice. Without the axiom of choice it is consistent that we can find a function whose domain is $\mathbb R$ and its range has cardinality strictly larger than $\mathbb R$. Furthermore we can find a surjective mapping from sets which are incomparable in their cardinalities as well. Assuming that $D$ is well-orderable (e.g. countable) ensures that indeed the image cannot have more elements than the domain. – Asaf Karagila May 13 '12 at 22:27
It is not true that in most naive approaches they assume AC. In our introduction course we have a strict no AC policy (don't ask me why). Naive approach would be assuming dependent choice or countable choice. Lebesgue style. My proof do not require choice because the assumption that he set is countable allows us to define the inverse function by hand. In the general proof you are using AC to choose from the fibers, if the domain is already assumed to be well orderable we can skip the assumption of choice altogether. – Asaf Karagila May 14 '12 at 6:19
No, your proof is not flawed but you have to agree that explicit assumptions are always better than implicit assumptions. Furthermore the fact that you did not hear about any basic courses not assuming the axiom of choice does not mean that there are none around the globe. In fact teaching the basic course with as little choice as possible is a good thing because it limits the students to things they can do by hand and improve the intuition. – Asaf Karagila May 14 '12 at 17:55
I also don't see where I used the "ordering of the cardinal numbers". I used the fact that countable sets are those in bijection with a subset of the natural numbers. The general results is that the image of a well-ordered set is well-orderable. This is true without the axiom of choice, however there might be sets which are not well-orderable to which the proof will not apply and might be counterexamples to the general statement that the image always have cardinality of at most as the domain. – Asaf Karagila May 14 '12 at 18:00
If $X$ can be well-ordered, fix $<$ to be a well-ordering of $X$ and then for every element in the image of $X$ pick the $<$-minimal element from that fiber. This defines an injective map into a well-ordered set, therefore the image itself is well-ordered. The axiom of choice is needed in the general case to say that the set can be well-ordered in the first place (in fact there is a choice principle which is weaker than AC, I think - not sure, saying that if $f\colon A\to B$ is a surjective function there exists $g\colon B\to A$ an injection. This is not necessarily the inverse map, though.) – Asaf Karagila May 14 '12 at 20:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9342324137687683, "perplexity": 164.460413667513}, "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-2015-40/segments/1443737913815.59/warc/CC-MAIN-20151001221833-00016-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/402680/what-should-the-form-of-error-be-on-crossentropy-or-kl-divergence-loss-function | # What should the form of error be on CrossEntropy or KL-divergence loss function across samples of distributions?
Suppose your model produces (discrete) probability distributions and you have some truth distributions you want to compare to.
For each sample $$i$$, you can compute the loss as the KL divergence or the CrossEntropy, something like this:
$$L_i = \sum_j p_{ij} \log \frac{p_{ij}}{q_{ij}}$$
But what are some motivations for picking the loss over the samples? i.e. what kind of error distribution do we expect in typical situations and why?
For example, you could minimize $$\sum_i |L_i|$$ or $$\sum_i L_i^2$$. They would have different weightings of errors. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8127549886703491, "perplexity": 708.7429397754427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316194.18/warc/CC-MAIN-20190821194752-20190821220752-00047.warc.gz"} |
http://scicomp.stackexchange.com/users/729/nicolas-essis-breton | # Nicolas Essis-Breton
less info
reputation
17
bio website nicolasessisbreton.com location Montreal, Canada age 29 member for 2 years, 1 month seen Feb 10 at 15:56 profile views 6
I'm a graduate math student at Concordia University, Montreal.
# 4 Questions
8 Are there finite element software who handles more than five dimensions? 5 How to compute the wavelet approximation of a function? 3 Is there an Implementation of the Hilbert curve from $[0,1]$ to $[0,1]^n$, where $n$ is large? ($n=10,000$, say) 2 Choosing good basis functions to approximate a Lipschitz function
# 227 Reputation
+30 How to compute the wavelet approximation of a function? +25 How to compute the wavelet approximation of a function? +10 Choosing good basis functions to approximate a Lipschitz function +5 Is there an Implementation of the Hilbert curve from $[0,1]$ to $[0,1]^n$, where $n$ is large? ($n=10,000$, say)
3 How to compute the wavelet approximation of a function?
# 9 Tags
3 fourier-analysis × 3 0 spectral-method 3 python × 2 0 high-dimensional 3 wavelet × 2 0 numerical-analysis 0 finite-element × 2 0 pde 0 data-visualization
# 15 Accounts
Mathematics 1,631 rep 516 TeX - LaTeX 349 rep 18 Stack Overflow 243 rep 313 MathOverflow 232 rep 27 Computational Science 227 rep 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": 0.8971277475357056, "perplexity": 2147.906932948539}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394021920399/warc/CC-MAIN-20140305121840-00066-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/variation-of-einstein-hilbert-action.278619/ | # Variation of Einstein-Hilbert action
1. Dec 10, 2008
### jdstokes
The Einstein field equations $\mathsf{G} = \kappa \mathsf{T}$ can be derived by considering stationary metric variations of the Einstein Hilbert action,
$S = \int \mathrm{d}^4x \sqrt{-g} (R/2\kappa + \mathcal{L}_\mathrm{M})$.
$0 = \delta S = \int\mathrm{d}^4 x\left(\frac{1}{2\kappa}\frac{\partial (\sqrt{-g} R)}{\partial g^{\alpha\beta}}+ \frac{\partial (\sqrt{-g}\mathcal{L}_\mathrm{M})}{\partial g^{\alpha\beta}}\right)\delta g^{\alpha\beta}$
etc.
In conventional field theory, however, we consider variations of the action integrand with respect to both the field $\varphi$ as well as its first derivative $\partial_\mu \varphi$.
Why can we avoid doing this when $\varphi = g^{\alpha\beta}$?.
2. Dec 10, 2008
### Fredrik
Staff Emeritus
The action is always a functional that takes the field(s) to a number. The Lagrangian on the other hand is just a (multi-variable) polynomial (it takes a bunch of numbers to a number). The variables that we plug into it are the field and its derivatives at some spacetime point x. Isn't that the case here too? Aren't there derivatives of g in the definition of R?
3. Dec 10, 2008
### jdstokes
Yes, there are derivative terms in e.g., the Christoffel symbols.
My question is why don't we treat the field derivatives as independent variables as we do in, for example scalar field theory (or even classical mechanics for that matter).
Let me give you an example. In electromagnetism the equation of motion is
$\frac{\partial \mathcal{L}}{\partial A_\mu} - \partial_\nu \frac{\partial \mathcal{L}}{\partial \partial_\nu A_\mu} = 0$.
In general relativity the equation of motion is
$\frac{\partial \mathcal{L}}{\partial g_{\mu\nu}} = 0$. I don't understand why is it not
$\frac{\partial \mathcal{L}}{\partial g_{\mu \nu}} - \partial_\rho \frac{\partial \mathcal{L}}{\partial \partial_\rho g_{\mu\nu}}= 0$?
4. Dec 10, 2008
### Fredrik
Staff Emeritus
OK, I see what you mean. Unfortunately I don't know the answer.
5. Dec 10, 2008
### haushofer
I'm not sure if I understand your question correctly. If you want to use explicitly the EL equations instead of doing the variation directly, you consider the metric, it's first derivative and second derivative as independent fields. So t he EL equations become
$\frac{\delta\mathcal{L}}{\delta g_{\mu\nu}} = \frac{\partial \mathcal{L}}{\partial g_{\mu \nu}} - \partial_\rho \frac{\partial \mathcal{L}}{\partial (\partial_\rho g_{\mu\nu})} + \partial_{\rho}\partial_{\lambda}\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}\partial_{\lambda}g_{\mu\nu})}= 0$
in which I've set the boundary conditions to zero. Using now the Lagrangian
$\mathcal{L} = \sqrt{g}R$
gives you the right equations of motion after a very tedious calculation. In d'Inverno, chapter 11 you can find some more information on this.
Last edited: Dec 10, 2008
6. Dec 10, 2008
### haushofer
Maybe it's the confusing notation between partial derivatives and functional derivatives that disturbs you. In your
$0 = \delta S = \int\mathrm{d}^4 x\left(\frac{1}{2\kappa}\frac{\partial (\sqrt{-g} R)}{\partial g^{\alpha\beta}}+ \frac{\partial (\sqrt{-g}\mathcal{L}_\mathrm{M})}{\partial g^{\alpha\beta}}\right)\delta g^{\alpha\beta}$
I would have used functional derivatives $\delta$ instead of partial derivatives $\partial$.
7. Dec 10, 2008
### jdstokes
Hang on, the Einstein field equations are
$\frac{\partial \mathcal{L}}{\partial g_{\mu\nu}} = 0$,
where $\mathcal{L} = \sqrt{-g}(R/2\kappa + \mathcal{L}_{\rm matter})$
this is a partial derivative, not a functional derivative. Compare with with electromagnetism:
$\frac{\partial \mathcal{L}}{\partial A_\mu} - \partial_\nu \frac{\partial \mathcal{L}}{\partial \partial_\nu A_\mu} = 0$
these are also partial derivatives.
There is a discrepancy.
I don't understand where you got this equation from:
$\frac{\delta\mathcal{L}}{\delta g_{\mu\nu}} = \frac{\partial \mathcal{L}}{\partial g_{\mu \nu}} - \partial_\rho \frac{\partial \mathcal{L}}{\partial (\partial_\rho g_{\mu\nu})} + \partial_{\rho}\partial_{\lambda}\frac{\partial\ma thcal{L}}{\partial(\partial_{\rho}\partial_{\lambd a}g_{\mu\nu})}= 0$
It's not the Einstein field equations.
8. Dec 10, 2008
### jdstokes
I retract my last post.
Looking at the variation
$\delta(\sqrt{-g} g^{\alpha\beta}R_{\alpha\beta}) = (\delta \sqrt{-g})g^{\alpha\beta} R_{\alpha\beta} + \sqrt{-g}\delta g^{\alpha\beta}R_{\alpha\beta} + \sqrt{-g}g^{\alpha\beta}\delta R_{\alpha\beta}$
you can see that the first two terms come from partial derivatives with respect to the metric. In order for the last term to vanish, however, you need to consider variations with respect to the Christoffel symbols (or equivalently higher derivatives of the metric).
Thanks.
9. Dec 10, 2008
### George Jones
Staff Emeritus
For a careful treatment of variational principles of general relativity, including variation of the Christoffel symbols, I suggest that you look at chapter 4, Lagrangian and Hamiltonian formulations of general relativity, from the book A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics by Eric Poisson.
10. Dec 10, 2008
### haushofer
I think you've got your answer, but maybe this helps. Just write the whole calculation explicitly from the start and try to look at how the functional derivative is derived in the first place.
So define
$$\delta\mathcal{L} = \frac{\partial\mathcal{L}}{\partial g_{\mu\nu}}\delta g_{\mu\nu} + \frac{\partial\mathcal{L}}{\partial(\partial_{\lambda}g_{\mu\nu})}\delta(\partial_{\lambda}g_{\mu\nu}) + \frac{\partial\mathcal{L}}{\partial (\partial_{\lambda}\partial_{\rho}g_{\mu\nu})}\delta(\partial_{\lambda}\partial_{\rho}g_{\mu\nu})$$
Here you explicitly use that the field and it's derivative are independent fields. Some partial integrations give you the Euler Lagrange equations plus the boundary terms,
$$\delta\mathcal{L} = \Bigl(\frac{\partial \mathcal{L}}{\partial g_{\mu \nu}} - \partial_\rho \frac{\partial \mathcal{L}}{\partial (\partial_\rho g_{\mu\nu})} + \partial_{\rho}\partial_{\lambda}\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}\partial_{\lambda}g_{\mu\nu})}\Bigr)\delta g_{\mu\nu} + BC$$
Imposing boundary conditions on a hyperplane enables you to define the initial conditions with which you can describe the evolution of the metric to another hyperplane. So if BC=0 this makes people write
$$\frac{\delta\mathcal{L}}{\delta g_{\mu\nu}} \equiv \frac{\partial \mathcal{L}}{\partial g_{\mu \nu}} - \partial_\rho \frac{\partial \mathcal{L}}{\partial (\partial_\rho g_{\mu\nu})} + \partial_{\rho}\partial_{\lambda}\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}\partial_{\lambda}g_{\mu\nu})}$$
I must say that I'm also sometimes confused between the different notations :)
11. Apr 17, 2009
### schieghoven
Hello,
The similarity is closer than it first looks... you can integrate (by parts) all of the second derivatives of $g_{\mu\nu}$ in the E-H action, leaving only $g_{\mu\nu}$ and its first partial derivatives. Then you can use Euler-Lagrange equations analogous to the electromagnetic case you've mentioned.
Just beware that the integration by parts breaks general covariance of the integrand (but obviously not the integral as a whole). I presume this is why it is rarely mentioned in introductory courses.
Dave
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https://www.physicsforums.com/threads/mapping-functions.580292/ | # Mapping Functions
1. Feb 22, 2012
### taylor81792
1. The problem statement, all variables and given/known data
Let F be the set of all functions f mapping ℝ into ℝ that have derivatives of all orders. Determine whether the given map ∅ is an isomorphism of the first binary structure with the second.
{F,+} with {ℝ,+} where ∅(f)=f'(0)
2. Relevant equations
None
3. The attempt at a solution
If I have to determine it is isomorphism, I have to prove it is onto and 1-1, but I'm not sure how to do that with ∅(f)=f'(0)
2. Feb 22, 2012
### SteveL27
Is ∅ injective? (Same as asking if it's one-to-one). That is, does ∅ always send different functions to different reals?
3. Feb 22, 2012
### Deveno
i suggest taking a handful of functions, and calculating f'(0) for each of them.
possible candidates:
f(x) = xn (don't forget the special cases n = 0, and n = 1)
f(x) = ax + b
f(x) = sin(x)
f(x) = cos(x)
do any of these have the property that θ(f1) = θ(f2) but f1 ≠ f2?
4. Feb 22, 2012
### taylor81792
so I would try doing f(x)=X^n. and then get 0^0 and 0^1? so the answers would be 1 and 0. I'm still a little bit confused
5. Feb 22, 2012
### Dick
If that's confusing you, start with f(x)=x and f(x)=sin(x).
6. Feb 22, 2012
### Staff: Mentor
If n = 0, then f(x) = 1, so f(0) = 1.
If n = 1, then f(x) = x, so f(0) = 0.
7. Feb 22, 2012
### Deveno
unless i am mistaken, the problem is asking for the derivative of f at 0, not f(0).
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https://en.wikiversity.org/wiki/Theoretical_radiation_astronomy/Quiz | # Theoretical radiation astronomy/Quiz
The images show a relativistic jet emanating from the active galactic nucleus of M87 using X-ray, radio, and optical astronomy. Credit: NASA/CXC/VLA/HST.
Theoretical radiation astronomy is the key theory lecture for the course on the principles of radiation astronomy.
This is a quiz based on the lecture that you are free to take at any time or knowledge level.
Once you’ve read and studied the lecture itself, the links contained within the article/lecture, listed under See also, External links and in the `{{principles of radiation astronomy}}` template, you should have adequate background to take the quiz and score highly.
As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.
This quiz may need up to an hour to take and is equivalent to an hourly.
Suggestion: Have the lecture available in a separate window.
Enjoy learning by doing!
## Quiz
Point added for a correct answer: Points for an incorrect answer: Ignore the questions' coefficients:
1
True or False, An action or process of throwing or sending out a traveling ray in a line, beam, or stream of small cross section may be called radiation.
TRUE FALSE
2
Which of the following is not a characteristic of radiation?
throwing a beam a stream of charged or neutral particles calculating the energy of a beam sending out a traveling ray a secondary-object hazard
3
Which of the following are theoretical radiation astronomy phenomena associated with the Sun?
a core which emits neutrinos a solar wind which emanates out the polar coronal holes gravity the barycenter for the solar system polar coronal holes coronal clouds its position
4
Before the current era and perhaps before 6,000 b2k which classical planet may have been observed as a pole star for the Earth?
5
Complete the text:
A neutron star is a type of that can result from the of a during a supernova event.
6
True or False, Neutrinos emanate from a neutron star because an atomic nucleus the size hypothesized for a neutron star is unstable and the neutrons decompose giving off neutrinos.
TRUE FALSE
7
Which of the following is not a characteristic of a sky?
catching a beam a stream of charged or neutral particles it may appear as a dome over the Earth your standing on impervious to some traveling rays a secondary-object hazard
8
Which of the following are theoretical radiation astronomy phenomena associated with the Earth?
a core which emits neutrinos a charged particle wind which emanates out the polar ionosphere holes gravity near the barycenter for the Earth-Moon system the swirls of tan, green, blue, and white are most likely sediment in the water coronal clouds chlorophyll-containing phytoplankton aloft in the upper atmosphere
9
A possible solution to the discrepancy between the Spite plateau abundance and the predicted value of the primordial lithium abundance is lithium depletion through?
10
Complete the text:
The standard solar models have enjoyed tremendous success recently in terms of agreement between the predicted and the results from but some of the Sun still defy explanation, such as the degree of depletion.
11
Yes or No, The traditional equation ${\displaystyle E=mc^{2}}$ equates energy with matter in their interconversion.
Yes No
12
Which of the following is not a characteristic of a laboratory?
catching a beam one more degree of freedom than can be measured or controlled it may appear as a dome over a small portion of the Earth your standing on impervious to some traveling rays a secondary-object hazard
13
Which of the following are theoretical radiation astronomy phenomena associated with a laboratory on Earth?
a core which emits neutrinos a charged particle wind which emanates out of a beam line gravity near the barycenter for the Earth-Moon system swirls of tan, green, blue, and white in the water electric arcs chlorophyll-containing phytoplankton aloft in the upper atmosphere
14
A laboratory solution to the discrepancy between the Spite plateau abundance and the predicted value of the primordial lithium abundance is lithium depletion through?
15
Complete the text:
In the radiation physics laboratories here on Earth, the , , , , and of radiation is studied and relative to sources are proven.
16
True or False, The traditional equation ${\displaystyle E=mc^{2}}$ equates electromagnetism with mass in their interconversion.
TRUE FALSE
17
Which of the following is not a characteristic of a theory?
catching a beam one more degree of freedom than can be measured or controlled it may appear as a dome over a small portion of the Earth your standing on impervious to some traveling rays a secondary-object hazard
18
Which of the following are theoretical radiation astronomy phenomena associated with a satellite in orbit around the Earth?
background radiation a charged particle wind which emanates out of a beam line gravity near the barycenter for the Earth-Moon system swirls of tan, green, blue, and white in the water electric arcs chlorophyll-containing phytoplankton aloft in the upper atmosphere
19
Capital Greek letters are often used for?
20
Complete the text:
Let the English upper-case (capital) letter when juxtaposed to indicate radiation at wavelengths longer than those of radiation that emanates from atomic to those just above the extreme .
21
True or False, The traditional equation ${\displaystyle E=mc^{2}}$ equates electromagnetism with matter in their interconversion.
TRUE FALSE
22
Which of the following is not a characteristic of a control group?
catching a beam one more degree of freedom than can be measured or controlled it may appear as a dome over a small portion of the Earth your standing on impervious to some traveling rays a secondary-object hazard
23
Which of the following are theoretical radiation astronomy phenomena associated with a wanderer?
possible orbits a charged particle wind gravity near the barycenter its system swirls of tan, green, blue, and white in the liquid methane electric arcs chlorophyll-containing phytoplankton aloft in an upper atmosphere
24
Electromagnetic radiation emitted by accelerating charged particles that are moving at near the speed of light?
25
Complete the text:
Any of many in which something is related to something else by an equation of the form f(x) = a·xk is called a .
26
True or False, A theory may not be right but it should be testable.
TRUE FALSE
27
Which of the following is not a characteristic of background radiation?
naturally present ionizing radiation a continuum in the environment a one-time occurrence
28
Which of the following are theoretical radiation astronomy phenomena associated with a planet?
possible orbits a hyperbolic orbit nuclear fusion at its core nuclear fusion in its ionosphere near the barycenter of its stellar system accretion electric arcs impact craters
29
Electromagnetic radiation emitted by accelerating charged particles?
30
Complete the text:
The emissivity of a perfect equals 0, that of a perfect black body equals .
31
True or False, A theory may not be testable but it should be right.
TRUE FALSE
32
Which of the following is not a characteristic of a black body?
photons go in but they don't come out absorption a hole or entry much smaller than a cavity emissivity reflectance
33
Which of the following are theoretical radiation astronomy phenomena associated with a star?
possible orbits a hyperbolic orbit nuclear fusion at its core nuclear fusion in its chromosphere near the barycenter of its planetary system accretion electric arcs impact craters radar signature
34
Electromagnetic radiation emitted by decelerating charged particles?
35
Complete the text:
A surface's spectral and do not depend on wavelength [for a graybody], so that the emissivity is a .
36
True or False, The term meteor is usually associated with visual radiation from an object streaking across an Earth sky at night.
TRUE FALSE
37
Which of the following is not a theoretical characteristic of a meteor?
size of the object interpretation of associated phenomena a hot bow-wave ion wind forming ahead of the meteor
38
Which of the following are theoretical radiation astronomy phenomena associated with a megacryometeor?
possible orbits a hyperbolic orbit nuclear fusion at its core nuclear fusion in its chromosphere near the barycenter of its planetary system accretion electric arcs non-water ice cometary origin
39
Electromagnetic radiation emitted by passing charged particles?
40
Complete the text:
The Oort cloud is a hypothesized cloud of which may lie roughly , or nearly a light-year, from the Sun.
41
True or False, Particle radiation has three main origins: (1) galactic cosmic radiation, (2) solar particle radiation, and (3) geomagnetically trapped particle radiation.
TRUE FALSE
42
Which of the following is not a theoretical characteristic of space radiation?
spacetime interpretation of associated phenomena a hot bow-wave a steady rise forming ahead of the meteor
43
Which of the following are theoretical radiation astronomy phenomena associated with space radiation?
possible orbits a hyperbolic orbit nuclear fusion at a star's exposed core nuclear fusion in a star's chromosphere near the barycenter of its planetary system accretion electric arcs non-water ice cometary origin
44
Measurements from Voyager 1 revealed a steady rise since May in collisions with?
45
Complete the text:
At the same time, in late , there was a dramatic drop in collisions with , which are thought to originate from the .
46
True or False, Below EeV energies ultra high energy neutrons have boosted lifetimes.
TRUE FALSE
47
Which of the following is not a major source of protons within the solar system?
solar coronal clouds solar wind photosphere polar coronal holes coronal mass ejections
48
Which of the following may supply power into the Crab Nebula?
an outflowing wind particles into the pulsar particles from the pulsar electrons and positrons in the wind particles coming out of the pulsar very close to light speed
49
The Galactic distribution of 511 keV line emission distribution are the bulge + the thick disk or?
50
Complete the text:
Some neutrinos originating from the Sun may be produced by the reactions occurring in and above the . Differentiating these coronal cloud-induced neutrinos from the neutrino background and those theorized to be produced within the of the Sun may someday be possible with neutrino astronomy.
51
True or False, According to relativity theory, although the speed of light in a perfect vacuum cannot be exceeded, since there are no perfect vacuums anywhere in the universe, this speed of light can be exceeded in any medium and produce Cerenkov radiation.
TRUE FALSE
52
Which of the following is not a characteristic of outer space?
gaseous pressure much less than atmospheric pressure similar to a laboratory vacuum free space imperfect vacuum partial vacuum
53
Which of the following are characteristic of QED vacuum?
fluctuations no photons no matter particles relative permittivity relative permeability
54
The most accurate standard for the metre is conveniently defined so that there are exactly 299,792,458 of them to the distance travelled by light in a standard?
55
Complete the text:
According to relativity, c is the maximum speed at which all , , and in the universe can travel.
56
True or False, An electron beam furnace is a type of vacuum furnace employing a high-energy electron beam in vacuum as the means for delivery of heat to the material being melted.
TRUE FALSE
57
Which of the following is not a characteristic of the heliosphere?
outward speed of the solar wind diminishes to zero inward pressure from interstellar space is compacting the magnetic field the solar wind even blows back at us a 100-fold increase in the intensity of high-energy electrons from elsewhere in the galaxy diffuse into our solar system from outside the source of heat that brings the coronal cloud near the Sun hot enough to emit X-rays may be the photosphere
58
Which of the following are characteristic of radiation all around us?
fluctuations ionizing airborne biologically functional components of the human body a component of DNA
59
Background radiation may simply be any radiation that is?
60
Complete the text:
The cosmic microwave background radiation is a glow that fills the in the part of the .
61
True or False, In a laboratory, background radiation refers to the measured value from any sources that affect an instrument when a radiation source sample is being measured.
TRUE FALSE
62
Which of the following is not a characteristic of radiation sensitivity?
susceptibility a material inert to change physical changes from radiation chemical changes by radiation radiation induced change to a material
63
Which of the following are characteristic of radiation damage?
harmful changes induced radioactivity proton activation photodisintegration transmutation of elements lead into gold
64
High-intensity ionizing radiation in air can produce a visible ionized air glow of telltale?
65
Complete the text:
Like gases, lack fixed structure; the effects of is therefore mainly limited to .
66
True or False, Although the Earth's field is generally well approximated by a magnetic dipole with its axis near the rotational axis, there are occasional dramatic events where the North and South geomagnetic poles trade places.
TRUE FALSE
67
Which of the following is not a characteristic of geomagnetic polar reversals?
the Earth becomes a monopole basalts record reversals seafloor magnetic anomalies reversals in ocean sediment cores reversals appear to occur at random intervals
68
Which of the following are characteristic of geomagnetic polar reversals?
the longer the period for reversal the longer the surface irradiation the ionosphere seems to reach the surface life-forms may suffer from radiation sickness asteroids may strike the Earth that otherwise would not the rotation of the Earth slows to a halt poles of ice completely melt
69
The diffuse extragalactic background radiation (DEBRA) refers to the diffuse photon field from extragalactic origin that fill our?
70
Complete the text:
DEBRA contains over ~ decades of from ~ eV to ~ GeV.
71
True or False, Terrestrial gamma-ray flashes pose a challenge to current theories of lightning, especially with the discovery of the clear signatures of neutrinos produced in lightning.
TRUE FALSE
72
Which of the following is not a characteristic of terrestrial gamma-ray flashes?
antimatter signatures atmospheric origin TGFs occur about 50 times per day globally gamma radiation fountains downward from high cloud-top sources higher energy gamma rays than come from the Sun
73
Which of the following are characteristic of interstellar extinction?
redder color indices closer stars more affected color excess observed color index minus intrinsic color index red shift blue shift
74
As of 1977, model calculations cannot reproduce the observed breadth of the Ca II λ3933 line in Da,F stars like Ross 627 without appealing to an?
75
Complete the text:
Regarding a blue haze layer near the south polar region of Titan, the difference in color above and nearer the could be due to of the haze.
76
True or False, Observations of X-rays have sometimes found the spectrum to have an upper portion and the lower portion suggesting a two stage acceleration process.
TRUE FALSE
77
Which of the following is not a characteristic of terrestrial X-ray flashes?
antimatter signatures atmospheric origin X-ray flashes occur at least 50 times per day globally X-radiation fountains upward from low cloud-top sources as high or higher energy X-rays as come from the Sun
78
Which of the following are characteristic of high-velocity stars?
moving faster than 65 km/s closer stars more affected may point away from a stellar association comet-like appearance red shift blue shift
79
A cyan color can result from a freshly excavated high-Ti?
80
Complete the text:
The swirls of tan, green, blue, and white are most likely in the water. Some of the color may come from .
81
True or False, Nitrogen/oxygen abundance ratios may decrease from outer-arm nebulae to inner-arm nebulae in spiral galaxies.
TRUE FALSE
82
Which of the following is not a characteristic of meteoritic lithium abundance?
light elements may have been formed by the irradiation of interstellar matter closely matches the solar abundance not diminished by nucleosynthesis not destroyed by nuclear fission reactions may have been produced by cosmic-ray spallation
83
Which of the following may be characteristic of hydrogen deficiency in stars?
may have been consumed by nucleosynthesis star formation in a cloud deficient in hydrogen may point away from a stellar association may have been formed by white dwarf mergers may have had transfer of helium from the secondary to the primary a possible massive convective event
84
Submillimeter radiances can be matched by models which include ice particles of?
85
Complete the text:
One of the reasons why detection of is controversial is that although (and some other methods like rotational spectroscopy) are good for the identification of simple species with large dipole moments, they are less sensitive to more molecules, even something relatively small like .
86
True or False, The weak speed of a charged particle can exceed the speed of light.
TRUE FALSE
87
Which of the following is not a characteristic of a globular cluster?
a spherical collection of stars moves together but unbound by gravity orbits a galactic core bound by gravity relatively high stellar density towards its center may have formed together
88
Which of the following may be characteristic of orbital theory?
a hyperbolic pass stellar association may point away from a stellar association eccentricity obliquity precession
89
A double star with an orbital plane that lies near enough to the line of sight to undergo eclipses is an?
90
Complete the text:
By comparing astronomical observations with laboratory measurements, astrochemists can infer the abundances, , and of stars and interstellar clouds.
91
True or False, A localized, transient volume that is observed may be called a region.
TRUE FALSE
92
Which of the following is not a characteristic of astrognosy?
internal structure element composition distributions of plasma, gases, liquids, or solids landscape spheres approximate concentricity
93
Which of the following may be characteristic of magnetohydrodynamics?
driven by current gradients neutral atoms closed tube loops twisted flux open field lines synchrotron radiation
94
The MINOS experiment uses Fermilab's NuMI beam, which is an intense beam of neutrinos, that travels 455 miles (732 km) through the Earth to the?
95
Complete the text:
Theoretical astrohistory concerns , , and experimental facts with respect to observations in the past of the of Earth.
96
True or False, Although no tubular telescope has been found at ancient archaeological sites, ancient observers may have used air telescopes.
TRUE FALSE
97
Which of the following is not a characteristic of the electrical theory of the corona?
repulsive force repulsive ejection auroral-like phenomenon around the Sun rays and streamers centrifugal ejection of particles from the solar limb coronal discharges
98
Which of the following may be characteristic of a universe?
an origin singularity chaos aether local steady-state uncountability irrational numbers
99
Astronomical radiation mathematics is the laboratory mathematics such as simulations that are generated to try to understand the observations of?
100
Complete the text:
The materials science of observatories involves the use of or materials that allow greater flexibility in the construction and deployment of observatories on, , and above rocky-object surfaces. The environment near the observatory can also be used to observation.
## Hypotheses
1. Questions leaning more on theoretical development may be better. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.8530744314193726, "perplexity": 1943.7273032495254}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684425.36/warc/CC-MAIN-20191018181458-20191018204958-00172.warc.gz"} |
https://www.physicsforums.com/threads/electric-field-of-a-moving-charge.974548/ | # I Electric Field of a moving charge
#### Zahid Iftikhar
110
21
Summary
Stationary charge has only electric field, a moving charge has a magnetic field, why not electric field?
When a charge is at rest, it has an electric field only. When the charge starts moving , it is said to have accompanied a magnetic field. My question relates to its electric field while in motion. Does it still exist or not? I know in electron guns electrons are deflected while passing thru the electric fields, so it seems electric field is still there. Please guide me on this.
High regards.
Related Classical Physics News on Phys.org
Staff Emeritus
22,994
5,266
The electric field is still there.
#### Zahid Iftikhar
110
21
Does it have the same intensity?
#### Ibix
5,138
3,440
Not in general.
The easiest thing to do if you want an algebraic answer is to write down the electromagnetic field tensor for the field of a stationary charge and then boost it.
#### Nugatory
Mentor
12,198
4,664
Many I-level treatments of E&M don’t use the electromagnetic tensor (which is a sadness because it’s not that hard and is by far the most elegant approach). If @Zahid Iftikhar isn’t comfortable working with tensors there are also the standard formulas for transforming electrical and magnetic fields between frames - Google will find these, and they’ll be in any first-year E&M textbook.
#### Zahid Iftikhar
110
21
Thanks for the time. I ll try the way you suggested.
Regards
#### vanhees71
Gold Member
12,692
4,882
Rather than doing the Lorentz boost, it's much simpler to write down the four-potential first. I'll work in Heaviside-Lorentz units.
For the particle at rest you have (in Lorenz gauge)
$$\bar{A}^{\mu}=\begin{pmatrix}\Phi_C \\0 \\0 \\0 \end{pmatrix},$$
where
$$\Phi_C=\frac{q}{4 \pi \bar{r}}.$$
Now let's write this in manifestly covariant form. The only four-vector we have in this problem is the four-velocity of the particle $u^{\mu}=\gamma (1,\vec{\beta})$ with $\beta=\vec{v}/c$ and $\gamma=(1-\beta^2)^{-1/2}$. In the rest frame $u^{\mu}=(1,0,0,0)$. Thus $A^{\mu}=\Phi_C u^{\mu}$. Now we only have to write $\Phi_C$ in a manifestly covariant way (which now is identified as a Lorentz scalar field!). Obviously we can write
$$\bar{r}^2=\vec{\bar{x}}^2=(\bar{u}_{\mu} \bar{x}^{\mu})^2 - \bar{x}_{\mu} \bar{x}^{\mu}$$. Thus we have in a general frame
$$A^{\mu}(x)=\frac{q}{4 \pi \sqrt{(u \cdot x)^2-x \cdot x}} u^{\mu}.$$
Now the electric and magnetic field components are given by (written in (1+3)-notation now)
$$\vec{E}=-\frac{1}{c} \partial_t \vec{A}-\vec{\nabla} A^0, \quad \vec{B}=\vec{\nabla} \times \vec{A}.$$
The result is
$$\vec{E}= -\frac{1}{c} \partial_t \vec{A}-\vec{\nabla} A^0 = \frac{q}{4 \pi} \frac{\gamma(\vec{x}-\vec{v} t)}{[(u \cdot x)^2-x^2]^{3/2}}, \quad \vec{B}=\vec{\nabla} \times \vec{A} = \frac{q}{4 \pi} \frac{\vec{u} \times \vec{x}}{[(u \cdot x)^2-x^2]^{3/2}}=\vec{\beta} \times \vec{E}.$$
#### Zahid Iftikhar
110
21
Thanks Sir for your effort and time. Mathematics is my downside. I find difficulty in drawing something from these equations. I am apologetic for this weakness of mine.
#### tom_
5
2
First you should realize that $\vec{B}$ and $\vec{E}$ cannot be measured directly. One can only measure their effects and these consist in the fact that the fields exert forces onto charges. By definition, E fields act on resting and moving charges, B fields act only on moving charges.
So your true question is, which force would a moving point charge (velocity $\vec{v}_s$) exert on another moving point charge (velocity $\vec{v}_d$)? The answer, derived from the Maxwell equations, is:
$$\vec{F}_M(\vec{r},\vec{v}_s,\vec{v}_d) = \frac{c\,\left(c^2-v_s^2\right)\,\left(\vec{r} + \frac{1}{c^2}\,\vec{r}\times\vec{v}_s\times\vec{v}_d\right)}{\sqrt{\left(c^2 - v_s^2\right) + \left(\frac{\vec{r}}{r}\,\vec{v}_s\right)^2}^{3}}\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}.$$
You can get to this formula by inserting E and B field (see answer above) into the formula of the Lorentz force. For low velocities $\vec{v}_s$ and $\vec{v}_d$ this formula turns into
$$\vec{F}_{M}(\vec{r},\vec{v}_s,\vec{v}_d) \,\approx\, \vec{F}_W(\vec{r},\vec{v}_s) + \frac{1}{c^2}\,\left(\vec{r}\times\vec{v}_s\times\vec{v}_d - \frac{v_s^2\,\vec{r}}{2}\right) \frac{q_s\,q_d}{4\,\pi\,\varepsilon_0\,r^3}.$$
with
$$\vec{F}_{W}(\vec{r},\vec{v}) = \left(1 + \frac{v^2}{c^2} - \frac{3}{2} \left(\frac{\vec{r}}{r}\,\frac{\vec{v}}{c}\right)^2\right)\,\frac{q_s\,q_d}{4\,\pi\,\varepsilon_0}\,\frac{\vec{r}}{r^3}$$
and the differential speed $\vec{v} = \vec{v}_d - \vec{v}_s$. This formula is called Weber force (Weber electrodynamics).
The Weber force formula works very well for its alone in practice. If the forces of all charge carriers are integrated in a current-carrying wire, one can see how the magnetic forces result from the speed differences between electrons/atoms in relation to a resting/moving test charge. The electric fields are cancelled out and only the portion of the field that results from the distortions remains.
#### Zahid Iftikhar
110
21
Thank you very much for the reply. I am proud of people at PF for being so helpful in clarifying concepts.
What I get out of your explanation is that once charges set into motion, their electric field is mainly cancelled out and but just a portion remains due to distortions. But the magnetic field is predominant in this situation. I am trying to improve upon use of Maxwell Equations, Lorentz Transformations etc.
High Regards.
Zahid
#### ndriana
3
1
No, as long as it has an electric charge, it will always have an electric field. Remember, motion is relative, so although your electron is moving, it is always at rest with respect to itself.
The only reason they don't talk about it when introducing magnetism is the equation gets complicated and instead of learning magnetism, the study jumps into electromagnetism.
#### tom_
5
2
What I get out of your explanation is that once charges set into motion, their electric field is mainly cancelled out and but just a portion remains due to distortions.
No. Take formula $\vec{F}_W$ and set the relative velocity $\vec{v}=\vec{0}$. You see that the Weber force becomes the Coulomb force, i.e. the net electromagnetic force is only caused by the E-field for $\vec{v}=\vec{0}$.
Set now $\vec{v}=(0.01,0,0)c$. From this follows $v^2=0.0001c^2$ and $v^2/c^2 = 0.0001$. The second term with $\vec{v}$ also contains a $c$ below the fraction line. You see, for small relative velocities, a tiny correction of the electric force occurs. This is magnetism.
In a current-carrying metallic conductor, the electrical components compensate each other (The metal ions have the same speed-independent field as the moving electrons). What remains is the distortion field of the electrons (terms with v and c), so only a force can be observed here when test charges move in relation to the wire.
Just think about it and play with the formulas. Is a simple effect.
Homework Helper
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#### vanhees71
Gold Member
12,692
4,882
The complete field is given in #7...
#### davidpeng949
2
3
Summary: Stationary charge has only electric field, a moving charge has a magnetic field, why not electric field?
When a charge is at rest, it has an electric field only. When the charge starts moving , it is said to have accompanied a magnetic field. My question relates to its electric field while in motion. Does it still exist or not? I know in electron guns electrons are deflected while passing thru the electric fields, so it seems electric field is still there. Please guide me on this.
High regards.
i think it is a great question, since it leads to an deeper question that, therefore, the magnetic field law should be able to be derived mathematically from the Coulomb law and velocity, nothing more.
#### vanhees71
Gold Member
12,692
4,882
There's an electromagnetic field. The split in electric and magnetic components is frame dependent. The purely "electric" Coulomb field of a static charge distribution in some frame of reference has also magnetic components in another frame, where the charge distribution moves.
#### Zahid Iftikhar
110
21
Really serious consideration. So matter is not that simple. It involves relativity.
Thanks for the time sir.
High Regards
#### vanhees71
Gold Member
12,692
4,882
Yes, indeed. As got unnoticed for about 50 years after its discovery Maxwell theory is the paradigmatic example for a relativistic classical field theory. Only when treated as such it becomes completely consistent with the rest of physics, particularly the space-time model describing all observations best (that's special relativity and Minkowski space as long as you can neglect gravity and general relativity and a pseudo-Riemannian manifold if gravity has to be included).
106
41
#### davidpeng949
2
3
The fundamental differences between static electric field and magnetic field are the following:
(1) The sources, “e-particle Q” vs. “e-current Qv”;
(2) The way the fields induced by source, “divE” vs. “curlB”;
(3) The nature of the fields, “vector field E”vs. “axial vector field B”;
(4) Effects of the fields on a test e-particles, “qE”vs. “qvxB”.
A physics student may ask questions: Why the motion of e-particles induces magnetic fields? Does the generation of magnetic fields relate with the Coulomb’s law? A teacher’s answer is that magnetism is the combination of electric field with Special Relativity and does not relate with the Coulomb’s law. The arguement is that the teacher’s answer is not sufficient to explain the above four fundamental differences.
On the other hand, if one can derive magnetic field B from coulomb law and velocity, then all of above 4 fundamental differences explained.
#### Ibix
5,138
3,440
A teacher’s answer is that magnetism is the combination of electric field with Special Relativity and does not relate with the Coulomb’s law. The arguement is that the teacher’s answer is not sufficient to explain the above four fundamental differences.
On the other hand, if one can derive magnetic field B from coulomb law and velocity, then all of above 4 fundamental differences explained.
I'm not at all sure what you are saying here, but it's easy enough to show that not all magnetic fields are just electric fields seen in another frame. The quantity $E^2-B^2$ is invariant (the same in all frames), so if there is a frame where $E=0$ there cannot be a frame where $B=0$, and vice versa.
#### vanhees71
Gold Member
12,692
4,882
The fundamental differences between static electric field and magnetic field are the following:
(1) The sources, “e-particle Q” vs. “e-current Qv”;
(2) The way the fields induced by source, “divE” vs. “curlB”;
(3) The nature of the fields, “vector field E”vs. “axial vector field B”;
(4) Effects of the fields on a test e-particles, “qE”vs. “qvxB”.
A physics student may ask questions: Why the motion of e-particles induces magnetic fields? Does the generation of magnetic fields relate with the Coulomb’s law? A teacher’s answer is that magnetism is the combination of electric field with Special Relativity and does not relate with the Coulomb’s law. The arguement is that the teacher’s answer is not sufficient to explain the above four fundamental differences.
On the other hand, if one can derive magnetic field B from coulomb law and velocity, then all of above 4 fundamental differences explained.
Well, as long as there is an inertial reference frame (Minkowski frame), where all charges are at rest you have a static field. In this special frame you have only electric components since in this frame the four-current simply is
$(j^{\mu})=(c \rho(\vec{x}),0,0,0)$. In this frame thus the solution for the four-potential simply is the Coulomb field, using Heaviside-Lorentz units
$$A^{\mu} = \phi(\vec{x}) (1,0,0,0), \quad \phi(\vec{x}) = \int_{\mathbb{R}^3} \mathrm{d}^3 r' \frac{\rho(\vec{x}')}{4 \pi|\vec{x}-\vec{x}'|}.$$
It's also very easy to give the potential in a general frame. Just write everything in manifestly covariant form:
$$\tilde{A}^{\mu}=\tilde{u}^{\mu} \tilde{\phi}(\tilde{x}).$$
Here $\tilde{u}^{\mu}$ is the four-velocity of the new frame. Of course you have to transform also the scalar Coulomb potential as a scalar potential,
$$\tilde{\phi}(\tilde{x})=\phi(x).$$
Now it's clear that in the general frame you have also magnetic components,
$$\tilde{\vec{B}}=\tilde{\vec{\nabla}} \times \tilde{\vec{A}} = -\vec{u} \times \tilde{\vec{\nabla}} \tilde{\phi}(\tilde{x}).$$
The electric field in the general frame is
$$\tilde{\vec{E}}=-\frac{1}{c} \partial_{\tilde{t}} \tilde{\vec{A}}-\tilde{\vec{\nabla}} \tilde{A}^0,$$
Since $\tilde{A}^0=\tilde{u}^0 \tilde{\phi}$ and $\tilde{\vec{A}}=\vec{u} \tilde{\phi}$ we have
$$\tilde{\vec{B}}=\frac{\tilde{\vec{u}}}{\tilde{u}^0} \times \tilde{\vec{E}}.$$
As I said before: There's one electromagnetic field, represented by the antisymmetric Faraday tensor, and whether or not in a given situation you have electric and/or magnetic components of the field is frame dependent.
#### Zahid Iftikhar
110
21
Yes, indeed. As got unnoticed for about 50 years after its discovery Maxwell theory is the paradigmatic example for a relativistic classical field theory. Only when treated as such it becomes completely consistent with the rest of physics, particularly the space-time model describing all observations best (that's special relativity and Minkowski space as long as you can neglect gravity and general relativity and a pseudo-Riemannian manifold if gravity has to be included).
Thanks for your time and help.
"Electric Field of a moving charge"
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https://nikhartman.github.io/NerdNite37/ | # Quantum Computing
... from transistors to quantum supremacy
Slides by Nik Hartman. Originally for Nerd Nite YVR #37.
### Quantum supremacy
• When a quantum computer can accomplish something a classical computer cannot.
• Likely to show up in your newsfeed in the next 1-5 years.
### First -- Let's talk classical computers
All the computer hardware we know and love today is based on the semiconductor transistor .
### Transistors = Semiconductor Junctions
Now we can do calculations with binary numbers!
### Before we do calculations... let's learn to count
$2^2=4$ $2^1=2$ $2^0=1$ SUM 0 0 0 0 0 0 1 1 0 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 1 1 1 7
AND-gate
A B OUTPUT
0 0 0
1 0 0
0 1 0
1 1 1
### many possible logic gates can be created
EXAMPLE: 2 + 3 = ??? (or 010 + 011 = ???)
2 + 3 = 5
### but computers do more than add numbers...
• Many of the hardest computational problems are simulations of nature.
• Often we can be simplify and accept approximate results
• What about exact simulations of nature?
• To describe nature exactly we need quantum mechanics.
### example: interesting (but difficult) simulation
• Simulating molecules for better/safer drugs
• Quanum mechanics determines molecule shape, interactions, and efficacy
### Superposition
A fundamental property of the very small world.
• The only results we can measure are $\uparrow$ or $\downarrow$
• Before measurement it can exist in a combination (superposition) of both states.
superposition = $a \cdot \downarrow + b \cdot \uparrow$
• $a$ and $b$ are the probabilities of measuring the atom pointing down or up
### Superposition -- An Analogy*
superposition = $\frac{1}{2} \cdot LAMB + \frac{1}{2} \cdot LOBSTER$
• What happens when the waiter asks (measures) what you would like?
• Your uncertain order (superposition) collapses into a single decision.
• If this scenario could be recreated over and over -- 50% of the time you choose lamb, 50% lobster
### more is different -- and may be impossible
• A system of N atoms has $2^N$ possible configurations
• $2 \cdot 2 \cdot 2 \ldots 2 \cdot 2 \cdot 2 = 2^N$
• Any simulation must keep track of all of these configurations to determine most probable measurement result
• Adding another atom doubles the processing power needed
### One limit on this plan
• 1 billion operations per second => smartphone
• Uses about as much power as an LED light bulb
• Today's supercomputers: 100 million-billion ($10^{17}$) operations per second
• About as much power as New York City
• $2.5 million CAD in electricity per year • That is roughly 1 million billion ($10^{15}$) Joules per year • World produces about 100 billion billion ($10^{21}$) Joules per year ### transistor size limits • Modern CPU has 10 billion ($10^{10}$) transistors. • Supercomputers use 100,000 processors in parallel. • About$10^{15}$transistors in total. • Number of cells in 10 people. • Probability of electron(s) spontaneously hopping across$\sim e^{-kL}$• L = channel length • k = electron energy ### What do we get for all that power? Simulate interactions between 45 atoms Using current technology and all of the world's electricity, we can get to about 70 atoms. ### Richard Feynman suggests a way out If simulating quantum mechanics on a classical computer is too costly, build a quantum computer. ### Transistors + Quantum Mechanics = quantum bits • Always 0 or 1 • Channel open or closed • Any superposition of 0 and 1 • Atom up or down ### A quantum processor needs more than one qubit • Qubits can be combine to make more complicated superpositions • Single atom superposition: $$superposition = a_0 \cdot \downarrow + a_1 \cdot \uparrow$$ • Many atoms forming a single superposition are entangled. • Multiple (3) atom entangled superposition: $$superposition = a_0 \cdot \downarrow\downarrow\downarrow + a_1 \cdot \downarrow\downarrow\uparrow + a_2 \cdot \downarrow\uparrow\downarrow + a_3 \cdot \uparrow\downarrow\downarrow + \\ a_4 \cdot \downarrow\uparrow\uparrow + a_5 \cdot \uparrow\downarrow\uparrow + a_6 \cdot \uparrow\uparrow\downarrow + a_7 \cdot \uparrow\uparrow\uparrow$$ • We can perform logical gates (if-then statements) on these quantum bits! • Connect quantum bits with a gates (if-then statements, e.g. AND, NAND, XOR, ...) • We invented a quantum computer! ### A sensibly sized simulation of nature • Entangled state of N-qubits is a superposition of all$2^N\$ configurations.
• $$superposition = a_0 \cdot \downarrow\downarrow\downarrow + a_1 \cdot \downarrow\downarrow\uparrow + a_2 \cdot \downarrow\uparrow\downarrow + a_3 \cdot \uparrow\downarrow\downarrow + \\ a_4 \cdot \downarrow\uparrow\uparrow + a_5 \cdot \uparrow\downarrow\uparrow + a_6 \cdot \uparrow\uparrow\downarrow + a_7 \cdot \uparrow\uparrow\uparrow$$
• Quantum computer needs only N qubits to simulate N quantum mechanical objects.
• About 50 qubits can do something impossible for a classical computer!
### Real Qubit
• Superconducting Ring
• CW current = 0
• CCW current = 1
• Built on pieces of semiconductor
• Easy to connect many qubits
• Just like connecting many transistors in our classical computer
### Real Quantum Computers
• Intel -- 49 qubits
• IBM -- 50 qubits
• Not quite enough... yet
• Superconducting states stable for around 100 micro-seconds
• Must stay very cold and isolated from environment
• Correcting errors requires additional qubits
• Need 100-500 for 50 atom simulation
### What about our hometown heros?!
D-Wave makes quantum annealing processors. Not universal quantum comptuers.
• Define interactions between qubits.
• Example: spins have lower energy when pointed in the same direction as neighbor(s). | {"extraction_info": {"found_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": 0.9676874279975891, "perplexity": 3458.092389541391}, "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/1593657146247.90/warc/CC-MAIN-20200713162746-20200713192746-00081.warc.gz"} |
http://mathhelpforum.com/calculus/169508-graphs-asymptotes.html | # Math Help - Graphs and asymptotes
1. ## Graphs and asymptotes
State the conditions on k such that the graph of $y=\frac{1}{x^2+kx+16}$ has:
a) two vertical asympototes
b) only one vertical asymptote
c) no vertical asymptotes.
a) $(-\infty ,-8)\cup(8,\infty)$
b) $k=\pm8$
c) $(-8,8)$
However if the equation above were to have two vertical asymptotes wouldn't the factorised denominator of the equation above be:
$\frac{1}{(x+8)(x+2)}$ or $\frac{1}{(x-8)(x-2)}$ or $\frac{1}{(x+1)(x+16)}$ or $\frac{1}{(x-1)(x-16)}$
So wouldn't $k=\pm10$ or $k=\pm17$? If not, why not?
And if it were to have only one vertical asymptote, the equation with the factorised denominator would be $\frac{1}{(x+4)^2}$ or $\frac{1}{(x-4)^2}$
Hence $k=\pm8$ makes sense.
If the equation were to have no vertical asymptote, wouldn't x simply equal x=0? If x did equal zero, how would you find the range of values for k with no vertical asymptote?
Any help regarding questions a and c above and how to get the answer would be appreciated!
2. Originally Posted by Joker37
However if the equation above were to have two vertical asymptotes wouldn't the factorised denominator of the equation above be:
$\frac{1}{(x+8)(x+2)}$ or $\frac{1}{(x-8)(x-2)}$ or $\frac{1}{(x+1)(x+16)}$ or $\frac{1}{(x-1)(x-16)}$
So wouldn't $k=\pm10$ or $k=\pm17$? If not, why not?
$k=\pm10$ or $k=\pm17$ certainly work. Note that these values belong to $(-\infty ,-8)\cup(8,\infty)$. $\pm10$ and $\pm17$ are not the whole answer because when the denominator factors into $(x-a)(x-b)$, $a$ and $b$ don't have to be integer.
If the equation were to have no vertical asymptote, wouldn't x simply equal x=0?
I am not sure I understand. Under what conditions would x be 0? The question was about the possible values of k.
The graph of $\frac{1}{x^2+kx+16}$ has $n$ vertical asymptotes, where $n=0,1,2$, iff $x^2+kx+16=0$ has $n$ roots, so all you need is to examine the discriminant of this quadratic equation.
3. Originally Posted by emakarov
$k=\pm10$ or $k=\pm17$ certainly work. Note that these values belong to $(-\infty ,-8)\cup(8,\infty)$.
But how do you know that the values belong to $(-\infty, -8)\cup(8,\infty)$ based on the information given? What are the steps to figuring this out?
Originally Posted by emakarov
The graph of $\frac{1}{x^2+kx+16}$ has $n$ vertical asymptotes, where $n=0,1,2$, iff $x^2+kx+16=0$ has $n$ roots, so all you need is to examine the discriminant of this quadratic equation.
But if I did this wouldn't that mean that there would be two unknown variables in the one equation. I'm sorry, I'm still a bit confused. I'm not very good at maths and need further assistance.
4. The discriminant of $x^2+kx+16$ is $k^2-4\cdot16=k^2-64$. There are two roots iff $k^2-64>0$, i.e., $k<-8$ or $k>8$. There is one root iff $k^2-64=0$, i.e., $k=\pm8$. There are no roots iff $k^2-64<0$, i.e., $-8.
5. Originally Posted by Joker37
State the conditions on k such that the graph of $y=\frac{1}{x^2+kx+16}$ has:
a) two vertical asympototes
b) only one vertical asymptote
c) no vertical asymptotes.
a) $(-\infty ,-8)\cup(8,\infty)$
b) $k=\pm8$
c) $(-8,8)$
However if the equation above were to have two vertical asymptotes wouldn't the factorised denominator of the equation above be:
$\frac{1}{(x+8)(x+2)}$ or $\frac{1}{(x-8)(x-2)}$ or $\frac{1}{(x+1)(x+16)}$ or $\frac{1}{(x-1)(x-16)}$
So wouldn't $k=\pm10$ or $k=\pm17$? If not, why not?
In order to have two asyptotes, the denominator must factor into two distinct factors, but not necessarily those two!
[tex]x^2+ kt+ 16= (x+ a)(x+ b)= x+ (a+b)x+ ab. Take a and b to be any two numbers, not necessarily integers, calculate k from that. Take a to be any number at all, and calculate b= 16/a. The only values of a that won't work is a= 4 or a= -4 because then b= 16/4= 4 or b= 16/-4= -4 so a and b are not different. If a and b are both 4, or a and b are both -4, then k= 8. For any other k, there will be two factors.
And if it were to have only one vertical asymptote, the equation with the factorised denominator would be $\frac{1}{(x+4)^2}$ or $\frac{1}{(x-4)^2}$
Hence $k=\pm8$ makes sense.
Yes, good!
If the equation were to have no vertical asymptote, wouldn't x simply equal x=0? If x did equal zero, how would you find the range of values for k with no vertical asymptote?
What? No, x is the independent variable. It can be any number (when there are no asymptotes). And there will be no asymptotes if and only the denominator is never 0. The values of x that make a quadratic, $ax^2+ bx+ c$, 0 are given by the quadratic formula: $\frac{-b\pm\sqrt{b^2- 4ac}}{2a}$. There will be NO real number x that makes that 0 if the "discriminant", $b^2- 4ac$ is negative. Here, your quadratic is $x^2+ kx+16$ so a= 1, b= k, c= 16. That means that the discriminant is $k^2- 4(1)(16)= k^2- 64$. There will be no real x that makes the quadrtic in the denominator 0, and so no vertical asymptote as long as $k^2- 64< 0$ which is the same as $-8< k< 8$.
Any help regarding questions a and c above and how to get the answer would be appreciated!
Thanks to Emakarov for pointing out a typo.
6. Originally Posted by Joker37
State the conditions on k such that the graph of
$y=\frac{1}{x^2+kx+16}$
has:
a) two vertical asympototes
b) only one vertical asymptote
c) no vertical asymptotes.
a) $(-\infty ,-8)\cup(8,\infty)$
b) $k=\pm8$
c) $(-8,8)$
However if the equation above were to have two vertical asymptotes wouldn't the factorised denominator of the equation above be:
$\frac{1}{(x+8)(x+2)},\;\; or\;\;\frac{1}{(x-8)(x-2)},\;\; or \;\;\frac{1}{(x+1)(x+16)},\;\; or\;\; \frac{1}{(x-1)(x-16)}$
So wouldn't
$k=\pm10,\;\; or \;\;k=\pm17\;\;?$
If not, why not ?
This problem is similar to your recent one containing roots.
$x^2+bx+c=(x-\alpha)(x-\beta)$
You have found the "integer roots".
However, 16 factorises in numerous ways... 32 and 0.5 being another example.
$(x-32)(x-0.5)=x^2-0.5x-32x+16=x^2-32.5x+16$
So you see, there are endless values of k possible.
You can also write the roots of the denominator being zero using the quadratic formula.
$\displaystyle\ x^2+kx+16=0\Rightarrow\alpha,\;\beta=\frac{-k\pm\sqrt{k^2-(4)16}}{2}=\frac{-k\pm\sqrt{k^2-64}}{2}$
$-\infty gives a positive value under the square root.
$8 also gives a positive value under the square root.
These values of "k" cause $\alpha,\;\beta$ to be distinct.
We cannot have a negative value under the square root.
That accounts for the solution for (a).
And if it were to have only one vertical asymptote, the equation with the factorised denominator would be
$\frac{1}{(x+4)^2}\;\;or \;\;\frac{1}{(x-4)^2}$
Hence
$k=\pm8$
makes sense.
Good!
If the equation were to have no vertical asymptote, wouldn't x simply equal x=0? If x did equal zero, how would you find the range of values for k with no vertical asymptote?
Any help regarding questions a and c above and how to get the answer would be appreciated!
Yes, x=0 works. However, the question refers to the possible values of "k".
However, if the value of k lies in between $-8$ and $8$, then $x^2+kx+16$ is never zero,
and so the denominator will not be zero and cause the function to have an asymptote.
If a quadratic equation has only complex roots (no real roots), then it never crosses
the x-axis, so it is never zero.
You have all of that to study.
7. Thanks for the help everybody!
I understand now. | {"extraction_info": {"found_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": 87, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9814884662628174, "perplexity": 463.85189426528393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464050955095.64/warc/CC-MAIN-20160524004915-00073-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/special-relativity-of-two-clocks.874524/ | # I Special relativity of two clocks
1. Jun 5, 2016
### Physgeek64
Why is it that for two clocks that are synchronised in one frame, S, but not in another, S', is there an offset in the time by a factor of $\frac{Lv}{c}$, as measured in S'. Where L is the proper length of the body, as measured in S. I'm confused as to why there is not a factor of $\gamma$ here
Many thanks
2. Jun 5, 2016
### Orodruin
Staff Emeritus
Why do you think there should be a factor $\gamma$? Have you looked at how the relation is derived?
3. Jun 5, 2016
### Battlemage!
I'm pretty sure the offset in time is Lv/c2, not Lv/c (since obviously Lv/c is in units of length).
Also section 11.3 of this link has a problem that comes up with your Lv/c involving synchronized clocks on a train. It has a nice picture too showing the distance the photon must travel, which gives those two factors.
http://www.people.fas.harvard.edu/~djmorin/chap11.pdf
4. Jun 6, 2016
### Physgeek64
How careless of me- I did mean over $c^2$. Funnily enough, this was the book that caused my confusion. I don't feel like he explains it very well. However, I have since worked it out- so all it good.
Thank you for replying though- it's very appreciated :)
5. Jun 7, 2016
### Battlemage!
Haha no it definitely could be clearer, but it does have a good picture I think.
Draft saved Draft deleted
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http://www.physics.umn.edu/events/calendar/spa.HETLS/2016?view=yes&print=yes&skel=yes | # semester, 2016
Friday, January 22nd 2016
12:30 pm:
Speaker: No Seminar
Friday, January 29th 2016
Speaker: Robert Caldwell (Dartmouth)
Subject: Cosmic Polarization Rotation
Friday, February 5th 2016
Subject: Instanton calculus in quantum mechanics: quartic double-well and sine-Gordon potentials.
In this talk we discuss two quantum mechanical problems, the
quartic double-well and the sine-Gordon potentials.
Feynman diagrams in the instanton background are used for the calculation
of the tunneling amplitude (the instanton density) in the three-loop
order. Unlike the two-loop contribution where all involved Feynman
integrals are rational numbers, we will show that in the three-loop case
they can contain irrational contributions as well.
Friday, February 12th 2016
Subject: Probing the Primordial Universe using Massive Fields
Friday, February 19th 2016
Speaker: Yuhsin Tsai (Maryland)
Subject: Exotic Signals in The Twin Higgs Model
Twin Higgs (TH) model gives a naturalness motivation to study the non-colored BSM particles, which usually have decay processes relating to the dark-hadronization and displaced signal. In this talk, I will use the exotic twin-quarks to discuss the unique collider phenomenology of the TH models. These quarks play a vital role in UV completing the TH model, and their decay products contain both SM particles and twin-hadrons, including twin-glueballs, mesons, and leptons. These twin-objects decay displacedly into SM b-quarks or leptons, and the striking signal allows the reach of twin UV-physics at the LHC and future collider. Interesting complimentarily constraints between collider and astrophysical search will also be discussed.
Friday, February 26th 2016
Speaker: Tamar Friedmann (Rochester)
Friday, March 4th 2016
Speaker: No Seminar
Friday, March 11th 2016
Speaker: Yuri Bonder (ICN - UNAM)
Subject: Using effective field theory to test Lorentz invariance
Friday, March 18th 2016
12:30 pm:
Speaker: No Seminar (Spring Break)
Friday, March 25th 2016
Speaker: Matthew Schwartz, (Harvard)
Subject: Tunneling in Quantum Field Theory and the Fate of the Universe
One of the most concrete implications of the discovery of the Higgs boson is that, in the absence of physics beyond the standard model, the long term fate of our universe can now be established through precision calculations. Are we in a metastable minimum of the Higgs potential or the true minimum? If we are in a metastable vacuum, what is its lifetime? To answer these questions, we need to understand tunneling in quantum field theory. As we delve into how tunneling works, we find many unusual elements: complex quantities which should be real, gauge-dependent quantities which should be physical, an hbar expansion which differs from the ordinary loop expansion, and ultraviolet degrees of freedom that don't decouple. This talk will discuss some of these elements and present some new perspectives on quantum tunneling.
Friday, April 1st 2016
Speaker: Kiel Howe (Fermilab)
Subject: Induced Electroweak Symmetry Breaking and the Composite (Twin) Higgs
In induced electroweak symmetry breaking models, the SM-like Higgs vev is triggered by a small hidden sector electroweak symmetry breaking tadpole. This structure decouples the physical Higgs mass from the quartic term in the Higgs potential, leading to interesting possibilities in UV completions of the Higgs sector as a composite pseudo-Nambu Goldstone boson (PNGB). This framework can allow the minimal ~v^2/f^2 tuning of some models to be evaded, and is particularly appealing in the case of the composite twin Higgs where neutral top partners cut off the leading contributions to the PNGB potential.
Friday, April 8th 2016
Speaker: Yasha Shnir (LTP, JINR)
Subject: Gravitating solitons and hairy black holes
We overview the pattern of evolution of self-gravitating solitons in the Einstein gravity coupled to matter field. We consider transformation of the matric and links between the Skyrmions and monopoles in flat space-time and Bartnik-McKinnon solitons and hairy black holes. We also briefly discuss corresponding solitons in asymptotycally AdS spacetime
Friday, April 15th 2016
Speaker: Jeremy Mardon, (Stanford)
Subject: Feeling the Pulse of Dark Matter
Over a huge mass range, from ~keV down to ~10^-22 eV, bosonic dark matter candidates can be described as oscillating classical fields. Through weak couplings to the Standard Model, they can induce in a variety of feint, time-oscillating classical signals. I will discuss ways to search for this dark matter “pulse”, including precision accelerometers, and the “Dark Matter Radio” being built at Stanford.
Friday, April 22nd 2016
Speaker: Brian Henning (Yale)
Subject: Operator Bases and Effective Field Theories
The operator basis of an effective field theory (EFT) is the set of independent operators which contribute to scattering processes. We embark on the first systematic studies of operator bases, aiming to elucidate the structure underlying what is meant by "independent operators". We show that operator bases are organized by the conformal algebra, allowing us to systematically account for redundancies associated with the use of equations of motion and integration by parts. As a means to study the operator basis, we introduce a partition function defined to count operators weighted by their field content. We provide a matrix integral formula that allows us to compute this partition function. This allows us to solve an outstanding problem in EFTs: determining the number of independent higher dimension operators in a given EFT. This solution is applied to the Standard Model EFT, where we enumerate the operator content up to dimension fifteen. The physical definition and rich structure underlying operator bases is suggestive that more physical information can be pulled from the operator basis, and we give a few speculative thoughts along these lines.
Friday, April 29th 2016
Speaker: Sebastian Ellis (Michigan)
Subject: TBA
Friday, May 6th 2016
Speaker: Chris Hill (Fermilab)
Subject: Some aspects of Axion Electrodynamics
Friday, September 9th 2016
There will be no seminar this week.
Friday, September 16th 2016
Speaker: Jared Kaplan (John Hopkins)
Subject: Information Loss in AdS_3 / CFT_2
We discuss information loss from black hole physics in
AdS_3, focusing on two sharp signatures infecting CFT_2 correlators at
large central charge c: 'forbidden singularities' arising from
Euclidean-time periodicity due to the effective Hawking temperature,
and late-time exponential decay in the Lorentzian region. We study an
infinite class of examples where forbidden singularities can be
resolved by non-perturbative effects at finite c, and we show that the
resolution has certain universal features that also apply in the
general case. Analytically continuing to the Lorentzian regime, we
find that the non-perturbative effects that resolve forbidden
singularities qualitatively change the behavior of correlators at
times t ∼ S_{BH}, the black hole entropy. This may resolve the
exponential decay of correlators at late times in black hole
backgrounds. By Borel resumming the 1/c expansion of exact examples,
we explicitly identify 'information-restoring' effects from heavy
states that should correspond to classical solutions in AdS_3. Our
results suggest a line of inquiry towards a more precise formulation
of the gravitational path integral in AdS_3.
Friday, September 23rd 2016
Speaker: Chi-Ming Chang (Berkeley)
Subject: Five Dimensional Superconformal Field Theories
This talk consists of two parts. In the first part, we discuss indices of 5d superconformal field theories (SCFTs) with emphasis on indices of 1d operators. We construct an index for BPS operators supported on a ray in 5d SCFTs with exceptional global symmetries. We compute the E_n representations (for n=2,\dots,7) of operators of low spin, thus verifying that while the expression for the index is only SO(2n-2)\timesU(1) invariant, the index itself exhibits the full E_n symmetry (at least up to the order we expanded). The ray operators we studied in 5d can be viewed as generalizations of operators constructed in a Yang-Mills theory with fundamental matter by attaching an open Wilson line to a quark. For n\le 7, in contrast to local operators, they carry nontrivial charge under the \mathbb{Z}_{9-n}\subset E_n center of the global symmetry. In the second part, we discuss an on going project of bootstrapping 5d SCFTs. We consider the four point function of the 1/2 BPS operators. We discuss the crossing symmetry and the superconformal blocks of the four point function. We present some preliminary results on the bounds of the operator dimensions and OPE coefficients obtained from analyzing the crossing equation numerically.
Friday, September 30th 2016
Speaker: Louis Yang (UCLA)
Subject: Leptogenesis via the Relaxation of Higgs and other Scalar Fields
During inflation, a scalar field with a shallow potential such as the Higgs field can develop a large vacuum expectation value (VEV) through quantum fluctuation. The relaxation of the scalar field from this large value to the minimum of its potential after inflation can lead to several interesting consequences in the Universe. In this talk, I will explore a possibility that the baryon number asymmetry of the Universe is generated via the Higgs field relaxation during reheating. I will also show that the same leptogenesis mechanism can be applied to other scalar fields explaining the matter-antimatter asymmetry of the Universe.
Friday, October 7th 2016
Speaker: Kurt Hinterbichler (Case Western Reserve)
Subject: Massive and Partially Massless Gravity
I will review recent developments in the non-linear theory of massive gravitons, or spin-2 fields. On de Sitter space, there exists a special value for the mass of a graviton for which the linear theory propagates 4 rather than 5 degrees of freedom. If a fully non-linear version of the theory exists and can be coupled to known matter, it would have interesting properties and could solve the cosmological constant problem. I will describe evidence for and obstructions to the existence of such a theory, and recent developments.
Friday, October 14th 2016
Speaker: Lena Funcke (Max Planck Institute for Physics and Ludwig Maximilian University of Munich)
Subject: Rethinking the Origin of Small Neutrino Masses
The observed small neutrino masses are one of the greatest mysteries in current theoretical particle physics. Many possible origins have been proposed so far, such as the see-saw mechanism, radiative corrections, or large extra dimensions. While all these models have been connected in some way to the Higgs condensate, we propose a substantially different mechanism based on nonperturbative gravity: assuming that gravity contains a topological theta-term analogous to the famous theta-term of QCD, we show that a neutrino condensate emerges and effectively generates the small neutrino masses. This neutrino mass generation mechanism implies numerous phenomenological consequences, such as the invalidity of the cosmological neutrino mass bound, enhanced neutrino-neutrino interactions, and neutrino decays.
Friday, October 21st 2016
Speaker: Bartosz Fornal (UCSD)
Subject: Is there a sign of new physics in beryllium transitions?
A 6.8 sigma anomaly in the invariant mass distribution of e+e- pairs produced via internal pair creation in 8Be nuclear transitions has been reported recently by Krasznahorkay et al. in Phys. Rev. Lett. 116 (2016) 042501. The data can be explained by a 17 MeV vector gauge boson X produced in the transition of an excited beryllium state to the ground state, 8Be* -> 8Be X, followed by the decay X -> e+e-. We find that the gauge boson X can be associated with a new protophobic fifth force (i.e. with a coupling to protons suppressed compared to its coupling to neutrons) with a characteristic range of 10 fm and milli-charged couplings to first generation quarks and electrons. We show that such a protophobic gauge boson is consistent with all available experimental constraints and we discuss several ways to embed this new particle into an anomaly-free extension of the Standard Model. One of the most appealing theories of this type is a model with gauged baryon number, in which the new gauge boson kinetically mixes with the photon, and provides a portal to the dark matter sector.
Friday, October 28th 2016
Speaker: Peter Koroteev (Perimeter)
Subject: Revisiting mirror symmetry in three dimensions
Three dimensional gauge theories with eight and four supercharges are known to enjoy three-dimensional mirror symmetry — a duality which can be thought of a supersymmetric generalization of particle-vortex duality. In the supersymmetric world theory it interchanges Coulomb and Higgs branches of the two dual theories. Originally 3d mirror symmetry was proposed for certain type of N=2 quiver gauge theories. We shall provide a generalization to a larger class of models which includes more complicated quiver and theories with Chern-Simons terms.
Friday, November 4th 2016
Speaker: Maulik Parikh (ASU)
Subject: The First and Second Laws of Gravity
A compelling idea in quantum gravity is that gravity is not in fact a fundamental force. Rather, in this view, gravity (and perhaps spacetime itself) is regarded as emerging from the coarse-graining of some as-yet-unidentified microscopic degrees of freedom. I will show that the first and second laws of thermodynamics, applied to the underlying degrees of freedom, give rise precisely to Einstein's equations and the null energy condition, respectively.
Friday, November 11th 2016
Speaker: Bogdan A Dobrescu (FNAL)
Subject: Vectorlike fermions and new gauge bosons
A fourth generation of chiral quarks and leptons is tightly constrained by LHC data. However, new fermions may exist if their mass is not generated by the Higgs field. Such "vectorlike quarks” are searched at the LHC under the assumption that they decay into a quark and a boson. I will show that other, more exotic decays of the vectorlike quarks may be the dominant ones. Furthermore, new gauge bosons may undergo cascade decays via vectorlike fermions, leading to novel collider signatures.
Friday, November 18th 2016
Speaker: Yue Zhang (Northwestern)
Subject: Search For Dark Matter In Terms of Dark Bound States
Understanding the nature of dark matter is an open question of central importance to particle physics and cosmology. In this talk, I discuss a model where the dark matter is a fermion charged under a dark U(1) gauge symmetry and its interactions are mediated by a massive dark photon. I will summarize the current status in the search for such a dark sector. The main focus of this talk is on the non-perturbative effects in particular dark matter bound states, which could have strong impact on the interpretation of existing experimental results and leads to new channels for the future search.
Friday, November 25th 2016
Speaker: No seminar- University closed for Thanksgiving break
Friday, December 2nd 2016
Subject: Quark-gluon jet discrimination and its applications at the LHC
The discrimination of quark- and gluon-initiated jets is a topic being actively explored by the ATLAS and CMS collaborations, with potential applications to physics searches at the LHC. In this talk, I shall give a brief overview of observables sensitive to quark and gluon jet sub-structure, the current understanding and challenges in predicting them using Monte Carlo tools, and how some of those challenges can be overcome using data-driven templates being derived by the LHC experiments. I shall then illustrate the potential impact of quark-gluon tagging in physics searches, taking the example of gluino pair production at the LHC.
Friday, December 9th 2016
Speaker: David Poland (Yale U.)
Subject: Mysteries at the Bootstrap Frontier
In this informal talk I will summarize various projects at the frontier of the conformal bootstrap, including studying 4-point functions of fermions, stress-energy tensors, and chiral multiplets in superconformal theories. The resulting bootstrap constraints both make contact with known theories and also reveal mysterious features that may give the first pieces of evidence for new previously unknown conformal field theories.
Friday, December 16th 2016
Speaker: Ian Lewis (University of Kansas)
Subject: New Physics in Double Higgs Production
With the discovery of a Higgs boson, the Standard Model of Particle Physics is complete. However, there is still much to learn about the Standard Model and we may yet expect new physics. This talk will discuss the various ways in which new physics can appear in measurements of processes involving Higgs bosons. There will be a particular focus on double Higgs production. Measurement of this process is important for determining the shape of the scalar potential and the mechanism of electroweak symmetry breaking in the Standard Model. I will give an overview of new physics that can appear in this channel, including resonant production in a simple extension of the Standard Model.
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