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https://cstheory.stackexchange.com/questions/35930/nondeterministic-communication-complexity-of-hamming-distance?noredirect=1 | # Nondeterministic communication complexity of Hamming distance
It is something that I think should be known: what is nondeterministic communication complexity of following task:
is $H(x,y) \geq k$?
There is an obvious upper bound $k \log(n)$. I would expect this to be tight up to a $o(\log(n))$ additive term (for small $k$). Is this known?
• A slightly better upper bound is $\log \binom{n}{k}$. – András Salamon Jun 8 '16 at 14:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874535202980042, "perplexity": 411.32431164022546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1596439735867.93/warc/CC-MAIN-20200804102630-20200804132630-00320.warc.gz"} |
https://physics.stackexchange.com/questions/252695/white-noise-in-the-langevin-model-and-its-autocorrelation-function | # White noise in the Langevin model and it's autocorrelation function
I am having some trouble understanding and interpreting the noise term in the Langevin equation for a colloidal particle in a fluid. By the Langevin model, I mean the following model as the equation of motion of a colloidal particle in a fluid:
$$m \frac{d^2 x}{dt^2}= - \gamma v + \eta (t)$$
where $x$, $v$ are the position and velocity of the particle respectively. The constant $\gamma$ depends on the size of the object and the viscosity of the medium, and $\eta (t)$ is 'Gaussian White noise' and is a stochastic process.
The autocorrelation function of $\eta(t)$ is written as: $$<\eta(t) \eta(t+\tau)>=k\delta(\tau)$$
What I understand from the term 'Gaussian noise' is that the function $\eta(t)$ will take random values at any time $t$, corresponding to a Gaussian distribution with zero mean and some variance, and the value of $\eta(t)$ at any other time is independent and identically distributed (identical to the distribution at t).
Is my understanding of the noise term correct? Or is the value of the noise term impulsive at every instant of time, unlike what I have described in the above paragraph? (The autocorrelation function of the noise term seems to indicate so).
If the noise term is impulsive in nature, then why is it called Gaussian noise? On the other hand, if the noise term is finite valued at every instant of time, then how can the autocorrelation function, which is the expectation value of the function taken at two instants of time be impulsive in nature?
• "...and the value of $\nu(t)$ at any other time is independent and identically distributed (identical to the distribution at t)." Can you explain where this statement comes from? I don't think $\nu$ is defined anywhere in the post. Is it supposed to be $v$? If so, I'm still not sure I understand where that statement is coming from. Apr 28 '16 at 18:30
• @DanielSank Sorry, I meant $\eta(t)$ instead of $\nu(t)$. And this is what I understand the force to be; not what I have read from some source. Thanks for correcting me. Apr 28 '16 at 18:32
The name "Gaussian noise" actually has to do with the higher order correlations in the noise, such as: $$\langle \eta(t) \eta(t+\tau_1) \eta(t+\tau_2) \rangle,$$ $$\langle \eta(t) \eta(t+\tau_1) \eta(t+\tau_2) \eta(t+\tau_3) \rangle,$$ and so on.
As to whether the $\eta(t)$ is finite, no, it is not finite. However if you smooth it with any nonzero smoothing timescale, then the result will be finite. From a frequency-domain point of view, if you look at the fourier transform of a segment of $\eta(t)$ over a small time interval, it will have finite fourier components but they will extend up to infinite frequency.
• Thank you for answering. So is the value taken by $\eta(t)$ finite valued at every instant of time, or is it impulsive? Apr 29 '16 at 8:39 | {"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": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9325507879257202, "perplexity": 113.95078477370278}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057416.67/warc/CC-MAIN-20210923013955-20210923043955-00703.warc.gz"} |
http://mathoverflow.net/questions/45934/proofs-of-rohlins-theorem-an-oriented-4-manifold-with-zero-signature-bounds-a?sort=votes | # Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)
A celebrated theorem of Rohlin states the following
An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero.
Simple homological arguments based on Lefschetz duality show that the vanishing of the signature is a necessary condition. Showing that it is also sufficient is however harder.
I know two proofs of this fact. Each is a variation of Rohlin's proof of the simpler 3-dimensional case, which says
An oriented closed 3-manifold $M^3$ bounds an oriented 4-manifold.
I was wondering if someone knows a more elementary proof, for instance based on Kirby calculus. The two proofs I know start as follows.
1. By Whitney's theorem we can embed any closed oriented n-manifold $M^n$ in $\mathbb R^{2n}$ and we can immerse it in $\mathbb R^{2n-1}$. The immersion self-intersects into circles, and by accurately surgerying $M^n$ we can eliminate these self-intersections. Surgerying changes $M^n$ via a $(n+1)$-dimensional cobordism, hence we can suppose that $M^n$ itself embeds in $\mathbb R^{2n-1}$.
2. As for knots in 3-space, any codimension-2 closed oriented manifold $M^n \subset \mathbb R^{n+2}$ bounds an oriented "Seifert" $(n+1)$-manifold $W^{n+1}$.
When $n=3$ these two facts imply that every closed oriented 3-manifold bounds an oriented 4-manifold. When $n=4$ we only obain that every closed oriented 4-manifold is cobordant to a codimension-3 embedded $M^4 \subset \mathbb R^7$ and more work has to be done.
• In his original proof, Rohlin shows that up to blowing up $M^4$ in some points (i.e. making connected sums with $\pm\mathbb {CP}^2$) we can suppose that $M^4$ bounds a 5-cycle in $\mathbb R^7$, which can be subsequently smoothed to an oriented 5-manifold (blow-ups are needed in both steps!). This proof is explained in A la recherche de la topologie perdue.
• In Kirby's book The topology of 4-manifolds, he proves that up to cobordism the 4-manifold $M^4$ can be immersed in $\mathbb R^6$. Such an immersion has double and triple points, like a surface in $\mathbb R^3$. Triple points have signs. He proves a nice theorem which says that the number of triple points counted with sign equals (up to a factor) the first Pontryagin number, which in turns equals (up to a factor 3) the signature thanks to the Hirzebruch formula! Therefore if $M$ has signature zero we can pair double points with opposite signs and destroy them by surgery. Finally we obtain an embedded cobordant 4-manifold $M^4 \subset \mathbb R^6$. Now codimension is two and there is a "Seifert" 5-manifold bounding $M^4$.
Finally, here is my question:
Do you know any other proof different from these ones? For instance, a proof which does not use embeddings in Euclidean space?
References are of course welcome.
-
Rene Thom, "Quelques propriétés globales des variétés différentiables", Commentarii Mathematici Helvetici 28, page 17–86?? In this well known and fields-medal-winning paper, Thom computes the unoriented cobordism completely, and also a large chunk of oriented cobordism, including the statement that $\Omega^{SO}_{4}=\mathbb{Z}$, which immediately implies the result you are asking for. But Thom's construction relies on embedding into euclidean space as well. – Johannes Ebert Nov 14 '10 at 13:26
Thank you very much, I though Thom only considered the non-oriented case. I am curious to see which techniques he used. – Bruno Martelli Nov 15 '10 at 17:39
It depends on what you consider elementary. Gompf-Stipcisz has something like this: Morse theory gives a handle decomposition. Surger circles (the trace gives a bordism) to kill the 1-handles (this introduces 2-handles). Turn upside down and kill the 3-handles. 2 handles are attached along a framed link whose surgery gives an $S^3$ since the 4-manifold is closed. Kirby calculus says your diagram is Kirby move equivalent to the empty framed link. So do the Kirby moves, but every time you blow up a +1 also blow up a -1 (off in a corner) and note that $CP^2 \# -CP^2$ bounds (it's the boundary of a $D^3$ bundle over $S^2$.). Handle slides are diffeos of the 4-manifold. Don't blow down, just move extra $\pm 1$ unknots aside. When you are done, your picture is a unlink with framings $\pm 1$, the same number of each since the signature is zero. This shows your manifold is bordant to a connected sum of $CP^2\# -CP^2$.
-
How can one prove Kirby calculus (namely, that two diagrams of the same 3-manifold are Kirby move equivalent) without using Rohlin's theorem? Gompf-Stipcisz' proof of Kirby calculus (in page 161) makes use of Rohlin's theorem, which is stated without proof in page 341. – Bruno Martelli Nov 13 '10 at 20:50
Oh, you're looking for a non circular proof! I don't know. Incidentally, on that page they attribute this fact to Thom, not Rohlin. – Paul Nov 13 '10 at 22:57
@Bruno: Kirby's first published proof just used Cerf's theorem on generic 1-parameter families of smooth functions connecting two Morse functions on a manifold, didn't it? – Ryan Budney Nov 14 '10 at 6:52
@Ryan: Kirby seems to use Rohlin's theorem at the beginning of page 37 of his paper. As in Gomps-Stipcisz, given two Kirby diagrams of the same 3-manifold he builds a closed 4-manifold by gluing the two resulting 4-dimensional handlebodies, he makes some blow-ups in order to kill the signature and he then uses that the resulting 4-manifold bounds a 5-manifold. Then he applies Cerf's theory to this 5-manifold, as far as I can understand from a quick reading. – Bruno Martelli Nov 14 '10 at 11:54
The MCG proofs of Kirby's theorem don't use Rokhlin's theorem. See my answer here: mathoverflow.net/questions/16848/proofs-of-kirbys-theorem – Daniel Moskovich Nov 14 '10 at 12:39
show 1 more comment | {"extraction_info": {"found_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": 0.8981627225875854, "perplexity": 740.16650355407}, "config": {"markdown_headings": true, "markdown_code": false, "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-2014-10/segments/1394011126320/warc/CC-MAIN-20140305091846-00004-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.varsitytutors.com/gre_verbal-help/two-adjectives-or-adverbs-in-two-blank-texts | Example Questions
← Previous 1 3 4 5 6 7 8 9 12 13
Example Question #1 : Two Adjectives Or Adverbs In Two Blank Texts
The raft floated on the ocean waves, lulling the young man to sleep under the rays of the summer sun.
enjoyably . . . morning
ceaselessly . . . fleeting
lightly . . . warm
buoyantly . . . warm
placidly . . . undulating
placidly . . . undulating
Explanation:
"Buoyantly, warm" is a trap to try to get you to choose a seemingly close, "practice list" word—buoyant—with "warm," luring you because of the apparent connection to the sun rays. Likewise, "ceaselessly . . . fleeting" tries to trap you into overthinking the answer. The sense of the sentence wants you to consider the calmness of the waves because of the key word "lulling;" therefore, the best answer is "placidly" (calmly) and "undulating" (moving like a wave—with the overtone of gentleness, though not necessarily).
Example Question #2 : Two Adjectives Or Adverbs In Two Blank Texts
After fifty years of marriage, affairs still did not tire the old couple. They persevered in the often overwhelming duties of their life-long commitment.
diurnal . . . regularly
connubial . . . doggedly
amorous . . . joyfully
erotic . . . rapaciously
tedious . . . lovingly
connubial . . . doggedly
Explanation:
Several of these options are tempting. Perhaps "amorous" and "joyfully" seem to make sense, as does "tedious" and "lovingly." The key phrase, though, is "often overwhelming." This indicates that the perseverance is more than a minor affair of "pushing along;" therefore, it would be best to have a word to capture a certain tenacity in this regard. "Rapaciously" does not really fit the bill for this, but "doggedly" does. Likewise, "connubial" means related to marriage and thus fits the fact that the duties are "of" the life-long commitment. That is, they are "of a marital nature" (or at least related thereto).
Example Question #3 : Two Adjectives Or Adverbs In Two Blank Texts
Not used to the etiquette of high society, the common man behaved __________, much to the chagrin of the __________ and endlessly polite dinner guests.
tediously . . . boorish
tediously . . . glib
indecorously . . . boorish
aptly . . . glib
indecorously . . . prudish
indecorously . . . prudish
Explanation:
The man most likely did not follow the rules of high etiquette, or behaved indecorously. The dinner guests, on the other hand, seem to over-value the rules of politeness, being prudish.
Example Question #4 : Two Adjectives Or Adverbs In Two Blank Texts
__________ owls of that species have __________, fluff on their bellies, and don't lose their striped appearance until they molt and gain their adult feathers when they're a year old.
juvenile . . . pedantic
ambiguous . . . apathetic
fledgling . . . striated
precarious . . . phlegmatic
hegemonic . . . caustic
fledgling . . . striated
Explanation:
For the first blank, we need an adjective that means something like "young," because the owls don't lose their striped down until they're a year old. Possible choices include "fledgling" ("relating to a young bird") and "juvenile" ("of, for, or relating to someone or something young"). For the second blank, we need a word that means "striped;" since "striated" means "striped" and "pedantic" means "narrowly, stodgily, and often ostentatiously learned," "striated" is the better choice, and the answer is "fledgling, striated."
Example Question #5 : Two Adjectives Or Adverbs In Two Blank Texts
The __________ manner of the young man was in stark contrast with the __________ older businessman.
urbane ... phlegmatic
decorous . . . boorish
esurient ... penurious
tedious . . . timorous
pellucid . . . limpid
decorous . . . boorish
Explanation:
Here we have a contrast: there aren't any clue to what the words are, but they should be opposites. The best fit is "decorous" and "boorish"—the first meaning "characterized by proper manners" and the latter "unmannered and crude."
Example Question #6 : Two Adjectives Or Adverbs In Two Blank Texts
Suzanne never doubted the words of even the most __________ liar. Her __________ personality led her to fall into the plotting hands of even the most well known scoundrels.
repentant . . . charitable
loquacious . . . rapt
notable . . . forgiving
mendacious . . . ingenuous
fulminating . . . placid
mendacious . . . ingenuous
Explanation:
From the context, it would seem that Suzanne has a rather innocent and naïve personality. (She misses even the most well known scoundrels.) Now, we might call the liar "notable," but in this case, "forgiving" does not completely fit as well as does the correct answer. "Mendacious" does in a sense reduplicate the sense of lying, but in so doing, it strengthens the indictment against such persons. Particularly, "ingenuous" captures Suzanne's innocent and unsuspecting personality.
Example Question #1 : Two Adjectives Or Adverbs In Two Blank Texts
Choose the word or set of words that, when inserted into the sentence, best completes the sentence.
Dave took everything that was said to him seriously, even __________ comments, which often forced his friends to digress from the topic of conversation to explain jokes which were normally left __________
viscous . . . querulous
filial . . . luminous
facetious . . . tacit
pusillanimous . . . pithy
sportive . . . austere
facetious . . . tacit
Explanation:
For the first blank, we're looking for an adjective that means the opposite of "serious." Either "facetious," which means joking or jesting, often inappropriately or "sportive," which means playful or lighthearted,could work. For the second blank, we need an adjective that means not explained. In choosing between "tacit" (understood or implied without being directly stated) and "austere" (severe or strict in manner, attitude, or appearance), "tacit" is the better choice, so "facetious . . . tacit" is the correct answer.
Example Question #2 : Two Adjectives Or Adverbs In Two Blank Texts
Choose the word or set of words that, when inserted into the sentence, best completes the sentence.
The butler completed his tasks in a __________ manner, barely paying attention to his work as he overheard the household's __________ scandal in its earliest stages of hushed conversations and snide comments.
pervasive . . . aggrandized
perfunctory . . . nascent
hackneyed . . . implacable
desultory . . . resolute
sordid . . . boisterous
perfunctory . . . nascent
Explanation:
For the first blank, we're looking for an adjective that reflects how the butler worked without paying attention to his work. Either "perfunctory" (carried out with a minimum of effort or reflection) or "desultory" (lacking a plan, purpose, or enthusiasm) could work. For the second blank, we need an adjective that describes how the household scandal is "in its earliest stages." In choosing between "nascent" (just beginning to develop) and "resolute" (admirably determined), "nascent" is the better choice, so the answer is "perfunctory . . . nascent."
Example Question #3 : Two Adjectives Or Adverbs In Two Blank Texts
Choose the word or set of words that, when inserted into the sentence, best completes the sentence.
The river, which was normally so __________ that you could see the rocks at the bottom, had become __________ with sediment after the night's storm.
lucid . . . turgid
limpid . . . turbid
benign . . . opaque
luminous . . . torpid
clear . . . mercurial
limpid . . . turbid
Explanation:
For the first blank, we need an adjective that means "clear" and applies to liquids. While "lucid," "limpid," "luminous," and "clear" all sound like potential correct answers, "lucid" means expressed clearly or easy to understand and refers to ideas or texts, and "luminous" means reflecting or emitting light. This leaves us with "clear" and "limpid," which describes liquids and means free of anything that darkens; completely clear, as potential answers. So, we need to pick between "mercurial" and "turbid" for the second blank, which needs an adjective describing the river's cloudiness after the storm. Since "turbid" refers to liquids and means cloudy, opaque, or thick with suspended matter, it is the better choice, and the answer is "limpid . . . turbid."
Example Question #4 : Two Adjectives Or Adverbs In Two Blank Texts
Choose the word or set of words that, when inserted into the sentence, best completes the sentence.
The young monk was finally fed up with the __________ actions of his confrère, whose sweet-seeming piety was a mask for a judgmental, indeed __________, attitude.
sanctimonious . . . acrid
hidden . . . outlandish
questionable . . . ostentatious
clandestine . . . unquestionable
two-faced . . . equivocal
sanctimonious . . . acrid
Explanation:
The key phrase is the intensifying "indeed . . ." Here, the only option is "acrid," which can mean not only bitter and angry, but also sarcastic in tone. The first word, "sanctimonious," fits as well, for it indicates being showy with one's holiness—from the Latin word sanctus, whence we derive other words like "sancity" and "sanctify."
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https://www.physicsforums.com/threads/torque-is-same-all-along-lever.776411/ | # Torque is same all along lever?
1. Oct 15, 2014
### yosimba2000
Here's my background so you know where I'm having trouble. I've attached a picture.
I have to find the force produced by the muscle to keep the arm and ball at its current position in the picture. Now, I originally assumed Torque was Force in rotation, and intuitively thought you would need to use more force if you're closer to the pivot (shorter lever arm) than compared to the end (longer lever arm). I was right about force needed being greater as the lever arm got shorter, but I'm having touble understanding why the Torque along the lever must be the same. So, now I know Torque and Force are not the same.
Mathematically, I know there's only one right answer, but how about conceptually? What is torque if it's not force? What does it represent? Why isn't the torque increasing as the lever arm length decreases?
Thanks!
#### Attached Files:
• ###### Untitled.png
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Last edited: Oct 15, 2014
2. Oct 15, 2014
### elegysix
Torque is the perpendicular force times the distance to the pivot point. It is completely different from force. it is F*d.
Given that, you can see that as d decreases, so does the torque.
In this case, in order to hold the weight steady, the two torques must be equal. Because d for the muscle is much less than the other, it requires a much larger force.
does that help?
Last edited: Oct 15, 2014
3. Oct 15, 2014
### yosimba2000
sorry, it still hasn't clicked. I mean, mathematically I agree with what you've said, but I still can't understand what Torque represents. What is it that makes it the same everywhere along the lever?
4. Oct 15, 2014
### elegysix
it is not the same everywhere along the lever. torque is only defined about a pivot point. For the arm to hold the weight steady, the two torques about that point must be equal.
I am at a loss for offering an easy conceptual understanding. I just think of it as F*d.
5. Oct 15, 2014
### yosimba2000
wait I think I've got it now. I was originally thinking of the upward force that the forearm would provide to support the ball, but now I'm thinking of the ball dragging down the forearm. And to balance that we need an upward force and the torques have to equal. Thanks a bunch!
6. Oct 15, 2014
### elegysix
you're welcome :)
Similar Discussions: Torque is same all along lever? | {"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": 0.9002643823623657, "perplexity": 876.8054033701551}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102307.32/warc/CC-MAIN-20170816144701-20170816164701-00506.warc.gz"} |
http://math.stackexchange.com/questions/6629/large-deviation-properties-of-a-function-of-a-geometric-random-variable?answertab=votes | # Large Deviation Properties of a function of a geometric random variable
Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is
$p_s = (1 - x)^{s - 1} x$,
Consider the following function of $s$
$V(s) = \delta^s$,
where $\delta < 1$. Then the expected value of this function is given by
$\sum_ 1^{\infty} V(s) p_s = \frac{\delta x} {1 - \delta (1 - x)}$.
Consider now the deviation of V(s) from its mean :
$(V(s) - \frac{\delta x} {1 - \delta (1 - x)})$
What distribution does this have?
What I am really interested in is the behavior of the follow ordinary stochastic differential equation:
$\dot{V} = \delta^s - V$,
stochastic because the law of motion depends on the random variable $s$. The stochastic approximation technique allows me to focus on the mean dynamics, which are given by
$\dot{V} = \frac{\delta x} {1 - \delta (1 - x)} - V$,
whose equilibrium is simply $\hat{V} = \frac{\delta x} {1 - \delta (1 - x)}$.
I would like to characterize the large deviation properties of the original ODE, IE calculate the exponential likelihood $V$ crosses a particular boundary $c$.
For example, if $\delta = .95$ and $x= .01$, $\hat{V} = 0.159664$. If we set $V_0 = 0.159664$, and let $V_t = V_{t-1} + .2 (\delta^s - V_{t-1})$, and $s$ has the above distribution, how do I calculate the expected time to $V_t$ crossing, say, $c=.4$? What is the associated rate function?
Editing to give the background to the problem:
I am interested in the following stochastic dynamic system:
$V_t = V_{t-1} + \gamma (\delta^{S_{t-1}} - V_{t-1})$,
where $\delta <1$, $\gamma<1$, both positive, and each $S_t$ is an i.i.d. geometric random variable with success parameter $x$. What this models is agents who have to wait a geometric length of time to get a reward of value 1, and they have time discount factor $\delta$, and they are learning the expected value of their reward using a constant gain adaptive learning procedure, with gain $\gamma$.
So considering $S$ as following a Poisson, rather than Geometric, makes little difference to the sense of the problem. The goal is
1. Find the equilibria of the learning dynamics
2. Characterize the mean time to escape from this equilibrium, the large deviation properties.
So, for large deviation properties, I should be able to use something like Cramer's theorem, I think. As Mike says below, iterating the dynamics gives
$V_t = (1−γ)^tV_0+γ\sum_{k=0}^{t−1}(1−γ)^{t−1−k}δ^{S_k}$
So it all depends on $\sum_{k=0}^{t−1}(1−γ)^{t−1−k}δ^{S_k}$, the discounted sum of some random variables.
The Probability that $V_t >c$ is then given by
$Pr(V_t > c ) = Pr(\sum_{k=0}^{t−1}(1−γ)^{t−1−k}δ^{S_k} > \frac{c-(1−γ)^tV_0}{\gamma})$
Let $Z:=\frac{c-(1−γ)^tV_0}{\gamma}$. so we need the distribution of $\sum_{k=0}^{t−1}(1−γ)^{t−1−k}δ^{S_k}$, which as Mike says is a discounted sum of independent random variables, and so we have that this is approximately normal, for large $t$.
Turning now to the question of large deviations, if things were not discounted by powers of $(1-\gamma)$, we could appeal directly to Cramer's Theorem;
$Pr(\sum \delta^{S_k} > n a) \leq Exp[-n(r^* a - Log(E[Exp(r^* \delta^S)])]$
where $r^*$ is chosen to maximize $r a - Log(E[Exp(r \delta^S)]$. The problem would be to simply calculate the moment generating function of $\delta^S$. But, I don't exactly want $Pr(\sum \delta^{S_k} > n a)$, I need $Pr(\sum (1-\gamma)^{t-1-k} \delta^{S_k} > a)$; so not an average of $\delta^{S_k}$s, but a discounted sum. so it is not quite a direct application.
-
If a random variable $X$ has distribution $P_X$, then $Y = X-E(X)$ is merely a translated version of $X$. It has the distribution $P_Y(Y = y) = P_X(X = E(X)+y)$. This answers the first part of your question. – Dinesh Oct 12 '10 at 22:10
Since $\sum (1-\gamma)^{t-1-k} \delta^{S_k}$ is approximately normal, you can get a good approximation to $P(\sum (1-\gamma)^{t-1-k} \delta^{S_k} > a)$ from software or from a table of normal distribution values. – Mike Spivey Oct 19 '10 at 20:02
If $\delta > 0$, then the transformations you are applying to the random variable $S$ to get the quantity $T = \delta^S - \frac{\delta x}{1 - \delta(1-x)}$ are all one-to-one. Since $S$ is discrete, this means you can construct the probability distribution of $T$ by inverting those transformations. This would yield
$$P(T = t) = P\left(\delta^S - \frac{\delta x}{1 - \delta(1-x)} = t\right) = P\left(S = \frac{\ln\left(\frac{\delta x}{1 - \delta(1-x)} + t\right)}{\ln \delta}\right) = (1 - x)^{y-1} x,$$
where $$y = \frac{\ln\left(\frac{\delta x}{1 - \delta(1-x)} + t\right)}{\ln \delta},$$
for any value of $t$ such that $y$ is in the set $\{1, 2, \ldots\}$.
$$V_t = (1- \gamma)^t V_0 + \gamma \sum_{k=0}^{t-1} (1 - \gamma)^{t-1-k} \delta^{S_k}.$$
So you have a sum of discounted random variables. If they were not discounted the distribution would be approximately normal for large $t$. It turns out, though, that there is an analogous result for the distribution of a sum of discounted random variables -- a discounted central limit theorem, as it were. You might have some success using this discounted central limit theorem as an approximation.
Here are the specifics. Let $Y_v = \sum_{k=0}^{\infty} v^k X_k$, where the $X_k$'s are independent and have a common distribution function. Let
$$Z_v = \frac{\sqrt{1-v}}{\sigma} \left(Y_v - \frac{\mu}{1-v}\right).$$
Then $Z_v$ is asymptotically normal with mean $0$ and variance $\frac{1}{1+v}$, for $v \to 1$, where $\mu$ and $\sigma$ are the common mean and standard deviation of the $X_k$'s. (The expected value of $|X_k - \mu|^3$ must exist, too.)
Moreover, there is a bound on the error in this approximation. Let $F_v(x)$ and $N_v(x)$ be the cdf's of the $Z_v$ random variable and of a normal with mean $0$ and variance $\frac{1}{1+v}$, respectively. Then
$$|F_v(x) - N_v(x)| \leq \frac{C \rho \sqrt{1-v}}{\sigma^3},$$ where $\rho = E[|X_k - \mu|^3]$ and $C = 5.4$.
The reference is Hans Gerber, "The Discounted Central Limit Theorem and Its Berry-Esseen Analogue," Annals of Mathematical Statistics 42(1), pp. 389-392, 1971.
-
I feel like it should be reducible to a statement about a negative binomial distribution - allowing $V_t$ to wander far from $\hat{V}$ really requires getting enough realizations of $S$ that are from from \emph{its} mean. So it ought to be reducible to a statement about sums of realizations of $S$, and the sum of geometric variables has a negative binomial distribution, whose large deviation bounds are quite sharp, since it is exponential in the tail. For some reason I am having trouble putting it into that form, however. – Dennis Oct 13 '10 at 14:04
After working a bit, it definitely seems that what I need is to say something about $\sum_{t=1}^n \delta^{S_t}$, that is, my deviation properties depend on the distribution of sums $\delta^{S_t}$. Your work above gives the distribution of each term of this sum, and it is something like a rescaled geometric. Does it follow that the sum is some kind of negative binomial? – Dennis Oct 13 '10 at 19:46
I don't think so. The reason has to do with the support of a geometric distribution (which is closed under addition) vs. that of the transformed distribution you want (which is not necessarily). For example, take $n = 2$. If $S_1$ and $S_2$ are geometric, then you can achieve $S_1 + S_2 = 2$ via any of $(0,2), (1,1), (0,1)$, which is where the negative binomial aspect comes from. However, with your transformed geometrics and $\delta = 0.5$, then you can achieve $\delta^{S_1} + \delta^{S_2} = 1.25$ only with $(0,2)$ and $(2,0)$. The value $(1,1)$ yields $\delta^{S_1} + \delta^{S_2} = 1$. – Mike Spivey Oct 13 '10 at 20:30
Can you approximate the geometric distribution with an exponential one? You wouldn't run into the same kinds of problems with the support if you were working with a continuous random variable. – Mike Spivey Oct 13 '10 at 20:31
I think that could be easily justified, in this context. I will write up something to add to the question, to give a little more background in the problem. – Dennis Oct 13 '10 at 20:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9735720753669739, "perplexity": 152.21502123019854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1418802774894.154/warc/CC-MAIN-20141217075254-00091-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/singularities-complex-analysis.779592/ | # Homework Help: Singularities Complex Analysis
1. Nov 2, 2014
### Darth Frodo
1. The problem statement, all variables and given/known data
Determine the location and type of singularity of f(z) = 1/sin^2(z)
2. Relevant equations
3. The attempt at a solution
I'm not really sure how to calculate this. At this point, we don't have explicit formulae for the coefficients of a Laurent series so I really don't know what to do. Taylor series?
Any help would be much appreciated. Thanks.
2. Nov 2, 2014
### Dick
You could just use the definition. A function 1/f(z) has a singularity at z=a if f(a)=0. If (z-a)^n/f(z-a) has a finite limit as z->a then then the singularity is order n. Where does sin^(z)=0 and what power n do you need?
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https://www.physicsforums.com/threads/help-solve-this-probability-question.38267/ | # Help solve this probability question
1. Aug 5, 2004
### denian
im during direct traslation from my language to english. hope you can understand the question.
Orchard A produces red mangoes. the mass of the red mango is distributed normally with mean 500g and standard deviation 10g
Orchard B produces yellow mangoes. the mass of the yellow mango is distributed normally with mean 490g and standard deviation 6g
(i) show that probability that the mass of a red mango chosen at random is less than 472g = 0.212
# i have proven this part.
(ii) a lorry collects mangoes from both orchard. amount of red mangoes collected is two times than the amount of yellow mangoes collected. if one mangoes is picked at random from the mangoes collected and its mass is less than 492g, find the probability that the mango is red mango.
# plz help me with this one. thx.
2. Aug 5, 2004
### denian
i dont have the answer for this question actually. and i just hope to know the way to solve it. thx.
3. Aug 5, 2004
### Galileo
Use the formula for conditional probability:
Given the mass of the mango is less than 492g, what is the probability it will be red?
In general the probability of an event A conditioned on C is:
$$P(A|C)=\frac{P(A \cup C)}{P(C)}$$
In this problem, A would be the event of picking a red mango and
C would be the event of picking a mango with a mass less than 492g.
4. Aug 5, 2004
### denian
sorry. im still blur.
how to find $$P(A \cup C)$$?
the marks given for this question is 5. so, i think the working is long.
5. Aug 5, 2004
### Galileo
I`m sorry, that should've been:
$$P(A|C)=\frac{P(A \cap C)}{P(C)}$$
$P(A \cap C)$ is the probability of getting a red mango which weighs less than 492g.
You could use:
$$P(C|A)=\frac{p(A \cap C)}{P(A)}$$
and combine with the above to give:
$$P(A|C)=\frac{P(C|A)P(A)}{P(C)}$$
Hope that helps
6. Aug 8, 2004
### denian
thanks. i have try first.
7. Aug 11, 2004
### denian
probability question. pls help to check
im sorry. i still not able to solve it. can u show me ur whole working.. sorry..
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https://www.physicsforums.com/threads/magnitude-direction-of-magnetic-field.261082/ | # Magnitude & Direction of Magnetic Field
1. Oct 2, 2008
### yayirunin2car
1. The problem statement, all variables and given/known data
An electron moving with a speed of 9.0 multiplied by 105 m/s in the positive y direction experiences zero magnetic force. When it moves in the positive z direction it experiences a force of 1.7 multiplied by 10-13 N that points in the negative x direction. What is the magnitude and direction of the magnetic field?
2. Relevant equations
B = F/(|q|vsinθ)
3. The attempt at a solution
I plugged in the force, the velocity, the charge as 1.60e-19, and guessed that the angle was 90 degrees between the velocity and magnetic field vectors. My answer was wrong, and my guess of the direction of the magnetic field in the positive y direction was also wrong. So, I'm hoping it's in the negative y direction.
Any help?
Can you offer guidance or do you also need help?
Draft saved Draft deleted
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https://towardbrightspark.info/average-acceleration/ | ## Average acceleration …
Toward bright spark; The average acceleration can be defined as below:
The ratio of the velocity vector change to the time spent on this change. The unit of average acceleration in SI is m/s2:
$\vec{a_{avg}}=\frac{\Delta&space;\vec{v}}{\Delta&space;t}=\frac{\vec{v_{2}}-\vec{v_{1}}}{t_{2}-t_{1}}$
For instance, in the following figure the body undergoes acceleration because the velocity vector is changing:
It should be noted that the above figure is about a two-dimensional movement which is beyond the scope of the current chapter.
### Points:
1. Acceleration is non-zero when the velocity vector changes. To produce acceleration the velocity vector should change due to one of the following reasons:
• The magnitude of the velocity vector (speed) changes.
• The direction of the velocity vector changes.
• Both the magnitude and direction of the velocity vector change.
Now you can install Toward bright spark app and review More tips as a video tutorial. | {"extraction_info": {"found_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": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9707126021385193, "perplexity": 557.8003854429558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038072366.31/warc/CC-MAIN-20210413122252-20210413152252-00016.warc.gz"} |
http://physics.stackexchange.com/questions/13845/depth-of-sea-and-point-at-which-waves-break | Depth of sea and point at which waves break
Following up on this answer, is the point at which waves break on the sea shore a guide to the depth of the sea at that point? Could it indicate eg hidden rocks?
Explain the direction of waves on sea shore
-
yes it does. Wikipedia quotes the critical number for the ratio of wave height to wavelength to be .17: en.wikipedia.org/wiki/Water_waves#Wave_breaking – luksen Aug 22 '11 at 14:34
@luksen, this was an interesting article. Also states that the waves break when the depth is down to just 25% more than the wave height . – storabjorn Aug 24 '11 at 7:34 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9263495802879333, "perplexity": 1222.5721978478061}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783398216.41/warc/CC-MAIN-20160624154958-00102-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/288331/paper-request-constructing-joint-mass-distributions-with-constraints | # Paper request: Constructing Joint Mass Distributions with Constraints
I am seeking papers that construct or prove the existence of joint mass distributions (couplings) of given discrete variables (preferably at least one of which is infinite) with the condition that certain entries in the joint mass table must be 0.
Some entries are 0: In my example, I have two discrete (integer-valued) random variables $A,B$, with $1\le A\le n$ and $1\le B$. I am placing a restriction on the coupling $(A',B')$ (if such a coupling exists) of $A,B$ by forcing some entries to be 0: $$P(A'=i,B'=j)=0 \text{ when } \frac{i}{\gcd(i,j)} \text{ is composite}.$$.
Recently, (Reference Request for Couplings with Conditions) I learned that Hall's Marriage Theorem may provide a technique to prove the existence of a joint distribution with such contraints.
I would like to see examples (not necessarily using Hall's Theorem) which successfully construct or prove the existence such joint distributions. | {"extraction_info": {"found_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": 0.9048006534576416, "perplexity": 461.43847659433646}, "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-2021-25/segments/1623487625967.33/warc/CC-MAIN-20210616155529-20210616185529-00296.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/intermediate-algebra-12th-edition/chapter-6-section-6-1-rational-expressions-and-functions-multiplying-and-dividing-6-1-exercises-page-373/13 | ## Intermediate Algebra (12th Edition)
a. {$x$|$x$ is a real number} b. $(-∞,∞)$
We are given the function $f(x)=\frac{2x^{2}-3x+4}{3x^{2}+8}$. The domain of the function will be all values of $x$ such that the denominator does not equal 0. Therefore, we can set the denominator equal to 0. We will exclude from the domain all values of $x$ that make the denominator equal 0. $3x^{2}+8=0$ Subtract 8 from both sides. $3x^{2}=-8$ Divide by 3. $x^{2}=-\frac{8}{3}$ There is no real number value for $x$ such that $x^{2}=-\frac{8}{3}$, so we know that the domain includes all real numbers. Therefore, the domain is {$x$|$x$ is a real number} in set-builder notation and $(-∞,∞)$ in interval notation. | {"extraction_info": {"found_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": 0.9295244216918945, "perplexity": 117.13475523141116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496670729.90/warc/CC-MAIN-20191121023525-20191121051525-00555.warc.gz"} |
https://www.intechopen.com/chapters/52985 | Open access peer-reviewed chapter
# Simulated Annealing of Constrained Statistical Functions
Written By
Barry Smith and Augustine Wong
Submitted: March 29th, 2016 Reviewed: September 28th, 2016 Published: April 26th, 2017
DOI: 10.5772/66069
From the Edited Volume
## Computational Optimization in Engineering
Edited by Hossein Peyvandi
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## Abstract
In 1987, Corana et al. published a simulated annealing (SA) algorithm. Soon thereafter in 1993, Goffe et al. coded the algorithm in FORTRAN and showed that SA could uncover global optima missed by traditional optimization software when applied to statistical modeling and estimation in economics (econometrics). This chapter shows how and why SA can be used successfully to perform likelihood-based statistical inference on models where likelihood is constrained by often very complicated functions defined on a compact parameter space. These constraints arise because likelihood-based inference involves comparing the maxima of constrained versus unconstrained statistical optimization models. The chapter begins with a review of the relevant literature on SA and constrained optimization using penalty techniques. Next, a constrained optimization problem based in maximum likelihood stress-strength modeling is introduced, and its statistical and numerical properties are summarized. SA is then used to solve a sequence of penalty-constrained optimization problems, and the results are used to construct a confidence interval for the parameter of interest in the statistical model. The chapter concludes with a brief summary of the results and some ways we were able to enhance the performance of SA in this setting.
### Keywords
• SA
• constrained optimization
• penalty
• likelihood
## 1. SA and penalty-constrained optimization
Several chapters in this book consider the foundations and development of the simulated annealing (SA) algorithm. In this chapter, we focus on just one version of the algorithm given in Ref. [1] and one FORTRAN implementation of the algorithm presented in Ref. [2]. The reasons for this are efficiency and familiarity. The algorithm in Ref. [1] is tight and effective and the FORTRAN implementation in [2] is dependable, easily extensible and sufficiently fast that it can be applied to both complicated small-scale problems and to very large Monte Carlo studies such as the one in Ref. [3].
### 1.1. Background
The SA algorithm in Ref. [1] was developed as an approach to find the unconstrained global optimum of functions with a potential multiplicity of optima, some of which may lie on the boundaries of the function’s domain. The function being optimized need not be differentiable or even continuous, but it must be bounded. As well, the domain of the function must be compact. The search algorithm in Ref. [1] depends upon function evaluations. Derivatives play no role. The reader is referred to the careful statement of the algorithm on pages 266–269 in Ref. [1]. Here we provide an overview of the algorithm and its implementation.
In a multivariate setting, the domain of the function is initially defined as a (perhaps very) large hypercube. Random paths of individual choice variable values are searched at each stage. The number of such searches is controlled by the user. The step sizes that partially define the random paths are part of the algorithm and will in general be different for each choice variable at each stage. The algorithm cycles through and individually changes (increases or decreases) all choice variables. Each time a variable change leads to a “better” value of the function being optimized, this point is accepted. Sometimes “worse” points are temporarily accepted. That is, sometimes, the algorithm deliberately allows a search path at a stage to contain “worse” points. By allowing the paths to meander through better and worse points in the domain, the algorithm allows the domain to be searched for better optima that can only be reached by first passing through the worse regions. In this way, the search process can escape from local optima that are dominated by one or more global optima. This distinguishes SA from traditional hill-climbing algorithms that use the local properties of the function to move always in better directions. Movements in worse directions are governed by a Metropolis decision. In a sense, the Metropolis decision can be thought of as the algorithm giving permission to the search process to move in a worse direction, depending upon the roll of a (weighted) die. As the algorithm progresses through subsequent (“cooling”) stages, the chance that a worse point will be permitted/accepted decreases. In addition, as the algorithm progresses, the effective domain of the function (that part of the domain to which the search is effectively restricted) is contracted. In part, the search evolves in a fashion consistent with the overall (global versus local) topography of the surface of the function being optimized. Asymptotically, the algorithm converges to a domain that contains the best optimum encountered and which has a user-specified (small) volume. Convergence criteria and the number and length of searches at each stage are determined by parameters set by the user.
The FORTRAN implementation in Ref. [2] is faithful to the algorithm in Ref. [1], but it does include some features and suggestions that tend to help in deciding whether a global optimum has been reached and, in the initial stages of optimization, the features allow the researcher to search individual subsets of the domain of the function. The researcher can control the number of searches, the initial “temperature” of the model and the rate at which temperature decreases. Uphill and rejected downhill moves are balanced by changes to the upper and lower bounds on parameter changes. In Ref. [2], the suggestion is made that starting the SA algorithm from a variety of randomly selected domain points may provide information about whether a global optimum has been found. This is balanced, to some extent, by the realization that the properties of SA that make it a global optimizer also tend to make it independent of initial values of the choice variables. Any indication of sensitivity to starting values is, therefore, a strong suggestion that the SA user-determined parameters should be changed to provide a more thorough search.
As with any mathematical tool, becoming adept at using SA to solve optimization problems requires practice. FORTRAN code for the SA implementation in Ref. [2] is widely available. This code, written in FORTRAN 77, is carefully documented and contains an example problem that illustrates many of the features of SA. These features, such as convergence criteria, cooling rate, lengths of search paths, and the like can be adjusted to study how the implementation works. It is straightforward to code and solve new problems. The advice in Ref. [2] and in the code is helpful in finding the set of search parameters that solves the problem at hand.
Finally, Goffe et al. [2] address the issue of the speed of SA and recommend the ways that the user can tune the implementation to run more quickly. The implementation was published in 1994, and since then multicore very fast processors with at least 64-bit single precision have become the norm. At the same time, though, there are optimization problems that are now being addressed at the limit of current technology.
### 1.2. Penalty-constrained optimization
The foregoing discussion has provided an overview of SA and how it can be applied to optimize a function. We now turn to a discussion of how SA can be used to find the global optimum of functions when there are constraints on the choice variables. Our principal concern is how to find the optimum of the statistical functions when the choice variables must satisfy an integral equality constraint. Our approach, however, is quite general and it can be adapted to solve optimization problems subject to several inequality as well as equality constraints that may or may not involve integrals. At this point, it is helpful to introduce some notation.
Our choice variables are represented by the vector: θ. The objective function to be maximized is given by: l(θ). As well, there is a constraint given by: R(θ) = ψ0. In the statistical problem considered in the next section, we examine the unconstrained problem.
#### 1.2.1. (Unconstrained problem) U: choose θ to maximize l(θ)
(Unconstrained Problem) U: Choose θto maximize l(θ) is an important part of the analysis. It is almost always the case in statistical settings that the unconstrained problem can be solved in a straightforward manner using SA.
#### 1.2.2. (Constrained problem) C: choose θ to maximize l(θ) subject to R(θ) = ψ0
(Constrained Problem) C: Choose θto maximize l(θ) subject to R(θ) = ψ0 can also be solved using simulated annealing. Indeed, as we will see, SA is a very natural approach to solving problem C.
Within a statistical setting, substitution and Lagrange multiplier techniques tend not to work well when attempting to solve C. Typically, the objective function, l(θ), and the constraint, R(θ) = ψ0 will not have properties that guarantee a straightforward solution to the optimization problem. For example, the constraint can reasonably be expected to be a highly nonlinear function of the choice variables, θ. In both the substitution and Lagrange approaches, it is necessary at each iteration to solve the constraint equation to express one choice variable in terms of all of the others. If one or more of the choice variables takes on an extreme value (very large or very small) during the iteration process, then this can lead in turn to an extreme constraint solution and such extreme values tend to perpetuate themselves through subsequent iterations. The traditional absence of some form of textbook concavity or convexity on the objective function and/or the constraint function typically results in a failed optimization attempt. Standard derivative-based optimizers often get lost when derivatives take extreme values or when the Hessian of the function is indefinite. In part, this is due to their strong dependence upon local properties of the function being optimized.
The penalty function approach provides an alternative way of dealing with the constraint, which does not require exact satisfaction of the constraint equation at each “iteration.” In fact, the constraint equation only holds asymptotically. To clarify this, we begin by introducing the penalty function PL:
PLθψ0=lθkRθψ02,k>0.E1
This is the penalty function associated with the constrained optimization problem Cintroduced above. In this case, kis a positive parameter controlled by the researcher.
In a recent paper, Byrne [4] studied how the penalty functions like PLcould be used to solve constrained optimization problems such as C. The following three conditions are introduced:
1. θis chosen from a compact set.
2. The functions l(θ) and R(θ) are continuous.
3. Let θk*be the vector that corresponds to the global maximum of PLin Eq. (1) when the parameter multiplying the squared term is k. We assume that each element in the sequence θk*,k=1,2,exists.
Then, based on these conditions, it is proved in Ref. [4] that the sequence θk*,k=1,2,converges to the θ* that solves problem C.
This result is extremely important for applying simulated annealing to solve constrained optimization problems. First, SA chooses candidate optimizers from a compact set. Second, continuity of the penalty function PLis more than what is required for SA to reach a global optimum. The complicating point is that the result in Ref. [4] is expressed in terms of the limit of a sequence of SA optimizations. But, in our experience, this is not a complication of considerable practical importance. In the first place, even for large values of k, SA is not helped by starting the iterations for the (k + 1)st solution at the optimal values from the kth solution. In practice, given that SA searches (“cools”) sufficiently slowly and follows long enough paths in the domain, it tends to find the global solution of the (k + 1) problem regardless of the starting values it is given. Second, there is a practical issue of how much accuracy can be expected. The SA algorithm terminates when successive improvements in the value of the objective function are all less that a user-specified threshold. This means that the contribution of the term k[R(θ) − ψ0]2 must also be small in absolute value. In all of the problems we have considered, setting k = 100, 000 is certainly enough to get a high-quality estimate of θ*. That is, it is reasonable to truncate the sequence at this value of k.
In concluding this section, we reconsider the question: “Why does a penalty approach work when the Lagrange approach fails?” In our experience, the Lagrange approach, which requires solutions of an equation to a given level of accuracy, is prone to problems of numerical accuracy and their propagation. Alternatively, the penalty approach never requires the constraint to be exactly satisfied. Instead, it increasingly discourages large squared deviations of the constraint function R(θ) from its constraint value ψ0 as kincreases. As a final point, when conditions are satisfied for the Lagrange multiplier formally to exist, it can be obtained as the limit of the partial derivative of the penalty function with respect to the parameter ψ0 as kincreases.
## 2. Modeling reliability using SA penalized likelihood
### 2.1. Background on reliability
We consider two variables Xand Y. We refer to them respectively as Strength and Stress. For example, Strength could refer to the “time before failure” of a component such as a digital storage device. Alternatively, Stress might measure the total time that the device is used. From the standpoint of a manufacturer, Xand Yare both random variables with distributions that can, in principle, be estimated from available breakdown and usage data. Reliability, R, is defined as the probability that the component will withstand the stress it faces in use. In particular,
R=PY<X.E2
A variety of distributions have been used for Xand Yin the literature. The actual choice depends upon the process being studied. It is standard and reasonable to suppose that Xand Yare independent.
That is, the probability distribution of Ydoes not depend upon any realized value of Xand vice versa. If dependence is possible in a given setting, it is easily accommodated. Formally, (Eq. (2)) will continue to hold, but additional parameters, associated with the interdependence, may appear in both the objective function and the constraint. In a formal sense, the penalty approach is unchanged.
In this section, we suppose that both Xand Yare distributed as exponentiated exponential distributions. Exponentiated exponential distributions, EE(αβ), have two parameters: α > 0 controls shape and β > 0 controls scale. Adopting the notation introduced in Ref. [5], the cumulative distribution function is:
Fxαβ=1eβxα,α>0,β>0,x>0,E3
As in Ref. [6], we assume that Xis distributed as EE(α1β1) and Yis distributed as EE(α2β2). We do not, however, constrain the scale parameters, β1 and β2, to be equal. As a result, the expression for reliability is:
R=PY<X=0α1β11eβ1xα11eβ1x1eβ2xα2dx.E4
There is no known closed-form solution for this integral. As noted in Ref. [7], introducing the change of variables: z = β1xallows one to see that Ris homogeneous of degree 0 in (β1β2). The contours of Rare, therefore, all constant along a line in a parameter space defined by β2 = β1.
### 2.2. The unconstrained EE likelihood equation and it properties
Following Ref. [7], we let x = (x1, …, xn)′ and y = (y1, …, ym)′ denote the realizations of random samples from EE(α1β1) and EE(α2β2), respectively. The log-likelihood function of the above model can be written:
lα1β1α2β2xy=nlogα1+nlogβ1+α11i=1nlog1eβ1xiβ1i=1nxi+mlogα2+mlogβ2+α21j=1mlog1eβ2yjβ2j=1myj.E5
We denote the parameter vector as θ = (α1α2β1β2)′.
The Appendix in Ref. [7] contains a derivation of the properties of l(θ) = l(α1β1α2β2xy). In particular, l(θ) is not a concave function of θnor is it quasi- or pseudo-concave. There is a small region around the point where the gradient of l(θ) vanishes and in that region, the Hessian matrix is negative definite. Elsewhere in the parameter space, the determinant of the Hessian matrix changes sign frequently. Thus, extreme care must be taken in trying to maximize l(θ), using a derivative-based algorithm. We found that a variable-metric algorithm would work as long as an approximate Hessian matrix, constrained to be negative definite, is used over a restricted parameter space.
One example, which we consider in greater detail later in the paper, uses the following data with sample sizes of 11 and 9: x= (2.1828, 0.5911, 1.0711, 0.9007, 1.7814, 1.3616, 0.8629, 0.2301, 1.5183, 0.8481, 1.0845) and y= (0.8874, 1.1482, 0.8227, 0.4086, 0.5596, 1.1978, 1.1324, 0.5625, 1.0679). Our SA program quickly and easily solved the associated unconstrained maximum likelihood optimization problem.
### 2.3 Constrained likelihood maximization
As will be discussed in the next section, inference for the “parameter” R(θ) in our reliability model requires that the likelihood function l(θ) be maximized subject to the constraint R(θ) = ψ0 for a range of values of the constraint parameter ψ0. These constrained optimization problems are all solved using the penalty function approach introduced in Section 1.2 and using the penalty function PL(θψ0) given in Eq. (1). The functions l(θ) and R(θ) can be thought of now as the unconstrained EElikelihood function and the reliability function, respectively.
## 3. Likelihood-based inference and penalty functions
In Eq. (5), we introduced the statistical log-likelihood function primarily as an example of a function that needs to be maximized (with and without constraint) in a statistical setting. In this section, we provide more details about likelihood functions and inference. Our brief discussion is not intended as a complete explanation of the underlying statistical notions. Rather, it is intended only to motivate some importance of constrained and unconstrained optimization within statistics. Our discussion is rooted in the example of Eq. (5).
### 3.1. Background on likelihood models in statistics
Likelihood is akin to probability. The difference in the notions for our purposes is that likelihood is measured in terms of the probability density governing the realizations of a continuous random variable. Technically, the probability of any one outcome, say x0, of a continuous random variable is 0. The value of the density, say h(x), evaluated at x0 and multiplied by dx, that is, h(x0) dx, can be thought of as approximately the probability that there will be a realization of the random variable Xin a very small interval containing x0. It is common to have situations where the realized (observed) values of a random variable Xarise from a process of random sampling where the outcomes are independent of each other yet are governed by identical probability density functions, h(x). The likelihood of a given sample of realized values is defined as the product of the densities corresponding to each of the outcomes in the sample; so the likelihood of a given sample is, up to a scaling factor, a notion similar to the probability of the sample. The likelihood of a sample of realizations of Xand Yis, given our assumptions, the product of the likelihoods of the Xand the Ysamples. For a variety of reasons, it is often easier to work with a positive monotonic transformation of the sample likelihood. In particular, we work with the log-likelihood of the sample. In Eq. (5), we are given the log-likelihood of a sample where the densities come from possibly different exponentiated exponential distributions.
The derivation of the log-likelihood associated with a sample of realizations is just the beginning of the modeling process. Extensions include forecasting the next realization of a random variable or perhaps finding an interval where one can be 95% confident that the next realization of a function of the random variables will fall. For example, we could ask for a 95% confidence interval of the measure of reliability in Eq. (4), incorporating the information in the sample given at the end of Section 2.2. These are questions of statistical inference. We answer these questions by solving the optimization problems.
The likelihood function given in Eq. (5) can be combined with the sample of 11 realizations of Xand 9 realizations of Ygiven in Section 2.2. We can use the information in the data to estimate the unknown vector of parameters: θ = (α1α2β1β2). One set of estimates of the parameters of the model is obtained by maximizing the likelihood function with the choice variables being the parameters. These maximum likelihood parameter estimates can be thought of as the parameter values that yield specific density functions that are most likely to have generated the data. Of course, 20 observations are not enough to achieve certainty, so there is a related theory about where the true (population) parameters lie in relation to their estimates. Indeed, there are probability distributions associated with the maximum likelihood parameter estimation process, and the parameter values that maximize the log-likelihood for a given sample of data realizations are themselves just realizations. These probability distributions or their approximations allow us to estimate how close the parameter realizations are to the true parameter values.
There is also a probability distribution for the maximized value of the log-likelihood function. This allows us to ask questions such as the following: do I induce a “significant” change in the maximized likelihood value when I constrain the parameter estimates (choice variables) to satisfy an additional condition or set of conditions. This leads back to the constrained optimization problem Cin Section 1.2, and the penalty function in Eq. (1).
In the subsection that follows, we present the process of inference for our reliability model. The presentation is more technical.
### 3.2. Inference in the reliability model
Given l(θ) is the log likelihood function, we denote the unconstrained maximum likelihood estimator θ^when l(θ) alone is maximized. As well, we define
jθ^=2lθθθθ^E6
as the observed information matrix evaluated at θ^. Finally, we let θ^ψbe the constrained maximum likelihood estimator of θgiven by maximizing l(θ) subject to R(θ) = ψ. Formally, θ^ψcan be obtained for any ψin the range of R(θ) by maximizing l(θ) subject to the constraint R(θ) = ψusing the penalty function approach within SA.
The aim next is to obtain inference concerning R = R(θ), where dim(R) = 1. Two widely used likelihood-based methods for obtaining confidence interval for Rare based on the asymptotic distribution of the maximum likelihood estimator θ^and the (log) likelihood ratio statistic.
Taking θas the true population parameter vector, θ^θvarθ^1θ^θis asymptotically distributed as chi-square with degrees of freedom equal to dim(θ), and variance-covariance matrix var^θ^j1θ^. Since R = R(θ) depends upon the entire vector of parameters, we approximate its variance by applying the Delta method to R^=Rθ^and obtain:
var^R^Rθθ^var^θ^Rθθ^E7
where
Rθθ^=Rθθθ^.E8
Since dimR^=1, we have
R^Rvar^R^E9
asymptotically distributed as standard normal. An approximate (1 − α)100 % confidence interval for Rbased on θ^is
R^zα/2var^R^,R^+zα/2var^R^E10
where zα/2 is the (1 − α/2)100th percentile of the standard normal distribution. This is our first confidence interval.
Alternatively, with regularity conditions stated in Refs. [8, 9], the log likelihood ratio statistic:
Wψ=2θ^θ^ψE11
is asymptotically distributed as chi square with 1 degree of freedom. Therefore, an approximate (1 − α)100 % confidence interval of Rbased on the likelihood ratio statistic is:
ψ:Wψχ1,α2.E12
The set of all constrained values ψin the domain of Rthat cannot be rejected at the (1 − α)100 % as the true value R(θ) is defined in Eq. (10). These values form our second confidence interval.
It should be noted that both methods have rates of convergence O(n− 1/2). While the MLE-based interval is often preferred because of simplicity in calculation, the log-likelihood ratio method has the advantage that it is invariant to reparameterization and the MLE-based method is not. The results presented in Ref. [10] suggest that, in terms of coverage, the confidence interval based on the log-likelihood ratio statistic should be preferred to the MLE-based interval. In particular, when 95% confidence intervals for both statistics are compared, the interval from the log-likelihood ratio statistic is shorter and therefore more precise.
The results are summarized in Table 1. We note that, consistent with Ref. [10], the χ2 interval is indeed shorter than the MLE interval.
95% Confidence interval
MLE(0.306, 0.934)
χ2(0.378, 0.823)
### Table 1.
Interval estimates of R.
## 4. Conclusion
This chapter has considered how SA can play an important role as a global optimizer of constrained likelihood-based statistical models. SA is naturally paired with the penalty function approach to constrained optimization. SA and the penalty approach both require compact domains and bounded functions. Penalty functions must be continuous and, within a statistical setting, this almost always holds. SA supplies the global optimization property that guarantees that the penalty function approach converges to the global constrained optimum. Even though our implementation of SA does not make use of derivatives, the converged penalty function will often be differentiable and Lagrange multipliers, gradients, and Hessian matrices can be calculated. The extension of these results to multiple constraints is computationally straightforward.
In this chapter, we have motivated the pairing of SA and penalty functions in a statistical setting. But the approach can be used to solve many other types of numerical constrained optimization problems. Over time, we have accumulated a considerable amount of experience solving constrained problems using SA and the penalty functions. One lesson stands out: SA is a global optimizer and, for the most part, it should be independent of initial conditions such as starting values of parameters (choice variables). If you find that you get a different optimum after changing the starting values, then it is likely that neither solution is the true global optimum. You can usually remedy this by increasing the initial temperature, slowing the rate of cooling, and/or increasing the length and number of search paths.
## References
1. 1. Corana, A., Marchese, M., Martini, C. and Ridella, C. Minimizing multimodal functions of continuous variables with the simulated annealing algorithm. ACM Transactions on Mathematical Software. 1987; 13: 262–280.
2. 2. Goffe, W.L., Ferrier, G.D. and Rogers, J. Global optimization of statistical functions with simulated annealing. Journal of Econometrics. 1994; 60: 65–99.
3. 3. Li, X., and Smith, B. Diagnostic analysis and computational strategies for estimating discrete time duration models—a Monte Carlo study. Journal of Econometrics. 2015; 187(1): 275–292.
4. 4. Byrne, C.L. Sequential unconstrained minimization: a survey. [Internet]. 2013. Available from: http://faculty.uml.edu/cbyrne/SUM.pdf.
5. 5. Gupta, R.D. and Kundu, D. Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biometrical Journal. 2001; 43(1): 117–130.
6. 6. Kundu, D. and Gupta, R.D. Estimation of P[X < Y] for generalized exponential distributions. Metrika. 2005; 61: 291–308.
7. 7. Smith, B., Wang, S., Wong, A. and Zhou, X. A penalized likelihood approach to parameter estimation with integral reliability constraints. Entropy. 2015; 17: 4040–4063. DOI:10.3390/e17064040.
8. 8. Severeni, T. Likelihood Methods in Statistics. New York: Oxford University Press; 2000.
9. 9. Cox, D.R. and Hinkley, D.V. Thoeretical Statistics. London: Chapman and Hall; 1974.
10. 10. Doganaksoy, N. and Schmee, J. Comparisons of approximate confidence intervals for distributions used in life-data analysis. Technometrics. 1993; 35: 175–184.
## Notes
• That is, the probability distribution of Y does not depend upon any realized value of X and vice versa. If dependence is possible in a given setting, it is easily accommodated. Formally, (Eq. (2)) will continue to hold, but additional parameters, associated with the interdependence, may appear in both the objective function and the constraint. In a formal sense, the penalty approach is unchanged.
Written By
Barry Smith and Augustine Wong
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http://math.stackexchange.com/questions/52471/every-3d-rotation-must-have-an-axis | # Every 3D rotation must have an axis?
For a 1D object 1D space we can translate $x\mapsto x+a$ but cannot define a rotation as $x\mapsto ax$ would not leave the distance invariant between two points and hence the onyl rigid tranformation is the translation. In 2D and 3D we can choose an axis and define a rotation using the rotation matrix. However, it seems that algebraically a rigid motion cannot be defined (as the sin and cos occur in pairs) unless we have one coordinate which remains unchanged $z=z'$ in order for $x^2+y^2+z^2 = x'^2+y'^2+z'^2$. I cannot prove this.
Prove that there exists or does not exist a rotation such that $x\neq x'$ , $y\neq y'$ and $z\neq z'$. Or, I think equivalently, prove or disprove that any rotation in 2D or 3D must have an axis.
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What's an "axis" in 2d? Just a point? Usually a rotation is defined as a distance-preserving map that fixes a point (and fixes orientation), so that would seem to immediately give an answer for 2d. – MartianInvader Jul 19 '11 at 19:28
@MartianInvader a rotation in 2D has an axis in 3D space. A rotation in 3D has an axis in 3D space. Is it possible to have a rotation which does not have an axis in 3D space, but maybe in 4D? – kuch nahi Jul 19 '11 at 19:33
@KuchNahi I think I see what you mean. In odd dimensions, rotations must always have a fixed direction, whereas in even dimensions this is not true (with the 2-dimensional case being an obvious example). – Vhailor Jul 19 '11 at 19:36
The question is rather vague, and the title borders the absurd: a rotation 'is' about a point. – leonbloy Jul 19 '11 at 19:40
@leonbloy could you suggest a better title. – kuch nahi Jul 19 '11 at 19:50
It is true that any three-dimensional rotation must have a fixed line -- this is sometimes called Euler's Principal Axis Theorem.
[Note that my answer presumes that we are talking about linear maps. It is a classical theorem -- which I believe has been discussed on this site before -- that an isometry of $\mathbb{R}^n$ is linear iff it fixes the origin. If you have an isometry with any fixed points, then -- by translating your coordinate system -- you may assume that the isometry fixes the origin.]
I will show that this holds for any matrix $A \in \operatorname{SO}_n(\mathbb{R})$ when $n$ is odd: that is, $A$ is an orthogonal matrix of determinant $1$.
Step 1: In particular, $A$ viewed as a matrix over $\mathbb{C}$ is unitary, i.e., it preserves the standard Hermitian inner product $\langle, \rangle$ on $\mathbb{C}^n$. In particular we have for all $v \in \mathbb{C}^n$, $|Av|^2 = \langle Av, Av \rangle = \langle v, v \rangle = |v|^2$, so $|Av| = |v|$. In particular if $v$ is a nonzero eigenvector for $A$ -- so that $A v = \lambda$ -- we get $|v| = |\lambda v| = |\lambda| |v|$ and thus $|\lambda| = 1$. That is, every eigenvalue of $\lambda$ has complex absolute value $1$.
Step 2: The eigenvalues of $A$ are the roots of a polynomial with real coefficients -- the characteristic polynomial -- hence the complex roots come in conjugate pairs $\lambda, \overline{\lambda}$ and thus $\lambda \cdot \overline{\lambda} = |\lambda|^2 = 1$. Therefore the product of all of the complex eigenvalues is equal to $1$. Since the determinant of $A$ is $1$, the product of all the eigenvalues is equal to $1$, and since $n$ is odd and the number of complex eigenvalues is even, the number of real eigenvalues is odd and in particular positive. Moreover each real eigenvalue is either $\pm 1$ and the product of an odd number of them is equal to $1$, so $1$ occurs as an eigenvalue of $A$. That is, the $1$-eigenspace of $A$ is nonzero, so there is (at least) a one-dimensional subspace that $A$ leaves pointwise fixed.
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This is surprisingly clear answer for a question I could not express clearly. Thank you. – kuch nahi Jul 19 '11 at 20:01
This effectively shows that any rotation in odd dimensions have an axis - and if we think the rotation as applied to the points of a (hyper)sphere surface, it shows that any rotation in 3 (5,7...) dimensions (or any composition of rotations, that is also a rotation) has at least one fixed point. Which it's obviously not true in 2D. – leonbloy Jul 19 '11 at 20:08
@Pete: Yes, we totally agree, I was not objecting anything. Just pointing out the equivalence (perhaps too obvious, but as the OP was not very clear...) of saying that a rotation "has an axis" (thinking in the full 3D space, for a 3D rotation) or that it has "a fixed point" (thinking in the sphere surface). – leonbloy Jul 19 '11 at 20:23
In 3 dimensions, the composition of rotations is always a rotation about some axis. This is due to a theorem of Euler.
One of your statements is not true, at least not in the way you formulated it. You can easily construct a rotation in 3D that doesn't fix any of the three coordinate axes x,y and z, simply compose a rotation about one of the axes with another one about another axis and you will get another rotation (by Euler's Theorem that I referenced above) which is not about a coordinate axis most of the time.
-
A simple explicit example is the map $(x,y,z) \mapsto (z,x,y)$, which is a rotation by $2\pi/3$ about the axis $(1,1,1)$ and does not fix any of the three coordinate axes. – Rahul Jul 19 '11 at 19:41
@Rahul But it does fix one axis (just not the original ones). The question was if it must always fix an axis. – kuch nahi Jul 19 '11 at 20:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9500300884246826, "perplexity": 211.1095661660216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009751.93/warc/CC-MAIN-20141125155649-00233-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.diagramlink.com/probability-tree-diagrams/ | # Probability Tree Diagrams
Printable examples of Probability Tree Diagrams are available for you to understand better about the probability theory. Probability theory can be studied using tree diagrams, and the following images show you some examples of the probability theory applied using tree diagrams and vice versa. Discover the diagrams in the following examples below. Click to view the full-size.
Probability tree diagram is used in strategic decision making, valuation or probability calculations. A probability tree has two main parts: the branches and the ends. Each branch is generally written on the branches, while the outcome is written on the ends of the branches. Study the following Probability Tree Diagrams and try to understand each of the given probability.
Probability trees are useful for calculating combined probabilities. The probability is a measure of the possibility that any event will occur or not. This probability can be expressed using tree diagrams. Tree diagram shows the outcome of any event. This is used for giving all possible possibilities which is also called the sample space. And, each leaf in the tree represents an outcome of any events.
The diagrams that we provided above are probability tree diagram. All of the diagrams are ready to be downloaded and printed in good quality. We hope these diagram examples and brief explanation help you in studying and understanding tree diagrams! Find more interesting and educative diagrams in our site by browsing up our category! | {"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": 0.8563969731330872, "perplexity": 454.8993088602826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575751.84/warc/CC-MAIN-20190922221623-20190923003623-00318.warc.gz"} |
https://economics.stackexchange.com/questions/18063/crra-utility-function-with-a-scale-parameter/18073 | # CRRA utility function with a scale parameter
I am wondering if it is possible to write down a following CRRA utility function ;
$$u\left(c(t)\right)=a\frac{c\left(t\right)^{1-\sigma}}{1-\sigma}$$ where $a>0$ is a constant scale parameter. I need this $a$ to have some numerical results but I am not sure if $a$ can be justified. In a growh model, I am trying to show the existence of Hopf bifurcation and limit cycles. Then, with a scale parameter like that, I can show that it exists.
Normally, it does not change the usual assumptions on a CRRA utility function, (an increasing concave function)
Is there any way to justify it (are there some examples of this kind ?) or are there any types of utility function with constant scale parameters ?
• "Numerical results" at what level? Please elaborate. For example, in the usual intertemporal model, growth rates are not affected since the constant cancels out. – Alecos Papadopoulos Aug 29 '17 at 20:01
• @AlecosPapadopoulos In fact, I am trying to show that for some parameter set, there is a Hopf bifurcation and limit cycles. I edit the question. You are right that growh model does not change but steady state levels change. – optimal control Aug 29 '17 at 20:32
• I trying to understand : if you have a model with a steady state in growth rates (a "balanced growth path") then you cannot have a periodic solution since it would imply a change in the growth rate (and alpha does not affect the growth rate). If instead you have a model with a steady state in levels, certain alpha values transform the long-run constant level-value to a periodic solution? – Alecos Papadopoulos Aug 30 '17 at 0:06
• This is exactly what you said. It is not a model with balanced growth path. I have a model with a steady state in levels and certain values are likely to make periodic solution and with the scale parameter bifurcation occurs. – optimal control Aug 30 '17 at 6:00
So any linear transformation of $u(c)$ solved the differential equation.
In section 3 the author derives a linear transformation that allows for a balanced growth path under bounded consumption. In particular, the author computes an expression for $a$ above. This is not necessarily what you are interested on, but gives an idea of the values of $a$ given assumptions on $\sigma$, which might give you an idea about how to justify the value of $a$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9491996765136719, "perplexity": 320.1549987080477}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487629209.28/warc/CC-MAIN-20210617041347-20210617071347-00266.warc.gz"} |
https://tex.stackexchange.com/questions/329611/tikz-override-fill-of-shape-based-on-one-pgfkey-with-another-in-command | # TikZ: Override fill of shape based on one pgfkey with another in command
I'm creating a package that generates a tikz picture based on a command with pgf key/value parameters. I want to have two parameters, one that specifies a base shape with a particular fill colour, and another that turns that fill "on" when false (to the specified colour), and "off" (to white) when true. I've created a simplified example below:
\documentclass[border=5pt]{standalone}
\usepackage{xifthen}
\usepackage{tikz}
\newboolean{shapecolour}
\pgfkeys{/tikz/.cd,
circle/.style={fill=red},
square/.style={fill=green},
triangle/.style={fill=yellow},
blank/.style={fill=white},
colourshape/.store in=\colourshape,
colourshape=circle,
nocolour/.store in=\nocolour,
nocolour=false,
circle/.pic={\draw [fill] circle(0.5); },
square/.pic={\draw [fill] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;},
triangle/.pic={\draw [fill] (0,0) -- (1,0) -- (0.5, 1) -- cycle;},
}
\begin{document}
\newcommand{\TestCommand}[1][]{
\tikzset{colourshape=circle, nocolour=false, #1}
\setboolean{shapecolour}{\nocolour}
\pic[\ifthenelse{\boolean{shapecolour}}{blank}{\colourshape}]{\colourshape};
}
\begin{tikzpicture}
\TestCommand[colourshape=triangle, nocolour=true]
\end{tikzpicture}
\end{document}
This is the desired outcome of my code, based on the pgfkeys arguments put into TestCommand:
So the code above should produce a triangle with a white fill. Instead I get the error ! Argument of \boolean has an extra }. I know I can achieve the same result in a more verbose way by specifying a separate pic for each variant of the shape, but I would like to have a more compact solution, especially as there will be more style options to follow.
• Ouch, you are overriding a lot of default TikZ keys. Sep 14 '16 at 19:15
• @percusse I know, the real thing doesn't do this. It's a quick and dirty example. It works if I type in the style parameters manually rather than using the conditional, so it's OK. Sep 14 '16 at 19:21
It is hard to be certain without more information about the overall project. However, from your example, I would not use a conditional here at all. If you do use a conditional, I would look into PGF's .is if handler, which will make things much easier.
First some keys:
\tikzset{%
We can afford to use verbose names here because the user interface won't require them at all.
Crazymoomin circle/.pic={\draw [fill=Crazymoomin@fill] (.5,.5) circle (0.5);},
Crazymoomin square/.pic={\draw [fill=Crazymoomin@fill] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;},
Crazymoomin triangle/.pic={\draw [fill=Crazymoomin@fill] (0,0) -- (1,0) -- (0.5, 1) -- cycle;},
So that we can use simple key names in the user interface, we'll put them on a custom path, but we'll make sure that standard TikZ keys work here, too.
/Crazymoomin/.search also={/tikz},
Switch paths.
/Crazymoomin/.cd,
Now we can use simple names without overwriting the defaults.
fill/.code={%
This will hold a custom colour.
\colorlet{Crazymoomin@fill}{#1}%
},
And a shape.
shape/.store in=\Crazymoomin@shape,
To handle the colour/shape combo neatly, let's make colour shape a choice key.
colour shape/.is choice,
Now for the options, which each switch to our path and set the pic shape and fill.
colour shape/triangle/.style={/Crazymoomin/.cd, shape=triangle, fill=yellow},
colour shape/circle/.style={/Crazymoomin/.cd, shape=circle, fill=red},
colour shape/square/.style={/Crazymoomin/.cd, shape=square, fill=green},
no colour can just set the fill to white.
no colour/.style={/Crazymoomin/fill=white},
Make sure we have a default for everything.
fill=gray,
shape=circle,
}
Now the command.
\newcommand{\TestCommand}[1][]{%
\tikzset{%
Switch to our path.
/Crazymoomin/.cd,
Default setting. This sets the filling colour anyway, so no colour is effectively false by default.
colour shape=circle,
User options.
#1,
}%
And the pic.
\pic {Crazymoomin \Crazymoomin@shape};
}
Then we can write, for example,
\begin{tikzpicture}
\TestCommand[colour shape=triangle, no colour]
\scoped[xshift=12.5mm]{\TestCommand[colour shape=circle, no colour]}
\scoped[xshift=25mm]{\TestCommand[colour shape=square, no colour]}
\scoped[yshift=12.5mm]{\TestCommand[colour shape=triangle]}
\scoped[xshift=12.5mm, yshift=12.5mm]{\TestCommand[colour shape=circle]}
\scoped[xshift=25mm, yshift=12.5mm]{\TestCommand[colour shape=square]}
\end{tikzpicture}
to produce
Obviously this is awkward because I didn't want to mess around too much with \TextCommand which doesn't have any positioning information (eek!?).
Complete code:
\documentclass[border=10pt,multi,tikz]{standalone}
\makeatletter
\tikzset{%
Crazymoomin circle/.pic={\draw [fill=Crazymoomin@fill] (.5,.5) circle (0.5);},
Crazymoomin square/.pic={\draw [fill=Crazymoomin@fill] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;},
Crazymoomin triangle/.pic={\draw [fill=Crazymoomin@fill] (0,0) -- (1,0) -- (0.5, 1) -- cycle;},
/Crazymoomin/.search also={/tikz},
/Crazymoomin/.cd,
fill/.code={%
\colorlet{Crazymoomin@fill}{#1}%
},
shape/.store in=\Crazymoomin@shape,
colour shape/.is choice,
colour shape/triangle/.style={/Crazymoomin/.cd, shape=triangle, fill=yellow},
colour shape/circle/.style={/Crazymoomin/.cd, shape=circle, fill=red},
colour shape/square/.style={/Crazymoomin/.cd, shape=square, fill=green},
no colour/.style={/Crazymoomin/fill=white},
fill=gray,
shape=circle,
}
\newcommand{\TestCommand}[1][]{%
\tikzset{%
/Crazymoomin/.cd,
colour shape=circle,
#1,
}%
\pic {Crazymoomin \Crazymoomin@shape};
}
\makeatother
\begin{document}
\begin{tikzpicture}
\TestCommand[colour shape=triangle, no colour]
\scoped[xshift=12.5mm]{\TestCommand[colour shape=circle, no colour]}
\scoped[xshift=25mm]{\TestCommand[colour shape=square, no colour]}
\scoped[yshift=12.5mm]{\TestCommand[colour shape=triangle]}
\scoped[xshift=12.5mm, yshift=12.5mm]{\TestCommand[colour shape=circle]}
\scoped[xshift=25mm, yshift=12.5mm]{\TestCommand[colour shape=square]}
\end{tikzpicture}
\end{document}
• Thankyou for the answer, very helpful. I do change the outline pattern and colour using keys as well, but I have managed to get this to work because I'm changing separate style attributes. However, your method looks a lot more attractive so i'll see if I can do something similar for those. I wasn't planning on adding positioning information as they will probably either be drawn on their own or as part of a larger Tikz picture where I would expect positioning would be controlled out of scope of the command. I haven't decided yet if I'm going to include it. Sep 15 '16 at 8:16 | {"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": 0.8013669848442078, "perplexity": 3265.310985425343}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587593.0/warc/CC-MAIN-20211024173743-20211024203743-00521.warc.gz"} |
http://atozmath.com/Default.aspx?q1=compare%20fraction%203%2F4%2C5%2F6%60118&do=1 | Home
Solve any problem (step by step solutions) Input table (Matrix, Statistics)
Mode :
SolutionHelp
Solution Find compare fraction 3/4,5/6Solution:Your problem -> compare fraction 3/4,5/6Step-1 : Find the LCD of denominatorsHere, LCD of 4, 6 = 12Step-2 : Convert each fraction into its equivalent with the LCD in the denominatorFor 3/4, multiply numerator and denominator by 3 to have LCD = 12 in the denominator.3/4 = 3/4 xx 3/3 = 9/12For 5/6, multiply numerator and denominator by 2 to have LCD = 12 in the denominator.5/6 = 5/6 xx 2/2 = 10/12Step-3 : Arrange fractions: If denominators are the same, then we can arrange the numerators.Here 9 < 10 :. 9/12 < 10/12 So, we conclude 3/4< 5/6 Solution provided by AtoZmath.com Any wrong solution, solution improvement, feedback then Submit Here Want to know about AtoZmath.com and me | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8599043488502502, "perplexity": 4255.575592818566}, "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-2019-18/segments/1555578584186.40/warc/CC-MAIN-20190423015050-20190423041050-00267.warc.gz"} |
http://mathhelpforum.com/calculus/231141-prove-each-statement-mathematical-induction.html | # Thread: prove each statement by mathematical induction
1. ## prove each statement by mathematical induction
1. $5+10+15+...+5n=\frac{5n(n+1)}{2}$
2. $\frac{4}{5}+\frac{4}{5^2}+\frac{4}{5^3}+...+\frac {4}{5^n}=1-\frac{1}{5^n}$
3. $2^n>n^2$, for $n \ge 5$, $n \varepsilon \mathbb{N}$
4. $5.6+5.6^2+2.6^3+....+5.6^n=6(6^n-1)$
i would appreciate any help so thanks in advance!
2. ## Re: prove each statement by mathematical induction
Do you know how proof by induction works? I.e., the logic behind it and its process?
3. ## Re: prove each statement by mathematical induction
i understand the steps $S_n, S_1, S_k$ and that you assume that $S_k$ is true so that you may prove $S_{k+1}$. I just have trouble proving $S_{k+1}$
4. ## Re: prove each statement by mathematical induction
Hello, tinspire!
$\displaystyle 2.\;\;\frac{4}{5}+\frac{4}{5^2}+\frac{4}{5^3}+ \text{ . . . }+\frac{4}{5^n}\:=\:1-\frac{1}{5^n}$
Verify $S_1\!:\;\;\dfrac{4}{5} \:=\:1- \dfrac{1}{5} \quad\Rightarrow\quad \dfrac{4}{5} \:=\:\dfrac{4}{5}$ . True!
Assume $S_k\!:\;\;\dfrac{4}{5} + \dfrac{4}{5^2} + \dfrac{4}{5^3} + \text{ . . . }+\dfrac{4}{5^k} \;=\;1 - \dfrac{1}{5^k}$
Add $\dfrac{4}{5^{k+1}}$ to both sides:
$\qquad \dfrac{4}{5} + \dfrac{4}{5^2} + \dfrac{4}{5^3} + \text{ . . . }+ \dfrac{4}{5^k} + \dfrac{4}{5^{k+1}} \;=\;1 - \dfrac{1}{5^k} + \dfrac{4}{5^{k+1}}\;\;[1]$
The right side is: $\:1 - \dfrac{1}{5^k} + \dfrac{4}{5^{k+1}} \;=\;1 - \dfrac{5}{5^{k+1}} + \dfrac{4}{5^{k+1}} \;=\; 1 - \dfrac{1}{5^{k+1}}$
Then [1] becomes: $\:\dfrac{4}{5} + \dfrac{4}{5^2} + \dfrac{4}{5^3} +\text{ . . . }+ \dfrac{4}{5^{k+1}} \;=\;1 - \dfrac{1}{5^{k+1}}$
This is $S_{k+1}.$
The inductive proof is complete.
5. ## Re: prove each statement by mathematical induction
Originally Posted by tinspire
1. $5+10+15+...+5n=\frac{5n(n+1)}{2}$
2. $\frac{4}{5}+\frac{4}{5^2}+\frac{4}{5^3}+...+\frac {4}{5^n}=1-\frac{1}{5^n}$
3. $2^n>n^2$, for $n \ge 5$, $n \varepsilon \mathbb{N}$
4. $5.6+5.6^2+2.6^3+....+5.6^n=6(6^n-1)$
i would appreciate any help so thanks in advance!
Originally Posted by tinspire
i understand the steps $S_n, S_1, S_k$ and that you assume that $S_k$ is true so that you may prove $S_{k+1}$. I just have trouble proving $S_{k+1}$
well let's look at the first one
let n=1
$P(1)=[5(1)=\dfrac {5(1)(1+1)}{2}]$
$5\overset{?}{=}5$
Yes. $5=5$ So $P(1)=True$
Now assume $P(n)=True$ and consider $P(n+1)$
$\displaystyle{\sum_{k=1}^{n+1}}5k=$
$5(n+1)+\displaystyle{\sum_{k=1}^{n}}5k=$
$5(n+1)+\dfrac {5n(n+1)}{2}=$
$\dfrac{2\cdot 5(n+1)+5n(n+1)}{2}=\dfrac{5(n+1)(n+2)}{2}=\dfrac{5 (n+1)((n+1)+1)}{2}$
and thus given the statement is true for $n$ it is true for $n+1$
You can work the 3 others. The key is to try and get the $P(n+1)$ expression to include $P(n)$ so you can substitute in the known simplification for $P(n)$.
6. ## Re: prove each statement by mathematical induction
3. 2^n > n^2, for n≥5, nεN
Base case:
n=5, 2^5=32 and 5^2= 25. Hence, lhs > rhs (Base case is true.)
Now, for any k > 5, assume that 2^k > k^2 is true.
Then, we shall prove that 2^(k+1) > (k+1)^2 is also true.
(But first, I am going to digress for a bit.)
2k^2 - (k+1)^2 = 2k^2 - k^2 - 2k -1 = k^2-2k - 1 = (k-1)^2 -2: since k > 5, it's clear that (k-1)^2 - 2 > 0.
My point? for any k > 5, 2k^2 - (k+1)^2 > 0, or better put, 2k^2 > (k+1)^2 starting from 5.
Then, 2^(k+1) = 2*2^k > 2* k^2 = 2k^2 > (k+1)^2 as needed.
7. ## Re: prove each statement by mathematical induction
And for 4, is it supposed to be 5(6+6^2+6^3...+6^n) on the left hand side? [I factored out the 5.]
8. ## Re: prove each statement by mathematical induction
Thanks for everyone's response, I really appreciate all the help and yes I made a mistake. That is supposed to a multipcation sign not a decimal.
9. ## Re: prove each statement by mathematical induction
For the third one don't I have to plug in five for n
10. ## Re: prove each statement by mathematical induction
Oh I didn't realize that's what you did. Never mind
11. ## Re: prove each statement by mathematical induction
In the first step of number 3 for proving S_{k+1} where did 2k^2 come from?
12. ## Re: prove each statement by mathematical induction
What can I do for the fourth one?
13. ## Re: prove each statement by mathematical induction
So that is $5\cdot 6^3$ and not $2\cdot 6^3$ as you have?
Then it is $5(6+ 6^2+ 6^3+ \cdot\cdot\cdot+ 6^n)= 6(6^n- 1)$.
I presume you see that when n= 1 that is 5(6)= 30 and 6(6- 1)= 6(5)= 30
Now if it is true that for n= k, $5(6+ 6^2+ 6^3+ \cdot\cdot\cdot+ 6^k)= 6(6^k- 1)$
then $5(6+ 6^2+ 6^3+ \cdot\cdot\cdot+ 6^k+ 6^{k+1})= 6(6^k- 1)+ 5(6^{k+ 1})= 6^{k+ 1}- 6+ 5(6^{k+1})$
14. ## Re: prove each statement by mathematical induction
yeah thats a five not a two
15. ## Re: prove each statement by mathematical induction
$6\cdot 6^k-6+5(6^k \cdot 6)$
what can i do now?
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https://www.physicsforums.com/threads/what-does-the-t-s-diagram-look-like-for-an-ammonia-heat-pump.914274/ | # What does the T-S diagram look like for an ammonia heat pump?
1. May 10, 2017
### swampwiz
I'm a bit confused as to what the Temperature-Entropy diagram looks like for an ammonia heat pump. I understand perfectly how a freon type of refrigerant heat pump works: starting with working fluid at a state of a low-Temperature & high-Entropy gas (which is at low pressure), do work on the working fluid by compressing it, thereby resulting in an increase in Pressure & Temperature (and ideally no change in Entropy, although in "real life", there is always some small increase) such that the Temperature is higher than the high-Temperature ambient environment; then allow that working fluid to lose heat to that ambient environment, thereby resulting in a loss of Entropy such that the working fluid loses pressure until it hits the gas-liquid saturation state, and then continuing on by condensing the working fluid into a liquid; then allow the working fluid to depressurize, typically via a throttling device that achieves depressurization back to (or nearly back to) the initial pressure by friction (i.e., of the working fluid itself, or the friction with the containing vessel), which achieves a drop in Temperature (albeit that the friction actually causes a small rise in Entropy, but nowhere near the initial state) that is lower than the low-Temperature ambient environment; then allow the working fluid to gain heat from that ambient environment, thereby increasing the entropy to get back the initial state. The net effect is that the state of the working fluid moves in a counter-clockwise path such that there are 2 functional paths in which one of the paths has a lower Entropy than the other, and with the heat that is lost to the high-Temperature ambient environment is the area under the high-Temperature path, the heat that is gained from the low-Temperature ambient environment is the area under the low-Temperature path, and thus the difference of the two is the amount of work that must be done in the compression, and which also is the area bounded by the 2 paths.
What I don't understand is how to achieve this divergence of these 2 paths in the ammonia system that doesn't seem to use any pressurization. Certainly, the only way that an ammonia system could work is to somehow have a pair of state paths with some difference in between, but how is this done without pressurization?
2. May 15, 2017
### PF_Help_Bot
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http://ywukakyzin.ml/legend/surveys-on-solution-methods-for-inverse-problems.php | Tackling power potency at huge Scale explores seminal study in large-scale eco-friendly computing. It starts off with low-level, hardware-based techniques after which traverses up the software program stack with more and more higher-level, software-based methods. This e-book covers finite aspect equipment for numerous ordinary eigenvalues that come up from technology and engineering. The historical past for common eigenvalue difficulties is integrated in addition to useful research instruments, finite point discretization tools, convergence research, thoughts for matrix assessment difficulties, and laptop implementation.
Louis,Joyce McLaughlin,William Rundell Inverse difficulties are eager about deciding upon explanations for saw or wanted results. The mathematical modelling of inverse difficulties often results in ill-posed difficulties, i. This quantity comprises twelve survey papers approximately answer tools for inverse and ill-posed difficulties and approximately their software to precise kinds of inverse difficulties, e.
The papers were written by means of top specialists within the box and supply an up to date account of resolution tools for inverse problems. Iterative methods for ill-posed equations need to be stopped after a finite number of iterations to keep the ill-posedness under control; compare, for example, [ 23 , 85 ].
The main idea of the decomposition methods is to split full nonlinear shape reconstruction problem given by 18 into a linear ill-posed equation which is solved first and a nonlinear well-posed equation to be solved in a second step. In order to discuss the Kirsch-Kress method [ 56 — 58 ] via potential approach let us employ an initial boundary as an approximation of the actual boundary.
We assume that is in the interior of the true scatterer. In this case, the approximate total field can be approximated by the sum of the incident wave and a single-layer potential, where is the fundamental solution to the Helmholtz equation in two dimensions and is a continuous density source function defined over. The far field pattern of the single-layer potential should be measured far field pattern , which leads to the linear equation with some constant depending on the dimension of the space under consideration.
The first step consists of solving the ill-posed linear equation 23 to calculate. Then, when we consider a Dirichlet boundary condition, the shape is found in a second nonlinear step as the zero curve of the total field. To find this zero curve, we introduce an operator , which maps to the values of the approximate total field , on such that Then the problem is reduced to the solution of the following optimization problem: Linear sampling [ 70 — 74 ] is based on solving the far field equation Then, the density satisfies that is, when the source point approaches the boundary.
This behaviour can be used to visualize the shape of the scatterer from the knowledge of the far field pattern for all directions. We also refer to the orthogonality sampling method, which has recently been suggested in [ 86 ]. It is particularly suited to deal with multifrequency data as is naturally obtained when acoustic pulses are used to probe an object or region in space. The method has been independently suggested by Ito et al.
A convergence analysis of the method in the limit of small scatterers has recently been achieved by Griesmaier [ 89 ]. The inverse problem considered in this section is finding a continuous function which is defined on the boundary of the obstacle from the knowledge of the scattered acoustic waves for a given shape in two dimensions.
We discuss the methods whose aims are to reconstruct , in the impedance boundary condition 6 , and , in the conductive boundary condition 7. Note that conductivity function is employed to model the inhomogeneity which might exist on the boundary of an object in a more realistic way. In [ 90 — 92 ], 3D obstacles with impedance boundary condition are studied for acoustic case, where in [ 91 ] electromagnetic case is included. In the same field, the papers [ 64 , 93 — 96 ] are focused on 2D geometries for the less complexity of governing numerical experiments. To this aim, approximate Green's functions are used to reduce the nonlinear problem to two linear moment problems.
On the other hand, the study [ 92 ] is devoted for the reconstruction of impedance functions via the Kirsch-Kress and Colton-Monk decomposition methods. Furthermore, some interesting papers appeared on the impedance reconstructions, recently [ 97 , 98 ]. The paper [ 93 ] introduced a new method for impedance reconstructions in the spirit of the Kirsch-Kress decomposition method. That is, the scattered field is represented via single-layer potential over the known boundary of the impedance cylinder , instead of defining an auxiliary initially guessed curve.
From the knowledge of the density function now the total field and the normal derivative of the total field can be computed on the boundary of the obstacle via jump relations [ 22 ]. Finally, is obtained from 6 in the least squares sense.
This method is also extended for the reconstructions of the conductivity functions of the obstacles in free space [ 99 ], for the obstacles buried in penetrable cylinders [ ] and for a combination of a shape and conductivity function reconstruction problem [ ], firstly by Yaman [ 6 ]. Moreover, [ 64 , 95 , 96 ] are devoted for the shape and impedance reconstructions of 2D obstacles in acoustics.
To do this, the hybrid method is employed by Serranho [ 64 , 66 ]. In [ 95 ], a level set algorithm is combined with boundary integral equations in acoustic case to reconstruct the shape and impedance of 2D obstacles from multi-illuminations, and in [ 96 ], it has been shown that the knowledge of the scattered fields corresponding to three incident waves can be used for the determination of the shape and the impedance via integral equation methods and conformal mapping techniques.
Let be a 3D space variable, and let be a 1D time variable; let be the displacement vector function of an inhomogeneous anisotropic elastic material characterizing by density and the elastic moduli. The density and elastic moduli are varying functions of position. Combining the properties of the strain-energy function with Hooke's law we find [ ] that satisfy the following property and strong convexity for any nonzero real symmetric matrix.
Equations for motion in inhomogeneous anisotropic elastic materials are, in our notation see, e. Let us examine as an approximate solution of 30 for. Near a wavefront , we assume that components of are fluctuating much more rapidly than or , and the successive derivatives and are fluctuating still more rapidly. Substituting 31 into 30 we find see, e. Thus, the left-hand side of 32 must be much smaller than.
We conclude that the matrix of coefficients of must be singular: This equation determines the possible wavefronts in an elastic medium, since it gives a constraint on the function. In an inhomogeneous isotropic medium, where is the Kronecker symbol; that is, if and if ; moreover, , are known as the Lame functions. In an inhomogeneous isotropic medium, the special form 34 of makes it possible to get 33 as follows: This is, satisfies the eikonal equation or eikonal equation where is the local -wave speed and is the local -wave speed.
Suppose that a point source at position becomes active at a time chosen to be the origin,. In homogeneous isotropic medium, wavefronts emanate from the source as ever-expanding spheres, with radius for -waves and for -waves, arriving at the general position at time and. We introduce the function as the travel time required for the wavefront to reach from. The function satisfies 36 for -waves and 37 for -waves.
### Submission history
One of the first inverse problem, stated and studied in geophysics, was the inverse kinematic problem. The physical interpretation of this problem is the following. Let us assume that Earth is an isotropic inhomogeneous elastic medium and the measurements of the seismic waves, arising from a point source and propagating in Earth, are given for points on its surface.
These measurements contain data of the travel time of seismic waves between the point of the source and any point of the Earth's surface. The inverse kinematic problem is to find the speed of the seismic waves inside of Earth using the measurement data. Mathematically we can state the inverse kinematic problem as follows. Let be a domain bounded by the surface , and let be the function of the travel time required by a signal with unknown speed to reach from. Find for all from if the function is given for all points and , where and are subsets of.
Herglotz [ ] and Wiechert and Zoeppritz [ ] were the first who studied the inverse kinematic problem in assumptions if is known for any from. Gerver and Markushevich [ ] have showed that the condition can be illiminated, but in this case the inverse kinematic problems have many solutions and the set of these solutions has been described. The first theoretical study of the inverse kinematic problem for a horizontal inhomogeneous medium has been made by Lavrenti'ev and Romanov [ ]. The first result of the study of the multidimentional inverse kinematic problem in a linear approximation, when the function depends on 3D space variable , was obtained by Romanov [ ].
In a recent time, the Earth is modeled as an anisotropic elastic medium which is located in the given 3D domain. The wave speed is given by a symmetric positive definite matrix , that is, a Riemannian metric in mathematical terms see, e. The problem is to determine the metric in a given domain from the lengths of geodesics joining points on the boundary of the domain. The linearization of this problem leads to a problem of the integral geometry [ — ].
The regular study of the problems of finding the isotropic and anisotropic Riemannian metrics and integral geometry problems has been made in the works [ , — ]. The modern numerical algorithms for the computation of the inverse kinematic problems of seismic have been developed in the works [ , ].
Let us note that isotropic inhomogeneous elastic medium is completely characterized by three functions: Using the point source at position of the boundary of the given domain , which becomes active at the time , we measure the function of the travel time required for the fronts of - and -waves to reach from. We use these information for solving two inverse kinematic problems for - and -waves. The solutions of these problems are speeds ,.
To complete the identification of unknown isotropic inhomogeneous medium we need to determine the last unknown function after finding ,. An inverse problem to recover in a given bounded domain , containing an isotropic inhomogeneous elastic medium, has been solved by Yakhno [ — ]. In these works, the displacement fields have been measured for all points and running the boundary of for all times from a time interval containing the time of arriving of the -waves.
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The vertical inhomogeneous model of Earth is one of the popular models of geophysics [ ], and the inverse problems of recovering the density and Lame functions and , depending on one variable and appearing in equations of elastodynamics 30 for the case of inhomogeneous isotropic elastic medium, have been studied by many authors [ 1 , 2 , 7 — 15 , 21 , 22 , 24 , 26 — 28 , 36 — 46 , 53 — 55 , 59 — 61 , 70 — 73 , 78 — 81 , 85 , 87 — 91 , 97 , 98 , , , , , , , , — ]. Because the unknown functions depend on one variable, the inverse problems of their recovering are called one-dimensional inverse problems although all differential equations of elasticity contain 3D space variable and 1D time variable.
Alekseev and Dobrinsky [ , ] were the first who described the importance of one-dimensional inverse problems of elastodynamics in geophysics and studied them as problems of the recovery of smooth functions , , and of one variable. The uniqueness of the solutions of these inverse problems has been studied firstly by Blagoveschenskii [ ] and Romanov [ ] for the isotropic elastic media and then by Volkova and Romanov [ ] for anisotropic elastic media. The regular study of the theory, methods, and applications of one-dimensional inverse problems for dynamical differential equations of isotropic and anisotropic elastic media has been made in works [ , , , , — ] and others.
The recent development of theory, methods, and applications of one-dimensional inverse problems of dynamic elasticity can be found in the works [ , , , ]. We note that a model of Earth as a composite medium consisting of a finite number of different elastic layers is very popular in geophysics.
## A Survey on Inverse Problems for Applied Sciences
In this case, the one-dimensional inverse problem consists of finding and the Lame functions and as functions of one variable with piecewise constant values. The computation of solutions of this type of one-dimensional inverse problems has been studied in [ , , ]. The modern theory and methods of the construction of solutions of one-dimensional inverse problems for the equations of elastodynamics in elastic composite media can be found in the works [ , , — ].
Linearized multidimensional inverse dynamic problems or inverse problems in the Born approximation take an important place through all statements of multidimensional inverse problems for equations of elastodynamics. The statements of these problems have natural physical and mathematical sense. From the physical point of view, an isotropic inhomogeneous elastic body, which is characterized by the Lame functions and and density , is included in a vertical inhomogeneous or homogeneous elastic medium.
Let, for example, the half space contain this medium, and let the characteristics , , and of the elastic body be unknown functions. The linearized inverse problem is to find these unknown functions if we measure the first act of scattering the displacement field on the surface arising from the forces located on the same surface. From the mathematical point of view, we consider the differential equations of elastodynamics in a half space with boundary conditions on. We assume that the Lame functions and and density appearing in differential equations and boundary conditions can be presented in the form where , , and are functions depending on and characterizing vertical inhomogeneous medium and , , and are functions of 3D space variable characterizing the elastic body which is included in the vertical inhomogeneous medium.
We assume that the displacement field is presented in the form , where is the displacement field of the vertical inhomogeneous medium arising from the given forces, and is the first act of scattering on the inhomogeneous inclusion with characteristics , , and. The equations of elastodynamics with boundary conditions are linearized around , , , and.
The unknown functions , , and appear in inhomogeneous terms of linearized equations for. We need to recover , , and if we know for see, e. The uniqueness of the solution of a linearized multidimensional inverse problem has been studied by Romanov [ ]. The existence theorem and computation of a solutions of a linearized multidimensional inverse problems of elastodynamics have been obtained in the works [ , ].
The recovery of the function characteristics of an elastic body in linear approximation was a subject of the works [ , , , ]. The linearized inverse problems of determining the function characteristics of transversally isotropic elastic media from the measurements of reflected waves have been developed by Sharafutdinov [ , ].
The linearized inverse problems for nonhomogeneous bodies have been stated and developed by Steinberg [ ]. The inverse problems of determining the elastic moduli and density as functions of the space variables in a bounded domain from observed data of the solution on the boundary or a part of the boundary of this domain are geophysical motivated. One important class of these problems is inverse problems for equations of elastodynamics in terms of the Dirichlet-to-Neumann map.
1. [] Optimization Methods for Inverse Problems!
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4. The Dirichlet-to-Neumann map models surface measurements by giving the correspondence between a displacement at the boundary of the given bounded domain and surface traction where is the unit outer normal to , is the observed time interval, and , are components of the displacement vector function satisfying The details of the use of the Dirichlet-to-Neumann map in modeling surface measurements in inverse problems can be found in [ ]. The inverse problems in the Dirichlet-to-Neumann map statements are successfully applied to study the unique determination of the solutions of the inverse problems of elasticity as in the static isotropic and anisotropic cases [ — ] as well as in dynamic case [ ].
For the study of the inverse problems for the scalar partial differential equations with a finite number of observation, Bukhgeim and Klibanov [ ] proposed a remarkable method based on a Carleman estimate. Later, the Carleman estimate method has been generalized to study the uniqueness and stability estimate of the solutions of the inverse problems for equations of elastodynamics by Isakov [ ], Ikehata et al.
In the papers [ — , ] the authors assume some geometric constraints on the surface under observation for proving the uniqueness and stability of the solutions of the multidimensional inverse problems using the Carleman estimates.
## Mathematics > Optimization and Control
Later, the stability estimate theorem for solutions of a multidimensional inverse problem for equations of elastodynamics has been proven for an arbitrary subboundary by Bellassoued et al. In the following, we present a general overview on some inversion-based application areas. We then give more details about particular selected applications in the subsequent sections. Further, different applications whose details are skipped in this section such as remote sensing, nuclear science, and geophysics can be found in [ , , — ].
Underwater Acoustics and Traveltime Tomography. Inverse problems related to underwater acoustics are a critical research area due to their wide range of important practical applications. In water-type medium, propagating acoustic waves are collected at a selected number of separate hydrophones to obtain measured field data. Generally, reconstructions of desired parameters from the knowledge of scattered acoustic waves lead to nonlinear and ill-posed inverse problems.
Therefore, it is always a complicated issue to find unique and stable solutions, and one has to apply some additional techniques, that is, regularizations, for the proper treatment of the ill-posedness. The inspection of an object without touching or without changing its characteristic properties is a general definition of nondestructive testing NDT in the literature.
Therefore, approaches which are used for inverse problems, that is, synthetic aperture focussing technique SAFT, [ , ] , diffraction tomography DT, [ — , — ] , multiple signal classification MUSIC, [ , , , , ] , linear sampling method LSM, [ 70 — 74 ] , factorization [ 75 , 76 , ], point source [ 23 , 68 , 69 ], no response test [ 82 — 84 ], and so forth, are also employed for NDT problems [ ].
More specifically, SAFT is an algorithm which uses the collection of echo signals over a specific aperture to obtain a reconstruction by performing time shifting and superposition of adjacent signals. DT is based on a linear solution of the wave equation which can be obtained via the Born or Rytov approximations. Here, the linearization approaches define mainly the success and the solution space of the inverse problem. MUSIC was initially employed as a direct imaging algorithm to obtain locations of point scatterers [ ] and extended to find also the geometry of targets [ ].
The method employs the eigenvalue structure of time-reversal matrix which is obtained from measured data at different receiver antennas. The main idea of linear sampling method is to find an indicator function such that its value provides whether an arbitrarily tested space coordinate lies inside or outside of the object.
Nowadays medical imaging, which can be considered as one of the most developed area of inverse problems in practice, provides high-resolution reconstructions in the order of millimeters. In the last decays, the main effort is given for the implementation of harmless, fast, cheap, robust, and reliable techniques to use in practice for obtaining high-resolution images in real time. In the similar direction, early studies of bioimaging started with the reconstructions of 2D images of human body parts via the inverse Radon transform [ , , ] of measured X-rays which were attenuated inside the body [ , ].
Afterwards, 3D images were assembled via computed tomography CT-scan from a series of X-ray data measured on different planes sinograms in 2D [ , , , , ]. Even though satisfactory results were obtained with the X-ray radiology especially for the bone structures [ , ] the method was found not sufficiently efficient due to attenuation characteristics of X-rays and harm risks of using ionized radiation on humans. On the other hand, electrical impedance tomography EIT , after its main idea and formulation were introduced by Calderon in [ ] and D.
Isaacson [ ], has gained high interest both from theoretical and physical point of view. In principle, in EIT low-frequency electrical currents are applied to the body part under investigation. Then electrical properties of body tissues are computed from the measurements of electric currents and voltage at the boundary [ , , , , , , , — ]. EIT is successfully applied for diagnosis of breast cancer, monitoring brain and gastrointestinal functions, detection of blood clots in the lungs, and so forth [ ].
Furthermore, electroencephalography EEG and magnetoencephalography MEG are used for passive monitoring of neuron activities in the brain from the weak electric or magnetic fields, respectively, [ , , , , , ].
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A different approach which is based on using properties of subatomic particles with the connection to electromagnetism opened a new area for obtaining high spatial resolutions in bioimaging, for example, magnetic resonance imaging MRI , positron emission tomography PET , and single-photon emission computed tomography SPECT.
In MRI, the patient stays in a tunnel under a strong magnetic field typically 0. This large static magnetic field aligns protons of many atoms either parallel or antiparallel existing in the body. In the meantime, weak radio frequency fields are applied systematically to the patient for altering the alignment of the magnetization. As a result of this procedure, rotating magnetic fields induce a voltage at the receiver coils of the magnet which is used to reconstruct the image of the scanning area [ , , — ].
Gamma rays, which occur when an electron and an emitted positron annihilate in PET, and photons which are released in SPECT can be visualized out of body by using scintigraphic detectors for clinical applications of oncology, cardiology, pharmacology, and so forth [ , , — ].
Another group of techniques such as microwave tomography, ultrasound, and optical imaging, which are commonly used for solutions of inverse problems in different areas, are also applied to biomedical applications especially for investigating soft body tissues [ , , , , , — ]. For instance, optical tomography is used for the detection of cancerous cells in breast and brain.
Acoustic waves are employed for the imaging of liver, kidney, fetus in pregnant women, and so forth, and microwaves are used in mammography and diagnosis of leukemia [ , ]. Over the past two decades, it has became feasible to simulate atmospheric and geophysical processes from large-scale atmospheric flow down to convective processes on a kilometer scale. This led to the need to determine initial conditions for simulations and forecasts from a collection of diverse direct and indirect measurements, and the field of data assimilation arose.
Inverse Problems in Biological and Environmental Applications.
### A Survey on Inverse Problems for Applied Sciences
Inverse problems are of growing importance in many parts of medicine or biology as well as in environmental applications. Here, we will provide a brief introduction into two areas, first into recent results of neural field theory and second into the basic setup of data assimilation as it is, for example, used in numerical weather prediction or for climate projections , which usually incorporates various inverse problems. Neural activity is often modelled by the activity potential in some domain. The activity potential satisfies some integro-differential equations, which in its simplest form has been suggested by Wilson and Cowan [ , ] and Amari [ ]: The kernel models the strength of the influence of an excitation at point to the neural field at point.
The first term of the right-hand side of 41 generates the decay of the activity potential in the absence of excitation or inhibition. Neural fields have been widely studied in recent years, with applications to a wide range of medical phenomena starting from electroencephalogram EEG and magnetoencephalogram MEG rhythms to robotic behaviour—for an extensive literature list we refer to [ ]. Neural network, that is, the discrete version of neural fields, has attracted strong interest over many years and is a standard tool today in many applied parts of science.
Neural networks have also been used to solve inverse problems, compare, for example, [ , ] for some recent papers and further citations. However, here we want to look into inverse problems which arise in the modelling of neural activities themselves. Training of neural networks or neural fields is, in general, an ill-posed inverse problem, as we will see in due course. Inverse neural field theory investigates the construction of connectivity kernels given some dynamics for and some time intervals.
This so-called full-field neural inverse problem is linear and ill-posed in the sense of Hadamard, as can be readily seen by the following transform. We define If we further define the operator equation 41 obtains the form The task is to find the operator given the family of states and corresponding images for. Changing our perspective slightly, introducing the operator we transform 41 or 44 into for each fixed. For each , 46 is an integral equation of the first kind for the function.
Usually we need some smoothness of the potential with respect to its arguments and. In this case, the operator is an integral operator with continuous kernel, which is known to be compact in either or cf. Thus, the inverse neural field problem is ill-posed. The particular form 46 provides a basis for kernel construction, that is, for the solution of the neural field problem. We may apply spectral regularization schemes as described in [ 1 , 22 , 85 ], for example, Tikhonov regularization in a similar sense of 21 with regularization parameter as regularized inverse to calculate from the knowledge of and.
For smooth dynamics, the problem is exponentially ill-posed. We refer to [ — , ] for the analysis and many examples for the inverse neural field problem. The problem of the ill-posedness of the neural inverse problem is addressed in [ ], where a dimensional reduction approach is suggested to decompose the large and strongly ill-posed full problem into more stable individual tasks.
In inverse problems, we are usually given measured data, and the task is to gain insight into some unknown parameter functions insight of an inaccessible body or area of space or to reconstruct the shape of scatterers or inclusions, but often, the quantities under reconstruction are not static but dynamic and change over time. Then, the inverse task is not only carried out once but the reconstruction is repeated over time with some cycling frequency given by a time interval.
Let for be an element of a Hilbert space representing our measurement data.
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The task is to reconstruct some state in a Hilbert space , where the measurement is described by an observation operator. An underlying dynamical system is given by some operator , mapping the state at time onto the state at time. | {"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": 0.9145314693450928, "perplexity": 544.9941731291401}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570987795403.76/warc/CC-MAIN-20191022004128-20191022031628-00287.warc.gz"} |
http://chemistry.stackexchange.com/questions/2743/classical-reason-for-good-heat-conductance-in-diamond | Classical reason for good heat conductance in diamond?
My teacher in my physics course attributed this effect to phonons, more here, but I am skeptical about this argument, it feels like he is overlooking the whole question -- what about with Silica that has similar structure to diamond with covalent bonds?
My friend attributed this to the small mass of atoms and large Young constant, considering the system to be made of springs -- $v_{sound}=\sqrt{\frac{Y}{\rho}}$. He suggested to think of heat as sound to use the formula.
Silica has less strong bonds to diamond so perhaps this is a reason why diamondis a good conductor of hear and silica is not. Air is good insulator so does it mean that carbon atoms are so closely packed and silicon atoms not? Is this the reason for the difference in conductance?
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I think that there is no really good classical reason. Heat conductivity in a solid depends on bonding in that solid and bonding is inherently a quantum mechanical phenomenon.
For instance, recall that graphite is a good conductor of heat (and electricity) in two dimensions and a fairly good insulator (heatwise and electrically) in the third. This is due to the layered structure in graphite and that is due to the bonding in graphite.
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+1 for the start but I think one could create some sort of mechanical-engineering-style approximations for the object to be counted as classical, interested how far they are off from the more real QM models. – hhh Feb 14 '13 at 17:45
I think the classical explanation for good heat-conductance in diamond is the spring-analogy. But it does not explain all elements such as metals where electron-gas in valence shell contribute to the good heat-conductance. In order to understand this deeper, you need to study the statistics outlined below. The phonon gas occurs between the bonds in diamond while in the electron gas with metals of blocks I/II.
Let's think about classical features such as heat capacity and heat-conductance. As said, they are different in nature depending on the location in periodic table.
I/II blocks in periodic table
Let's consider the first block and second block. The good heat-conductance in metals such as Li, Na and Mg is due to the electron-gas that forms fermi-balls according to Fermi-Dirac statistics. When you experience the heat-change, phonos are the interactive quantitative pieces of energy.
Only the electrons in the highlighted region contribute meaningfully to the specific heat capacity. The source of this picture is the page 166 here, a course booklet in 2062 Physics course in Aalto University.
In order to understand this, you must understand Fermi-Dirac statistics and Fourier Transforms by which you can get the location-frequency-spectrums (in Finnish paikkataajuus spektri): you have distribution of particles and then you use that distribution to deduce the frequency plot.
IV block elements such as Si and C
Now the heat conductance here with structures such as diamond or Silica cannot be due to electron-gas because there is no moving electrons in diamond or silica. The very good heat conductance in diamond is due to mechanical vibration and very strong bonds. Silica has much weaker bonds than carbon atoms with four covalent bonds. Young's modulus is analogical to the mechanical Hooke's constant. Einstein and Debyen explained this so that atoms act like massless springs (page 155 of the earlier lecture slide). Now you can calculate the internal energy with the help of Bose-Einstein statistics.
This plot shows the heat capacities as a function of temperature. The parameter $\theta_D$ is the Debye temperature, $R$ is the gas-constant and dots show real temperatures. The source is page 158 of the earlier lecture slides.
Now in Chemistry, people often use the Maxwell-Boltzman statistics. In Quantum mechanics, they say that Maxwell-Boltzman is like an approximation -- it only work well with very low-temperature or very high temperature -- otherwise it is a bad approximation even though useful in practice but not correct. The statistics Fermi-Dirac and Bose-Einstein are the more realistic description of nature. Depending on the spin-number of atom, you choose the statistics. If your spin is some integer (or zero with modulo 1), you have a boson. If your spin has half (or half with modulo 1), you have a fermion. Examples of fermions contain electron, neutron, proton -- and they all have spin of half.
Let's consider Rubidium 86 (atomic number 37 so 37 electrons, 37 protons and 49 neutrons) to its isotope 87 (37 electrons, 37 protons and 50 neutrons). Because 86's spin is odd and 87's spin is even, 86 is fermion while 87 is boson. Now use Boson-Einstein statistics with 87 and Fermi-Dirac with 86.
P.s. You could mathematically approach this problem this way:
1. calculate amount of particles $N$
2. derivate $N$ with respect to $E$ energy
3. find the point when the derivative $\frac{\partial N}{\partial E}$ is zero i.e. $\frac{\partial N}{\partial E}=0$.
-
All true, but the calculations and ideas are NOT classical. – Paul J. Gans Dec 25 '12 at 2:33
@PaulJ.Gans that is right but I thought it to be useful to contain an in-depth explanation also -- apparently there is no simpler way to explain this. We are still missing the in-depth classical explanation. The assumption that particles as springs leads to which kind of models? Strench-compression pictures? FEM? I don't know -- perhaps needing mechanical-engineering-approximation for the classical case, investigating. – hhh Feb 14 '13 at 17:43
You should correct neutrnos to neutrons. – user2617804 Dec 5 '13 at 3:46
@user2617804 thank you, fixed! – hhh Dec 5 '13 at 11:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.815500020980835, "perplexity": 1068.3343333878065}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775669.60/warc/CC-MAIN-20141217075255-00067-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.hzdr.de/db/!Publications?pNid=0&pSelTitle=6213&pSelMenu=0 | # Publications Repository - Helmholtz-Zentrum Dresden-Rossendorf
1 Publication
Ultrafast X-ray Computed Tomography with a Linearly Scanned Electron Beam Source
Hampel, U.; Speck, M.; Koch, D.; Menz, H.-J.; Mayer, H.-G.; Fietz, J.; Hoppe, D.; Schleicher, E.; Zippe, C.; Prasser, H.-M.;
We devised and tested a novel computed tomography approach that utilises a scanned electron beam X-ray source to produce fast sequences of tomographic images of transient density distributions. In contrary to classical electron beam tomography we used a linear deflection pattern for the electron beam and a non-annular detector arc to record transmission data of an object from different projection angles. This approach gives the highest achievable axial resolution and is comparatively moderate in effort and costs. For the inverse problem we applied iterative image reconstruction techniques to reconstruct the density distribution from a limited data set. The method has been experimentally tested on static and dynamic phantoms with a frame rate of 1000 images per second and a spatial resolution of approximately 1 mm in plane and axial.
Keywords: high-speed X-ray computed tomography, limited angle reconstruction
• Flow Measurement and Instrumentation 16 (2005), 65-72 | {"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": 0.867734432220459, "perplexity": 3869.679035098259}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574039.24/warc/CC-MAIN-20190920134548-20190920160548-00364.warc.gz"} |
http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=FileTools/Text/ReadNextInteger | FileTools[Text] - Maple Programming Help
Home : Support : Online Help : Programming : Input and Output : File Manipulation : FileTools : FileTools/Text/ReadNextInteger
FileTools[Text]
read the next integer from a file
Parameters
file - file descriptor or filename
Description
• The ReadNextInteger(file) command attempts to read an integer from file. The ReadNextInteger function skips non-numeric characters until a valid integer is found. The integer is read from file and returned. If no characters form a valid integer, then NULL is returned. In this case, the file is read to the end of the file.
• If file is the name of a file that has not been opened, Maple attempts to open the file before attempting to read the integer.
• An error is raised if file is not a valid descriptor or if it is the name of a file that cannot be opened.
Examples
> $\mathrm{FileTools}[\mathrm{Text}][\mathrm{WriteString}]\left("testfile","1 two 3 four"\right)$
${12}$ (1)
> $\mathrm{FileTools}[\mathrm{Text}][\mathrm{Close}]\left("testfile"\right):$
> $\mathrm{FileTools}[\mathrm{Text}][\mathrm{ReadNextInteger}]\left("testfile"\right)$
${1}$ (2)
> $\mathrm{FileTools}[\mathrm{Text}][\mathrm{ReadNextInteger}]\left("testfile"\right)$
${3}$ (3)
> $\mathrm{FileTools}[\mathrm{Text}][\mathrm{ReadNextInteger}]\left("testfile"\right)$
> $\mathrm{FileTools}[\mathrm{AtEndOfFile}]\left("testfile"\right)$
${\mathrm{true}}$ (4) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 10, "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": 0.800331175327301, "perplexity": 2116.1738865173766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982950827.61/warc/CC-MAIN-20160823200910-00185-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://planetmath.org/HewittMarczewskiPondiczeryTheorem | # Hewitt-Marczewski-Pondiczery theorem
The Hewitt–Marczewski–Pondiczery Theorem is a result on the density of products of topological spaces. This theorem was arrived at independently by Hewitt[1], Marczewski[2] and Pondiczery[3] in the 1940s.
###### Theorem.
Let $\kappa$ be an infinite cardinal number and $S$ an index set of cardinality at most $2^{\kappa}$. If $X_{s}$ $(s\in S)$ are topological spaces with $d(X_{s})\leq\kappa$ then
$d\left(\prod_{s\in S}X_{s}\right)\leq\kappa.$
The special case $\kappa=\aleph_{0}$ says that the product of at most continuum many separable spaces is separable.
## References
• 1 Edwin Hewitt, A remark on density characters, Bull. Amer. Math. Soc. 52 (1946), 641–643. (This paper is available as a PDF file from the AMS website: http://www.ams.org/journals/bull/1946-52-08/home.htmlBull. Amer. Math. Soc., Volume 52, Number 8.)
• 2 Edward Marczewski, Séparabilité et multiplication cartésienne des espaces topologiques, Fund. Math. 34 (1947), 127–143. (This paper is available as a PDF file from the Polish Virtual Library of Science: http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=34&jez=enFundamenta Mathematicae, Volume 34.)
• 3 E. S. Pondiczery, Power problems in abstract spaces, Duke Math. J. 11 (1944), 835–837.
Title Hewitt-Marczewski-Pondiczery theorem HewittMarczewskiPondiczeryTheorem 2013-03-22 17:16:56 2013-03-22 17:16:56 yark (2760) yark (2760) 6 yark (2760) Theorem msc 54D65 msc 54A25 Dense Separable | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 8, "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": 0.934670090675354, "perplexity": 1916.903608660891}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1521257647768.45/warc/CC-MAIN-20180322034041-20180322054041-00442.warc.gz"} |
https://physics.stackexchange.com/questions/486147/change-in-magnetic-flux-eddy-currents-vs-moving-wire | # Change in Magnetic Flux: Eddy Currents vs Moving Wire
From what I have read, as a conducting rod moves perpendicular to a uniform magnetic field, this will lead to a change in magnetic flux and therefore an emf will be induced (area being swept as the conducting rod moves).
However, in terms of eddy currents, I have been told that when the metal plate is moving perpendicular to and within the uniform magnetic field, there is not change in magnetic flux and therefore no induced emf.
As the conducting rod moves and there is an induced emf (and therefore change in magnetic flux), how is it that there is no induced emf in the form of eddy currents as the metal plate is within the uniform magnetic field.
To me, I don't see where the difference lies between the two scenarios, they are both conductors moving within a uniform magnetic field.
Could someone please explain to me where my understanding is flawed.
• If an electric charge moves in a magnetic field the Lorenz forces (see your formula above) acts upon it. However, this is only true, if the motion is not parallel to the magnetic field. Therefore, could you please clarify the relative directions of the B-field and the motion of the metal plate. Jun 15, 2019 at 10:45
• @Semoi please see it now Jun 16, 2019 at 12:33
• Thank you for clarifying the directions. My second guess is, that the problem is due to wording. We usually use the term eddy currents if we like to address the induced currents which are produced by a change in the B-field, $dB/dt \propto I$. However, in the situation described above the B-field is constant. Therefore, although we obtain a voltage across the moving conductor, we usually do not speak of eddy currents. Nevertheless, the voltage across the plate exists. Jun 16, 2019 at 18:20
The flux of the magnetic field through any surface whose border lies inside the rod is defined as $$\int_S\vec{B}\cdot\hat{n}dS$$ where the integral is calculated over the whole surface and $$\hat{n}$$ is the versor perpendicular to the surface (point by point). For example if you consider a rectanguar border and the rectangle as the surface, the flux is simply given by $$\int_S\vec{B}\cdot\hat{n}dS=\vec{B}\cdot\hat{n}\cdot A$$ where $$A$$ is the area of the rectangle and $$\vec{B}\cdot\hat{n}$$ is the same in any point of the rectangle because the field is uniform. For this particular surface the flux is constant. Without actually calculating it but just by looking at the flux definition we can conclude that this is true for any surface with any border that moves with the rod because the field is uniform and if the rod moves perpendicularly to the field, the angle between $$\vec{B}$$ and $$\hat{n}$$ does not change over time. | {"extraction_info": {"found_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": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9500967264175415, "perplexity": 164.50536661136078}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571692.3/warc/CC-MAIN-20220812105810-20220812135810-00375.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2006_Romanian_NMO_Problems/Grade_8/Problem_1&oldid=47920 | # 2006 Romanian NMO Problems/Grade 8/Problem 1
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
## Problem
We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.
## Solution
We use the lemma that given two non-coplanar circles in space that intersect at two points, there exists a point P such that P is equidistant from any point on one circle and any point on the other circle.
Proof of lemma: Let the two circles be and and let them intersect at and . Draw the line through (the center of) perpendicular to the plane of the circle. We know that any point on that line is equidistant from any point on because for a point on the line and a point on the circle, , which does not depend on the point chosen. Similarly, we draw the line through . Since and lie on both circles, any point on or is equidistant from and . However, the locus of all points equidistant from and is the plane that perpendicularly bisects . Therefore, and lie on one plane. Since our two circles are not coplanar, and must intersect at our desired point.
Let the cube have vertices , , , , , , , , where all sides but are known to be cyclic quadrilaterals. First, we consider the circumcircles of quadrilaterals and . By our lemma, there exists a point equidistant from , , , , , . Let the perpendicular from to the plane intersect the plane at . By HL congruency, the triangles , , and are congruent. Since , O is the center of quadrilateral . By SAS congruency, is congruent to the aforementioned triangles, so . Similarly, if we focus on quadrilateral , we get that . Therefore, . Let the perpendicular from to the plane intersect the plane at . By HL congruency, the triangles , , , and are congruent. Thus, and is cyclic. | {"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": 0.946442723274231, "perplexity": 420.5306222083038}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780058450.44/warc/CC-MAIN-20210927120736-20210927150736-00134.warc.gz"} |
https://cs.stackexchange.com/questions/108745/is-there-a-path-of-length-k-between-given-vertex-to-a-subset-of-vertices-in-a | # Is there a path of length $k$ between given vertex to a subset of vertices in a connected directed graph
I am trying to find an efficient algorithm that on input $$(G,s,A,k)$$ returns true iff $$G$$ is a connected directed graph, $$s$$ is a vertex in $$G$$, $$A$$ is a set of vertices in $$G$$ and there is a path of length $$k$$ from $$s$$ to a vertex in $$A$$
What is the best run time we can get here?
I am currently stuck with this slow algorithm (there might be some minor problems but the concept is of the importance) $$T$$ that on input $$(G,s,A,k)$$:
for all vertices $$t$$ in $$A$$:
if $$k=1$$ and $$t$$ is a neighbor of $$s$$ return TRUE
if $$k=0$$ and $$s=t$$ return TRUE
for all neighbors $$r$$ of $$s$$ return $$T(G,r,\{t \},k-1)$$
this algorithm is rather slow because it exhausts all possibilities.
*cycles of any size (1 or more) are allowed in the graph and in the paths
• Are you looking for simple paths only? – orlp Apr 30 '19 at 12:31
• It should include possibility of self edges (i can't assume simple paths or simple graphs) – Oren Apr 30 '19 at 13:36
From the algorithm you describe and the comment you made on your question I understand that you are interested in walks and not in simple paths and that the graph is unweighted.
The algorithm you describe has at least exponential complexity since you examine each possible subset of $$A$$ (it has $$2^{|A|} = O(2^N)$$ subsets). We can use a much faster polynomial time algorithm.
One such algorithm has $$O(k(N + M))$$ computational complexity. You will need $$k + 1$$ boolean matrices, $$B_0, B_1, \dots, B_k$$, of size $$N$$. $$B_0$$ will be initialized to false, excluding its $$s$$-th position which will be initiallized to true. That is:
$$B_0[u] = 1$$, iff $$u = s$$
You will execute $$k$$ steps. In the $$i$$-th step you will calculate $$B_i$$ using the following formula:
$$B_i[u] = 1$$, iff $$\exists v: (v, u) \in E \wedge B_{i - 1}[v] = 1$$
This way $$B_i[u]$$ will be true iff there is a walk of length $$i$$ from $$s$$ to $$u$$. When you have calculated $$B_k$$ you can find the answer by checking $$B_k$$'s positions that match to $$A$$'s nodes' ids. Iff at least one of these array positions make $$B_k$$ true, then the answer is positive. The computational complexity of this algorithm is $$O(k(N + M))$$ and its space complexity is $$O(kN)$$. You can drop the space complexity to $$O(N + M)$$ if you observe that only the previous and the current boolean matrix $$B_{i - 1}, B_i, \forall i \in \{1, 2, \dots, k\}$$ are needed during each step. Bellow you can see a C++ implementation of this algorithm.
#include <iostream>
#include <vector>
using namespace std;
#define MAXN 100005 // Equal to maximum possible N + 5
vector<int> A; // Nodes that belong in set A
bool PB[MAXN]; // boolean array of the previous step
bool B[MAXN]; // current boolean array
int main()
{
int N, M, S_A, s, k;
// N: number of nodes
// M: number of directed edges
// S_A: size of set A
// s: starting node
// k: length of walk to be done
cin >> N >> M >> S_A >> s >> k;
for (int i = 0; i < M; i++)
{
int u, v;
cin >> u >> v;
E[u].push_back(v); // there is an edge from u to v
}
for (int i = 0; i < S_A; i++)
{
int a;
cin >> a;
A.push_back(a); // a belongs to A
}
// Create B_0
for (int i = 0; i < N; i++) B[i] = false;
B[s] = true;
// Perform the k repetitions
for (int i = 1; i <= k; i++)
{
// Copy B (B_i) to PB (B_{i-1}) and initialize B
for (int j = 0; j < N; j++)
{
PB[j] = B[j];
B[j] = false;
}
for (int j = 0; j < N; j++) if (PB[j])
{
// There is a walk from s to j of length i-1...
for (int l = 0; l < E[j].size(); l++)
{
// ...and an edge from j to E[j][l]...
B[E[j][l]] = true;
// ...so there is a walk of length i from s to E[j][l]
}
}
}
// Now B = B_k
for (int i = 0; i < S_A; i++)
{
int a = A[i];
if (B[a])
{
// There is a walk of length k from s to a
cout << "YES\n";
return(0);
}
}
// There is no walk of length k from k to any node of A
cout << "NO\n";
return(0);
}
This algorithm is fast if $$k$$ is small but when $$k$$ becomes larger its computational complexity increases significantly. We will now describe an algorithm which is faster for large values of $$k$$.
You can use the fact that the $$(i, j)$$ cell of $$Adj^p$$ (where $$Adj$$ is the adjacency matrix of the graph) is equal to the number of walks from $$i$$ to $$j$$ in the graph with length equal to $$p$$ (easy to prove using mathematical induction). In your case you are interested in walks of length equal to $$k$$ so you have to calculate $$Adj^k$$. You can efficiently calculate $$Adj^k$$ in $$\lceil log_2k \rceil$$ steps using exponentiation by squaring. Each step of the above calculation requires squaring an $$N \times N$$ matrix. This can be done in $$O(N^3)$$ using the naive algorithm or in $$O(N^{2.373})$$ using the fastest known algorithm. The fact that you want to know if a path between two nodes exists and not the number of different paths allows you to avoid large numbers by making the integer matrix boolean after each multiplication. When you have $$Adj^k$$ (in boolean format) you can check if there is a path of length $$k$$ from $$s$$ to a vertex in $$A$$ by checking the $$s$$-th row of $$Adj^k$$ and specifically the columns regarding nodes of $$A$$. Formally the result is positive iff $$\exists u \in A: Adj^k(s, u) = 1$$ The algorithm is the following:
1. Read input and calculate adjacency matrix of the graph: $$O(N^2)$$
2. Calculate $$Adj^k$$ using exponentiation by squaring, fast matrix multiplication and making matrix boolean after each multiplication: $$O(N^{2.373}logk)$$
3. Check if there exists a cell $$Adj^k(s, u) = 1$$ where $$u \in A$$: $$O(N)$$
4. Output the reslt: $$O(1)$$
The total computational complexity of this algorithm is $$O(N^{2.373}logk)$$ and it is better than the first algorithm for large values of $$k$$. Both of them also include possibility of self edges as you mention in your comment on your question.
You are able to combine the two algorithms and run the fastest one based on the values of the parameters and achive $$O(\min \{k(N + M), N^{2.373}logk\})$$ computational complexity.
• Nice solution :) Before I +1 though, please correct the time complexity of the OP's current solution, which is much worse than $O(k(N+M))$. I think it's exponential in $M$ but it may be even worse (since the exact same subproblem instance can be generated a large number of times). – j_random_hacker Apr 30 '19 at 16:42
• @j_random_hacker Thank you for the observation. I mistakenly thought that OP described another algorithm which I included in my edited answer, because it is faster for smaller values of $k$. – George Vidalakis Apr 30 '19 at 18:08
• Actually I think the OP's algorithm is $O(|A|(N-1)^{(k-1)})$, since in the case where $G$ is a complete graph, for a particular choice of $t \in A$, each subproblem $(G, s, \{t\}, k)$ spawns $N-1$ subproblems $(G, u_i, \{t\}, k-1)$, $1 \le i \le N-1$ so the collection of all subproblems of any $k$-value for a given starting $t$ is an $(N-1)$-ary tree of height $k-1$. Otherwise great answer! – j_random_hacker May 1 '19 at 1:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 82, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8220428228378296, "perplexity": 348.4094522181851}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046154878.27/warc/CC-MAIN-20210804142918-20210804172918-00022.warc.gz"} |
http://math.stackexchange.com/questions/64492/automorphisms-inducing-automorphisms-of-quotient-groups | # Automorphisms inducing automorphisms of quotient groups
Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow \operatorname{Aut}(G/N)$. I was therefore wondering,
Under what conditions is the induced homomorphism $\operatorname{Aut}(G)\rightarrow \operatorname{Aut}(G/N)$
• a monomorphism?
• an epimorphism?
• an isomorphism?
I believe it should work for (semi-?)direct products $N\times H$ where $\operatorname{Aut}(N)$ is trivial and $N\not\cong H$ (for example, $C_2\times C_3$, $N=C_2$). But I can't prove even that!
-
@Michael Hardy: (I ask this question out of interest, and is not meant to be hostile in any way, shape or form!) Why did you edit my question to add \operatorname before each operator? My understanding of \operatorname in Latex is that it is purely asthetic...was this the reason you changed it, or is there something more complex going on? (I would have sent you a private message to ask this, but I cannot seem to work out how too...) – user1729 Sep 14 '11 at 14:09
What's wrong with aesthetics? Writing the operators in the default font (which is kerned to make juxtaposed letters look like they're variables being multiplied) rather than an upright font with word kerning is akin to a spelling error. Usually, editing not-own questions to correct trivial typos is frowned upon, but an exception is generally recognized for TeXnicalities, because it's thought to be helpful to the asker (and future readers) to show how things ought to be formatted. – Henning Makholm Sep 14 '11 at 15:07
Nothing is wrong with aesthetics. As I said, I was just wondering if there was something more complex going on. – user1729 Sep 14 '11 at 15:09
To add to @Henning 's point: there is also a semantic aspect to it. While these purely aesthetic things may seem like quibbles, there is the point that the engines like css, markdown, mathjax, and whatever else is used on this site have no clue what they are displaying, they're just trying to make some sense out of what you feed them. By using proper formatting you ensure that whatever design tweaks are incorporated in the future, the results should still remain human readable. So, for example, your list would be better displayed if you added a blank after the minus signs. – t.b. Sep 14 '11 at 15:32
@jug: That obvious map has an interesting kernel (1-cocycles from G to H=G/N). A very well written and interesting paper is that of Curran: 'Automorphisms of Semidirect Products' (link: jstor.org/discover/10.2307/…) There you actually get a bit more of a generalization: if the subgroup N is characteristic, Curran's construction gives you the full automorphism group of the product G=HN, but if it is just normal and G fixes it set-wise, his construction also gives you the full automorphism groups. – user34168 Jun 21 '12 at 13:28
Let $G=N\times K$ and assume that both $N$ and $K$ are characteristic in $G$. It is easy to show $Aut(G) \cong Aut(N)\times Aut(K)$ since both $N$ and $K$ are characteristic in G. Since $G/N \cong K$ then $Aut(G/N) \cong Aut(K)$. Thus there is a natural epimorphism $\phi:Aut(G) \to Aut(G/N)$ with $ker(\phi) \cong Aut(K)$.
Now you can ask when are they both characteristic in $G$? Actually, one simple condition provide this: Let $N$ and $K$ be finite groups with relatively prime orders, and set $G=N\times K$. Then both $N$ and $K$ are characteristic in $G$.
And you offer an example $G=C_2\times C_3$ and $N=C_2$ then set $K=C_3$ since order of N and K are relatively prime, the result is immediate from above construction. | {"extraction_info": {"found_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": 0.8984573483467102, "perplexity": 437.81110415137954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1469257828282.32/warc/CC-MAIN-20160723071028-00087-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://rd.springer.com/article/10.1186/1754-0429-1-13 | PMC Physics B
, 1:13
# Ab-initiocalculations of spin tunneling through an indirect barrier
Open Access
Research article
## Abstract
We use a fully relativistic layer Green's functions approach to investigate spin-dependent tunneling through a symmetric indirect band gap barrier like GaAs/AlAs/GaAs heterostructure along [100] direction. The method is based on Linear Muffin Tin Orbitals and it is within the Density Functional Theory (DFT) in the Local Density Approximation (LDA). We find that the results of our ab-initio calculations are in good agreement with the predictions of our previous empirical tight binding model [Phys. Rev. B, 075313 (2006)]. In addition we show the k||-dependence of the spin polarization which we did not previously include in the model. The ab-initio calculations indicate a strong k||-dependence of the transmission and the spin polarization due to band non-parabolicity. A large window of 25–50% spin polarization was found for a barrier of 8 AlAs monolayers at k|| = 0.03 2π/a. Our calculations show clearly that the appearance of energy windows with significant spin polarization depends mostly on the location of transmission resonances and their corresponding zeros and not on the magnitude of the spin splitting in the barrier.
PACS Codes: 71.70.Ej, 71.15.Mb, 71.55.Eq
## Keywords
GaAs Spin Polarization Local Density Approximation Fano Resonance Conduction Band Minimum
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
## Background
The possibility of spin-polarized transmission through a zinc-blende semiconductor symmetric barrier was investigated in a few previous articles [1, 2, 3]. In Ref. [1] only direct band gap materials were considered for the barrier. Nevertheless, many zinc-blende wide band gap semiconductors are indirect, therefore the indirect tunneling through a barrier must also be considered. This feature was addressed in Ref. [2]. The authors argued that the linear-k spin-orbit splitting at X point will induce larger effect than the k3-term at Γ point [2]. They found an energy window around the top of the barrier, where the effect arising from the linear-k term at X point is larger than the effect stemming from the k3-term at Γ point.
Mishra et al. made no reference to the importance of the interaction between the discrete states at the X valley in the barrier and the continuum [2]. In their work, the authors considered the electrons close to the top of the indirect barrier, where the indirect tunneling becomes dominant [2]. In our previous work [3] we have shown that, at these energies, the Γ-X mixing plays a greater role than Mishra et al. suggested [2]. We have also considered the Fano resonances that occur because of the interaction between the discrete states at the X point in the barrier and the continuum at the Γ point in the contacts. To be more precise, in such systems an incident electron can tunnel either directly through evanescent Γ state in the barrier or through quasi-bound X state in the barrier. The resonance occurs due to resonant tunneling through quasi-bound X state, while the anti-resonance occurs due to destructive interference between the two channels of electron transmission [4]. Below or above the resonance the two channels are out of phase. The zero in the transmission occurs whenever the amplitude magnitudes of the two channels become equal. In our previous paper [3] we have used a realistic empirical tight binding model to give a unified description of the spin dependent tunneling through an indirect symmetric barrier with the GaAs/AlAs/GaAs heterostructure as an example. We showed several interesting aspects of such process, with particular emphasis on the large energy windows of spin polarization that can be obtained when appropriate conditions are satisfied.
Although our study presented quantitative results, several simplifications were made. The tight-binding Hamiltonian assumes parabolic conduction bands and it does not distinguish between X1 and X3 states [3], whereas the conduction band minimum is at Δ, along Γ-X line. Moreover, the complex structure of spin-dependent evanescent states in the band gap was completely ignored. This complicated structure of spin-dependent evanescent states might play an important role in the spin-dependent transport due to the meV scale on which the spin splittings occur [5, 6, 7]. In the present paper we extend the previous work [3] by reporting the results of the first ab-initio multiband study of spin-dependent tunneling through a zinc-blende barrier. The calculations are performed at the LDA level of the DFT with its well-known shortcomings regarding the band gaps and the effective masses. From the point of view of device modeling this may constitute a handicap; tools like sp3d5s*[8] full band semi-empirical tight-binding would perform a better job in reproducing band gaps, effective masses, and complex and imaginary bands. In this work we are checking if the main points made in the previous work [3] (namely, the criteria needed for obtaining large energy windows with significant spin polarization) are still valid in the context of truly multi-band calculations. The current calculations take into account the full band structure of the GaAs/AlAs/GaAs [100] heterostructure (including the spin dependency of the imaginary bands in the barrier) and, unlike our previous two-band tight-binding calculations, the ab-initio multiband calculations show that the nonparabolicity should be also considered in order to obtain large spin polarization.
## Results and discussion
Our approach is based on the fully-relativistic first-principles transport method. The method is based on the Green's function representation of the tight-binding linear muffin-tin orbital (TB-LMTO) basis in the atomic spheres approximation [9]. Within the relativistic formulation of the Local Spin Density Approximation (LSDA) in which only the spin component of the current density is taken into account [10], inside each atomic sphere we solve the Kohn-Sham Dirac equation [11]
(1)
where,
(2)
Here, is the vector of Pauli matrices, is the momentum operator, and is the unit vector in the direction of the effective magnetic field inside the Muffin-Tin (MT) sphere. The energy E is referenced to the total relativistic energy W = mc2+E. The effective magnetic field in equation (1) can be found as B(r) = (V(r)+V)(r)/2 [12]. The solutions of the Kohn-Sham Dirac equation are linear combinations of bispinors:
(3)
(4)
Here, Ω κμ () are the spin spherical harmonics, μ is the projection of the total angular momentum and κ is the relativistic quantum number: κ2 = J(J + 1) + .
Within the fully relativistic LMTO method the boundary conditions at the MT radius s are given in the form of two matrix equations [13, 14]
(5)
(6)
where, N(E), P(E), D(E) and g(E) are 2 × 2 matrices for each value of κ, μ and site R. N(E) and P(E) are arbitrary matrices defined by the boundary conditions, with elements the κ, μ components of the so called potential and normalization functions defined in the scalar relativistic LMTO method [15]. D(E, s) and g(E, s) are matrices with elements the κ, μ components of logarithmic derivative and large component of the wave function. The primary difference of this work with previous fully relativistic formulations of LMTO method is that we use the third-order parametrization of the potential functions [16, 17]: the radial amplitudes are expanded up to quadratic terms in linearization energy εν,
(7)
The Green's function of the layered system is constructed by the principal-layer technique [13]. The layers from -∞ to 0 and from N + 1 to ∞ are the contacts, while the layers from 1 to N are the active layers. Then, the transmission coefficient within the Landauer-Büttiker approach [18] can be calculated as
(8)
where,
B p (k||, E) = i p (k||, z+) - Γ p (k||, z-)]
with p = 1,N, z± = E ± and
(10)
The surface Green's functions of the electrodes are constructed scalar-relativistically, which allows us to decompose the conductance into spin-conserving and spin-flip components [19]. g1,Nand gN,1 are the upper right corner and lower left corner components of the auxiliary Green's function matrix
g = [P-S]-1
Here, P is the fully relativistic potential function (6) and S the tridiagonal matrix of scalar relativistic structure constants [15].
The heterostructure is 'grown' in the [100] direction. We have tried two different geometries: one where the left semi-infinite GaAs is separated from the right semi-infinite GaAs by 4 monolayers of AlAs and another where the separation is 8 monolayers. The self-consistent charge distribution is achieved within scalar-relativistic TB-LMTO calculations for GaAs/AlAs heterostructures treated using supercells with 6 monolayers of GaAs separated by the number of AlAs monolayers that correspond to the above mentioned geometries. The transmission is calculated at zero bias with lateral k|| taken in the [001] direction.
In GaAs we included only Ga 4d orbitals, which provide a slightly larger fundamental band gap than it is normally predicted within LDA (when Ga 3d orbitals are included in the basis). The important part of the calculated band structures of GaAs and AlAs are shown in Fig. 1. The GaAs direct band gap is 0.53 eV compared to the experimental value of 1.52 eV, the AlAs direct (Γ-Γ) and indirect (fundamental) gaps are 2.22 eV and 1.25 eV, respectively, compared to the experimental values of 3.13 eV and 2.23 eV, respectively. We would like to point out that the energies examined in this work are in the vicinity of conduction band minimum. Even though the band gaps are smaller than the experimental they are big enough to ensure that the complex band structure around conduction minimum has the correct character [19]. Based on this justification, LDA based transport calculations through heterostructures that include semiconductors have been used before [20] and revealed important information that is impossible to obtain from a model calculation. The effective mass of incoming electrons is smaller than the experimental value and this is an inherent problem of the LDA; to this date the proper treatment of electron-electron correlations in ab-initio transport methods has not been addressed and to our knowledge and experience it is a very difficult problem on it's own. In fact, even the most sophisticated techniques, like GW (G = Green's function, W = screened Coulomb interaction) approximation, cannot match the experimental effective mass and ad-hoc adjustments are required [21]; moreover these sophisticated techniques require heavy computational resources hence they are used only in bulk geometry. The use of such techniques in layered or supercell geometry is currently out of question. Also, ad-hoc techniques are known to have difficulty in matching simultaneously both the band gap and effective mass to the experiment. For example, in Ref. [22], while the band gap of GaAs is correct, the conduction band minimum effective mass is off by 33%. In the problem we are currently examinining the GaAs electron effective mass controls only the width of the Fano resonances. As we will explain straightaway, several other parameters that are related to our problem are predicted well. The valence band offset is 0.36 eV in good agreement with previously calculated from the individual bulk materials [23]. The conduction band offset (Γ-X) is 0.37 eV. Many important features of the tunneling relevant to our discussion are controlled by the conduction band offset and the Dresselhaus splitting in the vicinity of X point of AlAs [3]. In the heterostructure there is a charge transfer that increases the valence band offset to 0.53 eV and therefore we obtain a value of 0.2 eV for the conduction band offset, which is very close to the value used in our empirical tight-binding calculations (0.16 eV) [3]. The coefficient β for the linear Dresselhaus splitting at X point (ΔE = βk||)comes out at 0.108 eVÅ. There are no available experimental estimates for β. In Ref. [21] the LDA bands and Dresselhaus splittings where compared to the predictions of the 'scaled' Quasiparticle self-consistent GW (QSGW) method which gives very accurate predictions of band eigenvalues and eigenvectors. It was found that both methods agree very well at the X point. Not surprisingly, the agreement is also good for the transverse and longitudinal effective masses for AlAs. The LDA values for transverse and longitudinal masses are 0.232 m0 and 0.836 m0, respectively, while the QSGW method predicts 0.225 m0 and 0.738 m0, respectively. While β dictates the magnitude of the spin splitting of the quasi-bound levels at X, the effective masses together with the barrier width dictate their energy position (hence, separation). Therefore, despite the apparent shortcoming of the LDA to give an overall reliable prediction of the conduction band structure of the materials involved here, separate aspects that are important in the process of indirect tunneling are predicted well.
In Fig. 2 we show the transmission for a heterostructure with 4 monolayers of AlAs. For k|| = 0.03 2π/a the tunneling starts at approximately -0.4 eV (for k|| = 0 it would start at approximately -0.5 eV). Below -0.3 eV the tunneling transmission exhibits a typical direct tunneling behavior, and the spin polarization is insignificantly small. Between -0.3 and -0.2 eV the tunneling electron experiences the first Fano resonance-antiresonance due to the interaction with the discrete levels in the X valley well; the resonances and zeros for the two spin channels occur at slightly different energies, resulting in the sharp peaks of spin polarization seen in the second panel of Fig. 2. A second resonance-antiresonance occurs just below 0.05 eV, resulting in another spin polarization 'hot spot'. Generally, for the given values of k|| and barrier size we find only 'hot spots' of spin polarization in the vicinity of Fano resonances, but no energy windows of large spin polarization. In Fig. 3 we show the transmission for the same values of k|| but for a heterostructure with 8 monolayers of AlAs. For k|| = 0.03 2π/a, there is a 0.05 eV energy window, around -0.2 eV, where the spin polarization of the transmission is approximately 25%, but no such window is observed for k|| = 0.06 2π/a. The Dresselhaus splitting is bigger for the latter value of k||, but the spin polarization of the transmission coefficient is smaller. This non-trivial behavior was predicted by the empirical tight-binding model that we used previously: the well separated resonances below and above -0.2 eV for k|| = 0.03 2π/a make the polarization window relatively wide [3]. As it was predicted by Ref. [3], wider barriers provide better conditions for well separated resonances and a proper order of corresponding antiresonances. In contrast, for k|| = 0.06 2π/a the transmission coefficient is very different due to nonparabolicity of the bands, (parabolic bands would provide more or less similar transmission coefficient for different k||) thereby the spin polarization is less efficiently filtered in the barrier. The same pattern is also observed in Fig. 2, i.e., there are very different transmission coefficients for different values of k||. Therefore, our ab-initio calculations show that the full k||-dependence should be considered in order to have a more realistic picture of the spin polarization.
We make a comment on the ab-initio method that we use in this paper. The surface Green functions of the left/right semi-infinite GaAs regions are scalar-relativistic, thus the Dresselhaus term is not taken into account in there. We consider also GaAs layers in the active region where the Hamiltonian is fully relativistic, but the transmission is for an electron that starts at the left semi-infinite and ends at the right semi-infinite region. However, we already questioned the importance of the Dresselhaus term in contacts [24]. The related calculations [25] show that this term has a minor effect on the overall spin polarization with a much greater role played by the current operator. In our ab-initio calculations the current operator is properly taken into account.
Finally, we comment about experimental aspects of the present subject. An experimental setup would measure currents with contributions from all other energy and momentum values that, in principles, will wash out the spin polarization. For instance, electronic states with the same energy, corresponding to k|| and -k||, will have opposite spins, thus the total current will carry the same amount of spin-up and spin-down. To fix these aspects we already suggested [3] that the electrons should be injected from a resonant tunneling diode in order to focus the electrons in the k||-plane [26]. Moreover, by applying a voltage bias in the k||-plane of the emitter [27, 28], the isotropy of k|| will be further broken, thus a net spin polarization will be induced in the energy window.
## Conclusion
In conclusion, we applied a fully-relativistic first-principles transport method to the spin-dependent tunneling through a GaAs/AlAs/GaAs [100] heterostructure. The method is based on the Green's function representation of the Tight-Binding Linear Muffin-Tin Orbital basis in the Atomic Spheres Approximation. The calculations were performed in the Local Spin Density Approximation within the Density Functional Theory.
Considering the full band structure of the GaAs/AlAs/GaAs [100] system, these calculations confirm previous general predictions made with a simplified empirical tight-binding method. Namely, in order to have windows with large spin polarizations, two conditions need to be satisfied: the first is to have well separated resonances such that their corresponding anti-resonances do not interact with each other and the second is that the energy order of the resonances in the spin channels have to be the same as the energy order of their corresponding zeros.
We found a large energy window of 25–50% spin polarization for a barrier of 8 AlAs monolayers at k|| = 0.03 2π/a, but there is no such energy window at k|| = 0.06 2π/a. Our study suggests that, in order to find energy windows with large spin polarization, a detailed knowledge of the energy dependence on k|| and spin must be considered.
## Notes
### Acknowledgements
The authors gratefully acknowledge the financial support from the Office of Naval Research and from Romanian Ministry of Education and Research.
## Supplementary material
13067_2007_13_MOESM1_ESM.pdf (41 kb)
Authors’ original file for figure 1
13067_2007_13_MOESM2_ESM.pdf (28 kb)
Authors’ original file for figure 2
13067_2007_13_MOESM3_ESM.pdf (29 kb)
Authors’ original file for figure 3
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## Authors and Affiliations
• Athanasios N Chantis
• 1
• Titus Sandu
• 2
• Jialei L Xu
• 3
1. 1.Theoretical Division, Los Alamos National LaboratoryLos AlamosUSA
2. 2.International Centre of BiodynamicsDistrict 6Romania
3. 3.Arizona State UniversityTempeUSA | {"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": 0.8658689260482788, "perplexity": 1962.5861917691734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589980.6/warc/CC-MAIN-20180718002426-20180718022426-00585.warc.gz"} |
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[1] Abdulla Eid and Iwan Duursma. Smooth Embeddings for the Suzuki and Ree Curves. Contemporary Mathematics 637 (2015) 251-291. Book volume table of contents View Article: PDF [2] Adam James, Kay Magaard and Sergey Shpectorov. The lift invariant distinguishes components of Hurwitz spaces for $A_5$. Proc. Amer. Math. Soc. 143 (2015) 1377-1390. Abstract, references, and article information View Article: PDF [3] Evan M. Bullock. Irreducibility and stable rationality of the loci of curves of genus at most six with a marked Weierstrass point. Proc. Amer. Math. Soc. 142 (2014) 1121-1132. Abstract, references, and article information View Article: PDF [4] Antonio F. Costa, Milagros Izquierdo and Ana M. Porto. Maximal and Non-maximal NEC and Fuchsian groups uniformizing Klein and Riemann surfaces. Contemporary Mathematics 629 (2014) 107-118. Book volume table of contents View Article: PDF [5] Marco Matone and Roberto Volpato. Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula. Proc. Amer. Math. Soc. 141 (2013) 2575-2587. Abstract, references, and article information View Article: PDF [6] Evan M. Bullock. Subcanonical points on algebraic curves. Trans. Amer. Math. Soc. 365 (2013) 99-122. Abstract, references, and article information View Article: PDF [7] Enrico Arbarello and Gabriele Mondello. Two remarks on the Weierstrass flag. Contemporary Mathematics 564 (2012) 137-144. Book volume table of contents View Article: PDF [8] Dror Varolin. The Riemann-Roch Theorem. Graduate Studies in Mathematics 125 (2011) 211-221. Book volume table of contents View Article: PDF [9] Dror Varolin. Uniformization. Graduate Studies in Mathematics 125 (2011) 165-176. Book volume table of contents View Article: PDF [10] Dror Varolin. Embedding Riemann surfaces. Graduate Studies in Mathematics 125 (2011) 197-210. Book volume table of contents View Article: PDF [11] Dror Varolin. Potential theory. Graduate Studies in Mathematics 125 (2011) 115-131. Book volume table of contents View Article: PDF [12] Dror Varolin. H\"ormander's Theorem. Graduate Studies in Mathematics 125 (2011) 177-196. Book volume table of contents View Article: PDF [13] Dror Varolin. Functions on Riemann surfaces. Graduate Studies in Mathematics 125 (2011) 37-60. Book volume table of contents View Article: PDF [14] Dror Varolin. Complex analysis. Graduate Studies in Mathematics 125 (2011) 1-20. Book volume table of contents View Article: PDF [15] Dror Varolin. Riemann surfaces. Graduate Studies in Mathematics 125 (2011) 21-36. Book volume table of contents View Article: PDF [16] Dror Varolin. Abel's Theorem. Graduate Studies in Mathematics 125 (2011) 223-232. Book volume table of contents View Article: PDF [17] Dror Varolin. Complex differential forms. Graduate Studies in Mathematics 125 (2011) 87-100. Book volume table of contents View Article: PDF [18] Dror Varolin. Harmonic forms. Graduate Studies in Mathematics 125 (2011) 145-164. Book volume table of contents View Article: PDF [19] Dror Varolin. Riemann Surfaces by Way of Complex Analytic Geometry. Graduate Studies in Mathematics 125 (2011) MR MR2798295. Book volume table of contents [20] Dror Varolin. Complex line bundles. Graduate Studies in Mathematics 125 (2011) 61-86. Book volume table of contents View Article: PDF [21] Dror Varolin. Calculus on line bundles. Graduate Studies in Mathematics 125 (2011) 101-113. Book volume table of contents View Article: PDF [22] Dror Varolin. Solving $\overline {\partial }$ with smooth data. Graduate Studies in Mathematics 125 (2011) 133-143. Book volume table of contents View Article: PDF [23] Scott Ahlgren, Nadia Masri and Jeremy Rouse. Vanishing of modular forms at infinity. Proc. Amer. Math. Soc. 137 (2009) 1205-1214. MR 2465641. Abstract, references, and article information View Article: PDF This article is available free of charge [24] William M. Goldman and Eugene Z. Xia. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces. Memoirs of the AMS 193 (2008) MR 2400111. Book volume table of contents [25] Eduardo Esteves and Parham Salehyan. Limit Weierstrass points on nodal reducible curves. Trans. Amer. Math. Soc. 359 (2007) 5035-5056. MR 2320659. Abstract, references, and article information View Article: PDF This article is available free of charge [26] Robert M. Guralnick and John Shareshian. Symmetric and alternating groups as monodromy groups of Riemann surfaces. I. Generic covers and covers with many branch points. Memoirs of the AMS 189 (2007) MR 2343794. Book volume table of contents [27] Ta Thi Hoai An and Julie Tzu-Yueh Wang. Unique range sets and uniqueness polynomials for algebraic curves. Trans. Amer. Math. Soc. 359 (2007) 937-964. MR 2262838. Abstract, references, and article information View Article: PDF This article is available free of charge [28] Martine Girard. The group of Weierstrass points of a plane quartic with at least eight hyperflexes. Math. Comp. 75 (2006) 1561-1583. MR 2219046. Abstract, references, and article information View Article: PDF This article is available free of charge [29] Marco Matone. Eigenfunctions of the Laplacian acting on degree zero bundles over special Riemann surfaces. Trans. Amer. Math. Soc. 356 (2004) 2989-3004. MR 2052938. Abstract, references, and article information View Article: PDF This article is available free of charge [30] Philip L. Bowers and Kenneth Stephenson. Uniformizing dessins and Bely\u \i \ maps via circle packing. Memoirs of the AMS 170 (2004) MR 2053391. Book volume table of contents
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Comments: Email Webmaster © Copyright , American Mathematical Society Contact Us · Sitemap · Privacy Statement | {"extraction_info": {"found_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": 0.8838441967964172, "perplexity": 3622.3692992189085}, "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-2016-26/segments/1466783397797.77/warc/CC-MAIN-20160624154957-00001-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-equations/132492-ordinary-de-problem.html | # Math Help - Ordinary DE Problem:
1. ## Ordinary DE Problem:
So I think I have found the solution to the following problem, but it feels like I just barely patched it together, if anyone could take a quick peek and make sure I wasn't making up my own math, I would be very appreciative!
When a liquid drop falls, it increases in mass. Mass at time $t$ is $m(t)$. Rate of growth of the mass of the liquid is $km(t)$ for positive constant $k$. When we apply the Laws of Motion, we get $(mv)' = gm(t)$, where $v$ is the velocity of the liquid drop, and $g$ is the acceleration due to gravity. The terminal velocity of the reaindrop is $\lim_{t \rightarrow \infty} v(t)$. Find an expression for the terminal velocity in terms of $g$ and $k$.
So I started with the knowledge that the rate of change of mass is km(t).
$\frac{dm}{dt} = km(t)$
$\frac{dm}{m(t)}=kdt$
$\ln|m(t)| = kt + c$
$m(t) = Ce^{kt}$
So now I used this result in conjunction with the information : $(m(t)v(t))'=gm(t)$.
$(m(t)v(t))' = gm(t)$
$v(t)m(t) = \int gm(t) dt = g \int m(t) dt$
Antiderivative of $m(t): \ \int m(t) dt = \int Ce^{kt} dt = \frac{C}{k}e^{kt} + Z$
$v(t)m(t) = \frac{gC}{k}e^{kt} + gZ$
$v(t) = \frac{\frac{gC}{k}e^{kt} + gZ}{Ce^{kt}}$
$v(t) = \frac{g}{k} + \frac{gZ}{Ce^{kt}}$
Now to find the terminal velocity, we take the limit: $\lim_{t \rightarrow \infty} v(t)$.
$Terminal \ Velocity = \lim_{t \rightarrow \infty} v(t) = \lim_{t \rightarrow \infty} (\frac{g}{k} + \frac{gZ}{Ce^{kt}}) = \frac{g}{k}$
Does this make good sense? Thanks!
2. Originally Posted by Kasper
So I think I have found the solution to the following problem, but it feels like I just barely patched it together, if anyone could take a quick peek and make sure I wasn't making up my own math, I would be very appreciative!
So I started with the knowledge that the rate of change of mass is km(t).
$\frac{dm}{dt} = km(t)$
$\frac{dm}{m(t)}=kdt$
$\ln|m(t)| = kt + c$
$m(t) = Ce^{kt}$
So now I used this result in conjunction with the information : $(m(t)v(t))'=gm(t)$.
$(m(t)v(t))' = gm(t)$
$v(t)m(t) = \int gm(t) dt = g \int m(t) dt$
Antiderivative of $m(t): \ \int m(t) dt = \int Ce^{kt} dt = \frac{C}{k}e^{kt} + Z$
$v(t)m(t) = \frac{gC}{k}e^{kt} + gZ$
$v(t) = \frac{\frac{gC}{k}e^{kt} + gZ}{Ce^{kt}}$
$v(t) = \frac{g}{k} + \frac{gZ}{Ce^{kt}}$
Now to find the terminal velocity, we take the limit: $\lim_{t \rightarrow \infty} v(t)$.
$Terminal \ Velocity = \lim_{t \rightarrow \infty} v(t) = \lim_{t \rightarrow \infty} (\frac{g}{k} + \frac{gZ}{Ce^{kt}}) = \frac{g}{k}$
Does this make good sense? Thanks!
Without checking your algebra it looks OK (the fact that the terminal velocity does not depend on the constants of integration is encouraging)
The only comment I would make is that you made hard work of the first part, you should recognise the form of the solution of y'=ky straight off without having to go through seperation of variables.
CB
3. Originally Posted by CaptainBlack
Without checking your algebra it looks OK (the fact that the terminal velocity does not depend on the constants of integration is encouraging)
The only comment I would make is that you made hard work of the first part, you should recognise the form of the solution of y'=ky straight off without having to go through seperation of variables.
CB
Ah, I've been doing it all the time just to practice the method of seperation, but I think your right. Regardless, thanks for the confirmation! | {"extraction_info": {"found_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": 36, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8711492419242859, "perplexity": 244.75058574811908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398450715.63/warc/CC-MAIN-20151124205410-00071-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://cs.stackexchange.com/questions/6456/how-many-different-max-heaps-exist-for-a-list-of-n-integers | How many different max-heaps exist for a list of n integers?
How many different max-heaps exist for a list of $$n$$ integers?
Example: list [1, 2, 3, 4]
The max-heap can be either 4 3 2 1:
4
/ \
3 2
/
1
or 4 2 3 1:
4
/ \
2 3
/
1
You can find a not-so-nice recursion in the OEIS database. Basically the idea is as follows. The root of an $n$-ary heap is always the maximum. The two subtrees hanging off the root are again maxheaps. Their size depends on $n$, is a bit tedious to compute the sizes $n_1,n_2$ (see the OEIS entry), clearly $n_1+n_2=n-1$. We can now pick, which elements go to the left heap and which go to the right heap. There are ${n-1 \choose n_1}$ ways how to distribute the elements. This gives the recurrence
$$a(n)= {n-1 \choose n_1} a(n_1)a(n_2).$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8298258781433105, "perplexity": 615.8157292156731}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571987.60/warc/CC-MAIN-20220813202507-20220813232507-00614.warc.gz"} |
https://www.physicsforums.com/threads/finding-the-force-to-break-static-friction.809510/ | # Finding the force to break static friction
1. Apr 19, 2015
### nevererdofit
1. The problem statement, all variables and given/known data
Samantha and Rebekah each push on a stack of doughnuts equally. If the coefficient of static friction is .28 and the box has a mass of 18kg, what force does each push individually to break static friction? Once it's moving, the coefficient of kinetic friction is .17, how much will the doughnuts be accelerated onto the floor?
mass of object = 18kg
coefficient of friction (static) = .28
coefficient of friction (kinetic) = .17
2. Relevant equations
The only one I know that is obvious to use is f = ma, not sure how to apply it to this context
3. The attempt at a solution
On the x and y plane, I know the force diagram reads normal force traveling up, force of gravity traveling down, Applied force traveling right, and force of friction traveling left.
Last edited: Apr 19, 2015
2. Apr 19, 2015
### paisiello2
Do you know the formula for static and kinetic friction?
3. Apr 19, 2015
### nevererdofit
I was never taught the formulas for each type of friction no
4. Apr 19, 2015
### paisiello2
5. Apr 19, 2015
### nevererdofit
So after trying this is how I attempt it
(I assume gravity as 10m/s/s)
I know normal force = mg so fn = (18)(10) fn = 180N
then
f = (.28)x(180)
f = 50.4N
then I assume you divide the answer in two as it asks for the individual force, so
f = 25.2N
to find the acceleration you just use f = ma substituting your found force
25.2 = (18)a
a = 1.4m/s/s
am I correct?
6. Apr 19, 2015
### paisiello2
Not quite. Is the box sliding or not?
7. Apr 19, 2015
### nevererdofit
they both push on the stack of doughnuts, so I believe it is sliding. Really though you are just trying to find the force that was needed to break the static force, or when it wasn't in motion. Then the second part of the question just wanted the acceleration of their push
8. Apr 19, 2015
### paisiello2
Yes, so when you found the first answer did you assume the box was sliding or not?
9. Apr 19, 2015
### nevererdofit
yes I did assume the box was sliding as there was force being emitted on it
10. Apr 19, 2015
### paisiello2
Well, you are contradicting yourself because you also said there was no motion. Sliding= motion.
11. Apr 19, 2015
### billy_joule
That looks correct.
A couple problems with this part. You've ignore kinetic friction force and only used one of the applied forces.
Draft saved Draft deleted
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http://math.libretexts.org/Core/Calculus/Vector_Calculus/4%3A_Integration_in_Vector_Fields/4.1%3A_Differentiation_and_Integration_of_Vector_Valued_Functions | $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
# 4.1: Differentiation and Integration of Vector Valued Functions
### Calculus of Vector Valued Functions
The formal definition of the derivative of a vector valued function is very similar to the definition of the derivative of a real valued function.
Definition
The Derivative of a Vector Valued Function
Let $$r(t)$$ be a vector valued function, then
$r'(t) = \lim_{h \rightarrow 0} \dfrac{r(t+h)-r(t)}{h}$
Because the derivative of a sum is the sum of the derivative, we can find the derivative of each of the components of the vector valued function to find its derivative.
Example 1
$\dfrac{d}{dt} (3 \hat{\text{i}} + \sin t \hat{\text{j}}) = \cos t \hat{\text{j}}$
$\dfrac{d}{dt} \left(3t^2\, \hat{\text{i}} + \cos{(4t)}\, \hat{\text{j}} + te^t \, \hat{\text{k}} \right) = 6t \, \hat{\text{i}} -4\sin{(t)}\,\hat{\text{j}} + (e^t + te^t)\, \hat{\text{k}}$
### Properties of Vector Valued Functions
All of the properties of differentiation still hold for vector values functions. Moreover because there are a variety of ways of defining multiplication, there is an abundance of product rules.
Suppose that $$\text{v}(t)$$ and $$\text{w}(t)$$ are vector valued functions, $$f(t)$$ is a scalar function, and $$c$$ is a real number then
1. $$\dfrac{d}{dt} \left( \text{v}(t) + \text{w}(t) \right) = \dfrac{d}{dt}\text{v}(t) + \dfrac{d}{dt} \text{w}(t)$$
2. $$\dfrac{d}{dt} c\text{v}(t) = c\, \dfrac{d}{dt} \text{v}(t)$$
3. $$\dfrac{d}{dt}(f(t) \text{v}(t)) = f '(t) \text{v}(t) + f(t) \text{v}(t)'$$
4. $$\left( v(t) \cdot \text{w}(t) \right)' = \text{v}'(t) \cdot \text{w}(t)+ \text{v}(t) \cdot \text{w}'(t)$$
5. $$(v(t) \times \text{w}(t))' = \text{v}'(t) \times \text{w}(t) + \text{v}(t) \times \text{w}'(t)$$
6. $$\dfrac{d}{dt} v(f(t)) = \text{v}(t)'(f(t)) f '(t)$$
Example 2
Show that if $$r$$ is a differentiable vector valued function with constant magnitude, then
$r \cdot r' = 0$
Solution
Since $$r$$ has constant magnitude, call its magnitude $$k$$,
$k^2 = |r|^2 = r \cdot r$
Taking derivatives of the left and right sides gives
$0 = (r \cdot r)' = r' \cdot r + r \cdot r'$
$= r \cdot r' + r \cdot r' = 2r \cdot r'$
Divide by two and the result follows
### Integration of vector valued functions
We define the integral of a vector valued function as the integral of each component. This definition holds for both definite and indefinite integrals.
Example 2
Evaluate
$\int (\sin t)\, \hat{\textbf{i}} + 2t\, \hat{\textbf{j}} - 8t^3 \, \hat{\textbf{k}} \; dt$
Solution
Just take the integral of each component
$\int (\sin t)\,dt \, \hat{\textbf{i}} + \int 2\,t \, dt \, \hat{\textbf{j}} - \int 8\,t^3 \,dt \, \hat{\textbf{k}}$
$= (-\cos t + c_1)\, \hat{\textbf{i}} + (t^2 + c_2)\, \hat{\textbf{j}} + (2\,t^4 + c_3)\, \hat{\textbf{k}}$
Notice that we have introduce three different constants, one for each component. | {"extraction_info": {"found_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": 0.9930139183998108, "perplexity": 186.89538896479266}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560280746.40/warc/CC-MAIN-20170116095120-00336-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.media4math.com/NY-8.EE.4 | ## NY-8.EE.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
There are 43 resources.
Title Description Thumbnail Image Curriculum Topics
## Definition--Scientific Notation
### Definition--Scientific Notation This is part of a collection of definitions on various math topics.
Numerical Expressions, Variable Expressions
## INSTRUCTIONAL RESOURCE: Algebra Application: Why Are Wildfires So Dangerous?
### INSTRUCTIONAL RESOURCE: Algebra Application: Why Are Wildfires So Dangerous? In this Algebra Application, students learn about wildfires and the measurement of air quality. The math topics covered include: Scientific notation, Rates, Density, Data Analysis. The specific focus of this investigation is the health hazards from wildfire smoke.
Laws of Exponents, Applications of Ratios, Proportions, and Percents
## Math in the News: Issue 77--Voyager Breaks Free
### Math in the News: Issue 77--Voyager Breaks Free September 2013. In this issue we look at Voyager 1's amazing milestone: Leaving our Solar System and entering interstellar space. This provides an excellent opportunity to explore large numbers using scientific notation.
Numerical Expressions, Variable Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 1
### Worksheet: Converting Numbers from Scientific Notation, Set 1 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 10
### Worksheet: Converting Numbers from Scientific Notation, Set 10 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 11
### Worksheet: Converting Numbers from Scientific Notation, Set 11 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 12
### Worksheet: Converting Numbers from Scientific Notation, Set 12 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 13
### Worksheet: Converting Numbers from Scientific Notation, Set 13 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 14
### Worksheet: Converting Numbers from Scientific Notation, Set 14 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 15
### Worksheet: Converting Numbers from Scientific Notation, Set 15 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 16
### Worksheet: Converting Numbers from Scientific Notation, Set 16 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 17
### Worksheet: Converting Numbers from Scientific Notation, Set 17 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 18
### Worksheet: Converting Numbers from Scientific Notation, Set 18 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 19
### Worksheet: Converting Numbers from Scientific Notation, Set 19 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 2
### Worksheet: Converting Numbers from Scientific Notation, Set 2 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 20
### Worksheet: Converting Numbers from Scientific Notation, Set 20 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 3
### Worksheet: Converting Numbers from Scientific Notation, Set 3 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 4
### Worksheet: Converting Numbers from Scientific Notation, Set 4 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 5
### Worksheet: Converting Numbers from Scientific Notation, Set 5 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 6
### Worksheet: Converting Numbers from Scientific Notation, Set 6 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 7
### Worksheet: Converting Numbers from Scientific Notation, Set 7 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 8
### Worksheet: Converting Numbers from Scientific Notation, Set 8 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Converting Numbers from Scientific Notation, Set 9
### Worksheet: Converting Numbers from Scientific Notation, Set 9 This is part of a collection of math worksheets on the topic of converting numbers from scientific notation.
Numerical Expressions
## Worksheet: Writing Numbers in Scientific Notation, Set 1
### Worksheet: Writing Numbers in Scientific Notation, Set 1 This is part of a collection of math worksheets on the topic of scientific notation.
Numerical Expressions
## Worksheet: Writing Numbers in Scientific Notation, Set 10
### Worksheet: Writing Numbers in Scientific Notation, Set 10 This is part of a collection of math worksheets on the topic of scientific notation.
Numerical Expressions | {"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": 0.9258008003234863, "perplexity": 3374.6015668267096}, "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-2023-14/segments/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00033.warc.gz"} |
http://www.computer.org/csdl/trans/ts/2006/09/e0642-abs.html | Subscribe
Issue No.09 - September (2006 vol.32)
pp: 642-663
ABSTRACT
This paper proposes a methodology and instrumentation infrastructure toward the reverse engineering of UML (Unified Modeling Language) sequence diagrams from dynamic analysis. One motivation is, of course, to help people understand the behavior of systems with no (complete) documentation. However, such reverse-engineered dynamic models can also be used for quality assurance purposes. They can, for example, be compared with design sequence diagrams and the conformance of the implementation to the design can thus be verified. Furthermore, discrepancies can also suggest failures in meeting the specifications. Due to size constraints, this paper focuses on the distribution aspects of the methodology we propose. We formally define our approach using metamodels and consistency rules. The instrumentation is based on Aspect-Oriented Programming in order to alleviate the effort overhead usually associated with source code instrumentation. A case study is discussed to demonstrate the applicability of the approach on a concrete example.
INDEX TERMS
UML, sequence diagram, reverse engineering, distribution, RMI, AspectJ, OCL.
CITATION
Lionel C. Briand, Yvan Labiche, Johanne Leduc, "Toward the Reverse Engineering of UML Sequence Diagrams for Distributed Java Software", IEEE Transactions on Software Engineering, vol.32, no. 9, pp. 642-663, September 2006, doi:10.1109/TSE.2006.96 | {"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": 0.8587272763252258, "perplexity": 2796.876560863724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096156.35/warc/CC-MAIN-20150627031816-00150-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.ucl.ac.uk/astrophysics/research/cosmology/euclid | Close
UCL Astrophysics Group
Home
# Euclid
Euclid is a proposed space telescope being developed with the European Space Agency (ESA) for launch around 2018. It is planned to observe the whole sky with the same image quality as the Hubble Space Telescope and measure the rainbow of colours (spectra) within millions of galaxies. These very high quality imaging observations will be used to investigate dark energy using cosmic lensing. The precise three dimensional positions of galaxies obtained from the spectra will be used to study the clustering of galaxies. The imaging science part of the mission is described in detail in the Euclid Imaging Consortium Science Book.
At the UCL Department of Physics and Astronomy, we are heavily involved in preparations for the imaging survey. We lead studies to determine the requirements cosmic lensing places on the optical design of the instrument. We are also studying the impact of choice of optical filters on the accuracy of the photometric redshift measurements and working on the requirements for spectroscopic training sets. Among other work, we have made forecasts for the cosmological constraints on dark energy in the presence of intrinsic alignments between galaxies that arise during galaxy formation.
Much of the current activity focuses on the core science which drives the mission design. However, it is clear that high resolution imaging of the entire extragalactic sky has tremendous value beyond the study of dark energy using cosmic lensing and galaxy clustering. This includes high resolution imaging of over two billion galaxies in the visible and infra-red parts of the spectrum which could, for example, be used to classify galaxies into ellipticals and spirals. | {"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": 0.8187820911407471, "perplexity": 921.5204356734358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585449.31/warc/CC-MAIN-20211021230549-20211022020549-00438.warc.gz"} |
https://rin.io/minutephysics-is-incorrect/ | # MinutePhysics is Naively Incorrect: Finding the Limit of a Geometric Series
I recently watched a video by MinutePhysics where he seems to prove that infinity = -1.
He took the sum of the infinite series, beginning with 1, where each term is double the previous term.
He claims that S = infinity (so far, so good). I would argue that S approaches infinity, but this is purely technical tricky business.
Then he doubles S, thus doubling all of it’s terms (still okay).
He then subtracts the two,
terms cancel, leaving S = -1.
This is a geometric series (each term gets multiplied by a common factor; in this case, the common factor is 2). If you take the common factor to be 1/2 instead, then you’ll have
Each term is getting smaller, approaching zero. If you calculate S, you’ll see that the partial sums approach 2:
Thus, the value 2 is the limit of the series. We call this type of series a convergent series, because the sum of the series converges to one number.
However, the example used by MinutePhysics is not a convergent series (the terms double each time). We call this a divergent series.
A divergent series needn’t sum to infinity. Divergent is an umbrella term that covers all series that are non-convergent. This includes periodic series (series that flip between two values), such as
The terms of this series will flip between the two values -1 and 1. Its sum will alternate between 0 and -1.
However, just because a series is divergent doesn’t mean you can’t give its sum a value. This is not the limit, for there is no limit that the sum approaches.
Let’s say you have a general geometric series; each term is being multiplied by a common factor, r.
You can use the same tricks as before to find a general limit for this series:
This reduces to
This trick only finds the limit of convergent series, because the limit of a divergent series doesn’t exist.
In other words, the limit has to exist in order to find it!
For the limit to exist, each term must be smaller than the last (the common ratio must be less than 1).
If you apply this formula to a convergent series, you’ll get the limit. When the common ratio was 1/2, the limit = 2.
If you apply this formula to a divergent series, you’ll get a value. But it won’t be the limit of the series, because the limit doesn’t exist.
Side Note: When you apply to the original example (the common ratio was 2), you’ll find the value -1. This isn’t the limit, because the limit doesn’t exist, it’s what the limit would be if it was a convergent series.
If the common ratio is -1, the sequence will flip between two values, for example, 6 and -6. When you add up the partial sums, these flip as well. You’ll get 6, then 0, then 6, then 0… If you apply , you’ll get the average of the partial sums (in this case, 3). Finding the average of partial sums is applicable to all series:
When you find the average of the partial sums of a convergent series, you’ll find that it coincides with the limit.
If you apply it to divergent series, you don’t get the limit; you get this average of partial sums.
Either way, this value has meaning.
So, what was the problem with the video Adding Past Infinity?
He used a method that can be applied to all geometric series.
When you apply this method to a convergent geometric series, you get the limit of the series.
MinutePhysics applied this method to a divergent geometric series (which is fine).
He got a value (which is fine).
But he claimed that this value was the limit of the series (which is not fine, because the limit doesn’t exist)! | {"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": 0.9646397233009338, "perplexity": 318.31688739333083}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571056.58/warc/CC-MAIN-20220809155137-20220809185137-00049.warc.gz"} |
https://mathoverflow.net/questions/231553/euler-characteristic-on-flat-families-of-quasi-projective-schemes | # Euler characteristic on flat families of quasi-projective schemes
Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent sheaf on $X$, flat over $\mathrm{Spec}(A)$. What can we say about the Euler characteristic of $\mathcal{F} \otimes \mathcal{O}_{X_t}$, $\chi(\mathcal{F} \otimes \mathcal{O}_{X_t})$ as $t$ varies over points in $\mathrm{Spec}(A)$? Is it upper-semi continuous, lower-semi continuous or constant? We know that if $\pi$ is projective then the Euler characteristic remains constant.
• It is typically neither upper semicontinuous nor lower semicontinuous. – Jason Starr Feb 18 '16 at 21:47
• @JasonStarr Is there any condition we can add on $\pi$ (not properness) which will ensure one of them? – user45397 Feb 18 '16 at 22:03
• @JasonStarr Is there any text or literature which deals with a similar question? – user45397 Feb 18 '16 at 22:25
• I am unaware of any reference that studies this without a properness hypothesis. – Jason Starr Feb 18 '16 at 23:57
I am not aware of such results in full generality, but I know that working without the properness assumtpion was in part the main motivation for Grothendieck to write SGA 2.
Let me focus on a related question (but not exactly the same). Let $\pi : X \rightarrow Y$ be a flat morphism of finite type with $Y$ a smooth variety over a field. Given a coherent sheaf $F$ on $X$ flay over $Y$, you would like to know if $h^i(X_t, F_t)$ might be upper semi-continous in some cases.
Essentially, what you have to answer are the following question:
Is the sheaf $R^i \pi_* F$ coherent?
If the answer to this question is yes, then I believe the semi-continuity of $h^i(X_t, F_t)$ is true under the flatness hypothesis (this becomes a linear algebra question if I remember correctly).
As Jason Starr mentionned, the well-known result is that the answer to this question is yes, if you assume $\pi$ proper.
If you want to drop the properness assumption, you have to add some other hypotheses for the coherence of $R^i \pi_* F$ to hold. (though I don't have a counter-example, I am pretty sure the cohrence does not hold without any assumption). In fact, many interesting results in SGA 2 address the coherence issue if you withdraw the properness hypothesis.
The price you have to pay is to add a depth hypothesis. In fact you will "compactify" $\pi$ from $\tilde{X}$ to $Y$. But you don't assume that $F$ comes from a coherent sheaf on $\tilde{X}$. What you want to know is when the sheaf $R^i j_* F$ is coherent (where $j : X \rightarrow \tilde{X}$ is the open immersion).
You have to make some assumptions on the depth of $F_x$ for $x \in X$.
Corollary $2.3$ and Theorem $3.1$ of expose $VIII$ in SGA 2 tell you what depth hypothesis you have to add to get some coherence results | {"extraction_info": {"found_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": 0.9365336298942566, "perplexity": 247.45666307345843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247492825.22/warc/CC-MAIN-20190219203410-20190219225410-00277.warc.gz"} |
http://raweb.inria.fr/rapportsactivite/RA2015/mistis/uid5.html | Application Domains
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
PDF e-Pub
## Section: Research Program
### Mixture models
Participants : Alexis Arnaud, Jean-Baptiste Durand, Florence Forbes, Aina Frau Pascual, Alessandro Chiancone, Stéphane Girard, Marie-José Martinez.
Key-words: mixture of distributions, EM algorithm, missing data, conditional independence, statistical pattern recognition, clustering, unsupervised and partially supervised learning.
In a first approach, we consider statistical parametric models, $\theta$ being the parameter, possibly multi-dimensional, usually unknown and to be estimated. We consider cases where the data naturally divides into observed data $y={y}_{1},...,{y}_{n}$ and unobserved or missing data $z={z}_{1},...,{z}_{n}$. The missing data ${z}_{i}$ represents for instance the memberships of one of a set of $K$ alternative categories. The distribution of an observed ${y}_{i}$ can be written as a finite mixture of distributions,
These models are interesting in that they may point out hidden variable responsible for most of the observed variability and so that the observed variables are conditionally independent. Their estimation is often difficult due to the missing data. The Expectation-Maximization (EM) algorithm is a general and now standard approach to maximization of the likelihood in missing data problems. It provides parameter estimation but also values for missing data.
Mixture models correspond to independent ${z}_{i}$'s. They have been increasingly used in statistical pattern recognition. They enable a formal (model-based) approach to (unsupervised) clustering. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "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": 0.9393220543861389, "perplexity": 2163.792598631277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188891.62/warc/CC-MAIN-20170322212948-00087-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.ideals.illinois.edu/handle/2142/71201 | ## Files in this item
FilesDescriptionFormat
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8209564.pdf (3MB)
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## Description
Title: Limit Theorems for Weakly Dependent Random Vectors Author(s): Dhompongsa, Sompong Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: In Chapter I we improve upon results on the almost sure approximation of the empirical process of weakly dependent random vectors, recently obtained by Berkes and Philipp and Philipp and Pinzur. For strongly mixing sequences we relax the bounds on the mixing rates, and for absolutely regular sequences we improve the error term. We also extend these results to random vectors which are functions of the given sequence as well as to random variables which are evaluated at lacunary sequences.In Chapter 2 we extend the uniform law of the iterated logarithm for classes of functions in Lip (alpha) ((alpha) > 1/2) evaluated at lacunary sequences and functions of mixing processes, due to Kaufman and Philipp. Here we relax the lacunarity condition and the mixing rates. Issue Date: 1982 Type: Text Description: 142 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982. URI: http://hdl.handle.net/2142/71201 Other Identifier(s): (UMI)AAI8209564 Date Available in IDEALS: 2014-12-16 Date Deposited: 1982
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http://www.12000.org/my_notes/l2hwin/ | ## Installation of Latex2html
June 22, 2013
### Contents
A plain text version is also available of this page which was generated using the following catdvi command 1
catdvi -e 1 -U index.dvi | sed -re "s/$U\+2022$/*/g" | sed -re "s/([^^[:space:]])\s+/\1 /g" > index.txt
The file is index.txt
### 1 Installation on Linux
Installation of Latex2html is very easy on Linux. Used package manager (GUI interface) and select latex2html package. It will install everything needed making sure that texlive is also installed using the package manager so that latex2html style files can be seen by latex.
Once latex2html is installed, I then created a file called $HOME/.latex2html-init This is my copy of my .latex2html-init latex2html-init which fixes some issues mentioned below. >which latex2html /usr/bin/latex2html In addition to the above, I had to edit the file /usr/share/latex2html/styles/more_amsmath.perl for the rendering bug described below. That is all! now latex2html is ready to be used on linux. ### 2 Installation on windows with MikTex 2.8 After struggling with installation of Latex2html (l2h) on windows for many days, I have decided to write down the final layout showing graphically what the tree looks like. I think a graphical illustration to describe the installation is easier for me, and I suspect for many others, to see where things are. The diagram below shows the final installed tree. This tree layout can be used as a guide for anyone who wants to install l2h on their windows system. click to enlarge #### 2.1 Installation steps The following are the main points to watch out for in order to minimize installation problems 1. Install the software components needed on the same drive, where your data is and where you will be working, because if you install the software on the C: drive, and your data is on another driver, then you will find that things might not work (this was in my case). 2. Start by downloading all the software components and install them one after the other. These are the components needed and where I downloaded them from 3. Now install MikTex, perl, netpbm, ghostcript, gsview into the directories as shown in the tree. 4. untar latex2html into folder latex2html-initial/ 5. Edit latex2hml-initial/prefs.m and add/change the corresponding lines as the following$prefs{'EXTRAPATH'} = 'C:\\texmf\\gsAFLP\\gs8.54\\bin;C:\\texmf\\GnuWin32\\bin';
$prefs{'PREFIX'} = 'C:\\texmf\\latex2html';$prefs{'TEXPATH'} = 'C:\\texmf\\MiKTeX2.8\\tex';
$prefs{'TMPSPACE'} = 'C:\\texmf\\tmp'; 6. Edit latex2html-initial/config.bat and add/change the corresponding lines as the following set PERL=c:\texmf\perl\bin\perl.exe 7. Add the following to the PATH environment variable on windows (Using Computer->Properties->Advanced) c:\texmf\latex2html\bin; c:\texmf\perl\bin; c:\texmf\GnuWin32\bin; c:\texmf\MiKTeX2.8\miktex\bin 8. Add a new environment variable GS_LIB and give it the following value .;C:\texmf\gsAFLP\gs8.54\lib;C:\texmf\gsAFLP\fonts Notice that the above is sufficient to make latex2html happy as far as GS_LIB is concerned, and hence there is no need to edit the file cfgcache.pm and set this value there as well, which if you check the file now, most likely this variable will be empty$cfg{'GS_LIB'} = q'';
but this was OK on my system, since it is now defined in an environment variable. But it should do no harm to also replace the above with
$cfg{'GS_LIB'} = q'.;C:\\texmf\\gsAFLP\\gs8.54\\lib;C:\\texmf\\gsAFLP\\fonts'; Even though I did not have to do it on my system as I mentioned above. 9. Add a new environment variable RGBDEF and give it the following value C:\texmf\latex2html\styles\rgb.txt 10. There is no need to define an environment variable TEXINPUTS since install.bat when run will copy the latex2html style files and put them in location within MikTex folder where they can be located by MikTex latex.exe without the need to define such an environment variable. 11. Start a new DOS terminal to make sure the effect of setting the above environment variable is now in effect. 12. There is one bug that shows up if you are using the package amsmath which causes the generated HTML to contain extra text showing up inside the images. To fix this 2, cd to the folder G:\texmf\latex2html-initial\styles note: the this file on my linux box was located in /usr/share/latex2html/styles/more_amsmath.perl and edit the file called more_amsmath.perl as follows ----- From here ----- *** styles/more_amsmath.perl.ORG Sat Dec 2 15:15:01 2000 --- styles/more_amsmath.perl Fri Oct 1 08:42:49 2004 *************** *** 95,100 **** --- 95,101 ---- } } else {$tag = ';SPMnbsp;;SPMnbsp;;SPMnbsp;' }
$*=0; +$scan =~ s/($comment_mark\d+) /$1\n/g;
if ($labels) {$labels =~ s/$anchor_mark/$tag/o;
($labels ,$scan);
----- To here -----
13. cd back to latex2html-initial/, run config.bat. If no errors then go to the next step.
14. run install.bat. Now latex2html will be installed to C:\texmf\latex2html
15. run latex2html-initial/test.bat and look for any errors.
16. If all is well, that is all. Now you are ready to use latex2html. Go to next step
17. Perform post installation configuration. Edit the file c:\texmf\latex2html\l2hconf.pm and make any changes needed. The following are the changes I made to my l2hconf.pm to make the generate HTML and images look better for me.
$FONT_SIZE = "12pt";$WHITE_BACKGROUND = 1;
$LOCAL_ICONS = 1;$MAX_SPLIT_DEPTH = 4;
$SHORTEXTN = 1;$ANTI_ALIAS = 1;
$ANTI_ALIAS_TEXT = 1;$HTML_VERSION = '4.0';
$MATH_SCALE_FACTOR = 1.8;$DISP_SCALE_FACTOR = 1.0;
$FIGURE_SCALE_FACTOR = 1.0;$TRANSPARENT_FIGURES = 1;
$NO_SUBDIR = 1;$DISCARD_PS = 0;
18. On Linux/Unix system, the above commands can be put in $HOME/.latex2html-init Make sure to end the file with "1;" This is my file as an example cat .latex2html-init$FONT_SIZE = "12pt";
$WHITE_BACKGROUND = 1;$LOCAL_ICONS = 1;
$MAX_SPLIT_DEPTH = 4;$SHORTEXTN = 1;
$ANTI_ALIAS = 1;$ANTI_ALIAS_TEXT = 1;
$HTML_VERSION = '4.0';$MATH_SCALE_FACTOR = 1.8;
$DISP_SCALE_FACTOR = 1.0;$FIGURE_SCALE_FACTOR = 1.0;
$TRANSPARENT_FIGURES = 1;$NO_SUBDIR = 1;
$DISCARD_PS = 0;$DVIPSOPT = '-E';
$LATEX_COLOR = ""; 1; 19. one final step is important. This step is needed to allow you to run miktex on latex files that include the package html. Since html.sty comes with latex2html, trying to compile latex file that uses the html package using miktex itself will fail, since miktex does now know where these style files are. Hence we need to tell miktex where these files are and register the root using miktex GUI. To do that, create a folder such as c:\myStyleFiles\tex\latex\misc It is very important that the path contain tex\latex in it. The folder misc can be names anything, as this is where you will copy the latex2html style files to. Once you create the above directories, the copy all the style files from latex2html tree, and they are located in the folder texinput in your latex2html installation tree. Copy all these style files to the directoy c:\myStyleFiles\tex\latex\misc You should now have C:\myStyleFiles\tex\latex\misc\$ dir
frames.sty
hthtml.sty
htmllist.sty
latin9.def
techexplHTML.tex
verbatimfiles.sty
heqn.sty
html.sty
justify.sty
ldump.sty
url.sty
20. Now that the latex2html style files are put in a TDS compliant tree structure, we can register this tree with Miktex. Start the Miktex GUI options, Using Miktex->Maintenance->Setting and now click on Roots in the menu, and click on Add... and select the folder c:\myStyleFiles making sure you just select this root, and do not select the whole path below it. Just stop at the root when selecting. Click OK
21. Now that latex2html style files are registered with Miktex, we can use programs such as TexMaker or any other program to compile the latex files. We can also use Latex2html to compile the same latex files to html.
#### 2.2 Some errors and possible solution
1. If you get this error when running test.bat that looks like something as the following
pstoimg.bat: Error: Ghostscript returned error status 1
pstoimg.bat: Error: Couldn't find pnm output of G:\texmf\tmp\l2h1716\image002.ps
Then check that you have defined GS_LIB environment variable correctly to point to the ghostscript lib/ and font/ directly as shown above.
2. if you get an error from test.bat that looks like something as the following
pstoimg.bat: Error: pnmtopng.exe -interlace -trans gray85 < p3704.pnm > img1.png" failed: No such file or directory
Then make sure that you have defined the env. variable RGBDEF as described above.
3. some of the bitmap images produced for the mathematics in the document have a solid dark bar, usually at the bottom or the side of the bitmap image. The reason for this is unknown. But I found by trial and error that setting the following values in my l2hconf.pm eliminated most if not all of these
$MATH_SCALE_FACTOR = 1.8;$DISP_SCALE_FACTOR = 1.0;
On my system, any value less than 1.8 for the above, with everything else is fixed, produced the side solid edges again.
This problem needs to be fixed.
4. Some mathematics equations are still underlined, even after doing the above. I found this was the case when I was using linux. If you still see the underlines, then try the following: edit l2hconf.pm and set the value of DVIPSOPT as follows
$DVIPSOPT = '-E' ; 5. Some mathematics formula have gray color in background. I noticed this on some equations having gray background. In this case, edit l2hconf.pm and set the following:$LATEX_COLOR = "";
The above should remove the gray background. If the above does not remove the gray background then try the following: In your latex document itself, add the following 2 lines 3 in the document preamble
\usepackage{color}
\pagecolor{white}
Then try again. This should, hopefully, remove the gray background. The above trick did it for me when I moved to new Linux OS and found the gray background came back, even though I was using the same .latex2html-init file as before.
6. Watch out for the -no_resuse option to latex2html. If one is also using -no_subdir, then Latex2html will ask the user if they want the images in the current folder deleted:
latex2html -no_reuse -no_subdir foo.tex
This is LaTeX2HTML Version 2008 (1.71)
....
Cannot create directory .\: File exists
(r) Reuse the images in the old directory OR
(d) *** DELETE *** the images in .\ OR
(q) Quit ?
And if you select option (d) then it will delete all the images in the current folder. This can include any images that were not related to latex2html earlier runs at all, and it could be your own images that you did not want deleted. This happened to me, but I had a backup copy. So, if you intend on using -no_subdir then it is safer to not use -no_reuse.
7. l2h does not support the package ragged2e. I use HTML CSS style to change the alignment of the table cells. The following code will not work as expected in l2h
\documentclass{article}
\usepackage{html}
\usepackage{array}
\usepackage{ragged2e}
\newcolumntype{P}[1]{>{\RaggedRight\hspace{0pt}}p{#1}}
\begin{document}
\begin{tabular}{|P{2in}|P{2in}|} %notice UPPER case P here
\hline
\end{tabular}
\end{document}
8. An error ! LaTeX Error: File `html.sty' not found. is generated when compiling the latex file due to including the html package \usepackage{html} using another program such as texmaker's pdflatex.
html.sty file is part of latex2html, and hence some programs that run your latex file will not know where it is. First, try to run texhash
>sudo texhash
texhash: Updating /usr/local/share/texmf/ls-R...
texhash: Updating /var/lib/texmf/ls-R-TEXLIVEMAIN...
texhash: Updating /var/lib/texmf/ls-R-TEXLIVEDIST...
texhash: Updating /var/lib/texmf/ls-R-TEXMFMAIN...
texhash: Updating /var/lib/texmf/ls-R...
texhash: Done.
and try again. If you still have the error, it could mean that you have installed texlive from one source, and latex2html from another source and texmf is not coordinated. When I had this problem on linux, the reason was that l2h was installed using apt and latex was installed directly from a tar file.
To resolve this, apt was used to install both texlive and also latex2html. Now the problem went away. Hence the suggestion is, on linux, to use apt or the package manager to install both latex and latex2html. This way, all the path and texmf setting is correct, and this error should be not show any more.
#### 2.3 Making your latex code latex2html friendly
These are few things that I found that helped in using latex2html
1. Always include html package
\usepackage{html}
2. l2h does not support tabularx package. Either just use tabular or as a work around add the following to the preamble
\begin{htmlonly}
\newenvironment{tabularx}[2]{\begin{tabular}{#2}}{\end{tabular}}
\end{htmlonly}
3. In addition, l2h will not recognize this
\raggedright
4. l2h does not support listings.sty, therefore if the latex code already does something as
\lstinputlisting{file.txt}
then it will not work with l2h. The solution is to add the following4
...
\begin{htmlonly}
\usepackage{verbatim}
\providecommand{\lstinputlisting}[2][]{\verbatiminput{#2}}
\end{htmlonly}
.....
\begin{document}
...
\lstinputlisting{file.txt}
...
\end{document}
I found one problem in the above. If the file has an absolute path on it, then it will not be included. But if the file is in the same folder as the latex file, then it will be included. So, the following will not work
...
\begin{htmlonly}
\usepackage{verbatim}
\providecommand{\lstinputlisting}[2][]{\verbatiminput{#2}}
\end{htmlonly}
.....
\begin{document}
...
\lstinputlisting{/home/my_files/file.txt}
...
\end{document}
5. Some images that I use were too large for inclusion in PDF without being scaled, even though the size looked OK for html.
So I wanted a way to use conditional logic so that when the latex file is run by pdflatex, it will scale the image, but when it is run by latex2html, the image is not scaled. I settled at this solution.
\documentclass[]{article}%
\usepackage{html}
\begin{document}
\begin{htmlonly}
\includegraphics[]{image.png}
\end{htmlonly}
\begin{latexonly}
\includegraphics[scale=0.75]{image.png}
\end{latexonly}
\end{document}
6. Wanted a way to modify the <HEAD> META tags in the HTML file generated by latex2html. I settled on this method for now. May be there is a better way? Thanks for old posts I found on the net by Thomas Anders and Ross Moore which helped in finding this solution.
1. created a file called l2h_init.pl in the same folder where my latex file is. This file looks like this
$MY_KEYWORDS = "Latex2html, HTML, MikTex";$MY_DESCRIPTION = "This describes Latex2html";
sub meta_information {
local($_) = @_; if (not defined$MY_KEYWORDS) { $MY_KEYWORDS = "$FILE"; }
if (not defined $MY_DESCRIPTION) {$MY_DESCRIPTION = "$_"; } do { s/<[^>]*>//g; "<!-- Do not edit here - edit TeX file$FILE.tex instead -->\n" .
"<META NAME=\"description\" CONTENT=\"$MY_DESCRIPTION\">\n" . "<META NAME=\"keywords\" CONTENT=\"$MY_KEYWORDS\">\n" .
"<META NAME=\"resource-type\" CONTENT=\"document\">\n" .
"<META NAME=\"distribution\" CONTENT=\"global\">\n" .
"$MY_META" } if$_;
}
1; # This must be the last line
2. Now run latex2html to convert my latex file to html. Suppose my latex file is called foo.tex, then the command I used is
latex2html -init_file l2h_init.pl foo.tex
3. now I looked at the generated HTML file foo.htm, and I see the header as follows
<TITLE>index</TITLE>
<!-- Do not edit here - edit TeX file index.tex instead -->
<META NAME="description" CONTENT="This describes Latex2html">
<META NAME="keywords" CONTENT="Latex2html, HTML, MikTex">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META NAME="Generator" CONTENT="LaTeX2HTML v2008">
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
7. do not use the following \href instead, use the following
And remember to always include the html package using
\usepackage{html}
8. do not enclose the image in a frame using \fbox. This will cause the image not to show up correctly. i.e. do not do something as the following
\fbox{\includegraphics[]{image.jpg}}
\includegraphics[]{image.jpg}
9. I had the following problem when using l2h when it creates a subdirectory to put all the resulting HTML and image into. This is because the hyperlinks inside the HTML would now all be off by one level.
Assuming you are at folder called A/ and you run l2h on a latex file called B.tex which contains hyperlinks to other places in your site, and these links are added so that they are relative hyperlinks. i.e. the hyperlinks are relative the folder A. When l2h process the file B.tex, it would by default create a sub folder and put the result into it. This folder is called A/B/.
However, from B/ the hyperlinks are now not correct as the case would be if the file was still in A/. Therefore, I now add the following option to my l2h command -no_subdir. This means the current folder A/ will contain all the output from the l2h run, which can make the folder too messy.
However, this for me was a better solution than having to change all the hyperlinks in the latex file or to make them all absolute links (which I do not think is a good idea).
I think the correct solution is to make l2h behave the same way as when Microsoft WORD saves a document as a web page, where it would leave the html file at the same level as the word document, but would then create a special folder called document_files which contains all the images and other html files generated.
10. Use a css file to customize the look of the resulting HTML instead of adding HTML code in the latex file using l2h special latex commands. Once you create a css file, you can tell l2h to use it with the option
latex2html -style="myfile.css" myfile.tex
11. When running latex2html on a file called say foo.tex, it will create also a file index.htm in addition to foo.htm and link to foo.htm from index.htm. This can cause a problem if one already has an index.htm in the same folder and hence it will be overwritten.
To make latex2html create only foo.htm the option -no_auto_link can be used which is what I currently do.
12. Including animated GIF files. To include an animated gif file, one can not include a gif file in latex, i.e. one can not just type
\includegraphics[]{file.gif}
I have a file called foo.gif, an animated file, which I want to have as a thumbnail, and run as animated, and wanted to also click on it if I wanted to see the animation in actual size. By trial and error, I found the following solution which worked for me
\documentclass[]{article}%
\usepackage{html}
\usepackage{graphicx}
\begin{document}
\begin{htmlonly}
\end{htmlonly}
When running the above through latex2html, I would get a warning about a "bad file descriptor" and "could not copy foo.gif to tree" type messages, but the latex2html did complete, and I ignored these. The generated HTML did work, and that is what important.
13. to support both latex2html and pdflatex do the following
\documentclass{article}%
\usepackage{html}
\usepackage{ifpdf}
\begin{document}
\begin{htmlonly}
%l2h only
\end{htmlonly}
%\begin{latexonly} % WARNING, SPECIAL COMMENT DO NOT REMOVE
\ifpdf
%for pdflatex only
\fi
%\end{latexonly} % WARNING, SPECIAL COMMENT DO NOT REMOVE
\end{document}
#### 2.4 log files
In here I have links to the output generated from running config.bat, install.bat and test.bat.
I also have link to my l2hconf.pm. Notice that on my PC, I have installed everything on the G:\ drive under a folder I called LATEX and not under C:\texmf as I showed in the diagram, but as long as you have installed everything on the same drive, it does not have to be the C: drive and can be anything else.
### 3 installation on cygwin
These are the steps I did to install latex2html under cygwin.
1. I downloaded cygwin from http://www.cygwin.com and installed it on windows. I installed EVERYTHING. Used Unix mode. (not DOS mode).
2. made sure the following is installed after the above is completed
> which latex
/usr/bin/latex
> latex -v
pdfeTeX 3.141592-1.21a-2.2 (Web2C 7.5.4)
kpathsea version 3.5.4
> which pdflatex
/usr/bin/pdflatex
> pdflatex -v
pdfeTeX 3.141592-1.21a-2.2 (Web2C 7.5.4)
kpathsea version 3.5.4
> cygcheck -c | grep -i tex
tetex-base 3.0.0-3 OK
> which perl
/usr/bin/perl
> perl -v
This is perl, v5.10.1 (*) built for i686-cygwin-thread-multi-64int
(with 12 registered patches, see perl -V for more detail)
> cygcheck -c | grep -i perl
perl 5.10.1-3 OK
> which gs
/usr/bin/gs
> gs -v
GPL Ghostscript 8.63 (2008-08-01)
Copyright (C) 2008 Artifex Software, Inc. All rights reserved.
3. Looked to make sure netpbm package allready installed from above installation of cygwin, to check, I lookeded for one of its programs
> which pngtopnm
/usr/bin/pngtopnm
> cygcheck -c | grep -i netpbm
libnetpbm-devel 10.49.2-1 OK
libnetpbm10 10.49.2-1 OK
netpbm 10.49.2-1 OK
4. downloaded latex2html version 2008 from http://www.ctan.org/tex-archive/support/latex2html/ and untar it in some directoy.
5. cd to latexhtml and edit the file L2hos.pm, replace the line (near the end)
@ISA = load('L2hos', $^O); with @ISA = load('L2hos', 'unix'); 6. edited prefs.pm and set location of tmporary disk space. This step is not needed, but I like to be more sure where temporary disk space for latex2html is.$prefs{'TMPSPACE'} = '/cygdrive/G/LATEX/TMP';
7. run configure
./configure
Noticed that it did not use TMP defined above in prefs.pm, which is not I want TMP to point to. So I edited the file generated by running configure above, which is called cfgcache.pm located in the same folder, and forced it to use my TMPSPACE by changing the line as follows:
$cfg{'TMPSPACE'} = q'/cygdrive/G/LATEX/TMP'; 8. run the command make 9. run the command make install 10. run the command make test cd to the test directory and check the output to make sure it is OK. As of today (June 6, 2010), there is a problem with cygwin installation which is not yet resolved. Cygwin generates an error coming from one of the perl file (L2h/Unix.pm). I send a bug report to cygwin on this. Until this is fixed, I am not able to run Latex2html under cygwin Installed cygwin 1.7.5 on windows 7 (64 bit OS), and when I run some command which uses perl, I get the following error: 0 [main] perl 2528 C:\cygwin\bin\perl.exe: *** fatal error - Internal error: TP_NUM_W_BUFS too small. DETAILS: I installed cygwin, all of it on windows 7, 64 bit os. All went ok. Then I installed latex2html, and I am trying to use latex2html under cygwin, which uses perl. (latex2hml is a PERL script) When I run latex2html command on some latex file, I get many of the above errors each time perl is called:$ uname -a
CYGWIN_NT-6.1-WOW64 me-PC 1.7.5(0.225/5/3) 2010-04-12 19:07 i686 Cygwin
$cygcheck -c | grep perl perl 5.10.1-3 OK perl-Error 0.17016-1 OK perl-ExtUtils-Depends 0.302-1 OK perl-ExtUtils-PkgConfig 1.12-1 OK perl-Graphics-Magick 1.3.7-2 OK perl-Image-Magick 6.4.0.6-2 OK perl-libwin32 0.28-3 OK perl-Locale-gettext 1.05-11 OK perl-ming 0.4.3-1 Incomplete perl-SGMLSpm 1.03ii-2 OK perl-Tk 804.028-3 OK perl-Win32-GUI 1.06-3 OK perl-XML-Simple 2.18-10 OK perl_manpages 5.10.1-3 OK postgresql-plperl 8.2.11-1 OK subversion-perl 1.6.11-1 OK$
$make test ..... Converting image #2 1 [main] perl 3104 C:\cygwin\bin\perl.exe: *** fatal error - Internal error: TP_NUM_W_BUFS too small. Error while converting image This is the file where this error comes from Unix.pm This is a more detailed error message: > 262: my ($self,$cmd,$in,$out,$err) = @_;
> 263: carp qq{Debug (syswait): Running "$cmd"\n} if($Verbose);
> 265: my $status; > 266: my$child_pid;
> 267: if ($child_pid = fork) { > 268:$status = waitpid($child_pid, 0); > 274: unless(exec($cmd)) {
> 0 [main] perl 4524 C:\cygwin\bin\perl.exe: *** fatal error - Internal error: TP_NUM_W_BUFS too small.
> 269: carp "Debug (syswait): Finished child process: #$child_pid\n" > 270: if($Verbose);
> 271: $child_pid = 0; > 272: return($?);
> exited L2hos::Unix::syswait
#### 3.1 log files
In here I have links to the output generated from running above commands:
1. This is output from cygwin configure
2. This is output from make install
### 4 the problem with the verb command
When using Latex2html from Linux hosted inside a virtual machine such as Oracle VBox, and when the latex files sits on windows shared folder, then I found that Latex2html does not generate code for the \verb command.
This note describes this problem. As of now, there is no solution to this other than copying the latex source file to the Linux file system.
Here is another note on the same problem above that I wrote before
### 5 Using graphics in latex2html
These notes are really my cheat sheet on how to including graphics in Latex, and not specific to Latex2html. I put them here so they are all in the same place.
Generally, I use graphicsx package for everything. To go from .tex file to .dvi file, one uses the latex command. To go from .tex to .pdf file, one uses the pdflatex command. But depending on which is used, different graphics are supported as show in this diagram
foo.tex ---- latex----------------------> foo.dvi
^
| \usepackage[]{graphicx}
| \includegraphics[]{a}
|
a.ps
a.eps
foo.tex ---- pdflatex ------------------> foo.pdf
^
| \usepackage[pdftex]{graphicx}
| \includegraphics[]{a.png}
|
a.png
a.jpg
a.pdf
Note the use of \usepackage[pdftex]{graphicx} when the input image is .png or .jpg type. Without this, then pdflatex will complain that it does not know the image size (no BoundingBox) error.
Unless I put an explicit size. Like this
foo.tex ---- pdflatex ------------------> foo.pdf
^
| \usepackage[]{graphicx}
| \includegraphics[natheight=1in,natwidth=5in,height=2in,width=5in]{a.png}
|
a.png
a.jpg
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http://math.stackexchange.com/users/31281/xxxxx | # XxXxX
less info
reputation
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bio website location age member for 1 year, 9 months seen May 17 '13 at 13:00 profile views 27
Self Study, Analysis & PDE & Probability
# 5 Questions
12 Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic? 4 sequence of decreasing compact sets 3 Question missing condition in Royden Exercise 7.42 b, about Baire Category 2 About continuous linear functional on the topology generated by linear functionals 2 About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$
# 124 Reputation
+20 Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic? +10 About continuous linear functional on the topology generated by linear functionals +10 About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$ +5 Question missing condition in Royden Exercise 7.42 b, about Baire Category
This user has not answered any questions
# 6 Tags
0 general-topology × 3 0 locally-convex-spaces 0 functional-analysis × 2 0 baire-category 0 real-analysis × 2 0 metric-spaces
# 2 Accounts
Mathematics 124 rep 6 TeX - LaTeX 21 rep 1 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9165121912956238, "perplexity": 3967.8475939098334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1393999649814/warc/CC-MAIN-20140305060729-00053-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=653339 | # Thermodynamic Derivation of Wien's Law?
by Collisionman
Tags: derivation, thermodynamic, wien
P: 36 Can someone tell me how I can derive Wien's law, i.e., $\lambda_{max} T = constant$ where $\lambda_{max}$ is the peak wavelength and $T$ is the absolute temperature of the black body, using the equation, $P=\frac{U^{*}}{3}$ where $U^{*}$ is the energy density. Note: I'm not looking for the derivation using Plank's formula. I'm looking for a purely thermodynamic derivation. Thanks in advance!!
P: 1,200 what is P?
P: 861 There is a thermodynamic derivation of Wien's Law in Heat and Thermodynamics by Roberts and Miller. It invokes considering slow expansion of a cavity, Doppler shift of reflected radiation, and so on. It is long and complicated, Maybe slicker derivations exist. These day, most textbook writers don't bother with this sort of derivation, but derive it from Planck's law. But I know you don't want this.
P: 36
## Thermodynamic Derivation of Wien's Law?
Quote by MikeyW what is P?
$P$ is the Radiation Pressure. It relates to the first law of termodynamics definition of work, $PdV$. Basically, I'm looking to derive Wien's law from the first law.
Related Discussions Advanced Physics Homework 8 Advanced Physics Homework 3 Materials & Chemical Engineering 2 Advanced Physics Homework 0 Classical Physics 0 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9023219347000122, "perplexity": 710.0981143241896}, "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-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00321-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/relative-velocity.45240/ | # Relative Velocity
1. Sep 29, 2004
### Lava
Hi, I'm struggling through this question and can't seem to come up with an answer :(
The question is: A student in a boat wants to cross the river from point A to point B (see diagram). The current of the river is v (2.0km/h, flowing east); speed of the boat in still water is u (5.0km/h). The point B is at an angle θ (60˚) from the distance L (0.25km) measuring the width of the river.
pt B
--------------------
..\........|
....\......|.L.......V
......\....|....... ---->
........\θ|
..........\
--------------------
pt A
(diagonal line = path of boat, the perpendicular line to the river side is the length L) Ignore the dots...I don't know how to set empty spaces
Last edited: Sep 29, 2004
2. Sep 29, 2004
### Tide
So, what exactly is the question?
3. Sep 29, 2004
### Lava
Sorry about that..the question is how long would it take the student to travel from point A to point B. Thanks
Have something to add?
Similar Discussions: Relative Velocity | {"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": 0.8931071758270264, "perplexity": 1369.6904928080744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541896.91/warc/CC-MAIN-20161202170901-00322-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.ideals.illinois.edu/handle/2142/16340/browse?value=Amir%2C+Abdelmadjid&type=author | # Browse Dissertations and Theses - Mathematics by Author "Amir, Abdelmadjid"
• (1990)
We show the almost sure uniform pathwise convergence of birth and death processes and random walks to Brownian motion with drift and sticky Brownian motion. We thus provide constructive definitions of such diffusions. In ...
application/pdf
PDF (2MB) | {"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": 0.9569586515426636, "perplexity": 2536.3388278163975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1466783395621.98/warc/CC-MAIN-20160624154955-00055-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1-common-core-15th-edition/chapter-1-foundations-for-algebra-1-5-adding-and-subtracting-real-numbers-practice-and-problem-solving-exercises-page-36/65 | ## Algebra 1: Common Core (15th Edition)
The negative (opposite) of an absolute value will always be negative, while an absolute value will never be negative. The only time this equation is true is when m is 0, since 0 and -0 have the same value. For example, let m=-5. −|−5|$=^{?}$|−(−5)| −5≠5 For example, let m=0. −|0|$=^{?}$|−0| −0=0 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8335682153701782, "perplexity": 1104.6179279147327}, "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-2022-49/segments/1669446710916.40/warc/CC-MAIN-20221202183117-20221202213117-00299.warc.gz"} |
https://www.queryhome.com/puzzle/16932/calling-all-electrical-engineers | # Calling all electrical engineers
+1 vote
59 views
Problem: a uniform electric field of magnitude 250 V per meter is directed in the positive x direction. A 12 uC (micro-coulomb) charge moves from the origin to the point (x,y) = (20 cm, 50 cm).
Question: what was the change in potential energy of this charge?
posted Sep 16, 2016
+1 vote
Distance the charge travelled = sqrt (20^2 + 50^2)
= sqrt (2900)cm = sqrt (0.29) m.
Now if the charge is moving against the field it's potential energy decreases else it will increase.
In both cases the change is
(Charge)*(potential difference)
(12*10^-6)*(250*sqrt (0.29)) = 0.001616 joules of potential difference.
Yes, you are correct.
Now for the next question:
Through what potential difference (expressed in volts) did the charge move?
Actually, I see you've already answered that as part of your equation: 250 volts*sqrt (0.29) = 134.63 volts
Thank you !
Similar Puzzles | {"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": 0.9341586232185364, "perplexity": 3219.510428736448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247494424.70/warc/CC-MAIN-20190220024254-20190220050254-00421.warc.gz"} |
http://tex.stackexchange.com/questions/14145/how-do-i-create-a-macro-which-reads-the-content-of-a-file-when-the-macro-is-defi/14147 | # How do I create a macro which reads the content of a file when the macro is defined?
I am trying to do something like:
`\newcommand{\sometext}{\input{somefile.txt}}`
This works, but requires inputting the file every time the macro is used in the document.
I would like to know a way of forcing the file to be read when the macro is defined, so that when the macro is called in the document, it only represents the text, and not the command to read the text in.
-
You can read the content of a file into a macro with Heiko Oberdiek's `catchfile` package:
``````\CatchFileDef{\sometext}{somefile.txt}{<setup>}
``````
This will read the file like a normal TeX file, i.e. it can include macros etc. The `<setup>` argument can be empty for files read normally but can include special code to e.g. read the file content verbatim or with special handling of line endings etc. A good candidate here is `\makeatletter` if the file contains macros with `@` in their names.
Verbatim mode can be set using `\let\do\@makeother\dospecials`. You might want also to add `\@noligs` to disable ligatures. The text should then be typeset using `\verbatim@font` (which is identical to `\normalfont\ttfamily`). Otherwise some symbols (like `_`) will not be displayed correctly. The above commands need `\makeatletter` to be used before the `\Catchfile` and `\makeatother` afterwards.
If you want to preserve line endings use `\obeylines`. You should also add `\obeyspaces` if you want the spaces being print normally not with a special symbol. Also `\frenchspacing` can be added to avoid the larger space after dots.
``````\makeatletter
\CatchFileDef{\sometext}{somefile.txt}{\let\do\@makeother\dospecials\@noligs\obeyspaces\frenchspacing\relax}
\makeatother
% Verbatim text requires a suitable font:
\texttt{\sometext}
``````
Make sure that the last command in `<setup>` isn't an assignment like `...\endlinechar=-1}`. You need to add an `\relax` then, otherwise TeX will expand the internal commands of `\CatchFileDef` to look for the rest of the number. I mentioned that to the author already and an internal `\relax` will be added in the next version.
-
Thanks- that works well. But could you provide some more details on reading the content verbatim? What would one use as the setup argument in that case? – Mark Mar 24 '11 at 19:24
@Mark: I updated my answer to include information about verbatim mode. – Martin Scharrer Mar 24 '11 at 21:30 | {"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": 0.9664614200592041, "perplexity": 854.1820259870502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397865.91/warc/CC-MAIN-20160624154957-00096-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://en.m.wikibooks.org/wiki/Biological_Physics/The_Zeroth_%26_First_Laws | Biological Physics/The Zeroth & First Laws
The zeroth law of thermodynamics came as an historical afterthought. It is often phrased in terms of thermal equilibrium (equal temperature), however, the basic idea may be applied to all types of equilibrium.
If system A is in equilibrium with system B, and system B is in equilibrium with system C, then system A is in equilibrium with system C.
Consider an ice chest full of ice and a soda can in the chest. The inside of the ice chest is system A. The can is system B. The soda is system C. If the ice has had time to bring the can to 0oC, system A is in thermal equilibrium with system B. If the can has had time to bring the soda to 0oC, system B is in thermal equilibrium with system C. We could also say this system is in mechanical equilibrium since there is no volume change occurring once everything is 0oC.
The first law of thermodynamics deals with the conservation of energy.
${\displaystyle \Delta U=W+Q}$
Its essence is that all changes in energy, ${\displaystyle \Delta U}$ of a system can be tracked by work ${\displaystyle W}$ and heat energy ${\displaystyle Q}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.8941242098808289, "perplexity": 478.30686436269235}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305288.57/warc/CC-MAIN-20220127193303-20220127223303-00143.warc.gz"} |
http://www.waynetippetts.com/?tag=androgenous-style | ## Posts Tagged ‘Androgenous Style’
### London – Street Life
Tuesday, January 7th, 2014
Tags: , , , , , , , , , , , , , , , , , , , , , , , ,
Posted in Street Style | 2 Comments »
### Paris – Parisien
Wednesday, April 4th, 2012
Tags: , , , , , , ,
Posted in Street Style | 1 Comment »
### London – Bohemian Gypsy
Tuesday, November 22nd, 2011
Takahiro gotta cool androgynous boho look going on. The levis Takahiro customised himself.
Tags: , , , , , , , , , , , , , , , , ,
Posted in Street Style | 1 Comment »
### London – Bebop Redux
Friday, September 9th, 2011
Fashion photographer Georgie told that it is rare that she should happen to be wearing all women’s clothes when I took this photo of her. Georgie says she generally prefers to mix her apparel up with mens fashions for a more androgynous look.
Tags: , , , , , , , , , , , , , , , , , , , , , , , , , , ,
Posted in Street Style | 2 Comments »
### Paris – Attitudes
Thursday, June 30th, 2011
Tags: , , , , , , , ,
Posted in Street Style | No Comments »
### London – Androgenous Style
Friday, December 31st, 2010
Tags: , , , , , , ,
Posted in Street Style | 3 Comments » | {"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": 0.9593099355697632, "perplexity": 4879.408459689485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1466783393442.26/warc/CC-MAIN-20160624154953-00138-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://testbook.com/question-answer/the-minimum-rate-at-which-a-signal-can-be-sampled--5e97c2dcf60d5d437c20ef52 | The minimum rate at which a signal can be sampled and still be reconstructed from its samples is known as:
Free Practice With Testbook Mock Tests
Options:
1. Spectrum
2. Nyquist rate
3. Anti-aliasing
4. Sampling
Correct Answer: Option 2 (Solution Below)
This question was previously asked in
CIL MT Electrical: 2020 Official Paper
Solution:
• The sampling theorem states that “a signal can be exactly reproduced if it is sampled at the rate fs which is greater than twice the maximum frequency W in the modulating signal."
• The minimum rate at which a signal can be sampled and still be reconstructed from its samples is known as Nyquist rate.
• If the sampled frequency is less than the Nyquist frequency, overlapping of lower and upper sidebands known as aliasing takes place.
• The main reason for aliasing is undersampling $$i.e.\;{f_s} < 2{f_m}$$
fs = Sampling frequency and fm = Modulating frequency
Aliasing is explained with the help of the spectrum as shown: | {"extraction_info": {"found_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": 0.8957176208496094, "perplexity": 1504.059115034867}, "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-2021-31/segments/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00361.warc.gz"} |
http://mathhelpforum.com/differential-geometry/190765-uniform-convergence-print.html | # Uniform Convergence
• October 19th 2011, 02:56 AM
hmmmm
Uniform Convergence
Am I right in saying that if I have $f_n=\frac{x^n}{x^n+1}$ and
$f(x) = \left\{\begin{array}{c l} 0 & x \in [0,1)\\ 0.5 & x =1 \\ 1 & x\in(1,\infty)\end{array}\right.$
definied on $(0,\infty)$ is not uniform convergent to f(x) (the problem being near the 1)?
Thanks for any help
• October 19th 2011, 03:08 AM
emakarov
Re: Uniform Convergence
According to the uniform limit theorem, the uniform limit of a sequence of continuous functions is continuous, so you are right.
• October 19th 2011, 03:13 AM
hmmmm
Re: Uniform Convergence
Cool thanks (I have proved it from the definition, which is a lot longer, feel a bit foolish now for asking!) it is pointwise convergent to f(x) though right?
• October 19th 2011, 03:18 AM
emakarov
Re: Uniform Convergence
Yes.
Hint: To avoid <br/> in LaTeX formulas, remove all line breaks between $$and$$ (not LaTeX line breaks \\; just put the whole formula on one editor line).
• October 19th 2011, 03:23 AM
hmmmm
Re: Uniform Convergence
Thanks very much for all the help (sorry about the poor LaTex and thanks for editing it)
• October 19th 2011, 03:25 AM
emakarov
Re: Uniform Convergence
Ha, it was Plato who edited it! | {"extraction_info": {"found_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": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9787169694900513, "perplexity": 2951.790045664344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00216-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/98161/fft-for-solving-poisson-equation-in-3-dimensions-with-periodic-boundary-conditio/98207 | # FFT for solving Poisson equation in 3 dimensions with periodic boundary conditions
In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ created by a charge distribution $\rho(\mathbf{r})$ is
$$\Delta V(\mathbf{r})=-\frac{\rho(\mathbf{r})}{\epsilon_0},$$
where $\epsilon_0$ is the vacuum permitivity.
I would like to use Fast Fourier Transforms (FFT) to solve this PDE. My only choice is to use FFTW, which computes the Discrete Fourier Transform (DFT) of a 1D complex array $X$ of size $N$ computes an array $Y$ where: $$Y_{k}\triangleq\sum_{n=0}^{N-1}X_{n}e^{-2i\pi nk/N},$$ and the backward DFT computes $$Y_{k}=\sum_{n=0}^{N-1}X_{n}e^{2i\pi nk/N}.$$
It is important to note these definitions. As far as I understand it, the sampling interval in these definitions is equal to 1. Furthermore, $FFT^{-1}(FFT(X))=N\cdot X$, which is not usual. I think it is said this definition is not unitary.
My problem is that I have a sampling interval in each direction $L_1$, $L_2$, $L_3 \neq 1$.
My question : could you help me get an expression of $\Delta V$ in k-space ? With a simplest definition of FFTs, it would be for instance $-\mathbf{k}^2 V(\mathbf{k})$. Here ... I can't get it.
Sorry if the question is a bit messy. I'm not a mathematician and this is quite new for me.
-
I think $\Delta V$ should be $-k^2 V$ in $k$-space. – Fabian Jan 11 '12 at 16:35
@Fabian corrected. Thanks. – max Jan 11 '12 at 17:36
You define $L_1, L_2, L_3$ to be the sampling interval in the 3 directions. Thus a point $\mathbf{r}$ in real space is given by $$\mathbf{r} = (L_1 n_1, L_2 n_2, L_3 n_3)$$ where $n_j \in \{0,\dots, N-1\}$.
Let us understand how $\partial V(\mathbf r)/\partial r_1$ is represented in Fourier space. As you sample your space, the derivative has to be replaced by a finite difference. So we have $$\frac{\partial V(\mathbf r)}{\partial r_1} \approx \frac{V(\mathbf{r} + L_1 \mathbf{e}_1) - V(\mathbf{r})}{L_1}= \frac{V_{n_1 +1, n_2, n_3} - V_{n_1-1,n_2,n_3}}{2L_1}$$ where I introduced the notation $V_{n_1,n_2,n_3} = V(\mathbf{r}=(L_1 n_1, L_2 n_2, L_3 n_3))$.
Given the definition of the Fourier transform in the question, you have $$\hat{V}_{k_1,k_2,k_3} = \sum_{n_1,n_2,n_3} V_{n_1,n_2,n_3} \exp\left(-2\pi i \sum_{j=1}^3 n_j k_j/N_j\right). \qquad\qquad(1)$$ Then $$V_{n_1,n_2,n_3} = \frac{1}{N_1 N_2 N_3} \sum_{k_1,k_2,k_3} \hat V_{k_1,k_2,k_3} \exp\left(2\pi i \sum_{j=1}^3 n_j k_j/N_j\right); \qquad\qquad(2)$$ Eq. (1) is the definition of $\hat{V}$ and $\hat{V}$ will be what the library will hopefully calculate for you. Eq. (2) then just follows from (1) --- and sadly does not coincide with the backtransform which the library offers.
Plugging (2) into the expression for $\partial V(\mathbf r)/\partial r_1$ yields $$\frac{\partial V(\mathbf r)}{\partial r_1} = \frac{1}{N_1 N_2 N_3} \sum_{k_1,k_2,k_3} \frac{i\sin(2\pi k_1)}{L_1} \hat V_{k_1,k_2,k_3} \exp\left(2\pi i \sum_{j=1}^3 n_j k_j/N_j\right).$$ Thus $\partial/\partial k_1$ is represented in the Fourier domain as $i\sin(2\pi k_1)/L_1$. Analogue, you can easily show that $$\frac\partial{\partial k_j} \mapsto \frac{i\sin(2\pi k_j)}{L_j}.$$
Using these results, we obtain $$\Delta \mapsto - \sum_{j=1}^3\frac{\sin(2\pi k_j)^2}{L_j^2}.$$
-
That's very nice I'll now be able to demonstrate the Froueri transform (whatever the definition) of $\Delta$. Thank you @Fabian. What is unclear to me is considering my practical problem and the actual definition of backward FFT in FFTW, what should I divide the forward FFT of $-\rho(\mathbf{r})/\epsilon_0$ by in order to calculate $V(\mathbf{r})$? – max Jan 11 '12 at 17:48
@MaximilienLevesque: keep the forward transform as defined in the package, so just solve $[\sum_j \sin^2(2\pi k_j)/L_j^2] \hat{V} = \hat{\rho}/\epsilon_0$. Just make sure that you include the factor $1/N_1 N_2 N_3$ when (at the end) transforming back to $V$. – Fabian Jan 11 '12 at 18:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9748085141181946, "perplexity": 245.07081518733006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1393999652921/warc/CC-MAIN-20140305060732-00055-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://physicsoverflow.org/34814/proving-killing-contractions-geodesics-constants-motion?show=34816 | # Proving that Killing form contractions with geodesics are constants of motion
+ 3 like - 0 dislike
2630 views
I want to prove the fundamental theorem of Killing forms, namely that
$$\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) = 0$$
If $P(\lambda)$ is a Geodesic curve, which implies that $\dot{P}^{\mu} \xi_{\mu}(P(\lambda))$ are constants of geodesic motion
This should be straightforward to prove, basically expanding the derivative expression
\begin{align*} \frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) &= \\ &=\frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))+\frac{d P^{\mu}}{d \lambda}\partial_{;\nu} \xi_{\mu}(P(\lambda)) \frac{d P^{\nu}}{d \lambda} \\ \end{align*}
We now use the fact that $\xi_{\nu}$ is a Killing form, that is:
$$\partial_{;\nu} \xi_{\mu}(P(\lambda)) = - \partial_{;\mu} \xi_{\nu}(P(\lambda))$$
And we expand the covariant derivative:
\begin{align*} \frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) &= \\ &=\frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))-\frac{d P^{\mu}}{d \lambda}\partial_{;\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\nu}}{d \lambda} \\ &= \frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))-\frac{d P^{\mu}}{d \lambda} \Big[ \partial_{\mu} \xi_{\nu}(P(\lambda)) - \Gamma^{\theta}_{\mu \nu} \xi_{\theta}(P(\lambda)) \Big] \frac{d P^{\nu}}{d \lambda} \\ &= \frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))+ \Gamma^{\theta}_{\mu \nu} \xi_{\theta}(P(\lambda))\frac{d P^{\mu}}{d \lambda}\frac{d P^{\nu}}{d \lambda} - \partial_{\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\mu}}{d \lambda} \frac{d P^{\nu}}{d \lambda} \end{align*}
But
$$\Gamma^{\theta}_{\mu \nu}\frac{d P^{\mu}}{d \lambda}\frac{d P^{\nu}}{d \lambda}= - \frac{d^2 P^{\theta}}{d \lambda^2}$$
Since the curve is a geodesic, which means that the expression simplifies:
\begin{align*} \frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu}(P(\lambda)) \Big) &= \\ &= \frac{d^2 P^{\mu}}{d \lambda^2}\xi_{\mu}(P(\lambda))- \frac{d^2 P^{\theta}}{d \lambda^2} \xi_{\theta}(P(\lambda)) - \partial_{\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\mu}}{d \lambda} \frac{d P^{\nu}}{d \lambda} \\ &= - \partial_{\mu} \xi_{\nu}(P(\lambda)) \frac{d P^{\mu}}{d \lambda} \frac{d P^{\nu}}{d \lambda} \end{align*}
So, I am able to get rid of those two terms, but there is still an uncancelled term with the coordinate derivative of the $\xi$ form. I don't know how to proceed next
Suggestions?
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
Suggestion to the question (v1): Replace the word Killing form with Killing vector field.
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user Qmechanic
In the very last expression if you change $\mu$ and $\nu$ the expression will be the same (since there is a summation over them). On the other hand $\xi$ is Killing so it will change sign. Thus the expression is equal to its negative, so it must be zero.
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user MBN
And all the calculation, from where you expanded the covarient derivative till the end, are unnecessary.
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user MBN
@Qmechanic, the reason to call it a form is that symmetrization of indices with the covariant index of a derivative ought to be a covariant index as well. But I agree that it might lead to confusion with the other Killing form used in Lie algebras
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
@MBN, no, you are missing the $\frac{d^2 P^{\mu}}{d \lambda^2} \xi_{\mu}$ term
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
+ 3 like - 0 dislike
Your $d/d\lambda$ should be understood in terms of parallel transport $$\frac{d}{d\lambda}\equiv \frac{dP^\mu}{d\lambda}\partial_{;\mu}$$ It is not an ordinary derivative when it acts on a vector.
So writing your third line in more transparent notation: $$\frac{d}{d \lambda} \Big( \frac{d P^{\mu}}{d \lambda} \xi_{\mu} \Big) =\left(\frac{dP^\nu}{d\lambda}\partial_{;\nu} \frac{dP^\mu}{d\lambda}\right)\xi_{\mu}+\left(\frac{dP^\nu}{d\lambda}\partial_{;\nu} \,\xi_{\mu}\right)\frac{dP^\mu}{d\lambda}$$ The second term vanishes since $\partial_{;\mu}\xi_\nu$ is antisymmetric in the indices as MBN points out in his comment (although it only applies to the covariant derivative). The first term vanishes by the geodesic equation, as should be clear in this notation.
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user octonion
answered Dec 29, 2015 by (145 points)
A few observations: 1) your expression has the index $\mu$ repeated four times, which might be confusing
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
2) What is the point of using two notations for the covariant derivative? $\nabla_{\mu}=\partial_{;\mu}$
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
3) $\partial_{;\nu} \frac{d P^{\mu}}{d\lambda}$ is not meaningful since $P^{\mu}$ is not a function of coordinates but of $\lambda$, so there is no point in complicating things here; $\frac{d}{d\lambda}( \frac{d P^{\mu}}{d \lambda})$ is just $\frac{d^2 P^{\mu}}{d \lambda^2}$
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
I copy pasted your answer and made clear the part where I think your notation is confusing you. By all means change the indices or use semicolons, I am just trying to communicate where the error is to you. It is not just a complication, to use the product rule over a contraction like you did you need to use the covariant derivative. Since you are dotting the index with the tangent vector it does not matter if $dP^\mu/d\lambda$ is only defined on the path.
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user octonion
How do you evaluate the expression $\partial_{;\nu} \frac{dP^\mu}{d\lambda}$? (or $\nabla_{\nu} \frac{dP^\mu}{d\lambda}$ in your previous notation)
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
You can only evaluate it in the tangent direction. That's ok because it is contracted with the tangent vector, and $P^\mu_{,\lambda}\partial_\mu = \partial_\lambda$. Expand it out in terms of the Christoffel symbols and you'll recognize the geodesic equation and see why the first term is zero.
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user octonion
ok that makes sense
This post imported from StackExchange Physics at 2015-12-31 08:11 (UTC), posted by SE-user lurscher
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http://mathhelpforum.com/calculus/1674-green-s-theorem-proof-misunderstanding.html | # Math Help - Green's Theorem Proof Misunderstanding
1. ## Green's Theorem Proof Misunderstanding
By the way, I also posted this in the Urgent Help section.
Okay, I get the half that is normally explained in the books which is the same as on wikipedia (http://en.wikipedia.org/wiki/Green's_theorem).
This is the best that I can explain this:
I understand the part of the explaination where integral of (P dx) = - double integral of (partial derivative of P with respect to y) dA.
I don't, however, understand why on the other half, integral of (Q dy) = double integral of (partial derivative of Q with respect to x) dA, does not have a negative sign in front of one of the two halves.
If essetially you are doing the same process twice, once with a type I region and the other time with a type II region, why does one half of the proof end up with a negative (which I understand) but not the other half?
2. Depends on the orientation of the curve, i.e which point is the initial one. Here the minus is needed to travel along the boundary in the positive direction (anti-clockwise). | {"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": 0.9536930322647095, "perplexity": 350.91221952123374}, "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-2014-41/segments/1410657114926.36/warc/CC-MAIN-20140914011154-00300-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://www.fz-juelich.de/SharedDocs/Termine/PGI/PGI-1/DE/2013/2013-01-25PGIKolloquiumKirschner.html | # PGI-Kolloquium
Spin-dependent two-electron emission from ferromagnetic Fe(001)
Prof. Dr. Jürgen Kirschner, Max-Planck-Institut für Mikrostrukturphysik, Halle
Anfang
25.01.2013 11:00 Uhr
Veranstaltungsort
PGI-Hörsaal
# Abstract
We present a joint experimental and theoretical study of correlated electron pair emission from a ferromagnetic Fe(001) surface induced by spin-polarized low-energy electrons. Spin-dependent angular and energy distributions of the emitted pairs have been measured and calculated. They are analyzed with the aid of the spin-, momentum-, symmetry-, and layer-resolved valence electron density of states, which we obtained by an ab-initio density functional theory calculation. The observed spectra are found to arise almost completely from only three surface parallel atomic layers. Momentum distributions for parallel spins of the emitted electrons exhibit an exchange-correlation hole, which is larger than the correlation hole in the antiparallel spin case. By comparing experimental antiparallel-spin pair spectra with their theoretical counterparts we determine an effective screening strength of the Coulomb interaction in the surface region.
## Kontakt
Prof. Dr. Stefan Blügel
Telefon: +49 2461 61-4249
Fax: +49 2461 61-2850
E-Mail: [email protected] | {"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": 0.9279378652572632, "perplexity": 3974.6950180314616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689192.26/warc/CC-MAIN-20170922202048-20170922222048-00078.warc.gz"} |
https://cs.stackexchange.com/questions/105863/what-is-wrong-with-this-solution-for-mathcalo-logn-choose-fracn2 | # What is wrong with this solution for $\mathcal{O}({\log({n \choose \frac{n}{2}})})$?
In this recitation on MIT OCW, the instructor uses Stirling's approximation to calculate that
$$\mathcal{O}({\log({n \choose \frac{n}{2}})}) = \mathcal{O}(n)$$.
However, I went through the following steps to conclude that $$\mathcal{O}({\log({n \choose \frac{n}{2}})}) = \mathcal{O}(\log{n})$$. Where did I go wrong?
First, note that $${n \choose \frac{n}{2}} = \frac{n!}{\frac{n}{2}!\frac{n}{2}!}$$. By basic logarithm laws, we get that this is equal to $$\log{(n!)} - \log{(\frac{n}{2}!\frac{n}{2}!)}$$. From this it follows that:
$$\mathcal{O}(\log{(n!)} - \log{(\frac{n}{2}!\frac{n}{2}!)})\\ = \mathcal{O}(\log{(n!)}) \\ = \mathcal{O}\Big(\log{\big(n(n-1)(n-2)\cdots(1)\big)}\Big) \\ = \mathcal{O}\Big(\log{n} + \log{(n-1)} + \ldots + \log{(1))}\Big) = \mathcal{O}(\log{n})$$
So, what is wrong here? I've gone over it for a while and I can't see any mistakes. Plotting these functions, though, I can see quite clearly that there must be a mistake.
$$\mathcal{O}\Big(\log{n} + \log{(n-1)} + \ldots + \log{(1))}\Big) = \mathcal{O}(\log{n})$$
That is not right. When $$n$$ is large enough,
\begin{align} \log{n} + \log{(n-1)} + \ldots + \log(1) &\ge \log{n} + \log{(n-1)} + \ldots + \log(n/2)\\ &\ge n/2 \log(n/2)\\ &=\Theta(n\log n) \end{align}
More precisely, since $$n!\sim {\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}$$ by Stirling's approximation,
\begin{align} \log{n} + \log{(n-1)} + \ldots + \log(1) &=\log(n!)\\ &\sim \log\left({\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}\right) \\ &\sim n(\log(n) -1) \\ &\sim n\log(n) \end{align}
You can take a moment to try understanding intuitively why $$\lim_{n\to\infty}\frac {\log(n!)}{\log (n^n)} = 1.$$
• Thank you. However I am a little confused how you got from the 2nd->3rd->4th steps for the first set of equations/inequalities. Could you clarify? – ubadub Mar 21 '19 at 4:12
• Actually I think I figured it out, leaving here for others: the second step comes from the fact that there are $n/2$ log terms before $\log(n/2)$, and $log(n/2)$ is less than or equal to all of them, so $(n/2)log(n/2)$ must be less than or equal to the sum of all the log terms. The third follows from that. However, why are the third and fourth steps necessary? Isn't $(n/2)\log(n/2)$ already $\Theta(n\log n)$? – ubadub Mar 21 '19 at 4:18
• Yes, $(n/2)\log(n/2)=\Theta(n\log n)$. I must have been overly detailed. I was doubling details when you asked. Upon with you "isn't" feedback, I just reduced my verbosity. – John L. Mar 21 '19 at 4:22
The last step is incorrect. $$\mathcal{O}\Big(\log{n} + \log{(n-1)} + \ldots + \log{(1))}\Big)$$ is not $$\mathcal{O}(\log{n})$$. You made the same mistake as What goes wrong with sums of Landau terms?. See also What is the asymptotic runtime of this nested loop?.
• Thanks. That's not me in "What goes wrong with sums of Landau terms?" but I see that it is the same mistake I made. – ubadub Mar 21 '19 at 4:05
• @ubadub, sorry, I phrased that wrong. I didn't mean it was you, I meant they made the same mistake you did (it's an easy mistake to make). I edited my answer. Sorry about that! – D.W. Mar 21 '19 at 4:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9894424676895142, "perplexity": 533.2713381461317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1581875145767.72/warc/CC-MAIN-20200223093317-20200223123317-00002.warc.gz"} |
https://www.graduate.technion.ac.il/Theses/Abstracts.asp?Id=23394 | M.Sc Thesis
M.Sc Student Rotem Sagi Light Controlled Self Assembly of Structures from Silicon Bricks Department of Physics Professor Uri Sivan
Abstract
This research presents a technique for fabricating nonspherical colloidal particles using photolithography that has been developed. The fabricated colloids are silicon boxes with typical sizes of 5x5x10 µm. These anisotropic “bricks” are studied at an air-water interface, where they self assemble into open structures or chains. Each brick consists of a p-n junction, which is localized away from the brick's edges. The p-n junction is overall neutral and hardly affects the potential presented by the brick to the solution. However, the p and n edges are opposite charged and hence create an attractive dipole-dipole interaction with other bricks. In case of absence of light the bricks are expected to build a filament. However, light generates electrons and holes that drift in the built-in junction and tend to flatten the brick's band bending. This flattening is equivalent to forward biasing of a diode which tends to reduce the charge presented to the other brick and, hence, reduces the brick's dipole moment. Since the bricks are overall charged, the elimination of the dipole moment should lead to brick-brick repulsion resulting in the melting of the filaments.
The “bricks” experience strong, anisotropic, and long ranged attractive capillary interactions which greatly exceed thermal energy of kBT . ‘Capillary forces’ are interactions between particles mediated by fluid interfaces and originate in the overlap of the menisci formed around each separate particle. In the case of the anisotropic “bricks” the capillary force rises from irregular wetting at the particle surface. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8415333032608032, "perplexity": 2550.8829132059286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347407667.28/warc/CC-MAIN-20200530071741-20200530101741-00254.warc.gz"} |
https://thoughtstreams.io/jtauber/note-quantization/4475/ | # Note Quantization
48 thoughts
last posted April 25, 2014, 2:23 a.m.
45 earlier thoughts
0
What's particularly compelling about the above model for swing is that, as long as actual note placement is relative to the grid, we can easily swing a straight time rhythm or de-swing a swing rhythm back to a straight time just by changing the grid parameters τmgb1 and τmgb2.
2 later thoughts | {"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": 0.9428573250770569, "perplexity": 4509.925733691198}, "config": {"markdown_headings": true, "markdown_code": false, "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-2023-14/segments/1679296950528.96/warc/CC-MAIN-20230402105054-20230402135054-00219.warc.gz"} |
https://www.physicsforums.com/threads/temperature-of-something-is-a-measure-of-the-average-kinetic-energy.75824/ | # Temperature of something is a measure of the average kinetic energy
1. May 16, 2005
If the temperature of something is a measure of the average kinetic energy of its particles then why aren't white water rapids hot?
2. May 16, 2005
### El Hombre Invisible
I don't think the increase in kinetic energy of a single molecule from being in rapids will be that much compared to the same molecule in, say, a glass. Anyone know the average KE of a water molecule at room temperature/atmos. pressure, etc?
3. May 16, 2005
### FredGarvin
If I understand your question, it is due to the rapid mixing of air into the water that causes the white appearance.
4. May 16, 2005
### Staff: Mentor
Just because something 'appears' white, that does not mean that something is 'white' hot. White simply means that a sufficient variety of photons are incident upon the eyes so that the color is not red, orange, yellow, green, blue or violet, i.e. the full visible spectrum is represented.
Ti dioxide is very white - even a room temperature. It simply reflects much of the incident light.
5. May 16, 2005
### Staff: Mentor
I think you guys may have misinterpreted: I don't think madness is asking about the white color, but rather the energy of the rapids.
madness - the energy contained in moving water is insignificant compared to the molecular energy of its temperature. It would take a massive waterfall to measure a difference in temperature from the top to the bottom. You can calculate how much if you like: the energy required to raise a gram of water by 1 degree C is 1 joule. Potential energy is mass times height: 1g of water raised 1000m is 1 joule. So a 1000m waterfall will have a temperature in the basin below 1 degree C higher than in the river above.
6. May 16, 2005
### efebest
i am also wqrking mordern techniques on temrature measurements can u help
7. May 16, 2005
### Integral
Staff Emeritus
I am not entirely satisfied with this conclusion. I think it over simplifies to much. It totally neglects the fact that there will be energy exchange between water and air. It totally neglects that the velocity of the falling droplet must be transfered to molecular motion, it is not clear to me that this happens with 100% efficiency.
To understand this we need to understand the difference between the velocity of the center of mass of a droplet and the average molecular velocity of the molecules in the droplet. It is not clear to me that the two are as closely related as your argument requires.
What is the frame of reference for the average velocity of a molecule? The only answer is that it must be measured with respect to the Center of mass of the body (or droplet in this case). If this is the case then clearly any motion of the C.M. has NO effect on the temperature of the body.
Consider for a moment the contents of the space shuttle as it is launched into space, does not every atom on board see a large change in velocity? Does the temperature change due to this change in velocity? Sure the skin temp increases due to air resistance but that is an entirely different issue.
Now, if Russ's argument holds, and 1 deg K is equivalent to a velocity change corresponding to a fall of 1000m, then what would the velocity of a molecule at 273K be? Seems like that is pretty fast. Once again I have reservations about the validity of this argument.
Last edited: May 16, 2005
8. May 16, 2005
### Staff: Mentor
Well, it certainly neglects air resistance, but regarding conversion of the falling droplet kinetic energy to heat, there is nowhere else that it can go. Besides - air resistance is all converted to heat as well.
I think you may have misinterpreted my point, in any case - I'm not talking about measuring the falling dropplet's temperature (I don't know how or if that would work - I suspect it wouldn't), I'm talking only about the temperature of the basin underneath. When water hits the basin, a lot of things happen (sound, splashes, churning with viscous friction, etc.), but one way or another, all of that kinetic energy becomes heat. It has to, otherwise it would violate the 1st law of thermo.
The fact that impact energy becomes heat can be seen in such devices as a needle gun or a pneumatic drill. Due to the impacts alone, they get very hot.
Last edited: May 16, 2005
9. May 16, 2005
### Integral
Staff Emeritus
Russ,
This question is about the temperature of the WATER DROPLET not the rock it hits. Perhaps the impact may increase the temperature of the water as well as that of the rock, that would provide the energy transfer mechanism necessary to increase the velocity of a water molecule. It does not imply that the temperature of the water has increased simply because the C.M. droplet has a velocity.
Considering the rock, I would guess that most of the energy of the fall can be accounted for in the velocity of the rebounding droplets created by the splash, that energy will then be lost to air resistance.
I would expect the rock to be in a thermal equilibrium with the water so would expect very little if any transfer of thermal energy between rock and water.
But, again, the effects of the collision between the rock and the water is not the topic of this thread.
10. May 16, 2005
### Staff: Mentor
I didn't mention any rocks, Integral. I'll await clarification from the original poster before going further with this.
11. May 16, 2005
### Gokul43201
Staff Emeritus
Temperature is the mean kinetic energy in the rest frame of the body involved. It is defined to be independent of the motion of the body. The energy from the motion of the body is accounted for in its kinetic energy. Saying that its temperature is higher from the motion means you are increasing the thermal energy (internal energy) as well. That is double-counting - you are accounting for the same quantity twice, and that is just bad math.
12. May 16, 2005
### jdavel
russ_waters,
That would be quite a coincidence! You meant 1 calorie to raise 1 gm 1 degree.
13. May 17, 2005
### Staff: Mentor
Oops - I realize that was a coincidence - I thought the SI system was designed that way. In fact, a calorie is 4.2 joule. Thanks for the correction.
14. May 17, 2005
### dextercioby
Russ,that's the definition of calorie.The heat needed to increase the temperature of 1 gr of water from 14.5°C to 15.5°C at normal atmosferical pressure.
Daniel.
15. May 1, 2009
### Roger44
Re: temperature
Am I therefore right in saying that a person sitting astride a molecule of an ideal monoatomic gas would feel the molecule go cold when it enters a Bernoulli type restriction in a tube?
I say this because the molecule would have lost part of its random kinetic energy, converted into increased velocity of the body along the tube.
16. May 1, 2009
### Staff: Mentor
Re: temperature
No, air does not lose/convert thermal energy when going through a venturi tube, according to Bernoulli. Bernoulli's principle/equation is actually a conservation of energy statement: the total energy of the flow remains constant because while the velocity increases, the pressure decreases.
So Bernoulli's equation/principle assumes no energy loss in the flow due to friction, viscocity, etc. The OP (and I with my waterfall) was describing a scenario where flow energy is lost. If you wanted to, you could just add an "E" to one side of your conservation of energy statement, but it is extremely difficult to actually calculate the energy - if you knew (ie, if you measured) the energy, you could go back and figure out how it affected the flow.
This is similar to what an engineer does when they design a pipe, though the tables they use to select the pipe already have that calculated, providing a pressure drop over a certain distance of flow at a certain velocity for a certain pipe. The total pressure loss is calculated and the pump is then sized/selected to provide the necessary flow energy.
Last edited: May 1, 2009
17. May 1, 2009
### mgb_phys
Re: temperature
Joule supposedly spent his honeymoon in Switzerland trying to measure the temperature difference at the top and bottom of a waterfall.
In theory the water at the bottom should be hotter because of the loss of PE.
18. May 1, 2009
### Roger44
Re: temperature
The molecules lose energy in the radial direction, which explains why the pressure against the walls of the tube decreases. This energy goes into increasning the velocity along the tube.
Like the man in the spacefraft mentioned above, sitting astride a molecule he would not be aware of this velocity increase. But he would feel the energy drop in the random radial motion. Wouldn't he?
19. May 1, 2009
### Staff: Mentor
Re: temperature
If the spacecraft were equipped with inertial navigation (all are, afaik), he'd be able to measure his change in velocity.
20. May 1, 2009
### sylas
Re: temperature
The mean kinetic energy of molecules in a substance at temperature T in Kelvin is 1.5 kT, where k = 1.38e-23 is Boltzmann's constant.
Water molecules have an atomic weight of about 18; and so 1 molecule of water weighs 18 / A, where A = 6.02e26; the number of atomic weight units in one kilogram. (Avogadro's number, up by 1000 so I can work in SI units of kilogram rather than gram)
Suppose you have cold water in the rapids, about 280K. (That's 7 degrees C).
The mean square of velocity is given by 0.5 mv^2 = 1.5 kT, where m = 18/A.
Hence v^2 = 3kTA/18 = kTA/6 ~ 400000.
That is, the molecules in cold water have velocities by virtue of temperature of about 630 m/s. The random bulk motions of turbulent water in the rapids is much much smaller than this. Adding energy to water by agitating it vigorously will increase the temperature, but measuring that is hard.
Put another way. At any given moment, a large proportion of the water molecules in the Niagara falls are moving upwards.
Conclusion. There must be something better to do on your honeymoon. Let me think...
Cheers -- sylas | {"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": 0.8736857175827026, "perplexity": 681.5028847300403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1480698542712.49/warc/CC-MAIN-20161202170902-00193-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/60981/latex-3-read-write-to-file-like-toc | LaTeX 3 read/write to file like toc
I tried to reimplement some kind of a toc facility. I write wouts argument into a file and try to reinclude this. So the Problem is, that \ior_str_gto:NN seems to stop at linebreaks. Please consider the following example:
\documentclass{report}
\usepackage[latin1]{inputenc}
\usepackage{ expl3, l3keys2e, xfrac, xparse}
\ExplSyntaxOn
\iow_new:N\iow_out
\ior_new:N\ior_out
\file_if_exist:nTF {myout.out} {
\ior_open:Nn\ior_out{myout.out}
\ior_str_gto:NN\ior_out\tl_get_out
\ior_close:N\ior_out }
{ \typeout{no file! rerun}
}
\iow_open:Nn\iow_out{myout.out}
\DeclareDocumentCommand\wout { m } {
\iow_now:Nx\iow_out{#1}
}
\DeclareDocumentCommand\rout { } {
\tl_use:N\tl_get_out
}
\ExplSyntaxOff
%
%
\begin{document}
\wout{bla}
\wout{blub}
\rout
\end{document}
The Question is: How can i make \ior_to_gstr to read in the whole file? and: is there a better way to implement such a function using l3?
-
\ior_to:Nn, etc. read one line at a time. If you look at LaTeX2e, the TOC concept is implemented by \inputing the entire file. We have some experimental stuff for doing whole file loops, but not yet anything 'release ready'. Perhaps suggest what might work for you! – Joseph Wright Jun 23 '12 at 18:20
I thought that you might answer this. My TeX knowledge is far away from writing s.th. for the kernel. – bloodworks Jun 23 '12 at 18:41
The LaTeX2e kernel reads files for tables, etc. using \input rather than line by line. However, it is possible to set up a line-by-line read here using epxl3 to work in a 'string' manner. There is an experimental function called \ior_str_map_inline:Nn which does more or less what seems to be wanted here:
\documentclass{article}
\usepackage{expl3}
\ExplSyntaxOn
\iow_new:N \g_my_out_iow
\ior_new:N \g_my_out_ior
\tl_new:N \g_my_out_tl
\file_if_exist:nTF { myout.out }
{
\ior_open:Nn \g_my_out_ior { myout.out }
\ior_str_map_inline:Nn \g_my_out_ior
{ \tl_gput_right:Nn \g_my_out_tl {#1 \par } }
\ior_close:N \g_my_out_ior
}
{ \typeout{no file! rerun} }
\iow_open:Nn \g_my_out_iow { myout.out }
\DeclareDocumentCommand\wout { m }
{ \iow_now:Nn \g_my_out_iow {#1} }
\DeclareDocumentCommand\rout { }
{ \tl_use:N \g_my_out_tl }
\ExplSyntaxOff
%
%
\begin{document}
\wout{bla}
\wout{blub}
\rout
\end{document}
(I have tided up a few variable names.) I've added \par to each line as it's not clear to me quite what is wanted. I've also written unexpanded to file, again as the wider context is not clear.
-
There are a few changes I will be making to the I/O code before the next CTAN update. That should not impact here, but I do notice I need to address the 'always global' nature of streams! – Joseph Wright Jun 23 '12 at 18:58
Thats very fine. I´m definitively going to look up this function. One thing still worries me: i do open the output stream for the entire compile time. I wonder if there is a tex function which allows opening out streams w/o recreating the whole file (such as >> in the shell). – bloodworks Jun 23 '12 at 19:19
@bloodworks There is no >> equivalent in TeX: one can read a file and write back, but there are issues. The LaTeX3 recommendation is not to keep the file open unless you need \iow_shipout:Nn or similar. If you want \iow_now:Nn then you can save everything to a tl and do a one-shot write at the end of the document. (We may well look at the shipout issue again, as one can solve this too, but it's more tricky and at the moment not something we need.) – Joseph Wright Jun 23 '12 at 19:21
@JosephWright: Would it make sense to add a function (in \ior or \file) to read the contents of a whole file? – Bruno Le Floch Jun 24 '12 at 9:22
Whats about tl_extract_between – bloodworks Jun 24 '12 at 19:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8881750106811523, "perplexity": 3097.728514197766}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861848830.49/warc/CC-MAIN-20160428164408-00077-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.openpipeflow.org/index.php?title=File:GMRESm.f90&oldid=836 | # File:GMRESm.f90
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
GMRESm.f90(file size: 5 KB, MIME type: application/acad)
$\renewcommand{\vec}[1]{ {\bf #1} } \newcommand{\bnabla}{\vec{\nabla}} \newcommand{\Rey}{Re} \def\vechat#1{\hat{\vec{#1}}} \def\mat#1{#1}$
## The GMRES(m) Method
This implements the classic GMRES(m) method for solving the system $A\vec{x}=\vec{b}$ for $\vec{x}$. This implementation minimises the error $\|A\vec{x}-\vec{b}\|$ subject to the additional constraint $\|\vec{x}\|<\delta$. (This constraint may be ignored by supplying $\delta<0$ in the implementation.)
The main advantage of the GMRES method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself. This means that $A$ need not even be stored, and could correspond to a very complex linear 'action' on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$. For a given starting vector $\vec{x}_0$, the method seeks solutions for $\vec{x}$ in $\mathrm{span}\{\vec{x}_0,\,A\vec{x}_0,\,A^2\vec{x}_0,...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of this basis set. The set of orthogonalised vectors is called the Krylov-subspace, and m is the maximum number of vectors stored.
Whereas m is traditionally a small number, e.g. 3 or 4, the added constraint renders restarts difficult. If the constraint is important, then m should be chosen sufficiently large to solve within m iterations.
GMRES is closely related to the Arnoldi method. See the remarks at File:Arnoldi.f on improved suitability via timestepping or exponentiation of the matrix.
## The code
This constraint $\|\vec{x}\|<\delta$ may be ignored by supplying negative 'del'.
In addition to scalar and array variables, the routine needs to be passed
• an external function that calculates dot products,
• an external subroutine that calculates the action of multiplication by $A$,
• an external subroutine that replaces a vector $\vec{x}$ with the solution of $M\vec{x}'=\vec{x}$ for $\vec{x}'$, where $M$ is a preconditioner matrix. This may simply be an empty subroutine if no preconditioner is required, i.e. $M=I$.
The functions above may require auxiliary data in addition to $\vec{x}$ or $\vec{\delta x}$. Place this data in a module and access via 'use mymodule' in the function/subroutine.
## Parallel use
It is NOT necessary to edit this code for parallel (MPI) use:
• let each thread pass its subsection for the vector $\vec{x}$,
• make the dot product function mpi_allreduce the result of the dot product.
• to avoid multiple outputs to the terminal, set info=1 on rank 0 and info=0 for the other ranks.
## File history
Click on a date/time to view the file as it appeared at that time.
Date/Time Dimensions User Comment current 04:30, 13 December 2016 (5 KB) Apwillis (Talk | contribs) Solve Ax=b for x, subject to constraint |x|
• You cannot overwrite this file.
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http://mathoverflow.net/questions/90317/can-you-name-these-orthogonal-polynomials?sort=oldest | # Can you name these orthogonal polynomials?
I have a collection of orthogonal polynomials in infinitely commuting variables $x_1, x_2, x_3, \ldots$. I think they must be well known (perhaps Schur or Hermite polynomials or some variant thereof), but I haven't succeeded in finding them in the literature in a form that's recognizable to me. If anyone can point me to an appropriate reference I would be grateful. I suspect the answer to this must be very familiar to many people, but I'm not one of those people.
The polynomials are indexed by Young diagrams (partitions) of all sizes (i.e. [], [1], [2], [1,1], [3], [2,1], [1,1,1], [4], ...).
The measure respect to which they are orthogonal is $$\prod_{k=1}^\infty \frac{1}{\sqrt{2\pi k}}e^{-\frac{x_k^2}{2k}}dx_k$$ In other words, a product of gaussian measures, with the width proportional to $\sqrt{k}$. (From some points of view it is more natural to replace $x_k$ with $x_k-1$ when $k$ is even; i.e. shift the gaussian to be centered at 1 instead of 0 when $k$ is even.)
The multiplication rule for the polynomials is more complicated than Littlewood-Richardson. Multiplying polynomials corresponding to Young diagrams of sizes $a$ and $b$ results in Young diagrams of sizes ranging from $|a-b|$ to $a+b$. (The highest order part of the multiplication rule is Littlewood-Richardson.) For example $[1] * [2,1] = [2] + [1,1] + [3,1] + [2,2] + [2,1,1]$.
Empirically, it seems to be true that if you sum the polynomials for all Young diagrams of size $n$, weighted by the dimension of the Young diagram, you get the $n$-th Hermite polynomial in the variable $x_1$. (Hat-tip to Suvrit for suggesting that I look at Hermite polynomials.)
-
@Kevin: for the curious onlookers like me, could you provide some additional references / context that could aim me in self-edification regarding the background material for this question. (also, are your polynomials some kind of Hermite polynomials?) – Suvrit Mar 6 '12 at 1:25
Thanks for the tip on Hermite polynomials. These seem to be some sort of higher dimensional variant. As for references or context for self-edification... I'm not really an expert on orthogonal polynomials or representation theory, which is why I'm asking a potentially elementary question here. From my point of view the context is TQFTs and skein modules. The above polynomials bear the same relation to the Birman-Wenzl-Murikami category as Chebyshev polynomials bear to the Temperley-Leb category. A good starting point for this subject is... – Kevin Walker Mar 6 '12 at 2:52
...the book "Temperley-Lieb recoupling theory and invariants of 3-manfolds" by Kauffman and Lins. After that one might want to look at various papers of Hugh Morton from the 1990s (or 2000s?). But that's my own peculiar point of view on the subject, not the standard one. – Kevin Walker Mar 6 '12 at 2:55
Have you tried weighting by a different column of the character table of the symmetric group? – John Wiltshire-Gordon Mar 6 '12 at 8:04
There is a good chance that they are special cases of Macdonald polynomials of type $A_n$. Check Macdonald's book on symmetric functions and Hall polynomials. – Richard Borcherds Mar 6 '12 at 21:47
Let $H^{[k]}_n(x)$ denote the variant of Hermite polynomials which are orthogonal with respect to the measure $$\frac{1}{\sqrt{2\pi k}}e^{-x^2/2k}dx .$$ Since the measure in the question is a product of the above measures (over all positive integers $k$), we have a family of orthogonal multivariable polynomials $$H^{[1]}_{n_1}(x_1) H^{[2]}_{n_2}(x_2) \cdots H^{[j]}_{n_j}(x_j) ,$$ indexed by tuples $(n_1,\ldots,n_l)$. The orthogonal polynomials of the questions are linear combinations of these. More specifically, let $N = \sum_i i\cdot n_i$. Think of $(n_1,\ldots,n_l)$ as encoding a conjugacy class in the symmetric group $S_N$, where $n_i$ is the number of $i$-cycles in a permutation. We can use the character table of $S_N$ to change basis from conjugacy-class-bump-functions to characters-of-representations. Applying an analogous change of basis to the above products of Hermite polynomials (separately for each $N$) yields the polynomials described in the question. There are some normalization factors I have not mentioned, related to the fact that some of the bases mentioned above are orthogonal but not orthonormal. | {"extraction_info": {"found_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": 0.920055627822876, "perplexity": 280.3145708327542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1393999677208/warc/CC-MAIN-20140305060757-00067-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/how-can-you-use-triode-as-detector-in-radio-wave-receivers.377367/ | # Homework Help: How Can you use triode as detector in radio wave receivers?
1. Feb 11, 2010
### aliz_khanz
1. The problem statement, all variables and given/known data
How Can you use triode as detector in radio wave receivers?
2. Relevant equations
N/A
3. The attempt at a solution
when the radio receivers were invented , triode was used to detect them but what function enables this is our of my mind!
2. Feb 11, 2010
### Phrak
Triode/Diode Detector
See the picture. It's just a peak detector for AM transmission.
Last edited by a moderator: May 4, 2017 | {"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": 0.9468993544578552, "perplexity": 4569.790857030164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2018-39/segments/1537267157028.10/warc/CC-MAIN-20180921092215-20180921112615-00372.warc.gz"} |
http://mathoverflow.net/questions/123535/a-submodule-of-a-constant-d-module-is-constant/123558 | A submodule of a constant D-module is constant
Hello,
Let $X$ be a smooth variety in char. 0. Let us call a $D$-module on $X$ constant, if it is isomorphic to a finite direct sum of the $D$-modules $O$ (the sheaf of regular functions with the usual $D$-module structure). Then a subquotient of a constant $D$-module should be constant. But how to show it? The reason why it should be true is that Riemann-Hilbert correspondence (when over the complex numbers) translates as to representations of a the fundamental group, and there such a statement is trivial.
Thanks, Sasha
-
If your base field $k$ is algebraically closed (or if $X$ has a rational point), then this follows directly from the fact that the category of $\mathcal{O}_X$-coherent flat connections is neutral Tannakian, i.e. equivalent to the category of finite dimensional $k$-representations of some affine $k$-group scheme $G$. For this fact the Riemann-Hilbert correspondence is not needed, it just gives you additional information about $G$.
Here is a more direct argument: If $E$ is a $D$-module which is coherent as an $\mathcal{O}_X$-module, then $E$ is automatically locally free. In particular, if $E$ is a sub $D$-module of the "constant" $D$-module $\mathcal{O}_X^n$, then the short exact sequence of $D$-modules
$$0\rightarrow E \rightarrow \mathcal{O}^n_X \rightarrow \mathcal{O}^n_X/E\rightarrow 0$$ is locally split as a sequence of $\mathcal{O}_X$-modules.
We see that if $n=1$, then $E=0$ or $E=\mathcal{O}_X$. We proceed by induction.
Since a $D$-module is trivial if and only if there exists a dense open subset of $X$ on which it is trivial, we may work in the local ring of a closed point $x\in X$ (lets assume $X$ to be connected…).
If $e_1,\ldots, e_n$ is a horizontal basis, then the horizontal sections of $\mathcal{O}_X^n$ are precisely those sections which are in the $k$-span of $e_1,\ldots, e_n$. The group of $D$-automorphisms of $\mathcal{O}_X^n$ then identifies with $GL_n(k)$. Thus if $E$ contains a horizontal section, we are done by induction. The case $n=1$ implies that if $E\cap e_i\mathcal{O}_X\neq 0$ for some $i$, then $e_i\in E$. If $f_1e_1+\ldots+f_n e_n$ is a section of $E$, which is not horizontal, then some $f_i\in \mathcal{O}_{X,x}$ is nonconstant, say $f_1$. One finds a differential operator $\partial$, such that $\partial(f_1)\in \mathcal{O}_{X,x}^\times$, and concludes that $E$ intersects the $\mathcal{O}_X$-span of $e_2,\ldots, e_n$ nontrivially. Hence, either $e_1\in E$, or $E\subset \bigoplus_{i\geq 2} e_i\mathcal{O}_X$, and in both cases we are done.
-
Suppose that $E\subset\mathcal O_X^n=V\otimes\mathcal O_X$, where $V$ is a finite-dimensional vector space, is a D-module. It is well known that both $E$ and $V\otimes\mathcal O_X/E$, being both D-modules and coherent $\mathcal O_X$-modules, must be locally free. Hence, if $\mathrm{rank}E=k$, there exists a morphism $f\colon X\to G(k,V)$ (the Grassmann variety) s.t. $f^*S\cong E$, where $S$ is the tautological subbundle. Since the subsheaf $E\subset V\otimes\mathcal O_X$ is mapped into itself by vector fields on $X$ (i.e., local derivations of $\mathcal O_X$), the derivative of the morphism $f$ is identically zero. Sionce characteristic is $0$, it follows that the mapping $f$ is constant, whence $E=f^*S$ is constant
-
Thank you for the answer; I did not understand your final conclusion, that $E$ is constant. You got that $E$ is constant as an $O$-module, but what about the $D$-module structure? – Sasha Mar 5 '13 at 9:25
The D-module structure is inherited from the constant D-module $V\otimes\mathcal O_X$. Actually, it means that there exists a $k$-dimensional linear subspace $F\subet V$ s.t. $E$ is the sheaf of $F$-valued functions, and derivations action on these functions by differentiating, as they act on local sections of $V\otimes \mathcal O_X$. – Serge Lvovski Mar 5 '13 at 13:25
Thank you, now I understand. – Sasha Mar 5 '13 at 14:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9926863312721252, "perplexity": 111.81222986155475}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207930109.71/warc/CC-MAIN-20150521113210-00310-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/solid-solution-hardening-concentration-dependence.791083/ | # Solid solution hardening, concentration dependence
Tags:
1. Jan 7, 2015
### CarlJose
When you add impurity atoms to a material, the yield strength often increases by a process known as solid solution hardening. This is because the impurity atoms create a barrier to dislocation motion. The literature describing this phenomenon dates back to the 1960s with some famous papers by R.L. Fleischer.
Anyway, this hardening is supposedly dependent on the spacing between the impurity atoms. Since the spacing between impurities should be proportional to the impurity concentration in some way (i.e., more impurities, closer spacing between them), increase in yield strength is often plotted against impurity concentration. Experimentally, this increase has been shown to vary with the square-root of impurity concentration.
*My question is this: Why is it a square-root dependence, and not a cube-root dependence? Atomic concentration is per unit volume, so if you increase impurity concentration, the spacing between these impurity atoms should scale with the cube root of the concentration, right? Every resource I have found states that impurity spacing varies with the square-root of impurity concentration but I have yet to find a good explanation for this. Am I missing something? Can someone explain?
2. Jan 7, 2015
### Staff: Mentor
You could argue that atoms in a material rarely move along a line. You can have different parts moving against each other, where the whole area is affected. The number of impurity atoms in a specific area goes with $n^{2/3}$. The true dependence could be between those effects.
3. Jan 7, 2015
### CarlJose
Thanks for the reply.
Why would the number of impurity atoms in a specific area go with $n^{2/3}$? (and I'm assuming you are using '$n$' to represent atomic concentration?) Is there some sort of geometrical justification for this?
Also, I guess I'd be willing to accept that the $n^{1/2}$ dependence is just an empirical observation (and between two functional limits), but I've found a few sources that state it as if it has some sort of theoretical justification. I'm suspicious that it has something to do with the impurities necessarily being within a crystalline lattice, but I can't figure out why that would make a difference...
4. Jan 8, 2015
### Staff: Mentor
Sorry, my previous post did not make sense. A surface cutting through a volume of N atoms will have N2/3 atoms (neglecting prefactors), but the number of impurity atoms within that surface will scale linearly with the total number of impurity atoms in the material.
The mean area without impurities (as part of a larger area we consider) should scale with the inverse square of the average linear distance, which gives n-2/3, but that is harder to translate to material properties.
Last edited: Jan 8, 2015
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Disclaimer: Not medical or professional advice. | {"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": 0.8726485371589661, "perplexity": 2050.112534900897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104204514.62/warc/CC-MAIN-20220702192528-20220702222528-00275.warc.gz"} |
https://www.ebooksread.com/authors-eng/s-l-sidney-luxton-loney/the-elements-of-coordinate-geometry-eno/page-19-the-elements-of-coordinate-geometry-eno.shtml | S. L. (Sidney Luxton) Loney.
# The elements of coordinate geometry online
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Online LibraryS. L. (Sidney Luxton) LoneyThe elements of coordinate geometry → online text (page 19 of 26)
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37. If a rectangular hyperbola circumscribe a triangle, shew that
it meets the circle circumscribing the triangle in a fourth point, which
is at the other end of the diameter of the hyperbola which passes
through the orthocentre of the triangle.
Hence prove that the locus of the centre of a rectangular hyper-
bola which circumscribes a triangle is the nine-point circle of the
triangle.
38. Two rectangular hyperbolas are such that the asymptotes of
one are parallel to the axes of the other and the centre of each lies on
the other. If any circle through the centre of one cut the other again
in the points P, Q, and jR, prove that PQR is a triangle such that each
side is the polar of the opposite vertex with respect to the first
hyperbola.
20
CHAPTER XIV.
POLAR EQUATION OF A CONIC SECTION, ITS FOCUS
BEING THE POLE.
335. Let S be the focus, A the vertex, and ZM the
directrix ; draw SZ perpendicular to ZM.
Let ZS be chosen as the positive direction of the
initial line, and produce it to X,
Take any point P on the
curve, and let its polar co-
ordinates be r and 6, so that
we have
SP = r, and iXSP^O.
Draw PN perpendicular
to the initial line, and PM
perpendicular to the directrix.
Let SL be the semi-latus-
rectum, and let SL = I.
Since SL = e , SZ^ we have
szJ-,
e
Hence
r = SP = e.PM=e,ZN
= e{ZS + SN)
I
= ei- + SP. cosO
Therefore
r =
= l + e.r. cos 0.
1
1 — 6 008 6^
.(1).
THE POLAR EQUATION, FOCUS BEING POLE. 307
This, being the relation holding between the polar
coordinates of any point on the curve, is, by Art. 42, the
required polar equation.
Cor. If SZ be taken as the positive direction of the initial line and
the vectorial angle measured clockwise, the equation to the curve is
I
r= .
1 + e cos d
336. If the conic be a parabola, we have e = l, and the equation
I I I ^e
IS r = z = — ^— = - cosec^ - .
1-cos^ ^ . ^d 2 2
2 sm^ -
If the initial line, instead of being the axis, be such that the axis
is inclined at an angle y to it, then, in the previous article, instead of
6 we must substitute d -y.
The equation in this case is then
-=l-ecos(^-7).
337. To trace the curve — = \—e cos 6.
r
Case I. e = 1^ so that the equation is - = 1 - cos 6.
When is zero, we have - — 0, so that r is infinite. As
r
increases from 0° to 90°, cos^ decreases from 1 to 0,
and hence - increases from to 1, i.e. r decreases from
r
infinity to I.
As increases from 90° to 180°, cos^ decreases from
to — 1, and hence - increases from 1 to 2, i.e. r decreases
r
from I to ^l.
Similarly, as changes from 180° to 270°, r increases
from - to I, and, as changes from 270° to 360°, r increases
from ^ to CO .
The curve is thus the parabola co FPLAL'P'F' oo of
Art. 197.
20—2
308 COORDINATE GEOMETRY.
Case II. e<l. When is zero, we have - = 1— e,
r
i.e. r— r . This gives the point A' in the figure of Art.
247.
As 6 increases from 0° to 90°, cos B decreases from 1 to
0, and therefore 1— ecos^ increases from 1 -e to 1, i.e. -
r
increases from 1— e to 1, i.e. r decreases from to I.
1— e
"We thus obtain the portion A'PBL.
As 6 increases from 90° to 180°, cos^ decreases from
to — 1, and therefore 1—e cos increases from 1 to 1 + e,
i.e. - increases from 1 to 1 + e, i.e.r decreases from I to r .
r 1 +e
We thus obtain the portion LA of the curve, where
SA = ^.
Similarly, as increases from 180° to 270° and then to
360°, we have the portions AL' and L'B'P'A'.
Since cos 6 = cos (— 6) = cos (360° — 6), the curve is sym-
Case III. e > 1. When is zero, 1 -e cos 6 is equal
to 1—e, i.e. — (e— 1), and is therefore a negative quantity,
since e > 1. This zero value of gives r = — I — (e — 1).
We thus have the point A' in the figure of Art. 295.
Let 6 increase from 0° to cos~^ ( " ) • Thus 1—e cos
increases algebraically from — (e — 1) to — 0,
i.e. — increases algebraically from — (e — 1) to — 0,
i.e. r decreases algebraically from = to — cc .
For these values of 6 the radius vector is therefore
negative and increases in numerical length from to oo .
THE POLAR EQUATION, FOCUS BEING POLE. 309
We thus have the portion A'P^R oo of the curve. For
this portion r is negative.
If 6 be very slightly greater than cos~^ - , then cos 6 is
slightly less than - , so that 1 -e cos is small and positive,
and therefore r is very great and is positive. Hence, as 6
increases through the angle cos~^ - , the value of r changes
from — cx) to + 00 .
As 6 increases from cos~^— to tt, 1— ecos^ increases
e
from to 1 + e and hence r decreases from oo to q .
1 +e
Now is < — - . Hence the point A, which corresponds
to 6^ = TT, is such that SA < SA'.
For values of 9 between cos~^- and tt we therefore
e
have the portion, ao RPA^ of the curve. For this portion
r is positive.
As increases from tt to 27r — cos~^ - , e cos 9 increases
e
from — e to 1, so that 1 — ecos^ decreases from 1 +e to 0,
and therefore r increases from — to oo . Corresponding
to these values of 6 we have the portion AL'R^ oo of the
curve, for which r is positive.
Finally, as increases from Stt-cos"^- to 27r, ecos^
e
increases from 1 to e, so that 1 — e cos 6 decreases algebraic-
ally from to 1 — e, i.e. - is negative and increases
r
numerically from to e— 1, and therefore r is negative and
decreases from go to . Corresponding to these values
of 6 we have the portion, oo R{A\ of the curve. For this
portion r is negative.
310 COORDINATE GEOMETRY.
r is therefore always positive for the right-hand branch
of the curve and negative for the left-hand branch.
It will be noted that the curve is described in the order
A'P^R' 00 00 RPAL'R^ oo oo R^A'.
338. In Case III. of the last article, let any straight line be
dravm through S to meet the nearer branch in ^, and the further
branch in q.
The vectorial angle of p is XS-p, and we have
I
^~l-e cos XS;p '
The vectorial angle of q is not XSq but the angle that qS produced
makes with SX, i.e. it is XSq^ir. Also for the point q the radius
vector is negative so that the relation (1) of Art. 335 gives, for the
point a,
^ l-ecos{XSq-f^7r) 1 + ecosXSq'
*-^* ^^^~l + ecosXSq'
This is the relation connecting the distance, Sq, of any point on
the further branch of the hyperbola with the angle XSq that it makes
with the initial line.
339. Equation to the directrices.
Considering the figure of Art. 295, the numerical values
of the distances SZ and SZ' are - and - + 2CZ,
e e
I.e. - and - + 2
e e e(e2-l)'
since ^^=-e-^^)' [Art. 300.]
The equations to the two directrices are therefore
r cos o = — ,
e
. rl 21 -] le^+1
and r cos c^ = — - + — — - — — = — - .
\je e(e^ — 1)J e e^—l
The same equations would be found to hold in the case
of the ellipse.
POLAR EQUATION TO A CONIC. 311
340. Equation to the asymptotes.
The perpendicular distance from S upon an asymptote
(Fig., Art. 315)
= OS sin ACK^ = as . . ^ = h.
s/a" + }p
Also the asymptote CQ makes an angle cos~^ — with the
axis. The perpendicular on it from S therefore makes an
angle ^ + cos-i - .
Hence, by Art. 88, the polar equation to the asymptote
CQ is
h = r cos ^ — 9 — cos~^ - = r sin 6 — cos~^ - .
The polar equation to the other asymptote is similarly
h — r cos 6- y-^ cos "^ - ) = — r sin {d + cos~* - j .
341. Ex. 1. In any conic, prove that
(1) the sum of the reciprocals of the segments of any focal chord
is constant, and
(2) the sum of the reciprocals of two perpendicular focal chords is
constant.
Let PSP' be any focal chord, and let the vectorial angle of P be a,
so that the vectorial angle of P' is ir + a.
(1) By equation (1)
of Art. 335, we have
^p=l ecosa,
and
A =1
SP'
- e cos (tt + a) = 1 + e (
Hence
SP^SP'~ '
so that
112
SP'^ SP'~ I'
The semi-latus-rectum is therefore the harmonic mean between
the segments of any focal chord.
.312 COORDINATE GEOMETRY.
(2) Let QSQ' be the focal chord perpendicular to PSP', so that the
vectorial angles of Q and Q' are - + a and -^ + a. We then have
7^7^ = 1 -e cos ( - + a | = l + esina,
and — - = l-ecos (-|^ + aj = l + ecos(^ + aj=l-esina.
Hence
7 7 27
PF'=SP + SP' = , +
1- e cos a 1 + e cos a 1 - e^ cos^ a '
Z Z 21
^^ ^ ^ l + esma 1-e sin a 1-e^ sm^ a
Therefore
1 1 1-e^ cos^ a 1 - e^ sin^ a _2-e^
PP''^'QQ'^ 2Z "^ 21 ~ ^2F'
and is therefore the same for all such pairs of chords.
Ex. 2. Prove that the locus of the middle jioints of focal chords of
a conic section is a conic section.
Let PSQ be any chord, the angle PSX being d, so that
I
SP
and SQ=
Let .
r and d.
1-e cos d '
I I
1-e cos [it + 6) 1 + e cos '
Let iJ be the middle point ot PQ, and let its polar coordinates be
Then .=SP-BP=SP_«^+^= «^-««
^ Ll-«cos^ 1 + e COS ^J 1-
2
ecos^
e'-^ cos 20'
i.e. r^-eVcos^^rrZe. ?'cos0.
Transforming to Cartesian coordinates this equation becomes
x^ + y^- e^x^ =lex (1) .
If the original conic be a parabola, we have e = l, and equation (1)
becomes y^ = lx, so that the locus is a parabola whose vertex is S and
latus-rectum I.
If e be not equal to unity, equation (1) may be written in the form
^ (i-')[-ir^J+'/=4-(^j
and therefore represents an ellipse or a hyperbola according as the
original conic is an ellipse or a hyperbola.
POLAR EQUATION TO THE TANGENT. 313
342. To find the 'polar equation of the tangent at any
point P of the conic section - — \~e cos B.
Let P be the point (rj, a), and let Q be another point
on the curve, whose coordinates are (r^, /?), so that we have
1 — ecosa (1),
T
and - = 1 - e cos i3 (2).
By Art. 89, the polar equation of the line PQ is
sin(;g-a) _ sin(6>-a) sin (j8 - 6)
r r^ ^1 '
By means of equations (1) and (2) this equation becomes
- sin (^ - a) = sin (^ — a) {1 — e cos /3} + sin {(i-6){\—e cos a}
= {sin(^ -a) + sin {(3-0)} -e {sin (^ -a) cos^ + sin (/3-^) cos a}
^ . B-a 20-a-B
= 2 sm ^— ^r — cos
— e{(sin^ cos a - cos ^ sin a) cos ^ + (sin /5 cos ^— cos ^ sin ^) cos a}
2 sin ^—^ — cos [B ~ J — e cos sin ifi - a),
^^_^A_ecos^ ..,(3).
I B-a //,a + y8
».e. - = sec — - — cos ' "
r A
This is the equation to the straight line joining two
points, P and Q^ on the curve whose vectorial angles, a and
^, are given.
To obtain the equation of the tangent at P we take Q
indefinitely close to P, i.e. we put /5 = a, and the equation
(3) then becomes
- =cos (^ — a) — ecos^ (4).
This is the required equation to the tangent at the
point a.
314 COOKDINATE GEOMETKY.
343. If we assume a suitable form for the equation to the
joining chord we can more easily obtain the required equation.
Let the required equation be
-=Lcos(^-7) -ecos^ (1).
[On transformation to Cartesian coordinates this equation is
easily seen to represent a straight line ; also since it contains two
arbitrary constants, L and 7, it can be made to pass through any two
points.]
If it pass through the point {r^^ a), we have
1 - e cos a =— =Xi cos (a - 7) - e cos a,
i.e. I, cos (a -7) = ! (2).
Similarly, if it pass through the point {r^, /3) on the curve, we have
Lcos(/3-7) = l (3).
Solving these, we have, [since a and ^ are not equal]
Substituting this value m (3), we obtam i = sec— ^ .
The equation (1) is then
(-^0-
-=sec— — COS ( d ^ ) -ecos^.
As in the last article, the equation to the tangent at the point a is
then
I
-=QOS{d-a)-eGosd.
r
^344. To find the polar equation of the polar of any
point (r^ , ^1) with respect to the conic section ~ — \—e cos 0.
Let the tangents at the points v^hose vectorial angles
are a and /? meet in the point (r^, ^1).
The coordinates r^ and 61 must therefore satisfy equation
(4) of Art. 342, so that
— — cos(^i — a)-ecos^i (1).
Similarly,
- = cos{6^- 13) -ecose^ (2).
• POLAR EQUATION TO THE POLAR. 315
Subtracting (2) from (1), we have
cos (\$1 — a) = cos (^1 — /?),
and therefore
^1 - a = — (^1 — /?), [since a and /? are not equal],
2
Substituting this value in (1), we have
t.e
= ^1 (3).
— = COS \ — ^r- ay —e cos d-i .
n [ 2 )
%.e. cos ^-^r — = — hecos^, (4).
Also, by equation (3) of Art. 342, the equation of the
line joining the points a and fi is
- + e cos d = sec -— — cos ' "
T 2
( - + e cos ij \ cos — - — = cos (6^ — ^ J ,
(- + ecos^ j f- + ecos^ij = cos(^-^i) (5).
%,e.
This therefore is the required polar equation to the polar
of the point (r^, 6-^.
^345. To find the equation to the normal at the point
whose vectorial angle is a.
The equation to the tangent at the point a is
- = cos (0 — a) — e cos 0,
i.e., in Cartesian coordinates,
a? (cos a — e) + yBina = l (1).
Let the equation to the normal be
^cos^ + ^sin^ = - (2),
i.e. Ax + Bi/ = l (3).
316 COORDINATE GEOMETRY.
Since (1) and (3) are perpendicular, we have
A (cos a-e) + Bs>ma = (4).
Since (2) goes through the point ( ^"^ , a j we have
A cos a + ^ sin a = 1 — e cos a (5).
Solving (4) and (5), we have
1— ecosa , „ (1 — ecosa) (e — cosa)
A = , and B = /^— .
e e sm a
The equation (2) then becomes
Zesina
sm a cos + {e — cos a) sm =
r(l — e cos a) '
e sin a I
I.e. sui(0 — a)~esuid = — ^ — .-.
^ ^ l—e cos a r
346. If the axis of the conic be inclined at an angle y to the
initial line, so that the equation to the conic is
- = 1-6 cos (^-7),
r
the equation to the tangent at the point a is obtained by substituting
a - 7 and ^ - 7 for a and 6 in the equation of Art. 342.
The tangent is therefore
-=e cos {d-a)-e cos [d - 7).
The equation of the line joining the two points a and p is, by the
same article,
I B-a
- = sec^-^r- cos
r 2
The equation to the polar of the point {i\ , 6^) is, by Art. 344,
j- + gcos(^-7)l j- + e cos (^1-7)1 =cos{d-d{).
Also the equation to the normal at the point a
■,« V . , ^x-> elsm(a-y)
r {e sm ^ - 7 + sin a -6)} = , , \ •
*- \ 1/ \ 'i l-ecos(a-7)
347. Ex. 1. If the tangents at any two points P and Q of a
conic meet in a point T, and if the straight line PQ meet the directrix
corresponding to S in a point K, then the angle KST is a right angle.
POLAR EQUATION. EXAMPLES. 3l7
If the vectorial angles of P and Q be a and /3 respectively, the
equation to PQ is, by equation (3) of Art. 342,
I ^-
- = sec^
r 2
^cosT^-'^Vgcos^ (1).
Also the equation to the directrix is, by Art. 339,
-= -ecos^ (2).
r
If we solve the equations (1) and (2), we shall obtain the polar
coordinates of K.
But, by subtracting (2) from (1), we have
so that SK bisects the exterior angle between SP and SQ.
Also, by equation (3) of Art. 344, we have the vectorial angle of T
equal to ^— , i.e. L TSX=
2 ' 2
Hence Z KST= L KSX - z TSX='^ .
Ex. 2. S is the focus and P and Q tioo points on a conic such that
the angle PSQ is constant and equal to 25; prove that
(1) the locus of the intersection of tangents at P and Q is a conic
section ivhose focus is S,
and (2) the line PQ always touches a conic ivhose focus is S.
(1) Let the vectorial angles of P and Q be respectively 7 + 5 and
7 - S, where 7 is variable.
By equation (4) of Art. 342, the tangents at P and Q are therefore
- = cos(6*-7-5)-ecos^ (1),
and - = cos(6'-7 + 5)-ecos^ .(2).
If, between these two equations, we eliminate the variable quantity
7, we shall have the locus of the point of intersection of the two
tangents.
Subtracting (2) from (1), we have
cos {e - y-8) = cos (^ - 7 + 8).
Hence, (since 5 is not zero) we have 7=^.
318 COORDINATE GEOMETRY.
Substituting for 7 in (1), we have
-=cos5-gcos^,
r
i e. =1 - esec ocos^.
r
Hence the required locus is a conic whose focus is 8, whose latus
rectum is 21 sec 5, and whose eccentricity is e sec 5.
It is therefore an ellipse, parabola, or hyperbola, according as
e sec 5 is < = >1, i.e. according as cos 8> = <:e.
(2) The equation to PQ is, by equation (3) of Art. 342,
- = sec5cos(^- 7) -ecos^,
T
i,e. =cos(&-7) -ecos dcos^ (3).
T
Comparing this with equation (4) of Art. 342, we see that it always
touches a conic whose latus rectum is 21 cos 5 and whose eccentricity
is ecosS.
Also the directrix is in each case the same as that of the original
^ , ^, Z sec 5 - i cos 5 ^ ^ I
conic. For both r and are equal to - .
e sec 8 e cos 5 e
Ex. 3. A circle passes through the focus S of a conic and meets it
in four points ivhose distances from S are r^, ?2, r^, and r^. Prove that
dH^
(1) r-^r^r^r^ = —^ , lohere 21 and e are the latus rectum and
eccentricity of the conic, and d is the diameter of the circle,
. /«x 1 1 1 1 2
and (2) - + - + - + - = 7.
J-i r^ rg 7-4 I
Take the focus as pole, and the axis of the conic as initial line, so
that its equation is
- = l-ecos^ (1).
If the diameter of the circle, which passes through S, be inclined
at an angle 7 to the axis, its equation is, by Art. 172,
r=dcos{d-y) (2).
If, between (1) and (2), we eliminate 6, we shall have an equation
in r, whose roots are ?i, r^, r^, and r^.
r — l I (r — 1\^
From (1) we have cos 6= , and hence sin ^= \/ ^~\ ) ♦
and then (2) gives
r=d cos 7 cos 9 + d sin 7 sin 6,
i.e. {er^ -dco8y{r - l)}^=d^ sin2 7 [eV _ (^ _ 1^2^^
i.e. eV- 2edcos7 . r3+r2 {d^ + 2eldGosy- e2d2sin27) - 2ld^r+dH^=sO.
POLAR EQUATION. EXAMPLES. 319
Hence, by Art. 2, we have
'V2^3^4 = -^ (%
and »'2''3''4 + r3r^»i + ?4rir2 + rir2r3=-^ (4).
Dividing (4) by (3), we have
11112
-+-+- + - = 7. •
rj ra r. r^ I
EXAMPLES. XXXIX.
1. In a parabola, prove that the length of a focal chord which is
incHned at 30° to the axis is four times the length of the latus-rectum.
The tangents at two points, P and Q, of a conic meet in T, and S
is the focus ; prove that
2. if the conic be a parabola, then ST^=SP . SQ.
3. if the conic be central, then — - -^7^;^ = —^ , where b is
the semi-minor axis.
4. The vectorial angle of T is the semi-sum of the vectorial
angles of P and Q.
Hence, by reference to Art. 338, prove that, if P and Q be on
different branches of a hyperbola, then ST bisects the supplement of
the angle PSQ, and that in other cases, whatever be the conic, ST
bisects the angle PSQ.
5. A straight line drawn through the common focus *S of a
number of conies meets them in the points Pj, P^, ... ; on it is taken
a point Q such that the reciprocal of SQ is equal to the sum of the
reciprocals of /SPj, SP^,.... Prove that the locus of Q is a conic
section whose focus is 0, and shew that the reciprocal of its latus-
rectum is equal to the sum of the reciprocals of the latera recta of the
given conies.
6. Prove that perpendicular focal chords of a rectangular hyper-
bola are equal.
7. PSP' and QSQ' are two perpendicular focal chords of a conic ;
prove that ^^ + ^-^-^-^ is constant.
8. Shew that the length of any focal chord of a conic is a third
proportional to the transverse axis and the diameter parallel to the
chord.
9. If a straight line drawn through the focus S oi a. hyperbola,
parallel to an asymptote, meet the curve in P, prove that SP is one
quarter of the latus rectum.
320 COORDINATE GEOMETRY. tExS.
10 Prove that the equations - = l-ecos6 and -=-ecos0-l
r r
represent the same conic.
11. Two conies have a common focus; prove that two of their
common chords pass through the intersection of their directrices.
12. P is any point on a conic, whose focus is S, and a straight
line is drawn through 5^ at a given angle with SP to meet the tangent
at P in T ; prove that the locus of T is a conic whose focus and
directrix are the same as those of the original conic.
13. If a chord of a conic section subtend a constant angle at the
focus, prove that the locus of the point where it meets the internal
bisector of the angle 2a is the conic section
ZC0S5 ^ r. n
= l-e cos 8 COS 6.
r
14. Two conic sections have a common focus about which one of
them is turned ; prove that the common chord is always a tangent to
another conic, having the same focus, and whose eccentricity is the
ratio of the eccentricities of the given conies.
15. Two ellipses have a common focus ; two radii vectores, one to
each ellipse, are drawn from the focus at right angles to one another
and tangents are drawn at their extremities ; prove that these tangents
meet on a fixed conic, and find when it is a parabola.
16. Prove that the sum of the distances from the focus of the
points in which a conic is intersected by any circle, whose centre is at
a fixed point on the transverse axis, is constant.
17. Shew that the equation to the circle circumscribing the triangle
2a
formed by the three tangents to the parabola r = z drawn at
•^ 1 - cos d
the points whose vectorial angles are a, j3, and y, is
a 8 y . fa+B+y
r=a cosec - cosec ~ cosec ^ sm '
-)
2 2 2 V 2
and hence that it always passes through the focus.
18. If tangents be drawn to the same parabola at points whose
vectorial angles are a, /3, y, and 8, shew that the centres of the circles
circumscribing the four triangles formed by these four Hnes all lie on
the circle whose equation is
a B y 8 r^ a + ^ + y + S-
r= - a cosec ^ cosec ^- cosec ^ cosec - cos ' ^
z A d a
fg a + P+y + 8 -\
19. The circle circumscribing the triangle formed by three tangents
to a parabola is drawn; prove that the tangent to it at the focus
makes with the axis an angle equal to the sum of the angles made
with the axis by the three tangents.
XXXIX.] POLAR EQUATION. EXAMPLES. 321
20. Shew that the equation to the circle, which passes through
the focus and touches the curve - = 1 - ecos 6 at the point ^ = a, is
ril-e cos af = l cos {d -a) - el cos {9 - 2a).
21. A given circle, whose centre is on the axis of a parabola,
passes through the focus S and is cut in four points A, B, C, and D by
any conic, of given latus-rectum, having S as focus and a tangent to-
the parabola for directrix ; prove that the sum of the distances of the
points A, B, G, and D from S is constant.
22. Prove that the locus of the vertices of all parabolas that can be
drawn touching a given circle of radius a and having a fixed point on
the circumference as focus is r=2acos^-, the fixed point being the
pole and the diameter through it the initial line.
23. Two conic sections have the same focus and directrix. Shew
that any tangent from the outer curve to the inner one subtends a
constant angle at the focus.
.24. Two equal ellipses, of eccentricity e, are placed with their
axes at right angles and they have one focus S in common ; if PQ be
g
a common tangent, shew that the angle PSQ is equal to 2 sin-^ —p, •
25. Prove that the two conies — ^l-e^cos^ and - = l-e2Cos(^-a)
will touch one another, if
Zi2 (1 _ e^^) + 1^ (1 - e^^) + ^l^l^e-^e^ cos a = 0.
26. An ellipse and a hyperbola have the same focus S and
intersect in four real points, two on each branch of the hyperbola ; if
rj and r^ be the distances from S of the two points of intersection on
the nearer branch, and r^ and r^^ be those of the two points on the
further branch, and if I and V be the semi-latera-recta of the two
conies, prove that
«+r)(Ui)+(z-.)(i + i)=4.
[Make use of Art. 338.]
a
27. If the normals at three points of the parabola r=a cosec^-,
whose vectorial angles are a, /3, and 7, meet in a point whose vectorial
angle is 5, prove that 25=a + /3 + 7-7r.
L. 21
CHAPTER XV.
GENERAL EQUATION OF THE SECOND DEGREE.
TRACING OF CURVES.
348. Particular cases of Conic Sections. The
general definition of a Conic Section in Art. 196 was that
it is the locus of a point P which moves so that its distance
from a given point S is in a constant ratio to its perpen-
dicular distance FM from a given straight line ZK.
When S does not lie on the straight line ZK, we have
found that the locus is an ellipse, a parabola, or a hyperbola
according as the eccentricity e is <= or > 1.
The Circle is a sub-case of the Ellipse. For the
equation of Art. 139 is the same as the equation (6) of
Art. 247 when h^ = a^, i.e. when e = 0. In this case
The Circle is therefore a
CS=0, and SZ=: - ae = oo
e
Conic Section, whose eccentricity is zero, and whose direc-
trix is at an infinite distance.
Next, let S lie on the straight line ZK, so that S and Z
coincide.
In this case, since
SF=e.FM,
we have
. „^,, FM 1
Online LibraryS. L. (Sidney Luxton) LoneyThe elements of coordinate geometry → online text (page 19 of 26) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9308596253395081, "perplexity": 4941.199019447072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710534.53/warc/CC-MAIN-20221128171516-20221128201516-00023.warc.gz"} |
http://tex.stackexchange.com/questions/28658/frutiger-font-in-texlive-2011-and-lion-os | # Frutiger font in TexLive 2011 and Lion OS
I tried to install the Frutiger font on TexLive 2011 using Lion OS. I followed this guide:
1. Copy the file frutiger.tar.gz to the folder `/usr/local/texlive/texmf-local`.
2. Unzip the file.
3. Open a shell and execute:
`sudo updmap-sys --enable Map=/usr/local/texlive/texmf-local/dvips/base/frutiger.map`
When I try to generate a PDF that uses Frutiger font, I get the following error:
```!pdfTeX error: /usr/texbin/pdflatex (file pfrr8r): Font pfrr8r at 540 not found
==> Fatal error occurred, no output PDF file produced!```
Any hints how to solve that?
-
Did you also execute the command `sudo mktexlsr` or, equivalently, `sudo texhash`?
Third and final suggestion :-) Have you specified the `T1` font encoding in your TeX program? (I believe the Frutiger fonts from the CTAN only work with T1, and maybe TS1.) If this doesn't work, I give up! Hopefully, others will come up with better suggestions. OK: here's a final suggestion: You should provide a MWE of your problems, so that others can at least try to replicate it.
On the Mac, `/usr/texbin` is (roughly) a symbolic link to the currently chosen tex distribution. It will ultimately resolve correctly to the relevant texlive directory. So seeing `/usr/texbin` in the log is correct. – Alan Munn Sep 16 '11 at 12:24
One specifies the "T1" font encoding by issuing the command `\usepackage[T1]{fontenc}` in the document's preamble. – Mico Sep 16 '11 at 14:10 | {"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": 0.9930353164672852, "perplexity": 4027.831135442317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398445033.85/warc/CC-MAIN-20151124205405-00332-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://escholarship.org/uc/item/26b1889v | A simple tool for bounding the deviation of random matrices on geometric sets
Open Access Publications from the University of California
A simple tool for bounding the deviation of random matrices on geometric sets
• Author(s): Liaw, C
• Mehrabian, A
• Plan, Y
• Vershynin, R
• et al.
Abstract
Let $A$ be an isotropic, sub-gaussian $m \times n$ matrix. We prove that the process $Z_x := \|Ax\|_2 - \sqrt m \|x\|_2$ has sub-gaussian increments. Using this, we show that for any bounded set $T \subseteq \mathbb{R}^n$, the deviation of $\|Ax\|_2$ around its mean is uniformly bounded by the Gaussian complexity of $T$. We also prove a local version of this theorem, which allows for unbounded sets. These theorems have various applications, some of which are reviewed in this paper. In particular, we give a new result regarding model selection in the constrained linear model.
Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you. | {"extraction_info": {"found_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": 0.8298490643501282, "perplexity": 551.4781630613081}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314353.10/warc/CC-MAIN-20190818231019-20190819013019-00039.warc.gz"} |
https://worldwidescience.org/topicpages/a/acoustic+decay+instability.html | Sample records for acoustic decay instability
1. Electron heating caused by the ion-acoustic decay instability in a finite-length system
International Nuclear Information System (INIS)
Rambo, P.W.; Woo, W.; DeGroot, J.S.; Mizuno, K.
1984-01-01
The ion-acoustic decay instability is investigated for a finite-length plasma with density somewhat below the cutoff density of the electromagnetic driver (napprox.0.7n/sub c/). For this regime, the heating in a very long system can overpopulate the electron tail and cause linear saturation of the low phase velocity electron plasma waves. For a short system, the instability is nonlinearly saturated at larger amplitude by ion trapping. Absorption can be significantly increased by the large-amplitude ion waves. These results compare favorably with microwave experiments
2. The Influence of Trapped Particles on the Parametric Decay Instability of Near-Acoustic Waves
Science.gov (United States)
Affolter, M.; Anderegg, F.; Dubin, D. H. E.; Driscoll, C. F.
2017-10-01
We present quantitative measurements of a decay instability to lower frequencies of near-acoustic waves. These experiments are conducted on pure ion plasmas confined in a cylindrical Penning-Malmberg trap. The axisymmetric, standing plasma waves have near-acoustic dispersion, discretized by the axial wave number kz =mz(π /Lp) . The nonlinear coupling rates are measured between large amplitude mz = 2 (pump) waves and small amplitude mz = 1 (daughter) waves, which have a small frequency detuning Δω = 2ω1 -ω2 . Classical 3-wave parametric coupling rates are proportional to pump wave amplitude as Γ (δn2 /n0) , with oscillatory energy exchange for Γ Δω / 2 . Experiments on cold plasmas agree quantitatively for oscillatory energy exchange, and agree within a factor-of-two for decay instability rates. However, nascent theory suggest that this latter agreement is merely fortuitous, and that the instability mechanism is trapped particles. Experiments at higher temperatures show that trapped particles reduce the instability threshold below classical 3-wave theory predictions. Supported by NSF Grant PHY-1414570, and DOE Grants DE-SC0002451 and DE-SC0008693. M. Affolter is supported by the DOE FES Postdoctoral Research Program administered by ORISE for the DOE. ORISE is managed by ORAU under DOE Contract Number DE-SC0014664.
3. Character of decay instability
International Nuclear Information System (INIS)
Polovin, R.V.; Demutskii, V.P.
1981-01-01
If the initial wave is unstable in the upper half plane Im ω>0 and there are no branch points of the quasiwave number, or if waves traveling in the same direction coalesce at a branch point, the instability is convective. On the other hand, if a branch point k(ω) does exist in the upper half-plane Im ω>0, and not all the waves that merge at this point travel in the same direction, the instability is absolute. A Green's function that describes the evolution of the perturbations of the initial wave in space and in time is constructed. The growth rates of the decay instability of the harmonics are determined. The produced waves are richer in harmonics than the initial waves. It is shown that the decay instability of an Alfven wave is absolute
4. Electron/electron acoustic instability
International Nuclear Information System (INIS)
Gary, S.P.
1987-01-01
The electron acoustic wave becomes a normal mode of an unmagnetized collisionless plasma in the presence of two electron components with similar densities, but strongly disparate temperatures. The characteristic frequency of this mode is the plasma frequency of the cooler electron component. If these two electron components have a relative drift speed several times the thermal speed of the cooler component, the electron/electron acoustic instability may arise. This paper describes the parametric dependences of the threshold drift speed and maximum growth rate of this instability, and compares these with the same properties of the electron/ion acoustic instability. Under the condition of zero current, the electron/ion acoustic instability typically has the lower threshold drift speed, so that observation of the electron/electron acoustic instability is a strong indication of the presence of an electrical current in the plasma
5. Decay instability of a whistler in a plasma
International Nuclear Information System (INIS)
Tewari, D.P.; Sharma, R.R.
1982-01-01
The parametric instabilities of a high power whistler in a high density plasma possess large growth rate when the scattered sideband is an electrostatic lower hybrid mode. The efficient channels of decay include oscillating two stream instability, nonlinear Landau damping and resonant decay involving ion acoustic and ion cyclotron modes. The processes of nonlinear scattering, i.e., the ones possessing whistler sidebands are relatively less significant. (author)
6. Parametric decay instabilities in ECR heated plasmas
International Nuclear Information System (INIS)
Porkolab, M.
1982-01-01
The possibility of parametric excitation of electron Bernstein waves and low frequency ion oscillations during ECR heating at omega/sub o/ approx. = l omega/sub ce/, l = 1,2 is examined. In particular, the thresholds for such instabilities are calculated. It is found that Bernstein waves and lower hybrid quasi-modes have relatively low homogeneous where T/sub e/ approx. = T/sub i/. Thus, these processes may lead to nonlinear absorption and/or scattering of the incident pump wave. The resulting Bernstein waves may lead to either more effective heating (especially during the start-up phase) or to loss of microwave energy if the decay waves propagate out of the system before their energy is absorbed by particles. While at omega/sub o/ = omega/sub UH/ the threshold is reduced due to the WKB enhancement of the pump wave, (and this instability may be important in tokamaks) in EBT's and tandem mirrors the instability at omega /sub o/ greater than or equal to 2 omega/sub ce/ may be important. The instability may persist even if omega > 2 omega/sub ce/ and this may be the case during finite beta depression of the magnetic field in which case the decay waves may be trapped in the local magnetic well so that convective losses are minimized. The excited fluctuations may lead to additional scattering of the ring electrons and the incident microwave fields. Application of these calculations to ECR heating of tokamaks, tandem mirrors, and EBT's will be examined
7. Simulation of the electron acoustic instability for a finite-size electron beam system
International Nuclear Information System (INIS)
Lin, C.S.; Winske, D.
1987-01-01
Satellite observations at midlatitudes (≅20,000 km) near the earth's dayside polar cusp boundary layer indicate that the upward electron beams have a narrow latitudinal width up to 0.1 0 . In the cusp boundary layer where the electron population consists of a finite-size electron beam in a background of uniform cold and hot electrons, the electron acoustic mode is unstable inside the electron beam but damped outside the electron beam. Simulations of the electron acoustic instability for a finite-size beam system are carried out with a particle-in-cell code to investigate the heating phenomena associated with the instability and the width of the heating region. The simulations show that the finite-size electron beam radiates electrostatic electron acoustic waves. The decay length of the electron acoustic waves outside the beam in the simulation agrees with the spatial decay length derived from the linear dispersion equation
8. Threshold of decay instability in an inhomogeneous plasma (Leningrad 1973)
International Nuclear Information System (INIS)
Piliia, A.D.
It is shown that in a spatially inhomogeneous plasma there can exist an absolute decay instability with a threshold lower than that found earlier. This instability arises when two parametrically coupled waves have turning points inside the plasma layer. The cause of the instability is a positive inverse coupling, caused by a nonlinear conversion and a reflection of the waves
9. Evidence for the electromagnetic decay instability driven by two plasmon decay
International Nuclear Information System (INIS)
Baker, K.L.; Afeyan, B.B.; Estabrook, K.G.; Drake, R.P.
1997-01-01
This paper examines the electromagnetic decay instability (EDI) and its role in laser-produced plasmas. The electromagnetic decay instability provides another channel through which parametric instabilities involving Langmuir waves can saturate. In the case where EDI is pumped by the Langmuir waves associated with two plasmon decay, EDI is shown to present an explanation for ω o /2 emission from laser-produced plasmas which is consistent with experimental observations
10. Light-induced ion-acoustic instability of rarefied plasma
International Nuclear Information System (INIS)
Krasnov, I.V.; Sizykh, D.V.
1987-01-01
A new method of ion-acoustic instability excitation under the effect of coherent light, resonance to ion quantum transitions on collisionless plasma, is suggested. The light-induced ion-acoustic instability (LIIAI) considered is based on the induced progressive nonequilibrium resonance particles in the field of travelling electromagnetic wave. Principal possibility to use LIIAI in high-resolution spectroscopy and in applied problems of plasma physics, related to its instability, is pointed out
11. Absolute decay parametric instability of high-temperature plasma
International Nuclear Information System (INIS)
Zozulya, A.A.; Silin, V.P.; Tikhonchuk, V.T.
1986-01-01
A new absolute decay parametric instability having wide spatial localization region is shown to be possible near critical plasma density. Its excitation is conditioned by distributed feedback of counter-running Langmuir waves occurring during parametric decay of incident and reflected pumping wave components. In a hot plasma with the temperature of the order of kiloelectronvolt its threshold is lower than that of a known convective decay parametric instability. Minimum absolute instability threshold is shown to be realized under conditions of spatial parametric resonance of higher orders
12. About the magneto-acoustic instabilities in mirrors
International Nuclear Information System (INIS)
Zvonkov, A.V.; Timofeev, A.V.
1984-01-01
It is shown that the characteristic of a plasma in mirrors anisotropy of io on distribution function versus velocities may results in the drive of magneto-acoustic instabilities. This instability, in contast to the well known Alyven oscillation instability, is driven on ion cyclotron frequency harmonics The instability in question has been possibly observed during the experiments a at the tmx device, where the oscillations have been excited both at the ion cycl tron frequency and harmonics
13. Multifragmentation: Surface instabilities or statistical decay
International Nuclear Information System (INIS)
Moretto, L.G.; Tso, K.; Delis, D.; Colonna, N.; Wozniak, G.J.
1992-11-01
Boltzmann-Nordheim-Vlasov calculations show multifragmentation that seems to originate from surface instabilities. These instabilities are traced to a sheet instability caused by the proximity interaction. Experimental data, on the other hand, suggest that multifragmentation may be dominated by phase space
14. Multifragmentation: surface instabilities or statistical decay?
International Nuclear Information System (INIS)
Moretto, L.G.; Tso, K.; Delis, D.; Colonna, N.; Wozniak, G.J.
1993-01-01
Boltzmann-Nordheim-Vlasov calculations show multifragmentation that seems to originate from surface instabilities. These instabilities are traced to a sheet instability caused by the proximity interaction. Experimental data, on the other hand, suggest that multifragmentation may be dominated by phase space. (author)
15. Longitudinal acoustic instabilities in slender solid propellant rockets : linear analysis
OpenAIRE
García Schafer, Juan Esteban; Liñán Martínez, Amable
2001-01-01
To describe the acoustic instabilities in the combustion chambers of laterally burning solid propellant rockets the interaction of the mean flow with the acoustic waves is analysed, using multiple scale techniques, for realistic cases in which the combustion chamber is slender and the nozzle area is small compared with the cross-sectional area of the chamber. Associated with the longitudinal acoustic oscillations we find vorticity and entropy waves, with a wavelength typically small compared ...
16. Role of parametric decay instabilities in generating ionospheric irregularities
International Nuclear Information System (INIS)
Kuo, S.P.; Cheo, B.R.; Lee, M.C.
1983-01-01
We show that purely growing instabilities driven by the saturation spectrum of parametric decay instabilities can produce a broad spectrum of ionospheric irregularities. The threshold field Vertical BarE/sub th/Vertical Bar of the instabilities decreases with the scale lengths lambda of the ionospheric irregularities as Vertical BarE/sub th/Vertical Barproportionallambda -2 in the small-scale range ( -2 with scale lengths larger than a few kilometers. The excitation of kilometer-scale irregularities is strictly restricted by the instabilities themselves and by the spatial inhomogeneity of the medium. These results are drawn from the analyses of four-wave interaction. Ion-neutral collisions impose no net effect on the instabilities when the excited ionospheric irregularities have a field-aligned nature
17. Propellant injection strategy for suppressing acoustic combustion instability
Science.gov (United States)
Diao, Qina
Shear-coaxial injector elements are often used in liquid-propellant-rocket thrust chambers, where combustion instabilities remain a significant problem. A conventional solution to the combustion instability problem relies on passive control techniques that use empirically-developed hardware such as acoustic baffles and tuned cavities. In addition to adding weight and decreasing engine performance, these devices are designed using trial-and-error methods, which do not provide the capability to predict the overall system stability characteristics in advance. In this thesis, two novel control strategies that are based on propellant fluid dynamics were investigated for mitigating acoustic instability involving shear-coaxial injector elements. The new control strategies would use a set of controlled injectors allowing local adjustment of propellant flow patterns for each operating condition, particularly when instability could become a problem. One strategy relies on reducing the oxidizer-fuel density gradient by blending heavier methane with the main fuel, hydrogen. Another strategy utilizes modifying the equivalence ratio to affect the acoustic impedance through mixing and reaction rate changes. The potential effectiveness of these strategies was assessed by conducting unit-physics experiments. Two different model combustors, one simulating a single-element injector test and the other a double-element injector test, were designed and tested for flame-acoustic interaction. For these experiments, the Reynolds number of the central oxygen jet was kept between 4700 and 5500 making the injector flames sufficiently turbulent. A compression driver, mounted on one side of the combustor wall, provided controlled acoustic excitation to the injector flames, simulating the initial phase of flame-acoustic interaction. Acoustic excitation was applied either as band-limited white noise forcing between 100 Hz and 5000 Hz or as single-frequency, fixed-amplitude forcing at 1150 Hz
18. Theory of 'strong' turbulence - Application to the ion acoustic instability
International Nuclear Information System (INIS)
1984-01-01
In this thesis, we apply the techniques recently developed in the theory of turbulence to study the evolution of the current-driven ion acoustic instability. We present a method allow to describe analytically and with a self-coherent manner the dynamic of the deformation of the distribution function of particles in the same time as the evolution of the turbulent energy. We have also discerned the saturation mechanisms of the instability as well as their domain of validity. (author) [fr
19. One-dimensional acoustic modeling of thermoacoustic instabilities (on cd)
NARCIS (Netherlands)
van Kampen, J.F.; Huls, R.A.; Kok, Jacobus B.W.; van der Meer, Theodorus H.; Nilsson, A.; Boden, H.
2003-01-01
In this paper the acoustic stability of a premixed turbulent natural gas flame confined in a combustor is investigated. Specifically when the flame is operated in a lean premixed mode, the thermoacoustic system is known to exhibit instabilities. These arise from a feedback mechanism between the
20. Graphical analysis of electron inertia induced acoustic instability
International Nuclear Information System (INIS)
Karmakar, P.K.; Deka, U.; Dwivedi, C.B.
2005-01-01
Recently, the practical significance of the asymptotic limit of m e /m i →0 for electron density distribution has been judged in a two-component plasma system with drifting ions. It is reported that in the presence of drifting ions with drift speed exceeding the ion acoustic wave speed, the electron inertial delay effect facilitates the resonance coupling of the usual fluid ion acoustic mode with the ion-beam mode. In this contribution the same instability is analyzed by graphical and numerical methods. This is to note that the obtained dispersion relation differs from those of the other known normal modes of low frequency ion plasma oscillations and waves. This is due to consideration of electron inertial delay in derivation of the dispersion relation of the ion acoustic wave fluctuations. Numerical calculations of the dispersion relation and wave energy are carried out to depict the graphical appearance of poles and positive-negative energy modes. It is found that the electron inertia induced ion acoustic wave instability arises out of linear resonance coupling between the negative and positive energy modes. Characterization of the resonance nature of the instability in Mach number space for different wave numbers of the ion acoustic mode is presented
1. Controlling chaos in the current-driven ion acoustic instability
International Nuclear Information System (INIS)
Fukuyama, T.; Taniguchi, K.; Kawai, Y.
2002-01-01
Control of intermittent chaos caused by the current-driven ion acoustic instability is attempted and the controlling mechanism is investigated. When a small negative dc voltage is applied to the chaotic system as a perturbation, the system changes from a chaotic state to a periodic state while maintaining the instability, indicating that the chaotic state caused by the ion acoustic instability is well controlled by applying a small negative dc voltage. A hysteresis structure is observed on the V-I curve of the mesh grid to which the negative dc voltage to control is applied. Furthermore, when a negative dc voltage is applied to the state which shows a laminar structure existing under same experimental conditions, the system becomes chaotic via a bifurcation. Driven-chaos is excited when a negative dc voltage is applied to the laminar state. Applying a small negative dc voltage leads to controlling intermittent chaos while exciting driven-chaos
2. Parametric decay instabilities in an infinite, homogeneous, weakly anisotropic plasma
International Nuclear Information System (INIS)
Grandal, B.
1976-01-01
The parametric decay of a transverse electromagnetic (em) wave with a frequency close to, but larger than, the electron plasma frequency is investigated for an infinite, homogeneous, weakly magnetoactive plasma. A two-component fluid description is employed, and the damping of the linear plasma waves is introduced phenomenologically to include both Landau and collisional damping. The transverse em wave will decay into a longitudinal electron plasma wave and an em ion-acoustic wave. Only the latter wave is assumed to be affected by the weak, constant magnetic field. The threshold expression for growth of electron plasma waves is equal to that of the isotropic plasma when the em ion-acoustic wave's direction of propagation lies inside a wide double cone, whose axis is along the constant magnetic field. When the em ion-acoustic wave propagates outside this double cone, an additional factor, which depends directly upon the magnetic field, appears in the threshold expression. This factor can, under certain conditions, reduce the threshold for growth of electron plasma waves below that of the isotropic plasma
3. Acoustic tomography for decay detection in black cherry trees
Science.gov (United States)
Xiping Wang; Jan Wiedenbeck; Shanqing Liang
2009-01-01
This study investigated the potential of using acoustic tomography for detecting internal decay in high-value hardwood trees in the forest. Twelve black cherry (Prunus serotina) trees that had a wide range of physical characteristics were tested in a stand of second-growth hardwoods in Kane, PA, using a PiCUS Sonic Tomograph tool. The trees were felled after the field...
4. Electron-acoustic Instability Simulated By Modified Zakharov Equations
Science.gov (United States)
Jásenský, V.; Fiala, V.; Vána, O.; Trávnícek, P.; Hellinger, P.
We present non-linear equations describing processes in plasma when electron - acoustic waves are excited. These waves are present for instance in the vicinity of Earth's bow shock and in the polar ionosphere. Frequently they are excited by an elec- tron beam in a plasma with two electron populations, a cold and hot one. We derive modified Zakharov equations from kinetic theory for such a case together with numer- ical method for solving of this type of equations. Bispectral analysis is used to show which non-linear wave processes are of importance in course of the instability. Finally, we compare these results with similar simulations using Vlasov approach.
5. The magnetized electron-acoustic instability driven by a warm, field-aligned electron beam
International Nuclear Information System (INIS)
Sooklal, A.; Mace, R.L.
2004-01-01
The electron-acoustic instability in a magnetized plasma having three electron components, one of which is a field-aligned beam of intermediate temperature, is investigated. When the plasma frequency of the cool electrons exceeds the electron gyrofrequency, the electron-acoustic instability 'bifurcates' at sufficiently large propagation angles with respect to the magnetic field to yield an obliquely propagating, low-frequency electron-acoustic instability and a higher frequency cyclotron-sound instability. Each of these instabilities retains certain wave features of its progenitor, the quasiparallel electron-acoustic instability, but displays also new magnetic qualities through its dependence on the electron gyrofrequency. The obliquely propagating electron-acoustic instability requires a lower threshold beam speed for its excitation than does the cyclotron-sound instability, and for low to intermediate beam speeds has the higher maximum growth rate. When the plasma is sufficiently strongly magnetized that the plasma frequency of the cool electrons is less than the electron gyrofrequency, the only instability in the electron-acoustic frequency range is the strongly magnetized electron-acoustic instability. Its growth rate and real frequency exhibit a monotonic decrease with wave propagation angle and it grows at small to intermediate wave numbers where its parallel phase speed is approximately constant. The relevance of the results to the interpretation of cusp auroral hiss and auroral broadband electrostatic noise is briefly discussed
6. The parametric decay of dust ion acoustic waves in non-uniform quantum dusty magnetoplasmas
International Nuclear Information System (INIS)
Jamil, M.; Ali, Waris; Shah, H. A.; Shahid, M.; Murtaza, G.; Salimullah, M.
2011-01-01
The parametric decay instability of a dust ion acoustic wave into low-frequency electrostatic dust-lower-hybrid and electromagnetic shear Alfven waves has been investigated in detail in an inhomogeneous cold quantum dusty plasma in the presence of external/ambient uniform magnetic field. The quantum magnetohydrodynamic model of plasmas with quantum effect arising through the Bohm potential and Fermi degenerate pressure has been employed in order to find the linear and nonlinear responses of the plasma particles for three-wave nonlinear coupling in a dusty magnetoplasma. A relatively high frequency electrostatic dust ion acoustic wave has been taken as the pump wave. It couples with two other low-frequency internal possible modes of the dusty magnetoplasma, viz., the dust-lower-hybrid and shear Alfven waves. The nonlinear dispersion relation of the dust-lower-hybrid wave has been solved to obtain the growth rate of the parametric decay instability. The growth rate is at a maximum for a small value of the external magnetic field B 0 . It is noted that the growth rate is proportional to the unperturbed electron number density n oe and is independent of inhomogeneity beyond L e =2 cm. An extraordinary growth rate is observed with the quantum effect.
7. The grain charging and the dust acoustic wave instability
International Nuclear Information System (INIS)
Varma, Ram K.
2001-01-01
The stability of the steady charging state of the assembly of dust grains in a plasma is analyzed using, besides the equations of continuity and momentum balance, also the equations of thermal energy balance with the grain charging terms for both the electron and ion species. The grain charging terms account for the energy exchange between the dust grains and the electron and ion fluids. The grains are taken to be immobile for the purpose of this analysis. Two limiting cases are analyzed: (i) f(≡4πn d λ D 2 a) >1 (n d is the dust number density, λ D plasma Debye length, and a, the grain radius). The steady grain charge state is found to be stable in the case f o is unaffected. On the other hand, in the limit f>>1, the state is found to be unstable provided γ q (≡q o e/aT e ) e -T i )/T e (T e , T i are electron and ion temperatures). A coherent charging of the dust grains results as a consequence of this instability until γ q ≅(1/2) (T e -T i )/T i . Next, by letting the grain charges be mobile, so that the perturbation of dust number density is nonzero, we examine the stability of the dust-acoustic wave (DAW). The DAW is found to be unstable, also in the f>>1 case, while stable in the f<<1. The instability of the DAW also implies a concomitant grain charge growth, which would again be of a coherent nature
8. The influence of the group delay of digital filters on acoustic decay measurements
DEFF Research Database (Denmark)
Sobreira-Seoane, Manuel A.; Cabo, David Pérez; Jacobsen, Finn
2012-01-01
In this paper the error due to the phase response of digital filters on acoustic decay measurements is analyzed. There are two main sources of errors when an acoustic decay is filtered: the error due to the bandwidth of the filters related to their magnitude response, and the error due to their p...
9. Thermo-acoustic instabilities in lean premixed swirl-stabilized combustion and their link to acoustically coupled and decoupled flame macrostructures
KAUST Repository
Taamallah, Soufien; LaBry, Zachary A.; Shanbhogue, Santosh J.; Ghoniem, Ahmed F.
2015-01-01
10. Theory of the acoustic instability and behavior of the phase velocity of acoustic waves in a weakly ionized plasma
International Nuclear Information System (INIS)
Torosyan, O.S.; Mkrtchyan, A.R.
2003-01-01
The amplification of acoustic waves due to the transfer of thermal energy from electrons to the neutral component of a glow discharge plasma is studied theoretically. It is shown that, in order for acoustic instability (sound amplification) to occur, the amount of energy transferred should exceed the threshold energy, which depends on the plasma parameters and the acoustic wave frequency. The energy balance equation for an electron gas in the positive column of a glow discharge is analyzed for conditions typical of experiments in which acoustic wave amplification has been observed. Based on this analysis, one can affirm that, first, the energy transferred to neutral gas in elastic electron-atom collisions is substantially lower than the threshold energy for acoustic wave amplification and, second, that the energy transferred from electrons to neutral gas in inelastic collisions is much higher than that transferred in elastic collisions and thus may exceed the threshold energy. It is also shown that, for amplification to occur, there should exist some heat dissipation mechanism more efficient than gas heat conduction. It is suggested that this may be convective radial mixing within a positive column due to acoustic streaming in the field of an acoustic wave. The features of the phase velocity of sound waves in the presence of acoustic instability are investigated
11. Dynamics of beam-driven Langmuir and ion-acoustic waves including electrostatic decay
International Nuclear Information System (INIS)
Li, B.; Willes, A.J.; Robinson, P.A.; Cairns, I.H.
2003-01-01
The evolution of Langmuir waves and ion-acoustic waves stimulated by a hot electron beam in an initially homogeneous plasma is investigated numerically in time, position, and wave number space. Quasilinear interactions between the beam particles and Langmuir waves, nonlinear interactions between the Langmuir and ion-acoustic waves through Langmuir decay processes, and spontaneous emission are taken into account in the kinetic theory employed. For illustrative parameters of those in the solar wind near 1 a.u., nonlinear Langmuir decays are observed to transfer the beam-driven Langmuir waves rapidly out of resonance. The scattered Langmuir waves then undergo further decays, moving sequentially toward small wave numbers, until decay is kinematically prohibited. The main features of the evolution of Langmuir and ion-acoustic waves are spatially inhomogeneous. The scattered Langmuir spectra increase and eventually reach or exceed the beam-driven Langmuir spectra at a given spatial location (except in regions where further decays proceed). The ion-acoustic waves are relatively weak and subject to damping at the later stages of their evolution. The development of fine structures in the product Langmuir and ion-acoustic waves are observed, due to depletion of their energy by decay and dominant damping effects, respectively. The propagation of the beam is essentially unaffected by the operation of the decay process. The decay process is thus slaved to the primary beam-plasma evolution, as assumed in previous studies. A variation of the ratio of electron temperature to ion temperature is found to affect not only the ion-acoustic wave levels through effects on the damping rate, but also the dynamics of decay via effects on the decay rate. The latter was not addressed in previous studies. Furthermore, spontaneous emission of ion-acoustic waves is found to affect the dynamics of decay, thus its inclusion is necessary to correctly model the Langmuir and ion-acoustic spectra
12. Incompressible Modes Excited by Supersonic Shear in Boundary Layers: Acoustic CFS Instability
Energy Technology Data Exchange (ETDEWEB)
Belyaev, Mikhail A., E-mail: [email protected] [Astronomy Department, University of California, Berkeley, CA 94720 (United States)
2017-02-01
We present an instability for exciting incompressible modes (e.g., gravity or Rossby modes) at the surface of a star accreting through a boundary layer. The instability excites a stellar mode by sourcing an acoustic wave in the disk at the boundary layer, which carries a flux of energy and angular momentum with the opposite sign as the energy and angular momentum density of the stellar mode. We call this instability the acoustic Chandrasekhar–Friedman–Schutz (CFS) instability, because of the direct analogy to the CFS instability for exciting modes on a rotating star by emission of energy in the form of gravitational waves. However, the acoustic CFS instability differs from its gravitational wave counterpart in that the fluid medium in which the acoustic wave propagates (i.e., the accretion disk) typically rotates faster than the star in which the incompressible mode is sourced. For this reason, the instability can operate even for a non-rotating star in the presence of an accretion disk. We discuss applications of our results to high-frequency quasi-periodic oscillations in accreting black hole and neutron star systems and dwarf nova oscillations in cataclysmic variables.
13. The Current-Driven, Ion-Acoustic Instability in a Collisionless Plasma
DEFF Research Database (Denmark)
Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens
1979-01-01
The current-driven, ion-acoustic instability was investigated by means of an experiment performed in a collisionless plasma produced in a single-ended Q-machine. Reflections at the ends of the plasma column gave rise to a standing wave. Parameters of the instability were investigated, and it was ......, and it was demonstrated that the fluctuations in the plasma column behave as a classical Van der Pol oscillator. Accurate measurements of the growth rate of the instability can be performed by making explicit use of the particular properties of such a system.......The current-driven, ion-acoustic instability was investigated by means of an experiment performed in a collisionless plasma produced in a single-ended Q-machine. Reflections at the ends of the plasma column gave rise to a standing wave. Parameters of the instability were investigated...
14. The influence of electron inertia on the modulational instability of ion-acoustic waves
International Nuclear Information System (INIS)
Parkes, E.J.
1993-01-01
The influence of electron inertia, ion streaming and weak relativistic effects on the modulational instability of ion-acoustic waves in a collisionless unmagnetized plasma is investigated. The derivative expansion method is used to derive a nonlinear Schroedinger equation, from which an instability criterion is deduced. When electron inertia is ignored, ion streaming and weak relativistic effects have little effect on the instability criterion. It is shown that when electron inertia is taken into account, the instability criterion is sensitive to weakly relativistic ion streaming, but not to the ratio of electron mass to ion mass. (Author)
15. A SIMPLE TOY MODEL OF THE ADVECTIVE-ACOUSTIC INSTABILITY. I. PERTURBATIVE APPROACH
International Nuclear Information System (INIS)
Foglizzo, T.
2009-01-01
Some general properties of the advective-acoustic instability are described and understood using a toy model, which is simple enough to allow for analytical estimates of the eigenfrequencies. The essential ingredients of this model, in the unperturbed regime, are a stationary shock and a subsonic region of deceleration. For the sake of analytical simplicity, the two-dimensional unperturbed flow is parallel and the deceleration is produced adiabatically by an external potential. The instability mechanism is determined unambiguously as the consequence of a cycle between advected and acoustic perturbations. The purely acoustic cycle, considered alone, is proven to be stable in this flow. Its contribution to the instability can be either constructive or destructive. A frequency cutoff is associated with the advection time through the region of deceleration. This cutoff frequency explains why the instability favors eigenmodes with a low frequency and a large horizontal wavelength. The relation between the instability occurring in this highly simplified toy model and the properties of standing accretion shock instability observed in the numerical simulations of stellar core collapse is discussed. This simple setup is proposed as a benchmark test to evaluate the accuracy, in the linear regime, of numerical simulations involving this instability. We illustrate such benchmark simulations in a companion paper.
16. The formation and dissipation of electrostatic shock waves: the role of ion–ion acoustic instabilities
Science.gov (United States)
Zhang, Wen-shuai; Cai, Hong-bo; Zhu, Shao-ping
2018-05-01
The role of ion–ion acoustic instabilities in the formation and dissipation of collisionless electrostatic shock waves driven by counter-streaming supersonic plasma flows has been investigated via two-dimensional particle-in-cell simulations. The nonlinear evolution of unstable waves and ion velocity distributions has been analyzed in detail. It is found that for electrostatic shocks driven by moderate-velocity flows, longitudinal and oblique ion–ion acoustic instabilities can be excited in the downstream and upstream regions, which lead to thermalization of the transmitted and reflected ions, respectively. For high-velocity flows, oblique ion–ion acoustic instabilities can develop in the overlap layer during the shock formation process and impede the shock formation.
17. 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
It is well known that acoustic waves exert forces on a boundary with which they interact; these forces can be so intense that they can compensate for the weight of small objects up to a few grams. In this way, it is possible to maintain solid or liquid samples levitating in a fluid, avoiding...... the use of containers, which may be undesirable for certain applications. Moreover, small samples can be manipulated by means of acoustic waves. In this paper, we report a study on the oscillational instabilities that can appear on a levitated solid sphere in single-axis acoustic devices. A theory...... 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...
18. Cross-field dust acoustic instability in a dusty negative ion plasma
International Nuclear Information System (INIS)
Rosenberg, M
2010-01-01
A cross-field dust acoustic instability in a dusty negative ion plasma in a magnetic field is studied using kinetic theory. The instability is driven by the ExB drifts of the ions. It is assumed that the negative ions are much heavier than the positive ions, and that the dust is negatively charged. The case where the positive ions and electrons are magnetized, the negative ions are marginally unmagnetized, and the dust is unmagnetized is considered. The focus is on a situation where Doppler resonances near harmonics of the positive ion gyrofrequency can affect the spectrum of unstable dust acoustic waves. Application to possible laboratory experimental parameters is discussed.
19. A Monte-Carlo investigation of the uncertainty of acoustic decay measurements
DEFF Research Database (Denmark)
Cabo, David Pérez; Seoane, Manuel A. Sobreira; Jacobsen, Finn
2012-01-01
Measurement of acoustic decays can be problematic at low frequencies: short decays cannot be evaluated accurately. Several effects influencing the evaluation will be reviewed in this paper. As new contribution, the measurement uncertainty due to one-third octave band pass filters will be analysed...
20. A Comment on Interaction of Lower Hybrid Waves with the Current-Driven Ion-Acoustic Instability
DEFF Research Database (Denmark)
Schrittwieser, R.; Juul Rasmussen, Jens
1985-01-01
Majeski et al. (1984) have investigated the interaction between the current-driven 'ion-acoustic' instability and high frequency lower hybrid waves. The 'ion-acoustic' instability was excited by drawing an electron current through the plasma column of a single-ended Q-machine by means...... of a positively biased cold plate. Schmittwieser et al. do not believe that the observed instability is of the ion-acoustic type but that it is rather the so-called potential relaxation instability....
1. From the advective-acoustic instability to the asymmetric explosions of Core Collapse Supernovae
International Nuclear Information System (INIS)
Galletti, Pascal
2005-01-01
The advective-acoustic cycle is a hydrodynamical mechanism fed by the coupling between advected waves (entropy, vorticity) and an acoustic feedback. Already studied in physics (rumble instability in ramjet, whistling tea kettle), it was introduced in astrophysics in the frame of the instability of the Bondi-Hoyle-Lyttleton accretion flow. In this thesis, we propose this cycle as an explanation for the asymmetry of the explosion of Core Collapse Supernovae. The evaluation of Eigenmodes for the classical accretion above a solid surface (white dwarfs, neutron stars) and the use of a toy-model reveal the importance of the advective-acoustic cycle in such an instable accretion flow. Following these results and the comparison with numerical simulations, a modelization of the flow when the shock stalls during a Core Collapse Supernova, shows that the advective-acoustic cycle is a natural mechanism to explain the non-spherical instability of the shock. The domination of l = 1 modes may be responsible for the observed pulsar kicks. (author) [fr
2. Development of beam instability in a plasma in the presence of ion-acoustic turbulence
International Nuclear Information System (INIS)
Popel', S.I.
1993-01-01
Effect of radiation-resonance interactions (RRI) of ion-acoustic waves and electrons is accounted for in consideration of the beam instability in a plasma in the presence of ion-acoustic turbulences. It is shown that variation of the superthermal part of the electron distribution function due to fast particle generation, conditioned by RRI of ion-acoustic waves and plasma electrons, leads to decreasing the increment of Langmuir wave swinging and may lead to beam instability stabilization. Conditions are obtained for excess of electron energy increase rate due to RRI over their energy increase rate due to nonlinear and quasi-linear interactions of resonant and nonresonant interactions with wave beam
3. Ion acoustic instability of HPT particles, FAC density, anomalous ...
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
electromagnetic field grow at the expense of the electron kinetic energy and as a result, vari- ous instabilities set in. The magnetometer data indicate that these waves are spatially corre- lated with perturbations; we interpret them as field aligned current (FAC) layers directed into and out of the auroral ionosphere. Kinetic the-.
4. Auroral ion beams and ion acoustic wave generation by fan instability
Energy Technology Data Exchange (ETDEWEB)
1996-04-01
Satellite observations indicate that efficient energy transport among various plasma particles and between plasma waves and plasma particles is taking place in auroral ion beam regions. These observations show that two characteristic wave types are associated with the auroral ion beam regions: electrostatic hydrogen cyclotron waves with frequencies above hydrogen gyrofrequency, and low frequency waves with frequencies below hydrogen gyrofrequency. We speculate that the low frequency waves can be ion acoustic waves generated through the fan instability. The presence of a cold background ion component is necessary for the onset of this instability. A cold ion component has been directly observed and has been indirectly suggested from observations of solitary wave structures. The wave-particle interaction during the development of the fan instability results in an efficient ion beam heating in the direction perpendicular to the ambient magnetic field. The fan instability development and the ion beam heating is demonstrated in a numerical particle simulation. 23 refs, 16 figs.
5. Method of determination of thermo-acoustic coolant instability boundaries in reactor core at NPPs with WWER
International Nuclear Information System (INIS)
Skalozubov, Volodymyr; Kolykhanov, Viktor; Kovryzhkin, Yuriy
2007-01-01
The regulatory body of Ukraine, the National Atomic Energy Company and the Scientific and Production Centre have led team-works concerned with previously unstudied factors or phenomena affecting reactor safety. As a result it is determined that the thermo-acoustic coolant instability conditions can appear in the core at definite operating WWER regimes. Considerable cyclic dynamic loads affect fuel claddings over thermo-acoustic pressure oscillations. These loads can result in inadmissible cassette design damage and containment damage. Taking into account calculation and experimental research authors submit a method of on-line assessment of WWER core state concerning thermo-acoustic coolant instability. According to this method, the thermo-acoustic coolant instability appearance conditions can be estimated using normal registered parameters (pressure, temperature, heat demand etc.). At operative modes, a WWER-1000 core is stable to tracheotomies oscillations, but reduction of coolant discharge through the core for some times can result in thermo-acoustic coolant instability. Thermo-acoustic instability appears at separate transitional modes concerned with reactor scram and unloading/loading at all power units. When thermo-acoustic instability begins in transitional modes, core elements are under influence of high-frequency coolant pressure pulsations for a long time (tens of hours)
6. On Long-Time Instabilities in Staggered Finite Difference Simulations of the Seismic Acoustic Wave Equations on Discontinuous Grids
KAUST Repository
Gao, Longfei; Ketcheson, David I.; Keyes, David E.
2017-01-01
We consider the long-time instability issue associated with finite difference simulation of seismic acoustic wave equations on discontinuous grids. This issue is exhibited by a prototype algebraic problem abstracted from practical application
7. On the role of acoustic feedback in boundary-layer instability.
Science.gov (United States)
Wu, Xuesong
2014-07-28
In this paper, the classical triple-deck formalism is employed to investigate two instability problems in which an acoustic feedback loop plays an essential role. The first concerns a subsonic boundary layer over a flat plate on which two well-separated roughness elements are present. A spatially amplifying Tollmien-Schlichting (T-S) wave between the roughness elements is scattered by the downstream roughness to emit a sound wave that propagates upstream and impinges on the upstream roughness to regenerate the T-S wave, thereby forming a closed feedback loop in the streamwise direction. Numerical calculations suggest that, at high Reynolds numbers and for moderate roughness heights, the long-range acoustic coupling may lead to absolute instability, which is characterized by self-sustained oscillations at discrete frequencies. The dominant peak frequency may jump from one value to another as the Reynolds number, or the distance between the roughness elements, is varied gradually. The second problem concerns the supersonic 'twin boundary layers' that develop along two well-separated parallel flat plates. The two boundary layers are in mutual interaction through the impinging and reflected acoustic waves. It is found that the interaction leads to a new instability that is absent in the unconfined boundary layer. © 2014 The Author(s) Published by the Royal Society. All rights reserved.
8. Observation of the low-frequency ion acoustic instability in the turbulently heated TRIAM-1 tokamak plasma
Energy Technology Data Exchange (ETDEWEB)
Mitarai, O; Watanabe, T; Nakamura, Y; Nakamura, K; Hiraki, N; Toi, K; Kawai, Y; Itoh, S [Kyushu Univ., Fukuoka (Japan). Research Inst. for Applied Mechanics
1980-12-01
Density fluctuations in the frequency range of several MHz are observed in the turbulently heated TRIAM-1 tokamak plasma by means of a 4 mm microwave scattering method. It is found from the measurement of the dispersion relation that this instability is considered to be the low-frequency ion acoustic instability propagating nearly perpendicular to the toroidal magnetic field.
9. Anomalous phenomena in ECRH experiments at toroidal devices and low-threshold parametric decay instabilities
Directory of Open Access Journals (Sweden)
Saveliev A.N.
2012-09-01
Full Text Available In the paper the possibility of total 3D trapping of electron Bernstein (EB waves in the tokamak equatorial plane in the vicinity of the local density maximum produced by electron pump-out-effect is demonstrated. Thresholds and growth rates of the associated absolute (temporally growing parametric decay instability (PDI leading to anomalous absorption is predicted in the range of less than 100 kW. Its possible role in explanation of ion acceleration observed in ECRH experiments as well as in redistribution of the deposited power is discussed.
10. Dust-acoustic instability in an inductive gas-discharge plasma
International Nuclear Information System (INIS)
Zobnin, A.V.; Usachev, A.D.; Petrov, O.F.; Fortov, V.E.
2002-01-01
Spontaneous excitation of a dust-particle density wave is observed in a dust cloud levitating in the region of the diffused edge of an rf inductive low-pressure gas-discharge plasma. The main physical parameters of this wave and of the background plasma are measured. The analytic model proposed for the observed phenomenon is based on the theory of dust sound and successfully correlates with experimental data in a wide range of experimental conditions. The effect of variable charge of dust particles on the evolution of the observed dust-plasma instability is studied analytically. It is shown that the necessary condition for the development of the dust-acoustic instability is the presence of a dc electric field in the dust cloud region
11. Analysis of oscillational instabilities in acoustic levitation using the finite-difference time-domain method
DEFF Research Database (Denmark)
Santillan, Arturo Orozco
2011-01-01
The aim of the work described in this paper has been to investigate the use of the finite-difference time-domain method to describe the interactions between a moving object and a sound field. The main objective was to simulate oscillational instabilities that appear in single-axis acoustic...... levitation devices and to describe their evolution in time to further understand the physical mechanism involved. The study shows that the method gives accurate results for steady state conditions, and that it is a promising tool for simulations with a moving object....
12. Impact of Relativistic Electron Beam on Hole Acoustic Instability in Quantum Semiconductor Plasmas
Science.gov (United States)
Siddique, M.; Jamil, M.; Rasheed, A.; Areeb, F.; Javed, Asif; Sumera, P.
2018-01-01
We studied the influence of the classical relativistic beam of electrons on the hole acoustic wave (HAW) instability exciting in the semiconductor quantum plasmas. We conducted this study by using the quantum-hydrodynamic model of dense plasmas, incorporating the quantum effects of semiconductor plasma species which include degeneracy pressure, exchange-correlation potential and Bohm potential. Analysis of the quantum characteristics of semiconductor plasma species along with relativistic effect of beam electrons on the dispersion relation of the HAW is given in detail qualitatively and quantitatively by plotting them numerically. It is worth mentioning that the relativistic electron beam (REB) stabilises the HAWs exciting in semiconductor (GaAs) degenerate plasma.
13. Instability of nonplanar modulated dust acoustic wave packets in a strongly coupled nonthermal dusty plasma
Energy Technology Data Exchange (ETDEWEB)
El-Labany, S. K., E-mail: [email protected]; Zedan, N. A., E-mail: [email protected] [Department of Physics, Faculty of Science, Damietta University, New Damietta, P.O. 34517 (Egypt); El-Taibany, W. F., E-mail: [email protected], E-mail: [email protected] [Department of Physics, Faculty of Science, Damietta University, New Damietta, P.O. 34517 (Egypt); Department of Physics, College of Science for Girls in Abha, King Khalid University, P.O. 960 Abha (Saudi Arabia)
2015-07-15
Cylindrical and spherical amplitude modulations of dust acoustic (DA) solitary wave envelopes in a strongly coupled dusty plasma containing nonthermal distributed ions are studied. Employing a reductive perturbation technique, a modified nonlinear Schrödinger equation including the geometrical effect is derived. The influences of nonthermal ions, polarization force, and the geometries on the modulational instability conditions are analyzed and the possible rogue wave structures are discussed in detail. It is found that the spherical DA waves are more structurally stable to perturbations than the cylindrical ones. Possible applications of these theoretical findings are briefly discussed.
14. Limit cycle behaviour of the bump-on-tail and ion-acoustic instability
International Nuclear Information System (INIS)
Janssen, P.A.E.M.; Rasmussen, J.J.
1980-12-01
The nonlinear dynamics of the bump-on-tail and current-driven ion-acoustic instability is considered. The eigenmodes have discrete k because of finite periodic boundary conditions. Increasing a critical parameter (the number density and the electron drift velocity respectively) above its neutral stable value by a small fractional amount Δ 2 , one mode becomes unstable. The nonlinear dynamics of the unstable mode is determined by means of the multiple time scale method. Usually, limit cycle behaviour is found. A short comparison with quasi-linear theory is given, and the results are compared with experiment. (Auth.)
15. Acoustic tweezing of particles using decaying opposing travelling surface acoustic waves (DOTSAW).
Science.gov (United States)
Ng, Jia Wei; Devendran, Citsabehsan; Neild, Adrian
2017-10-11
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.
16. Physics of the ion acoustic wave driven by the stimulated Brillouin scattering instability
International Nuclear Information System (INIS)
Clayton, C.E.
1984-01-01
The ion acoustic wave excited in the stimulated Brillouin scattering (SBS) instability is probed via collective ruby-laser Thomson scattering in order to understand the low saturation level observed in the instability. Many of the features observed in the Brillouin backscattered CO 2 laser light from the underdense gas-target plasma are also observed in the Thomson scattered ruby light - from which it is learned that the ion acoustic wave grows exponentially and then saturates as the CO 2 pump power is increased. The primary advantage of the ruby Thomson scattering diagnostic is in its capability of providing simultaneous space and time resolved measurements of the ion wave amplitude. From these first such detailed measurements, it was found that the ion wave grows exponentially in space at a rate that agrees with the linear convective SBS theory. However, at higher pump powers, the ion wave saturates at an inferred amplitude of anti-n/n 0 approx. = 5 to 10%. Further increases in the pump power appear to result in an increase in the length over which the ion wave is saturated. A nearly constant SBS reflectivity in this saturated regime, however, suggests that the saturated ion wave does not contribute as much to the scattered power as would be expected from Bragg scattering theory. This apparent contradiction can be resolved if ion trapping is responsible for the saturation of the ion wave
17. Landau quantization effects on hole-acoustic instability in semiconductor plasmas
Science.gov (United States)
Sumera, P.; Rasheed, A.; Jamil, M.; Siddique, M.; Areeb, F.
2017-12-01
The growth rate of the hole acoustic waves (HAWs) exciting in magnetized semiconductor quantum plasma pumped by the electron beam has been investigated. The instability of the waves contains quantum effects including the exchange and correlation potential, Bohm potential, Fermi-degenerate pressure, and the magnetic quantization of semiconductor plasma species. The effects of various plasma parameters, which include relative concentration of plasma particles, beam electron temperature, beam speed, plasma temperature (temperature of electrons/holes), and Landau electron orbital magnetic quantization parameter η, on the growth rate of HAWs, have been discussed. The numerical study of our model of acoustic waves has been applied, as an example, to the GaAs semiconductor exposed to electron beam in the magnetic field environment. An increment in either the concentration of the semiconductor electrons or the speed of beam electrons, in the presence of magnetic quantization of fermion orbital motion, enhances remarkably the growth rate of the HAWs. Although the growth rate of the waves reduces with a rise in the thermal temperature of plasma species, at a particular temperature, we receive a higher instability due to the contribution of magnetic quantization of fermions to it.
18. Ion-acoustic waves in ultracold neutral plasmas: Modulational instability and dissipative rogue waves
Energy Technology Data Exchange (ETDEWEB)
El-Tantawy, S.A., E-mail: [email protected]
2017-02-26
Progress is reported on the modulational instability (MI) of ion-acoustic waves (IAWs) and dissipative rogue waves (RWs) in ultracold neutral plasmas (UNPs). The UNPs consist of inertial ions fluid and Maxwellian inertialess hot electrons, and the presence of an ion kinematic viscosity is allowed. For this purpose, a modified nonlinear Schrödinger equation (NLSE) is derived and then solved analytically to show the occurrence of MI. It is found that the (in)stability regions of the wavepacks are dependent on time due to of the existence of the dissipative term. The existing regions of the MI of the IAWs are inventoried precisely. After that, we use a suitable transformation to convert the modified NLSE into the normal NLSE whose analytical solutions for rogue waves are known. The rogue wave propagation condition and its behavior are discussed. The impact of the relevant physical parameters on the profile of the RWs is examined. - Highlights: • UNPs are modeled by the phenomenological generalized hydrodynamic equations. • The derivative expansion method has been employed in order to derive a modified-NLSE. • A suitable transformation is used to transform the modified-NLSE into the standard NLSE. • The effect of the ion viscosity on the modulational instability and rogue waves is investigated.
19. Decay of Wannier-Mott excitons interacting with acoustic phonon in semiconductors with a degenerate valence band
International Nuclear Information System (INIS)
Nguyen Toan Thang; Nguyen Ai Viet; Nguyen Hong Quang
1987-06-01
Decay probabilities of light and heavy excitons interacting with acoustic phonons in cubic semiconductors with a degenerate valence band are calculated. The numerical results for GaAs showed that the decay probability of the light exciton is much greater than that of the heavy one. (author). 10 refs, 1 fig
20. Pulsational Pair-instability Model for Superluminous Supernova PTF12dam:Interaction and Radioactive Decay
Energy Technology Data Exchange (ETDEWEB)
Tolstov, Alexey; Nomoto, Ken’ichi; Blinnikov, Sergei; Quimby, Robert [Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583 (Japan); Sorokina, Elena [Sternberg Astronomical Institute, M.V.Lomonosov Moscow State University, 119991 Moscow (Russian Federation); Baklanov, Petr, E-mail: [email protected] [Institute for Theoretical and Experimental Physics (ITEP), 117218 Moscow (Russian Federation)
2017-02-01
Being a superluminous supernova, PTF12dam can be explained by a {sup 56}Ni-powered model, a magnetar-powered model, or an interaction model. We propose that PTF12dam is a pulsational pair-instability supernova, where the outer envelope of a progenitor is ejected during the pulsations. Thus, it is powered by a double energy source: radioactive decay of {sup 56}Ni and a radiative shock in a dense circumstellar medium. To describe multicolor light curves and spectra, we use radiation-hydrodynamics calculations of the STELLA code. We found that light curves are well described in the model with 40 M {sub ⊙} ejecta and 20–40 M {sub ⊙} circumstellar medium. The ejected {sup 56}Ni mass is about 6 M {sub ⊙}, which results from explosive nucleosynthesis with large explosion energy (2–3)×10{sup 52} erg. In comparison with alternative scenarios of pair-instability supernova and magnetar-powered supernova, in the interaction model, all the observed main photometric characteristics are well reproduced: multicolor light curves, color temperatures, and photospheric velocities.
1. Dissipation of Alfven Waves at Fluid Scale through Parametric Decay Instabilities in Low-beta Turbulent Plasma
Science.gov (United States)
Fu, X.; Li, H.; Guo, F.; Li, X.; Roytershteyn, V.
2017-12-01
The solar wind is a turbulent magnetized plasma extending from the upper atmosphere of the sun to the edge of the heliosphere. It carries charged particles and magnetic fields originated from the Sun, which have great impact on the geomagnetic environment and human activities in space. In such a magnetized plasma, Alfven waves play a crucial role in carrying energy from the surface of the Sun, injecting into the solar wind and establishing power-law spectra through turbulent energy cascades. On the other hand, in compressible plasmas large amplitude Alfven waves are subject to a parametric decay instability (PDI) which converts an Alfven wave to another counter-propagating Alfven wave and an ion acoustic wave (slow mode). The counter-propagating Alfven wave provides an important ingredient for turbulent cascade, and the slow-mode wave provides a channel for solar wind heating in a spatial scale much larger than ion kinetic scales. Growth and saturation of PDI in quiet plasma have been intensively studied using linear theory and nonlinear simulations in the past. Here using 3D hybrid simulations, we show that PDI is still effective in turbulent low-beta plasmas, generating slow modes and causing ion heating. Selected events in WIND data are analyzed to identify slow modes in the solar wind and the role of PDI, and compared with our simulation results. We also investigate the validity of linear Vlasov theory regarding PDI growth and slow mode damping in turbulent plasmas. Since PDI favors low plasma beta, we expect to see more evidence of PDI in the solar wind close to the Sun, especially from the upcoming NASA's Parker Solar Probe mission which will provide unprecedented wave and plasma data as close as 8.5 solar radii from the Sun.
2. Simultaneous measurement of surface tension and viscosity using freely decaying oscillations of acoustically levitated droplets
Science.gov (United States)
Kremer, J.; Kilzer, A.; Petermann, M.
2018-01-01
Oscillations of small liquid drops around a spherical shape have been of great interest to scientists measuring physical properties such as interfacial tension and viscosity, over the last few decades. A powerful tool for contactless positioning is acoustic levitation, which has been used to simultaneously determine the surface tension and viscosity of liquids at ambient pressure. In order to extend this acoustic levitation measurement method to high pressure systems, the method is first evaluated under ambient pressure. To measure surface tension and viscosity using acoustically levitated oscillating drops, an image analysis method has to be developed and factors which may affect measurement, such as sound field or oscillation amplitude, have to be analyzed. In this paper, we describe the simultaneous measurement of surface tension and viscosity using freely decaying shape oscillations of acoustically levitated droplets of different liquids (silicone oils AK 5 and AK 10, squalane, 1-propanol, 1-butanol, 1-pentanol, 1-hexanol, 1-heptanol, and 1-octanol) in air. These liquids vary in viscosity from 2 to about 30 mPa s. An acoustic levitation system, including an optimized standing wave acoustic levitator and a high-speed camera, was used for this study. An image analysis was performed with a self-written Matlab® code. The frequency of oscillation and the damping constant, required for the determination of surface tension and viscosity, respectively, were calculated from the evolution of the equatorial and polar radii. The results and observations are compared to data from the literature in order to analyze the accuracy of surface tension and viscosity determination, as well as the effect of non-spherical drop shape or amplitude of oscillation on measurement.
3. Investigation on thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing in scramjet cooling channel based on wavelet entropy method
Science.gov (United States)
Zan, Hao; Li, Haowei; Jiang, Yuguang; Wu, Meng; Zhou, Weixing; Bao, Wen
2018-06-01
As part of our efforts to find ways and means to further improve the regenerative cooling technology in scramjet, the experiments of thermo-acoustic instability dynamic characteristics of hydrocarbon fuel flowing have been conducted in horizontal circular tubes at different conditions. The experimental results indicate that there is a developing process from thermo-acoustic stability to instability. In order to have a deep understanding on the developing process of thermo-acoustic instability, the method of Multi-scale Shannon Wavelet Entropy (MSWE) based on Wavelet Transform Correlation Filter (WTCF) and Multi-Scale Shannon Entropy (MSE) is adopted in this paper. The results demonstrate that the developing process of thermo-acoustic instability from noise and weak signals is well detected by MSWE method and the differences among the stability, the developing process and the instability can be identified. These properties render the method particularly powerful for warning thermo-acoustic instability of hydrocarbon fuel flowing in scramjet cooling channels. The mass flow rate and the inlet pressure will make an influence on the developing process of the thermo-acoustic instability. The investigation on thermo-acoustic instability dynamic characteristics at supercritical pressure based on wavelet entropy method offers guidance on the control of scramjet fuel supply, which can secure stable fuel flowing in regenerative cooling system.
4. 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.
5. Nonlinear ion-acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves
International Nuclear Information System (INIS)
Guo, Shimin; Mei, Liquan; Sun, Anbang
2013-01-01
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
6. Experimental observation of pulsating instability under acoustic field in downward-propagating flames at large Lewis number
KAUST Repository
Yoon, Sung Hwan
2017-10-12
According to previous theory, pulsating propagation in a premixed flame only appears when the reduced Lewis number, β(Le-1), is larger than a critical value (Sivashinsky criterion: 4(1 +3) ≈ 11), where β represents the Zel\\'dovich number (for general premixed flames, β ≈ 10), which requires Lewis number Le > 2.1. However, few experimental observation have been reported because the critical reduced Lewis number for the onset of pulsating instability is beyond what can be reached in experiments. Furthermore, the coupling with the unavoidable hydrodynamic instability limits the observation of pure pulsating instabilities in flames. Here, we describe a novel method to observe the pulsating instability. We utilize a thermoacoustic field caused by interaction between heat release and acoustic pressure fluctuations of the downward-propagating premixed flames in a tube to enhance conductive heat loss at the tube wall and radiative heat loss at the open end of the tube due to extended flame residence time by diminished flame surface area, i.e., flat flame. The thermoacoustic field allowed pure observation of the pulsating motion since the primary acoustic force suppressed the intrinsic hydrodynamic instability resulting from thermal expansion. By employing this method, we have provided new experimental observations of the pulsating instability for premixed flames. The Lewis number (i.e., Le ≈ 1.86) was less than the critical value suggested previously.
7. Nonlinear development of the two-plasmon decay instability in three dimensions
Energy Technology Data Exchange (ETDEWEB)
Vu, H. X. [University of California, San Diego, La Jolla, California 92093 (United States); DuBois, D. F.; Russell, D. A. [Lodestar Research Corporation, Boulder, Colorado 80301 (United States); Myatt, J. F.; Zhang, J. [Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623 (United States); Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627 (United States)
2014-04-15
Most recent experiments on the excitation of the two plasmon-decay (TPD) instability involve a three-dimensional (3D) array of overlapping laser beams. Our recent two dimensional (2D) simulations suggested that Langmuir cavitation and collapse are important nonlinear saturation mechanisms for TPD. There are important quantitative differences in the Langmuir collapse process in 2D and 3D. To address these and other issues, we have developed a 3D Zakharov code. It has been applied to study the evolution of TPD from absolute instabilities (arising from 3D laser geometries) to the nonlinear state (J. Zhang et al., Phys. Rev. Lett. (submitted)). The present paper concentrates on the nonlinear saturated state excited by the collective action of two crossed laser beams with arbitrary polarizations. Remarkable agreement between 3D and 2D simulations is found for several averaged physical quantities when the beams are polarized in their common plane. As in the previous 2D simulations, we find: (a) the collective, initially convectively unstable triad modes dominate after a sub-picosecond burst, (b) Langmuir cavitation and collapse are important nonlinearities, and (c) that the statistics of intense cavitons are characteristic of a Gaussian random process. The 3D steady-state saturated Langmuir energy level is about 30% higher than in 2D. The auto-correlation functions of the Langmuir envelope field and of the low-frequency electron density field yield the spatial shape of the strongest collapsing cavitons which are 3D ellipsoids whose orientation depends on the laser polarizations. This tilting of the caviton's strongest electric field direction away from the normal to the target surface is a major new 3D result. This tilting may deflect the hot electron flux and thereby mitigate target preheat.
8. Ionization and acoustical instability of a low temperature magnetized plasma in a combined (direct and alternating) electrical field
International Nuclear Information System (INIS)
Andropov, V.G.; Sinkevich, O.A.
1983-01-01
It is shown that the ionization front which moves through a gas along a magnetic field in a combined electrical field, which lies in the plane of the front, may be unstable, as a result of the development of an ionization instability in the plasma behind the front. The criterion of instability of the ionization front does not greatly differ from the criterion of instability of an infinite plasma. The ionization front in the magnetic field is stable only in an electrical field of circular polarization or in a combined field in which the direct and alternating electrical fields are orthogonal and the Joule heat liberation from them is equal. The generation of sound is possible in a magnetized plasma in an alternating electrical field orthogonal to a magnetic due to the parametric acoustical instability at the frequency of the external electrical field. 8 refs
9. Universal instability of dust ion-sound waves and dust-acoustic waves
International Nuclear Information System (INIS)
Tsytovich, V.N.; Watanabe, K.
2002-01-01
It is shown that the dust ion-sound waves (DISW) and the dust-acoustic waves (DAW) are universally unstable for wave numbers less than some critical wave number. The basic dusty plasma state is assumed to be quasi-neutral with balance of the plasma particle absorption on the dust particles and the ionization with the rate proportional to the electron density. An analytical expression for the critical wave numbers, for the frequencies and for the growth rates of DISW and DAW are found using the hydrodynamic description of dusty plasma components with self-consistent treatment of the dust charge variations and by taking into account the change of the ion and electron distributions in the dust charging process. Most of the previous treatment do not take into account the latter process and do not treat the basic state self-consistently. The critical lengths corresponding to these critical wave numbers can be easily achieved in the existing experiments. It is shown that at the wave numbers larger than the critical ones DISW and DAW have a large damping which was not treated previously and which can be also measured. The instabilities found in the present work on their non linear stage can lead to formation of different types of dust self-organized structures. (author)
10. Dark matter component decaying after recombination: Sensitivity to baryon acoustic oscillation and redshift space distortion probes
Science.gov (United States)
Chudaykin, A.; Gorbunov, D.; Tkachev, I.
2018-04-01
It has been recently suggested [1] that a subdominant fraction of dark matter decaying after recombination may alleviate tension between high-redshift (CMB anisotropy) and low-redshift (Hubble constant, cluster counts) measurements. In this report, we continue our previous study [2] of the decaying dark matter (DDM) model adding all available recent baryon acoustic oscillation (BAO) and redshift space distortions (RSD) measurements. We find that the BAO/RSD measurements generically prefer the standard Λ CDM and combined with other cosmological measurements impose an upper limit on the DDM fraction at the level of ˜5 %, strengthening by a factor of 1.5 limits obtained in [2] mostly from CMB data. However, the numbers vary from one analysis to other based on the same Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 (DR12) galaxy sample. Overall, the model with a few percent DDM fraction provides a better fit to the combined cosmological data as compared to the Λ CDM : the cluster counting and direct measurements of the Hubble parameter are responsible for that. The improvement can be as large as 1.5 σ and grows to 3.3 σ when the CMB lensing power amplitude AL is introduced as a free fitting parameter.
11. Modulational instability of ion-acoustic soliton in a multicomponent plasma
International Nuclear Information System (INIS)
Tsukabayashi, I.; Yagishita, T.; Nakamura, Y.
1986-01-01
An experiment has been performed in a multi-dipole double plasma device. The inner diameter is 80 cm and its total length is 150 cm. The chamber is evacuated down to 8x10/sup -7/ Torr. Argon and sulfur hexafluoride are introduced independently into the chamber under continuous pumping. The pressure of Ar is 2 x 10/sup -4/ Torr and the partial pressure of SF/sub 6/ is changed 0 to 3 x 10/sup -8/ Torr. The plasma includes several species of positive and negative ions, SF/sub 6//sup -/. However, since ions of lighter mass dominate the ion-acoustic wave, the plasma is considered to be effectively composed of AR/sup +/, F/sup -/ and electrons. Initial modulated sinusoidal signals, the absolute amplitude 1.5 to 0.1 V, the percentage modulation 0 to 100%, the duration of the train 200μsec, the carrier frequency f/sub o/ = w/2π = 200 to 300 kHz and the modulation frequency Ω/2π=15 to 20 kHz, are applied to the driver plasma. The detected signals increase the percentage modulation with the distance from the separation grid, and the growth rate is proportional to the amplitude of applied signal. The measurement of the power spectra show that the frequency of the carrier wave shifts to the lower side-bands (f/sub o/ -Ω/2π and f/sub o/ -2Ω/2π) as development of the amplitude modulation instability. These results can be explained by the analysis of the N-S equation
12. Electron acoustic waves and parametric instabilities in a 4-component relativistic quantum plasma with Thomas-Fermi distributed electrons
Science.gov (United States)
Ikramullah, Ahmad, Rashid; Sharif, Saqib; Khattak, Fida Younus
2018-01-01
The interaction of Circularly Polarized Electro-Magnetic (CPEM) waves with a 4-component relativistic quantum plasma is studied. The plasma constituents are: relativistic-degenerate electrons and positrons, dynamic degenerate ions, and Thomas-Fermi distributed electrons in the background. We have employed the Klein-Gordon equations for the electrons as well as for the positrons, while the ions are represented by the Schrödinger equation. The Maxwell and Poisson equations are used for electromagnetic waves. Three modes are observed: one of the modes is associated with the electron acoustic wave, a second mode at frequencies greater than the electron acoustic wave mode could be associated with the positrons, and the third one at the lowest frequencies could be associated with the ions. Furthermore, Stimulated Raman Scattering (SRS), Modulational, and Stimulated Brillouin Scattering (SBS) instabilities are studied. It is observed that the growth rates of both the SRS and SBS instabilities decrease with increase in the quantum parameter of the plasma. It is also observed that the scattering spectra in both the SRS and SBS get restricted to very small wavenumber regions. It is shown that for low amplitude CPEM wave interaction with the quantum plasma, the positron concentration has no effect on the SRS and SBS spectra. In the case of large amplitude CPEM wave interaction, however, one observes spectral changes with varying positron concentrations. An increase in the positron concentration also enhances the scattering instability growth rates. Moreover, the growth rate first increases and then decreases with increasing intensity of the CPEM wave, indicating an optimum value of the CPEM wave intensity for the growth of these scattering instabilities. The modulational instability also shows dependence on the quantum parameter as well as on the positron concentration.
13. Thermo-acoustic instabilities in lean premixed swirl-stabilized combustion and their link to acoustically coupled and decoupled flame macrostructures
KAUST Repository
Taamallah, Soufien
2015-01-01
14. Parametric decay instability near the upper hybrid resonance in magnetically confined fusion plasmas
DEFF Research Database (Denmark)
Hansen, Søren Kjer; Nielsen, Stefan Kragh; Salewski, Mirko
2017-01-01
In this paper we investigate parametric decay of an electromagnetic pump wave into two electrostatic daughter waves, particularly an X-mode pump wave decaying into a warm upper hybrid wave (a limit of an electron Bernstein wave) and a warm lower hybrid wave. We describe the general theory...
15. Modified electron-acoustic and lower-hybrid drift dissipative instability in a two-electron temperature plasma
International Nuclear Information System (INIS)
Bose, M.
1989-01-01
It is often found, in fusion devices as well as in the auroral ionosphere, that the electrons consist of two distinct group, viz., hot and cold. These two-temperature electron model is sometimes convenient for analytical purposes. Thus the authors have considered a two-temperature electron plasma. In this paper, they investigated analytically the drift dissipative instabilities of modified electron-acoustic and lower-hybrid wve in a two-electron temperature plasma. It is found that the modified electron-acoustic drift dissipative mode are strongly dependent on the number density of cold electrons. From the expression of the growth rate, it is clear that these cold electrons can control the growth of this mode as well
16. On Long-Time Instabilities in Staggered Finite Difference Simulations of the Seismic Acoustic Wave Equations on Discontinuous Grids
KAUST Repository
Gao, Longfei
2017-10-26
We consider the long-time instability issue associated with finite difference simulation of seismic acoustic wave equations on discontinuous grids. This issue is exhibited by a prototype algebraic problem abstracted from practical application settings. Analysis of this algebraic problem leads to better understanding of the cause of the instability and provides guidance for its treatment. Specifically, we use the concept of discrete energy to derive the proper solution transfer operators and design an effective way to damp the unstable solution modes. Our investigation shows that the interpolation operators need to be matched with their companion restriction operators in order to properly couple the coarse and fine grids. Moreover, to provide effective damping, specially designed diffusive terms are introduced to the equations at designated locations and discretized with specially designed schemes. These techniques are applied to simulations in practical settings and are shown to lead to superior results in terms of both stability and accuracy.
17. On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids
Science.gov (United States)
Gao, Longfei; Ketcheson, David; Keyes, David
2018-02-01
We consider the long-time instability issue associated with finite difference simulation of seismic acoustic wave equations on discontinuous grids. This issue is exhibited by a prototype algebraic problem abstracted from practical application settings. Analysis of this algebraic problem leads to better understanding of the cause of the instability and provides guidance for its treatment. Specifically, we use the concept of discrete energy to derive the proper solution transfer operators and design an effective way to damp the unstable solution modes. Our investigation shows that the interpolation operators need to be matched with their companion restriction operators in order to properly couple the coarse and fine grids. Moreover, to provide effective damping, specially designed diffusive terms are introduced to the equations at designated locations and discretized with specially designed schemes. These techniques are applied to simulations in practical settings and are shown to lead to superior results in terms of both stability and accuracy.
18. Instability of dust ion-acoustic waves in a dusty plasma containing elongated and rotating charged dust grains
International Nuclear Information System (INIS)
Shukla, P.K.; Tskhakaya, D.D.
2001-01-01
The dispersion properties of the dust ion-acoustic waves (DIAWs) in an unmagnetized dusty plasma is examined when the plasma constituents are electrons, ions, and charged dust grains which are elongated and rotating. Since the dipole moment of elongated and rotating dust grains is nonzero, significant modifications of the DIAW spectrum emerge. It is found that the DIAWs are subjected to an instability when the DIAW frequency approximately equals the angular rotation frequency of the elongated dust grains. The relevance of our investigation to enhanced fluctuations in space and laboratory dusty plasmas is pointed out
19. The Effect of Acoustic Forcing on Instabilities and Breakdown in Swept-Wing Flow over a Backward-Facing Step
Science.gov (United States)
Eppink, Jenna L.; Shishkov, Olga; Wlezien, Richard W.; King, Rudolph A.; Choudhari, Meelan
2016-01-01
Instability interaction and breakdown were experimentally investigated in the flow over a swept backward-facing step. Acoustic forcing was used to excite the Tollmien-Schlichting (TS) instability and to acquire phase-locked results. The phase-averaged results illustrate the complex nature of the interaction between the TS and stationary cross flow instabilities. The weak stationary cross flow disturbance causes a distortion of the TS wavefront. The breakdown process is characterized by large positive and negative spikes in velocity. The positive spikes occur near the same time and location as the positive part of the TS wave. Higher-order spectral analysis was used to further investigate the nonlinear interactions between the TS instability and the traveling cross flow disturbances. The results reveal that a likely cause for the generation of the spikes corresponds to nonlinear interactions between the TS, traveling cross flow, and stationary cross flow disturbances. The spikes begin at low amplitudes of the unsteady and steady disturbances (2-4% U (sub e) (i.e. boundary layer edge velocity)) but can achieve very large amplitudes (20-30 percent U (sub e) (i.e. boundary layer edge velocity)) that initiate an early, though highly intermittent, breakdown to turbulence.
20. Amplitude modulation of quantum-ion-acoustic wavepackets in electron-positron-ion plasmas: Modulational instability, envelope modes, extreme wavesa)
Science.gov (United States)
Rahman, Ata-ur-; Kerr, Michael Mc; El-Taibany, Wael F.; Kourakis, Ioannis; Qamar, A.
2015-02-01
A semirelativistic fluid model is employed to describe the nonlinear amplitude modulation of low-frequency (ionic scale) electrostatic waves in an unmagnetized electron-positron-ion plasma. Electrons and positrons are assumed to be degenerated and inertialess, whereas ions are warm and classical. A multiscale perturbation method is used to derive a nonlinear Schrödinger equation for the envelope amplitude, based on which the occurrence of modulational instability is investigated in detail. Various types of localized ion acoustic excitations are shown to exist, in the form of either bright type envelope solitons (envelope pulses) or dark-type envelope solitons (voids, holes). The plasma configurational parameters (namely, the relativistic degeneracy parameter, the positron concentration, and the ionic temperature) are shown to affect the conditions for modulational instability significantly, in fact modifying the associated threshold as well as the instability growth rate. In particular, the relativistic degeneracy parameter leads to an enhancement of the modulational instability mechanism. Furthermore, the effect of different relevant plasma parameters on the characteristics (amplitude, width) of these envelope solitary structures is also presented in detail. Finally, the occurrence of extreme amplitude excitation (rogue waves) is also discussed briefly. Our results aim at elucidating the formation and dynamics of nonlinear electrostatic excitations in superdense astrophysical regimes.
1. Amplitude modulation of quantum-ion-acoustic wavepackets in electron-positron-ion plasmas: Modulational instability, envelope modes, extreme waves
Energy Technology Data Exchange (ETDEWEB)
Rahman, Ata-ur-, E-mail: [email protected] [Department of Physics, University of Peshawar, Peshawar 25000 (Pakistan); Department of Physics, Islamia College Peshawar, Khyber Pakhtunkhwa (Pakistan); Kerr, Michael Mc, E-mail: [email protected]; Kourakis, Ioannis, E-mail: [email protected] [Centre for Plasma Physics, Department of Physics and Astronomy, Queen' s University Belfast, BT7 1NN Northern Ireland (United Kingdom); El-Taibany, Wael F., E-mail: [email protected] [Department of Physics, Faculty of Science, Damietta University, New Damietta, P.O. Box 34517 (Egypt); Department of Physics, College of Science for Girls in Abha, King Khalid University, P.O. Box 960, Abha (Saudi Arabia); Qamar, A., E-mail: [email protected] [Department of Physics, University of Peshawar, Peshawar 25000 (Pakistan)
2015-02-15
A semirelativistic fluid model is employed to describe the nonlinear amplitude modulation of low-frequency (ionic scale) electrostatic waves in an unmagnetized electron-positron-ion plasma. Electrons and positrons are assumed to be degenerated and inertialess, whereas ions are warm and classical. A multiscale perturbation method is used to derive a nonlinear Schrödinger equation for the envelope amplitude, based on which the occurrence of modulational instability is investigated in detail. Various types of localized ion acoustic excitations are shown to exist, in the form of either bright type envelope solitons (envelope pulses) or dark-type envelope solitons (voids, holes). The plasma configurational parameters (namely, the relativistic degeneracy parameter, the positron concentration, and the ionic temperature) are shown to affect the conditions for modulational instability significantly, in fact modifying the associated threshold as well as the instability growth rate. In particular, the relativistic degeneracy parameter leads to an enhancement of the modulational instability mechanism. Furthermore, the effect of different relevant plasma parameters on the characteristics (amplitude, width) of these envelope solitary structures is also presented in detail. Finally, the occurrence of extreme amplitude excitation (rogue waves) is also discussed briefly. Our results aim at elucidating the formation and dynamics of nonlinear electrostatic excitations in superdense astrophysical regimes.
2. Correspondence Between “Stable” Flame Macrostructure and Thermo-acoustic Instability in Premixed Swirl-Stabilized Turbulent Combustion
KAUST Repository
Taamallah, Soufien; LaBry, Zachary A.; Shanbhogue, Santosh J.; Habib, Mohamed A. M.; Ghoniem, Ahmed F.
2014-01-01
Copyright © 2015 by ASME. In this paper, we conduct an experimental investigation to study the link between the flame macroscale structure - or flame brush spatial distribution - and thermo-acoustic instabilities, in a premixed swirl-stabilized dump combustor. We operate the combustor with premixed methane-air in the range of equivalence ratio (Φ) from the lean blowout limit to Φ = 0. 75. First, we observe the different dynamic modes in this lean range as Φ is raised. We also document the effect of Φ on the flame macrostructure. Next, we examine the correspondence between dynamic mode transitions and changes in flame macrostructure. To do so, we modify the combustor length - by downstream truncation - without changing the underlying flow upstream. Thus, the resonant frequencies of the geometry are altered allowing for decoupling the heat release rate fluctuations and the acoustic feedback. Mean flame configurations in the modified combustor and for the same range of equivalence ratio are examined, following the same experimental protocol. It is found that not only the same sequence of flame macrostructures is observed in both combustors but also that the transitions occur at a similar set of equivalence ratio. In particular, the appearance of the flame in the outside recirculation zone (ORZ) in the long combustor - which occurs simultaneously with the onset of instability at the fundamental frequency - happens at similar Φ when compared to the short combustor, but without being in latter case accompanied by a transition to thermo-acoustic instability. Then, we interrogate the flow field by analyzing the streamlines, mean, and rms velocities for the nonreacting flow and the different flame types. Finally, we focus on the transition of the flame to the ORZ in the acoustically decoupled case. Our analysis of this transition shows that it occurs gradually with an intermittent appearance of a flame in the ORZ and an increasing probability with Φ. The spectral
3. Correspondence Between “Stable” Flame Macrostructure and Thermo-acoustic Instability in Premixed Swirl-Stabilized Turbulent Combustion
KAUST Repository
Taamallah, Soufien
2014-12-23
Copyright © 2015 by ASME. In this paper, we conduct an experimental investigation to study the link between the flame macroscale structure - or flame brush spatial distribution - and thermo-acoustic instabilities, in a premixed swirl-stabilized dump combustor. We operate the combustor with premixed methane-air in the range of equivalence ratio (Φ) from the lean blowout limit to Φ = 0. 75. First, we observe the different dynamic modes in this lean range as Φ is raised. We also document the effect of Φ on the flame macrostructure. Next, we examine the correspondence between dynamic mode transitions and changes in flame macrostructure. To do so, we modify the combustor length - by downstream truncation - without changing the underlying flow upstream. Thus, the resonant frequencies of the geometry are altered allowing for decoupling the heat release rate fluctuations and the acoustic feedback. Mean flame configurations in the modified combustor and for the same range of equivalence ratio are examined, following the same experimental protocol. It is found that not only the same sequence of flame macrostructures is observed in both combustors but also that the transitions occur at a similar set of equivalence ratio. In particular, the appearance of the flame in the outside recirculation zone (ORZ) in the long combustor - which occurs simultaneously with the onset of instability at the fundamental frequency - happens at similar Φ when compared to the short combustor, but without being in latter case accompanied by a transition to thermo-acoustic instability. Then, we interrogate the flow field by analyzing the streamlines, mean, and rms velocities for the nonreacting flow and the different flame types. Finally, we focus on the transition of the flame to the ORZ in the acoustically decoupled case. Our analysis of this transition shows that it occurs gradually with an intermittent appearance of a flame in the ORZ and an increasing probability with Φ. The spectral
4. Experimental observation of microwave absorption and electron heating due to the two plasmon decay instability and resonance absorption
International Nuclear Information System (INIS)
Rasmussen, D.A.
1981-01-01
The interaction of intense microwaves with an inhomogeneous plasma is studied in two experimental devices. In the first device an investigation was made of microwave absorption and electron heating due to the parametric decay of microwaves into electron plasma waves (Two Plasmon Decay instability, TPDI), modeling a process which can occur near the quarter critical surface in laser driven pellets. P-polarized microwave (f = 1.2 GHz, P 0 less than or equal to 12 kW) are applied to an essentially collisionless, inhomogeneous plasma, in an oversized waveguide, in the U.C. Davis Prometheus III device. The initial density scale length near the quarter critical surface is quite long (L/lambda/sub De/ approx. = 3000 or k 0 L approx. = 15). The observed threshold power for the TPDI is quite low (P/sub T/approx. = 0.1 kW or v/sub os//v/sub e/ approx. = 0.1). Near the threshold the decay waves only occur near the quarter critical surface. As the incident power is increased above threshold, the decay waves spread to lower densities, and for P 0 greater than or equal to lkW, (v/sub os//v/sub e/ greater than or equal to 0.3) suprathermal electron heating is strong for high powers (T/sub H/ less than or equal to 12 T/sub e/ for P 0 less than or equal to 8 kW or v/sub os//v/sub e/ less than or equal to 0.9)
5. The Parametric Decay Instability of Alfvén Waves in Turbulent Plasmas and the Applications in the Solar Wind
Energy Technology Data Exchange (ETDEWEB)
Shi, Mijie; Xiao, Chijie; Wang, Xiaogang [State Key Laboratory of Nuclear Physics and Technology, Fusion Simulation Center, School of Physics, Peking University, Beijing 100871 (China); Li, Hui, E-mail: [email protected] [Los Alamos National Laboratory, Los Alamos, NM 87545 (United States)
2017-06-10
We perform three-dimensional ideal magnetohydrodynamic (MHD) simulations to study the parametric decay instability (PDI) of Alfvén waves in turbulent plasmas and explore its possible applications in the solar wind. We find that, over a broad range of parameters in background turbulence amplitudes, the PDI of an Alfvén wave with various amplitudes can still occur, though its growth rate in turbulent plasmas tends to be lower than both the theoretical linear theory prediction and that in the non-turbulent situations. Spatial–temporal FFT analyses of density fluctuations produced by the PDI match well with the dispersion relation of the slow MHD waves. This result may provide an explanation of the generation mechanism of slow waves in the solar wind observed at 1 au. It further highlights the need to explore the effects of density variations in modifying the turbulence properties as well as in heating the solar wind plasmas.
6. Wavelength scaling of the two-plasmon decay and stimulated Raman-scattering instabilities. Annual progress report, September 10, 1981-September 9, 1982
International Nuclear Information System (INIS)
Chen, F.F.; Joshi, C.; Ebrahim, N.A.
1983-03-01
This report contains description of the joint work done by the UCLA-Yale Users' Group at NLUF, LLE Rochester on the two plasmon decay and stimulated Raman scattering instabilities at 0.35 μm laser wavelength. A brief summary of the theoretical work on how SBS influences SRS is also given
7. Modulational instability: Conservation laws and bright soliton solution of ion-acoustic waves in electron-positron-ion-dust plasmas
Science.gov (United States)
EL-Kalaawy, O. H.
2018-02-01
We consider the nonlinear propagation of non-planar (cylindrical and spherical) ion-acoustic (IA) envelope solitary waves in an unmagnetized plasma consisting of electron-positron-ion-dust plasma with two-electron temperature distributions in the context of the non-extensive statistics. The basic set of fluid equations is reduced to the modified nonlinear Schrödinger (MNLS) equation in cylindrical and spherical geometry by using the reductive perturbation method (RPM). It is found that the nature of the modulational instabilities would be significantly modified due to the effects of the non-extensive and other plasma parameters as well as cylindrical and spherical geometry. Conservation laws of the MNLS equation are obtained by Lie symmetry and multiplier method. A new exact solution (envelope bright soliton) is obtained by the extended homogeneous balance method. Finally, we study the results of this article.
8. Ion temperature anisotropy effects on threshold conditions of a shear-modified current driven electrostatic ion-acoustic instability in the topside auroral ionosphere
Directory of Open Access Journals (Sweden)
P. J. G. Perron
2013-03-01
Full Text Available Temperature anisotropies may be encountered in space plasmas when there is a preferred direction, for instance, a strong magnetic or electric field. In this paper, we study how ion temperature anisotropy can affect the threshold conditions of a shear-modified current driven electrostatic ion-acoustic (CDEIA instability. In particular, this communication focuses on instabilities in the context of topside auroral F-region situations and in the limit where finite Larmor radius corrections are small. We derived a new fluid-like expression for the critical drift which depends explicitly on ion anisotropy. More importantly, for ion to electron temperature ratios typical of F-region, solutions of the kinetic dispersion relation show that ion temperature anisotropy may significantly lower the drift threshold required for instability. In some cases, a perpendicular to parallel ion temperature ratio of 2 and may reduce the relative drift required for the onset of instability by a factor of approximately 30, assuming the ion-acoustic speed of the medium remains constant. Therefore, the ion temperature anisotropy should be considered in future studies of ion-acoustic waves and instabilities in the high-latitude ionospheric F-region.
9. Instabilities and prediction of the acoustic resonance of flows with wall injection; Instabilites et prevision de l'accrochage acoustique des ecoulements avec injection parietale
Energy Technology Data Exchange (ETDEWEB)
Avalon, G. [Office National d' Etudes et de Recherches Aerospatiales (ONERA), 91 - Palaiseau (France); Casalis, G. [Office National d' Etudes et de Recherches Aerospatiales (ONERA), 91 - Palaiseau (France)
1998-07-01
Aero-acoustic coupling that occurs inside solid propellant rocket engines can lead to a longitudinal acoustic mode resonance of the combustion chamber. This phenomenon, which can have various origins, in analyzed using the Vecla test facility and the theory of linear stability of flows. Different comparisons between the hot-wire measurements performed and the theory of stability confirm the presence of intrinsic instabilities for this type of flow. The instability allows to selectively amplify a given range of frequencies which depends on the injection velocity and on the conduit height. The results obtained seem to indicate that when this frequency range does not comprise the longitudinal acoustic mode or the first harmonics, the flow becomes turbulent downstream. (J.S.)
10. Modulational instability of the obliquely modulated ion acoustic waves in a warm ion plasma
International Nuclear Information System (INIS)
Saxena, M.K.; Arora, A.K.; Sharma, S.R.
1981-01-01
Using KBM. perturbation technique, it is shown that the modulationally unstable domain in the (kappa - phi) plane for the obliquely modulated ion acoustic waves is appreciably modified due to the finite ion temperature. It is also shown that in a collisionless plasma having small TAUsub(i)/TAUsub(e) ( 0 approximately 0.1) may exceed the Landau damping rate provided the modulation is sufficiently oblique. (author)
11. Improvement in decay ratio calculation in LAPUR5 methodology for BWR instability
International Nuclear Information System (INIS)
Li Hsuannien; Yang Tzungshiue; Shih Chunkuan; Wang Jongrong; Lin Haotzu
2009-01-01
LAPUR5, based on frequency domain approach, is a computer code that analyzes the core stability and calculates decay ratios (DRs) of boiling water nuclear reactors. In current methodology, one set of parameters (three friction multipliers and one density reactivity coefficient multiplier) is chosen for LAPUR5 input files, LAPURX and LAPURW. The calculation stops and DR for this particular set of parameters is obtained when the convergence criteria (pressure, mass flow rate) are first met. However, there are other sets of parameters which could also meet the same convergence criteria without being identified. In order to cover these ranges of parameters, we developed an improved procedure to calculate DR in LAPUR5. First, we define the ranges and increments of those dominant input parameters in the input files for DR loop search. After LAPUR5 program execution, we can obtain all DRs for every set of parameters which satisfy the converge criteria in one single operation. The part for loop search procedure covers those steps in preparing LAPURX and LAPURW input files. As a demonstration, we looked into the reload design of Kuosheng Unit 2 Cycle 22. We found that the global DR has a maximum at exposure of 9070 MWd/t and the regional DR has a maximum at exposure of 5770 MWd/t. It should be noted that the regional DR turns out to be larger than the global ones for exposures less than 5770 MWd/t. Furthermore, we see that either global or regional DR by the loop search method is greater than the corresponding values from our previous approach. It is concluded that the loop search method can reduce human error and save human labor as compared with the previous version of LAPUR5 methodology. Now the maximum DR can be effectively obtained for a given plant operating conditions and a more precise stability boundary, with less uncertainty, can be plotted on plant power/flow map. (author)
12. Effective collision frequency due to ion-acoustic instability: Theory and simulations
Czech Academy of Sciences Publication Activity Database
Hellinger, Petr; Trávníček, Pavel; Menietti, J. D.
2004-01-01
Roč. 31, č. 10 (2004), L10806 ISSN 0094-8276 R&D Projects: GA MŠk ME 500; GA AV ČR IAA3042403 Grant - others:ESA PRODEX(XE) 14529/00/NL/SFe; NASA (US) NAG5-11942 Institutional research plan: CEZ:AV0Z3042911 Keywords : Magnetospheric Physics: Plasma waves and instabilities * Space Plasma Physics: Kinetic and MHD theory * Space Plasma Physics: Magnetic reconnection Subject RIV: BL - Plasma and Gas Discharge Physics Impact factor: 2.378, year: 2004
13. Current-driven ion-acoustic and potential-relaxation instabilities excited in plasma plume during electron beam welding
Energy Technology Data Exchange (ETDEWEB)
Trushnikov, D. N., E-mail: [email protected] [The department for Applied Physics, Perm National Research Polytechnic University, Perm, 614990 (Russian Federation); Mladenov, G. M., E-mail: [email protected]; Koleva, E. G., E-mail: [email protected] [Institute of Electronics, Bulgarian Academy of Sciences, 72 Tzarigradsko Shose, 1784, Sofia (Bulgaria); Technology Centre of Electron Beam and Plasma Technologies and Techniques, 68-70 Vrania, ap.10, Banishora,1309, Sofia (Bulgaria); Belenkiy, V. Ya., E-mail: [email protected]; Varushkin, S. V., E-mail: [email protected] [The department for Welding Production and Technology of Constructional Materials, Perm National Research Polytechnic University, Perm, 614990 (Russian Federation)
2014-04-15
Many papers have sought correlations between the parameters of secondary particles generated above the beam/work piece interaction zone, dynamics of processes in the keyhole, and technological processes. Low- and high-frequency oscillations of the current, collected by plasma have been observed above the welding zone during electron beam welding. Low-frequency oscillations of secondary signals are related to capillary instabilities of the keyhole, however; the physical mechanisms responsible for the high-frequency oscillations (>10 kHz) of the collected current are not fully understood. This paper shows that peak frequencies in the spectra of the collected high-frequency signal are dependent on the reciprocal distance between the welding zone and collector electrode. From the relationship between current harmonics frequency and distance of the collector/welding zone, it can be estimated that the draft velocity of electrons or phase velocity of excited waves is about 1600 m/s. The dispersion relation with the properties of ion-acoustic waves is related to electron temperature 10 000 K, ion temperature 2 400 K and plasma density 10{sup 16} m{sup −3}, which is analogues to the parameters of potential-relaxation instabilities, observed in similar conditions. The estimated critical density of the transported current for creating the anomalous resistance state of plasma is of the order of 3 A·m{sup −2}, i.e. 8 mA for a 3–10 cm{sup 2} collector electrode. Thus, it is assumed that the observed high-frequency oscillations of the current collected by the positive collector electrode are caused by relaxation processes in the plasma plume above the welding zone, and not a direct demonstration of oscillations in the keyhole.
14. A comparative study of plasma heating by ion acoustic and modified two-stream instabilities at subcritical quasi-perpendicular shocks
International Nuclear Information System (INIS)
Winske, D.; Giacalone, J.; Thomsen, M.F.; Mellott, M.M.
1987-01-01
Plasma heating due to the ion instability and the modified two-stream instability is examined for quasi-perpendicular subcritical shocks. Electron and ion heating is investigated as a function of upstream electron to ion temperature ratio and plasma beta using second-order heating rates. A simple shock model is employed in which the cross-field electron-ion drift speed is adjusted until the total (adiabatic plus anomalous) heating matches that required by the Rankine-Hugoniot relations. Quantities such as the width of the shock and the maximum electric field fluctuations are also calculated, and the results are compared with the ISEE data set of subcritical box shock crossings. The observed width of the shock, the amount of plasma heating, and the low-frequency electric field intensity are in reasonably good agreement with the calculations for the modified two-stream instability. On the other hand, the wave intensities at higher frequency are about 4 orders of magnitude smaller than those predicted for the ion acoustic instability at saturation, consistent with the fact that the measured shock widths imply cross-field drift speeds that are below threshold for this instability. It is therefore concluded that the dissipation at these shocks is most likely due to the lower frequency, modified two-stream instability
15. Frequency-dependent Alfvén-wave Propagation in the Solar Wind: Onset and Suppression of Parametric Decay Instability
Science.gov (United States)
Shoda, Munehito; Yokoyama, Takaaki; Suzuki, Takeru K.
2018-06-01
Using numerical simulations we investigate the onset and suppression of parametric decay instability (PDI) in the solar wind, focusing on the suppression effect by the wind acceleration and expansion. Wave propagation and dissipation from the coronal base to 1 au is solved numerically in a self-consistent manner; we take into account the feedback of wave energy and pressure in the background. Monochromatic waves with various injection frequencies, f 0, are injected to discuss the suppression of PDI, while broadband waves are applied to compare the numerical results with observation. We find that high-frequency ({f}0≳ {10}-3 {Hz}) Alfvén waves are subject to PDI. Meanwhile, the maximum growth rate of the PDI of low-frequency ({f}0≲ {10}-4 {Hz}) Alfvén waves becomes negative due to acceleration and expansion effects. Medium-frequency ({f}0≈ {10}-3.5 {Hz}) Alfvén waves have a positive growth rate but do not show the signature of PDI up to 1 au because the growth rate is too small. The medium-frequency waves experience neither PDI nor reflection so they propagate through the solar wind most efficiently. The solar wind is shown to possess a frequency-filtering mechanism with respect to Alfvén waves. The simulations with broadband waves indicate that the observed trend of the density fluctuation is well explained by the evolution of PDI while the observed cross-helicity evolution is in agreement with low-frequency wave propagation.
16. The application of the acoustic emission technique to stone decay by sodium sulphate in laboratory tests
Directory of Open Access Journals (Sweden)
Grossi, C. M.
1997-03-01
Full Text Available Acoustic emission was monitored during salt crystallisation cycles in order to study the mechanisms of rock deterioration by sodium sulphate in laboratory tests. Some porous carbonate stones used in Spanish monuments (Cathedral of Oviedo, Murcia and Seo Vella of Lérida were selected for this study. The acoustic emission detected during the different stages of the cycles (immersion, drying and cooling was interpreted to be the result of the salt behaviour inside the stone. The use of this technique has confirmed that this behaviour depends on salt characteristics (solubility, hydration state and polymorphism of anhydrous sodium sulphate and stone porosity and pore network.
17. Thermo-acoustic instabilities of high-frequency combustion in rocket engines; Instabilites thermo-acoustiques de combustion haute-frequence dans les moteurs fusees
Energy Technology Data Exchange (ETDEWEB)
Cheuret, F
2005-10-15
Rocket motors are confined environments where combustion occurs in extreme conditions. Combustion instabilities can occur at high frequencies; they are tied to the acoustic modes of the combustion chamber. A common research chamber, CRC, allows us to study the response of a turbulent two-phase flame to acoustic oscillations of low or high amplitudes. The chamber is characterised under cold conditions to obtain, in particular, the relative damping coefficient of acoustic oscillations. The structure and frequency of the modes are determined in the case where the chamber is coupled to a lateral cavity. We have used a powder gun to study the response to a forced acoustic excitation at high amplitude. The results guide us towards shorter flames. The injectors were then modified to study the combustion noise level as a function of injection conditions. The speed of the gas determines whether the flames are attached or lifted. The noise level of lifted flames is higher. That of attached flames is proportional to the Weber number. The shorter flames whose length is less than the radius of the CRC, necessary condition to obtain an effective coupling, are the most sensitive to acoustic perturbations. The use of a toothed wheel at different positions in the chamber allowed us to obtain informations on the origin of the thermo-acoustic coupling, main objective of this thesis. The flame is sensitive to pressure acoustic oscillations, with a quasi-zero response time. These observations suggest that under the conditions of the CRC, we observe essentially the response of chemical kinetics to pressure oscillations. (author)
18. Parametric dependence of two-plasmon decay in homogeneous plasma
International Nuclear Information System (INIS)
Dimitrijevic, Dejan R
2010-01-01
A hydrodynamic model of two-plasmon decay in a homogeneous plasma slab near the quarter-critical density is constructed in order to improve our understanding of the spatio-temporal evolution of the daughter electron plasma waves in plasma in the course of the instability. The scaling of the amplitudes of the participating waves with laser and plasma parameters is investigated. The secondary coupling of two daughter electron plasma waves with an ion-acoustic wave is assumed to be the principal mechanism of saturation of the instability. The impact of the inherently nonresonant nature of this secondary coupling on the development of two-plasmon decay is researched and it is shown to significantly influence the electron plasma wave dynamics. Its inclusion leads to nonuniformity of the spatial profile of the instability and causes the burst-like pattern of the instability development, which should result in the burst-like hot-electron production in homogeneous plasma.
19. TOWARD A MAGNETOHYDRODYNAMIC THEORY OF THE STATIONARY ACCRETION SHOCK INSTABILITY: TOY MODEL OF THE ADVECTIVE-ACOUSTIC CYCLE IN A MAGNETIZED FLOW
International Nuclear Information System (INIS)
Guilet, Jerome; Foglizzo, Thierry
2010-01-01
The effect of a magnetic field on the linear phase of the advective-acoustic instability is investigated as a first step toward a magnetohydrodynamic (MHD) theory of the stationary accretion shock instability taking place during stellar core collapse. We study a toy model where the flow behind a planar stationary accretion shock is adiabatically decelerated by an external potential. Two magnetic field geometries are considered: parallel or perpendicular to the shock. The entropy-vorticity wave, which is simply advected in the unmagnetized limit, separates into five different waves: the entropy perturbations are advected, while the vorticity can propagate along the field lines through two Alfven waves and two slow magnetosonic waves. The two cycles existing in the unmagnetized limit, advective-acoustic and purely acoustic, are replaced by up to six distinct MHD cycles. The phase differences among the cycles play an important role in determining the total cycle efficiency and hence the growth rate. Oscillations in the growth rate as a function of the magnetic field strength are due to this varying phase shift. A vertical magnetic field hardly affects the cycle efficiency in the regime of super-Alfvenic accretion that is considered. In contrast, we find that a horizontal magnetic field strongly increases the efficiencies of the vorticity cycles that bend the field lines, resulting in a significant increase of the growth rate if the different cycles are in phase. These magnetic effects are significant for large-scale modes if the Alfven velocity is a sizable fraction of the flow velocity.
20. Adaptative control of aero-acoustic instabilities. Application to propulsion systems; Controle adaptatif des instabilites aeroacoustiques. Application aux systemes de propulsion
Energy Technology Data Exchange (ETDEWEB)
Mettenleiter, M.
2000-02-15
This work treats active adaptive control of aero-acoustic instabilities. In particular, we are interested in an application to solid propellant rockets. The study is part of the research program ASSM coordinated by CNES and ONERA and the aim is to increase the performance of the P230 segmented solid propellant boosters of the Ariane 5 rocket. The work has been carried out in collaboration with other partners of this program. The main objective of this study is the development of control algorithms, able to diminish low frequency instabilities encountered in propulsion systems. First, the instability phenomenon is analyzed in a simplified experimental setup and similarity is shown with instabilities observed in real propulsion systems. This study enables us to conceive adaptive control strategies, which have been tested on three different levels: - In a simplified dynamical simulation; - During an experimental study; - Using full numerical simulations. The three levels of application made it possible to study the behaviour of the different control strategies. We could show that the actuator signal modifies the behaviour of the system on the acoustic level. But as there is a strong interaction between the pressure fluctuations and the hydrodynamic behaviour, the flow structure is also modified by active control. This behaviour corresponds to the simplified model of the phenomenon, which has been used to define the control algorithms. The control action 'at the noise source' makes it possible to distinguish this kind of algorithms from schemes based on the anti-noise principle. After this first part, where we showed the feasibility of control, we particularly considered algorithms which can act in an unknown environment. The information about the system behaviour. which is necessary for convergence of the controller is now obtained in parallel during control. An identification off-line, used at the beginning of the research, is no longer necessary. Self
1. Experimental observation of pulsating instability under acoustic field in downward-propagating flames at large Lewis number
KAUST Repository
Yoon, Sung Hwan; Hu, Longhua; Fujita, Osamu
2017-01-01
by interaction between heat release and acoustic pressure fluctuations of the downward-propagating premixed flames in a tube to enhance conductive heat loss at the tube wall and radiative heat loss at the open end of the tube due to extended flame residence time
2. Experimental investigation of two-dimensional critical surface structure, stimulated Raman scattering, and two-plasmon decay instability. Annual report, January 1, 1981-April 30, 1982
International Nuclear Information System (INIS)
Wong, A.Y.; Eggleston, D.L.; Tanikawa, T.; Qian, S.J.
1982-11-01
Experimental observations of the space and time evolution of resonantly enhanced electrostatic electric fields and plasma density in cylindrical geometry demonstrate the development of two-dimensional caviton structure when an initial density perturbation is imposed on the plasma in the direction perpendicular to the driver field. This two-dimensional structure is observed after the development of profile modification and grows on the ion time scale. The existence of a large azimuthal electric field component is an observational signature of two-dimensional structure. Enhanced electric field maxima are found to be azimuthally correlated with the density minima. Both the density cavities and electric field peaks exhibit increased azimuthal location with the growth of two-dimensional structure. The two-dimensional development exhibits a strong dependence on both perturbation wavenumber and driver power. The related theoretical literature is reviewed and numerical, analytical, and qualitative hybrid models for a driven, two-dimensional, inhomogeneous plasma are presented. Preliminary work is presented in the following additional areas: weak magnetic field effects on critical surface physics, optical measurements of fast electron production, two-dimensional effects in microwave-plasma interactions, Langmuir wave trapping, stimulated Raman scattering and two-plasmon decay instability
3. Type I intermittency related to the spatiotemporal dynamics of double layers and ion-acoustic instabilities in plasma
International Nuclear Information System (INIS)
Chiriac, S.; Dimitriu, D. G.; Sanduloviciu, M.
2007-01-01
Anodic double layer instabilities occur in low-temperature diffusion filament-type discharge plasma by applying a certain positive bias with respect to the plasma potential to an additional electrode. Periodic nonlinear regimes, characterized by proper dynamics of double layers, are sustained if excitation and ionization rates in front of the electrode reach the value for which current limitation effects appear in the static current-voltage characteristic. It was experimentally shown that under specific experimental conditions these ordered spatiotemporal phenomena can evolve into chaotic states by type I intermittency. This transition was verified by the evolution of time series, fast Fourier transform amplitude plots, three-dimensional reconstructed state spaces, power laws, and flickering phenomena spectrum, as well as by the return map and tangent bifurcation
4. Plastic Instabilities Induced by the Portevin - Le Châtelier Effect and Fracture Character of Deformed Mg-Li Alloys Investigated Using the Acoustic Emission Method
Directory of Open Access Journals (Sweden)
Pawełek A.
2016-06-01
Full Text Available The results of the investigation of both mechanical and acoustic emission (AE behaviors of Mg4Li5Al and Mg4Li4Zn alloys subjected to compression and tensile tests at room temperature are compared with the test results obtained using the same alloys and loading scheme but at elevated temperatures. The main aim of the paper is to investigate, to determine and to explain the relation between plastic flow instabilities and the fracture characteristics. There are discussed the possible influence of the factors related with enhanced internal stresses such as: segregation of precipitates along grain boundaries, interaction of solute atoms with mobile dislocations (Cottrell atmospheres as well as dislocation pile-ups which may lead to the microcracks formation due to the creation of very high stress concentration at grain boundaries. The results show that the plastic flow discontinuities are related to the Portevin-Le Châtelier phenomenon (PL effect and they are correlated with the generation of characteristic AE pulse trains. The fractography of broken samples was analyzed on the basis of light (optical, TEM and SEM images.
5. Experimental analysis of thermo-acoustic instabilities in a generic gas turbine combustor by phase-correlated PIV, chemiluminescence, and laser Raman scattering measurements
Science.gov (United States)
Arndt, Christoph M.; Severin, Michael; Dem, Claudiu; Stöhr, Michael; Steinberg, Adam M.; Meier, Wolfgang
2015-04-01
A gas turbine model combustor for partially premixed swirl flames was equipped with an optical combustion chamber and operated with CH4 and air at atmospheric pressure. The burner consisted of two concentric nozzles for separately controlled air flows and a ring of holes 12 mm upstream of the nozzle exits for fuel injection. The flame described here had a thermal power of 25 kW, a global equivalence ratio of 0.7, and exhibited thermo-acoustic instabilities at a frequency of approximately 400 Hz. The phase-dependent variations in the flame shape and relative heat release rate were determined by OH* chemiluminescence imaging; the flow velocities by stereoscopic particle image velocimetry (PIV); and the major species concentrations, mixture fraction, and temperature by laser Raman scattering. The PIV measurements showed that the flow field performed a "pumping" mode with varying inflow velocities and extent of the inner recirculation zone, triggered by the pressure variations in the combustion chamber. The flow field oscillations were accompanied by variations in the mixture fraction in the inflow region and at the flame root, which in turn were mainly caused by the variations in the CH4 concentration. The mean phase-dependent changes in the fluxes of CH4 and N2 through cross-sectional planes of the combustion chamber at different heights above the nozzle were estimated by combining the PIV and Raman data. The results revealed a periodic variation in the CH4 flux by more than 150 % in relation to the mean value, due to the combined influence of the oscillating flow velocity, density variations, and CH4 concentration. Based on the experimental results, the feedback mechanism of the thermo-acoustic pulsations could be identified as a periodic fluctuation of the equivalence ratio and fuel mass flow together with a convective delay for the transport of fuel from the fuel injector to the flame zone. The combustor and the measured data are well suited for the validation of
6. Droplet behaviour in an acoustic field: application to high frequency instability in liquid propellant rocket engines; Comportement de gouttes dans un champ acoustique: applications aux instabilites hautes-frequences dans les moteurs de fusees a ergols liquides
Energy Technology Data Exchange (ETDEWEB)
Boisneau, O.; Lecourt, R.; Grisch, F.; Orain, M.
2002-07-01
A setup has been developed at ONERA in the scope of studying interaction between calibrated droplets and a transversal acoustic wave in the scope of high frequency instabilities in liquid rocket engines. First, the setup has been checked acoustically by hot-wire anemometer and microphone. We present an analytical solution of the Stokes' droplet motion equation in an acoustic field. The trajectory equation can be split into three different parts: a sinusoidal part (negligible in liquid rocket engines), a transient part and a final mean position (only function of the loudspeaker characteristics but never reached). Some kind of vibrational breakup at low Weber's number has been observed using line-of-sight visualization of acoustic/droplet interactions. However, preponderant phenomena observed were jet oscillations and droplet coalescence. For ambient temperature, PLIF visualization has shown a coupling between the created vapor cylinder and the acoustic induced jet position. For hot temperature, some unsteady phenomena seem to appear but further processing are needed. (authors)
7. Observational Signatures of Parametric Instability at 1AU
Science.gov (United States)
Bowen, T. A.; Bale, S. D.; Badman, S.
2017-12-01
Observations and simulations of inertial compressive turbulence in the solar wind are characterized by density structures anti-correlated with magnetic fluctuations parallel to the mean field. This signature has been interpreted as observational evidence for non-propagating pressure balanced structures (PBS), kinetic ion acoustic waves, as well as the MHD slow mode. Recent work, specifically Verscharen et al. (2017), has highlighted the unexpected fluid like nature of the solar wind. Given the high damping rates of parallel propagating compressive fluctuations, their ubiquity in satellite observations is surprising and suggests the presence of a driving process. One possible candidate for the generation of compressive fluctuations in the solar wind is the parametric instability, in which large amplitude Alfvenic fluctuations decay into parallel propagating compressive waves. This work employs 10 years of WIND observations in order to test the parametric decay process as a source of compressive waves in the solar wind through comparing collisionless damping rates of compressive fluctuations with growth rates of the parametric instability. Preliminary results suggest that generation of compressive waves through parametric decay is overdamped at 1 AU. However, the higher parametric decay rates expected in the inner heliosphere likely allow for growth of the slow mode-the remnants of which could explain density fluctuations observed at 1AU.
8. Gravitational instability in a multicomponent expanding medium
International Nuclear Information System (INIS)
Solov'eva, L.V.; Nurgaliev, I.S.
1985-01-01
In the Newtonian approximation we consider the gravitational instability of a two- or N-component medium in an expanding universe. The system of density-perturbation equations is solved in the short- and long-wave limits. For small values of the wave vector k, a result obtained for the stationary case continues to hold true: at most there can exist only one unstable mode. If k is kept fixed, the introduction of a perturbation component delta/sub i/ will speed the growth of fluctuations delta/sub j/, provided the adiabatic indices γ/sub i/>γ/sub j/. In the large-k limit, ordinary acoustic waves result. Other components will begin to manifest themselves in the first-order terms when the oscillation amplitude is expanded in powers of k -1 : provided γ/sub j/>γ/sub i/> or =4/3, the ith-component amplitude will decay more slowly than otherwise
9. Parametric decay below the upper hybrid frequency
Energy Technology Data Exchange (ETDEWEB)
Albers, E; Krause, K; Schlueter, H [Bochum Univ. (Germany, F.R.). Inst. fuer Experimentalphysik 2
1977-03-21
Parametric decay of the upper hybrid mode is observed between the electron cyclotron frequency and its first two harmonics. The decay products are identified as electron Bernstein and ion acoustic mode. The diagnostic results confirm the relevant dispersion relations.
10. Nonlinear instability and chaos in plasma wave-wave interactions
International Nuclear Information System (INIS)
Kueny, C.S.
1993-01-01
Conventional linear stability analysis may fail for fluid systems with an indefinite free energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes. Instability may then occur either via dissipation of the negative energy modes. Instability may then occur either via dissipitation of the negative energy modes, or nonlinearly via resonant wave-wave coupling, which leads to explosive growth. In the dissipationaless case, it is conjectured that intrinsic chaotic behavior may allow initially non-resonant systems to reach resonance by diffusion in phase space. This is illustrated for a simple equilibrium involving cold counter-streaming ions. The system is described in the fluid approximation by a Hamilitonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamilitonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, which occur generically for long enough wavelengths. Three-wave interactions which occur in isolated, but numerous, regions of parameter space can drive either decay instability or explosive instability. When the resonance for explosive growth is detuned, a stable region exists around the equilibrium point in phase space, while explosive growth occurs outside of a separatrix. These interactions may be described exactly if only one resonance is considered, while multiple nonlinear terms make the Hamiltonian nonintegradable. Simple Hamiltonians of two and three degrees of freedom are studied numerically using symplectic integration algorithms, including an explicit algorithm derived using Lie algebraic methods
11. Saturation of radiation-induced parametric instabilities by excitation of Langmuir turbulence
Energy Technology Data Exchange (ETDEWEB)
Dubois, D.F.; Rose, H.A. [Los Alamos National Lab., NM (United States); Russell, D. [Lodestar Research Inc., Boulder, CO (United States)
1995-12-01
Progress made in the last few years in the calculation of the saturation spectra of parametric instabilities which involve Langmuir daughter waves will be reviewed. These instabilities include the ion acoustic decay instability, the two plasmon decay instability (TPDI), and stimulated Raman scattering (SRS). In particular I will emphasize spectral signatures which can be directly compared with experiment. The calculations are based on reduced models of driven Laugmuir turbulence. Thomson scattering from hf-induced Langmuir turbulence in the unpreconditioned ionosphere has resulted in detailed agreement between theory and experiment at early times. Strong turbulence signatures dominate in this regime where the weak turbulence approximation fails completely. Recent experimental studies of the TPDI have measured the Fourier spectra of Langmuir waves as well as the angular and frequency, spectra of light emitted near 3/2 of the pump frequency again permitting some detailed comparisons with theory. The experiments on SRS are less detailed but by Thomson scattering the secondary decay of the daughter Langmuir wave has been observed. Scaling laws derived from a local model of SRS saturation are compared with full simulations and recent Nova experiments.
12. Saturation of radiation-induced parametric instabilities by excitation of Langmuir turbulence
International Nuclear Information System (INIS)
DuBois, D.F.
1996-01-01
Progress made in the last few years in the calculation of the saturation spectra of parametric instabilities which involve Langmuir daughter waves will be reviewed. These instabilities include the ion acoustic decay instability, the two plasmon decay instability (TPDI), and stimulated Raman scattering (SRS). In particular we will emphasize spectral signatures which can be directly compared with experiment. The calculations are based on reduced models of driven Langmuir turbulence. Thomson scattering from hf-induced Langmuir turbulence in the unpreconditioned ionosphere has resulted in detailed agreement between theory and experiment at early times. Strong turbulence signatures dominate in this regime where the weak turbulence approximation fails completely. Recent experimental studies of the TPDI have measured the Fourier spectra of Langmuir waves as well as the angular and frequency spectra of light emitted near 3/2 of the pump frequency again permitting some detailed comparisons with theory. Thomson scattering measurements of the Langmuir wave spectra from SRS are consistent with the saturation by secondary and tertiary decay of the primary SRS Langmuir waves. Scaling laws derived from a local model of SRS saturation are compared with full simulations and recent Nova experiments. (orig.)
13. A numerical study on acoustic behavior in gas turbine combustor with acoustic resonator
International Nuclear Information System (INIS)
Park, I Sun; Sohn, Chae Hoon
2005-01-01
Acoustic behavior in gas turbine combustor with acoustic resonator is investigated numerically by adopting linear acoustic analysis. Helmholtz-type resonator is employed as acoustic resonator to suppress acoustic instability passively. The tuning frequency of acoustic resonator is adjusted by varying its length. Through harmonic analysis, acoustic-pressure responses of chamber to acoustic excitation are obtained and the resonant acoustic modes are identified. Acoustic damping effect of acoustic resonator is quantified by damping factor. As the tuning frequency of acoustic resonator approaches the target frequency of the resonant mode to be suppressed, mode split from the original resonant mode to lower and upper modes appears and thereby complex patterns of acoustic responses show up. Considering mode split and damping effect as a function of tuning frequency, it is desirable to make acoustic resonator tuned to broad-band frequencies near the maximum frequency of those of the possible upper modes
14. Measurements of parametric instability near the critical density and the resultant electron heating: Final report
International Nuclear Information System (INIS)
Mizuno, K.; De Groot, J.S.; Seka, W.
1986-01-01
Detailed studies of the ion acoustic parametric decay instability have been made. Theoretical and particle simulation results indicate these instabilities are important in long scale length plasma irradiated by moderate intensity laser light (10'' ≤ Iλ 2 /T/sub e/ (W/cm 2 ) (μm 2 )/(keV) ≤ 5 x 10 14 ). Laser light (λ 0 ≅ 1/2 μm) is focused onto a CH target. The parametric decay instability has been measured by detecting the emission spectrum at frequencies near 2ω 0 . The experimental results clearly indicate that this parametric instability is important for short wavelength (1/2 μm) laser light irradiation. The threshold of the parametric instability (λ 0 = 1/2 μm) was only slightly higher than that of 1 μm laser case. The measured wavelength shift of the Stokes component (λ 0 = 1/2 μm) compared very well with the 1 μm laser results
15. Instability and star evolution
International Nuclear Information System (INIS)
Mirzoyan, L.V.
1981-01-01
The observational data are discussed which testify that the phenomena of dynamical instability of stars and stellar systems are definite manifestations of their evolution. The study of these phenomena has shown that the instability is a regular phase of stellar evolution. It has resulted in the recognition of the most important regularities of the process of star formation concerning its nature. This became possible due to the discovery in 1947 of stellar associations in our Galaxy. The results of the study of the dynamical instability of stellar associations contradict the predictions of classical hypothesis of stellar condensation. These data supplied a basis for a new hypothesis on the formation of stars and nebulae by the decay of superdense protostars [ru
16. Nonlinear instability and chaos in plasma wave-wave interactions, I., Introduction
International Nuclear Information System (INIS)
Kueny, C.S.; Morrison, P.J.
1994-11-01
Conventional linear stability analyses may fail for fluid systems with an indefinite free energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave-wave coupling, leading to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially nonresonant systems to reach resonance by diffusion in phase space. In this and a companion paper [submitted to Physics of Plasmas], this phenomenon is demonstrated for a simple equilibrium involving cold counterstreaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, and to either decay or explosive instability via three-wave interactions. These instabilities are described for various (integrable) systems of waves interacting via single nonlinear terms. This discussion provides the foundation for the treatment of nonintegrable systems in the companion paper
17. Nonlinear instability and chaos in plasma wave--wave interactions. I. Introduction
International Nuclear Information System (INIS)
Kueny, C.S.; Morrison, P.J.
1995-01-01
Conventional linear stability analyses may fail for fluid systems with an indefinite free-energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave--wave coupling, leading to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially nonresonant systems to reach resonance by diffusion in phase space. In this and a companion paper (submitted to Phys. Plasmas), this phenomenon is demonstrated for a simple equilibrium involving cold counterstreaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, and to either decay or explosive instability via three-wave interactions. These instabilities are described for various integrable systems of waves interacting via single nonlinear terms. This discussion provides the foundation for the treatment of nonintegrable systems in the companion paper. copyright 1995 American Institute of Physics
18. Ionospheric modification and parametric instabilities
International Nuclear Information System (INIS)
Fejer, J.A.
1979-01-01
Thresholds and linear growth rates for stimulated Brillouin and Raman scattering and for the parametric decay instability are derived by using arguments of energy transfer. For this purpose an expression for the ponderomotive force is derived. Conditions under which the partial pressure force due to differential dissipation exceeds the ponderomotive force are also discussed. Stimulated Brillouin and Raman scattering are weakly excited by existing incoherent backscatter radars. The parametric decay instability is strongly excited in ionospheric heating experiments. Saturation theories of the parametric decay instability are therefore described. After a brief discussion of the purely growing instability the effect of using several pumps is discussed as well as the effects of inhomogenicity. Turning to detailed theories of ionospheric heating, artificial spread F is discussed in terms of a purely growing instability where the nonlinearity is due to dissipation. Field-aligned short-scale striations are explained in terms of dissipation of the parametrically excited Langmuir waves (plasma oscillations): they might be further amplified by an explosive instability (except the magnetic equator). Broadband absorption is probably responsible for the 'overshoot' effect: the initially observed level of parametrically excited Langmuir waves is much higher than the steady state level
19. Ion-acoustic plasma turbulence
International Nuclear Information System (INIS)
Bychenkov, V.Y.; Silin, V.P.
1982-01-01
A theory is developed of the nonlinear state that is established in a plasma as a result of development of ion-acoustic instability. Account is taken simultaneously of the linear induced scattering of the waves by the ions and of the quasilinear relaxation of the electrons by the ion-acoustic pulsations. The distribution of the ion-acoustic turbulence in frequency and in angle is obtained. An Ohm's law is established and expressions are obtained for the electronic heat flux and for the relaxation time of the electron temperature in a turbulent plasma. Anomalously large absorption and scattering of the electromagnetic waves by the ion-acoustic pulsations is predicted
20. Size effects on cavitation instabilities
DEFF Research Database (Denmark)
Niordson, Christian Frithiof; Tvergaard, Viggo
2006-01-01
growth is here analyzed for such cases. A finite strain generalization of a higher order strain gradient plasticity theory is applied for a power-law hardening material, and the numerical analyses are carried out for an axisymmetric unit cell containing a spherical void. In the range of high stress...... triaxiality, where cavitation instabilities are predicted by conventional plasticity theory, such instabilities are also found for the nonlocal theory, but the effects of gradient hardening delay the onset of the instability. Furthermore, in some cases the cavitation stress reaches a maximum and then decays...... as the void grows to a size well above the characteristic material length....
1. Electrostatic ion acoustic waves
International Nuclear Information System (INIS)
Hasegawa, A.
1983-01-01
In this paper, certain aspects of plasma physics are illustrated through a study of electrostatic ion acoustic waves. The paper consists of three Sections. Section II deals with linear properties of the ion acoustic wave including derivation of the dispersions relation with the effect of Landau damping and of an ambient magnetic field. The section also introduces the excitation processes of the ion acoustic wave due to an electron drift or to a stimulated Brillouin scattering. The nonlinear properties are introduced in Section III and IV. In Section III, incoherent nonlinear effects such as quasilinear and mode-coupling saturations of the instability are discussed. The coherent nonlinear effects such as the generation of ion acoustic solitons, shocks and weak double layers are presented in Section IV. (Auth.)
2. Linearised dynamics and non-modal instability analysis of an impinging under-expanded supersonic jet
Science.gov (United States)
Karami, Shahram; Stegeman, Paul C.; Theofilis, Vassilis; Schmid, Peter J.; Soria, Julio
2018-04-01
Non-modal instability analysis of the shear layer near the nozzle of a supersonic under-expanded impinging jet is studied. The shear layer instability is considered to be one of the main components of the feedback loop in supersonic jets. The feedback loop is observed in instantaneous visualisations of the density field where it is noted that acoustic waves scattered by the nozzle lip internalise as shear layer instabilities. A modal analysis describes the asymptotic limit of the instability disturbances and fails to capture short-time responses. Therefore, a non-modal analysis which allows the quantitative description of the short-time amplification or decay of a disturbance is performed by means of a local far-field pressure pulse. An impulse response analysis is performed which allows a wide range of frequencies to be excited. The temporal and spatial growths of the disturbances in the shear layer near the nozzle are studied by decomposing the response using dynamic mode decomposition and Hilbert transform analysis. The short-time response shows that disturbances with non-dimensionalised temporal frequencies in the range of 1 to 4 have positive growth rates in the shear layer. The Hilbert transform analysis shows that high non-dimensionalised temporal frequencies (>4) are dampened immediately, whereas low non-dimensionalised temporal frequencies (analysis show that spatial frequencies between 1 and 3 have positive spatial growth rates. Finally, the envelope of the streamwise velocity disturbances reveals the presence of a convective instability.
3. Density Fluctuations in the Solar Wind Driven by Alfvén Wave Parametric Decay
Science.gov (United States)
Bowen, Trevor A.; Badman, Samuel; Hellinger, Petr; Bale, Stuart D.
2018-02-01
Measurements and simulations of inertial compressive turbulence in the solar wind are characterized by anti-correlated magnetic fluctuations parallel to the mean field and density structures. This signature has been interpreted as observational evidence for non-propagating pressure balanced structures, kinetic ion-acoustic waves, as well as the MHD slow-mode. Given the high damping rates of parallel propagating compressive fluctuations, their ubiquity in satellite observations is surprising and suggestive of a local driving process. One possible candidate for the generation of compressive fluctuations in the solar wind is the Alfvén wave parametric instability. Here, we test the parametric decay process as a source of compressive waves in the solar wind by comparing the collisionless damping rates of compressive fluctuations with growth rates of the parametric decay instability daughter waves. Our results suggest that generation of compressive waves through parametric decay is overdamped at 1 au, but that the presence of slow-mode-like density fluctuations is correlated with the parametric decay of Alfvén waves.
4. The large density electron beam-plasma Buneman instability
International Nuclear Information System (INIS)
Mantei, T.D.; Doveil, F.; Gresillon, D.
1976-01-01
The threshold conditions and growth rate of the Buneman (electron beam-stationary ion) instability are calculated with kinetic theory, including a stationary electronic population. A criteria on the wave energy sign is used to separate the Buneman hydrodynamic instability from the ion-acoustic kinetic instability. The stationary electron population raises the instability threshold and, for large beam velocities yields a maximum growth rate oblique to the beam. (author)
5. Stable And Oscillating Acoustic Levitation
Science.gov (United States)
Barmatz, Martin B.; Garrett, Steven L.
1988-01-01
Sample stability or instability determined by levitating frequency. Degree of oscillation of acoustically levitated object along axis of levitation chamber controlled by varying frequency of acoustic driver for axis above or below frequency of corresponding chamber resonance. Stabilization/oscillation technique applied in normal Earth gravity, or in absence of gravity to bring object quickly to rest at nominal levitation position or make object oscillate in desired range about that position.
6. Architectural acoustics
National Research Council Canada - National Science Library
Long, Marshall
2014-01-01
.... Beginning with a brief history, it reviews the fundamentals of acoustics, human perception and reaction to sound, acoustic noise measurements, noise metrics, and environmental noise characterization...
7. Acoustic emission
International Nuclear Information System (INIS)
Nichols, R.W.
1976-01-01
The volume contains six papers which together provide an overall review of the inspection technique known as acoustic emission or stress wave emission. The titles are: a welder's introduction to acoustic emission technology; use of acoustic emission for detection of defects as they arise during fabrication; examples of laboratory application and assessment of acoustic emission in the United Kingdom; (Part I: acoustic emission behaviour of low alloy steels; Part II: fatigue crack assessment from proof testing and continuous monitoring); inspection of selected areas of engineering structures by acoustic emission; Japanese experience in laboratory and practical applications of acoustic emission to welded structures; and ASME acoustic emission code status. (U.K.)
8. Generalized laser filamentation instability coupled to cooling instability
International Nuclear Information System (INIS)
Liang, E.P.; Wong, J.; Garrison, J.
1984-01-01
We consider the propagation of laser light in an initially slightly nonuniform plasma. The classical dispersion relation for the laser filamentation growth rate (see e.g., B. Langdon, in the 1980 Lawrence Livermore National Laboratory Laser Program Annual Report, pp. 3-56, UCRL-50021-80, 1981) can be generalized to include other acoustical effects. For example, we find that the inclusion of potential imbalances in the heating and cooling rates of the ambient medium due to density and temperature perturbations can cause the laser filamentation mode to bifurcate into a cooling instability mode at long acoustic wavelengths. We also attempt to study semi-analytically the nonlinear evolution of this and related instabilities. These results have wide applications to a variety of chemical gas lasers and phenomena related to laser-target interactions (e.g., jet-like behavior)
9. Carpal instability
International Nuclear Information System (INIS)
Schmitt, R.; Froehner, S.; Coblenz, G.; Christopoulos, G.
2006-01-01
This review addresses the pathoanatomical basics as well as the clinical and radiological presentation of instability patterns of the wrist. Carpal instability mostly follows an injury; however, other diseases, like CPPD arthropathy, can be associated. Instability occurs either if the carpus is unable to sustain physiologic loads (''dyskinetics'') or suffers from abnormal motion of its bones during movement (''dyskinematics''). In the classification of carpal instability, dissociative subcategories (located within proximal carpal row) are differentiated from non-dissociative subcategories (present between the carpal rows) and combined patterns. It is essential to note that the unstable wrist initially does not cause relevant signs in standard radiograms, therefore being ''occult'' for the radiologic assessment. This paper emphasizes the high utility of kinematographic studies, contrast-enhanced magnetic resonance imaging (MRI) and MR arthrography for detecting these predynamic and dynamic instability stages. Later in the natural history of carpal instability, static malalignment of the wrist and osteoarthritis will develop, both being associated with significant morbidity and disability. To prevent individual and socio-economic implications, the handsurgeon or orthopedist, as well as the radiologist, is challenged for early and precise diagnosis. (orig.)
Science.gov (United States)
Radioactive decay is the emission of energy in the form of ionizing radiation. Example decay chains illustrate how radioactive atoms can go through many transformations as they become stable and no longer radioactive.
11. Performance through Deformation and Instability
Science.gov (United States)
Bertoldi, Katia
2015-03-01
Materials capable of undergoing large deformations like elastomers and gels are ubiquitous in daily life and nature. An exciting field of engineering is emerging that uses these compliant materials to design active devices, such as actuators, adaptive optical systems and self-regulating fluidics. Compliant structures may significantly change their architecture in response to diverse stimuli. When excessive deformation is applied, they may eventually become unstable. Traditionally, mechanical instabilities have been viewed as an inconvenience, with research focusing on how to avoid them. Here, I will demonstrate that these instabilities can be exploited to design materials with novel, switchable functionalities. The abrupt changes introduced into the architecture of soft materials by instabilities will be used to change their shape in a sudden, but controlled manner. Possible and exciting applications include materials with unusual properties such negative Poisson's ratio, phononic crystals with tunable low-frequency acoustic band gaps and reversible encapsulation systems.
12. Parametric instabilities in advanced gravitational wave detectors
International Nuclear Information System (INIS)
Gras, S; Zhao, C; Blair, D G; Ju, L
2010-01-01
As the LIGO interferometric gravitational wave detectors have finished gathering a large observational data set, an intense effort is underway to upgrade these observatories to improve their sensitivity by a factor of ∼10. High circulating power in the arm cavities is required, which leads to the possibility of parametric instability due to three-mode opto-acoustic resonant interactions between the carrier, transverse optical modes and acoustic modes. Here, we present detailed numerical analysis of parametric instability in a configuration that is similar to Advanced LIGO. After examining parametric instability for a single three-mode interaction in detail, we examine instability for the best and worst cases, as determined by the resonance condition of transverse modes in the power and signal recycling cavities. We find that, in the best case, the dual recycling detector is substantially less susceptible to instability than a single cavity, but its susceptibility is dependent on the signal recycling cavity design, and on tuning for narrow band operation. In all cases considered, the interferometer will experience parametric instability at full power operation, but the gain varies from 3 to 1000, and the number of unstable modes varies between 7 and 30 per test mass. The analysis focuses on understanding the detector complexity in relation to opto-acoustic interactions, on providing insights that can enable predictions of the detector response to transient disturbances, and of variations in thermal compensation conditions.
13. Weak decays
International Nuclear Information System (INIS)
Wojcicki, S.
1978-11-01
Lectures are given on weak decays from a phenomenological point of view, emphasizing new results and ideas and the relation of recent results to the new standard theoretical model. The general framework within which the weak decay is viewed and relevant fundamental questions, weak decays of noncharmed hadrons, decays of muons and the tau, and the decays of charmed particles are covered. Limitation is made to the discussion of those topics that either have received recent experimental attention or are relevant to the new physics. (JFP) 178 references
14. Acoustic detection of pneumothorax
Science.gov (United States)
Mansy, Hansen A.; Royston, Thomas J.; Balk, Robert A.; Sandler, Richard H.
2003-04-01
This study aims at investigating the feasibility of using low-frequency (pneumothorax detection were tested in dogs. In the first approach, broadband acoustic signals were introduced into the trachea during end-expiration and transmitted waves were measured at the chest surface. Pneumothorax was found to consistently decrease pulmonary acoustic transmission in the 200-1200-Hz frequency band, while less change was observed at lower frequencies (ppneumothorax states (pPneumothorax was found to be associated with a preferential reduction of sound amplitude in the 200- to 700-Hz range, and a decrease of sound amplitude variation (in the 300 to 600-Hz band) during the respiration cycle (pPneumothorax changed the frequency and decay rate of percussive sounds. These results imply that certain medical conditions may be reliably detected using appropriate acoustic measurements and analysis. [Work supported by NIH/NHLBI #R44HL61108.
15. Communication Acoustics
DEFF Research Database (Denmark)
Blauert, Jens
Communication Acoustics deals with the fundamentals of those areas of acoustics which are related to modern communication technologies. Due to the advent of digital signal processing and recording in acoustics, these areas have enjoyed an enormous upswing during the last 4 decades. The book...... the book a source of valuable information for those who want to improve or refresh their knowledge in the field of communication acoustics - and to work their way deeper into it. Due to its interdisciplinary character Communication Acoustics is bound to attract readers from many different areas, such as......: acoustics, cognitive science, speech science, and communication technology....
16. Experimental Acoustic Evaluation of an Auditorium
Directory of Open Access Journals (Sweden)
Marina Dana Ţopa
2012-01-01
Full Text Available The paper presents a case history: the acoustical analysis of a rectangular auditorium. The following acoustical parameters were evaluated: early decay time, reverberation time, clarity, definition, and center time. The excitation signal was linear sweep sine and additional analysis was carried out: peak-to-noise ratio, reverberation time for empty and occupied room, standard deviation of acoustical parameters, diffusion, and just noticeable differences analysis. Conclusions about room’s destination and modeling were drawn in the end.
17. Collision and recombination driven instabilities in variable charged ...
The dust-acoustic instability driven by recombination of electrons and ions on the surface of charged and variably-charged dust grains as well as by collisions in dusty plasmas with significant pressure of background neutrals have been theoretically investigated. The recombination driven instability is shown to be dominant ...
18. Acoustic Neuroma
Science.gov (United States)
An acoustic neuroma is a benign tumor that develops on the nerve that connects the ear to the brain. ... can press against the brain, becoming life-threatening. Acoustic neuroma can be difficult to diagnose, because the ...
19. Tau decays
International Nuclear Information System (INIS)
Golutvin, A.
1994-09-01
The most recent experimental results of τ physics are reviewed. The covered topics include precision measurements of semihadronic τ decay and their impact on tau branching ratio budget, the current status of the tau consistency test, a determination of Michel parameters and τ neutrino helicity, and upper limits on lepton-number violating τ decays. (orig.)
20. Decay tank
International Nuclear Information System (INIS)
Matsumura, Seiichi; Tagishi, Akinori; Sakata, Yuji; Kontani, Koji; Sudo, Yukio; Kaminaga, Masanori; Kameyama, Iwao; Ando, Koei; Ishiki, Masahiko.
1990-01-01
The present invention concerns an decay tank for decaying a radioactivity concentration of a fluid containing radioactive material. The inside of an decay tank body is partitioned by partitioning plates to form a flow channel. A porous plate is attached at the portion above the end of the partitioning plate, that is, a portion where the flow is just turned. A part of the porous plate has a slit-like opening on the side close to the partitioning plate, that is, the inner side of the flow at the turning portion thereof. Accordingly, the primary coolants passed through the pool type nuclear reactor and flown into the decay tank are flow caused to uniformly over the entire part of the tank without causing swirling. Since a distribution in a staying time is thus decreased, the effect of decaying 16 N as radioactive nuclides in the primary coolants is increased even in a limited volume of the tank. (I.N.)
1. From instabilities to multifragmentation
International Nuclear Information System (INIS)
Chomaz, P.; Jacquot, B.; Colonna, M.; Guarnera, A.
1994-01-01
The main purpose of this article is to show that, in many physical situations, the spinodal decomposition of unstable systems can be correctly described by stochastic mean-field approaches. Such theories predict that the occurrence of spinodal instability leading the multifragmentation of an expended nuclear system, can be signed through the observation of time scales for the fragment formation of the order of 100 fm/c and of typical fragment size around A=20. We will finally discuss the fact that these fragments are formed at finite temperature and so can subsequently decay in flight. Finally, we will give some hints about possible experimental signals of such first order phase transitions. (authors). 12 refs., 5 figs
2. From instabilities to multifragmentation
Energy Technology Data Exchange (ETDEWEB)
Chomaz, P.; Jacquot, B. [Grand Accelerateur National dIons Lourds (GANIL), 14 - Caen (France); Colonna, M.; Guarnera, A. [Grand Accelerateur National dIons Lourds (GANIL), 14 - Caen (France)]|[Istituto Nazionale di Fisica Nucleare, Bologna (Italy)
1994-12-31
The main purpose of this article is to show that, in many physical situations, the spinodal decomposition of unstable systems can be correctly described by stochastic mean-field approaches. Such theories predict that the occurrence of spinodal instability leading the multifragmentation of an expended nuclear system, can be signed through the observation of time scales for the fragment formation of the order of 100 fm/c and of typical fragment size around A=20. We will finally discuss the fact that these fragments are formed at finite temperature and so can subsequently decay in flight. Finally, we will give some hints about possible experimental signals of such first order phase transitions. (authors). 12 refs., 5 figs.
3. Acoustic cloaking and transformation acoustics
International Nuclear Information System (INIS)
Chen Huanyang; Chan, C T
2010-01-01
In this review, we give a brief introduction to the application of the new technique of transformation acoustics, which draws on a correspondence between coordinate transformation and material properties. The technique is formulated for both acoustic waves and linear liquid surface waves. Some interesting conceptual devices can be designed for manipulating acoustic waves. For example, we can design acoustic cloaks that make an object invisible to acoustic waves, and the cloak can either encompass or lie outside the object to be concealed. Transformation acoustics, as an analog of transformation optics, can go beyond invisibility cloaking. As an illustration for manipulating linear liquid surface waves, we show that a liquid wave rotator can be designed and fabricated to rotate the wave front. The acoustic transformation media require acoustic materials which are anisotropic and inhomogeneous. Such materials are difficult to find in nature. However, composite materials with embedded sub-wavelength resonators can in principle be made and such 'acoustic metamaterials' can exhibit nearly arbitrary values of effective density and modulus tensors to satisfy the demanding material requirements in transformation acoustics. We introduce resonant sonic materials and Helmholtz resonators as examples of acoustic metamaterials that exhibit resonant behaviour in effective density and effective modulus. (topical review)
4. Electron acceleration during the decay of nonlinear Whistler waves in low-beta electron-ion plasma
International Nuclear Information System (INIS)
Umeda, Takayuki; Saito, Shinji; Nariyuki, Yasuhiro
2014-01-01
Relativistic electron acceleration through dissipation of a nonlinear, short-wavelength, and monochromatic electromagnetic whistler wave in low-beta plasma is investigated by utilizing a one-dimensional fully relativistic electromagnetic particle-in-cell code. The nonlinear (large-amplitude) parent whistler wave decays through the parametric instability which enhances electrostatic ion acoustic waves and electromagnetic whistler waves. These waves satisfy the condition of three-wave coupling. Through the decay instability, the energy of electron bulk velocity supporting the parent wave is converted to the thermal energy perpendicular to the background magnetic field. Increase of the perpendicular temperature triggers the electron temperature anisotropy instability which generates broadband whistler waves and heats electrons in the parallel direction. The broadband whistler waves are inverse-cascaded during the relaxation of the electron temperature anisotropy. In lower-beta conditions, electrons with a pitch angle of about 90° are successively accelerated by inverse-cascaded whistler waves, and selected electrons are accelerated to over a Lorentz factor of 10. The result implies that the nonlinear dissipation of a finite-amplitude and short-wavelength whistler wave plays an important role in producing relativistic nonthermal electrons over a few MeV especially at lower beta plasmas.
5. Topological Acoustics
Science.gov (United States)
Yang, Zhaoju; Gao, Fei; Shi, Xihang; Lin, Xiao; Gao, Zhen; Chong, Yidong; Zhang, Baile
2015-03-01
The manipulation of acoustic wave propagation in fluids has numerous applications, including some in everyday life. Acoustic technologies frequently develop in tandem with optics, using shared concepts such as waveguiding and metamedia. It is thus noteworthy that an entirely novel class of electromagnetic waves, known as "topological edge states," has recently been demonstrated. These are inspired by the electronic edge states occurring in topological insulators, and possess a striking and technologically promising property: the ability to travel in a single direction along a surface without backscattering, regardless of the existence of defects or disorder. Here, we develop an analogous theory of topological fluid acoustics, and propose a scheme for realizing topological edge states in an acoustic structure containing circulating fluids. The phenomenon of disorder-free one-way sound propagation, which does not occur in ordinary acoustic devices, may have novel applications for acoustic isolators, modulators, and transducers.
6. Acoustical Imaging
CERN Document Server
Litniewski, Jerzy; Kujawska, Tamara; 31st International Symposium on Acoustical Imaging
2012-01-01
The International Symposium on Acoustical Imaging is a unique forum for advanced research, covering new technologies, developments, methods and theories in all areas of acoustics. This interdisciplinary Symposium has been taking place continuously since 1968. In the course of the years the proceedings volumes in the Acoustical Imaging Series have become a reference for cutting-edge research in the field. In 2011 the 31st International Symposium on Acoustical Imaging was held in Warsaw, Poland, April 10-13. Offering both a broad perspective on the state-of-the-art as well as in-depth research contributions by the specialists in the field, this Volume 31 in the Series contains an excellent collection of papers in six major categories: Biological and Medical Imaging Physics and Mathematics of Acoustical Imaging Acoustic Microscopy Transducers and Arrays Nondestructive Evaluation and Industrial Applications Underwater Imaging
7. Acoustic textiles
CERN Document Server
Nayak, Rajkishore
2016-01-01
This book highlights the manufacturing and applications of acoustic textiles in various industries. It also includes examples from different industries in which acoustic textiles can be used to absorb noise and help reduce the impact of noise at the workplace. Given the importance of noise reduction in the working environment in several industries, the book offers a valuable guide for companies, educators and researchers involved with acoustic materials.
8. Buneman and ion two-stream instabilities in the foot of collisionless shocks
International Nuclear Information System (INIS)
Fumio Takahara
2008-01-01
Two-dimensional electrostatic PIC simulations as well as linear analysis have been made for double periodic boundary conditions mimicking the shock foot region of supernova remnants. We found that modes propagating obliquely to the beam direction grow fast enough so that no surfing acceleration occurs. We also found that a new type of instability called ion two-stream instability is excited after the Buneman instability saturated instead of the ion acoustic instability. Implications for electron heating are shortly discussed. (author)
9. Parametric instabilities excited by localized pumps near the lower-hybrid frequency
International Nuclear Information System (INIS)
Kuo, Y.Y.; Chen, L.
1976-04-01
Parametric instabilities excited in non-uniform plasmas by spatially localized pump fields oscillating near the local lower-hybrid frequency are analytically investigated. Corresponding threshold conditions, temporal growth rates, and spatial amplification factors are obtained for the oscillating-two-stream instability and the decay instabilities due to nonlinear electron and ion Landau dampings
10. B decays
CERN Document Server
Stone, Sheldon
1992-01-01
The study of b quarks has now reached a stage where it is useful to review what has been learned so far and also to look at the implications of future studies. The most important observations thus far - measurement of the "B" lifetime, B 0 - B 0 mixing, and the observation of b? u transitions, as well as more mundane results on hadronic and semileptonic transitions - are described in detail by experimentalists who have been closely involved with the measurements. Theoretical progress in understanding b quark decays, including the mechanisms of hadronic and semileptonic decays, are described. S
11. B decays
CERN Document Server
Stone, Sheldon
1994-01-01
This book reviews the study of b quarks and also looks at the implications of future studies. The most important observations thus far - including measurement of the ""B"" lifetime and observations of b -> u transitions - as well as the more mundane results of hadronic and semileptonic transitions are described in detail by experimentalists who have been closely involved with the measurements. Theoretical progress in understanding b quark decays, including the mechanisms of hadronic and semileptonic decays, are described. Synthesizing the experimental and theoretical information, the authors d
12. Role of collective effects in dominance of scattering off thermal ions over Langmuir wave decay: Analysis, simulations, and space applications
International Nuclear Information System (INIS)
Cairns, Iver H.
2000-01-01
Langmuir waves driven to high levels by beam instabilities are subject to nonlinear processes, including the closely related processes of scattering off thermal ions (STI) and a decay process in which the ion response is organized into a product ion acoustic wave. Calculations of the nonlinear growth rates predict that the decay process should always dominate STI, creating two paradoxes. The first is that three independent computer simulation studies show STI proceeding, with no evidence for the decay at all. The second is that observations in space of type III solar radio bursts and Earth's foreshock, which the simulations were intended to model, show evidence for the decay proceeding but no evidence for STI. Resolutions to these paradoxes follow from the realization that a nonlinear process cannot proceed when its growth rate exceeds the minimum frequency of the participating waves, since the required collective response cannot be maintained and the waves cannot respond appropriately, and that a significant number of e-foldings and wave periods must be contained in the time available. It is shown that application of these ''collective'' and ''time scale'' constraints to the simulations explains why the decay does not proceed in them, as well as why STI proceeds in specific simulations. This appears to be the first demonstration that collective constraints are important in understanding nonlinear phenomena. Furthermore, applying these constraints to space observations, it is predicted that the decay should proceed (and dominate STI) in type III sources and the high beam speed regions of Earth's foreshock for a specific range of wave levels, with a possible role for STI alone at slightly higher wave levels. Deeper in the foreshock, for slower beams and weaker wave levels, the decay and STI are predicted to become ineffective. Suggestions are given for future testing of the collective constraint and an explanation for why waves in space are usually much weaker than
13. KB-WOT Quality assurance acoustics: overview and protocols 2008 version
NARCIS (Netherlands)
Ybema, M.S.
2009-01-01
The quality of IMARES' acoustic surveys proved quite unstable in recent years despite extra effort in this field to bring this instability down. The amount of involved scientists in acoustics has been small compared to demersal survey work. Therefore scientific standards of acoustic surveys are
14. Thermo-acoustic cross-talk between cans in a can-annular combustor
NARCIS (Netherlands)
Farisco, Federica; Panek, Lukasz; Kok, Jim B.W.
2017-01-01
Thermo-acoustic instabilities in gas turbine engines are studied to avoid engine failure. Compared to the engines with annular combustors, the can-annular combustor design should be less vulnerable to acoustic burner-to-burner interaction, since the burners are acoustically coupled only by the
15. Observation of self-excited acoustic vortices in defect-mediated dust acoustic wave turbulence.
Science.gov (United States)
Tsai, Ya-Yi; I, Lin
2014-07-01
Using the self-excited dust acoustic wave as a platform, we demonstrate experimental observation of self-excited fluctuating acoustic vortex pairs with ± 1 topological charges through spontaneous waveform undulation in defect-mediated turbulence for three-dimensional traveling nonlinear longitudinal waves. The acoustic vortex pair has helical waveforms with opposite chirality around the low-density hole filament pair in xyt space (the xy plane is the plane normal to the wave propagation direction). It is generated through ruptures of sequential crest surfaces and reconnections with their trailing ruptured crest surfaces. The initial rupture is originated from the amplitude reduction induced by the formation of the kinked wave crest strip with strong stretching through the undulation instability. Increasing rupture causes the separation of the acoustic vortex pair after generation. A similar reverse process is followed for the acoustic vortex annihilating with the opposite-charged acoustic vortex from the same or another pair generation.
16. Aspects of electron acoustic wave physics in laser backscatter from plasmas
International Nuclear Information System (INIS)
Sircombe, N J; Arber, T D; Dendy, R O
2006-01-01
Recent experimental results from the Trident laser confirm the importance of kinetic effects in determining laser reflectivities at high intensities. Examples observed include scattering from low frequency electron acoustic waves (EAWs) and the first few stages of a cascade towards turbulence through the Langmuir decay instability. Interpretive and predictive computational capability in this area is assisted by the development of Vlasov codes, which offer high velocity space resolution in high energy regions of particle phase space and do not require analytical pre-processing of the fundamental equations. A direct Vlasov solver, capable of resolving these kinetic processes, is used here to address fundamental aspects of the existence and stability of the electron acoustic wave, together with its collective scattering properties. These simulations are extended to realistic laser and plasma parameters characteristic of single hot-spot experiments. Results are in qualitative agreement with experiments displaying both stimulated Raman and stimulated electron acoustic scattering. The amplitude of simulated EAWs is greater than that observed experimentally and is accompanied by a higher phase velocity. These minor differences can be attributed to the limitations of a one-dimensional collisionless model
17. Battlefield acoustics
CERN Document Server
Damarla, Thyagaraju
2015-01-01
This book presents all aspects of situational awareness in a battlefield using acoustic signals. It starts by presenting the science behind understanding and interpretation of sound signals. The book then goes on to provide various signal processing techniques used in acoustics to find the direction of sound source, localize gunfire, track vehicles, and detect people. The necessary mathematical background and various classification and fusion techniques are presented. The book contains majority of the things one would need to process acoustic signals for all aspects of situational awareness in one location. The book also presents array theory, which is pivotal in finding the direction of arrival of acoustic signals. In addition, the book presents techniques to fuse the information from multiple homogeneous/heterogeneous sensors for better detection. MATLAB code is provided for majority of the real application, which is a valuable resource in not only understanding the theory but readers, can also use the code...
18. Acoustics Research
Data.gov (United States)
National Oceanic and Atmospheric Administration, Department of Commerce — Fisheries acoustics data are collected from more than 200 sea-days each year aboard the FRV DELAWARE II and FRV ALBATROSS IV (decommissioned) and the FSV Henry B....
19. Acoustical Imaging
CERN Document Server
Akiyama, Iwaki
2009-01-01
The 29th International Symposium on Acoustical Imaging was held in Shonan Village, Kanagawa, Japan, April 15-18, 2007. This interdisciplinary Symposium has been taking place every two years since 1968 and forms a unique forum for advanced research, covering new technologies, developments, methods and theories in all areas of acoustics. In the course of the years the volumes in the Acoustical Imaging Series have developed and become well-known and appreciated reference works. Offering both a broad perspective on the state-of-the-art in the field as well as an in-depth look at its leading edge research, this Volume 29 in the Series contains again an excellent collection of seventy papers presented in nine major categories: Strain Imaging Biological and Medical Applications Acoustic Microscopy Non-Destructive Evaluation and Industrial Applications Components and Systems Geophysics and Underwater Imaging Physics and Mathematics Medical Image Analysis FDTD method and Other Numerical Simulations Audience Researcher...
20. Room Acoustics
Science.gov (United States)
Kuttruff, Heinrich; Mommertz, Eckard
The traditional task of room acoustics is to create or formulate conditions which ensure the best possible propagation of sound in a room from a sound source to a listener. Thus, objects of room acoustics are in particular assembly halls of all kinds, such as auditoria and lecture halls, conference rooms, theaters, concert halls or churches. Already at this point, it has to be pointed out that these conditions essentially depend on the question if speech or music should be transmitted; in the first case, the criterion for transmission quality is good speech intelligibility, in the other case, however, the success of room-acoustical efforts depends on other factors that cannot be quantified that easily, not least it also depends on the hearing habits of the listeners. In any case, absolutely "good acoustics" of a room do not exist.
1. Acoustic Localization with Infrasonic Signals
Science.gov (United States)
Threatt, Arnesha; Elbing, Brian
2015-11-01
Numerous geophysical and anthropogenic events emit infrasonic frequencies (<20 Hz), including volcanoes, hurricanes, wind turbines and tornadoes. These sounds, which cannot be heard by the human ear, can be detected from large distances (in excess of 100 miles) due to low frequency acoustic signals having a very low decay rate in the atmosphere. Thus infrasound could be used for long-range, passive monitoring and detection of these events. An array of microphones separated by known distances can be used to locate a given source, which is known as acoustic localization. However, acoustic localization with infrasound is particularly challenging due to contamination from other signals, sensitivity to wind noise and producing a trusted source for system development. The objective of the current work is to create an infrasonic source using a propane torch wand or a subwoofer and locate the source using multiple infrasonic microphones. This presentation will present preliminary results from various microphone configurations used to locate the source.
2. Anisotropic gravitational instability
International Nuclear Information System (INIS)
Polyachenko, V.L.; Fridman, A.M.
1988-01-01
Exact solutions of stability problems are obtained for two anisotropic gravitational systems of different geometries - a layer of finite thickness at rest and a rotating cylinder of finite radius. It is shown that the anisotropic gravitational instability which develops in both cases is of Jeans type. However, in contrast to the classical aperiodic Jeans instability, this instability is oscillatory. The physics of the anisotropic gravitational instability is investigated. It is shown that in a gravitating layer this instability is due, in particular, to excitation of previously unknown interchange-Jeans modes. In the cylinder, the oscillatory Jeans instability is associated with excitation of a rotational branch, this also being responsible for the beam gravitational instability. This is the reason why this instability and the anisotropic gravitational instability have so much in common
3. THE SATURATION OF SASI BY PARASITIC INSTABILITIES
International Nuclear Information System (INIS)
Guilet, Jerome; Sato, Jun'ichi; Foglizzo, Thierry
2010-01-01
The standing accretion shock instability (SASI) is commonly believed to be responsible for large amplitude dipolar oscillations of the stalled shock during core collapse, potentially leading to an asymmetric supernovae explosion. The degree of asymmetry depends on the amplitude of SASI, but the nonlinear saturation mechanism has never been elucidated. We investigate the role of parasitic instabilities as a possible cause of nonlinear SASI saturation. As the shock oscillations create both vorticity and entropy gradients, we show that both Kelvin-Helmholtz and Rayleigh-Taylor types of instabilities are able to grow on a SASI mode if its amplitude is large enough. We obtain simple estimates of their growth rates, taking into account the effects of advection and entropy stratification. In the context of the advective-acoustic cycle, we use numerical simulations to demonstrate how the acoustic feedback can be decreased if a parasitic instability distorts the advected structure. The amplitude of the shock deformation is estimated analytically in this scenario. When applied to the set up of Fernandez and Thompson, this saturation mechanism is able to explain the dramatic decrease of the SASI power when both the nuclear dissociation energy and the cooling rate are varied. Our results open new perspectives for anticipating the effect, on the SASI amplitude, of the physical ingredients involved in the modeling of the collapsing star.
4. Thermal instability in a stratified plasma
International Nuclear Information System (INIS)
Hermanns, D.F.M.; Priest, E.R.
1989-01-01
The thermal instability mechansism has been studied in connection to observed coronal features, like, e.g. prominences or cool cores in loops. Although these features show a lot of structure, most studies concern the thermal instability in an uniform medium. In this paper, we investigate the thermal instability and the interaction between thermal modes and the slow magneto-acoustic subspectrum for a stratified plasma slab. We fomulate the relevant system of equations and give some straightforward properties of the linear spectrum of a non-uniform plasma slab, i.e. the existence of continuous parts in the spectrum. We present a numerical scheme with which we can investigate the linear spectrum for equilibrium states with stratification. The slow and thermal subspectra of a crude coronal model are given as a preliminary result. (author). 6 refs.; 1 fig
5. Proton decay theory
International Nuclear Information System (INIS)
Marciano, W.J.
1983-01-01
Topics include minimal SU(5) predictions, gauge boson mediated proton decay, uncertainties in tau/sub p/, Higgs scalar effects, proton decay via Higgs scalars, supersymmetric SU(5), dimension 5 operators and proton decay, and Higgs scalars and proton decay
6. Plasmon band gap generated by intense ion acoustic waves
International Nuclear Information System (INIS)
Son, S.; Ku, S.
2010-01-01
In the presence of an intense ion acoustic wave, the energy-momentum dispersion relation of plasmons is strongly modified to exhibit a band gap structure. The intensity of an ion acoustic wave might be measured from the band gap width. The plasmon band gap can be used to block the nonlinear cascading channel of the Langmuir wave decay.
7. The study of waves, instabilities, and turbulence using Thomson scattering in laser plasmas
International Nuclear Information System (INIS)
Drake, R.P.
1995-01-01
Much basic work in plasma physics has been devoted to the study of wave properties in plasmas, one of the nonlinear development of driven waves, and of the instabilities in which such waves may participate. The use of laser-plasma techniques has allowed one to extend such studies into new regimes. Such techniques and their results are the subject here. Once one chooses a physical problem within this subject area, it is now possible to design a laser-plasma experiment that is optimized for the study of that problem. The plasma can be designed to have a variety of density and flow-velocity profiles, the damping of ion acoustic waves and of electron plasma waves can be independently controlled, and the waves can be driven weakly or strongly. By using Nd-glass lasers and their harmonics one can non-invasively drive and diagnose the waves, using separate laser beams to produce the plasma, drive the waves, and diagnose their properties. The author uses as examples some recent work with his collaborators, including the first experimental detection of ion plasma waves and the first direct observation of the plasma wave driven by the acoustic decay of laser light
8. Acoustic biosensors.
Science.gov (United States)
Fogel, Ronen; Limson, Janice; Seshia, Ashwin A
2016-06-30
Resonant and acoustic wave devices have been researched for several decades for application in the gravimetric sensing of a variety of biological and chemical analytes. These devices operate by coupling the measurand (e.g. analyte adsorption) as a modulation in the physical properties of the acoustic wave (e.g. resonant frequency, acoustic velocity, dissipation) that can then be correlated with the amount of adsorbed analyte. These devices can also be miniaturized with advantages in terms of cost, size and scalability, as well as potential additional features including integration with microfluidics and electronics, scaled sensitivities associated with smaller dimensions and higher operational frequencies, the ability to multiplex detection across arrays of hundreds of devices embedded in a single chip, increased throughput and the ability to interrogate a wider range of modes including within the same device. Additionally, device fabrication is often compatible with semiconductor volume batch manufacturing techniques enabling cost scalability and a high degree of precision and reproducibility in the manufacturing process. Integration with microfluidics handling also enables suitable sample pre-processing/separation/purification/amplification steps that could improve selectivity and the overall signal-to-noise ratio. Three device types are reviewed here: (i) bulk acoustic wave sensors, (ii) surface acoustic wave sensors, and (iii) micro/nano-electromechanical system (MEMS/NEMS) sensors. © 2016 The Author(s). Published by Portland Press Limited on behalf of the Biochemical Society.
9. Instability and degeneracy in the BMN correspondence
International Nuclear Information System (INIS)
Freedman, Daniel Z.; Guersoy, Umut
2003-01-01
Non-degenerate perturbation theory, which was used to calculate the scale dimension of operators on the gauge theory side of the correspondence, breaks down when effects of triple trace operators are included. We interpret this as an instability of excited single-string states in the dual string theory for decay into the continuum of degenerate 3-string states. We apply time-dependent perturbation theory to calculate the decay widths from gauge theory. These widths are new gauge theory data which can be compared with future calculations in light cone string field theory. (author)
10. Computer fan performance enhancement via acoustic perturbations
Energy Technology Data Exchange (ETDEWEB)
Greenblatt, David, E-mail: [email protected] [Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa (Israel); Avraham, Tzahi; Golan, Maayan [Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa (Israel)
2012-04-15
Highlights: Black-Right-Pointing-Pointer Computer fan effectiveness was increased by introducing acoustic perturbations. Black-Right-Pointing-Pointer Acoustic perturbations controlled blade boundary layer separation. Black-Right-Pointing-Pointer Optimum frequencies corresponded with airfoils studies. Black-Right-Pointing-Pointer Exploitation of flow instabilities was responsible for performance improvements. Black-Right-Pointing-Pointer Peak pressure and peak flowrate were increased by 40% and 15% respectively. - Abstract: A novel technique for increasing computer fan effectiveness, based on introducing acoustic perturbations onto the fan blades to control boundary layer separation, was assessed. Experiments were conducted in a specially designed facility that simultaneously allowed characterization of fan performance and introduction of the perturbations. A parametric study was conducted to determine the optimum control parameters, namely those that deliver the largest increase in fan pressure for a given flowrate. The optimum reduced frequencies corresponded with those identified on stationary airfoils and it was thus concluded that the exploitation of Kelvin-Helmholtz instabilities, commonly observed on airfoils, was responsible for the fan blade performance improvements. The optimum control inputs, such as acoustic frequency and sound pressure level, showed some variation with different fan flowrates. With the near-optimum control conditions identified, the full operational envelope of the fan, when subjected to acoustic perturbations, was assessed. The peak pressure and peak flowrate were increased by up to 40% and 15% respectively. The peak fan efficiency increased with acoustic perturbations but the overall system efficiency was reduced when the speaker input power was accounted for.
11. Computer fan performance enhancement via acoustic perturbations
International Nuclear Information System (INIS)
Greenblatt, David; Avraham, Tzahi; Golan, Maayan
2012-01-01
Highlights: ► Computer fan effectiveness was increased by introducing acoustic perturbations. ► Acoustic perturbations controlled blade boundary layer separation. ► Optimum frequencies corresponded with airfoils studies. ► Exploitation of flow instabilities was responsible for performance improvements. ► Peak pressure and peak flowrate were increased by 40% and 15% respectively. - Abstract: A novel technique for increasing computer fan effectiveness, based on introducing acoustic perturbations onto the fan blades to control boundary layer separation, was assessed. Experiments were conducted in a specially designed facility that simultaneously allowed characterization of fan performance and introduction of the perturbations. A parametric study was conducted to determine the optimum control parameters, namely those that deliver the largest increase in fan pressure for a given flowrate. The optimum reduced frequencies corresponded with those identified on stationary airfoils and it was thus concluded that the exploitation of Kelvin–Helmholtz instabilities, commonly observed on airfoils, was responsible for the fan blade performance improvements. The optimum control inputs, such as acoustic frequency and sound pressure level, showed some variation with different fan flowrates. With the near-optimum control conditions identified, the full operational envelope of the fan, when subjected to acoustic perturbations, was assessed. The peak pressure and peak flowrate were increased by up to 40% and 15% respectively. The peak fan efficiency increased with acoustic perturbations but the overall system efficiency was reduced when the speaker input power was accounted for.
12. Lower hybrid parametric instabilities nonuniform pump waves and tokamak applications
International Nuclear Information System (INIS)
Berger, R.L.; Chen, L.; Kaw, P.K.; Perkins, F.W.
1976-11-01
Electrostatic lower hybrid ''pump'' waves often launched into tokamak plasmas by structures (e.g., waveguides) whose dimensions are considerably smaller than characteristic plasma sizes. Such waves propagate in well-defined resonance cones and give rise to parametric instabilities driven by electron E x B velocities. The finite size of the resonance cone region determines the threshold for both convective quasimode decay instabilities and absolute instabilities. The excitation of absolute instabilities depends on whether a travelling or standing wave pump model is used; travelling wave pumps require the daughter waves to have a definite frequency shift. Altogether, parametric instabilities driven by E x B velocities occur for threshold fields significantly below the threshold for filamentation instabilities driven by pondermotive forces. Applications to tokamak heating show that nonlinear effects set in when a certain power-per-wave-launching port is exceeded
13. Radical conservatism and nucleon decay
International Nuclear Information System (INIS)
Wilczek, Frank
2000-01-01
Unification of couplings, observation of neutrino masses in the expected range, and several other considerations confirm central implications of straightforward gauge unification based on SO(10) or a close relative and incorporating low-energy supersymmetry. The remaining outstanding consequence of this circle of ideas, yet to be observed, is nucleon instability. Clearly, we should aspire to be as specific as possible regarding the rate and form of such instability. I argue that not only esthetics, but also the observed precision of unification of couplings, favors an economical symmetry-breaking (Higgs) structure. Assuming this, one can exploit its constraints to build reasonably economical, overconstrained yet phenomenologically viable models of quark and lepton masses. Putting it all together, one arrives at reasonably concrete, hopeful expectations regarding nucleon decay. These expectations are neither ruled out by existing experiments, nor hopelessly inaccessible. Furthermore, the branching fractions can discriminate among different possibilities for physics at the unification scale
14. Acoustic emission
International Nuclear Information System (INIS)
Straus, A.; Lopez Pumarega, M.I.; Di Gaetano, J.O.; D'Atellis, C.E.; Ruzzante, J.E.
1990-01-01
This paper is related to our activities on acoustic emission (A.E.). The work is made with different materials: metals and fibre reinforced plastics. At present, acoustic emission transducers are being developed for low and high temperature. A test to detect electrical discharges in electrical transformers was performed. Our experience in industrial tests to detect cracks or failures in tanks or tubes is also described. The use of A.E. for leak detection is considered. Works on pattern recognition of A.E. signals are also being performed. (Author)
15. Building Acoustics
Science.gov (United States)
Cowan, James
This chapter summarizes and explains key concepts of building acoustics. These issues include the behavior of sound waves in rooms, the most commonly used rating systems for sound and sound control in buildings, the most common noise sources found in buildings, practical noise control methods for these sources, and the specific topic of office acoustics. Common noise issues for multi-dwelling units can be derived from most of the sections of this chapter. Books can be and have been written on each of these topics, so the purpose of this chapter is to summarize this information and provide appropriate resources for further exploration of each topic.
16. Electromagnetic ion beam instability upstream of the earth's bow shock
International Nuclear Information System (INIS)
Gary, S.P.; Gosling, J.T.; Forslund, D.W.
1981-01-01
The linear theory of the electromagnetic ion beam instability for arbitrary angles of propagation has been studied. The parameters considered in the theory are typical of the solar wind upstream of the earth's bow shock when a 'reflected' proton beam is present. Maximum growth occurs for propagation parallel to the ambient field B, but this instability also displays significant growth at wave-vectors oblique to B, Oblique, unstable modes seem to be the likely source of the compressive magnetic fluctuations recently observed in conjunction with 'diffuse' ion population. An energetic ion beam does not directly give rise to linear growth of either ion acoustic or whistler mode instabilities
17. Instabilities in inhomogeneous plasma
International Nuclear Information System (INIS)
Mikhailovsky, A.B.
1983-01-01
The plasma inhomogeneity across the magnetic field causes a wide class of instabilities which are called instabilities of an inhomogeneous plasma or gradient instabilities. The instabilities that can be studied in the approximation of a magnetic field with parallel straight field lines are treated first, followed by a discussion of the influence of shear on these instabilities. The instabilities of a weakly inhomogeneous plasma with the Maxwellian velocity distribution of particles caused by the density and temperature gradients are often called drift instabilities, and the corresponding types of perturbations are the drift waves. An elementary theory of drift instabilities is presented, based on the simplest equations of motion of particles in the field of low-frequency and long-wavelength perturbations. Following that is a more complete theory of inhomogeneous collisionless plasma instabilities which uses the permittivity tensor and, in the case of electrostatic perturbations, the scalar of permittivity. The results are used to study the instabilities of a strongly inhomogeneous plasma. The instabilities of a plasma in crossed fields are discussed and the electromagnetic instabilities of plasma with finite and high pressure are described. (Auth.)
18. High Frequency Combustion Instabilities of LOx/CH4 Spray Flames in Rocket Engine Combustion Chambers
NARCIS (Netherlands)
Sliphorst, M.
2011-01-01
Ever since the early stages of space transportation in the 1940’s, and the related liquid propellant rocket engine development, combustion instability has been a major issue. High frequency combustion instability (HFCI) is the interaction between combustion and the acoustic field in the combustion
19. Critical condition for current-driven instability excited in turbulent heating of TRIAM-1 tokamak plasma
Energy Technology Data Exchange (ETDEWEB)
Nakamura, Y; Watanabe, T; Nagao, A; Nakamura, K; Kikuchi, M; Aoki, T; Hiraki, N; Itoh, S [Kyushu Univ., Fukuoka (Japan). Research Inst. for Applied Mechanics; Mitarai, O
1982-02-01
Critical condition for current-driven instability excited in turbulently heated TRIAM-1 tokamak plasma is investigated experimentally. Resistive hump in loop voltage, plasma density fluctuation and rapid increase of electron temperature in a skin layer are simultaneously observed at the time when the electron drift velocity amounts to the critical drift velocity for low-frequency ion acoustic instability.
20. Mirror Instability in the Turbulent Solar Wind
Energy Technology Data Exchange (ETDEWEB)
Hellinger, Petr [Astronomical Institute, CAS, Bocni II/1401,CZ-14100 Prague (Czech Republic); Landi, Simone; Verdini, Andrea; Franci, Luca [Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze Largo E. Fermi 2, I-50125 Firenze (Italy); Matteini, Lorenzo, E-mail: [email protected] [Department of Physics, Imperial College London, London SW7 2AZ (United Kingdom)
2017-04-01
The relationship between a decaying strong turbulence and the mirror instability in a slowly expanding plasma is investigated using two-dimensional hybrid expanding box simulations. We impose an initial ambient magnetic field perpendicular to the simulation box, and we start with a spectrum of large-scale, linearly polarized, random-phase Alfvénic fluctuations that have energy equipartition between kinetic and magnetic fluctuations and a vanishing correlation between the two fields. A turbulent cascade rapidly develops, magnetic field fluctuations exhibit a Kolmogorov-like power-law spectrum at large scales and a steeper spectrum at sub-ion scales. The imposed expansion (taking a strictly transverse ambient magnetic field) leads to the generation of an important perpendicular proton temperature anisotropy that eventually drives the mirror instability. This instability generates large-amplitude, nonpropagating, compressible, pressure-balanced magnetic structures in a form of magnetic enhancements/humps that reduce the perpendicular temperature anisotropy.
1. Micromachined silicon acoustic delay line with improved structural stability and acoustic directivity for real-time photoacoustic tomography
Science.gov (United States)
Cho, Young; Kumar, Akhil; Xu, Song; Zou, Jun
2017-03-01
Recent studies have shown that micromachined silicon acoustic delay lines can provide a promising solution to achieve real-time photoacoustic tomography without the need for complex transducer arrays and data acquisition electronics. However, as its length increases to provide longer delay time, the delay line becomes more vulnerable to structural instability due to reduced mechanical stiffness. In addition, the small cross-section area of the delay line results in a large acoustic acceptance angle and therefore poor directivity. To address these two issues, this paper reports the design, fabrication, and testing of a new silicon acoustic delay line enhanced with 3D printed polymer micro linker structures. First, mechanical deformation of the silicon acoustic delay line (with and without linker structures) under gravity was simulated by using finite element method. Second, the acoustic crosstalk and acoustic attenuation caused by the polymer micro linker structures were evaluated with both numerical simulation and ultrasound transmission testing. The result shows that the use of the polymer micro linker structures significantly improves the structural stability of the silicon acoustic delay lines without creating additional acoustic attenuation and crosstalk. In addition, a new tapered design for the input terminal of the delay line was also investigate to improve its acoustic directivity by reducing the acoustic acceptance angle. These two improvements are expected to provide an effective solution to eliminate current limitations on the achievable acoustic delay time and out-of-plane imaging resolution of micromachined silicon acoustic delay line arrays.
2. Thermo-acoustic characterization of the burner-turbine interface in a can-annular combustor using CFD
NARCIS (Netherlands)
Farisco, Federica
2016-01-01
Thermo-acoustic instabilities in high power density gas turbine engines need to be understood to avoid unexpected shutdown events. This dissertation is focused on the combustor-turbine interaction for acoustic waves. The first part of the study is based on the acoustic reflection coefficient
3. Airy acoustical-sheet spinner tweezers
Science.gov (United States)
Mitri, F. G.
2016-09-01
The Airy acoustical beam exhibits parabolic propagation and spatial acceleration, meaning that the propagation bending angle continuously increases before the beam trajectory reaches a critical angle where it decays after a propagation distance, without applying any external bending force. As such, it is of particular importance to investigate its properties from the standpoint of acoustical radiation force, spin torque, and particle dynamics theories, in the development of novel particle sorting techniques and acoustically mediated clearing systems. This work investigates these effects on a two-dimensional (2D) circular absorptive structure placed in the field of a nonparaxial Airy "acoustical-sheet" (i.e., finite beam in 2D), for potential applications in surface acoustic waves and acousto-fluidics. Based on the characteristics of the acoustic field, the beam is capable of manipulating the circular cylindrical fluid cross-section and guides it along a transverse or parabolic trajectory. This feature of Airy acoustical beams could lead to a unique characteristic in single-beam acoustical tweezers related to acoustical sieving, filtering, and removal of particles and cells from a section of a small channel. The analysis developed here is based on the description of the nonparaxial Airy beam using the angular spectrum decomposition of plane waves in close association with the partial-wave series expansion method in cylindrical coordinates. The numerical results demonstrate the ability of the nonparaxial Airy acoustical-sheet beam to pull, propel, or accelerate a particle along a parabolic trajectory, in addition to particle confinement in the transverse direction of wave propagation. Negative or positive radiation force and spin torque causing rotation in the clockwise or the anticlockwise direction can occur depending on the nondimensional parameter ka (where k is the wavenumber and a is the radius) and the location of the cylinder in the beam. Applications in
4. Visualization of conventional and combusting subsonic jet instabilities
CERN Document Server
Kozlov, Victor; Litvinenko, Yury
2016-01-01
Based on new information obtained on free microjets, this book explains the latest phenomena in flame evolution in the presence of a transverse acoustic field with round and plane propane microjet combustion. It gives an overview of recent experimental results on instability and dynamics of jets at low Reynolds numbers and provides the reader, step by step, with the milestones and recent advances in jet flow stability and combustion. Readers will also discover a clarification of the differences between top-hat and parabolic round and plane jet instability. Chapters demonstrate features of the interaction between jet and crossflow, and how experimental data testify to similarities of the perturbed flow patterns of laminar and turbulent round jets. A similar response of the jets to external acoustic oscillations is shown, as well as the peculiarities of the effect of a transverse acoustic field on downstream evolution of round and plane macro- and microjets. Basic features of round and plane, macro and micro je...
5. Streaming instabilities in a collisional dusty plasma
International Nuclear Information System (INIS)
Mamun, A. A.; Shukla, P. K.
2000-01-01
A pair of low-frequency electrostatic modes, which are very similar to those experimentally observed by Praburam and Goree [Phys. Plasmas 3, 1212 (1996)], are found to exist in a dusty plasma with a significant background neutral pressure and background ion streaming. One of these two modes is the dust-acoustic mode and the other one is a new mode which is due to the combined effects of the ion streaming and ion--neutral collisions. It has been shown that in the absence of the ion streaming, the dust-acoustic mode is damped due to the combined effects of the ion--neutral and dust--neutral collisions and the electron--ion recombination onto the dust grain surface. This result disagrees with Kaw and Singh [Phys. Rev. Lett. 79, 423 (1997)], who reported collisional instability of the dust-acoustic mode in such a dusty plasma. It has also been found that a streaming instability with the growth rate of the order of the dust plasma frequency is triggered when the background ion streaming speed relative to the charged dust particles is comparable or higher than the ion--thermal speed. This point completely agrees with Rosenberg [J. Vac. Soc. Technol. A 14, 631 (1996)
6. Acoustic Territoriality
DEFF Research Database (Denmark)
Kreutzfeldt, Jacob
2011-01-01
Under the heading of "Gang i København" a number of initiatives was presented by the Lord Mayer and the Technical and Environmental Mayer of Copenhagen in May 2006. The aim of the initiative, which roughly translates to Lively Copenhagen, was both to make Copenhagen a livelier city in terms of city...... this article outline a few approaches to a theory of acoustic territoriality....
7. Acoustic lenses
International Nuclear Information System (INIS)
Kittmer, C.A.
1983-03-01
Acoustic lenses focus ultrasound to produce pencil-like beams with reduced near fields. When fitted to conventional (flat-faced) transducers, such lenses greatly improve the ability to detect and size defects. This paper describes a program developed to design acoustic lenses for use in immersion or contact inspection, using normal or angle beam mode with flat or curved targets. Lens surfaces are circular in geometry to facilitate machining. For normal beam inspection of flat plate, spherical or cylindrical lenses are used. For angle beam or curved surface inspections, a compound lens is required to correct for the extra induced aberration. Such a lens is aspherical with one radius of curvature in the plane of incidence, and a different radius of curvature in the plane perpendicular to the incident plane. The resultant beam profile (i.e., location of the acoustic focus, beam diameter, 6 dB working range) depends on the degree of focusing and the transducer used. The operating frequency and bandwidth can be affected by the instrumentation used. Theoretical and measured beam profiles are in good agreement. Various applications, from zone focusing used for defect sizing in thick plate, to line focusing for pipe weld inspection, are discussed
8. Joint Instability and Osteoarthritis
Directory of Open Access Journals (Sweden)
Darryl Blalock
2015-01-01
Full Text Available Joint instability creates a clinical and economic burden in the health care system. Injuries and disorders that directly damage the joint structure or lead to joint instability are highly associated with osteoarthritis (OA. Thus, understanding the physiology of joint stability and the mechanisms of joint instability-induced OA is of clinical significance. The first section of this review discusses the structure and function of major joint tissues, including periarticular muscles, which play a significant role in joint stability. Because the knee, ankle, and shoulder joints demonstrate a high incidence of ligament injury and joint instability, the second section summarizes the mechanisms of ligament injury-associated joint instability of these joints. The final section highlights the recent advances in the understanding of the mechanical and biological mechanisms of joint instability-induced OA. These advances may lead to new opportunities for clinical intervention in the prevention and early treatment of OA.
9. Joint instability and osteoarthritis.
Science.gov (United States)
Blalock, Darryl; Miller, Andrew; Tilley, Michael; Wang, Jinxi
2015-01-01
Joint instability creates a clinical and economic burden in the health care system. Injuries and disorders that directly damage the joint structure or lead to joint instability are highly associated with osteoarthritis (OA). Thus, understanding the physiology of joint stability and the mechanisms of joint instability-induced OA is of clinical significance. The first section of this review discusses the structure and function of major joint tissues, including periarticular muscles, which play a significant role in joint stability. Because the knee, ankle, and shoulder joints demonstrate a high incidence of ligament injury and joint instability, the second section summarizes the mechanisms of ligament injury-associated joint instability of these joints. The final section highlights the recent advances in the understanding of the mechanical and biological mechanisms of joint instability-induced OA. These advances may lead to new opportunities for clinical intervention in the prevention and early treatment of OA.
10. Acoustic Neuroma Association
Science.gov (United States)
... EVENTS DONATE NEWS Home Learn Back Learn about acoustic neuroma AN Facts What is acoustic neuroma? Diagnosing ... Brain Freeze ? READ MORE Read More What is acoustic neuroma? Identifying an AN Learn More Get Info ...
11. Oblique Modulation of Ion-Acoustic Waves in a Warm Plasma
International Nuclear Information System (INIS)
Xue Jukui; Tang Rongan
2003-01-01
The stability of oblique modulation of ion-acoustic waves in an unmagnetized warm plasma is studied. A nonlinear Schroedinger equation governing the slow modulation of the wave amplitude is derived. The effect of temperature on the oblique modulational instability of the ion-acoustic wave is investigated. It is found that the ion temperature significantly changes the domain of the modulational instability in the k-θ plane
12. Hydrodynamic instabilities in inertial fusion
International Nuclear Information System (INIS)
Hoffman, N.M.
1994-01-01
This report discusses topics on hydrodynamics instabilities in inertial confinement: linear analysis of Rayleigh-Taylor instability; ablation-surface instability; bubble rise in late-stage Rayleigh-Taylor instability; and saturation and multimode interactions in intermediate-stage Rayleigh-Taylor instability
13. Electroacoustic control of Rijke tube instability
Science.gov (United States)
Zhang, Yumin; Huang, Lixi
2017-11-01
Unsteady heat release coupled with pressure fluctuation triggers the thermoacoustic instability which may damage a combustion chamber severely. This study demonstrates an electroacoustic control approach of suppressing the thermoacoustic instability in a Rijke tube by altering the wall boundary condition. An electrically shunted loudspeaker driver device is connected as a side-branch to the main tube via a small aperture. Tests in an impedance tube show that this device has sound absorption coefficient up to 40% under normal incidence from 100 Hz to 400 Hz, namely over two octaves. Experimental result demonstrates that such a broadband acoustic performance can effectively eliminate the Rijke-tube instability from 94 Hz to 378 Hz (when the tube length varies from 1.8 m to 0.9 m, the first mode frequency for the former is 94 Hz and the second mode frequency for the latter is 378 Hz). Theoretical investigation reveals that the devices act as a damper draining out sound energy through a tiny hole to eliminate the instability. Finally, it is also estimated based on the experimental data that small amount of sound energy is actually absorbed when the system undergoes a transition from the unstable to stable state if the contrpaol is activated. When the system is actually stabilized, no sound is radiated so no sound energy needs to be absorbed by the control device.
14. Current filamentation caused by the electrochemical instability in a fully ionized plasma
International Nuclear Information System (INIS)
Haines, M.G.; Marsh, F.
1983-01-01
This chapter is primarily concerned with the non-linear development of electrothermal instabilities in a fully ionized plasma discharge in which the current is predominantly carried parallel to an applied magnetic field, as in the Tokamak configuration. Discusses instabilities with wave-number K perpendicular to magnetic field B and current J; the non-linear steady state; amplitude of the filaments; and runaway electrons and ion acoustic instabilities. Concludes that the steady non-linear amplitude of the fully developed instability shows a spiky filamentary structure with the possibility of the generation of runaway electrons and ion acoustic turbulence in the current maxima. Finds that the addition of bremsstrahlung radiation loss enhances the instability, reducing the critical ratio of T /SUB e/ to T /SUB i/ for its onset, and yielding a maximum ion temperature attainable by Joule heating and equipartition
International Nuclear Information System (INIS)
Hacker-Klom, U.B.; Goehde, W.
2001-01-01
16. Parametric Instability in Advanced Laser Interferometer Gravitational Wave Detectors
International Nuclear Information System (INIS)
Ju, L; Grass, S; Zhao, C; Degallaix, J; Blair, D G
2006-01-01
High frequency parametric instabilities in optical cavities are radiation pressure induced interactions between test mass mechanical modes and cavity optical modes. The parametric gain depends on the cavity power and the quality factor of the test mass internal modes (usually in ultrasonic frequency range), as well as the overlap integral for the mechanical and optical modes. In advanced laser interferometers which require high optical power and very low acoustic loss test masses, parametric instabilities could prevent interferometer operation if not suppressed. Here we review the problem of parametric instabilities in advanced detector configurations for different combinations of sapphire and fused silica test masses, and compare three methods for control or suppression of parametric instabilities-thermal tuning, surface damping and active feedback
17. A variational approach to parametric instabilities in inhomogeneous plasmas
Energy Technology Data Exchange (ETDEWEB)
Afeyan, B.B.
1993-12-31
A variational principle is constructed for the pump strength of a three-wave parametric instability in a spatially nonuniform medium. Using this expression together with appropriate trial functions, analytic estimates of the growth rate of the most unstable mode of a given parametric instability may be calculated. The usefullness of the variational method is first demonstrated on the Rosenbluth model problem with a power-law phase-mismatch, followed by a treatment of the Liu, Rosenbluth, and White sidescattering model equation. Two particular instabilities which are of interest in laser fusion and laser-plasma interaction experiments are treated next. These are Stimulated Raman Scattering and Two-Plasmon Decay. Various incidence and scattering geometries, and different density profiles are considered. Previously known results are reproduced in a unified manner and extended to cases where the usual local-expansion techniques do not apply. In particular, using the variational approach, the growth rate of the Two-Plasmon Decay instability occurring at or anywhere below the apex of a parabolic density profile is obtained for the first time. Similarly, Stimulated Raman Scattering from a density extremum at or anywhere below quarter critical, and for all scattering angles from backscattering to sidescattering inclusively is considered for the first time. The limit where the Two-Plasmon Decay and Stimulated Raman Scattering instabilities merge and become indistinguishable is also treated.
18. Panel acoustic contribution analysis.
Science.gov (United States)
Wu, Sean F; Natarajan, Logesh Kumar
2013-02-01
Formulations are derived to analyze the relative panel acoustic contributions of a vibrating structure. The essence of this analysis is to correlate the acoustic power flow from each panel to the radiated acoustic pressure at any field point. The acoustic power is obtained by integrating the normal component of the surface acoustic intensity, which is the product of the surface acoustic pressure and normal surface velocity reconstructed by using the Helmholtz equation least squares based nearfield acoustical holography, over each panel. The significance of this methodology is that it enables one to analyze and rank relative acoustic contributions of individual panels of a complex vibrating structure to acoustic radiation anywhere in the field based on a single set of the acoustic pressures measured in the near field. Moreover, this approach is valid for both interior and exterior regions. Examples of using this method to analyze and rank the relative acoustic contributions of a scaled vehicle cabin are demonstrated.
19. Thermal and fluid dynamic analysis of partially premixed turbulent combustion driven by thermo acoustic effects
NARCIS (Netherlands)
Shahi, Mina; Kok, Jacobus B.W.; Pozarlik, Artur Krzysztof; Sponfeldner, Thomas; Malcolm, M.J.; Pawelczyk, M.; Paosawatyangyong, B.
2013-01-01
Thermo-acoustic instability can be caused by the feedback mechanism between unsteady heat release, acoustic oscillations and flow perturbations. In a gas turbine combustor limit cycles of pressure oscillations at elevated temperatures generated by the unstable combustion process enhance the
20. Acoustic pressure oscillations induced by confined turbulent premixed natural glas flames
NARCIS (Netherlands)
van Kampen, J.F.
2006-01-01
The present study is concerned with the development and validation of efientt numerical algorithms to check combustion systems for their sensitivity to thermoacoustic instabilities. For this purpose, a good acoustic model is needed. Since the acoustics in combustion systems are essentially
1. Thermo-acoustic coupling in can-annular combustors : A numerical investigation
NARCIS (Netherlands)
Farisco, Federica; Panek, Lukasz; Kok, Jim B.W.; Pent, Jared; Rajaram, Rajesh
2015-01-01
Thermo-acoustic instabilities in modern, high power density gas turbines need to be predicted and understood in order to avoid unexpected damage and engine failure. While the annular combustor design is expected to suffer from the occurrence of transverse waves and burner-to-burner acoustic
2. Acoustic detection in superconducting magnets for performance characterization and diagnostics
International Nuclear Information System (INIS)
Marchevsky, M; Wang, X; Sabbi, G; Prestemon, S
2013-01-01
Quench diagnostics in superconducting accelerator magnets is essential for understanding performance limitations and improving magnet design. Applicability of the conventional quench diagnostics methods such as voltage taps or quench antennas is limited for long magnets or complex winding geometries, and alternative approaches are desirable. Here, we discuss acoustic sensing technique for detecting mechanical vibrations in superconducting magnets. Using LARP high-field Nb3Sn quadrupole HQ01, we show how acoustic data is connected with voltage instabilities measured simultaneously in the magnet windings during provoked extractions and current ramps to quench. Instrumentation and data analysis techniques for acoustic sensing are reviewed. (author)
3. Acoustic detection in superconducting magnets for performance characterization and diagnostics
CERN Document Server
Marchevsky, M.; Sabbi, G.; Prestemon, S.
2013-01-01
Quench diagnostics in superconducting accelerator magnets is essential for understanding performance limitations and improving magnet design. Applicability of the conventional quench diagnostics methods such as voltage taps or quench antennas is limited for long magnets or complex winding geometries, and alternative approaches are desirable. Here, we discuss acoustic sensing technique for detecting mechanical vibrations in superconducting magnets. Using LARP high-field Nb$_{3}$Sn quadrupole HQ01 [1], we show how acoustic data is connected with voltage instabilities measured simultaneously in the magnet windings during provoked extractions and current ramps to quench. Instrumentation and data analysis techniques for acoustic sensing are reviewed.
4. Acoustic transducer
Science.gov (United States)
Drumheller, Douglas S.
2000-01-01
An active acoustic transducer tool for use down-hole applications. The tool includes a single cylindrical mandrel including a shoulder defining the boundary of a narrowed portion over which is placed a sandwich-style piezoelectric tranducer assembly. The piezoelectric transducer assembly is prestressed by being placed in a thermal interference fit between the shoulder of the mandrel and the base of an anvil which is likewise positioned over the narrower portion of the mandrel. In the preferred embodiment, assembly of the tool is accomplished using a hydraulic jack to stretch the mandrel prior to emplacement of the cylindrical sandwich-style piezoelectric transducer assembly and anvil. After those elements are positioned and secured, the stretched mandrel is allowed to return substantially to its original (pre-stretch) dimensions with the result that the piezoelectric transducer elements are compressed between the anvil and the shoulder of the mandrel.
5. Acoustic cryocooler
International Nuclear Information System (INIS)
Swift, G.W.; Martin, R.A.; Radebaugh, R.
1990-01-01
This patent describes an acoustic cryocooler with no moving parts is formed from a thermoacoustic driver (TAD) driving a pulse tube refrigerator (PTR) through a standing wave tube. Thermoacoustic elements in the TAD are spaced apart a distance effect to accommodate the increased thermal penetration length arising from the relatively low TAD operating frequency in the range of 15--60 Hz. At these low operating frequencies, a long tube is required to support the standing wave. The tube may be coiled to reduce the overall length of the cryocooler. One or two PTR's are located on the standing wave tube adjacent antinodes in the standing wave to be driven by the standing wave pressure oscillations. It is predicted that a heat input of 1000 W at 1000 K will maintain a cooling load of 5 W at 80 K
6. Use of acoustic vortices in acoustic levitation
DEFF Research Database (Denmark)
Cutanda Henriquez, Vicente; Santillan, Arturo Orozco; Juhl, Peter Møller
2009-01-01
Acoustic fields are known to exert forces on the surfaces of objects. These forces are noticeable if the sound pressure is sufficiently high. Two phenomena where acoustic forces are relevant are: i) acoustic levitation, where strong standing waves can hold small objects at certain positions......, counterbalancing their weight, and ii) acoustic vortices, spinning sound fields that can impinge angular momentum and cause rotation of objects. In this contribution, both force-creating sound fields are studied by means of numerical simulations. The Boundary Element Method is employed to this end. The simulation...... of acoustical vortices uses an efficient numerical implementation based on the superposition of two orthogonal sound fields with a delay of 90° between them. It is shown that acoustic levitation and the use of acoustic vortices can be combined to manipulate objects in an efficient and controlled manner without...
7. Tearing instabilities in turbulence
International Nuclear Information System (INIS)
Ishizawa, A.; Nakajima, N.
2009-01-01
Full text: Effects of micro-turbulence on tearing instabilities are investigated by numerically solving a reduced set of two-fluid equations. Micro-turbulence excites both large-scale and small-scale Fourier modes through energy transfer due to nonlinear mode coupling. The energy transfer to large scale mode does not directly excite tearing instability but it gives an initiation of tearing instability. When tearing instability starts to grow, the excited small scale mode plays an important role. The mixing of magnetic flux by micro-turbulence is the dominant factor of non-ideal MHD effect at the resonant surface and it gives rise to magnetic reconnection which causes tearing instability. Tearing instabilities were investigated against static equilibrium or flowing equilibrium so far. On the other hand, the recent progress of computer power allows us to investigate interactions between turbulence and coherent modes such as tearing instabilities in magnetically confined plasmas by means of direct numerical simulations. In order to investigate effects of turbulence on tearing instabilities we consider a situation that tearing mode is destabilized in a quasi-equilibrium including micro-turbulence. We choose an initial equilibrium that is unstable against kinetic ballooning modes and tearing instabilities. Tearing instabilities are current driven modes and thus they are unstable for large scale Fourier modes. On the other hand kinetic ballooning modes are unstable for poloidal Fourier modes that are characterized by ion Larmor radius. The energy of kinetic ballooning modes spreads over wave number space through nonlinear Fourier mode coupling. We present that micro-turbulence affects tearing instabilities in two different ways by three-dimensional numerical simulation of a reduced set of two-fluid equations. One is caused by energy transfer to large scale modes, the other is caused by energy transfer to small scale modes. The former is the excitation of initial
8. Interior acoustic cloak
OpenAIRE
Wael Akl; A. Baz
2014-01-01
Acoustic cloaks have traditionally been intended to externally surround critical objects to render these objects acoustically invisible. However, in this paper, the emphasis is placed on investigating the application of the acoustic cloaks to the interior walls of acoustic cavities in an attempt to minimize the noise levels inside these cavities. In this manner, the acoustic cloaks can serve as a viable and efficient alternative to the conventional passive noise attenuation treatments which a...
9. Relativistic gravitational instabilities
International Nuclear Information System (INIS)
Schutz, B.F.
1987-01-01
The purpose of these lectures is to review and explain what is known about the stability of relativistic stars and black holes, with particular emphases on two instabilities which are due entirely to relativistic effects. The first of these is the post-Newtonian pulsational instability discovered independently by Chandrasekhar (1964) and Fowler (1964). This effectively ruled out the then-popular supermassive star model for quasars, and it sets a limit to the central density of white dwarfs. The second instability was also discovered by Chandrasekhar (1970): the gravitational wave induced instability. This sets an upper bound on the rotation rate of neutron stars, which is near that of the millisecond pulsar PSR 1937+214, and which is beginning to constrain the equation of state of neutron matter. 111 references, 5 figures
10. Spondylolisthesis and Posterior Instability
International Nuclear Information System (INIS)
Niggemann, P.; Beyer, H.K.; Frey, H.; Grosskurth, D.; Simons, P.; Kuchta, J.
2009-01-01
We present the case of a patient with a spondylolisthesis of L5 on S1 due to spondylolysis at the level L5/S1. The vertebral slip was fixed and no anterior instability was found. Using functional magnetic resonance imaging (MRI) in an upright MRI scanner, posterior instability at the level of the spondylolytic defect of L5 was demonstrated. A structure, probably the hypertrophic ligament flava, arising from the spondylolytic defect was displaced toward the L5 nerve root, and a bilateral contact of the displaced structure with the L5 nerve root was shown in extension of the spine. To our knowledge, this is the first case described of posterior instability in patients with spondylolisthesis. The clinical implications of posterior instability are unknown; however, it is thought that this disorder is common and that it can only be diagnosed using upright MRI
11. Spondylolisthesis and Posterior Instability
Energy Technology Data Exchange (ETDEWEB)
Niggemann, P.; Beyer, H.K.; Frey, H.; Grosskurth, D. (Privatpraxis fuer Upright MRT, Koeln (Germany)); Simons, P.; Kuchta, J. (Media Park Klinik, Koeln (Germany))
2009-04-15
We present the case of a patient with a spondylolisthesis of L5 on S1 due to spondylolysis at the level L5/S1. The vertebral slip was fixed and no anterior instability was found. Using functional magnetic resonance imaging (MRI) in an upright MRI scanner, posterior instability at the level of the spondylolytic defect of L5 was demonstrated. A structure, probably the hypertrophic ligament flava, arising from the spondylolytic defect was displaced toward the L5 nerve root, and a bilateral contact of the displaced structure with the L5 nerve root was shown in extension of the spine. To our knowledge, this is the first case described of posterior instability in patients with spondylolisthesis. The clinical implications of posterior instability are unknown; however, it is thought that this disorder is common and that it can only be diagnosed using upright MRI.
12. Streaming gravity mode instability
International Nuclear Information System (INIS)
Wang Shui.
1989-05-01
In this paper, we study the stability of a current sheet with a sheared flow in a gravitational field which is perpendicular to the magnetic field and plasma flow. This mixing mode caused by a combined role of the sheared flow and gravity is named the streaming gravity mode instability. The conditions of this mode instability are discussed for an ideal four-layer model in the incompressible limit. (author). 5 refs
Energy Technology Data Exchange (ETDEWEB)
Little, John B [Harvard School of Public Health, Boston, MA 02115 (United States)
2003-06-01
Genomic instability is a hallmark of cancer cells, and is thought to be involved in the process of carcinogenesis. Indeed, a number of rare genetic disorders associated with a predisposition to cancer are characterised by genomic instability occurring in somatic cells. Of particular interest is the observation that transmissible instability can be induced in somatic cells from normal individuals by exposure to ionising radiation, leading to a persistent enhancement in the rate at which mutations and chromosomal aberrations arise in the progeny of the irradiated cells after many generations of replication. If such induced instability is involved in radiation carcinogenesis, it would imply that the initial carcinogenic event may not be a rare mutation occurring in a specific gene or set of genes. Rather, radiation may induce a process of instability in many cells in a population, enhancing the rate at which the multiple gene mutations necessary for the development of cancer may arise in a given cell lineage. Furthermore, radiation could act at any stage in the development of cancer by facilitating the accumulation of the remaining genetic events required to produce a fully malignant tumour. The experimental evidence for such induced instability is reviewed. (review)
International Nuclear Information System (INIS)
Little, John B
2003-01-01
Genomic instability is a hallmark of cancer cells, and is thought to be involved in the process of carcinogenesis. Indeed, a number of rare genetic disorders associated with a predisposition to cancer are characterised by genomic instability occurring in somatic cells. Of particular interest is the observation that transmissible instability can be induced in somatic cells from normal individuals by exposure to ionising radiation, leading to a persistent enhancement in the rate at which mutations and chromosomal aberrations arise in the progeny of the irradiated cells after many generations of replication. If such induced instability is involved in radiation carcinogenesis, it would imply that the initial carcinogenic event may not be a rare mutation occurring in a specific gene or set of genes. Rather, radiation may induce a process of instability in many cells in a population, enhancing the rate at which the multiple gene mutations necessary for the development of cancer may arise in a given cell lineage. Furthermore, radiation could act at any stage in the development of cancer by facilitating the accumulation of the remaining genetic events required to produce a fully malignant tumour. The experimental evidence for such induced instability is reviewed. (review)
15. Experimental study of parametric decay close to the upper hybrid frequency
Energy Technology Data Exchange (ETDEWEB)
Albers, E; Krause, K; Schlueter, H [Ruhr-Univ., Bochum (Germany, F.R.)
1978-04-01
In He, Ne, and Ar plasmas the parametric decay of the electromagnetic upper hybrid mode is studied in the range between the electron cyclotron frequency and its first two harmonics. The pump wave is excited by outside antennae. The decay products are identified as electron Bernstein and ion acoustic modes.
16. Thresholds of parametric instabilities near the lower hybrid frequency
International Nuclear Information System (INIS)
Berger, R.L.; Perkins, F.W.
1975-06-01
Resonant decay instabilities of a pump wave with frequency ω 0 near the lower-hybrid frequency ω/sub LH/ are analyzed with respect to the wavenumber k of the decay waves and the ratio ω 0 /ω/sub LH/ to determine the decay process with the minimum threshold. It was found that the lowest thresholds are for decay into an electron plasma (lower hybrid) wave plus either a backward ion-cyclotron wave, an ion Bernstein wave, or a low frequency sound wave. For ω 0 less than (2ω/sub LH/)/sup 1 / 2 /, it was found that these decay processes can occur and have faster growth than ion quasimodes provided the drift velocity (cE 0 /B 0 ) is much less than the sound speed. In many cases of interest, electromagnetic corrections to the lower-hybrid wave rule out decay into all but short wavelength (k rho/sub i/ greater than 1) waves. The experimental results are consistent with the linear theory of parametric instabilities in a homogeneous plasma. (U.S.)
17. Dental Caries (Tooth Decay)
Science.gov (United States)
... Materials Contact Us Home Research Data & Statistics Dental Caries (Tooth Decay) Dental caries (tooth decay) remains the most prevalent chronic disease ... adults, even though it is largely preventable. Although caries has significantly decreased for most Americans over the ...
18. Dental Caries (Tooth Decay)
Science.gov (United States)
... Contact Us Home Research Data & Statistics Share Dental Caries (Tooth Decay) Dental caries (tooth decay) remains the most prevalent chronic disease ... adults, even though it is largely preventable. Although caries has significantly decreased for most Americans over the ...
19. Role of collective effects in dominance of scattering off thermal ions over Langmuir wave decay: Analysis, simulations, and space applications
Energy Technology Data Exchange (ETDEWEB)
Cairns, Iver H.
2000-12-01
Langmuir waves driven to high levels by beam instabilities are subject to nonlinear processes, including the closely related processes of scattering off thermal ions (STI) and a decay process in which the ion response is organized into a product ion acoustic wave. Calculations of the nonlinear growth rates predict that the decay process should always dominate STI, creating two paradoxes. The first is that three independent computer simulation studies show STI proceeding, with no evidence for the decay at all. The second is that observations in space of type III solar radio bursts and Earth's foreshock, which the simulations were intended to model, show evidence for the decay proceeding but no evidence for STI. Resolutions to these paradoxes follow from the realization that a nonlinear process cannot proceed when its growth rate exceeds the minimum frequency of the participating waves, since the required collective response cannot be maintained and the waves cannot respond appropriately, and that a significant number of e-foldings and wave periods must be contained in the time available. It is shown that application of these ''collective'' and ''time scale'' constraints to the simulations explains why the decay does not proceed in them, as well as why STI proceeds in specific simulations. This appears to be the first demonstration that collective constraints are important in understanding nonlinear phenomena. Furthermore, applying these constraints to space observations, it is predicted that the decay should proceed (and dominate STI) in type III sources and the high beam speed regions of Earth's foreshock for a specific range of wave levels, with a possible role for STI alone at slightly higher wave levels. Deeper in the foreshock, for slower beams and weaker wave levels, the decay and STI are predicted to become ineffective. Suggestions are given for future testing of the collective constraint and an explanation
20. Acoustic--nuclear permeability logging system
International Nuclear Information System (INIS)
Dowling, D.J.; Arnold, D.M.
1978-01-01
A down hole logging tool featuring a neutron generator, an acoustic disturbance generator, and a radiation detection system is described. An array of acoustic magnetostriction transducers is arranged about the target of a neutron accelerator. Two gamma ray sensors are separated from the accelerator target by shielding. According to the method of the invention, the underground fluid at the level of a formation is bombarded by neutrons which react with oxygen in the fluid to produce unstable nitrogen 16 particles according to the reaction 16 O(n,p) 16 N. Acoustic pulses are communicated to the fluid, and are incident on the boundary of the borehole at the formation. The resulting net flow of fluid across the boundary is determined from radiation detection measurements of the decaying 16 N particles in the fluid. A measure of the permeability of the formation is obtained from the determination of net fluid flow across the boundary
1. Springer Handbook of Acoustics
CERN Document Server
Rossing, Thomas D
2007-01-01
Acoustics, the science of sound, has developed into a broad interdisciplinary field encompassing the academic disciplines of physics, engineering, psychology, speech, audiology, music, architecture, physiology, neuroscience, and others. The Springer Handbook of Acoustics is an unparalleled modern handbook reflecting this richly interdisciplinary nature edited by one of the acknowledged masters in the field, Thomas Rossing. Researchers and students benefit from the comprehensive contents spanning: animal acoustics including infrasound and ultrasound, environmental noise control, music and human speech and singing, physiological and psychological acoustics, architectural acoustics, physical and engineering acoustics, signal processing, medical acoustics, and ocean acoustics. This handbook reviews the most important areas of acoustics, with emphasis on current research. The authors of the various chapters are all experts in their fields. Each chapter is richly illustrated with figures and tables. The latest rese...
2. Responsive acoustic surfaces
DEFF Research Database (Denmark)
Peters, Brady; Tamke, Martin; Nielsen, Stig Anton
2011-01-01
Acoustic performance is defined by the parameter of reverberation time; however, this does not capture the acoustic experience in some types of open plan spaces. As many working and learning activities now take place in open plan spaces, it is important to be able to understand and design...... for the acoustic conditions of these spaces. This paper describes an experimental research project that studied the design processes necessary to design for sound. A responsive acoustic surface was designed, fabricated and tested. This acoustic surface was designed to create specific sonic effects. The design...... was simulated using custom integrated acoustic software and also using Odeon acoustic analysis software. The research demonstrates a method for designing space- and sound-defining surfaces, defines the concept of acoustic subspace, and suggests some new parameters for defining acoustic subspaces....
3. Transit-time instability in Hall thrusters
International Nuclear Information System (INIS)
Barral, Serge; Makowski, Karol; Peradzynski, Zbigniew; Dudeck, Michel
2005-01-01
Longitudinal waves characterized by a phase velocity of the order of the velocity of ions have been recurrently observed in Hall thruster experiments and simulations. The origin of this so-called ion transit-time instability is investigated with a simple one-dimensional fluid model of a Hall thruster discharge in which cold ions are accelerated between two electrodes within a quasineutral plasma. A short-wave asymptotics applied to linearized equations shows that plasma perturbations in such a device consist of quasineutral ion acoustic waves superimposed on a background standing wave generated by discharge current oscillations. Under adequate circumstances and, in particular, at high ionization levels, acoustic waves are amplified as they propagate, inducing strong perturbation of the ion density and velocity. Responding to the subsequent perturbation of the column resistivity, the discharge current generates a standing wave, the reflection of which sustains the generation of acoustic waves at the inlet boundary. A calculation of the frequency and growth rate of this resonance mechanism for a supersonic ion flow is proposed, which illustrates the influence of the ionization degree on their onset and the approximate scaling of the frequency with the ion transit time. Consistent with experimental reports, the traveling wave can be observed on plasma density and velocity perturbations, while the plasma potential ostensibly oscillates in phase along the discharge
4. Low-frequency instabilities of a warm plasma in a magnetic field
International Nuclear Information System (INIS)
Smith, D.F.; Hollweg, J.V.
1977-01-01
The marginal stability of a plasma carrying current along the static magnetic field with isotropic Maxwellian ions and isotropic Maxwellian electrons drifting relative to the ions is investigated. The complete electromagnetic dispersion relation is studied using numerical techniques; the electron sums are restricted to three terms which limits the analysis to frequencies much less than the electron gyro-frequency, but includes frequencies somewhat above the ion gyro-frequency. A 'kink-like' instability and an instability of the Alfven mode are found to have the lowest threshold drift velocities in most cases. In fact the threshold drift for the kink-like instability can be significantly less than the ion thermal speed. Electrostatic and electromagnetic ion-cyclotron instabilities are also found as well as the electro-static ion-acoustic instability. No instability of the fast magnetosonic mode was found. The stability analysis provides only threshold drift velocities and gives no information about growth rates. (author)
Directory of Open Access Journals (Sweden)
R.I. Parovik
2012-06-01
Full Text Available In a model of radioactive decay of radon in the sample (222Rn. The model assumes that the probability of the decay of radon and its half-life depends on the fractal properties of the geological environment. The dependencies of the decay parameters of the fractal dimension of the medium.
6. Parametric decay of lower hybrid wave into drift waves
International Nuclear Information System (INIS)
Sanuki, Heiji.
1976-12-01
A dispersion relation describing the parametric decay of a lower hybrid wave into an electrostatic drift wave and a drift Alfven wave is derived for an inhomogeneous magnetized plasma. Particularly the stimulated scattering of a drift Alfven wave in such a plasma was investigated in detail. The resonance backscattering instability is found to yield the minimum threshold. (auth.)
7. Chaotic neoclassical separatrix dissipation in parametric drift-wave decay.
Science.gov (United States)
Kabantsev, A A; Tsidulko, Yu A; Driscoll, C F
2014-02-07
Experiments and theory characterize a parametric decay instability between plasma drift waves when the nonlinear coupling is modified by an electrostatic barrier. Novel mode coupling terms representing enhanced dissipation and mode phase shifts are caused by chaotic separatrix crossings on the wave-ruffled separatrix. Experimental determination of these coupling terms is in broad agreement with new chaotic neoclassical transport analyses.
8. Ion acoustic waves in the solar wind
International Nuclear Information System (INIS)
Gurnett, D.A.; Frank, L.A.
1978-01-01
Plasma wave measurements on the Helios I and 2 spacecraft have revealed the occurrence of electric field turbulence in the solar wind at frequencies between the electron and ion plasma frequencies. Wavelength measurements with the Imp 6 spacecraft now provide strong evidence that these waves are short-wavelength ion acoustic waves which are Doppler-shifted upward in frequency by the motion of the solar wind. Comparison of the Helios results with measurements from the earth-orbiting Imp 6 and 8 spacecraft shows that the ion acoustic turbulence detected in interplanetary space has characteristics essentially identical to those of bursts of electrostatic turbulence generated by protons streaming into the solar wind from the earth's bow shock. In a few cases, enhanced ion acoustic wave intensities have been observed in direct association with abrupt increases in the anisotropy of the solar wind electron distribution. This relationship strongly suggests that the ion acoustic waves detected by Helios far from the earth are produced by an electron heat flux instability, as was suggested by Forslund. Possible related mechanisms which could explain the generation of ion acoustic waves by protons streaming into the solar wind from the earth's bow shock are also considered
9. Acoustics an introduction
CERN Document Server
Kuttruff, Heinrich
2006-01-01
This definitive textbook provides students with a comprehensive introduction to acoustics. Beginning with the basic physical ideas, Acoustics balances the fundamentals with engineering aspects, applications and electroacoustics, also covering music, speech and the properties of human hearing. The concepts of acoustics are exposed and applied in:room acousticssound insulation in buildingsnoise controlunderwater sound and ultrasoundScientifically thorough, but with mathematics kept to a minimum, Acoustics is the perfect introduction to acoustics for students at any level of mechanical, electrical or civil engineering courses and an accessible resource for architects, musicians or sound engineers requiring a technical understanding of acoustics and their applications.
10. Acoustic source for generating an acoustic beam
Science.gov (United States)
Vu, Cung Khac; Sinha, Dipen N.; Pantea, Cristian
2016-05-31
An acoustic source for generating an acoustic beam includes a housing; a plurality of spaced apart piezo-electric layers disposed within the housing; and a non-linear medium filling between the plurality of layers. Each of the plurality of piezoelectric layers is configured to generate an acoustic wave. The non-linear medium and the plurality of piezo-electric material layers have a matching impedance so as to enhance a transmission of the acoustic wave generated by each of plurality of layers through the remaining plurality of layers.
11. Double Arc Instability in the Solar Corona
Energy Technology Data Exchange (ETDEWEB)
Ishiguro, N.; Kusano, K., E-mail: [email protected] [Institute for Space-Earth Environmental Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601 Japan (Japan)
2017-07-10
The stability of the magnetic field in the solar corona is important for understanding the causes of solar eruptions. Although various scenarios have been suggested to date, the tether-cutting reconnection scenario proposed by Moore et al. is one of the widely accepted models to explain the onset process of solar eruptions. Although the tether-cutting reconnection scenario proposes that the sigmoidal field formed by internal reconnection is the magnetic field in the pre-eruptive state, the stability of the sigmoidal field has not yet been investigated quantitatively. In this paper, in order to elucidate the stability problem of the pre-eruptive state, we developed a simple numerical analysis in which the sigmoidal field is modeled by a double arc electric current loop and its stability is analyzed. As a result, we found that the double arc loop is more easily destabilized than the axisymmetric torus, and it becomes unstable even if the external field does not decay with altitude, which is in contrast to the axisymmetric torus instability. This suggests that tether-cutting reconnection may well work as the onset mechanism of solar eruptions, and if so, the critical condition for eruption under a certain geometry may be determined by a new type of instability rather than by the torus instability. Based on them, we propose a new type of instability called double arc instability (DAI). We discuss the critical conditions for DAI and derive a new parameter κ , defined as the product of the magnetic twist and the normalized flux of the tether-cutting reconnection.
12. Distribution of Acoustic Power Spectra for an Isolated Helicopter Fuselage
Directory of Open Access Journals (Sweden)
Kusyumov A.N.
2016-01-01
Full Text Available The broadband aerodynamic noise can be studied, assuming isotropic flow, turbulence and decay. Proudman’s approach allows practical calculations of noise based on CFD solutions of RANS or URANS equations at the stage of post processing and analysis of the solution. Another aspect is the broadband acoustic spectrum and the distribution of acoustic power over a range of frequencies. The acoustic energy spectrum distribution in isotropic turbulence is non monotonic and has a maximum at a certain value of Strouhal number. In the present work the value of acoustic power peak frequency is determined using a prescribed form of acoustic energy spectrum distribution presented in papers by S. Sarkar and M. Y. Hussaini and by G. M. Lilley. CFD modelling of the flow around isolated helicopter fuselage model was considered using the HMB CFD code and the RANS equations.
13. Theoretical study of time-dependent, ultrasound-induced acoustic streaming in microchannels
DEFF Research Database (Denmark)
Muller, Peter Barkholt; Bruus, Henrik
2015-01-01
Based on first- and second-order perturbation theory, we present a numerical study of the temporal buildup and decay of unsteady acoustic fields and acoustic streaming flows actuated by vibrating walls in the transverse cross-sectional plane of a long straight microchannel under adiabatic...
14. Combustion instabilities in sudden expansion oxy-fuel flames
Energy Technology Data Exchange (ETDEWEB)
Ditaranto, Mario; Hals, Joergen [Department of Energy Processes, SINTEF Energy Research, 7465 Trondheim (Norway)
2006-08-15
An experimental study on combustion instability is presented with focus on oxy-fuel type combustion. Oxidants composed of CO{sub 2}/O{sub 2} and methane are the reactants flowing through a premixer-combustor system. The reaction starts downstream a symmetric sudden expansion and is at the origin of different instability patterns depending on oxygen concentration and Reynolds number. The analysis has been conducted through measurement of pressure, CH* chemiluminescence, and velocity. As far as stability is concerned, oxy-fuel combustion with oxygen concentration similar to that found in air combustion cannot be sustained, but requires at least 30% oxygen to perform in a comparable manner. Under these conditions and for the sudden expansion configuration used in this study, the instability is at low frequency and low amplitude, controlled by the flame length inside the combustion chamber. Above a threshold concentration in oxygen dependent on equivalence ratio, the flame becomes organized and concentrated in the near field. Strong thermoacoustic instability is then triggered at characteristic acoustic modes of the system. Different modes can be triggered depending on the ratio of flame speed to inlet velocity, but for all types of instability encountered, the heat release and pressure fluctuations are linked by a variation in mass-flow rate. An acoustic model of the system coupled with a time-lag-based flame model made it possible to elucidate the acoustic mode selection in the system as a function of laminar flame speed and Reynolds number. The overall work brings elements of reflection concerning the potential risk of strong pressure oscillations in future gas turbine combustors for oxy-fuel gas cycles. (author)
15. Parametric Decay during HHFW on NSTX
International Nuclear Information System (INIS)
Wilson, J.R.; Bernabei, S.; Biewer, T.; Diem, S.; Hosea, J.; LeBlanc, B.; Phillips, C.K.; Ryan, P.; Swain, D.W.
2005-01-01
High Harmonic Fast Wave (HHFW) heating experiments on NSTX have been observed to be accompanied by significant edge ion heating (T i >> T e ). This heating is found to be anisotropic with T perp > T par . Simultaneously, coherent oscillations have been detected with an edge Langmuir probe. The oscillations are consistent with parametric decay of the incident fast wave (ω > 13ω ci ) into ion Bernstein waves and an unobserved ion-cyclotron quasi-mode. The observation of anisotropic heating is consistent with Bernstein wave damping, and the Bernstein waves should completely damp in the plasma periphery as they propagate toward a cyclotron harmonic resonance. The number of daughter waves is found to increase with rf power, and to increase as the incident wave's toroidal wavelength increases. The frequencies of the daughter wave are separated by the edge ion cyclotron frequency. Theoretical calculations of the threshold for this decay in uniform plasma indicate an extremely small value of incident power should be required to drive the instability. While such decays are commonly observed at lower harmonics in conventional ICRF heating scenarios, they usually do not involve the loss of significant wave power from the pump wave. On NSTX an estimate of the power loss can be found by calculating the minimum power required to support the edge ion heating (presumed to come from the decay Bernstein wave). This calculation indicates at least 20-30% of the incident rf power ends up as decay waves
16. Interior acoustic cloak
Directory of Open Access Journals (Sweden)
Wael Akl
2014-12-01
Full Text Available Acoustic cloaks have traditionally been intended to externally surround critical objects to render these objects acoustically invisible. However, in this paper, the emphasis is placed on investigating the application of the acoustic cloaks to the interior walls of acoustic cavities in an attempt to minimize the noise levels inside these cavities. In this manner, the acoustic cloaks can serve as a viable and efficient alternative to the conventional passive noise attenuation treatments which are invariably heavy and bulky. The transformation acoustics relationships that govern the operation of this class of interior acoustic cloaks are presented. Physical insights are given to relate these relationships to the reasons behind the effectiveness of the proposed interior acoustic cloaks. Finite element models are presented to demonstrate the characteristics of interior acoustic cloaks used in treating the interior walls of circular and square cavities both in the time and frequency domains. The obtained results emphasize the effectiveness of the proposed interior cloaks in eliminating the reflections of the acoustic waves from the walls of the treated cavities and thereby rendering these cavities acoustically quiet. It is important to note here that the proposed interior acoustic cloaks can find applications in acoustic cavities such as aircraft cabins and auditoriums as well as many other critical applications.
17. Plasma physics and instabilities
International Nuclear Information System (INIS)
Lashmore-Davies, C.N.
1981-01-01
These lectures procide an introduction to the theory of plasmas and their instabilities. Starting from the Bogoliubov, Born, Green, Kirkwood, and Yvon (BBGKY) hierarchy of kinetic equations, the additional concept of self-consistent fields leads to the fundamental Vlasov equation and hence to the warm two-fluid model and the one-fluid MHD, or cold, model. The properties of small-amplitude waves in magnetized (and unmagnetized) plasmas, and the instabilities to which they give rise, are described in some detail, and a complete chapter is devoted to Landau damping. The linear theory of plasma instabilities is illustrated by the current-driven electrostatic kind, with descriptions of the Penrose criterion and the energy principle of ideal MHD. There is a brief account of the application of feedback control. The non-linear theory is represented by three examples: quasi-linear velocity-space instabilities, three-wave instabilities, and the stability of an arbitrarily largeamplitude wave in a plasma. (orig.)
18. Nonlinear full two-fluid study of m=0 sausage instabilities in an axisymmetric Z pinch
International Nuclear Information System (INIS)
Loverich, J.; Shumlak, U.
2006-01-01
A nonlinear full five-moment two-fluid model is used to study axisymmetric instabilities in a Z pinch. When the electron velocity due to the current J is greater than the ion acoustic speed, high wave-number sausage instabilities develop that initiate shock waves in the ion fluid. This condition corresponds to a pinch radius on the order of a few ion Larmor radii
19. System for detecting acoustic emissions in multianvil experiments: Application to deep seismicity in the Earth
International Nuclear Information System (INIS)
Jung, Haemyeong; Fei Yingwei; Silver, Paul G.; Green, Harry W.
2006-01-01
One of the major goals in the experimental study of deep earthquakes is to identify slip instabilities at high pressure and high temperature (HPHT) that might be responsible for the occurrence of earthquakes. Detecting acoustic emissions from a specimen during faulting provides unique constraints on the instability process. There are few experimental studies reporting acoustic emissions under HPHT conditions, due to technical challenges. And those studies have used only one or at most two acoustic sensors during the experiments. Such techniques preclude the accurate location of the acoustic emission source region and thus the ability to distinguish real signal from noise that may be coming from outside the sample. We have developed a system for detecting acoustic emissions at HPHT. Here we present a four-channel acoustic emission detecting system working in the HPHT octahedral multianvil apparatus. Each channel has high resolution (12 bits) and a sampling rate of 30 MHz. In experiments at the pressures up to 6 GPa and temperatures up to 770 deg. C, we have observed acoustic emissions under various conditions. Analyzing these signals, we are able to show that this system permits us to distinguish between signal and noise, locate the source of the acoustic emission, and obtain reliable data on the radiation pattern. This system has greatly improved our ability to study faulting instabilities under high pressure and high temperature
20. Control of the vertical instability in tokamaks
International Nuclear Information System (INIS)
Lazarus, E.A.; Lister, J.B.; Neilson, G.H.
1989-05-01
The problem of control of the vertical instability is formulated for a massless filamentary plasma. The massless approximation is justified by an examination of the role of inertia in the control problem. The system is solved using Laplace transform techniques. The linear system is studied to determine the stability boundaries. It is found that the system can be stabilized up to a critical decay index, which is predominantly a function of the geometry of the passive stabilizing shell. A second, smaller critical index, which is a function of the geometry of the control coils, determines the limit of stability in the absence of derivative gain in the control circuit. The system is also studied numerically in order to incorporate the non-linear effects of power supply dynamics. The power supply bandwidth requirement is determined by the open-loop growth rate of the instability. The system is studied for a number of control coil options which are available on the DIII-D tokamak. It is found that many of the coils will not provide adequate stabilization and that the use of inboard coils is advantageous in stabilizing the system up to the critical index. Experiments carried out on DIII-D confirm the appropriateness of the model. Using the results of the model study, we have stabilized DIII-D plasmas with decay indices up to 98% of the critical index. Measurement of the plasma vertical position is also discussed. (author) 27 figs., 6 refs
1. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Facts What is acoustic neuroma? Diagnosing Symptoms Side Effects Keywords Questions to ask Choosing a healthcare provider ... Surgery What is acoustic neuroma Diagnosing Symptoms Side effects Question To Ask Treatment Options Back Overview Observation ...
2. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Facts What is acoustic neuroma? Diagnosing Symptoms Side Effects Keywords World Language Videos Questions to ask Choosing ... Surgery What is acoustic neuroma Diagnosing Symptoms Side effects Question To Ask Treatment Options Back Overview Observation ...
3. Atlantic Herring Acoustic Surveys
Data.gov (United States)
National Oceanic and Atmospheric Administration, Department of Commerce — The NEFSC Advanced Sampling Technologies Research Group conducts annual fisheries acoustic surveys using state-of-the-art acoustic, midwater trawling, and underwater...
Data.gov (United States)
National Oceanic and Atmospheric Administration, Department of Commerce — The Tethys database houses the metadata associated with the acoustic data collection efforts by the Passive Acoustic Group. These metadata include dates, locations...
5. Fingerprints of dynamical instabilities
International Nuclear Information System (INIS)
Chomaz, Ph.; Colonna, M.; Guarnera, A.
1993-01-01
It is explained why any reduced descriptions, such as mean field approximation, are stochastic in nature. It is shown that the introduction of this stochastic dynamics leads to a predictive theory in a statistical sens whatever the individual trajectories are characterized by the occurrence of bifurcations, instabilities or phase transitions. Concerning nuclear matter, the spinodal instability is discussed. In such a critical situation, the possibility to replace the stochastic part of the collision integral in the Boltzmann-Langevin model by the numerical noise associated with the finite number of test particles in ordinary BUU treatment is studied. It is shown that the fingerprints of these instabilities are kept during the evolution because of the relatively long recombination time compared with the typical time scales imposed by the Coulomb repulsion and the possible collective expansion. (author) 5 refs., 12 figs
6. Causes of genome instability
DEFF Research Database (Denmark)
Langie, Sabine A S; Koppen, Gudrun; Desaulniers, Daniel
2015-01-01
function, chromosome segregation, telomere length). The purpose of this review is to describe the crucial aspects of genome instability, to outline the ways in which environmental chemicals can affect this cancer hallmark and to identify candidate chemicals for further study. The overall aim is to make......Genome instability is a prerequisite for the development of cancer. It occurs when genome maintenance systems fail to safeguard the genome's integrity, whether as a consequence of inherited defects or induced via exposure to environmental agents (chemicals, biological agents and radiation). Thus...
7. Instabilities and nonequilibrium structures
International Nuclear Information System (INIS)
Tirapegui, E.; Villarroel, D.
1987-01-01
Physical systems can be studied both near to and far from equilibrium where instabilities appear. The behaviour in these two regions is reviewed in this book, from both the theoretical and application points of view. The influence of noise in these situations is an essential feature which cannot be ignored. It is therefore discussed using phenomenological and theoretical approaches for the numerous problems which still remain in the field. This volume should appeal to mathematicians and physicists interested in the areas of instability, bifurcation theory, dynamical systems, pattern formation, nonequilibrium structures and statistical mechanics. (Auth.)
8. RINGED ACCRETION DISKS: INSTABILITIES
Energy Technology Data Exchange (ETDEWEB)
Pugliese, D.; Stuchlík, Z., E-mail: [email protected], E-mail: [email protected] [Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo náměstí 13, CZ-74601 Opava (Czech Republic)
2016-04-01
We analyze the possibility that several instability points may be formed, due to the Paczyński mechanism of violation of mechanical equilibrium, in the orbiting matter around a supermassive Kerr black hole. We consider a recently proposed model of a ringed accretion disk, made up by several tori (rings) that can be corotating or counter-rotating relative to the Kerr attractor due to the history of the accretion process. Each torus is governed by the general relativistic hydrodynamic Boyer condition of equilibrium configurations of rotating perfect fluids. We prove that the number of the instability points is generally limited and depends on the dimensionless spin of the rotating attractor.
9. Tutorial on architectural acoustics
Science.gov (United States)
Shaw, Neil; Talaske, Rick; Bistafa, Sylvio
2002-11-01
This tutorial is intended to provide an overview of current knowledge and practice in architectural acoustics. Topics covered will include basic concepts and history, acoustics of small rooms (small rooms for speech such as classrooms and meeting rooms, music studios, small critical listening spaces such as home theatres) and the acoustics of large rooms (larger assembly halls, auditoria, and performance halls).
10. Coherent acoustic phonon oscillation accompanied with backward acoustic pulse below exciton resonance in a ZnO epifilm on oxide-buffered Si(1 1 1)
International Nuclear Information System (INIS)
Lin, Ja-Hon; Shen, Yu-Kai; Lu, Chia-Hui; Chen, Yao-Hui; Chang, Chun-peng; Liu, Wei-Rein; Hsu, Chia-Hung; Lee, Wei-Chin; Hong, Minghwei; Kwo, Jueinai-Raynien; Hsieh, Wen-Feng
2016-01-01
Unlike coherent acoustic phonons (CAPs) generated from heat induced thermal stress by the coated Au film, we demonstrated the oscillation from c-ZnO epitaxial film on oxide buffered Si through a degenerate pump–probe technique. As the excited photon energy was set below the exciton resonance, the electronic stress that resulted from defect resonance was used to induce acoustic wave. The damped oscillation revealed a superposition of a high frequency and long decay CAP signal with a backward propagating acoustic pulse which was generated by the absorption of the penetrated pump beam at the Si surface and selected by the ZnO layer as the acoustic resonator. (paper)
11. Acoustic Virtual Vortices with Tunable Orbital Angular Momentum for Trapping of Mie Particles
Science.gov (United States)
Marzo, Asier; Caleap, Mihai; Drinkwater, Bruce W.
2018-01-01
Acoustic vortices can transfer angular momentum and trap particles. Here, we show that particles trapped in airborne acoustic vortices orbit at high speeds, leading to dynamic instability and ejection. We demonstrate stable trapping inside acoustic vortices by generating sequences of short-pulsed vortices of equal helicity but opposite chirality. This produces a "virtual vortex" with an orbital angular momentum that can be tuned independently of the trapping force. We use this method to adjust the rotational speed of particles inside a vortex beam and, for the first time, create three-dimensional acoustics traps for particles of wavelength order (i.e., Mie particles).
12. Micromachined silicon acoustic delay line with 3D-printed micro linkers and tapered input for improved structural stability and acoustic directivity
International Nuclear Information System (INIS)
Cho, Y; Kumar, A; Xu, S; Zou, J
2016-01-01
Recent studies have shown that micromachined silicon acoustic delay lines can provide a promising solution to achieve real-time photoacoustic tomography without the need for complex transducer arrays and data acquisition electronics. To achieve deeper imaging depth and wider field of view, a longer delay time and therefore delay length are required. However, as the length of the delay line increases, it becomes more vulnerable to structural instability due to reduced mechanical stiffness. In this paper, we report the design, fabrication, and testing of a new silicon acoustic delay line enhanced with 3D printed polymer micro linker structures. First, mechanical deformation of the silicon acoustic delay line (with and without linker structures) under gravity was simulated by using finite element method. Second, the acoustic crosstalk and acoustic attenuation caused by the polymer micro linker structures were evaluated with both numerical simulation and ultrasound transmission testing. The result shows that the use of the polymer micro linker structures significantly improves the structural stability of the silicon acoustic delay lines without creating additional acoustic attenuation and crosstalk. In addition, the improvement of the acoustic acceptance angle of the silicon acoustic delay lines was also investigated to better suppress the reception of unwanted ultrasound signals outside of the imaging plane. These two improvements are expected to provide an effective solution to eliminate current limitations on the achievable acoustic delay time and out-of-plane imaging resolution of micromachined silicon acoustic delay line arrays. (paper)
13. Instability of a planar expansion wave
International Nuclear Information System (INIS)
Velikovich, A.L.; Zalesak, S.T.; Metzler, N.; Wouchuk, J.G.
2005-01-01
An expansion wave is produced when an incident shock wave interacts with a surface separating a fluid from a vacuum. Such an interaction starts the feedout process that transfers perturbations from the rippled inner (rear) to the outer (front) surface of a target in inertial confinement fusion. Being essentially a standing sonic wave superimposed on a centered expansion wave, a rippled expansion wave in an ideal gas, like a rippled shock wave, typically produces decaying oscillations of all fluid variables. Its behavior, however, is different at large and small values of the adiabatic exponent γ. At γ>3, the mass modulation amplitude δm in a rippled expansion wave exhibits a power-law growth with time ∝t β , where β=(γ-3)/(γ-1). This is the only example of a hydrodynamic instability whose law of growth, dependent on the equation of state, is expressed in a closed analytical form. The growth is shown to be driven by a physical mechanism similar to that of a classical Richtmyer-Meshkov instability. In the opposite extreme γ-1 -1/2 , and then starts to decrease. The mechanism driving the growth is the same as that of Vishniac's instability of a blast wave in a gas with low γ. Exact analytical expressions for the growth rates are derived for both cases and favorably compared to hydrodynamic simulation results
14. Buneman instability and Pierce instability in a collisionless bounded plasma
International Nuclear Information System (INIS)
Iizuka, Satoru; Saeki, Koichi; Sato, Noriyoshi; Hatta, Yoshisuke
1983-01-01
A systematic experiment is performed on the Buneman instability and the Pierce instability in a bounded plasma consisting of beam electrons and stationary ions. Current fluctuations are confirmed to be induced by the Buneman instability. On the other hand, the Pierce instability gives rise to a current limitation. The phenomena are well explained by Mikhailovskii's theory taking account of ion motion in a bounded plasma. (author)
15. The influence of wall resonances on the levitation of objects in a single-axis acoustic processing chamber
Science.gov (United States)
Ross, B. B.
1980-01-01
Instabilities were observed in high temperature, single axis acoustic processing chambers. At certain temperatures, strong wall resonances were generated within the processing chamber itself and these transverse resonances were thought sufficient to disrupt the levitation well. These wall resonances are apparently not strong enough to cause instabilities in the levitation well.
16. Elbow joint instability
DEFF Research Database (Denmark)
Olsen, Bo Sanderhoff; Henriksen, M G; Søjbjerg, Jens Ole
1994-01-01
The effect of simultaneous ulnar and radial collateral ligament division on the kinematics of the elbow joint is studied in a cadaveric model. Severance of the anterior part of the ulnar collateral ligament and the annular ligament led to significant elbow joint instability in valgus and varus...
17. Structural and Material Instability
DEFF Research Database (Denmark)
Cifuentes, Gustavo Cifuentes
This work is a small contribution to the general problem of structural and material instability. In this work, the main subject is the analysis of cracking and failure of structural elements made from quasi-brittle materials like concrete. The analysis is made using the finite element method. Three...
18. Agricultural Markets Instability
NARCIS (Netherlands)
Garrido, A.; Brümmer, B.; M'Barek, R.; Gielen-Meuwissen, M.P.M.; Morales-Opazo, C.
2016-01-01
Since the financial and food price crises of 2007, market instability has been a topic of major concern to agricultural economists and policy professionals. This volume provides an overview of the key issues surrounding food prices volatility, focusing primarily on drivers, long-term implications of
19. Comment on critical instability
International Nuclear Information System (INIS)
King, S.F.; Suzuki, Mahiko
1992-01-01
We discuss the problem of the mass splitting between top and bottom quarks, within the context of Nambu-Jona-Lasinio type models involving top and bottom quark condensates. We interpret the phenomenon of 'critical instability' recently proposed to account for such a mass splitting as the fine-tuning of two vacuum expectation values in a composite two-Higgs doublet model. (orig.)
20. SYMPOSIUM: Rare decays
Energy Technology Data Exchange (ETDEWEB)
Anon.
1989-04-15
Late last year, a symposium entitled 'Rare Decays' attracted 115 participants to a hotel in Vancouver, Canada. These participants were particle physicists interested in checking conventional selection rules to look for clues of possible new behaviour outside today's accepted 'Standard Model'. For physicists, 'rare decays' include processes that have so far not been seen, explicitly forbidden by the rules of the Standard Model, or processes highly suppressed because the decay is dominated by an easier route, or includes processes resulting from multiple transitions.
1. Effective Majorana neutrino decay
Energy Technology Data Exchange (ETDEWEB)
Duarte, Lucia [Instituto de Fisica, Facultad de Ingenieria,Universidad de la Republica, Montevideo (Uruguay); Romero, Ismael; Peressutti, Javier; Sampayo, Oscar A. [Universidad Nacional de Mar del Plata, Departamento de Fisica, Instituto de Investigaciones Fisicas de Mar del Plata (IFIMAR) CONICET, UNMDP, Mar del Plata (Argentina)
2016-08-15
We study the decay of heavy sterile Majorana neutrinos according to the interactions obtained from an effective general theory. We describe the two- and three-body decays for a wide range of neutrino masses. The results obtained and presented in this work could be useful for the study of the production and detection of these particles in a variety of high energy physics experiments and astrophysical observations. We show in different figures the dominant branching ratios and the total decay width. (orig.)
2. Axigluon decays of toponium
International Nuclear Information System (INIS)
Faustov, R.N.; Vasilevskaya, I.G.
1990-01-01
Chiral-colour model predicts the existence of axigluons which is an octet of massive axial-vector gauge bosons. In this respect toponium decays into axigluons and gluons are of interest. The following toponium decays are considered: θ → Ag, θ → AAg, θ → ggg → AAg. The width of toponium S-state decays is calculated under various possible values of axigluon mass
3. Decay of 143La
International Nuclear Information System (INIS)
Blachot, J.; Dousson, S.; Monnand, E.; Schussler, F.
1976-01-01
The decay of 143 La has been investigated. Sources have been obtained from 2 isotope separators (ISERE, OSIRIS). 12 gamma rays, with the most intense at 620keV representing only 1.4% of decay, have been attributed to the 143 La decay. A level scheme has been found and compared with the one deduced from (d,p) and (n,γ) reactions on 142 Ce [fr
4. Some limitations on processing materials in acoustic levitation devices
Science.gov (United States)
Oran, W. A.; Witherow, W. K.; Ross, B. B.; Rush, J. E.
1979-01-01
The spot heating of samples, suspended in an acoustic field, was investigated to determine if the technique could be used to process materials. A single axis resonance device operating in air at 25 C with an rms pressure maximum of 160 to 170 db was used in the experiments. The heat flow from a hot object suspended in a levitation node is dominated by the effects of the field, with the heat loss approximately 20 times larger than that due to natural convection. The acoustic forces which suspend the body at a node also serve to eject the heated air. The coupling between the locally heated region around the body and the acoustic field results in instabilities in both the pressure wave and force field. The investigations indicated the extreme difficulties in developing a materials processing device based on acoustic/spot heating for use in a terrestrial environment.
5. Acoustic topological insulator and robust one-way sound transport
Science.gov (United States)
He, Cheng; Ni, Xu; Ge, Hao; Sun, Xiao-Chen; Chen, Yan-Bin; Lu, Ming-Hui; Liu, Xiao-Ping; Chen, Yan-Feng
2016-12-01
Topological design of materials enables topological symmetries and facilitates unique backscattering-immune wave transport. In airborne acoustics, however, the intrinsic longitudinal nature of sound polarization makes the use of the conventional spin-orbital interaction mechanism impossible for achieving band inversion. The topological gauge flux is then typically introduced with a moving background in theoretical models. Its practical implementation is a serious challenge, though, due to inherent dynamic instabilities and noise. Here we realize the inversion of acoustic energy bands at a double Dirac cone and provide an experimental demonstration of an acoustic topological insulator. By manipulating the hopping interaction of neighbouring ’atoms’ in this new topological material, we successfully demonstrate the acoustic quantum spin Hall effect, characterized by robust pseudospin-dependent one-way edge sound transport. Our results are promising for the exploration of new routes for experimentally studying topological phenomena and related applications, for example, sound-noise reduction.
6. Tracking Code for Microwave Instability
International Nuclear Information System (INIS)
Heifets, S.; SLAC
2006-01-01
To study microwave instability the tracking code is developed. For bench marking, results are compared with Oide-Yokoya results [1] for broad-band Q = 1 impedance. Results hint to two possible mechanisms determining the threshold of instability
7. Instabilities in thin tunnel junctions
International Nuclear Information System (INIS)
1978-01-01
Tunnel junctions prepared for inelastic electron tunneling spectroscopy are often plagued by instabilities in the 0-500-meV range. This paper relates the bias at which the instability occurs to the barrier thickness
8. Parametric Room Acoustic Workflows
DEFF Research Database (Denmark)
Parigi, Dario; Svidt, Kjeld; Molin, Erik
2017-01-01
The paper investigates and assesses different room acoustics software and the opportunities they offer to engage in parametric acoustics workflow and to influence architectural designs. The first step consists in the testing and benchmarking of different tools on the basis of accuracy, speed...... and interoperability with Grasshopper 3d. The focus will be placed to the benchmarking of three different acoustic analysis tools based on raytracing. To compare the accuracy and speed of the acoustic evaluation across different tools, a homogeneous set of acoustic parameters is chosen. The room acoustics parameters...... included in the set are reverberation time (EDT, RT30), clarity (C50), loudness (G), and definition (D50). Scenarios are discussed for determining at different design stages the most suitable acoustic tool. Those scenarios are characterized, by the use of less accurate but fast evaluation tools to be used...
9. Nonlinear evolution of MHD instabilities
International Nuclear Information System (INIS)
Bateman, G.; Hicks, H.R.; Wooten, J.W.; Dory, R.A.
1975-01-01
A 3-D nonlinear MHD computer code was used to study the time evolution of internal instabilities. Velocity vortex cells are observed to persist into the nonlinear evolution. Pressure and density profiles convect around these cells for a weak localized instability, or convect into the wall for a strong instability. (U.S.)
10. Linear and non-linear ion acoustic phenomena in magnetic multi-dipole discharges
International Nuclear Information System (INIS)
Ferreira, J.L.
1986-12-01
An experimental study of ion acoustic phenomena in a multi-magnetic-dipole plasma device is presented. The plasma is uniform and free from external field, permitting good observation of space and laboratory plasma phenomena. The major interest was in the observtion of the propagation characterics of solitions and ion acoustic waves in a double plasma configuration. In this experiment plane waves were studied in a plasma composed by a mixture of negative and positive ions. The most important result was the first observation of solitary waves with negative potential, that means rarefaction ion acoustic solitions. The formation of non neutral regions inside the plasma and its relations with the inhibition of electron thermal flux were studied. A bootstrap action enhances the ion acoustic instability which generates an anomalous resistivity self consistently with a potential step. It was observed that this is the mechanism of cold electron thermalization during diffusion through a warn collisionless plasma. The importance of the bootstrap action in ion acoustic double layer formation was experimentally verified by ion acoustic instability inhibition, obtained via induced Landau damping of the ion acoustic spectrum of the instability. (author) [pt
11. Acoustic tomography for decay detection in red oak trees
Science.gov (United States)
Xiping Wang; R. Bruce Allison; Lihai Wang; Robert J. Ross
2007-01-01
The science of tree stability analysis uses both biological and engineering principles in attempting to rate a treeâs structural soundness and make reasonable predictions of potential for failure. In such analysis, arborists are often challenged by internal structural defects hidden from view within the trunks. This paper reports the results of an investigation using...
12. Induced nuclear beta decay
International Nuclear Information System (INIS)
Reiss, H.R.
1986-01-01
Certain nuclear beta decay transitions normally inhibited by angular momentum or parity considerations can be induced to occur by the application of an electromagnetic field. Such decays can be useful in the controlled production of power, and in fission waste disposal
13. B decays to baryons
We note that two-body decays to baryons are suppressed relative to three- and four-body decays. In most of these analyses, the invariant baryon–antibaryon mass shows an enhancement near the threshold. We propose a phenomenological interpretation of this quite common feature of hadronization to baryons.
14. Multiple preequilibrium decay processes
International Nuclear Information System (INIS)
Blann, M.
1987-11-01
Several treatments of multiple preequilibrium decay are reviewed with emphasis on the exciton and hybrid models. We show the expected behavior of this decay mode as a function of incident nucleon energy. The algorithms used in the hybrid model treatment are reviewed, and comparisons are made between predictions of the hybrid model and a broad range of experimental results. 24 refs., 20 figs
15. Aspects of B decays
International Nuclear Information System (INIS)
Faller, Sven
2011-01-01
B-meson decays are a good probe for testing the flavour sector of the standard model of particle physics. The standard model describes at present all experimental data satisfactorily, although some ''tensions'' exist, i.e. two to three sigma deviations from the predictions, in particular in B decays. The arguments against the standard model are thus purely theoretical. These tensions between experimental data and theoretical predictions provide an extension of the standard model by new physics contributions. Within the flavour sector main theoretical uncertainties are related to the hadronic matrix elements. For exclusive semileptonic anti B → D (*) l anti ν decays QCD sum rule techniques, which are suitable for studying hadronic matrix elements, however, with substantial, but estimable hadronic uncertainties, are used. The exploration of new physics effects in B-meson decays is done in an twofold way. In exclusive semileptonic anti B → D (*) l anti ν decays the effect of additional right-handed vector as well as left- and right-handed scalar and tensor hadronic current structures in the decay rates and the form factors are studied at the non-recoil point. As a second approach one studied the non-leptonic B 0 s →J/ψφ and B 0 →J/ψK S,L decays discussing CP violating effects in the time-dependent decay amplitudes by considering new physics phase in the B 0 - anti B 0 mixing phase. (orig.)
16. Acoustic Metamaterials in Aeronautics
Directory of Open Access Journals (Sweden)
Giorgio Palma
2018-06-01
Full Text Available Metamaterials, man-made composites that are scaled smaller than the wavelength, have demonstrated a huge potential for application in acoustics, allowing the production of sub-wavelength acoustic absorbers, acoustic invisibility, perfect acoustic mirrors and acoustic lenses for hyper focusing, and acoustic illusions and enabling new degrees of freedom in the control of the acoustic field. The zero, or even negative, refractive sound index of metamaterials offers possibilities for the control of acoustic patterns and sound at sub-wavelength scales. Despite the tremendous growth in research on acoustic metamaterials during the last decade, the potential of metamaterial-based technologies in aeronautics has still not been fully explored, and its utilization is still in its infancy. Thus, the principal concepts mentioned above could very well provide a means to develop devices that allow the mitigation of the impact of civil aviation noise on the community. This paper gives a review of the most relevant works on acoustic metamaterials, analyzing them for their potential applicability in aeronautics, and, in this process, identifying possible implementation areas and interesting metabehaviors. It also identifies some technical challenges and possible future directions for research with the goal of unveiling the potential of metamaterials technology in aeronautics.
17. Decay Process in an Active Medium. An Example
International Nuclear Information System (INIS)
Atamanuk, B.; Volokitin, A.S.
1999-01-01
In the background of many plasma phenomena are wave-wave interactions, wave-particle interactions, energy transfers, and so on. In natural and laboratory plasma, very often there are situations when a plasma is weakly unstable, that is not far above a threshold of instability. In these cases, we can describe a perturbed state in terms of a small finite number of plasma waves. In the present work, we will consider non-linear stabilization of the current instability in isothermal magnetized plasma by a three-wave decay process. This problem was studied in the many works, but some questions remain unclear, e.g. the existence and properties of chaotic regimes for certain parameters. We will consider an interesting, realistic plasma system with current instability in isothermal plasmas. To assume that only three waves are involved in the process of stabilization we have to consider instability in the conditions very close to the threshold, when only one low hybrid mode is unstable. This wave decays on two other strongly damped low hybrids waves. The dynamic stabilization of instability and swapping of energy from a radiant in area of damping (heat of plasma) thus happens. This example shows a common situation and that represents practical interest at the analysis of natural processes in ionosphere. (author)
18. Instabilities in the 'on' phase of the plasma focus
International Nuclear Information System (INIS)
Kaeppeler, H.J.
1990-07-01
In the operation of large plasma focus devices, e.g. POSEIDON, there appear saturation phenomena in the neutron production when the charging energy of the condensor bank approaches its nominal value. This saturation is attributed to the action of impurities. It is assumed that there appear instabilities which are in part caused by impurities. In order to be able to answer this question, the linear dispersion relation was derived from a three-fluid theory (electrons, ions and neutrals) with the aid of the computer algebra (CA) code MACSYMA. The inversion of the 17x17 matrix (it is assumed that v a =v i and T a =T i ) and solution of the determinant was carried out on a CONVEX C 120 computer using the CA code MAPLE. The calculation of the zeros was done with a modified CPZERO program from the SLATEC library. There appear four instabilities in the rundown phase of the plasma focus, two of them gradient driven. The first two are unstable electrostatic waves with very high phase velocities, thus they do not contribute to anomalous dissipation. The third is identified as a gradient driven space charge instability which may possibly lead to current chopping. The electron acoustic wave instability, here gradient driven, is the fourth. It was found in a previous study of MPD thruster instabilities. (orig.)
19. Instabilities in strongly coupled plasmas
CERN Document Server
Kalman, G J
2003-01-01
The conventional Vlasov treatment of beam-plasma instabilities is inappropriate when the plasma is strongly coupled. In the strongly coupled liquid state, the strong correlations between the dust grains fundamentally affect the conditions for instability. In the crystalline state, the inherent anisotropy couples the longitudinal and transverse polarizations, and results in unstable excitations in both polarizations. We summarize analyses of resonant and non-resonant, as well as resistive instabilities. We consider both ion-dust streaming and dust beam-plasma instabilities. Strong coupling, in general, leads to an enhancement of the growth rates. In the crystalline phase, a resonant transverse instability can be excited.
20. Orphans and political instability.
Science.gov (United States)
Breuning, Marijke; Ishiyama, John
2011-01-01
This study investigates the security implications of growing orphan populations, particularly in Sub-Saharan Africa. Little has been written about the security implications of this especially vulnerable group of children. Are growing orphan populations associated with increases in political instability as has been suggested? Using data from several sources, we employ regression analysis to test whether Sub-Saharan African countries with larger proportions of orphans and those with increasing orphan populations experience higher rates of political instability. We find that the increase in the orphan population is related to an increasing incidence of civil conflict, but do not find a similar relationship for the proportion of orphans. In addition, we find that the causes of orphanhood matter. We conclude that increases in orphan populations (rather than simple proportions) are destabilizing. We suggest possible avenues for mediating the security risks posed by growing orphan populations.
1. A trickle instability
Science.gov (United States)
Bossa, Benjamin
2005-11-01
We address the problem of the free fall of a long, horizontal and narrow liquid layer squeezed in a vertical open Hele-Shaw cell. The layer destabilizes as it falls down, evolving into a series of liquid blobs linked together by thin bridges, which ultimately break, leaving the initially connex fluid layer as a set a disjointed drops. The mechanism of this instability is the onset of a vertical pressure gradient due to the curvature difference of the moving contact line between the advancing interface and the rear interface. This instability, whose growth rate scales with a non-trivial power of the capillary number, amplifies indifferently a broad band of wavenumbers because of the flat shape of its dispersion relation in the thin layer limit. We will finally comment on the nature of the final fragmentation process and drop size distributions.
2. Instability and internet design
Directory of Open Access Journals (Sweden)
Sandra Braman
2016-09-01
Full Text Available Instability - unpredictable but constant change in one’s environment and the means with which one deals with it - has replaced convergence as the focal problem for telecommunications policy in general and internet policy in particular. Those who designed what we now call the internet during the first decade of the effort (1969-1979, who in essence served simultaneously as its policy-makers, developed techniques for coping with instability of value for network designers today and for those involved with any kind of large-scale sociotechnical infrastructure. Analysis of the technical document series that was medium for and record of that design process reveals coping techniques that began with defining the problem and went on to include conceptual labour, social practices, and technical approaches.
3. Imaging of patellofemoral instability
International Nuclear Information System (INIS)
Waldt, S.; Rummeny, E.J.
2012-01-01
Patellofemoral instability remains a diagnostic and therapeutic challenge due to its multifactorial genesis. The purpose of imaging is to systematically analyze predisposing factors, such as trochlear dysplasia, patella alta, tibial tuberosity-trochlear groove (TT-TG) distance, rotational deformities of the lower limb and patellar tilt. In order to evaluate anatomical abnormalities with a sufficient diagnostic accuracy, standardized measurement methods and implementation of various imaging modalities are necessary. Diagnosis of acute and often overlooked lateral patellar dislocation can be established with magnetic resonance imaging (MRI) because of its characteristic patterns of injury. Damage to the medial patellofemoral ligament (MPFL) has a significance just as high as the predisposing risk factors in relation to the cause of chronic instability. (orig.) [de
4. Linear waves and instabilities
International Nuclear Information System (INIS)
Bers, A.
1975-01-01
The electrodynamic equations for small-amplitude waves and their dispersion relation in a homogeneous plasma are outlined. For such waves, energy and momentum, and their flow and transformation, are described. Perturbation theory of waves is treated and applied to linear coupling of waves, and the resulting instabilities from such interactions between active and passive waves. Linear stability analysis in time and space is described where the time-asymptotic, time-space Green's function for an arbitrary dispersion relation is developed. The perturbation theory of waves is applied to nonlinear coupling, with particular emphasis on pump-driven interactions of waves. Details of the time--space evolution of instabilities due to coupling are given. (U.S.)
5. Cosmic ray driven instability
International Nuclear Information System (INIS)
Dorfi, E.A.; Drury, L.O.
1985-01-01
The interaction between energetic charged particles and thermal plasma, which forms the basis of diffusive shock acceleration, leads also to interesting dynamical phenomena. For a compressional mode propagating in a system with homoeneous energetic particle pressure it is well known that friction with the energetic particles leads to damping. The linear theory of this effect has been analyzed in detail by Ptuskin. Not so obvious is that a non-uniform energetic particle pressure can in addition amplify compressional disturbances. If the pressure gradient is sufficiently steep this growth can dominate the frictional damping and lead to an instability. It is important to not that this effect results from the collective nature of the interaction between the energetic particles and the gas and is not connected with the Parker instability, nor with the resonant amplification of Alfven waves
6. Solar Wind Electron Scattering by Kinetic Instabilities and Whistler Turbulence
Science.gov (United States)
Gary, S. P.
2015-12-01
The expansion of the solar wind away from the Sun drives electron velocity distributions away from the thermal Maxwellian form, yielding distributions near 1 AU which typically can be characterized as consisting of three anisotropic components: a more dense, relatively cool core, a relatively tenuous , relatively warm halo and a similarly tenuous, warm strahl. Each of these nonthermal components are potential sources of kinetic plasma instabilities; the enhanced waves from each instability can scatter the electrons, acting to reduce the various anisotropies and making their overall velocity distribution more nearly (but not completely) thermal. In contrast, simulations are demonstrating that the forward decay of whistler turbulence can lead to the development of a T||> T_perp electron anisotropy. This presentation will review linear theories of electron-driven kinetic instabilities (following the presentation by Daniel Verscharen at the 2015 SHINE Workshop), and will further consider the modification of electron velocity distributions as obtained from particle-in-cell simulations of such instabilities as well as from the decay of whistler turbulence.
7. Decay of hypernuclei
International Nuclear Information System (INIS)
Bando, H.
1985-01-01
The pionic and non-mesonic decays of hypernuclei are discussed. In the first part, various decay processes which could be useful to obtain information of hypernuclear structure are discussed. The experimental data concerning the pionic and non-mesonic decays are discussed in the second part. As the experimental data, there are only few lifetime data and some crude data on the non-mesonic to π decay ratio. In the third and the fourth parts, some theoretical analyses are made on the pionic and the nonmesonic decays. DDHF calculation was performed for Λ and N systems by using Skyrme type ΛN and NN effective interactions. A suppression factor of the order of 10 -3 for A nearly equal 100 was obtained. (Aoki, K.)
8. Rare Decays at LHCb
CERN Document Server
Belyaev, Ivan
2006-01-01
Rare loop-induced decays are sensitive to New Physics in many Standard Model extensions. In this paper we discuss the reconstruction of the radiative penguin decays $B^0_d \\to K^{*0} \\gamma, B^0_s \\to \\phi \\gamma , B^0_d \\to \\omega \\gamma, \\Lambda_b \\to \\Lambda \\gamma$, the electroweak penguin decays $B^0_d \\to K^{*0} \\mu^+ \\mu^-, B^+_u \\to K^+ \\mu^+ \\mu^-$, the gluonic penguin decays $B^0_d \\to \\phi K^0_S, B^0_s \\to \\phi \\phi$, and the decay $B^0_s \\to \\mu^+\\mu^-$ at LHCb. The selection criteria, evaluated efficiencies, expected annual yields and $B/S$ estimates are presented.
9. Instability in dynamic fracture
Science.gov (United States)
Fineberg, J.; Marder, M.
1999-05-01
The fracture of brittle amorphous materials is an especially challenging problem, because the way a large object shatters is intimately tied to details of cohesion at microscopic scales. This subject has been plagued by conceptual puzzles, and to make matters worse, experiments seemed to contradict the most firmly established theories. In this review, we will show that the theory and experiments fit within a coherent picture where dynamic instabilities of a crack tip play a crucial role. To accomplish this task, we first summarize the central results of linear elastic dynamic fracture mechanics, an elegant and powerful description of crack motion from the continuum perspective. We point out that this theory is unable to make predictions without additional input, information that must come either from experiment, or from other types of theories. We then proceed to discuss some of the most important experimental observations, and the methods that were used to obtain the them. Once the flux of energy to a crack tip passes a critical value, the crack becomes unstable, and it propagates in increasingly complicated ways. As a result, the crack cannot travel as quickly as theory had supposed, fracture surfaces become rough, it begins to branch and radiate sound, and the energy cost for crack motion increases considerably. All these phenomena are perfectly consistent with the continuum theory, but are not described by it. Therefore, we close the review with an account of theoretical and numerical work that attempts to explain the instabilities. Currently, the experimental understanding of crack tip instabilities in brittle amorphous materials is fairly detailed. We also have a detailed theoretical understanding of crack tip instabilities in crystals, reproducing qualitatively many features of the experiments, while numerical work is beginning to make the missing connections between experiment and theory.
10. Nonextensive GES instability with nonlinear pressure effects
Directory of Open Access Journals (Sweden)
Munmi Gohain
2018-03-01
Full Text Available We herein analyze the instability dynamics associated with the nonextensive nonthermal gravito-electrostatic sheath (GES model for the perturbed solar plasma portraiture. The usual neutral gas approximation is herewith judiciously relaxed and the laboratory plasma-wall interaction physics is procedurally incorporated amid barotropic nonlinearity. The main motivation here stems from the true nature of the solar plasma system as a set of concentric nonlocal nonthermal sub-layers as evidenced from different multi-space satellite probes and missions. The formalism couples the solar interior plasma (SIP, bounded and solar wind plasma (SWP, unbounded via the diffused solar surface boundary (SSB formed due to an exact long-range gravito-electrostatic force-equilibration. A linear normal mode ansatz reveals both dispersive and non-dispersive features of the modified GES collective wave excitations. It is seen that the thermostatistical GES stability depends solely on the electron-to-ion temperature ratio. The damping behavior on both the scales is more pronounced in the acoustic domain, K→∞, than the gravitational domain, K→0; where, K is the Jeans-normalized angular wave number. It offers a unique quasi-linear coupling of the gravitational and acoustic fluctuations amid the GES force action. The results may be useful to see the excitation dynamics of natural normal modes in bounded nonextensive astero-environs from a new viewpoint of the plasma-wall coupling mechanism.
11. Nonextensive GES instability with nonlinear pressure effects
Science.gov (United States)
Gohain, Munmi; Karmakar, Pralay Kumar
2018-03-01
We herein analyze the instability dynamics associated with the nonextensive nonthermal gravito-electrostatic sheath (GES) model for the perturbed solar plasma portraiture. The usual neutral gas approximation is herewith judiciously relaxed and the laboratory plasma-wall interaction physics is procedurally incorporated amid barotropic nonlinearity. The main motivation here stems from the true nature of the solar plasma system as a set of concentric nonlocal nonthermal sub-layers as evidenced from different multi-space satellite probes and missions. The formalism couples the solar interior plasma (SIP, bounded) and solar wind plasma (SWP, unbounded) via the diffused solar surface boundary (SSB) formed due to an exact long-range gravito-electrostatic force-equilibration. A linear normal mode ansatz reveals both dispersive and non-dispersive features of the modified GES collective wave excitations. It is seen that the thermostatistical GES stability depends solely on the electron-to-ion temperature ratio. The damping behavior on both the scales is more pronounced in the acoustic domain, K → ∞ , than the gravitational domain, K → 0 ; where, K is the Jeans-normalized angular wave number. It offers a unique quasi-linear coupling of the gravitational and acoustic fluctuations amid the GES force action. The results may be useful to see the excitation dynamics of natural normal modes in bounded nonextensive astero-environs from a new viewpoint of the plasma-wall coupling mechanism.
12. Relativistic centrifugal instability
Science.gov (United States)
Gourgouliatos, Konstantinos N.; Komissarov, Serguei S.
2018-03-01
Near the central engine, many astrophysical jets are expected to rotate about their axis. Further out they are expected to go through the processes of reconfinement and recollimation. In both these cases, the flow streams along a concave surface and hence, it is subject to the centrifugal force. It is well known that such flows may experience the centrifugal instability (CFI), to which there are many laboratory examples. The recent computer simulations of relativistic jets from active galactic nuclei undergoing the process of reconfinement show that in such jets CFI may dominate over the Kelvin-Helmholtz instability associated with velocity shear (Gourgouliatos & Komissarov). In this letter, we generalize the Rayleigh criterion for CFI in rotating fluids to relativistic flows using a heuristic analysis. We also present the results of computer simulations which support our analytic criterion for the case of an interface separating two uniformly rotating cylindrical flows. We discuss the difference between CFI and the Rayleigh-Taylor instability in flows with curved streamlines.
13. Analyses of MHD instabilities
International Nuclear Information System (INIS)
Takeda, Tatsuoki
1985-01-01
In this article analyses of the MHD stabilities which govern the global behavior of a fusion plasma are described from the viewpoint of the numerical computation. First, we describe the high accuracy calculation of the MHD equilibrium and then the analysis of the linear MHD instability. The former is the basis of the stability analysis and the latter is closely related to the limiting beta value which is a very important theoretical issue of the tokamak research. To attain a stable tokamak plasma with good confinement property it is necessary to control or suppress disruptive instabilities. We, next, describe the nonlinear MHD instabilities which relate with the disruption phenomena. Lastly, we describe vectorization of the MHD codes. The above MHD codes for fusion plasma analyses are relatively simple though very time-consuming and parts of the codes which need a lot of CPU time concentrate on a small portion of the codes, moreover, the codes are usually used by the developers of the codes themselves, which make it comparatively easy to attain a high performance ratio on the vector processor. (author)
International Nuclear Information System (INIS)
1989-01-01
Anomalous ion thermal conductivity remains an open physics issue for the present generation of high temperature Tokamaks. It is generally believed to be due to Ion Temperature Gradient Instability (η i mode). However, it has been difficult, if not impossible to identify this instability and study the anomalous transport due to it, directly. Therefore the production and identification of the mode is pursued in the simpler and experimentally convenient configuration of the Columbia Linear Machine (CLM). CLM is a steady state machine which already has all the appropriate parameters, except η i . This parameter is being increased to the appropriate value of the order of 1 by 'feathering' a tungsten screen located between the plasma source and the experimental cell to flatten the density profile and appropriate redesign of heating antennas to steepen the ion temperature profile. Once the instability is produced and identified, a thorough study of the characteristics of the mode can be done via a wide range of variation of all the critical parameters: η i , parallel wavelength, etc
15. Three-dimensional stability of solitary kinetic Alfven waves and ion-acoustic waves
International Nuclear Information System (INIS)
Ghosh, G.; Das, K.P.
1994-01-01
Starting from a set of equations that lead to a linear dispersion relation coupling kinetic Alfven waves and ion-acoustic waves, three-dimensional KdV equations are derived for these waves. These equations are then used to investigate the three-dimensional stability of solitary kinetic Alfven waves and ion-acoustic waves by the small-k perturbation expansion method of Rowlands and Infeld. For kinetic Alfven waves it is found that there is instability if the direction of the plane-wave perturbation lies inside a cone, and the growth rate of the instability attains a maximum when the direction of the perturbation lies in the plane containing the external magnetic field and the direction of propagation of the solitary wave. For ion-acoustic waves the growth rate of instability attains a maximum when the direction of the perturbation lies in a plane perpendicular to the direction of propagation of the solitary wave. (Author)
16. Nonlinear self-modulation of ion-acoustic waves
International Nuclear Information System (INIS)
Ikezi, H.; Schwarzenegger, K.; Simons, A.L.; Ohsawa, Y.; Kamimura, T.
1978-01-01
The nonlinear evolution of an ion-acoustic wave packet is studied. Experimentally, it is found that (i) nonlinear phase modulation develops in the wave packet; (ii) the phase modulation, together with the dispersion effect, causes expansion and breaking of the wave packet; (iii) the ions trapped in the troughs of the wave potential introduce self-phase modulation; and (iv) the ion-acoustic wave is stable with respect to the modulational instability. Computer simulations have reproduced the experimental results. The physical picture and the model equation describing the wave evolution are discussed
17. Calculation of the acoustical properties of triadic harmonies.
Science.gov (United States)
Cook, Norman D
2017-12-01
The author reports that the harmonic "tension" and major/minor "valence" of pitch combinations can be calculated directly from acoustical properties without relying on concepts from traditional harmony theory. The capability to compute the well-known types of harmonic triads means that their perception is not simply a consequence of learning an arbitrary cultural "idiom" handed down from the Italian Renaissance. On the contrary, for typical listeners familiar with diatonic music, attention to certain, definable, acoustical features underlies the perception of the valence (modality) and the inherent tension (instability) of three-tone harmonies.
18. Springer handbook of acoustics
CERN Document Server
2014-01-01
Acoustics, the science of sound, has developed into a broad interdisciplinary field encompassing the academic disciplines of physics, engineering, psychology, speech, audiology, music, architecture, physiology, neuroscience, and electronics. The Springer Handbook of Acoustics is also in his 2nd edition an unparalleled modern handbook reflecting this richly interdisciplinary nature edited by one of the acknowledged masters in the field, Thomas Rossing. Researchers and students benefit from the comprehensive contents. This new edition of the Handbook features over 11 revised and expanded chapters, new illustrations, and 2 new chapters covering microphone arrays and acoustic emission. Updated chapters contain the latest research and applications in, e.g. sound propagation in the atmosphere, nonlinear acoustics in fluids, building and concert hall acoustics, signal processing, psychoacoustics, computer music, animal bioacousics, sound intensity, modal acoustics as well as new chapters on microphone arrays an...
19. Vibro-acoustics
CERN Document Server
Nilsson, Anders
2015-01-01
This three-volume book gives a thorough and comprehensive presentation of vibration and acoustic theories. Different from traditional textbooks which typically deal with some aspects of either acoustic or vibration problems, it is unique of this book to combine those two correlated subjects together. Moreover, it provides fundamental analysis and mathematical descriptions for several crucial phenomena of Vibro-Acoustics which are quite useful in noise reduction, including how structures are excited, energy flows from an excitation point to a sound radiating surface, and finally how a structure radiates noise to a surrounding fluid. Many measurement results included in the text make the reading interesting and informative. Problems/questions are listed at the end of each chapter and the solutions are provided. This will help the readers to understand the topics of Vibro-Acoustics more deeply. The book should be of interest to anyone interested in sound and vibration, vehicle acoustics, ship acoustics and inter...
20. Charm Decays at BABAR
International Nuclear Information System (INIS)
Charles, M.
2004-01-01
The results of several studies of charmed mesons and baryons at BABAR are presented. First, searches for the rare decays D 0 → l + l - are presented and new upper limits on these processes are established. Second, a measurement of the branching fraction of the isospin-violating hadronic decay D* s (2112) + → D s + π 0 relative to the radiative decay D* s (2112) + → D s + γ is made. Third, the decays of D* sJ (2317) + and D sJ (2460) + mesons are studied and ratios of branching fractions are measured. Fourth, Cabibbo-suppressed decays of the Λ c + are examined and their branching fractions measured relative to Cabibbo-allowed modes. Fifth, the Χ c 0 is studied through its decays to Χ - π + and (Omega) - K + ; in addition to measuring the ratio of branching fractions for Χ c 0 produced from the c(bar c) continuum, the uncorrected momentum spectrum is measured, providing clear confirmation of Χ c 0 production in B decays
1. Iconic decay in schizophrenia.
Science.gov (United States)
Hahn, Britta; Kappenman, Emily S; Robinson, Benjamin M; Fuller, Rebecca L; Luck, Steven J; Gold, James M
2011-09-01
Working memory impairment is considered a core deficit in schizophrenia, but the precise nature of this deficit has not been determined. Multiple lines of evidence implicate deficits at the encoding stage. During encoding, information is held in a precategorical sensory store termed iconic memory, a literal image of the stimulus with high capacity but rapid decay. Pathologically increased iconic decay could reduce the number of items that can be transferred into working memory before the information is lost and could thus contribute to the working memory deficit seen in the illness. The current study used a partial report procedure to test the hypothesis that patients with schizophrenia (n = 37) display faster iconic memory decay than matched healthy control participants (n = 28). Six letters, arranged in a circle, were presented for 50 ms. Following a variable delay of 0-1000 ms, a central arrow cue indicated the item to be reported. In both patients and control subjects, recall accuracy decreased with increasing cue delay, reflecting decay of the iconic representation of the stimulus array. Patients displayed impaired memory performance across all cue delays, consistent with an impairment in working memory, but the rate of iconic memory decay did not differ between patients and controls. This provides clear evidence against faster loss of iconic memory representations in schizophrenia, ruling out iconic decay as an underlying source of the working memory impairment in this population. Thus, iconic decay rate can be added to a growing list of unimpaired cognitive building blocks in schizophrenia.
2. Acoustic and categorical dissimilarity of musical timbre: Evidence from asymmetriesbetween acoustic and chimeric sounds
Directory of Open Access Journals (Sweden)
Kai eSiedenburg
2016-01-01
Full Text Available This paper investigates the role of acoustic and categorical information in timbre dissimilarity ratings. Using a Gammatone-filterbank-based sound transformation, we created tones that were rated as less familiar than recorded tones from orchestral instruments and that were harder to associate with an unambiguous sound source (Exp. 1. A subset of transformed tones, a set of orchestral recordings, and a mixed set were then rated on pairwise dissimilarity (Exp. 2A. We observed that recorded instrument timbres clustered into subsets that distinguished timbres according to acoustic and categorical properties. For the subset of cross-category comparisons in the mixed set, we observed asymmetries in the distribution of ratings, as well as a stark decay of inter-rater agreement. These effects were replicated in a more robust within-subjects design (Exp. 2B and cannot be explained by acoustic factors alone. We finally introduced a novel model of timbre dissimilarity based on partial least-squares regression that compared the contributions of both acoustic and categorical timbre descriptors. The best model fit (R^2 = .88 was achieved when both types of descriptors were taken into account. These findings are interpreted as evidence for an interplay of acoustic and categorical information in timbre dissimilarity perception.
3. Acoustic and Categorical Dissimilarity of Musical Timbre: Evidence from Asymmetries Between Acoustic and Chimeric Sounds
Science.gov (United States)
Siedenburg, Kai; Jones-Mollerup, Kiray; McAdams, Stephen
2016-01-01
This paper investigates the role of acoustic and categorical information in timbre dissimilarity ratings. Using a Gammatone-filterbank-based sound transformation, we created tones that were rated as less familiar than recorded tones from orchestral instruments and that were harder to associate with an unambiguous sound source (Experiment 1). A subset of transformed tones, a set of orchestral recordings, and a mixed set were then rated on pairwise dissimilarity (Experiment 2A). We observed that recorded instrument timbres clustered into subsets that distinguished timbres according to acoustic and categorical properties. For the subset of cross-category comparisons in the mixed set, we observed asymmetries in the distribution of ratings, as well as a stark decay of inter-rater agreement. These effects were replicated in a more robust within-subjects design (Experiment 2B) and cannot be explained by acoustic factors alone. We finally introduced a novel model of timbre dissimilarity based on partial least-squares regression that compared the contributions of both acoustic and categorical timbre descriptors. The best model fit (R2 = 0.88) was achieved when both types of descriptors were taken into account. These findings are interpreted as evidence for an interplay of acoustic and categorical information in timbre dissimilarity perception. PMID:26779086
4. Handbook of Engineering Acoustics
CERN Document Server
Möser, Michael
2013-01-01
This book examines the physical background of engineering acoustics, focusing on empirically obtained engineering experience as well as on measurement techniques and engineering methods for prognostics. Its goal is not only to describe the state of art of engineering acoustics but also to give practical help to engineers in order to solve acoustic problems. It deals with the origin, the transmission and the methods of the abating different kinds of air-borne and structure-borne sounds caused by various mechanisms – from traffic to machinery and flow-induced sound. In addition the modern aspects of room and building acoustics, as well as psychoacoustics and active noise control, are covered.
5. Localized Acoustic Surface Modes
KAUST Repository
Farhat, Mohamed
2015-08-04
We introduce the concept of localized acoustic surface modes (ASMs). We demonstrate that they are induced on a two-dimensional cylindrical rigid surface with subwavelength corrugations under excitation by an incident acoustic plane wave. Our results show that the corrugated rigid surface is acoustically equivalent to a cylindrical scatterer with uniform mass density that can be represented using a Drude-like model. This, indeed, suggests that plasmonic-like acoustic materials can be engineered with potential applications in various areas including sensing, imaging, and cloaking.
6. Acoustic Technology Laboratory
Data.gov (United States)
Federal Laboratory Consortium — This laboratory contains an electro-magnetic worldwide data collection and field measurement capability in the area of acoustic technology. Outfitted by NASA Langley...
7. Shallow Water Acoustic Laboratory
Data.gov (United States)
Federal Laboratory Consortium — FUNCTION: Supports experimental research where high-frequency acoustic scattering and surface vibration measurements of fluid-loaded and non-fluid-loaded structures...
8. Laboratory for Structural Acoustics
Data.gov (United States)
Federal Laboratory Consortium — FUNCTION: Supports experimental research where acoustic radiation, scattering, and surface vibration measurements of fluid-loaded and non-fluid-loaded structures are...
9. Studies on the parametric decay of waves in fusion plasmas
International Nuclear Information System (INIS)
Paettikangas, T.
1992-08-01
Parametric instabilities of large-amplitude electromagnetic waves are investigated in fusion applications. In laser fusion, the electromegnetic wave reflected from the overdense plasma can act as a secondary pump wave and exite parametric instabilities. In double simulated Brilloun scattering (DSBS), both the incoming and the reflected pump wave scatter from a common ion sound wave. The stationary states and the dynamics of DSBS are investigated by using a simple envelope model. The ion sound wave that is exited in DSBS is shown to have soliton-like properties. The simulated Raman scattering (SRS) of free-electron-laser radiation can be applied to current drive in tokamaks. SRS generates fast longitudinal electron plasma waves which accelerate electrons to relativistic energies. Since the energetic current-carrying electrons are almost collisionless, the current decays very slowly. The feasibility of the Raman current drive in tokamaks is investigated theoretically. The current drive efficiency and the optimum free-electron-laser parameters are determined. The energy transfer to the fast electrons from the electrostatic wave is studied with relativistic Vlasov-Maxwell simulations. The parametric decay of a wave to half-harmonics is investigated. It is shown that the growth rate of the decay vanishes in the limit of a long wavelenght of the pump wave even for general electromagnetic or electrostatic decay models. The results are applied to the decay of a fast magnetosonic waves in tokamak plasmas. (orig.)
10. An acoustic energy framework for predicting combustion-driven acoustic instabilities in premixed gas-turbines
Science.gov (United States)
Ibrahim, Zuhair M. A.
The purpose of this study was to discover and assess student financial services delivered to students enrolled at East Tennessee State University. The research was undertaken for institutional self-improvement. The research explored changes that have occurred in student financial services in the dynamic higher education market. The research revealed universities pursued best practices for the delivery of student financial services through expanded employee knowledge, restructured organizations, and integrated information technologies. The research was conducted during October and November, 2006. The data were gathered from an online student survey of student financial services. The areas researched included: the Bursar office, the Financial Aid office, and online services. The results of the data analysis revealed problems with the students' perceived quality of existing financial services and the additional services students desire. The research focused on student perceptions of the quality of financial services by age and gender classifications and response categories. Although no statistically significant difference was found between the age-gender classifications on the perception of the quality of the financial services studied, the research adds to our understanding of student financial services at East Tennessee State University. Recommendation for continued research included annual surveys of segmented student populations that include ethnicity, age, gender, and educational level. The research would be used for continuous improvement efforts and student relationship management. Also additional research was recommended for employee learning in relation to the institution's mission, goals, and values.
International Nuclear Information System (INIS)
Roberts, B.L.; Booth, E.C.; Gall, K.P.; McIntyre, E.K.; Miller, J.P.; Whitehouse, D.A.; Bassalleck, B.; Hall, J.R.; Larson, K.D.; Wolfe, D.M.; Fickinger, W.J.; Robinson, D.K.; Hallin, A.L.; Hasinoff, M.D.; Measday, D.F.; Noble, A.J.; Waltham, C.E.; Hessey, N.P.; Lowe, J.; Horvath, D.; Salomon, M.
1990-01-01
New measurements of the Σ + and Λ weak radiative decays are discussed. The hyperons were produced at rest by the reaction K - p → Yπ where Y = Σ + or Λ. The monoenergetic pion was used to tag the hyperon production, and the branching ratios were determined from the relative amplitudes of Σ + → pγ to Σ + → pπ 0 and Λ → nγ to Λ → nπ 0 . The photons from weak radiative decays and from π 0 decays were detected with modular NaI arrays. (orig.)
12. SYMPOSIUM: Rare decays
International Nuclear Information System (INIS)
Anon.
1989-01-01
Late last year, a symposium entitled 'Rare Decays' attracted 115 participants to a hotel in Vancouver, Canada. These participants were particle physicists interested in checking conventional selection rules to look for clues of possible new behaviour outside today's accepted 'Standard Model'. For physicists, 'rare decays' include processes that have so far not been seen, explicitly forbidden by the rules of the Standard Model, or processes highly suppressed because the decay is dominated by an easier route, or includes processes resulting from multiple transitions
13. Acoustic Levitation With Less Equipment
Science.gov (United States)
Barmatz, M. B.; Jacobi, N.
1983-01-01
Certain chamber shapes require fewer than three acoustic drivers. Levitation at center of spherical chamber attained using only one acoustic driver. Exitation of lowest spherical mode produces asymmetric acoustic potential well.
14. What Is an Acoustic Neuroma
Science.gov (United States)
... CALENDAR DONATE NEWS Home Learn Back Learn about acoustic neuroma AN Facts What is acoustic neuroma? Diagnosing ... Italian Japanese Korean Portuguese Romanian Spanish What is Acoustic Neuroma? Each heading slides to reveal information. Important ...
15. Thin-wall approximation in vacuum decay: A lemma
Science.gov (United States)
2018-05-01
The "thin-wall approximation" gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two approximations converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for nonperturbative vacuum instability.
16. Instabilities of microstate geometries with antibranes
International Nuclear Information System (INIS)
Bena, Iosif; Pasini, Giulio
2016-01-01
One can obtain very large classes of horizonless microstate geometries corresponding to near-extremal black holes by placing probe supertubes whose action has metastable minima inside certain supersymmetric bubbling solutions http://dx.doi.org/10.1007/JHEP12(2012)014. We show that these minima can lower their energy when the bubbles move in certain directions in the moduli space, which implies that these near-extremal microstates are in fact unstable once one considers the dynamics of all their degrees of freedom. The decay of these solutions corresponds to Hawking radiation, and we compare the emission rate and frequency to those of the corresponding black hole. Our analysis supports the expectation that generic non-extremal black holes microstate geometries should be unstable. It also establishes the existence of a new type of instabilities for antibranes in highly-warped regions with charge dissolved in fluxes.
17. Raman sidescatter instability in a nonuniform plasma
Energy Technology Data Exchange (ETDEWEB)
Mostrom, M.A.
1977-07-15
In the various laser-fusion concepts, an intense electromagnetic wave (the laser) must propagate through an underdense plasma region where it could decay, via the stimulated Raman instability, into a Langmuir plasma wave and a scattered electromagnetic wave. Results are obtained by evaluating the ''Green's function'' response in time and space for the scattered electromagnetic waves assuming they are initiated by a ''delta-function'' source. We consider the case where the temporal growth dominates the plasma wave convection. Then the scattered electromagnetic waves are governed by a single second-order Helmholtz differential equation, in the position variable along the density gradient, with a complex potential having two simple zeros (turning points) and one simple pole.
18. Raman sidescatter instability in a nonuniform plasma
International Nuclear Information System (INIS)
Mostrom, M.A.
1977-01-01
In the various laser-fusion concepts, an intense electromagnetic wave (the laser) must propagate through an underdense plasma region where it could decay, via the stimulated Raman instability, into a Langmuir plasma wave and a scattered electromagnetic wave. Results are obtained by evaluating the ''Green's function'' response in time and space for the scattered electromagnetic waves assuming they are initiated by a ''delta-function'' source. We consider the case where the temporal growth dominates the plasma wave convection. Then the scattered electromagnetic waves are governed by a single second-order Helmholtz differential equation, in the position variable along the density gradient, with a complex potential having two simple zeros (turning points) and one simple pole
19. Instabilities of microstate geometries with antibranes
Energy Technology Data Exchange (ETDEWEB)
Bena, Iosif; Pasini, Giulio [Institut de physique théorique, Université Paris Saclay, CEA, CNRS,F-91191 Gif-sur-Yvette (France)
2016-04-29
One can obtain very large classes of horizonless microstate geometries corresponding to near-extremal black holes by placing probe supertubes whose action has metastable minima inside certain supersymmetric bubbling solutions http://dx.doi.org/10.1007/JHEP12(2012)014. We show that these minima can lower their energy when the bubbles move in certain directions in the moduli space, which implies that these near-extremal microstates are in fact unstable once one considers the dynamics of all their degrees of freedom. The decay of these solutions corresponds to Hawking radiation, and we compare the emission rate and frequency to those of the corresponding black hole. Our analysis supports the expectation that generic non-extremal black holes microstate geometries should be unstable. It also establishes the existence of a new type of instabilities for antibranes in highly-warped regions with charge dissolved in fluxes.
20. Instability of warped discs
Science.gov (United States)
Doǧan, S.; Nixon, C. J.; King, A. R.; Pringle, J. E.
2018-05-01
Accretion discs are generally warped. If a warp in a disc is too large, the disc can break' apart into two or more distinct planes, with only tenuous connections between them. Further, if an initially planar disc is subject to a strong differential precession, then it can be torn apart into discrete annuli that precess effectively independently. In previous investigations, torque-balance formulae have been used to predict where and when the disc breaks into distinct parts. In this work, focusing on discs with Keplerian rotation and where the shearing motions driving the radial communication of the warp are damped locally by turbulence (the diffusive' regime), we investigate the stability of warped discs to determine the precise criterion for an isolated warped disc to break. We find and solve the dispersion relation, which, in general, yields three roots. We provide a comprehensive analysis of this viscous-warp instability and the emergent growth rates and their dependence on disc parameters. The physics of the instability can be understood as a combination of (1) a term that would generally encapsulate the classical Lightman-Eardley instability in planar discs (given by ∂(νΣ)/∂Σ < 0) but is here modified by the warp to include ∂(ν1|ψ|)/∂|ψ| < 0, and (2) a similar condition acting on the diffusion of the warp amplitude given in simplified form by ∂(ν2|ψ|)/∂|ψ| < 0. We discuss our findings in the context of discs with an imposed precession, and comment on the implications for different astrophysical systems.
1. Teleportation via decay
therefore normally plays a negative role in quantum information processing [1]. ... of a decay be used in a fruitful way for quantum information process- ing? ..... The model independent portions of the analysis of communication through a noisy.
2. Decay of Hoyle state
2014-11-02
Nov 2, 2014 ... T K RANA, C BHATTACHARYA, S KUNDU, ... of various direct 3α decay mechanisms of the Hoyle state. ... Pramana – J. Phys., Vol. ... FMD predicts a compact triangle shape and LEFT predicts a bent arm chain structure,.
3. RARE KAON DECAYS
International Nuclear Information System (INIS)
LITTENBERG, L.
2005-01-01
Lepton flavor violation (LFV) experiments have probed sensitivities corresponding to mass scales of well over 100 TeV, making life difficult for models predicting accessible LFV in kaon decay and discouraging new dedicated experiments of this type
4. Neutrinoless double beta decay
2012-10-06
Oct 6, 2012 ... Anyhow, the 'multi-isotope' ansatz is needed to compensate for matrix element ... The neccessary half-life requirement to touch this ... site energy depositions (like double beta decay) and multiple site interactions (most of.
5. Cavities/Tooth Decay
Science.gov (United States)
... milk, ice cream, honey, sugar, soda, dried fruit, cake, cookies, hard candy and mints, dry cereal, and ... teeth can wear down and gums may recede, making teeth more vulnerable to root decay. Older adults ...
6. Inflaton decay in supergravity
Energy Technology Data Exchange (ETDEWEB)
Endo, M.; Takahashi, F. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Yanagida, T.T. [Tokyo Univ. (Japan). Dept. of Physics]|[Tokyo Univ. (Japan). Research Center for the Early Universe
2007-06-15
We discuss inflaton decay in supergravity, taking account of the gravitational effects. It is shown that, if the inflaton has a nonzero vacuum expectation value, it generically couples to any matter fields that appear in the superpotential at the tree level, and to any gauge sectors through anomalies in the supergravity. Through these processes, the inflaton generically decays into the supersymmetry breaking sector, producing many gravitinos. The inflaton also directly decays into a pair of the gravitinos. We derive constraints on both inflation models and supersymmetry breaking scenarios for avoiding overproduction of the gravitinos. Furthermore, the inflaton naturally decays into the visible sector via the top Yukawa coupling and SU(3){sub C} gauge interactions. (orig.)
7. Inflaton decay in supergravity
International Nuclear Information System (INIS)
Endo, M.; Takahashi, F.; Yanagida, T.T.; Tokyo Univ.
2007-06-01
We discuss inflaton decay in supergravity, taking account of the gravitational effects. It is shown that, if the inflaton has a nonzero vacuum expectation value, it generically couples to any matter fields that appear in the superpotential at the tree level, and to any gauge sectors through anomalies in the supergravity. Through these processes, the inflaton generically decays into the supersymmetry breaking sector, producing many gravitinos. The inflaton also directly decays into a pair of the gravitinos. We derive constraints on both inflation models and supersymmetry breaking scenarios for avoiding overproduction of the gravitinos. Furthermore, the inflaton naturally decays into the visible sector via the top Yukawa coupling and SU(3) C gauge interactions. (orig.)
8. Double beta decay: experiments
International Nuclear Information System (INIS)
Fiorini, Ettore
2006-01-01
The results obtained so far and those of the running experiments on neutrinoless double beta decay are reviewed. The plans for second generation experiments, the techniques to be adopted and the expected sensitivities are compared and discussed
9. System Detects Vibrational Instabilities
Science.gov (United States)
Bozeman, Richard J., Jr.
1990-01-01
Sustained vibrations at two critical frequencies trigger diagnostic response or shutdown. Vibration-analyzing electronic system detects instabilities of combustion in rocket engine. Controls pulse-mode firing of engine and identifies vibrations above threshold amplitude at 5.9 and/or 12kHz. Adapted to other detection and/or control schemes involving simultaneous real-time detection of signals above or below preset amplitudes at two or more specified frequencies. Potential applications include rotating machinery and encoders and decoders in security systems.
10. Evaporation and Antievaporation Instabilities
Directory of Open Access Journals (Sweden)
2017-10-01
Full Text Available We review (antievaporation phenomena within the context of quantum gravity and extended theories of gravity. The (antievaporation effect is an instability of the black hole horizon discovered in many different scenarios: quantum dilaton-gravity, f ( R -gravity, f ( T -gravity, string-inspired black holes, and brane-world cosmology. Evaporating and antievaporating black holes seem to have completely different thermodynamical features compared to standard semiclassical black holes. The purpose of this review is to provide an introduction to conceptual and technical aspects of (antievaporation effects, while discussing problems that are still open.
11. Streamer chamber: pion decay
CERN Multimedia
1992-01-01
The real particles produced in the decay of a positive pion can be seen in this image from a streamer chamber. Streamer chambers consist of a gas chamber through which a strong pulsed electric field is passed, creating sparks as a charged particle passes through it. A magnetic field is added to cause the decay products to follow curved paths so that their charge and momentum can be measured.
12. Aspects of B decays
Energy Technology Data Exchange (ETDEWEB)
Faller, Sven
2011-03-04
B-meson decays are a good probe for testing the flavour sector of the standard model of particle physics. The standard model describes at present all experimental data satisfactorily, although some ''tensions'' exist, i.e. two to three sigma deviations from the predictions, in particular in B decays. The arguments against the standard model are thus purely theoretical. These tensions between experimental data and theoretical predictions provide an extension of the standard model by new physics contributions. Within the flavour sector main theoretical uncertainties are related to the hadronic matrix elements. For exclusive semileptonic anti B {yields} D{sup (*)}l anti {nu} decays QCD sum rule techniques, which are suitable for studying hadronic matrix elements, however, with substantial, but estimable hadronic uncertainties, are used. The exploration of new physics effects in B-meson decays is done in an twofold way. In exclusive semileptonic anti B {yields} D{sup (*)}l anti {nu} decays the effect of additional right-handed vector as well as left- and right-handed scalar and tensor hadronic current structures in the decay rates and the form factors are studied at the non-recoil point. As a second approach one studied the non-leptonic B{sup 0}{sub s}{yields}J/{psi}{phi} and B{sup 0}{yields}J/{psi}K{sub S,L} decays discussing CP violating effects in the time-dependent decay amplitudes by considering new physics phase in the B{sup 0}- anti B{sup 0} mixing phase. (orig.)
13. Tau decays into kaons
International Nuclear Information System (INIS)
Finkemeier, M.; Mirkes, E.
1995-04-01
Predictions for semi-leptonic decay rates of the τ lepton into two meson final states and three meson final states are derived. The hadronic matrix elements are expressed in terms of form factors, which can be predicted by chiral Lagrangians supplemented by informations about all possible low-lying resonances in the different channels. Isospin symmetry relations among the different final states are carefully taken into account. The calculated brancing ratios are compared with measured decay rates where data are available
14. Iconic Decay in Schizophrenia
OpenAIRE
Hahn, Britta; Kappenman, Emily S.; Robinson, Benjamin M.; Fuller, Rebecca L.; Luck, Steven J.; Gold, James M.
2010-01-01
Working memory impairment is considered a core deficit in schizophrenia, but the precise nature of this deficit has not been determined. Multiple lines of evidence implicate deficits at the encoding stage. During encoding, information is held in a precategorical sensory store termed iconic memory, a literal image of the stimulus with high capacity but rapid decay. Pathologically increased iconic decay could reduce the number of items that can be transferred into working memory before the info...
15. Annihilation decays of bottomonium
International Nuclear Information System (INIS)
Monteiro, Antony Prakash; Bhat, Manjunath; D'Souza, Praveen P.; Vijaya Kumar, K.B.
2016-01-01
The bound state of a bottom quark b and its anti quark b-bar known as bottomonium was first seen in the spectrum of μμ"- pairs produced in 400 GeV proton-nucleus collisions at Fermilab. It was discovered as spin triplet states ϒ(1S), ϒ(2S) and ϒ(3S) by E288 collaboration at Fermilab. We have calculated annihilation decay widths of bottomonium states. The calculated decay widths are presented
16. Resonant Drag Instabilities in protoplanetary disks: the streaming instability and new, faster-growing instabilities
Science.gov (United States)
Squire, Jonathan; Hopkins, Philip F.
2018-04-01
We identify and study a number of new, rapidly growing instabilities of dust grains in protoplanetary disks, which may be important for planetesimal formation. The study is based on the recognition that dust-gas mixtures are generically unstable to a Resonant Drag Instability (RDI), whenever the gas, absent dust, supports undamped linear modes. We show that the "streaming instability" is an RDI associated with epicyclic oscillations; this provides simple interpretations for its mechanisms and accurate analytic expressions for its growth rates and fastest-growing wavelengths. We extend this analysis to more general dust streaming motions and other waves, including buoyancy and magnetohydrodynamic oscillations, finding various new instabilities. Most importantly, we identify the disk "settling instability," which occurs as dust settles vertically into the midplane of a rotating disk. For small grains, this instability grows many orders of magnitude faster than the standard streaming instability, with a growth rate that is independent of grain size. Growth timescales for realistic dust-to-gas ratios are comparable to the disk orbital period, and the characteristic wavelengths are more than an order of magnitude larger than the streaming instability (allowing the instability to concentrate larger masses). This suggests that in the process of settling, dust will band into rings then filaments or clumps, potentially seeding dust traps, high-metallicity regions that in turn seed the streaming instability, or even overdensities that coagulate or directly collapse to planetesimals.
17. Rare psi decays
International Nuclear Information System (INIS)
Partridge, R.
1986-01-01
Slightly more than ten years have passed since the psi was discovered, yet the study of psi decays continues to be an active and fruitful area of research. One reason for such longevity is that each successive experiment has increased their sensitivity over previous experiments either by improving detection efficiency or by increasing statistics. This has allowed the observation and, in some cases, detailed studies of rare psi decays. Branching ratios of ≅10-/sup 4/ are now routinely studied, while certain decay channels are beginning to show interesting effects at the 10-/sup 5/ level. Future experiments at the Beijing Electron Positron Collider (BEPC) have the potential for increasing sensitivities by one or two orders of magnitude, thus enabling many interesting studies impossible with current data samples. The author first examines the extent to which psi decays can be used to study electroweak phenomena. The remainder of this work is devoted to the more traditional task of using the psi to study quarks, gluons, and the properties of the strong interaction. Of particular interest is the study of radioactive psi decays, where a number of new particles have been discovered. Recent results regarding two of these particles, the θ(1700) and iota(1450), are discussed, as well as a study of the quark content of the eta and eta' using decays of the psi to vector-pseudoscalar final states
18. Decays of supernova neutrinos
International Nuclear Information System (INIS)
Lindner, Manfred; Ohlsson, Tommy; Winter, Walter
2002-01-01
Supernova neutrinos could be well-suited for probing neutrino decay, since decay may be observed even for very small decay rates or coupling constants. We will introduce an effective operator framework for the combined description of neutrino decay and neutrino oscillations for supernova neutrinos, which can especially take into account two properties: one is the radially symmetric neutrino flux, allowing a decay product to be re-directed towards the observer even if the parent neutrino had a different original direction of propagation. The other is decoherence because of the long baselines for coherently produced neutrinos. We will demonstrate how to use this effective theory to calculate the time-dependent fluxes at the detector. In addition, we will show the implications of a Majoron-like decay model. As a result, we will demonstrate that for certain parameter values one may observe some effects which could also mimic signals similar to the ones expected from supernova models, making it in general harder to separate neutrino and supernova properties
19. Rare and forbidden decays
CERN Document Server
Trampetic, Josip
2002-01-01
In these lectures I first cover radiative and semileptonic B decays, including the QCD corrections for the quark subprocesses. The exclusive modes and the evaluation of the hadronic matrix elements, i.e. the relevant hadronic form factors, are the second step. Small effects due to the long-distance, spectator contributions, etc. are discussed next. The second section we started with non-leptonic decays, typically $B \\to \\pi\\pi, K\\pi, \\rho\\pi,...$ We describe in more detail our prediction for decays dominated by the $b\\to s \\eta_c$ transition. Reports on the most recent experimental results are given at the end of each subsection. In the second part of the lectures I discuss decays forbidden by the Lorentz and gauge invariance, and due to the violation of the angular moment conservation, generally called the Standard Model-forbiden decays. However, the non-commutative QED and/or non-commutative Standard Model (NCSM), developed in a series of works in the last few years allow some of those decay modes. These ar...
20. Topological instability of a semi-bounded magnetic fluid drop under influence of magnetic and ultrasound fields
Energy Technology Data Exchange (ETDEWEB)
Bashtovoi, V., E-mail: [email protected] [Belarussian National Technical University, 65 Nezavisimosti Ave., Minsk 220013 (Belarus); Reks, A. [Belarussian National Technical University, 65 Nezavisimosti Ave., Minsk 220013 (Belarus); Baev, A. [Institute of Applied Physics of NAS of Belarus, 16 Akademicheskaya str., Minsk 220072 (Belarus); Mansoor, Al-Jhaish Taha Malik [Belarussian National Technical University, 65 Nezavisimosti Ave., Minsk 220013 (Belarus)
2017-06-01
Theoretical and experimental results on deformation and disintegration on parts (topological instability) of semi-bounded magnetic fluid drop placed on horizontal plate in the presence of gravity and vertical external uniform magnetic field, and the influence of acoustic wave on these processes, as well as an experimental results of acoustic fountain on free surface of magnetic fluid are presented. The role of individual mechanisms leading to disintegration is analyzed, and analytical relationships and experimental dependences for critical parameters are established.
1. Feedback stabilization of plasma instabilities
International Nuclear Information System (INIS)
Cap, F.F.
1977-01-01
This paper reviews the theoretical and experimental aspects of feedback stabilization. After giving an outline of a general theoretical model for electrostatic instabilities the author provides a theoretical analysis of the suppression of various types of instability. Experiments which have been carried out on the feedback stabilization of various types of plasma instability are reported. An extensive list of references is given. (B.R.H.)
2. Thermal Shrinkage for Shoulder Instability
OpenAIRE
Toth, Alison P.; Warren, Russell F.; Petrigliano, Frank A.; Doward, David A.; Cordasco, Frank A.; Altchek, David W.; O’Brien, Stephen J.
2010-01-01
Thermal capsular shrinkage was popular for the treatment of shoulder instability, despite a paucity of outcomes data in the literature defining the indications for this procedure or supporting its long-term efficacy. The purpose of this study was to perform a clinical evaluation of radiofrequency thermal capsular shrinkage for the treatment of shoulder instability, with a minimum 2-year follow-up. From 1999 to 2001, 101 consecutive patients with mild to moderate shoulder instability underwent...
3. Political Instability and Economic Growth
OpenAIRE
Alberto Alesina; Sule Ozler; Nouriel Roubini; Phillip Swagel
1992-01-01
This paper investigates the relationship between political instability and per capita GDP growth in a sample of 113 countries for the period 1950-1982. We define ?political instability? as the propensity of a government collapse, and we estimate a model in which political instability and economic growth are jointly determined. The main result of this paper is that in countries and time periods with a high propensity of government collapse, growth is significantly lower than otherwise. This ef...
4. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... 30041 770-205-8211 [email protected] The world’s #1 acoustic neuroma resource Click to learn more... ... is acoustic neuroma? Diagnosing Symptoms Side Effects Keywords World Language Videos Questions to ask Choosing a healthcare ...
Science.gov (United States)
Ballard, Kenny
2010-01-01
This presentation reviews the status of the acoustic equipment from the medical operations perspective. Included is information about the acoustic dosimeters, sound level meter, and headphones that are planned for use while on orbit. Finally there is information about on-orbit hearing assessments.
6. Instability of a planar expansion wave.
Science.gov (United States)
Velikovich, A L; Zalesak, S T; Metzler, N; Wouchuk, J G
2005-10-01
An expansion wave is produced when an incident shock wave interacts with a surface separating a fluid from a vacuum. Such an interaction starts the feedout process that transfers perturbations from the rippled inner (rear) to the outer (front) surface of a target in inertial confinement fusion. Being essentially a standing sonic wave superimposed on a centered expansion wave, a rippled expansion wave in an ideal gas, like a rippled shock wave, typically produces decaying oscillations of all fluid variables. Its behavior, however, is different at large and small values of the adiabatic exponent gamma. At gamma > 3, the mass modulation amplitude delta(m) in a rippled expansion wave exhibits a power-law growth with time alpha(t)beta, where beta = (gamma - 3)/(gamma - 1). This is the only example of a hydrodynamic instability whose law of growth, dependent on the equation of state, is expressed in a closed analytical form. The growth is shown to be driven by a physical mechanism similar to that of a classical Richtmyer-Meshkov instability. In the opposite extreme gamma - 1 gas with low . Exact analytical expressions for the growth rates are derived for both cases and favorably compared to hydrodynamic simulation results.
7. Raman sidescatter instability in a nonuniform plasma
International Nuclear Information System (INIS)
Mostrom, M.A.
1977-01-01
In the various laser-fusion concepts, an intense electromagnetic wave (the laser) must propagate through an under-dense plasma region where it could decay, via the stimulated Raman instability, into a Langmuir plasma wave and a scattered electromagnetic wave. This process could, therefore, scatter a significant fraction of the laser energy before it could be deposited in the plasma. A density gradient, in the direction of laser incidence, localizes the instability to a narrow resonance zone where the local plasma wave frequency approximately equals the difference-frequency between the incident and scattered electromagnetic waves. The narrowness of this zone can strongly inhibit the growth of back- or oblique-scattered electromagnetic waves since they quickly propagate out of their resonance region; however, the density gradient has a much weaker effect on side-scattered waves (which propagate perpendicular to the density gradient) since they remain in their resonance zone until refraction bends them out or they exit through the side of the finite diameter laser beam. Thus, we place particular emphasis on evaluating, in a manner valid for the side scattered electromagnetic waves (which are at their turning point), the level of exponentiation at which the growth is linearly saturated due to convection of the waves out of their resonance zone. We also determine the general nature and propagation of the scattered electromagnetic waves and obtain approximate values for the resonance zone size and the time required for the above saturation
8. Dust acoustic waves in a dc glow-discharge plasma
International Nuclear Information System (INIS)
Molotkov, V.I.; Nefedov, A.P.; Torchinskii, V.M.; Fortov, V.E.; Khrapak, A.G.
1999-01-01
The spontaneous excitation of low-frequency oscillations of the macroparticle density in ordered dust structures levitating in standing striations of a dc glow discharge is discovered. It is concluded on the basis of a simplified linear model of an ideal collisionless plasma that the observed instability is caused by the drift motion of ions relative to the dust, which leads to the excitation of dust acoustic oscillations of the plasma
9. Electron acoustic vortices in the presence of inhomogeneous current
Energy Technology Data Exchange (ETDEWEB)
Haque, Q; Masood, W; Saleem, H [Theoretical Plasma Physics Division, PINSTECH, P O Nilore, Islamabad (Pakistan)], E-mail: [email protected]
2008-03-15
Linear and nonlinear dynamics of an electron acoustic wave in an inhomogeneous magnetized plasma are studied in the presence of non-uniform background current. The modified Rayleigh instability condition is found due to shear in the magnetic field and the current. A nonlinear stationary solution is also obtained in the form of tripolar vortices. The relevance of the present study to auroral and magnetotail plasmas is pointed out.
10. Instabilities in mimetic matter perturbations
Energy Technology Data Exchange (ETDEWEB)
Firouzjahi, Hassan; Gorji, Mohammad Ali [School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran (Iran, Islamic Republic of); Mansoori, Seyed Ali Hosseini, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Physics Department, Shahrood University of Technology, P.O. Box 3619995161 Shahrood (Iran, Islamic Republic of)
2017-07-01
We study cosmological perturbations in mimetic matter scenario with a general higher derivative function. We calculate the quadratic action and show that both the kinetic term and the gradient term have the wrong sings. We perform the analysis in both comoving and Newtonian gauges and confirm that the Hamiltonians and the associated instabilities are consistent with each other in both gauges. The existence of instabilities is independent of the specific form of higher derivative function which generates gradients for mimetic field perturbations. It is verified that the ghost instability in mimetic perturbations is not associated with the higher derivative instabilities such as the Ostrogradsky ghost.
11. Acoustic Signals and Systems
DEFF Research Database (Denmark)
2008-01-01
The Handbook of Signal Processing in Acoustics will compile the techniques and applications of signal processing as they are used in the many varied areas of Acoustics. The Handbook will emphasize the interdisciplinary nature of signal processing in acoustics. Each Section of the Handbook...... will present topics on signal processing which are important in a specific area of acoustics. These will be of interest to specialists in these areas because they will be presented from their technical perspective, rather than a generic engineering approach to signal processing. Non-specialists, or specialists...... from different areas, will find the self-contained chapters accessible and will be interested in the similarities and differences between the approaches and techniques used in different areas of acoustics....
12. Computational Ocean Acoustics
CERN Document Server
Jensen, Finn B; Porter, Michael B; Schmidt, Henrik
2011-01-01
Since the mid-1970s, the computer has played an increasingly pivotal role in the field of ocean acoustics. Faster and less expensive than actual ocean experiments, and capable of accommodating the full complexity of the acoustic problem, numerical models are now standard research tools in ocean laboratories. The progress made in computational ocean acoustics over the last thirty years is summed up in this authoritative and innovatively illustrated new text. Written by some of the field's pioneers, all Fellows of the Acoustical Society of America, Computational Ocean Acoustics presents the latest numerical techniques for solving the wave equation in heterogeneous fluid–solid media. The authors discuss various computational schemes in detail, emphasizing the importance of theoretical foundations that lead directly to numerical implementations for real ocean environments. To further clarify the presentation, the fundamental propagation features of the techniques are illustrated in color. Computational Ocean A...
13. Calibration of acoustic emission transducers
International Nuclear Information System (INIS)
Leschek, W.C.
1976-01-01
A method is described for calibrating an acoustic emission transducer to be used in a pre-set frequency range. The absolute reception sensitivity of a reference transducer is determined at frequencies selected within the frequency range. The reference transducer and the acoustic emission transducer are put into acoustic communication with the surface of a limited acoustic medium representing an equivalent acoustic load appreciably identical to that of the medium in which the use of the acoustic emission transducer is intended. A blank random acoustic noise is emitted in the acoustic medium in order to establish a diffuse and reverberating sound field, after which the output responses of the reference transducer and of the acoustic emission transducer are obtained with respect to the diffuse and reverberating field, for selected frequencies. The output response of the acoustic emission transducer is compared with that of the reference transducer for the selected frequencies, so as to determine the reception sensitivity of the acoustic emission transducer [fr
14. Neutron decay, semileptonic hyperon decay and the Cabibbo model
International Nuclear Information System (INIS)
Siebert, H.W.
1989-01-01
The decay rates and formfactor ratios of neutron decay and semileptonic hyperon decays are compared in the framework of the Cabibbo model. The results indicate SU(3) symmetry breaking. The Kobayashi-Maskawa matrix element V us determined from these decays is in good agreement with the value determined from K→πeν decays, and with unitarity of the KM-matrix. (orig.)
15. Instability characteristics of fluidelastic instability of tube rows in crossflow
International Nuclear Information System (INIS)
Chen, S.S.; Jendrzejczyk, J.A.
1986-04-01
An experimental study is reported to investigate the jump phenomenon in critical flow velocities for tube rows with different pitch-to-diameter ratios and the excited and intrinsic instabilities for a tube row with a pitch-to-diameter ratio of 1.75. The experimental data provide additional insights into the instability phenomena of tube arrays in crossflow. 9 refs., 10 figs
16. Parametric Room Acoustic workflows with real-time acoustic simulation
DEFF Research Database (Denmark)
Parigi, Dario
2017-01-01
The paper investigates and assesses the opportunities that real-time acoustic simulation offer to engage in parametric acoustics workflow and to influence architectural designs from early design stages......The paper investigates and assesses the opportunities that real-time acoustic simulation offer to engage in parametric acoustics workflow and to influence architectural designs from early design stages...
17. CP violation in B decay
OpenAIRE
Yamamoto, Hitoshi
2001-01-01
We review the physics of CP violation in B decays. After introducing the CKM matrix and how it causes CP violation, we cover three types of CP violation that can occur in B decays: CP violation in mixing, CP violation by mixing-decay interference, and CP violation in decay.
18. Radioactive decay and labeled compounds
International Nuclear Information System (INIS)
Anon.
1991-01-01
This chapter on radioactive decay and labeled compounds has numerous intext equations and worked, sample problems. Topics covered include the following: terms and mathematics of radioactive decay; examples of calculations; graphs of decay equations; radioactivity or activity; activity measurements; activity decay; half-life determinations; labeled compounds. A 20 problem set is also included. 1 ref., 4 figs., 1 tab
19. Strength loss in decayed wood
Science.gov (United States)
Rebecca E. Ibach; Patricia K. Lebow
2014-01-01
Wood is a durable engineering material when used in an appropriate manner, but it is susceptible to biological decay when a log, sawn product, or final product is not stored, handled, or designed properly. Even before the biological decay of wood becomes visually apparent, the decay can cause the wood to become structurally unsound. The progression of decay to that...
20. Kinetic instabilities in relativistic plasmas: the Harris instability revisited
International Nuclear Information System (INIS)
Tautz, R.C.
2008-01-01
Plasma instabilities that generate aperiodic fluctuations are of outstanding importance in the astrophysical context. Two prominent examples are the electromagnetic Weibel instability and the electrostatic Harris instability, which operate in initially non-magnetized and magnetized plasmas, respectively. In this talk, the original formulation of the Harris instability will be reviewed and generalizations will be presented such as the inclusion of (1) relativistic effects, (2) ion effects, and (3) mode coupling. It will be shown that, with these modifications, a powerful method has been developed for the determination of both the existence and the growth rate of low-frequency instabilities. Applications can be found in astrophysical jets, where the rest frame can be used and so no parallel motion is present. At the end of the talk, how the particle composition of gamma-ray burst jets can be predicted using the Harris technique. (author)
1. Control of thermoacoustic instability with a drum-like silencer
Science.gov (United States)
Zhang, Guangyu; Wang, Xiaoyu; Li, Lei; Jing, Xiaodong; Sun, Xiaofeng
2017-10-01
Theoretical investigation is carried out by a novel method of controlling thermoacoustic instability with a drum-like silencer. It is shown that by decreasing the frequency of thermoacoustic system, the instability can be suppressed with the help of drum-like silencer. The purely reactive silencer, which is composed of a flexible membrane and a backing cavity, is usually known as a noise control device that works effectively in low frequency bandwidth without any aerodynamic loss. In present research, the silencer is exploited in a Rijke tube, as a means of decreasing the natural frequency of the system, and consequently changing the resonance period of the system. The "transfer element method" (TEM) is used to consider the interactions between the acoustic waves and the flexible membranes of the silencer. The effects of all possible properties of the silencer on the growth rate and resonance frequency of the thermoacoustic system are explored. According to the calculation results, it is found that for some properties of the silencer, the resonance frequencies are greatly decreased and then the phase difference between the unsteady heat release and the pressure fluctuation is increased. Consequently, the instability is suppressed with some dissipation that can not be able to control its onset in the original system. Therefore, when the damping is low, but not zero, it is effective to control thermoacoustic instability with this technique.
2. Acoustic Emission Analysis Applet (AEAA) Software
Science.gov (United States)
Nichols, Charles T.; Roth, Don J.
2013-01-01
NASA Glenn Research and NASA White Sands Test Facility have developed software supporting an automated pressure vessel structural health monitoring (SHM) system based on acoustic emissions (AE). The software, referred to as the Acoustic Emission Analysis Applet (AEAA), provides analysts with a tool that can interrogate data collected on Digital Wave Corp. and Physical Acoustics Corp. software using a wide spectrum of powerful filters and charts. This software can be made to work with any data once the data format is known. The applet will compute basic AE statistics, and statistics as a function of time and pressure (see figure). AEAA provides value added beyond the analysis provided by the respective vendors' analysis software. The software can handle data sets of unlimited size. A wide variety of government and commercial applications could benefit from this technology, notably requalification and usage tests for compressed gas and hydrogen-fueled vehicles. Future enhancements will add features similar to a "check engine" light on a vehicle. Once installed, the system will ultimately be used to alert International Space Station crewmembers to critical structural instabilities, but will have little impact to missions otherwise. Diagnostic information could then be transmitted to experienced technicians on the ground in a timely manner to determine whether pressure vessels have been impacted, are structurally unsound, or can be safely used to complete the mission.
3. Nonlinear Longitudinal Mode Instability in Liquid Propellant Rocket Engine Preburners
Science.gov (United States)
Sims, J. D. (Technical Monitor); Flandro, Gary A.; Majdalani, Joseph; Sims, Joseph D.
2004-01-01
Nonlinear pressure oscillations have been observed in liquid propellant rocket instability preburner devices. Unlike the familiar transverse mode instabilities that characterize primary combustion chambers, these oscillations appear as longitudinal gas motions with frequencies that are typical of the chamber axial acoustic modes. In several respects, the phenomenon is similar to longitudinal mode combustion instability appearing in low-smoke solid propellant motors. An important feature is evidence of steep-fronted wave motions with very high amplitude. Clearly, gas motions of this type threaten the mechanical integrity of associated engine components and create unacceptably high vibration levels. This paper focuses on development of the analytical tools needed to predict, diagnose, and correct instabilities of this type. For this purpose, mechanisms that lead to steep-fronted, high-amplitude pressure waves are described in detail. It is shown that such gas motions are the outcome of the natural steepening process in which initially low amplitude standing acoustic waves grow into shock-like disturbances. The energy source that promotes this behavior is a combination of unsteady combustion energy release and interactions with the quasi-steady mean chamber flow. Since shock waves characterize the gas motions, detonation-like mechanisms may well control the unsteady combustion processes. When the energy gains exceed the losses (represented mainly by nozzle and viscous damping), the waves can rapidly grow to a finite amplitude limit cycle. Analytical tools are described that allow the prediction of the limit cycle amplitude and show the dependence of this wave amplitude on the system geometry and other design parameters. This information can be used to guide corrective procedures that mitigate or eliminate the oscillations.
4. Naturally enhanced ion-acoustic spectra and their interpretation
DEFF Research Database (Denmark)
Sedgemore-Schulthess, K.J.F.; St. Maurice, J.P.
2001-01-01
acceleration, wave-particle and wave-wave interactions in the ionosphere, and their association with magnetospheric processes. There is now a substantial body of literature documenting observations of enhanced ion-acoustic spectra, but there remains controversy over generation mechanisms. We present a review...... years there has been much interest in naturally occurring (as opposed to artificially stimulated) enhanced ion-acoustic spectra seen in the auroral zone and cusp/cleft region. A study of the plasma instability processes that lead to such spectra will help us to better understand auroral particle...... of literature documenting observations of naturally enhanced ion-acoustic spectra, observed mainly along the geomagnetic field direction, along with a discussion of the theories put forward to explain such phenomena....
5. Sigma beta decay
International Nuclear Information System (INIS)
Newman, D.E.
1975-01-01
Describes an experiment to measure beta decays of the sigma particle. Sigmas produced by stopping a K - beam in a liquid hydrogen target decayed in the following reactions: Kp → Σπ; Σ → Neν. The electron and pion were detected by wire spark chambers in a magnetic spectrometer and by plastic scintillators, and were differentiated by a threshold gas Cherenkov counter. The neutron was detected by liquid scintillation counters. The data (n = 3) shell electrons or the highly excited electrons decay first. Instead, it is suggested that when there are two to five electrons in highly excited states immediately after a heavy ion--atom collision the first transitions to occur will be among highly excited Rydberg states in a cascade down to the 4s, 4p, and 3d-subshells. If one of the long lived states becomes occupied by electrons promoted during the collision or by electrons falling from higher levels, it will not decay until after the valence shell decays. LMM rates calculated to test the methods used are compared to previous works. The mixing coefficients are given in terms of the states 4s4p, 45sp+-, and 5s5p. The applicability of Cooper, Fano, and Prats' discussion of the energies and transition rates of doubly excited states is considered
6. Transmission acoustic microscopy investigation
Science.gov (United States)
Maev, Roman; Kolosov, Oleg; Levin, Vadim; Lobkis, Oleg
The nature of acoustic contrast, i.e. the connection of the amplitude and phase of the output signal of the acoustic microscope with the local values of the acoustic parameters of the sample (density, elasticity, viscosity) is a central problem of acoustic microscopy. A considerable number of studies have been devoted to the formation of the output signal of the reflection scanning acoustic microscope. For the transmission acoustic microscope (TAM) this problem has remained almost unstudied. Experimental investigation of the confocal system of the TAM was carried out on an independently manufactured laboratory mockup of the TAM with the working frequency of the 420 MHz. Acoustic lenses with the radius of curvature of about 500 microns and aperture angle of 45 deg were polished out in the end faces of two cylindrical sound conductors made from Al2O3 single crystals with an axis parallel to the axis C of the crystal (the length of the sound conductor is 20 mm; diameter, 6 mm). At the end faces of the sound conductor, opposite to the lenses, CdS transducers with a diameter of 2 mm were disposed. The electric channel of the TAM provided a possibility for registering the amplitude of the microscope output signal in the case of the dynamic range of the 50 dB.
7. The accidental (acoustical) tourist
Science.gov (United States)
Van Kirk, Wayne
2002-11-01
The acoustical phenomenon observed at an ancient temple in the Great Ball Court at Chichen Itza was described as ''little short of amazing--an ancient whispering gallery'' by Silvanus G. Morley, leader of the Carnegie Institute's archaeological team that excavated and restored these structures in the 1920s. Since then, many others have experienced the extraordinary acoustics at Chichen Itza and other Maya sites. Despite these reports, archaeologists and acousticians have until recently shown little interest in understanding these phenomena. After experiencing Chichen Itza's remarkable acoustics as a tourist in 1994, the author commenced collecting and disseminating information about acoustical phenomena there and at other Mayan sites, hoping to stimulate interest among archaeologists and acousticians. Were these designs accidental or intentional? If intentional, how was the knowledge obtained? How were acoustical features used? This paper highlights the author's collection of anecdotal reports of mysterious Mayan acoustics (http://http://www.ianlawton.com/pa1.htm), recommended reading for scientists and engineers who wish to pursue this fascinating study. Also recounted are some of the reactions of archaeologists-ranging from curious, helpful, and insightful to humorous and appalling--to outsiders' efforts to bring serious scientific attention to the new field of acoustical archaeology.
8. Translational illusion of acoustic sources by transformation acoustics.
Science.gov (United States)
Sun, Fei; Li, Shichao; He, Sailing
2017-09-01
An acoustic illusion of creating a translated acoustic source is designed by utilizing transformation acoustics. An acoustic source shifter (ASS) composed of layered acoustic metamaterials is designed to achieve such an illusion. A practical example where the ASS is made with naturally available materials is also given. Numerical simulations verify the performance of the proposed device. The designed ASS may have some applications in, e.g., anti-sonar detection.
9. Instabilities in the aether
International Nuclear Information System (INIS)
Carroll, Sean M.; Dulaney, Timothy R.; Gresham, Moira I.; Tam, Heywood
2009-01-01
We investigate the stability of theories in which Lorentz invariance is spontaneously broken by fixed-norm vector 'aether' fields. Models with generic kinetic terms are plagued either by ghosts or by tachyons, and are therefore physically unacceptable. There are precisely three kinetic terms that are not manifestly unstable: a sigma model (∂ μ A ν ) 2 , the Maxwell Lagrangian F μν F μν , and a scalar Lagrangian (∂ μ A μ ) 2 . The timelike sigma-model case is well defined and stable when the vector norm is fixed by a constraint; however, when it is determined by minimizing a potential there is necessarily a tachyonic ghost, and therefore an instability. In the Maxwell and scalar cases, the Hamiltonian is unbounded below, but at the level of perturbation theory there are fewer degrees of freedom and the models are stable. However, in these two theories there are obstacles to smooth evolution for certain choices of initial data.
10. Posterolateral elbow joint instability
DEFF Research Database (Denmark)
Olsen, Bo Sanderhoff; Søjbjerg, Jens Ole; Nielsen, K K
1998-01-01
Thirty-five osteoligamentous elbows were included in a study on the kinematics of posterolateral elbow joint instability during the pivot shift test (PST) before and after separate ligament cuttings in the lateral collateral ligament complex (LCLC). Division of the annular ligament or the lateral...... ulnar collateral ligament caused no laxity during the PST. Division of the lateral collateral ligament caused maximal laxity of 4 degrees and 23 degrees during forced PST in valgus and external rotation (supination), respectively. Cutting of the LCLC at the ulnar or the humeral insertion was necessary...... for any PST stressed elbow joint laxity to occur. Total division of the LCLC induced a maximal laxity of 7.9 degrees and 37 degrees during forced PST in valgus and external rotation (supination), respectively. This study suggests the lateral collateral ligament to be the primary soft tissue constraint...
11. Instabilities in electromagnetic quasilevitation.
Science.gov (United States)
Spragg, Kirk; Letout, Sebastien; Ernst, R; Sneyd, Alfred; Fautrelle, Yves
2014-05-01
We investigate free-surface instabilities occurring in various industrial processes involving liquid metal. Of particular interest is the behavior of the free surface of a pool of liquid metal when it is submitted to an alternating magnetic field. Experimentally, we study the effect of a vertical alternating medium-frequency magnetic field on an initially circular pool. We observe various types of behavior according to magnetic field amplitude, e.g., axisymmetric deformations, azimuthal mode structures, slow radial oscillation of the pool perimeter, and random rotation of the pool around its center. Drop rotation could be attributed to nonsymmetric shape deformations. The effect of oxidation leads to drastic changes in pool behavior. The experimental results are then compared to a linear stability analysis of the free surface of a circular liquid drop.
12. Acoustic building infiltration measurement system
Science.gov (United States)
Muehleisen, Ralph T.; Raman, Ganesh
2018-04-10
Systems and methods of detecting and identifying a leak from a container or building. Acoustic pressure and velocity are measured. Acoustic properties are acquired from the measured values. The acoustic properties are converted to infiltration/leakage information. Nearfield Acoustic Holography (NAH) may be one method to detect the leakages from a container by locating the noise sources.
13. Saturation of equatorial inertial instability
NARCIS (Netherlands)
Kloosterziel, R.C.; Orlandi, P.; Carnevale, G.F.
2015-01-01
Inertial instability in parallel shear flows and circular vortices in a uniformly rotating system ( $f$f-plane) redistributes absolute linear momentum or absolute angular momentum in such a way as to neutralize the instability. In previous studies we showed that, in the absence of other
14. Internal rotor friction instability
Science.gov (United States)
Walton, J.; Artiles, A.; Lund, J.; Dill, J.; Zorzi, E.
1990-01-01
The analytical developments and experimental investigations performed in assessing the effect of internal friction on rotor systems dynamic performance are documented. Analytical component models for axial splines, Curvic splines, and interference fit joints commonly found in modern high speed turbomachinery were developed. Rotor systems operating above a bending critical speed were shown to exhibit unstable subsynchronous vibrations at the first natural frequency. The effect of speed, bearing stiffness, joint stiffness, external damping, torque, and coefficient of friction, was evaluated. Testing included material coefficient of friction evaluations, component joint quantity and form of damping determinations, and rotordynamic stability assessments. Under conditions similar to those in the SSME turbopumps, material interfaces experienced a coefficient of friction of approx. 0.2 for lubricated and 0.8 for unlubricated conditions. The damping observed in the component joints displayed nearly linear behavior with increasing amplitude. Thus, the measured damping, as a function of amplitude, is not represented by either linear or Coulomb friction damper models. Rotordynamic testing of an axial spline joint under 5000 in.-lb of static torque, demonstrated the presence of an extremely severe instability when the rotor was operated above its first flexible natural frequency. The presence of this instability was predicted by nonlinear rotordynamic time-transient analysis using the nonlinear component model developed under this program. Corresponding rotordynamic testing of a shaft with an interference fit joint demonstrated the presence of subsynchronous vibrations at the first natural frequency. While subsynchronous vibrations were observed, they were bounded and significantly lower in amplitude than the synchronous vibrations.
15. Ion cyclotron instability saturation and turbulent plasma heating in the presence of ions moving across the magnetic field
International Nuclear Information System (INIS)
Mikhajlenko, V.S.; Stepanov, K.N.
1981-01-01
Ion cyclotron instability saturation is considered in terms of the turbulence theory when there is a beam of heavy ions with large thermal longitudinal velocity spread. The instability excitation is due to a cyclotron interaction with ions of the beam under the anomalous Doppler effect. The instability is shown to be saturated due to an induced plasma ion scattering of ion cyclotron waves when the beam ion charge number Zsub(b) is approximately 1. Decay processes, wave scattering by virtual wave polarization clouds and resonance broadening due to random walk of plasma ions in turbulent instability fields appear to be unimportant. For Zsub(b)>>1 the induced wave scattering by the beam ions is the main process determining the nonlinear stage of the instability. Estimates are given for the oscillation energy density in the instability saturation state and for the turbulent heating rate of plasma and beam ions [ru
16. Saturation regime of the collisionless drift instability in a hydrogen plasma column
International Nuclear Information System (INIS)
Boissier, R.
1982-09-01
The saturation regime of the collisionless drift instability is observed in a steady state hydrogen column. The steady state parameters are observed to relax around the average values. A quasilinear model is proposed to describe the dynamics of wave growth and density gradient decay
17. Large eddy simulation and combustion instabilities; Simulation des grandes echelles et instabilites de combustion
Energy Technology Data Exchange (ETDEWEB)
Lartigue, G.
2004-11-15
The new european laws on pollutants emission impose more and more constraints to motorists. This is particularly true for gas turbines manufacturers, that must design motors operating with very fuel-lean mixtures. Doing so, pollutants formation is significantly reduced but the problem of combustion stability arises. Actually, combustion regimes that have a large excess of air are naturally more sensitive to combustion instabilities. Numerical predictions of these instabilities is thus a key issue for many industrial involved in energy production. This thesis work tries to show that recent numerical tools are now able to predict these combustion instabilities. Particularly, the Large Eddy Simulation method, when implemented in a compressible CFD code, is able to take into account the main processes involved in combustion instabilities, such as acoustics and flame/vortex interaction. This work describes a new formulation of a Large Eddy Simulation numerical code that enables to take into account very precisely thermodynamics and chemistry, that are essential in combustion phenomena. A validation of this work will be presented in a complex geometry (the PRECCINSTA burner). Our numerical results will be successfully compared with experimental data gathered at DLR Stuttgart (Germany). Moreover, a detailed analysis of the acoustics in this configuration will be presented, as well as its interaction with the combustion. For this acoustics analysis, another CERFACS code has been extensively used, the Helmholtz solver AVSP. (author)
18. Understanding Aero-Fractures using optics and acoustics
Science.gov (United States)
Turkaya, Semih; Toussaint, Renaud; Kvalheim Eriksen, Fredrik; Zecevic, Megan; Daniel, Guillaume; Grude Flekkøy, Eirik; Jørgen Måløy, Knut
2016-04-01
In this work, analogue models are developed in a linear geometry, with confinement and at low porosity to study the instabilities that develop during fast motion of fluid in dense porous materials: fracturing, fingering, and channeling. We study these complex fluid/solid mechanical systems using two imaging techniques: optical imaging using a high speed camera (1000 fps) and high frequency resolution accelerometers. Additionally, we develop physical models rendering for the fluid mechanics in the channels and the propagation of microseismic waves around the fracture. We then compare a numerical resolution of this physical system with the observed experimental system. The experimental setup consists of a rectangular Hele-Shaw cell with three closed boundaries and one semi-permeable boundary which enables the flow of the fluid but not the solid particles. During the experiments, the fluid is injected into the system, with a constant injection pressure, from the point opposite to the semi-permeable boundary. At large enough injection pressures, the fluid also displaces grains and creates large channels and thin fractures towards the semi-permeable boundary. In the analysis phase, we compute the power spectrum of the acoustic signal in time windows of 5 ms, recorded by shock accelerometers Brüel & Kjaer 4374 (Frq. Range 1 Hz - 26 kHz) with 1 MHz sampling rate. The evolution of the power spectrum is compared with the optical recordings. The power spectrum initially follows a power law trend and when the channel network is developed, stick-slip events generating peaks with characteristic frequencies at 10, 30, 60 and 180 kHz are seen. These peaks are strongly influenced by the size and branching of the channels, compaction of the medium, vibration of air in the pores and the fundamental frequency of the plate. Furthermore, the number of these stick-slip events, similar to the data obtained in hydraulic fracturing operations, follows a Modified Omori Law decay with an
19. Dynamical Instability and Soliton Concept
International Nuclear Information System (INIS)
Kartavenko, V.G.
1994-01-01
The problem of dynamical instability and clustering (stable fragments formation) in a breakup of excited nuclear systems are considered from the points of view of the soliton concept. It is noted that the volume (spinodal) instability can be associated with nonlinear terms, and the surface (Rayleigh-Taylor type) instability, with the dispersion terms in the evolution equations. The spinodal instability and the Rayleigh-Taylor instability may compensate each other and lead to stable quasi-soliton type objects. The simple analytical model is presented to illustrate this physical picture. The time evolution of an initially compressed cold nuclear system is analysed in the framework of the inverse mean-field method. It is demonstrated that the nonlinearity and dispersion terms of the evolution equations can lead to clusterization in the final channel. 8 p
20. Beam instability Workshop - plenary sessions
International Nuclear Information System (INIS)
2001-01-01
The purpose of this workshop was to provide a review of the mechanisms of limiting beam instabilities, their cures, including feedback, and beam measurement for synchrotron radiation light sources. 12 plenary sessions took place whose titles are: 1) challenging brilliance and lifetime issues with increasing currents; 2) limiting instabilities in multibunch; 3) experience from high currents in B factories; 4) longitudinal dynamics in high intensity/bunch; 5) Transverse instabilities for high intensity/bunch; 6) working group introduction from ESRF experience; 7) impedance modelling: simulations, minimization; 8) report on the broadband impedance measurements and modelling workshop; 9) feedback systems for synchrotron light sources; 10) beam instabilities diagnostics; 11) harmonic cavities: the pros and cons; and 12) experimental study of fast beam-ion instabilities at PLS. This document gathers the 12 articles that were presented during these sessions
1. Beam instability Workshop - plenary sessions
Energy Technology Data Exchange (ETDEWEB)
NONE
2001-07-01
The purpose of this workshop was to provide a review of the mechanisms of limiting beam instabilities, their cures, including feedback, and beam measurement for synchrotron radiation light sources. 12 plenary sessions took place whose titles are: 1) challenging brilliance and lifetime issues with increasing currents; 2) limiting instabilities in multibunch; 3) experience from high currents in B factories; 4) longitudinal dynamics in high intensity/bunch; 5) Transverse instabilities for high intensity/bunch; 6) working group introduction from ESRF experience; 7) impedance modelling: simulations, minimization; 8) report on the broadband impedance measurements and modelling workshop; 9) feedback systems for synchrotron light sources; 10) beam instabilities diagnostics; 11) harmonic cavities: the pros and cons; and 12) experimental study of fast beam-ion instabilities at PLS. This document gathers the 12 articles that were presented during these sessions.
2. Acoustical heat pumping engine
Science.gov (United States)
Wheatley, J.C.; Swift, G.W.; Migliori, A.
1983-08-16
The disclosure is directed to an acoustical heat pumping engine without moving seals. A tubular housing holds a compressible fluid capable of supporting an acoustical standing wave. An acoustical driver is disposed at one end of the housing and the other end is capped. A second thermodynamic medium is disposed in the housing near to but spaced from the capped end. Heat is pumped along the second thermodynamic medium toward the capped end as a consequence both of the pressure oscillation due to the driver and imperfect thermal contact between the fluid and the second thermodynamic medium. 2 figs.
3. Deep Water Acoustics
Science.gov (United States)
2016-06-28
the Deep Water project and participate in the NPAL Workshops, including Art Baggeroer (MIT), J. Beron- Vera (UMiami), M. Brown (UMiami), T...Kathleen E . Wage. The North Pacific Acoustic Laboratory deep-water acoustic propagation experiments in the Philippine Sea. J. Acoust. Soc. Am., 134(4...estimate of the angle α during PhilSea09, made from ADCP measurements at the site of the DVLA. Sim. A B1 B2 B3 C D E F Prof. # 0 4 4 4 5 10 16 20 α
4. N+ρ decay of baryons in a flux-tube-breaking mechanism
International Nuclear Information System (INIS)
Stassart, P.; Stancu, F.
1990-01-01
A flux-tube-breaking mechanism motivated by QCD is extended to the analysis of the decay of nonstrange resonances into N+ρ. A proper threshold behavior is obtained by taking into account the instability of the ρ meson. The only parameter of the model has previously been fixed to adjust the decay of Δ into N+π. We find a good agreement with the few available data and make predictions for many other resonances where data are needed
5. Beta and muon decays
International Nuclear Information System (INIS)
Galindo, A.; Pascual, P.
1967-01-01
These notes represent a series of lectures delivered by the authors in the Junta de Energia Nuclear, during the Spring term of 1965. They were devoted to graduate students interested in the Theory of Elementary Particles. Special emphasis was focussed into the computational problems. Chapter I is a review of basic principles (Dirac equation, transition probabilities, final state interactions.) which will be needed later. In Chapter II the four-fermion punctual Interaction is discussed, Chapter III is devoted to the study of beta-decay; the main emphasis is given to the deduction of the formulae corresponding to electron-antineutrino correlation, electron energy spectrum, lifetimes, asymmetry of electrons emitted from polarized nuclei, electron and neutrino polarization and time reversal invariance in beta decay. In Chapter IV we deal with the decay of polarized muons with radiative corrections. Chapter V is devoted to an introduction to C.V.C. theory. (Author)
6. Decay of superdeformed bands
International Nuclear Information System (INIS)
Carpenter, M.P.; Khoo, T.L.; Lauritsen, T.
1995-01-01
One of the major challenges in the study of superdeformation is to directly connect the large number of superdeformed bands now known to the yrast states. In this way, excitation energies, spins and parities can be assigned to the levels in the second well which is essential to establish the collective and single-particle components of these bands. This paper will review some of the progress which has been made to understand the decay of superdeformed bands using the new arrays including the measurement of the total decay spectrum and the establishment of direct one-step decays from the superdeformed band to the yrast line in 194 Hg. 42 refs., 5 figs
7. Beta and muon decays
Energy Technology Data Exchange (ETDEWEB)
Galindo, A; Pascual, P
1967-07-01
These notes represent a series of lectures delivered by the authors in the Junta de Energia Nuclear, during the Spring term of 1965. They were devoted to graduate students interested in the Theory of Elementary Particles. Special emphasis was focussed into the computational problems. Chapter I is a review of basic principles (Dirac equation, transition probabilities, final state interactions.) which will be needed later. In Chapter II the four-fermion punctual Interaction is discussed, Chapter III is devoted to the study of beta-decay; the main emphasis is given to the deduction of the formulae corresponding to electron-antineutrino correlation, electron energy spectrum, lifetimes, asymmetry of electrons emitted from polarized nuclei, electron and neutrino polarization and time reversal invariance in beta decay. In Chapter IV we deal with the decay of polarized muons with radiative corrections. Chapter V is devoted to an introduction to C.V.C. theory. (Author)
8. Suppressed Charmed B Decay
Energy Technology Data Exchange (ETDEWEB)
Snoek, Hella Leonie [Vrije Univ., Amsterdam (Netherlands)
2009-06-02
This thesis describes the measurement of the branching fractions of the suppressed charmed B0 → D*- a0+ decays and the non-resonant B0 → D*- ηπ+ decays in approximately 230 million Υ(4S) → B$\\bar{B}$ events. The data have been collected with the BABAR detector at the PEP-II B factory at the Stanford Linear Accelerator Center in California. Theoretical predictions of the branching fraction of the B0 → D*- a{sub 0}+ decays show large QCD model dependent uncertainties. Non-factorizing terms, in the naive factorization model, that can be calculated by QCD factorizing models have a large impact on the branching fraction of these decay modes. The predictions of the branching fractions are of the order of 10-6. The measurement of the branching fraction gives more insight into the theoretical models. In general a better understanding of QCD models will be necessary to conduct weak interaction physics at the next level. The presence of CP violation in electroweak interactions allows the differentiation between matter and antimatter in the laws of physics. In the Standard Model, CP violation is incorporated in the CKM matrix that describes the weak interaction between quarks. Relations amongst the CKM matrix elements are used to present the two relevant parameters as the apex of a triangle (Unitarity Triangle) in a complex plane. The over-constraining of the CKM triangle by experimental measurements is an important test of the Standard Model. At this moment no stringent direct measurements of the CKM angle γ, one of the interior angles of the Unitarity Triangle, are available. The measurement of the angle γ can be performed using the decays of neutral B mesons. The B0 → D*- a0+ decay is sensitive to the angle γ and, in comparison to the current decays that are being employed, could significantly
9. Weak interactions: muon decay
International Nuclear Information System (INIS)
Sachs, A.M.; Sirlin, A.
1975-01-01
The traditional theory of the dominant mode of muon decay is presented, a survey of the experiments which have measured the observable features of the decay is given, and those things which can be learned about the parameters and nature of the theory from the experimental results are indicated. The following aspects of the theory of muon decay are presented first: general four-fermion theory, two-component theory of the neutrino, V--A theory, two-component and V--A theories vs general four-fermion theory, intermediate-boson hypothesis, radiative corrections, radiative corrections in the intermediate-boson theory, and endpoint singularities and corrections of order α 2 . Experiments on muon lifetime, isotropic electron spectrum, total asymmetry and energy dependence of asymmetry of electrons from polarized muons, and electron polarization are described, and a summary of experimental results is given. 7 figures, 2 tables, 109 references
10. Radiation acoustics and its applications
International Nuclear Information System (INIS)
Lyamshev, L.M.
1992-01-01
Radiation acoustics is a new branch of acoustics, developing on the boundary of acoustics, nuclear physics, elementary particles and high-energy physics. Its fundamentals are laying in the research of acoustical effects due to the interaction of penetrating radiation with matter. The study of radiation-acoustical effects leads to the new opportunities in the penetration radiation research (acoustical detection, radiation-acoustical dosimetry), study of the physical parameters of matter, in a solution of some applied problems of nondestructive testing, and also for the radiation-acoustical influence on physical and chemical structure of the matter. Results of theoretical and experimental investigations are given. Different mechanisms of the sound generation by penetrating radiation of liquids and solids are considered. Some applications - the radiation acoustical microscopy and visualisation, the acoustical detection of high energy X-ray particles and possibility of using of high energy neutrino beams in geoacoustics - are discussed
11. Acoustic pollution in hospital environments
International Nuclear Information System (INIS)
Olivera, J M; Rocha, L A; Rotger, V I; Herrera, M C
2011-01-01
There are many different services within a hospital. This means different types of noise which can be considered as acoustic pollution. Knowing that preterm infants exposed to high amounts of noise in the NICU are at a much higher risk because of their neurologic immaturity and physiologic instability, that excessive levels of noise also affect the persons and it can also impede some studies on patients, it was proposed to evaluate the Sound Pressure Level in some services of the Instituto de Maternidad, Tucumán, Argentina. There were evaluated the Level III NICU, the laundry service, a physical space destined for a service of evoked potential and a neonatal incubator under working conditions. The measurements were performed with a type II sonometer (CENTER 322) and it was also used an incubator analyzer (FLUKE INCU) for the incubator. The average values obtained were of 63.6 dBA for the NICU, 82.5dBA for the laundry room, 52.7 dBA for the evoked potential room and 62.8 dBA in the inside of the incubator under 64 dBA in the outside. The reports were documented in compliance with the appropriate standards.
12. Size Effect on Acoustic Emission Characteristics of Coal-Rock Damage Evolution
Directory of Open Access Journals (Sweden)
Zhijie Wen
2017-01-01
Full Text Available Coal-gas outburst, rock burst, and other mine dynamic disasters are closely related to the instability and failure of coal-rock. Coal-rock is the assemblies of mineral particles of varying sizes and shapes bonded together by cementing materials. The damage and rupture process of coal-rock is accompanied by acoustic emission (AE, which can be used as an effective means to monitor and predict the instability of coal-rock body. In this manuscript, considering the size effect of coal-rock, the influence of different height to diameter ratio on the acoustic emission characteristics of coal-rock damage evolution was discussed by microparticle flow PFC2D software platform. The results show that coal-rock size influences the uniaxial compressive strength, peak strain, and elastic modulus of itself; the size effect has little effect on the acoustic emission law of coal-rock damage and the effects of the size of coal-rock samples on acoustic emission characteristics are mainly reflected in three aspects: the triggering time of acoustic emission, the strain range of strong acoustic emission, and the intensity of acoustic emission; the damage evolution of coal-rock specimen can be divided into 4 stages: initial damage, stable development, accelerated development, and damage.
13. Gravitational Instabilities in Circumstellar Disks
Science.gov (United States)
Kratter, Kaitlin; Lodato, Giuseppe
2016-09-01
Star and planet formation are the complex outcomes of gravitational collapse and angular momentum transport mediated by protostellar and protoplanetary disks. In this review, we focus on the role of gravitational instability in this process. We begin with a brief overview of the observational evidence for massive disks that might be subject to gravitational instability and then highlight the diverse ways in which the instability manifests itself in protostellar and protoplanetary disks: the generation of spiral arms, small-scale turbulence-like density fluctuations, and fragmentation of the disk itself. We present the analytic theory that describes the linear growth phase of the instability supplemented with a survey of numerical simulations that aim to capture the nonlinear evolution. We emphasize the role of thermodynamics and large-scale infall in controlling the outcome of the instability. Despite apparent controversies in the literature, we show a remarkable level of agreement between analytic predictions and numerical results. In the next part of our review, we focus on the astrophysical consequences of the instability. We show that the disks most likely to be gravitationally unstable are young and relatively massive compared with their host star, Md/M*≥0.1. They will develop quasi-stable spiral arms that process infall from the background cloud. Although instability is less likely at later times, once infall becomes less important, the manifestations of the instability are more varied. In this regime, the disk thermodynamics, often regulated by stellar irradiation, dictates the development and evolution of the instability. In some cases the instability may lead to fragmentation into bound companions. These companions are more likely to be brown dwarfs or stars than planetary mass objects. Finally, we highlight open questions related to the development of a turbulent cascade in thin disks and the role of mode-mode coupling in setting the maximum angular
14. Thermal shrinkage for shoulder instability.
Science.gov (United States)
Toth, Alison P; Warren, Russell F; Petrigliano, Frank A; Doward, David A; Cordasco, Frank A; Altchek, David W; O'Brien, Stephen J
2011-07-01
Thermal capsular shrinkage was popular for the treatment of shoulder instability, despite a paucity of outcomes data in the literature defining the indications for this procedure or supporting its long-term efficacy. The purpose of this study was to perform a clinical evaluation of radiofrequency thermal capsular shrinkage for the treatment of shoulder instability, with a minimum 2-year follow-up. From 1999 to 2001, 101 consecutive patients with mild to moderate shoulder instability underwent shoulder stabilization surgery with thermal capsular shrinkage using a monopolar radiofrequency device. Follow-up included a subjective outcome questionnaire, discussion of pain, instability, and activity level. Mean follow-up was 3.3 years (range 2.0-4.7 years). The thermal capsular shrinkage procedure failed due to instability and/or pain in 31% of shoulders at a mean time of 39 months. In patients with unidirectional anterior instability and those with concomitant labral repair, the procedure proved effective. Patients with multidirectional instability had moderate success. In contrast, four of five patients with isolated posterior instability failed. Thermal capsular shrinkage has been advocated for the treatment of shoulder instability, particularly mild to moderate capsular laxity. The ease of the procedure makes it attractive. However, our retrospective review revealed an overall failure rate of 31% in 80 patients with 2-year minimum follow-up. This mid- to long-term cohort study adds to the literature lacking support for thermal capsulorrhaphy in general, particularly posterior instability. The online version of this article (doi:10.1007/s11420-010-9187-7) contains supplementary material, which is available to authorized users.
15. Instability timescale for the inclination instability in the solar system
Science.gov (United States)
Zderic, Alexander; Madigan, Ann-Marie; Fleisig, Jacob
2018-04-01
The gravitational influence of small bodies is often neglected in the study of solar system dynamics. However, this is not always an appropriate assumption. For example, mutual secular torques between low mass particles on eccentric orbits can result in a self-gravity instability (inclination instability'; Madigan & McCourt 2016). During the instability, inclinations increase exponentially, eccentricities decrease (detachment), and orbits cluster in argument of perihelion. In the solar system, the orbits of the most distant objects show all three of these characteristics (high inclination: Volk & Malhotra (2017), detachment: Delsanti & Jewitt (2006), and argument of perihelion clustering: Trujillo & Sheppard (2014)). The inclination instability is a natural explanation for these phenomena.Unfortunately, full N-body simulations of the solar system are unfeasible (N ≈ O(1012)), and the behavior of the instability depends on N, prohibiting the direct application of lower N simulations. Here we present the instability timescale's functional dependence on N, allowing us to extrapolate our simulation results to that appropriate for the solar system. We show that ~5 MEarth of small icy bodies in the Sedna region is sufficient for the inclination instability to occur in the outer solar system.
16. A real space calculation of absolutely unstable modes for two-plasmon decay in inhomogeneous plasma
International Nuclear Information System (INIS)
Powers, L.V.; Berger, R.L.
1986-01-01
Growth rates for absolute modes of two-plasmon decay are obtained by solving for eigenmodes of the coupled mode equations for obliquely scattered Langmuir waves in real space. This analysis establishes a connection both to previous analysis in Fourier transform space and to other parametric instabilities, the analysis of which is commonly done in real space. The essential feature of the instability which admits absolute modes in an inhomogeneous plasma is the strong spatial dependence of the coupling coefficients. Landau damping limits the perpendicular wavenumbers of the most unstable modes and raises the instability thresholds for background plasma temperatures above 1 keV. (author)
17. Sequential decay of Reggeons
International Nuclear Information System (INIS)
Yoshida, Toshihiro
1981-01-01
Probabilities of meson production in the sequential decay of Reggeons, which are formed from the projectile and the target in the hadron-hadron to Reggeon-Reggeon processes, are investigated. It is assumed that pair creation of heavy quarks and simultaneous creation of two antiquark-quark pairs are negligible. The leading-order terms with respect to ratio of creation probabilities of anti s s to anti u u (anti d d) are calculated. The production cross sections in the target fragmentation region are given in terms of probabilities in the initial decay of the Reggeons and an effect of manyparticle production. (author)
18. Do protons decay
International Nuclear Information System (INIS)
Litchfield, P.J.
1984-09-01
The experimental status of proton decay is reviewed after the Leipzig International conference, July 1984. A brief comparative description of the currently active experiments is given. From the overall samples of contained events it can be concluded that the experiments are working well and broadly agree with each other. The candidates for proton decay from each experiment are examined. Although several experiments report candidates at a higher rate than expected from background calculations, the validity of these calculations is still open to doubt. (author)
19. 103Pd decay
International Nuclear Information System (INIS)
Belyavenko, V.S.; Borozenets, G.P.; Vishnevskij, I.N.; Zheltonozhskij, V.A.
1986-01-01
103 Pd decay in different chemical states has been investigated. The change of the partial half-life period equal to 0.67±0.15% has been detected. The γ-spectrum has been measured to a high precision. The new data have been obtained on population probabilities of 103 Rh excited states and the total energy of decay for 103 Pd has been determined to a high precision (543.0±0.8). The values of log ft have been determined
20. Decay of 99Mo
International Nuclear Information System (INIS)
Dickens, J.K.; Love, T.A.
1976-01-01
Relative intensities for K x-rays and gamma rays emanating from 99 Mo in equilibrium with its 99 Tc* daughter have been measured using several Ge photon detectors. Combining these intensities with an evaluated set of electron-conversion coefficients has provided a set of absolute intensities for the observed gamma rays. The absolute intensity for the dominant 140.5-keV gamma ray in 99 Tc was determined to be 90.7 +- 0.6/100 99 Mo disintegrations for 99 Mo decay in equilibrium with decay of the 99 Tc* daughter
1. Supersymmetry in Z' decays
International Nuclear Information System (INIS)
Corcella, G.
2014-01-01
I study the phenomenology of new heavy neutral gauge bosons Z', predicted by Grand Unification Theories-driven U(1)' gauge groups and by the sequential standard model. BSM (Beyond Standard Model) decays into supersymmetric final states are accounted for, besides the SM (Standard Model) modes usually investigated. I give an estimate of the number of supersymmetric events in Z' decays possibly expected at LHC, as well as of the product of the Z' cross section times the branching fraction into electron and muon pairs. (author)
2. Functional Instability of the Ankle Joint: Etiopathogenesis
Directory of Open Access Journals (Sweden)
Aydan ÖRSÇELİK
2016-09-01
Full Text Available Ankle sprain is one of the most common sports injuries. Chronic ankle instability is a common complication of ankle sprains. Two causes of chronic ankle instability are mechanical instability and functional instability. It is important to understand functional instability etiopathogenesis of the ankle joint in order to guide diagnosis and treatment. This article aims to understand the etiopathogenesis of functional ankle instability.
3. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... a healthcare provider Request a patient kit Treatment Options Overview Observation Radiation Surgery What is acoustic neuroma Diagnosing Symptoms Side effects Question To Ask Treatment Options Back Overview Observation Radiation Surgery Choosing a healthcare ...
4. Acoustic-Levitation Chamber
Science.gov (United States)
Barmatz, M. B.; Granett, D.; Lee, M. C.
1984-01-01
Uncontaminated environments for highly-pure material processing provided within completely sealed levitation chamber that suspends particles by acoustic excitation. Technique ideally suited for material processing in low gravity environment of space.
5. Acoustic Casimir Effect
National Research Council Canada - National Science Library
Homes, Christopher
1997-01-01
...). When the indirect manifestations of the ZPF are interpreted as due to radiation pressure, acoustic noise can provide an excellent analog to investigate the Casimir effect as well as other effects due to the ZPF...
6. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... a patient kit Treatment Options Overview Observation Radiation Surgery What is acoustic neuroma Diagnosing Symptoms Side effects ... To Ask Treatment Options Back Overview Observation Radiation Surgery Choosing a healthcare provider Request a patient kit ...
7. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Choosing a healthcare provider Request a patient kit Treatment Options Overview Observation Radiation Surgery What is acoustic neuroma Diagnosing Symptoms Side effects Question To Ask Treatment Options Back Overview Observation Radiation Surgery Choosing a ...
8. Acoustic ambient noise recorder
Digital Repository Service at National Institute of Oceanography (India)
Saran, A.K.; Navelkar, G.S.; Almeida, A.M.; More, S.R.; Chodankar, P.V.; Murty, C.S.
with a robust outfit that can withstand high pressures and chemically corrosion resistant materials. Keeping these considerations in view, a CMOS micro-controller-based marine acoustic ambient noise recorder has been developed with a real time clock...
9. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Learn more about ANA About ANA Mission, Vision & Values Shop ANA Leadership & Staff Annual Reports Acoustic Neuroma ... 8211 [email protected] About ANA Mission, Vision & Values Leadership & Staff Annual Reports Shop ANA Home Learn ...
10. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... ANA About ANA Mission, Vision & Values Shop ANA Leadership & Staff Annual Reports Acoustic Neuroma Association 600 Peachtree ... [email protected] About ANA Mission, Vision & Values Leadership & Staff Annual Reports Shop ANA Home Learn Educational ...
11. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... patient kit Treatment Options Overview Observation Radiation Surgery What is acoustic neuroma Diagnosing ... Back Community Patient Stories Share Your Story Video Stories Caregivers Milestones Gallery Submit Your Milestone Team ANA Volunteer ...
12. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Connections Overview Find a Meeting Host a Meeting Volunteer Become a Volunteer Opportunities Support Overview Patient Events ... ANA About ANA Mission, Vision & Values Shop ANA Leadership & Staff Annual Reports Acoustic Neuroma Association 600 Peachtree ...
Data.gov (United States)
Federal Laboratory Consortium — FUNCTION: Collects underwater acoustic data and oceanographic data. Data are recorded onboard an ocean buoy and can be telemetered to a remote ship or shore station...
14. Acoustic MIMO signal processing
CERN Document Server
Huang, Yiteng; Chen, Jingdong
2006-01-01
A timely and important book addressing a variety of acoustic signal processing problems under multiple-input multiple-output (MIMO) scenarios. It uniquely investigates these problems within a unified framework offering a novel and penetrating analysis.
15. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Back Learn more about ANA About ANA Mission, Vision & Values Shop ANA Leadership & Staff Annual Reports Acoustic ... 205-8211 [email protected] About ANA Mission, Vision & Values Leadership & Staff Annual Reports Shop ANA Home ...
16. Thermal Acoustic Fatigue Apparatus
Data.gov (United States)
Federal Laboratory Consortium — The Thermal Acoustic Fatigue Apparatus (TAFA) is a progressive wave tube test facility that is used to test structures for dynamic response and sonic fatigue due to...
17. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Spanish Washington Support Group Leslie of Stone Mountain, ... Providers Acoustic Neuroma Association Donate Now Newly Diagnosed What is AN? Request a Patient Kit Treatment Options Get Support Find a Provider Discussion Forum ...
18. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Mission, Vision & Values Shop ANA Leadership & Staff Annual Reports Acoustic Neuroma Association 600 Peachtree Parkway Suite 108 ... About ANA Mission, Vision & Values Leadership & Staff Annual Reports Shop ANA Home Learn Educational Video English English ...
19. Acoustic Igniter, Phase I
Data.gov (United States)
National Aeronautics and Space Administration — An acoustic igniter eliminates the need to use electrical energy to drive spark systems to initiate combustion in liquid-propellant rockets. It does not involve the...
20. Department of Cybernetic Acoustics
Science.gov (United States)
The development of the theory, instrumentation and applications of methods and systems for the measurement, analysis, processing and synthesis of acoustic signals within the audio frequency range, particularly of the speech signal and the vibro-acoustic signal emitted by technical and industrial equipments treated as noise and vibration sources was discussed. The research work, both theoretical and experimental, aims at applications in various branches of science, and medicine, such as: acoustical diagnostics and phoniatric rehabilitation of pathological and postoperative states of the speech organ; bilateral ""man-machine'' speech communication based on the analysis, recognition and synthesis of the speech signal; vibro-acoustical diagnostics and continuous monitoring of the state of machines, technical equipments and technological processes.
1. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... 1 acoustic neuroma resource Click to learn more... LOGIN CALENDAR DONATE NEWS Home Learn Back Learn about ... Webinar Library Newsletter Library Patient Info Booklets Member Login Research ANA Survey/Registry AN Research Patient Registry ...
2. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... About ANA Mission, Vision & Values Shop ANA Leadership & Staff Annual Reports Acoustic Neuroma Association 600 Peachtree Parkway ... ANAUSA.org About ANA Mission, Vision & Values Leadership & Staff Annual Reports Shop ANA Home Learn Educational Video ...
3. Acoustic Neuroma Educational Video
Medline Plus
Full Text Available ... Click to learn more... LOGIN CALENDAR DONATE NEWS Home Learn Back Learn about acoustic neuroma AN Facts ... Vision & Values Leadership & Staff Annual Reports Shop ANA Home Learn Educational Video English English Arabic Catalan Chinese ( ...
4. Acoustic imaging system
Science.gov (United States)
Smith, Richard W.
1979-01-01
An acoustic imaging system for displaying an object viewed by a moving array of transducers as the array is pivoted about a fixed point within a given plane. A plurality of transducers are fixedly positioned and equally spaced within a laterally extending array and operatively directed to transmit and receive acoustic signals along substantially parallel transmission paths. The transducers are sequentially activated along the array to transmit and receive acoustic signals according to a preestablished sequence. Means are provided for generating output voltages for each reception of an acoustic signal, corresponding to the coordinate position of the object viewed as the array is pivoted. Receptions from each of the transducers are presented on the same display at coordinates corresponding to the actual position of the object viewed to form a plane view of the object scanned.
5. Principles of musical acoustics
CERN Document Server
Hartmann, William M
2013-01-01
Principles of Musical Acoustics focuses on the basic principles in the science and technology of music. Musical examples and specific musical instruments demonstrate the principles. The book begins with a study of vibrations and waves, in that order. These topics constitute the basic physical properties of sound, one of two pillars supporting the science of musical acoustics. The second pillar is the human element, the physiological and psychological aspects of acoustical science. The perceptual topics include loudness, pitch, tone color, and localization of sound. With these two pillars in place, it is possible to go in a variety of directions. The book treats in turn, the topics of room acoustics, audio both analog and digital, broadcasting, and speech. It ends with chapters on the traditional musical instruments, organized by family. The mathematical level of this book assumes that the reader is familiar with elementary algebra. Trigonometric functions, logarithms and powers also appear in the book, but co...
6. Unstable decay and state selection
International Nuclear Information System (INIS)
McKane, Alan; Tarlie, Martin
2001-01-01
The decay of unstable states when several metastable states are available for occupation is investigated using path-integral techniques. Specifically, a method is described that enables the probabilities with which the metastable states are occupied to be calculated by finding optimal paths, and fluctuations about them, in the weak-noise limit. The method is illustrated on a system described by two coupled Langevin equations, which are found in the study of instabilities in fluid dynamics and superconductivity. The problem involves a subtle interplay between nonlinearities and noise, and a naive approximation scheme that does not take this into account is shown to be unsatisfactory. The use of optimal paths is briefly reviewed and then applied to finding the conditional probability of ending up in one of the metastable states, having begun in the unstable state. There are several aspects of the calculation that distinguish it from most others involving optimal paths: (i) the paths do not begin and end on an attractor, and moreover, the final point is to a large extent arbitrary, (ii) the interplay between the fluctuations and the leading-order contribution are at the heart of the method, and (iii) the final result involves quantities that are not exponentially small in the noise strength. This final result, which gives the probability of a particular state being selected in terms of the parameters of the dynamics, is remarkably simple and agrees well with the results of numerical simulations. The method should be applicable to similar problems in a number of other areas, such as state selection in lasers, activationless chemical reactions, and population dynamics in fluctuating environments
7. Anal acoustic reflectometry
DEFF Research Database (Denmark)
Mitchell, Peter J; Klarskov, Niels; Telford, Karen J
2011-01-01
Anal acoustic reflectometry is a new technique of assessing anal sphincter function. Five new variables reflecting anal canal function are measured: the opening and closing pressure, the opening and closing elastance, and hysteresis.......Anal acoustic reflectometry is a new technique of assessing anal sphincter function. Five new variables reflecting anal canal function are measured: the opening and closing pressure, the opening and closing elastance, and hysteresis....
8. Enhanced ion acoustic fluctuations and ion outflows
Directory of Open Access Journals (Sweden)
F. R. E. Forme
1999-02-01
Full Text Available A number of observations showing enhanced ion acoustic echoes observed by means of incoherent scatter radars have been reported in the literature. The received power is extremely enhanced by up to 1 or 2 orders of magnitude above usual values, and it is mostly contained in one of the two ion acoustic lines. This spectral asymmetry and the intensity of the received signal cannot be resolved by the standard analysis procedure and often causes its failure. As a result, and in spite of a very clear spectral signature, the analysis is unable to fit the plasma parameters inside the regions of ion acoustic turbulence. We present European Incoherent Scatter radar (EISCAT observations of large ion outflows associated with the simultaneous occurrence of enhanced ion acoustic echoes. The ion fluxes can reach 1014 m-2 s-1 at 800 km altitude. From the very clear spectral signatures of these echoes, a method is presented to extract estimates of the electron temperature and the ion drift within the turbulent regions. It is shown that the electron gas is strongly heated up to 11 000 K. Also electron temperature gradients of about 0.02 K/m exist. Finally, the estimates of the electron temperature and of the ion drift are used to study the possible implications for the plasma transport inside turbulent regions. It is shown that strong electron temperature gradients cause enhancement of the ambipolar electric field and can account for the observed ion outflows.Key words. Ionosphere (auroral ionosphere; ionosphere · magnetosphere interactions; plasma waves and instabilities.
9. Searching the beginning of BWR power instability events with the Hilbert Huang transform
International Nuclear Information System (INIS)
Blázquez, Juan; Montalvo, Cristina; García-Berrocal, Agustín; Balbás, Miguel
2013-01-01
Highlights: ► The report of the instability is enriched by including its beginning and its end. ► The Hilbert Huang transform (HHT) is used for indentifying both. ► The first Intrinsic Mode Function (IMF) detects both. ► The methodology is applied to neutron detector signals from two plants. ► The Decay Ratio of IMF 1 is calculated. - Abstract: When a BWR instability takes place, the Regulator usually demands a report which must include many aspects such as the initial time of the instability and also the measurements adopted by the operator at that time. This initial time normally is difficult to know from the available data. In this work, a methodology is proposed to determine accurately when the instability began based on the Hilbert–Huang transform. The Empirical Mode Decomposition is applied to neutron detector signals coming from two plants which have recorded them during real instability events. The first intrinsic mode function shows sharply the beginning and the end of the incident. Besides, through the instantaneous amplitude and frequency of the first mode a kind of Decay Ratio can be assigned allowing us to obtain a sharper description of the instability
10. Triton beta decay
International Nuclear Information System (INIS)
Saito, T.Y.; Wu, Y.; Ishikawa, S.; Sasakawa, T.
1990-01-01
Triton β-decay has been calculated using wave functions for 3 He and 3 H obtained from (Coulomb-modified) Faddeev equations for various interactions. We get a value for the Gamow-Teller matrix element of √3 (0.962±0.002) without regards to two- or three-nucleon inteactions. This value agrees with the experimental value. (orig.)
11. Unparticles and muon decay
International Nuclear Information System (INIS)
Choudhury, Debajyoti; Ghosh, Dilip Kumar; Mamta
2008-01-01
Recently Georgi has discussed the possible existence of 'Unparticles' describable by operators having non-integral scaling dimensions. With the interaction of these with the Standard Model particles being constrained only by gauge and Lorentz symmetries, it affords a new source for lepton flavour violation. Current and future muon decay experiments are shown to be very sensitive to such scenarios
12. Unparticles and muon decay
Energy Technology Data Exchange (ETDEWEB)
Choudhury, Debajyoti [Department of Physics and Astrophysics, University of Delhi, Delhi 110 007 (India); Ghosh, Dilip Kumar [Department of Physics and Astrophysics, University of Delhi, Delhi 110 007 (India)], E-mail: [email protected]; Mamta [Department of Physics, S.G.T.B. Khalsa College, University of Delhi, Delhi 110 007 (India)
2008-01-03
Recently Georgi has discussed the possible existence of 'Unparticles' describable by operators having non-integral scaling dimensions. With the interaction of these with the Standard Model particles being constrained only by gauge and Lorentz symmetries, it affords a new source for lepton flavour violation. Current and future muon decay experiments are shown to be very sensitive to such scenarios.
13. Gluons in quarkonium decay
International Nuclear Information System (INIS)
Koller, K.; Walsh, T.
1978-03-01
We discuss what can be learned of the 3 S 1 quarkonium decay QantiQ → 3 gluoans QantiQ → γ + 2 gluons. The former is a way to find gluon jets and test QCD. The latter also allows us to measure gluoan + gluon → hadrons and look for pure gluonic resonances (glueballs). (orig.) [de
14. Symmetry violating kaon decays
International Nuclear Information System (INIS)
Herczeg, P.
1979-01-01
An analysis of the muon number violating decay modes of the K-mesons is given. Subsequently, some new developments in the field of CP-violation are reviewed and the question of time-reversal invariance and the status of CPT-invariance are briefly considered. 42 references
15. Double Beta Decay Experiments
International Nuclear Information System (INIS)
Piepke, A.
2005-01-01
The experimental observation of neutrino oscillations and thus neutrino mass and mixing gives a first hint at new particle physics. The absolute values of the neutrino mass and the properties of neutrinos under CP-conjugation remain unknown. The experimental investigation of the nuclear double beta decay is one of the key techniques for solving these open problems
16. On the proton decay
International Nuclear Information System (INIS)
Fonda, L.; Ghirardi, G.C.; Weber, T.
1983-07-01
The problem of the proton decay is considered taking into account that in actual experiments there is an interaction of the proton with its environment which could imply an increase of its theoretical lifetime. It is seen that, by application of the time-energy uncertainty relation, no prolongation of the lifetime is obtained in this case. (author)
17. Cosmology with decaying particles
International Nuclear Information System (INIS)
Turner, M.S.
1984-09-01
We consider a cosmological model in which an unstable massive relic particle species (denoted by X) has an initial mass density relative to baryons β -1 identically equal rho/sub X//rho/sub B/ >> 1, and then decays recently (redshift z less than or equal to 1000) into particles which are still relativistic today (denoted by R). We write down and solve the coupled equations for the cosmic scale factor a(t), the energy density in the various components (rho/sub X/, rho/sub R/, rho/sub B/), and the growth of linear density perturbations (delta rho/rho). The solutions form a one parameter (β) family of solutions; physically β -1 approx. = (Ω/sub R//Ω/sub NR/) x (1 + z/sub D/) = (ratio today of energy density of relativistic to nonrelativistic particles) x (1 + redshift of (decay)). We discuss the observational implications of such a cosmological model and compare our results to earlier results computed in the simultaneous decay approximation. In an appendix we briefly consider the case where one of the decay products of the X is massive and becomes nonrelativistic by the present epoch. 21 references
18. Cosmology with decaying particles
Energy Technology Data Exchange (ETDEWEB)
Turner, M.S.
1984-09-01
We consider a cosmological model in which an unstable massive relic particle species (denoted by X) has an initial mass density relative to baryons ..beta../sup -1/ identically equal rho/sub X//rho/sub B/ >> 1, and then decays recently (redshift z less than or equal to 1000) into particles which are still relativistic today (denoted by R). We write down and solve the coupled equations for the cosmic scale factor a(t), the energy density in the various components (rho/sub X/, rho/sub R/, rho/sub B/), and the growth of linear density perturbations (delta rho/rho). The solutions form a one parameter (..beta..) family of solutions; physically ..beta../sup -1/ approx. = (..cap omega../sub R//..cap omega../sub NR/) x (1 + z/sub D/) = (ratio today of energy density of relativistic to nonrelativistic particles) x (1 + redshift of (decay)). We discuss the observational implications of such a cosmological model and compare our results to earlier results computed in the simultaneous decay approximation. In an appendix we briefly consider the case where one of the decay products of the X is massive and becomes nonrelativistic by the present epoch. 21 references.
19. Acoustic Levitation System
Science.gov (United States)
Gammell, P. M.; Wang, T. G.; Croonquist, A.; Lee, M. C.
1985-01-01
Dense materials, such as steel balls, continuously levitated with energy provided by efficient high-powered siren in combination with shaped reflector. Reflector system, consisting of curved top reflector and flat lower reflector, eliminates instability in spatial positioning of sample.
20. Peierls transition with acoustic phonons and twist deformation in carbon nanotubes
NARCIS (Netherlands)
Figge, M. T.; Mostovoy, M. V.; Knöster, J.
1999-01-01
Submitted to: Phys. Rev. Lett. Abstract: We consider the Peierls instability due to the interaction of electrons with both acoustic and optical phonons. We suggest that such a transition takes place in carbon nanotubes with small radius. The topological excitations and the temperature dependence of
1. A comphrehensive model for the amplification of acoustic pressure waves by single hole orifices
NARCIS (Netherlands)
Moussou, P.; Testud, Ph.; Auregan, Y.; Hirschberg, A.
2008-01-01
Using a parallel flow approximation, a simple model of hydrodynamic instability is proposed for describing the behavior of an orifice as an acoustic amplifier. It is shown that the growing of perturbations in the vena contracta can generate negative damping for Strouhal numbers of the order of
2. Classification of decays involving variable decay chains with convolutional architectures
CERN Multimedia
CERN. Geneva
2018-01-01
Vidyo contribution We present a technique to perform classification of decays that exhibit decay chains involving a variable number of particles, which include a broad class of $B$ meson decays sensitive to new physics. The utility of such decays as a probe of the Standard Model is dependent upon accurate determination of the decay rate, which is challenged by the combinatorial background arising in high-multiplicity decay modes. In our model, each particle in the decay event is represented as a fixed-dimensional vector of feature attributes, forming an $n \\times k$ representation of the event, where $n$ is the number of particles in the event and $k$ is the dimensionality of the feature vector. A convolutional architecture is used to capture dependencies between the embedded particle representations and perform the final classification. The proposed model performs outperforms standard machine learning approaches based on Monte Carlo studies across a range of variable final-state decays with the Belle II det...
3. Parametric instabilities in an electron beam plasma system
International Nuclear Information System (INIS)
Nakach, R.; Cuperman, S.; Gell, Y.; Levush, B.
1981-01-01
The excitation of low frequency parametric instabilities by a finite wave length pump in a system consisting of a warm electron plasma traversed by a warm electron beam is investigated in a fluid dissipationless model. The dispersion relation for the three-dimensional problem in a magnetized plasma with arbitrary directions for the waves is derived, and the one-dimensional case is analyzed numerically. For the one-dimensional back-scattering decay process, it is found that when the plasma-electron Debye length (lambda sub(D)sup(p)) is larger than the beam-electron Debye length (lambda sub(D)sup(b)), two low frequency electrostatic instability branches with different growth rates may simultaneously exist. When lambda sub(D)sup(p) approximately lambda sub(D)sup(b), the large growth rate instability found in the analysis depends strongly on the amplitude of the pump field. In the case (lambda sub(D)sup(p) < lambda sub(D)sup(b)) only one low frequency instability branch is generally excited
4. Effects of Various Architectural Parameters on Six Room Acoustical Measures in Auditoria.
Science.gov (United States)
Chiang, Wei-Hwa
The effects of architectural parameters on six room acoustical measures were investigated by means of correlation analyses, factor analyses and multiple regression analyses based on data taken in twenty halls. Architectural parameters were used to estimate acoustical measures taken at individual locations within each room as well as the averages and standard deviations of all measured values in the rooms. The six acoustical measures were Early Decay Time (EDT10), Clarity Index (C80), Overall Level (G), Bass Ratio based on Early Decay Time (BR(EDT)), Treble Ratio based on Early Decay Time (TR(EDT)), and Early Inter-aural Cross Correlation (IACC80). A comprehensive method of quantifying various architectural characteristics of rooms was developed to define a large number of architectural parameters that were hypothesized to effect the acoustical measurements made in the rooms. This study quantitatively confirmed many of the principles used in the design of concert halls and auditoria. Three groups of room architectural parameters such as the parameters associated with the depth of diffusing surfaces were significantly correlated with the hall standard deviations of most of the acoustical measures. Significant differences of statistical relations among architectural parameters and receiver specific acoustical measures were found between a group of music halls and a group of lecture halls. For example, architectural parameters such as the relative distance from the receiver to the overhead ceiling increased the percentage of the variance of acoustical measures that was explained by Barron's revised theory from approximately 70% to 80% only when data were taken in the group of music halls. This study revealed the major architectural parameters which have strong relations with individual acoustical measures forming the basis for a more quantitative method for advancing the theoretical design of concert halls and other auditoria. The results of this study provide
5. Acoustic calibration apparatus for calibrating plethysmographic acoustic pressure sensors
Science.gov (United States)
Zuckerwar, Allan J. (Inventor); Davis, David C. (Inventor)
1995-01-01
An apparatus for calibrating an acoustic sensor is described. The apparatus includes a transmission material having an acoustic impedance approximately matching the acoustic impedance of the actual acoustic medium existing when the acoustic sensor is applied in actual in-service conditions. An elastic container holds the transmission material. A first sensor is coupled to the container at a first location on the container and a second sensor coupled to the container at a second location on the container, the second location being different from the first location. A sound producing device is coupled to the container and transmits acoustic signals inside the container.
6. Instability Suppression in a Swirl-Stabilized Combustor Using Microjet Air Injection
KAUST Repository
LaBry, Zachary
2010-01-04
In this study, we examine the effectiveness of microjet air injection as a means of suppressing thermoacoustic instabilities in a swirl-stabilized, lean-premixed propane/air combustor. High-speed stereo PIV measurements, taken to explore the mechanism of combustion instability, reveal that the inner recirculation zone plays a dominant role in the coupling of acoustics and heat release that leads to combustion instability. Six microjet injector configurations were designed to modify the inner and outer recirculation zones with the intent of decoupling the mechanism leading to instability. Microjets that injected air into the inner recirculation zone, swirling in the opposite sense to the primary swirl were effective in suppressing combustion instability, reducing the overall sound pressure level by up to 17 dB within a certain window of operating conditions. Stabilization was achieved near an equivalence ratio of 0.65, corresponding to the region where the combustor transitions from a 40 Hz instability mode to a 110 Hz instability mode. PIV measurements made of the stabilized flow revealed significant modification of the inner recirculation zone and substantial weakening of the outer recirculation zone.
7. Positional instability analysis of tokamak plasmas by ERATO
International Nuclear Information System (INIS)
Kumagai, Michikazu; Tsunematsu, Toshihide; Tokuda, Shinji; Takeda, Tatsuoki
1983-06-01
The stability of axisymmetric modes of a tokamak plasma(positional instabilities) is analyzed for the Solov'ev equilibrium by using the linear ideal MHD code ERATO-J. The dependence of the stability on various parameters, i.e., the ellipticity and triangularity of the plasma cross-section, the aspect ratio, the safety factor at the magnetic axis, and the distance between the plasma and a conducting shell is investigated. Comparison of the results with those by the rigid model shows that the stability condition derived from the rigid model in terms of the decay index(n-index) of the external equilibrating field is a good approximation for the plasma with small triangular deformation. Also the results are compared with those of the rigid displacement model and applicability of the various models on the positional instability analyses is discussed. (author)
8. Resistive instabilities in tokamaks
International Nuclear Information System (INIS)
Rutherford, P.H.
1985-10-01
Low-m tearing modes constitute the dominant instability problem in present-day tokamaks. In this lecture, the stability criteria for representative current profiles with q(0)-values slightly less than unit are reviewed; ''sawtooth'' reconnection to q(0)-values just at, or slightly exceeding, unity is generally destabilizing to the m = 2, n = 1 and m = 3, n = 2 modes, and severely limits the range of stable profile shapes. Feedback stabilization of m greater than or equal to 2 modes by rf heating or current drive, applied locally at the magnetic islands, appears feasible; feedback by island current drive is much more efficient, in terms of the radio-frequency power required, then feedback by island heating. Feedback stabilization of the m = 1 mode - although yielding particularly beneficial effects for resistive-tearing and high-beta stability by allowing q(0)-values substantially below unity - is more problematical, unless the m = 1 ideal-MHD mode can be made positively stable by strong triangular shaping of the central flux surfaces. Feedback techniques require a detectable, rotating MHD-like signal; the slowing of mode rotation - or the excitation of non-rotating modes - by an imperfectly conducting wall is also discussed
9. Sheared Electroconvective Instability
Science.gov (United States)
Kwak, Rhokyun; Pham, Van Sang; Lim, Kiang Meng; Han, Jongyoon
2012-11-01
Recently, ion concentration polarization (ICP) and related phenomena draw attention from physicists, due to its importance in understanding electrochemical systems. Researchers have been actively studying, but the complexity of this multiscale, multiphysics phenomenon has been limitation for gaining a detailed picture. Here, we consider electroconvective(EC) instability initiated by ICP under pressure-driven flow, a scenario often found in electrochemical desalinations. Combining scaling analysis, experiment, and numerical modeling, we reveal unique behaviors of sheared EC: unidirectional vortex structures, its size selection and vortex propagation. Selected by balancing the external pressure gradient and the electric body force, which generates Hagen-Poiseuille(HP) flow and vortical EC, the dimensionless EC thickness scales as (φ2 /UHP)1/3. The pressure-driven flow(or shear) suppresses unfavorably-directed vortices, and simultaneously pushes favorably-directed vortices with constant speed, which is linearly proportional to the total shear of HP flow. This is the first systematic characterization of sheared EC, which has significant implications on the optimization of electrodialysis and other electrochemical systems.
10. Analysis of Wave Velocity Patterns in Black Cherry Trees and its Effect on Internal Decay Detection
Science.gov (United States)
Guanghui Li; Xiping Wang; Jan Wiedenbeck; Robert J. Ross
2013-01-01
In this study, we examined stress wave velocity patterns in the cross sections of black cherry trees, developed analytical models of stress wave velocity in sound healthy trees, and then tested the effectiveness of the models as a tool for tree decay diagnosis. Acoustic tomography data of the tree cross sections were collected from 12 black cherry trees at a production...
11. Treatment of early and late reflections in a hybrid computer model for room acoustics
DEFF Research Database (Denmark)
Naylor, Graham
1992-01-01
The ODEON computer model for acoustics in large rooms is intended for use both in design (by predicting room acoustical indices quickly and easily) and in research (by forming the basis of an auralization system and allowing study of various room acoustical phenomena). These conflicting demands...... preclude the use of both pure'' image source and pure'' particle tracing methods. A hybrid model has been developed, in which rays discover potential image sources up to a specified order. Thereafter, the same ray tracing process is used in a different way to rapidly generate a dense reverberant decay...
12. Rare B decays at LEP
CERN Document Server
Kluit, P M
2001-01-01
The results of the LEP experiments for rare B decays will be reviewed, covering hadronic final states, radiative and other rare decays and results for the inclusive charmless branching ratio. (8 refs).
13. Tunnelling instability via perturbation theory
Energy Technology Data Exchange (ETDEWEB)
Graffi, S. (Bologna Univ. (Italy). Dip. di Matematica); Grecchi, V. (Moderna Univ. (Italy). Dip. di Matematica); Jona-Lasinio, G. (Paris-11 Univ., 91 - Orsay (France). Lab. de Physique Theorique et Hautes Energies)
1984-10-21
The semiclassical limit of low lying states in a multiwell potential is studied by rigorous perturbative techniques. In particular tunnelling instability and localisation of wave functions is obtained in a simple way under small deformations of symmetric potentials.
14. Fluctuations and Instability in Sedimentation
KAUST Repository
Guazzelli, É lisabeth; Hinch, John
2011-01-01
This review concentrates on the fluctuations of the velocities of sedimenting spheres, and on the structural instability of a suspension of settling fibers. For many years, theoretical estimates and numerical simulations predicted the fluctuations
15. Edge instabilities of topological superconductors
Energy Technology Data Exchange (ETDEWEB)
Hofmann, Johannes S. [Institut fuer Theoretische Physik und Astrophysik, Universitaet Wuerzburg (Germany); Max-Planck-Institut fuer Festkoerperforschung, Stuttgart (Germany); Assaad, Fakher F. [Institut fuer Theoretische Physik und Astrophysik, Universitaet Wuerzburg (Germany); Schnyder, Andreas P. [Max-Planck-Institut fuer Festkoerperforschung, Stuttgart (Germany)
2016-07-01
Nodal topological superconductors display zero-energy Majorana flat bands at generic edges. The flatness of these edge bands, which is protected by time-reversal and translation symmetry, gives rise to an extensive ground state degeneracy and a diverging density of states. Therefore, even arbitrarily weak interactions lead to an instability of the flat-band edge states towards time-reversal and translation-symmetry broken phases, which lift the ground-state degeneracy. Here, we employ Monte Carlo simulations combined with mean-field considerations to examine the instabilities of the flat-band edge states of d{sub xy}-wave superconductors. We find that attractive interactions induce a complex s-wave pairing instability together with a density wave instability. Repulsive interactions, on the other hand, lead to ferromagnetism mixed with spin-triplet pairing at the edge. We discuss the implications of our findings for experiments on cuprate high-temperature superconductors.
16. Instability of ties in compression
DEFF Research Database (Denmark)
Buch-Hansen, Thomas Cornelius
2013-01-01
Masonry cavity walls are loaded by wind pressure and vertical load from upper floors. These loads results in bending moments and compression forces in the ties connecting the outer and the inner wall in a cavity wall. Large cavity walls are furthermore loaded by differential movements from...... the temperature gradient between the outer and the inner wall, which results in critical increase of the bending moments in the ties. Since the ties are loaded by combined compression and moment forces, the loadbearing capacity is derived from instability equilibrium equations. Most of them are iterative, since...... exact instability solutions are complex to derive, not to mention the extra complexity introducing dimensional instability from the temperature gradients. Using an inverse variable substitution and comparing an exact theory with an analytical instability solution a method to design tie...
17. Summary of longitudinal instabilities workshop
Energy Technology Data Exchange (ETDEWEB)
Chasman, R.
1976-01-01
A five-day ISABELLE workshop on longitudinal instabilities was held at Brookhaven, August 9-13, 1976. About a dozen outside accelerator experts, both from Europe and the U.S.A., joined the local staff for discussions of longitudinal instabilities in ISABELLE. An agenda of talks was scheduled for the first day of the workshop. Later during the week, a presentation was given on the subject ''A more rigorous treatment of Landau damping in longitudinal beam instabilities''. A few progress meetings were held in which disagreements regarding calculations of coupling impedances were clarified. A summary session was held on the last day. Heavy emphasis was put on single bunched beam instabilities in the microwave region extending above the cut-off frequency of the ISABELLE vacuum chamber.
18. Predicting Catastrophic BGP Routing Instabilities
National Research Council Canada - National Science Library
Nguyen, Lien
2004-01-01
.... Currently, this critical function is performed by the Border Gateway Protocol (BGP) version 4 RF01771. Like all routing protocols, BGP is vulnerable to instabilities that reduce its effectiveness...
19. WELLBORE INSTABILITY: CAUSES AND CONSEQUENCES
Directory of Open Access Journals (Sweden)
Borivoje Pašić
2007-12-01
Full Text Available Wellbore instability is one of the main problems that engineers meet during drilling. The causes of wellbore instability are often classified into either mechanical (for example, failure of the rock around the hole because of high stresses, low rock strength, or inappropriate drilling practice or chemical effects which arise from damaging interaction between the rock, generally shale, and the drilling fluid. Often, field instances of instability are a result of a combination of both chemical and mechanical. This problem might cause serious complication in well and in some case can lead to expensive operational problems. The increasing demand for wellbore stability analyses during the planning stage of a field arise from economic considerations and the increasing use of deviated, extended reach and horizontal wells. This paper presents causes, indicators and diagnosing of wellbore instability as well as the wellbore stresses model.
20. Acoustic comfort in eating establishments
DEFF Research Database (Denmark)
Svensson, David; Jeong, Cheol-Ho; Brunskog, Jonas
2014-01-01
The subjective concept of acoustic comfort in eating establishments has been investigated in this study. The goal was to develop a predictive model for the acoustic comfort, by means of simple objective parameters, while also examining which other subjective acoustic parameters could help explain...... the feeling of acoustic comfort. Through several layers of anal ysis, acoustic comfort was found to be rather complex, and could not be explained entirely by common subjective parameters such as annoyance, intelligibility or privacy. A predictive model for the mean acoustic comfort for an eating establishment...
1. Flow-excited acoustic resonance excitation mechanism, design guidelines, and counter measures
International Nuclear Information System (INIS)
2014-01-01
The excitation mechanism of acoustic resonances has long been recognized, but the industry continues to be plagued by its undesirable consequences, manifested in severe vibration and noise problems in a wide range of industrial applications. This paper focuses on the nature of the excitation mechanism of acoustic resonances in piping systems containing impinging shear flows, such as flow over shallow and deep cavities. Since this feedback mechanism is caused by the coupling between acoustic resonators and shear flow instabilities, attention is focused first on the nature of various types of acoustic resonance modes and then on the aero-acoustic sound sources, which result from the interaction of the inherently unstable shear flow with the sound field generated by the resonant acoustic modes. Various flow-sound interaction patterns are discussed, in which the resonant sound field can be predominantly parallel or normal to the mean flow direction and the acoustic wavelength can be an order of magnitude longer than the length scale of the separated shear flow or as short as the cavity length scale. Since the state of knowledge in this field has been recently reviewed by Tonon et al. (2011, 'Aero-acoustics of Pipe Systems With Closed Branches', Int. J. Aeroacoust., 10(2), pp. 201-276), this article focuses on the more practical aspects of the phenomenon, including various flow sound interaction patterns and the resulting aero-acoustic sources, which are relevant to industrial applications. A general design guide proposal and practical means to alleviate the excitation mechanism are also presented. These are demonstrated by two examples of recent industrial case histories dealing with acoustic fatigue failure of the steam dryer in a boiling water reactor (BWR) due to acoustic resonance in the main steam piping and acoustic resonances in the roll posts of the Short Take-Off and Vertical Lift Joint Strike Fighter (JSF). (authors)
2. Genomic instability and radiation effects
International Nuclear Information System (INIS)
Christian Streffer
2007-01-01
Complete text of publication follows. Cancer, genetic mutations and developmental abnormalities are apparently associated with an increased genomic instability. Such phenomena have been frequently shown in human cancer cells in vitro and in situ. It is also well-known that individuals with a genetic predisposition for cancer proneness, such as ataxia telangiectesia, Fanconi anaemia etc. demonstrate a general high genomic instability e.g. in peripheral lymphocytes before a cancer has developed. Analogous data have been found in mice which develop a specific congenital malformation which has a genetic background. Under these aspects it is of high interest that ionising radiation can increase the genomic instability of mammalian cells after exposures in vitro an in vivo. This phenomenon is expressed 20 to 40 cell cycles after the exposure e.g. by de novo chromosomal aberrations. Such effects have been observed with high and low LET radiation, high LET radiation is more efficient. With low LET radiation a good dose response is observed in the dose range 0.2 to 2.0 Gy, Recently it has been reported that senescence and genomic instability was induced in human fibroblasts after 1 mGy carbon ions (1 in 18 cells are hit), apparently bystander effects also occurred under these conditions. The instability has been shown with DNA damage, chromosomal aberrations, gene mutation and cell death. It is also transferred to the next generation of mice with respect to gene mutations, chromosomal aberrations and congenital malformations. Several mechanisms have been discussed. The involvement of telomeres has gained interest. Genomic instability seems to be induced by a general lesion to the whole genome. The transmission of one chromosome from an irradiated cell to an non-irradiated cell leads to genomic instability in the untreated cells. Genomic instability increases mutation rates in the affected cells in general. As radiation late effects (cancer, gene mutations and congenital
3. Enhancement and suppression of opto-acoustic parametric interactions using optical feedback
International Nuclear Information System (INIS)
Zhang Zhongyang; Zhao Chunnong; Ju, L.; Blair, D. G.
2010-01-01
A three mode opto-acoustic parametric amplifier (OAPA) is created when two orthogonal optical modes in a high finesse optical cavity are coupled via an acoustic mode of the cavity mirror. Such interactions are predicted to occur in advanced long baseline gravitational wave detectors. They can have high positive gain, which leads to strong parametric instability. Here we show that an optical feedback scheme can enhance or suppress the parametric gain of an OAPA, allowing exploration of three-mode parametric interactions, especially in cavity systems that have insufficient optical power to achieve spontaneous instability. We derive analytical equations and show that optical feedback is capable of controlling predicted instabilities in advanced gravitational wave detectors within a time scale of 13∼10 s.
4. JNDC FP decay data file
International Nuclear Information System (INIS)
Yamamoto, Tohru; Akiyama, Masatsugu
1981-02-01
The decay data file for fission product nuclides (FP DECAY DATA FILE) has been prepared for summation calculation of the decay heat of fission products. The average energies released in β- and γ-transitions have been calculated with computer code PROFP. The calculated results and necessary information have been arranged in tabular form together with the estimated results for 470 nuclides of which decay data are not available experimentally. (author)
5. Visible neutrino decay at DUNE
Energy Technology Data Exchange (ETDEWEB)
Coloma, Pilar [Fermilab; Peres, Orlando G. [ICTP, Trieste
2017-05-09
If the heaviest neutrino mass eigenstate is unstable, its decay modes could include lighter neutrino eigenstates. In this case part of the decay products could be visible, as they would interact at neutrino detectors via mixing. At neutrino oscillation experiments, a characteristic signature of such \\emph{visible neutrino decay} would be an apparent excess of events at low energies. We focus on a simple phenomenological model in which the heaviest neutrino decays as \ 6. Aerodynamic instability: A case history Science.gov (United States) Eisenmann, R. C. 1985-01-01 The identification, diagnosis, and final correction of complex machinery malfunctions typically require the correlation of many parameters such as mechanical construction, process influence, maintenance history, and vibration response characteristics. The progression is reviewed of field testing, diagnosis, and final correction of a specific machinery instability problem. The case history presented addresses a unique low frequency instability problem on a high pressure barrel compressor. The malfunction was eventually diagnosed as a fluidic mechanism that manifested as an aerodynamic disturbance to the rotor assembly. 7. Surgical treatment of chest instability International Nuclear Information System (INIS) Kitka, M.; Masek, M. 2015-01-01 Fractures of the ribs is the most common thoracic injury after blunt trauma. Chest wall instability (flail chest) is a common occurrence in the presence of multiple ribs fracture. Unilateral or bilateral fractures more ribs anteriorly or posteriorly will produce enough instability that paradoxical respiratory motion results in hypoventilation of an unacceptable degree. Open approach and surgical stabilisation of the chest preserved pulmonary function, improved pain control, minimized posttraumatic deformities and shorter back to work time. (author) 8. Beam Instabilities in Hadron Synchrotrons CERN Document Server Métral, E; Bartosik, H; Biancacci, N; Buffat, X; Esteban Muller, J F; Herr, W; Iadarola, G; Lasheen, A; Li, K; Oeftiger, A; Pieloni, T; Quartullo, D; Rumolo, G; Salvant, B; Schenk, M; Shaposhnikova, E; Tambasco, C; Timko, H; Zannini, C; Burov, A; Banfi, D; Barranco, J; Mounet, N; Boine-Frankenheim, O; Niedermayer, U; Kornilov, V; White, S 2016-01-01 Beam instabilities cover a wide range of effects in particle accelerators and they have been the subjects of intense research for several decades. As the machines performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all these intricate phenomena, as it is very often not possible to treat the different effects separately. The aim of this paper is to review the main mechanisms, discussing in particular the recent developments of beam instability theories and simulations. 9. Microsatellite instability in bladder cancer DEFF Research Database (Denmark) Gonzalez-Zulueta, M; Ruppert, J M; Tokino, K 1993-01-01 Somatic instability at microsatellite repeats was detected in 6 of 200 transitional cell carcinomas of the bladder. Instabilities were apparent as changes in (GT)n repeat lengths on human chromosome 9 for four tumors and as alterations in a (CAG)n repeat in the androgen receptor gene on the X...... or larger (> 2 base pairs) alterations in repeat length. All six tumors were low stage (Ta-T1), suggesting that these alterations can occur early in bladder tumorigenesis.... 10. Waves and instabilities in plasmas International Nuclear Information System (INIS) Chen, L. 1987-01-01 The contents of this book are: Plasma as a Dielectric Medium; Nyquist Technique; Absolute and Convective Instabilities; Landau Damping and Phase Mixing; Particle Trapping and Breakdown of Linear Theory; Solution of Viasov Equation via Guilding-Center Transformation; Kinetic Theory of Magnetohydrodynamic Waves; Geometric Optics; Wave-Kinetic Equation; Cutoff and Resonance; Resonant Absorption; Mode Conversion; Gyrokinetic Equation; Drift Waves; Quasi-Linear Theory; Ponderomotive Force; Parametric Instabilities; Problem Sets for Homework, Midterm and Final Examinations 11. Instability following total knee arthroplasty. Science.gov (United States) Rodriguez-Merchan, E Carlos 2011-10-01 Background Knee prosthesis instability (KPI) is a frequent cause of failure of total knee arthroplasty. Moreover, the degree of constraint required to achieve immediate and long-term stability in total knee arthroplasty (TKA) is frequently debated. Questions This review aims to define the problem, analyze risk factors, and review strategies for prevention and treatment of KPI. Methods A PubMed (MEDLINE) search of the years 2000 to 2010 was performed using two key words: TKA and instability. One hundred and sixty-five initial articles were identified. The most important (17) articles as judged by the author were selected for this review. The main criteria for selection were that the articles addressed and provided solutions to the diagnosis and treatment of KPI. Results Patient-related risk factors predisposing to post-operative instability include deformity requiring a large surgical correction and aggressive ligament release, general or regional neuromuscular pathology, and hip or foot deformities. KPI can be prevented in most cases with appropriate selection of implants and good surgical technique. When ligament instability is anticipated post-operatively, the need for implants with a greater degree of constraint should be anticipated. In patients without significant varus or valgus malalignment and without significant flexion contracture, the posterior cruciate ligament (PCL) can be retained. However, the PCL should be sacrificed when deformity exists particularly in patients with rheumatoid arthritis, previous patellectomy, previous high tibial osteotomy or distal femoral osteotomy, and posttraumatic osteoarthritis with disruption of the PCL. In most cases, KPI requires revision surgery. Successful outcomes can only be obtained if the cause of KPI is identified and addressed. Conclusions Instability following TKA is a common cause of the need for revision. Typically, knees with deformity, rheumatoid arthritis, previous patellectomy or high tibial osteotomy, and 12. Gauge-independent scales related to the Standard Model vacuum instability International Nuclear Information System (INIS) Espinosa, J.R.; Garny, M.; Konstandin, T.; Riotto, A. 2016-08-01 The measured (central) values of the Higgs and top quark masses indicate that the Standard Model (SM) effective potential develops an instability at high field values. The scale of this instability, determined as the Higgs field value at which the potential drops below the electroweak minimum, is about 10"1"1 GeV. However, such a scale is unphysical as it is not gauge invariant and suffers from a gauge-fixing uncertainty of up to two orders of magnitude. Subjecting our system, the SM, to several probes of the instability (adding higher order operators to the potential; letting the vacuum decay through critical bubbles; heating up the system to very high temperature; inflating it) and asking in each case physical questions, we are able to provide several gauge-invariant scales related with the Higgs potential instability. 13. Gauge-Independent Scales Related to the Standard Model Vacuum Instability CERN Document Server Espinosa, Jose R.; Konstandin, Thomas; Riotto, Antonio 2017-01-01 The measured (central) values of the Higgs and top quark masses indicate that the Standard Model (SM) effective potential develops an instability at high field values. The scale of this instability, determined as the Higgs field value at which the potential drops below the electroweak minimum, is about10^{11}$GeV. However, such a scale is unphysical as it is not gauge-invariant and suffers from a gauge-fixing uncertainty of up to two orders of magnitude. Subjecting our system, the SM, to several probes of the instability (adding higher order operators to the potential; letting the vacuum decay through critical bubbles; heating up the system to very high temperature; inflating it) and asking in each case physical questions, we are able to provide several gauge-invariant scales related with the Higgs potential instability. 14. Instability of enclosed horizons Science.gov (United States) Kay, Bernard S. 2015-03-01 We point out that there are solutions to the scalar wave equation on dimensional Minkowski space with finite energy tails which, if they reflect off a uniformly accelerated mirror due to (say) Dirichlet boundary conditions on it, develop an infinite stress-energy tensor on the mirror's Rindler horizon. We also show that, in the presence of an image mirror in the opposite Rindler wedge, suitable compactly supported arbitrarily small initial data on a suitable initial surface will develop an arbitrarily large stress-energy scalar near where the two horizons cross. Also, while there is a regular Hartle-Hawking-Israel-like state for the quantum theory between these two mirrors, there are coherent states built on it for which there are similar singularities in the expectation value of the renormalized stress-energy tensor. We conjecture that in other situations with analogous enclosed horizons such as a (maximally extended) Schwarzschild black hole in equilibrium in a (stationary spherical) box or the (maximally extended) Schwarzschild-AdS spacetime, there will be similar stress-energy singularities and almost-singularities—leading to instability of the horizons when gravity is switched on and matter and gravity perturbations are allowed for. All this suggests it is incorrect to picture a black hole in equilibrium in a box or a Schwarzschild-AdS black hole as extending beyond the past and future horizons of a single Schwarzschild (/Schwarzschild-AdS) wedge. It would thus provide new evidence for 't Hooft's brick wall model while seeming to invalidate the picture in Maldacena's Eternal black holes in AdS'. It would thereby also support the validity of the author's matter-gravity entanglement hypothesis and of the paper ` Brick walls and AdS/CFT' by the author and Ortíz. 15. History of shoulder instability surgery. Science.gov (United States) Randelli, Pietro; Cucchi, Davide; Butt, Usman 2016-02-01 The surgical management of shoulder instability is an expanding and increasingly complex area of study within orthopaedics. This article describes the history and evolution of shoulder instability surgery, examining the development of its key principles, the currently accepted concepts and available surgical interventions. A comprehensive review of the available literature was performed using PubMed. The reference lists of reviewed articles were also scrutinised to ensure relevant information was included. The various types of shoulder instability including anterior, posterior and multidirectional instability are discussed, focussing on the history of surgical management of these topics, the current concepts and the results of available surgical interventions. The last century has seen important advancements in the understanding and treatment of shoulder instability. The transition from open to arthroscopic surgery has allowed the discovery of previously unrecognised pathologic entities and facilitated techniques to treat these. Nevertheless, open surgery still produces comparable results in the treatment of many instability-related conditions and is often required in complex or revision cases, particularly in the presence of bone loss. More high-quality research is required to better understand and characterise this spectrum of conditions so that successful evidence-based management algorithms can be developed. IV. 16. Modification of the collective Thomson scattering radiometer in the search for parametric decay on TEXTOR DEFF Research Database (Denmark) Nielsen, Stefan Kragh; Salewski, Mirko; Bongers, W. 2012-01-01 Strong scattering of high-power millimeter waves at 140 GHz has been shown to take place in heating and current-drive experiments at TEXTOR when a tearing mode is present in the plasma. The scattering signal is at present supposed to be generated by the parametric decay instability. Here we descr... 17. Decay of 57Ni International Nuclear Information System (INIS) Santos Scardino, A.M. dos. 1987-01-01 The decay of 57 Ni to 57 Co was studied by gamma ray spectroscopy using both singles and coincidence spectra. The sources were obtained with the 58 Ni (Y,n) 57 Ni reaction. Natural metallic nickel was irradiated in the bremsstrahluhng beam of the linear accelerator of the Instituto de Fisica da Universidade de Sao Paulo with 30 MeV electrons. The singles espectra were taken with 104 cc HPGe detector and the coincidences espectra with 27 and 53cc Ge(Li) and 104 cc. HPGe detectors. The energies of transitions that follow the 57 Ni decay were measured using 56 Co as standard (which was obtained by (Y,np) reaction in 58 Ni) and taking into account the cascade cross-over relations. (author) [pt 18. Electroweak penguin B decays CERN Document Server Nikodem, Thomas 2016-01-01 Flavour Changing Neutral Currents (FCNC) are sensitive probes for physics beyond the Standard Model (SM), so-called New Physics. An example of a FCNC is the$b \\to s$quark transition described by the electroweak penguin Feynman diagram shown in Figure 1. In the SM such FCNC are only allowed with a loop structure (as e:g: shown in the figure) and not by tree level processes. In the loops heavy particles appear virtually and do not need to be on shell. Therefore also not yet discovered heavy particles with up to a mass$\\mathcal{O}$(TeV) could virtually contribute significantly to observables. Several recent measurements of electroweak penguin B decays exhibit interesting tensions with SM predictions, most prominently in the angular observable$P'_5$5 of the decay$B^0 \\to K^{*0} \\mu^+ \\mu^1\$[1], which triggered a lot of discussion in the theory community [2]-[14].
19. Decay /sup 133/Ba
Energy Technology Data Exchange (ETDEWEB)
Singh, K; Hasiza, M L; Grewal, B S; Sahota, H S
1982-07-01
The relative gamma ray intensities of transitions in the decay of /sup 133/Ba have been measured using an intrinsic Ge detector. The electron capture branching ratios have been determined for 81, 161, 384 and 437 keV levels. The attenuation effect of long half-life of 81 keV levels has been studied in solid and liquid media. The electron capture decay has been investigated by changing the concentration of ethylene-diamine-tetraacetic acid (EDTA) environment. The 5/2/sup +/ yields 5/2/sup +/ 79.67 keV transition has an E0 to E2 intensity qsub(k)sup(2) <= 0.31. 10 refs., 4 figures.
20. Magnetoactive Acoustic Metamaterials.
Science.gov (United States)
Yu, Kunhao; Fang, Nicholas X; Huang, Guoliang; Wang, Qiming
2018-04-11
Acoustic metamaterials with negative constitutive parameters (modulus and/or mass density) have shown great potential in diverse applications ranging from sonic cloaking, abnormal refraction and superlensing, to noise canceling. In conventional acoustic metamaterials, the negative constitutive parameters are engineered via tailored structures with fixed geometries; therefore, the relationships between constitutive parameters and acoustic frequencies are typically fixed to form a 2D phase space once the structures are fabricated. Here, by means of a model system of magnetoactive lattice structures, stimuli-responsive acoustic metamaterials are demonstrated to be able to extend the 2D phase space to 3D through rapidly and repeatedly switching signs of constitutive parameters with remote magnetic fields. It is shown for the first time that effective modulus can be reversibly switched between positive and negative within controlled frequency regimes through lattice buckling modulated by theoretically predicted magnetic fields. The magnetically triggered negative-modulus and cavity-induced negative density are integrated to achieve flexible switching between single-negative and double-negative. This strategy opens promising avenues for remote, rapid, and reversible modulation of acoustic transportation, refraction, imaging, and focusing in subwavelength regimes. © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
1. Hypernuclear weak decay puzzle
International Nuclear Information System (INIS)
Barbero, C.; Horvat, D.; Narancic, Z.; Krmpotic, F.; Kuo, T.T.S.; Tadic, D.
2002-01-01
A general shell model formalism for the nonmesonic weak decay of the hypernuclei has been developed. It involves a partial wave expansion of the emitted nucleon waves, preserves naturally the antisymmetrization between the escaping particles and the residual core, and contains as a particular case the weak Λ-core coupling formalism. The extreme particle-hole model and the quasiparticle Tamm-Dancoff approximation are explicitly worked out. It is shown that the nuclear structure manifests itself basically through the Pauli principle, and a very simple expression is derived for the neutron- and proton-induced decays rates Γ n and Γ p , which does not involve the spectroscopic factors. We use the standard strangeness-changing weak ΛN→NN transition potential which comprises the exchange of the complete pseudoscalar and vector meson octets (π,η,K,ρ,ω,K * ), taking into account some important parity-violating transition operators that are systematically omitted in the literature. The interplay between different mesons in the decay of Λ 12 C is carefully analyzed. With the commonly used parametrization in the one-meson-exchange model (OMEM), the calculated rate Γ NM =Γ n +Γ p is of the order of the free Λ decay rate Γ 0 (Γ NM th congruent with Γ 0 ) and is consistent with experiments. Yet the measurements of Γ n/p =Γ n /Γ p and of Γ p are not well accounted for by the theory (Γ n/p th p th > or approx. 0.60Γ 0 ). It is suggested that, unless additional degrees of freedom are incorporated, the OMEM parameters should be radically modified
International Nuclear Information System (INIS)
Edwards, B.J.; Kamal, A.N.
1979-04-01
The status of decays of the kind V → Pγ and P → Vγviewed with special emphasis on the work done by the authors in this field. The low experimental value of GAMMA(rho → πγ) remains the outstanding problem. The lastest preliminary numbers from a Fermi Laboratory experiment go in the right direction but not far enough. 15 references
3. Decay of 83Sr
International Nuclear Information System (INIS)
Yu Xiaohan; Shi Shuanghui; Gu Jiahui
1997-01-01
The decay of 83 Sr was reinvestigated using γ singles and γ-γ-t coincidence measurement. A new level scheme of Rb, which contains 41 excited levels and about 180 transitions, is constructed. 19 new levels were added to the old level scheme and 8 formerly adopted levels were denied. A new data set of branching ratio, log(ft) value and spin parity was obtained
4. Shear flow instability in a partially-ionized plasma sheath around a fast-moving vehicle
International Nuclear Information System (INIS)
Sotnikov, V. I.; Mudaliar, S.; Genoni, T. C.; Rose, D. V.; Oliver, B. V.; Mehlhorn, T. A.
2011-01-01
The stability of ion acoustic waves in a sheared-flow, partially-ionized compressible plasma sheath around a fast-moving vehicle in the upper atmosphere, is described and evaluated for different flow profiles. In a compressible plasma with shear flow, instability occurs for any velocity profile, not just for profiles with an inflection point. A second-order differential equation for the electrostatic potential of excited ion acoustic waves in the presence of electron and ion collisions with neutrals is derived and solved numerically using a shooting method with boundary conditions appropriate for a finite thickness sheath in contact with the vehicle. We consider three different velocity flow profiles and find that in all cases that neutral collisions can completely suppress the instability.
5. MULTIFLUID MAGNETOHYDRODYNAMIC TURBULENT DECAY
International Nuclear Information System (INIS)
Downes, T. P.; O'Sullivan, S.
2011-01-01
It is generally believed that turbulence has a significant impact on the dynamics and evolution of molecular clouds and the star formation that occurs within them. Non-ideal magnetohydrodynamic (MHD) effects are known to influence the nature of this turbulence. We present the results of a suite of 512 3 resolution simulations of the decay of initially super-Alfvenic and supersonic fully multifluid MHD turbulence. We find that ambipolar diffusion increases the rate of decay of the turbulence while the Hall effect has virtually no impact. The decay of the kinetic energy can be fitted as a power law in time and the exponent is found to be -1.34 for fully multifluid MHD turbulence. The power spectra of density, velocity, and magnetic field are all steepened significantly by the inclusion of non-ideal terms. The dominant reason for this steepening is ambipolar diffusion with the Hall effect again playing a minimal role except at short length scales where it creates extra structure in the magnetic field. Interestingly we find that, at least at these resolutions, the majority of the physics of multifluid turbulence can be captured by simply introducing fixed (in time and space) resistive terms into the induction equation without the need for a full multifluid MHD treatment. The velocity dispersion is also examined and, in common with previously published results, it is found not to be power law in nature.
6. 152Eu decay
International Nuclear Information System (INIS)
Artamonova, K.P.; Vinogradov, V.M.; Grigor'ev, E.P.; Zolotavin, A.V.; Makarov, V.M.; Sergeev, V.O.; Usynko, T.M.
1978-01-01
The purpose of this paper is the measurement of the relative intensities of the most intensive conversion lines of 152 Eu, the determination of as reliable as possible magnitudes of the intensities of γ-quanta using all the available data on γ-radiation of 152 Eu, the measurement of the interval conversion coefficients (ICC) for the most intensive γ-transitions, the determination of the probabilities of the 152 Eu β-decays to the 152 Sm and 152 Gd levels. The conversion lines of the most intensive γ-transitions in the 152 Eu decay are studied and the corresponding ICC are measured on the beta-spectrometers of π√2 and UMB type. The balance for the γ-transitions in the 152 Sm and 152 Gd daughter nuclei are presented. This balance is used to determine the absolute intensities of γ-rays (in terms of the percentage of the 152 Eu decays) and the probabilities of β-transitions to the levels of daughter nuclei. More accurate data on γ-rays and conversion electrons obtained can be used for the calibration of gamma and beta spectrometers
7. Preliminary Results on the Effects of Distributed Aluminum Combustion Upon Acoustic Growth Rates in a Rijke Burner
OpenAIRE
Newbold, Brian R.
1998-01-01
Distributed particle combustion in solid propellant rocket motors may be a significant cause of acoustic combustion instability. A Rijke burner has been developed as a tool to investigate the phenomenon. Previous improvements and characterization of the upright burner lead to the addition of a particle injection flame. The injector flame increases the burner's acoustic driving by about 10% which is proportional to the injector's additional 2 g/min of gas. Frequency remained fairly constant fo...
8. Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
9. Research Based on the Acoustic Emission of Wind Power Tower Drum Dynamic Monitoring Technology
Science.gov (United States)
Zhang, Penglin; Sang, Yuan; Xu, Yaxing; Zhao, Zhiqiang
Wind power tower drum is one of the key components of the wind power equipment. Whether the wind tower drum performs safety directly affects the efficiency, life, and performance of wind power equipment. Wind power tower drum in the process of manufacture, installation, and operation may lead to injury, and the wind load and gravity load and long-term factors such as poor working environment under the action of crack initiation or distortion, which eventually result in the instability or crack of the wind power tower drum and cause huge economic losses. Thus detecting the wind power tower drum crack damage and instability is especially important. In this chapter, acoustic emission is used to monitor the whole process of wind power tower drum material Q345E steel tensile test at first, and processing and analysis tensile failure signal of the material. And then based on the acoustic emission testing technology to the dynamic monitoring of wind power tower drum, the overall detection and evaluation of the existence of active defects in the whole structure, and the acoustic emission signals collected for processing and analysis, we could preliminarily master the wind tower drum mechanism of acoustic emission source. The acoustic emission is a kind of online, efficient, and economic method, which has very broad prospects for work. The editorial committee of nondestructive testing qualification and certification of personnel teaching material of science and technology industry of national defense, "Acoustic emission testing" (China Machine Press, 2005.1).
10. Decay Modes of a Dense Plasma in a Magnetic Well
Energy Technology Data Exchange (ETDEWEB)
Coensgen, F. H.; Cummins, W. F.; Ellis, R. E.; Nexsen, Jr., W. E. [Lawrence Radiation Laboratory, University of California, Livermore, CA (United States)
1969-03-15
Energetic deuterium plasmas of {beta} Almost-Equal-To 5% are formed in an open-ended magnetic well system using the techniques of plasma injection and magnetic compression. Containment in the quasi-dc field following compression is studied. Under ordinary vacuum wall conditions there are rapid plasma losses, accompanied by rf signals at ion-cyclotron frequencies, {omega}{sub ci}. This activity is tentatively identified as the ion-ion instability due to a ''double humped'' ion energy distribution. The loss has been suppressed by forming gas-free Ti surfaces throughout the chamber. Under these latter conditions, it was also shown that interchange instabilities are suppressed by the minimum-B field for densities as high as 5x10{sup 13} cm{sup -3}. The D{sup +} energy distribution as derived from analysis of chargeexchange fast atoms extends from 2 to 50 keV and, following the initial containment phase, remains essentially unchanged with time. The mean ion energy of {beta} keV derived from the distribution is in good agreement with the ion temperature deduced from the measured density and prompt neutron flux. The fact that the reaction rate decays as n{sup 2} is further evidence that the energetic D{sup +} ions are the primary plasma component. The decay rate is at all times substantially greater than that expected from ion-ion scattering, and thus is indicative of anomalous losses. Sporadic bursts of particles through the mirrors as well as fluctuations near wci and harmonics give direct evidence of cooperative effects at densities above 2 x 10{sup 12} cm{sup -3}. The density history is divided into three periods: After compression, the decay proceeds exponentially with a characteristic lifetime {tau} Almost-Equal-To 200 {mu}s down to a density near 1.5 x 10{sup 13} cm{sup -3} where the decay rate abruptly decreases so that r increases to approximately 400 us. At densities {<=} 2 x 10{sup 12} cm{sup -3} the decay rate decreases markedly, so that this density remains
11. Search for Nucleon Decays in Super-Kamiokande
International Nuclear Information System (INIS)
Miura, Makoto
2010-01-01
Grand Unified Theories (GUTs) is motivated by merging of the coupling constants of the strong, weak, and electromagnetic forces at a large energy scale (∼10 16 GeV), which is out of the reach of accelerators. One of the other general features of GUTs is that they allow lepton and baryon number violations and they predict instability of nucleons. Then nucleon decay experiments are the direct probe for GUTs. The Super-Kamiokande (SK) is a water Cherenkov detector which keeps running to detect nucleon decays with large mass. There are no other nucleon decay detectors which have as long exposure as SK. The results of nucleon decay search based on 173 kton year (1996-2008) will be presented in the conference.The favored decay mode in GUTs based on SU(5) symmetry is p→e + π 0 . On the other hand, p→ν K + is favored by SUSY GUTs model. Those two modes will be mainly discussed. (authors)
12. Evidence against solar influence on nuclear decay constants
Directory of Open Access Journals (Sweden)
S. Pommé
2016-10-01
Full Text Available The hypothesis that proximity to the Sun causes variation of decay constants at permille level has been tested and disproved. Repeated activity measurements of mono-radionuclide sources were performed over periods from 200 days up to four decades at 14 laboratories across the globe. Residuals from the exponential nuclear decay curves were inspected for annual oscillations. Systematic deviations from a purely exponential decay curve differ from one data set to another and are attributable to instabilities in the instrumentation and measurement conditions. The most stable activity measurements of alpha, beta-minus, electron capture, and beta-plus decaying sources set an upper limit of 0.0006% to 0.008% to the amplitude of annual oscillations in the decay rate. Oscillations in phase with Earth's orbital distance to the Sun could not be observed within a 10−6 to 10−5 range of precision. There are also no apparent modulations over periods of weeks or months. Consequently, there is no indication of a natural impediment against sub-permille accuracy in half-life determinations, renormalisation of activity to a distant reference date, application of nuclear dating for archaeology, geo- and cosmochronology, nor in establishing the SI unit becquerel and seeking international equivalence of activity standards.
13. Advanced Active Acoustics Lab (AAAL)
Data.gov (United States)
Federal Laboratory Consortium — The Advanced Active Acoustics Lab (AAAL) is a state-of-the-art Undersea Warfare (USW) acoustic data analysis facility capable of both active and passive underwater...
14. Sea Turtle Acoustic Telemetry Data
Data.gov (United States)
National Oceanic and Atmospheric Administration, Department of Commerce — Acoustic transmitters attached to sea turtles captured in various fishing gear enable the animals to be passively tracked. Acoustic receivers set up in an array...
15. Perspective: Acoustic metamaterials in transition
KAUST Repository
Wu, Ying; Yang, Min; Sheng, Ping
2017-01-01
Acoustic metamaterials derive their novel characteristics from the interaction between acoustic waves with designed structures. Since its inception seventeen years ago, the field has been driven by fundamental geometric and physical principles
16. A Century of Acoustic Metrology
DEFF Research Database (Denmark)
Rasmussen, Knud
1998-01-01
The development in acoustic measurement technique over the last century is reviewed with special emphasis on the metrological aspect.......The development in acoustic measurement technique over the last century is reviewed with special emphasis on the metrological aspect....
17. Acoustic Levitation Containerless Processing
Science.gov (United States)
Whymark, R. R.; Rey, C. A.
1985-01-01
This research program consists of the development of acoustic containerless processing systems with applications in the areas of research in material sciences, as well as the production of new materials, solid forms with novel and unusual microstructures, fusion target spheres, and improved optical fibers. Efforts have been focused on the containerless processing at high temperatures for producing new kinds of glasses. Also, some development has occurred in the areas of containerlessly supporting liquids at room temperature, with applications in studies of fluid dynamics, potential undercooling of liquids, etc. The high temperature area holds the greatest promise for producing new kinds of glasses and ceramics, new alloys, and possibly unusual structural shapes, such as very uniform hollow glass shells for fusion target applications. High temperature acoustic levitation required for containerless processing has been demonstrated in low-g environments as well as in ground-based experiments. Future activities include continued development of the signals axis acoustic levitator.
18. Practical acoustic emission testing
CERN Document Server
2016-01-01
This book is intended for non-destructive testing (NDT) technicians who want to learn practical acoustic emission testing based on level 1 of ISO 9712 (Non-destructive testing – Qualification and certification of personnel) criteria. The essential aspects of ISO/DIS 18436-6 (Condition monitoring and diagnostics of machines – Requirements for training and certification of personnel, Part 6: Acoustic Emission) are explained, and readers can deepen their understanding with the help of practice exercises. This work presents the guiding principles of acoustic emission measurement, signal processing, algorithms for source location, measurement devices, applicability of testing methods, and measurement cases to support not only researchers in this field but also and especially NDT technicians.
19. Topological Acoustic Delay Line
Science.gov (United States)
Zhang, Zhiwang; Tian, Ye; Cheng, Ying; Wei, Qi; Liu, Xiaojun; Christensen, Johan
2018-03-01
Topological protected wave engineering in artificially structured media is at the frontier of ongoing metamaterials research that is inspired by quantum mechanics. Acoustic analogues of electronic topological insulators have recently led to a wealth of new opportunities in manipulating sound propagation with strikingly unconventional acoustic edge modes immune to backscattering. Earlier fabrications of topological insulators are characterized by an unreconfigurable geometry and a very narrow frequency response, which severely hinders the exploration and design of useful devices. Here we establish topologically protected sound in reconfigurable phononic crystals that can be switched on and off simply by rotating its three-legged "atoms" without altering the lattice structure. In particular, we engineer robust phase delay defects that take advantage of the ultrabroadband reflection-free sound propagation. Such topological delay lines serve as a paradigm in compact acoustic devices, interconnects, and electroacoustic integrated circuits.
20. 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... | {"extraction_info": {"found_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": 0.8635754585266113, "perplexity": 2765.7546182417786}, "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-2018-30/segments/1531676589902.8/warc/CC-MAIN-20180717203423-20180717223423-00272.warc.gz"} |
http://mathoverflow.net/questions/107616/chern-number-of-a-sphere | # Chern number of a sphere
Hi everybody. I think I get a problem with the definitions of the connections 1-form of a vector bundle.
Let's consider the sphere $S^2$ with its tangent bundle as a vector bundle. Let's take a tangent vector field $A$ regular on the sphere and construct using local patches these connections 1-forms:
$\omega^{\alpha}_{\beta}=$ $\delta^{{\alpha},{\beta}}\sum_j A_jdx^j$,
where $\delta^{{\alpha},{\beta}}$ is the Kronecker delta. I supposed that the vector field is regular and defined in the whole sphere, so the connection 1-forms do vanish in a certain point, because of the hairy ball theorem. Is it a problem? Why? I don't find in the definitions that the connections 1- form can't be zero...
Anyway from these connections we can construct the curvature 2-form and the first Chern number integrating that curvature. But the 2-form to integrate in ordero obtain the first Chern number here is essentially an exact form ($\Omega$=$dA$) and so the integration through the compact surface is zero.
But the first Chern number of these vector bundle should not be zero...
-
Connection 1-forms do not transform in the same way as ordinary 1-forms, so the local expressions you have written do not patch up to a well-defined connection. Otherwise you could just set all the connection 1-forms to zero and get a flat connection on any vector bundle. – Johannes Nordström Sep 20 '12 at 9:33
I'm a bit confused. Somewhere I read that I need a principal G-bundle. In that case I could consider (having base manyfold $S^2$) the frame bundle with the group $GL(2,R)$ acting on it? | {"extraction_info": {"found_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": 0.9438221454620361, "perplexity": 321.43260668779175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463928.32/warc/CC-MAIN-20150226074103-00331-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/53403 | Formats
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### Abstract
Separated flow models are of great interest to model two-phase flow such as Stratified, Stratified-Wavy and Annular flow patterns. Taitel and Dukler proposed a model for Stratified flow assuming that the interfacial effects can be neglected and modelled the pressure drop assuming smooth walls. Based on a new definition of hydraulic diameter for two-phase flows in channels and including the effect of shear on the interface, it is possible to obtain a more general analytical solution for Stratified flow. An extension to Annular flow is possible with the same model, but here the roughness of the tube walls and the interface is taken into account. This model allows a comprehensive approach to modelling of two-phase flow phenomena in the form of partial Reynolds numbers, partial pressure drops, partial rates of dissipation and partial rates of interfacial entrainment, and allows a direct comparison to be made between Stratified and Annular flows. This new model is an interesting platform that can be related to experimental data by the friction factors of the four surfaces of contact. | {"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": 0.802914559841156, "perplexity": 621.6138498670969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038076819.36/warc/CC-MAIN-20210414034544-20210414064544-00016.warc.gz"} |
https://byjus.com/product-to-sum-formula/ | Product to Sum Formula
Product to sum formulas are the trigonometric identities. These identities are used to rewrite products of sine and cosine. Product to sum formulas are also used to simplify the critical trigonometry function.To solve the trigonometry functions use below given Product to sum formula.
Sum to product formulas are:
$\large cos\;a\;cos\;b=\frac{1}{2}\left(cos\left(a+b\right)+cos\left(a-b\right )\right)$
$\large sin\;a\;sin\;b=\frac{1}{2}\left(sin\left(a-b\right)-cos\left(a+b\right )\right)$
$\large sin\;a\;sin\;b=\frac{1}{2}\left(sin\left(a+b\right)+sin\left(a-b\right)\right)$
$\large cos\;a\;sin\;b=\frac{1}{2}\left(sin\left(a+b\right)-sin\left(a-b\right)\right)$
solved examples
Question: Simplify the cos(3x) sin (2x) using product to sum formula.
Solution:
Given cos(3x)sin(2x)
Using formula,
cos a sin b = $\frac{1}{2}$$(sin(a + b) – sin(a – b)) cos 3x sin 2x = \frac{1}{2}$$(sin(3x + 2x) – sin(3x – 2x))$
cos 3x sin 2x = $\frac{1}{2}$$(sin(5x) – sin(x))$
Practise This Question
Which one of the following is the correct sequence of events that follows in a synaptic transmission of nerve impulse?
1. The presynaptic membrane is depolarized.
2. Voltage gated calcium channels open
3. Release of synaptic Vesicles
4. Calcium enters the synaptic knob
5. The neurotransmitter diffuses across the synaptic cleft and binds to the receptor proteins on the postsynaptic membrane. | {"extraction_info": {"found_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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9758525490760803, "perplexity": 3521.3334590512236}, "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-2019-26/segments/1560627999040.56/warc/CC-MAIN-20190619184037-20190619210037-00324.warc.gz"} |
https://education.ti.com/en/customer-support/knowledge-base/ti-nspire-family/product-usage/25890 | Education Technology
Knowledge Base
Solution 25890: Exact and Approximate Solutions Using the TI-Nspire™ Handheld and Computer Software.
Why do I sometimes get a decimal approximation in a stacked fraction calculation on the TI-Nspire family handheld and computer software when I expect to see the answer in stacked fraction form?
Using the TI-Nspire handheld and computer software, if only stacked fractions are used in the input, the resulting answer will also be displayed in stacked fraction form. However, if something is added to the input that approximates to a decimal value, the resulting answer will always be in decimal form.
Example: 1/2 + 1/3 = 5/6
but
1/2 + 1/3 + tan(p) = 0.888219
Please Note: Even though tan(p) = 0, the answer will still be displayed as 0.888219 because the TI-Nspire handheld and computer software approximate the value of tan(p).
This approximation feature does not affect the TI-Nspire CAS handheld and computer software. If 1/2 + 1/3 + tan(p) is input using a TI-Nspire CAS handheld or computer software, the resulting answer is 5/6. | {"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": 0.9113224148750305, "perplexity": 1950.305686475329}, "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-2021-25/segments/1623487662882.61/warc/CC-MAIN-20210620114611-20210620144611-00046.warc.gz"} |
https://en.wikipedia.org/wiki/Canonical_transformation | # Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).
Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates qQ do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transformation into
${\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}.}$
Therefore, coordinate transformations (also called point transformations) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type).
For clarity, we restrict the presentation here to calculus and classical mechanics. Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. (Canonical transformations are a special case of a symplectomorphism.) However, a brief introduction to the modern mathematical description is included at the end of this article.
## Notation
Boldface variables such as q represent a list of N generalized coordinates that need not transform like a vector under rotation, e.g.,
${\displaystyle \mathbf {q} \equiv \left(q_{1},q_{2},\ldots ,q_{N-1},q_{N}\right).}$
A dot over a variable or list signifies the time derivative, e.g.,
${\displaystyle {\dot {\mathbf {q} }}\equiv {\frac {d\mathbf {q} }{dt}}.}$
The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g.,
${\displaystyle \mathbf {p} \cdot \mathbf {q} \equiv \sum _{k=1}^{N}p_{k}q_{k}.}$
The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value.
## Direct approach
The functional form of Hamilton's equations is
{\displaystyle {\begin{aligned}{\dot {\mathbf {p} }}&=-{\frac {\partial H}{\partial \mathbf {q} }}\\{\dot {\mathbf {q} }}&={\frac {\partial H}{\partial \mathbf {p} }}\end{aligned}}}
By definition, the transformed coordinates have analogous dynamics
{\displaystyle {\begin{aligned}{\dot {\mathbf {P} }}&=-{\frac {\partial K}{\partial \mathbf {Q} }}\\{\dot {\mathbf {Q} }}&={\frac {\partial K}{\partial \mathbf {P} }}\end{aligned}}}
where K(Q, P) is a new Hamiltonian (sometimes called the Kamiltonian[1]) that must be determined.
In general, a transformation (q, p, t) → (Q, P, t) does not preserve the form of Hamilton's equations. For time independent transformations between (q, p) and (Q, P) we may check if the transformation is restricted canonical, as follows. Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate Qm is
{\displaystyle {\begin{aligned}{\dot {Q}}_{m}&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}\\&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\frac {\partial H}{\partial \mathbf {p} }}-{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\frac {\partial H}{\partial \mathbf {q} }}\\&=\lbrace Q_{m},H\rbrace \end{aligned}}}
where {⋅, ⋅} is the Poisson bracket.
We also have the identity for the conjugate momentum Pm
${\displaystyle {\frac {\partial H}{\partial P_{m}}}={\frac {\partial H}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}}$
If the transformation is canonical, these two must be equal, resulting in the equations
{\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}}
The analogous argument for the generalized momenta Pm leads to two other sets of equations
{\displaystyle {\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}}
These are the direct conditions to check whether a given transformation is canonical.
## Liouville's theorem
The direct conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e.,
${\displaystyle \int \mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} =\int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} }$
By calculus, the latter integral must equal the former times the Jacobian J
${\displaystyle \int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} =\int J\,\mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} }$
where the Jacobian is the determinant of the matrix of partial derivatives, which we write as
${\displaystyle J\equiv {\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}}$
Exploiting the "division" property of Jacobians yields
${\displaystyle J\equiv {\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {P} )}}\left/{\frac {\partial (\mathbf {q} ,\mathbf {p} )}{\partial (\mathbf {q} ,\mathbf {P} )}}\right.}$
Eliminating the repeated variables gives
${\displaystyle J\equiv {\frac {\partial (\mathbf {Q} )}{\partial (\mathbf {q} )}}\left/{\frac {\partial (\mathbf {p} )}{\partial (\mathbf {P} )}}\right.}$
Application of the direct conditions above yields J = 1.
## Generating function approach
To guarantee a valid transformation between (q, p, H) and (Q, P, K), we may resort to an indirect generating function approach. Both sets of variables must obey Hamilton's principle. That is the Action Integral over the Lagrangian ${\displaystyle {\mathcal {L}}_{qp}=\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)}$ and ${\displaystyle {\mathcal {L}}_{QP}=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)}$ respectively, obtained by the Hamiltonian via ("inverse") Legendre transformation, both must be stationary (so that one can use the Euler–Lagrange equations to arrive at equations of the above-mentioned and designated form; as it is shown for example here):
{\displaystyle {\begin{aligned}\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]dt&=0\\\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt&=0\end{aligned}}}
One way for both variational integral equalities to be satisfied is to have
${\displaystyle \lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt}}}$
Lagrangians are not unique: one can always multiply by a constant λ and add a total time derivative dG/dt and yield the same equations of motion (see for reference: [1]).
In general, the scaling factor λ is set equal to one; canonical transformations for which λ ≠ 1 are called extended canonical transformations. dG/dt is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.
Here G is a generating function of one old canonical coordinate (q or p), one new canonical coordinate (Q or P) and (possibly) the time t. Thus, there are four basic types of generating functions (although mixtures of these four types can exist), depending on the choice of variables. As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation (q, p) → (Q, P) is guaranteed to be canonical.
### Type 1 generating function
The type 1 generating function G1 depends only on the old and new generalized coordinates
${\displaystyle G\equiv G_{1}(\mathbf {q} ,\mathbf {Q} ,t)}$
To derive the implicit transformation, we expand the defining equation above
${\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{1}}{\partial t}}+{\frac {\partial G_{1}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}}$
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
{\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{1}}{\partial \mathbf {q} }}\\\mathbf {P} &=-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}\\K&=H+{\frac {\partial G_{1}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
${\displaystyle \mathbf {p} ={\frac {\partial G_{1}}{\partial \mathbf {q} }}}$
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
${\displaystyle \mathbf {P} =-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}}$
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
${\displaystyle K=H+{\frac {\partial G_{1}}{\partial t}}}$
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
${\displaystyle G_{1}\equiv \mathbf {q} \cdot \mathbf {Q} }$
This results in swapping the generalized coordinates for the momenta and vice versa
{\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{1}}{\partial \mathbf {q} }}=\mathbf {Q} \\\mathbf {P} &=-{\frac {\partial G_{1}}{\partial \mathbf {Q} }}=-\mathbf {q} \end{aligned}}}
and K = H. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.
### Type 2 generating function
The type 2 generating function G2 depends only on the old generalized coordinates and the new generalized momenta
${\displaystyle G\equiv -\mathbf {Q} \cdot \mathbf {P} +G_{2}(\mathbf {q} ,\mathbf {P} ,t)}$
where the ${\displaystyle -\mathbf {Q} \cdot \mathbf {P} }$ terms represent a Legendre transformation to change the right-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above
${\displaystyle \mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{2}}{\partial t}}+{\frac {\partial G_{2}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial G_{2}}{\partial \mathbf {P} }}\cdot {\dot {\mathbf {P} }}}$
Since the old coordinates and new momenta are each independent, the following 2N + 1 equations must hold
{\displaystyle {\begin{aligned}\mathbf {p} &={\frac {\partial G_{2}}{\partial \mathbf {q} }}\\\mathbf {Q} &={\frac {\partial G_{2}}{\partial \mathbf {P} }}\\K&=H+{\frac {\partial G_{2}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
${\displaystyle \mathbf {p} ={\frac {\partial G_{2}}{\partial \mathbf {q} }}}$
define relations between the new generalized momenta P and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
${\displaystyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}}$
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
${\displaystyle K=H+{\frac {\partial G_{2}}{\partial t}}}$
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
${\displaystyle G_{2}\equiv \mathbf {g} (\mathbf {q} ;t)\cdot \mathbf {P} }$
where g is a set of N functions. This results in a point transformation of the generalized coordinates
${\displaystyle \mathbf {Q} ={\frac {\partial G_{2}}{\partial \mathbf {P} }}=\mathbf {g} (\mathbf {q} ;t)}$
### Type 3 generating function
The type 3 generating function G3 depends only on the old generalized momenta and the new generalized coordinates
${\displaystyle G\equiv \mathbf {q} \cdot \mathbf {p} +G_{3}(\mathbf {p} ,\mathbf {Q} ,t)}$
where the ${\displaystyle \mathbf {q} \cdot \mathbf {p} }$ terms represent a Legendre transformation to change the left-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above
${\displaystyle -\mathbf {q} \cdot {\dot {\mathbf {p} }}-H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{3}}{\partial t}}+{\frac {\partial G_{3}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}+{\frac {\partial G_{3}}{\partial \mathbf {Q} }}\cdot {\dot {\mathbf {Q} }}}$
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
{\displaystyle {\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{3}}{\partial \mathbf {p} }}\\\mathbf {P} &=-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}\\K&=H+{\frac {\partial G_{3}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
${\displaystyle \mathbf {q} =-{\frac {\partial G_{3}}{\partial \mathbf {p} }}}$
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
${\displaystyle \mathbf {P} =-{\frac {\partial G_{3}}{\partial \mathbf {Q} }}}$
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
${\displaystyle K=H+{\frac {\partial G_{3}}{\partial t}}}$
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple.
### Type 4 generating function
The type 4 generating function ${\displaystyle G_{4}(\mathbf {p} ,\mathbf {P} ,t)}$ depends only on the old and new generalized momenta
${\displaystyle G\equiv \mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} +G_{4}(\mathbf {p} ,\mathbf {P} ,t)}$
where the ${\displaystyle \mathbf {q} \cdot \mathbf {p} -\mathbf {Q} \cdot \mathbf {P} }$ terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above
${\displaystyle -\mathbf {q} \cdot {\dot {\mathbf {p} }}-H(\mathbf {q} ,\mathbf {p} ,t)=-\mathbf {Q} \cdot {\dot {\mathbf {P} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {\partial G_{4}}{\partial t}}+{\frac {\partial G_{4}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}+{\frac {\partial G_{4}}{\partial \mathbf {P} }}\cdot {\dot {\mathbf {P} }}}$
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
{\displaystyle {\begin{aligned}\mathbf {q} &=-{\frac {\partial G_{4}}{\partial \mathbf {p} }}\\\mathbf {Q} &={\frac {\partial G_{4}}{\partial \mathbf {P} }}\\K&=H+{\frac {\partial G_{4}}{\partial t}}\end{aligned}}}
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
${\displaystyle \mathbf {q} =-{\frac {\partial G_{4}}{\partial \mathbf {p} }}}$
define relations between the new generalized momenta P and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
${\displaystyle \mathbf {Q} ={\frac {\partial G_{4}}{\partial \mathbf {P} }}}$
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
${\displaystyle K=H+{\frac {\partial G_{4}}{\partial t}}}$
yields a formula for K as a function of the new canonical coordinates (Q, P).
## Motion as a canonical transformation
Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If ${\displaystyle \mathbf {Q} (t)\equiv \mathbf {q} (t+\tau )}$ and ${\displaystyle \mathbf {P} (t)\equiv \mathbf {p} (t+\tau )}$, then Hamilton's principle is automatically satisfied
${\displaystyle \delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt=\delta \int _{t_{1}+\tau }^{t_{2}+\tau }\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t+\tau )\right]dt=0}$
since a valid trajectory ${\displaystyle (\mathbf {q} (t),\mathbf {p} (t))}$ should always satisfy Hamilton's principle, regardless of the endpoints.
## Modern mathematical description
In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as
${\displaystyle \sum _{i}p_{i}\,dq^{i}}$
up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates q is written here as a superscript (${\displaystyle q^{i}}$), not as a subscript as done above (${\displaystyle q_{i}}$). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.
## History
The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system. This work resulted in the publication of a pair of large volumes as Mémoires by the French Academy of Sciences, in 1860 and 1867. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 58, "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": 0.9785836338996887, "perplexity": 1494.7098844158484}, "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-2021-21/segments/1620243990449.41/warc/CC-MAIN-20210514091252-20210514121252-00124.warc.gz"} |
http://tex.stackexchange.com/questions/95878/superscript-asterisk-on-the-left-side-of-different-symbols-math-mode | # superscript asterisk on the left side of different symbols (math mode)
I need to add superscript asterisks to the left side of different symbols like +,\cdot,\leq,\mathbb{R}.
I read several different solutions to similar questions, which recommended either \prescript, which is ugly, since there is a huge space between the asterisk and \leq or \sideset from amsmath, which refuses to work on symbols like +.
Does anyone know some universal pretty solution, which puts the asterisk neatly before this symbols?
-
This is not simply using a "presuperscript"; what we need is to add the asterisk and keeping the meaning of the symbols; moreover, the asterisk seems needing to be nearer the symbol.
\documentclass{article}
\usepackage{amssymb}
\newcommand{\hplus}{\mathbin{^*\!+}}
\newcommand{\hcdot}{\mathbin{^*\!\cdot}}
\newcommand{\hleq}{\mathrel{^*\!}\leq}
\newcommand{\hR}{{}^*\mathbb{R}}
\begin{document}
$a\hcdot b \hplus c \hleq d \in \hR$
\end{document}
(The "h" is for "hyper", as I suspect this is for nonstandard analysis.)
-
Thank you, I did not know about mathbin and mathrel before. And yes, this is indeed for non-standard analysis. – Tomas Jan 29 '13 at 17:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9354742169380188, "perplexity": 1536.9845465129538}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1404776401658.57/warc/CC-MAIN-20140707234001-00031-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://www.learncram.com/ml-aggarwal/ml-aggarwal-class-8-solutions-for-icse-maths-chapter-10-ex-10-3/ | # ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 10 Algebraic Expressions and Identities Ex 10.3
## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 10 Algebraic Expressions and Identities Ex 10.3
Question 1.
Multiply:
(i) (5x – 2) by (3x + 4)
(ii) (ax + b) by (cx + d)
(iii) (4p – 7) by (2 – 3p)
(iv) (2x2 + 3) by (3x – 5)
(v) (1.5a – 2.5b) by (1.5a + 2.56)
(vi) $$\left(\frac{3}{7} p^{2}+4 q^{2}\right) \text { by } 7\left(p^{2}-\frac{3}{4} q^{2}\right)$$
Solution:
Question 2.
Multiply:
(i) (x – 2y + 3) by (x + 2y)
(ii) (3 – 5x + 2 × 2) by (4x – 5)
Solution:
Question 3.
Multiply:
(i) (3x2 – 2x – 1) by (2x2 + x – 5)
(ii) (2 – 3y – 5y2) by (2y – 1 + 3y2)
Solution:
Question 4.
Simplify:
(i) (x2 + 3) (x – 3) + 9
(ii) (x + 3) (x – 3) (x + 4) (x – 4)
(iii) (x + 5) (x + 6) (x + 7)
(iv) (p + q – 2r) (2p – q + r) – 4qr
(v) (p + q) (r + s) + (p – q)(r – s) – 2(pr + qs)
(vi) (x + y + z) (x – y + z) + (x + y – z) (-x + y + z) – 4zx
Solution:
Question 5.
If two adjacent sides of a rectangle are 5x2 + 25xy + 4y2 and 2x2 – 2xy + 3y2, find its area.
Solution: | {"extraction_info": {"found_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": 0.9120052456855774, "perplexity": 2491.767052055875}, "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/1679296949701.56/warc/CC-MAIN-20230401063607-20230401093607-00064.warc.gz"} |
https://tel.archives-ouvertes.fr/tel-02286000 | # Étude d'un système quantique ouvert en interactions répétées de type maser à un atome.
Abstract : Open quantum systems describe the evolution of a system S in interaction with one or more other systems called environments. Two approaches in the literature to study such systems: the hamiltonian approach in which the entire system is considered, and the markovian approach in which one gives up the idea of describing the environment and only considers a so called effective dynamics of the system S which takes into account the effect of the environment.A particular class of such systems will interest us: the quantum systems with repeated interactions. The system S interacts successively with a series of independent subsystems. The approach of these systems is both Hamiltonian and Markovian. Their study plays a fundamental role in the understanding of light-matter interactions as well as in quantum optics (like one-atom maser experiment).In this thesis we study a repeated interaction system of the one-atom maser type. The model describes an electromagnetic field trapped in a cavity and a beam of atoms passing through it but with an additional reservoir interacting continuously with the electromagnetic field. The idea is that the cavity is not perfectly isolated and we describe the leaks in the cavity via the interaction with this reservoir. Thus the interaction between the electromagnetic field and the atoms is described by a quantum system with repeated interactions and the interaction between the electromagnetic field and the reservoir is described by a Hamiltonian approach of open quantum systems.The system "cavity+reservoir" has been studied by Konenberg, based on previous works by Arai. Usingan explicit diagonalization of the hamiltonian he proved some properties of return to equilibrium. In a first part we will give a new approach to it using recent results by Nam, Napiorkowski and Solovej about the diagonalization of quadratic bosonic hamiltonians.First we study the self-adjointness of some Hamiltonians which will play an important role in this thesis and we consider the diagonalization of one of them. In a second time, we study the long time behavior of the system, we obtain an explicit formula for the evolution at a given time of Weyl observables. These results will also allow us to study the total energy variation as well as the energy exchanges in the system. Finally we study the entropy production in the system and relate it to the energy variation. To do so we will need to slightly generalize the Jaksic-Pillet entropy production formula.
Keywords :
Document type :
Theses
Cited literature [69 references]
https://tel.archives-ouvertes.fr/tel-02286000
Contributor : Abes Star : Contact
Submitted on : Friday, September 13, 2019 - 12:13:06 PM
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### File
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### Identifiers
• HAL Id : tel-02286000, version 1
### Citation
Thibault Ebroussard. Étude d'un système quantique ouvert en interactions répétées de type maser à un atome.. Mathématiques générales [math.GM]. Université de Cergy Pontoise, 2018. Français. ⟨NNT : 2018CERG0975⟩. ⟨tel-02286000⟩
Record views | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8761188387870789, "perplexity": 870.6776376401892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178363809.24/warc/CC-MAIN-20210302095427-20210302125427-00615.warc.gz"} |
https://www.physicsforums.com/threads/oscillations-in-a-closed-pipe.13182/ | # Oscillations in a closed pipe
1. Jan 24, 2004
### nina
Why cant closed pipes produce even numbered harmonics? I have been given an explaination, but it isnt very detailed. I'm doing A2 (or just A level) physics, so an explaination suitable for this level would be greatly appreciated!
Thanks.
2. Jan 24, 2004
### himanshu121
What explanation have been given to you so far
3. Jan 24, 2004
### nina
Just that the 1st overtone has a frequency of 3 times the fundamental frequency, therefore the second harmonic is missing.
Is that all I need to know? It doesnt seem much of an explaination to give in an exam.
Thanks for your help...
4. Jan 24, 2004
### himanshu121
yup it not enough Can u calulate the wavelength in terms of L
5. Jan 24, 2004
### nina
Yes, I can calculate the wavelength from L. But why is there no second harmonic in a closed pipe?
6. Jan 24, 2004
### himanshu121
Coz for Even harmonics u need a arrangement like this
2f, 4f which is clearly not possible
And u never get L = even integer * Wavelength
7. Jan 24, 2004
### himanshu121
8. Jan 24, 2004
### nina
Thanks! I will look at that more closely. thanks for the link too.
9. Jan 24, 2004
### himanshu121
basically it is due to the fact that there is pressure antinode and node at closed and open ends
10. Jan 24, 2004
### nina
actually, it makes sense now.
thank you so much. i may just get a good grade in my exam now...
11. Jan 24, 2004
### himanshu121
Goodluck for exams
Have something to add?
Similar Discussions: Oscillations in a closed pipe | {"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": 0.9022159576416016, "perplexity": 3475.9967951586486}, "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-2017-09/segments/1487501170186.50/warc/CC-MAIN-20170219104610-00340-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://simbad.cds.unistra.fr/simbad/sim-ref?bibcode=2002A%26A...382..241M | other querymodes : Identifierquery Coordinatequery Criteriaquery Referencequery Basicquery Scriptsubmission TAP Outputoptions Help
2002A&A...382..241M - Astronomy and Astrophysics, volume 382, 241-255 (2002/1-4)
Crystalline silicate dust around evolved stars. III. A correlations study of crystalline silicate features.
MOLSTER F.J., WATERS L.B.F.M., TIELENS A.G.G.M., KOIKE C. and CHIHARA H.
Abstract (from CDS):
We have carried out a quantitative trend analysis of the crystalline silicates observed in the ISO spectra of a sample of 14 stars with different evolutionary backgrounds. We have modeled the spectra using a simple dust radiative transfer model and have correlated the results with other known parameters. We confirm the abundance difference of the crystalline silicates in disk and in outflow sources, as found by Molster et al. (1999Natur.401..563M). We found some evidence that the enstatite over forsterite abundance ratio differs, it is slightly higher in the outflow sources with respect to the disk sources. It is clear that more data is required to fully test this hypothesis. We show that the 69.0 micron feature, attributed to forsterite, may be a very suitable temperature indicator. We found that the enstatite is more abundant than forsterite in almost all sources. The temperature of the enstatite grains is about equal to that of the forsterite grains in the disk sources but slightly lower in the outflow sources. Crystalline silicates are on average colder than amorphous silicates. This may be due to the difference in Fe content of both materials. Finally we find an indication that the ratio of ortho to clino enstatite, which is about 1:1 in disk sources, shifts towards ortho enstatite in the high luminosity (outflow) sources. | {"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": 0.9324069023132324, "perplexity": 4366.160319088087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20230208155417-20230208185417-00376.warc.gz"} |
http://blog.chungyc.org/tag/probability/ | # The probability of a probability
The Shores of the Dirac Sea has a somewhat head-scratching puzzle about probabilities:
Let us say that someone gives you a lopsided bet. Say that with probability $r$ one gets heads, and with probability $1-r$ one gets tails, and you have to pick heads or tails. You only know the outcome of the first event. Let's say after the first toss it came out heads. What is the probability that $r > \frac{1}{2}$? | {"extraction_info": {"found_math": true, "script_math_tex": 3, "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": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9189018607139587, "perplexity": 271.94227427039544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218188717.24/warc/CC-MAIN-20170322212948-00403-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://forum.qt.io/topic/17280/create-object-o-files-in-the-same-folder-as-source | # Create object (.o) files in the same folder as source
• In the project I'm actually trying to compile, there's a physical folder structure like this one:
abc.c
misc.c
xyz.c
tools\misc.c
As you can see, there are two files called misc.c, but they are in different folders, then it shouldn't be a problem really.
However, at objects creation, the compile creates both misc.o in the same folder (release).The problem lies in the fact the first misc.o (source dir\misc.c) is replaced by the second misc.o (source dir\tools\misc.c).
The project is actually quite complex, in a way there's more than one file in the same situation, then what I need is a way to instruct Qmake to create the .o files in the same folder as their sources (Foe me, it should happen by default).That way, I would have the misc.o files created in their corresponding folders:
source dir\release\misc.o
and
source dir\release\tools\misc.o
Is there a way to do this on Qmake without having to rename the original files one by one?
Best regards
• Not exactly what you want, but you might go for qmake subdirs template (check out Qt sources to get an idea of what it is about). I think this would solve your problem (but will also require some work in redesigning qmake projects). | {"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": 0.8594609498977661, "perplexity": 1667.3851425740356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991288.0/warc/CC-MAIN-20210518160705-20210518190705-00433.warc.gz"} |
https://ai.stackexchange.com/questions/13080/understanding-the-proof-that-a-search-is-optimal | # Understanding the proof that A* search is optimal
I don't understand the proof that $$A^*$$ is optimal.
Assume $$A^*$$ returns $$p$$ but there exists a $$p'$$ that is cheaper. When $$p$$ is chosen from the frontier, assume $$p''$$ (Which is part of the path $$p'$$) is chosen from the frontier. Since $$p$$ was chosen before $$p''$$, then we have $$\text{cost}(p) + \text{heuristic}(p) \leq \text{cost}(p'') + \text{heuristic}(p'')$$. $$p$$ ends at goal, therefore the $$\text{heuristic}(p) = 0$$. Therefore $$\text{cost}(p) \leq \text{cost}(p'') + \text{heuristic}(p'') \leq \text{cost}(p')$$ because heuristics are admissible. Therefore we have a contradiction.
I am confused: can't we also assume there's a cheaper path that's in a frontier closer to the start node than $$p$$? Or is part of the proof that's not possible because $$A^*$$ would have examined that path because it is like BFS with lowest cost search, so, if there's a cheaper path, it'll be at a further frontier?
The key phrase here is
In other words, the heuristics never overestimate the path length:
$$cost(n) + heuristic(n) \le cost(\text{any path going through n})$$
And since the frontier is ordered by $$\textbf{cost + heuristic}$$, when a completed path $$p$$ is dequeued from the frontier, we know that it must necessarily be $$\le$$ any path going through some other frontier node $$q$$, because
$$cost(p) = cost(p) + heuristic(p) \le cost(q) + heuristic(q) \le cost(\text{any path going through q})$$
• However, $p'$ in the proof is not any path, it is a path assumed to be cheaper than $p$. So, how do you explain the conclusion "Therefore $\text{cost}(p) \leq \text{cost}(p'') + \text{heuristic}(p'') \leq \text{cost}(p')$"? By assumption, $\text{cost}(p') < \text{cost}(p)$. How come that you can conclude the opposite only because you say, in the proof, that you remove $p$ before $p''$ from the frontier? Even if the frontier is ordered, it does not imply that you have not removed $p'$ before $p$. I think this is what is confusing in the provided proof. – nbro Jun 27 '19 at 18:22
• @nbro: It is a proof by contradiction. That is the contradiction. You can in fact assume that $p'$ has not been dequeued yet, because if it had, the algorithm would have terminated and returned $p'$ – BlueRaja - Danny Pflughoeft Jun 27 '19 at 18:32
• I think that's the actual answer to the original question "can't we also assume there's a cheaper path that's in a frontier closer to the start node than $p$?". – nbro Jun 27 '19 at 18:36
• I mean, the answer to that question is really the proof given by OP. My answer is simply an alternative proof that more directly answers his question, hopefully in a way that makes more intuitive sense. Neither of these are the same as your original question, which should technically be a separate question on this site, but is easily answered in a comment. – BlueRaja - Danny Pflughoeft Jun 27 '19 at 18:47
• How is my question different than "can't we also assume there's a cheaper path that's in a frontier closer to the start node than 𝑝?", which, I suppose, it means "can't we also assume that $p'$ is dequeued before $p$"? The proof simply tells you, in an intricate way, that you remove $p$ before $p'$, hence $p'$ cannot be cheaper. By the way, how is yours an alternative proof? You just rephrased parts of the proof. – nbro Jun 27 '19 at 18:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9628151059150696, "perplexity": 407.8151549990183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1618038078021.18/warc/CC-MAIN-20210414185709-20210414215709-00562.warc.gz"} |
http://www.thespectrumofriemannium.com/2012/10/16/log046-the-cherenkov-effect/?replytocom=9 | # LOG#046. The Cherenkov effect.
The Cherenkov effect/Cherenkov radiation, sometimes also called Vavilov-Cherenkov radiation, is our topic here in this post.
In 1934, P.A. Cherenkov was a post graduate student of S.I.Vavilov. He was investigating the luminescence of uranyl salts under the incidence of gamma rays from radium and he discovered a new type of luminiscence which could not be explained by the ordinary theory of fluorescence. It is well known that fluorescence arises as the result of transitions between excited states of atoms or molecules. The average duration of fluorescent emissions is about and the transition probability is altered by the addition of “quenching agents” or by some purification process of the material, some change in the ambient temperature, etc. It shows that none of these methods is able to quench the fluorescent emission totally, specifically the new radiation discovered by Cherenkov. A subsequent investigation of the new radiation ( named Cherenkov radiation by other scientists after the Cherenkov discovery of such a radiation) revealed some interesting features of its characteristics:
1st. The polarization of luminescence changes sharply when we apply a magnetic field. Cherenkov radiation luminescence is then causes by charged particles rather than by photons, the -ray quanta! Cherenkov’s experiment showed that these particles could be electrons produced by the interaction of -photons with the medium due to the photoelectric effect or the Compton effect itself.
2nd. The intensity of the Cherenkov’s radiation is independent of the charge Z of the medium. Therefore, it can not be of radiative origin.
3rd. The radiation is observed at certain angle (specifically forming a cone) to the direction of motion of charged particles.
The Cherenkov radiation was explained in 1937 by Frank and Tamm based on the foundations of classical electrodynamics. For the discovery and explanation of Cherenkov effect, Cherenkov, Frank and Tamm were awarded the Nobel Prize in 1958. We will discuss the Frank-Tamm formula later, but let me first explain how the classical electrodynamics handle the Vavilov-Cherenkov radiation.
The main conclusion that Frank and Tamm obtained comes from the following observation. They observed that the statement of classical electrodynamics concerning the impossibility of energy loss by radiation for a charged particle moving uniformly and following a straight line in vacuum is no longer valid if we go over from the vacuum to a medium with certain refractive index . They went further with the aid of an easy argument based on the laws of conservation of momentum and energy, a principle that rests in the core of Physics as everybody knows. Imagine a charged partice moving uniformly in a straight line, and suppose it can loose energy and momentum through radiation. In that case, the next equation holds:
This equation can not be satisfied for the vacuum but it MAY be valid for a medium with a refractive index gretear than one . We will simplify our discussion if we consider that the refractive index is constant (but similar conclusions would be obtained if the refractive index is some function of the frequency).
By the other hand, the total energy E of a particle having a non-null mass and moving freely in vacuum with some momentum p and velocity v will be:
and then
Moreover, the electromagnetic radiation in vaccum is given by the relativistic relationship
From this equation, we easily get that
Since the particle velocity is , we obtain that
In conclusion: the laws of conservation of energy and momentum prevent that a charged particle moving with a rectilinear and uniform motion in vacuum from giving away its energy and momentum in the form of electromagnetic radiation! The electromagnetic radiation can not accept the entire momentum given away by the charged particle.
Anyway, we realize that this restriction and constraint is removed and given up when the aprticle moves in a medium with a refractive index . In this case, the velocity of light in the medium would be
and the velocity v of the particle may not only become equal to the velocity of light in the medium, but even exceed it when the following phenomenological condition is satisfied:
It is obvious that, when the condition
will be satisfied for electromagnetic radiation emitted strictly in the direction of motion of the particle, i.e., in the direction of the angle . If , this equation is verified for some direction along with , where
is the projection of the particle velocity v on the observation direction. Then, in a medium with , the conservation laws of energy and momentum say that it is allowed that a charged particle with rectilinear and uniform motion, can loose fractions of energy and momentum and , whenever those lost energy and momentum is carried away by an electromagnetic radiation propagating in the medium at an angle/cone given by:
with respect to the observation direction of the particle motion.
These arguments, based on the conservation laws of momenergy, do not provide any ide about the real mechanism of the energy and momentum which are lost during the Cherenkov radiation. However, this mechanism must be associated with processes happening in the medium since the losses can not occur ( apparently) in vacuum under normal circumstances ( we will also discuss later the vacuum Cherenkov effect, and what it means in terms of Physics and symmetry breaking).
We have learned that Cherenkov radiation is of the same nature as certain other processes we do know and observer, for instance, in various media when bodies move in these media at a velocity exceeding that of the wave propagation. This is a remarkable result! Have you ever seen a V-shaped wave in the wake of a ship? Have you ever seen a conical wave caused by a supersonic boom of a plane or missile? In these examples, the wave field of the superfast object if found to be strongly perturbed in comparison with the field of a “slow” object ( in terms of the “velocity of sound” of the medium). It begins to decelerate the object!
Question: What is then the mechanism behind the superfast motion of a charged particle in a medium wiht a refractive index producing the Cherenkov effect/radiation?
Answer: The mechanism under the Cherenkov effect/radiation is the coherent emission by the dipoles formed due to the polarization of the medium atoms by the charged moving particle!
The idea is as follows. Dipoles are formed under the action of the electric field of the particle, which displaces the electrons of the sorrounding atoms relative to their nuclei. The return of the dipoles to the normal state (after the particle has left the given region) is accompanied by the emission of an electromagnetic signal or beam. If a particle moves slowly, the resulting polarization will be distribute symmetrically with respect to the particle position, since the electric field of the particle manages to polarize all the atoms in the near neighbourhood, including those lying ahead in its path. In that case, the resultant field of all dipoles away from the particle are equal to zero and their radiations neutralize one to one.
Then, if the particle move in a medium with a velocity exceeding the velocity or propagation of the electromagnetic field in that medium, i.e., whenever , a delayed polarization of the medium is observed, and consequently the resulting dipoles will be preferably oriented along the direction of motion of the particle. See the next figure:
It is evident that, if it occurs, there must be a direction along which a coherent radiation form dipoles emerges, since the waves emitted by the dipoles at different points along the path of the particle may turn our to be in the same phase. This direction can be easiy found experimentally and it can be easily obtained theoretically too. Let us imagine that a charged particle move from the left to the right with some velocity in a medium with a refractive index, with . We can apply the Huygens principle to build the wave front for the emitted particle. If, at instant , the aprticle is at the point , the surface enveloping the spherical waves emitted by the same particle on its own path from the origin at to the arbitrary point . The radius of the wave at the point at such an instant t is equal to . At the same moment, the wave radius at the point is equal to . At any intermediate point x’, the wave radius at instant t will be . Then, the radius decreases linearly with increasing . Thus, the enveloping surface is a cone with angle , where the angle satisfies in addition
The normal to the enveloping surface fixes the direction of propagation of the Cherenkov radiation. The angle between the normal and the -axis is equal to , and it is defined by the condition
or equivalently
This is the result we anticipated before. Indeed, it is completely general and Quantum Mechanics instroudces only a light and subtle correction to this classical result. From this last equation, we observer that the Cherenkov radiation propagates along the generators of a cone whose axis coincides with the direction of motion of the particle an the cone angle is equal to . This radiation can be registered on a colour film place perpendicularly to the direction of motion of the particle. Radiation flowing from a radiator of this type leaves a blue ring on the photographic film. These blue rings are the archetypical fingerprints of Vavilov-Cherenkov radiation!
The sharp directivity of the Cherenkov radiation makes it possible to determine the particle velocity from the value of the Cherenkov’s angle . From the Cherenkov’s formula above, it follows that the range of measurement of is equal to
For , the radiation is observed at an angle , while for the extreme with , the angle reaches a maximum value
For instance, in the case of water, and . Therefore, the Cherenkov radiation is observed in water whenever . For electrons being the charged particles passing through the water, this condition is satisfied if
As a consequence of this, the Cherenkov effect should be observed in water even for low-energy electrons ( for instance, in the case of electrons produced by beta decay, or Compton electrons, or photoelectroncs resulting from the interaction between water and gamma rays from radioactive products, the above energy can be easily obtained and surpassed!). The maximum angle at which the Cherenkov effec can be observed in water can be calculated from the condition previously seen:
This angle (for water) shows to be equal to about . In agreement with the so-called Frank-Tamm formula ( please, see below what that formula is and means), the number of photons in the frequency interval and emitted by some particle with charge Z moving with a velocity in a medium with a refractive indez n is provided by the next equation:
This formula has some striking features:
1st. The spectrum is identical for particles with , i.e., the spectrum is exactly the same, irespectively the nature of the particle. For instance, it could be produced both by protons, electrons, pions, muons or their antiparticles!
2nd. As Z increases, the number of emitted photons increases as .
3rd. increases with , the particle velocity, from zero ( with ) to
with .
4th. is approximately independent of . We observe that .
5th. As the spectrum is uniform in frequency, and , this means that the main energy of radiation is concentrated in the extreme short-wave region of the spectrum, i.e.,
And then, this feature explains the bluish-violet-like colour of the Cherenkov radiation!
Indeed, this feature also indicates the necessity of choosing materials for practical applications that are “transparent” up to the highest frequencies ( even the ultraviolet region). As a rule, it is known that in the X-ray region and hence the Cherenkov condition can not be satisfied! However, it was also shown by clever experimentalists that in some narrow regions of the X-ray spectrum the refractive index is ( the refractive index depends on the frequency in any reasonable materials. Practical Cherenkov materials are, thus, dispersive! ) and the Cherenkov radiation is effectively observed in apparently forbidden regions.
The Cherenkov effect is currently widely used in diverse applications. For instance, it is useful to determine the velocity of fast charged particles ( e.g, neutrino detectors can not obviously detect neutrinos but they can detect muons and other secondaries particles produced in the interaction with some polarizable medium, even when they are produced by (electro)weak interactions like those happening in the presence of chargeless neutrinos). The selection of the medium fo generating the Cherenkov radiation depends on the range of velocities over which measurements have to be produced with the aid of such a “Cherenkov counter”. Cherenkov detectors/counters are filled with liquids and gases and they are found, e.g., in Kamiokande, Superkamiokande and many other neutrino detectors and “telescopes”. It is worth mentioning that velocities of ultrarelativistic particles are measured with Cherenkov detectors whenever they are filled with some special gasesous medium with a refractive indes just slightly higher than the unity. This value of the refractive index can be changed by realating the gas pressure in the counter! So, Cherenkov detectors and counters are very flexible tools for particle physicists!
Remark: As I mentioned before, it is important to remember that (the most of) the practical Cherenkov radiators/materials ARE dispersive. It means that if is the photon frequency, and is the wavenumber, then the photons propagate with some group velocity , i.e.,
Note that if the medium is non-dispersive, this formula simplifies to the well known formula . As it should be for vacuum.
Accodingly, following the PDG, Tamm showed in a classical paper that for dispersive media the Cherenkov radiation is concentrated in a thin conical shell region whose vertex is at the moving charge and whose opening half-angle is given by the expression
where is the critical Cherenkov angle seen before, is the central value of the small frequency range under consideration under the Cherenkov condition. This cone has an opening half-angle (please, compare with the previous convention with for consistency), and unless the medium is non-dispersive (i.e. , ), we get . Typical Cherenkov radiation imaging produces blue rings.
THE CHERENKOV EFFECT: QUANTUM FORMULAE
When we considered the Cherenkov effect in the framework of QM, in particular the quantum theory of radiation, we can deduce the following formula for the Cherenkov effect that includes the quantum corrections due to the backreaction of the particle to the radiation:
where, like before, , n is the refraction index, is the De Broglie wavelength of the moving particle and is the wavelength of the emitted radiation.
Cherenkov radiation is observed whenever (i.e. if ), and the limit of the emission is on the short wave bands (explaining the typical blue radiation of this effect). Moreover, corresponds to .
By the other hand, the radiated energy per particle per unit of time is equal to:
where is the angular frequency of the radiation, with a maximum value of .
Remark: In the non-relativistic case, , and the condition implies that . Therefore, neglecting the quantum corrections (the charged particle self-interaction/backreaction to radiation), we can insert the limit and the above previous equations will simplify into:
Remember: is determined with the condition , where represents the dispersive effect of the material/medium through the refraction index.
THE FRANK-TAMM FORMULA
The number of photons produced per unit path length and per unit of energy of a charged particle (charge equals to ) is given by the celebrated Frank-Tamm formula:
In terms of common values of fundamental constants, it takes the value:
or equivalently it can be written as follows
The refraction index is a function of photon energy , and it is also the sensitivity of the transducer used to detect the light with the Cherenkov effect! Therefore, for practical uses, the Frank-Tamm formula must be multiplied by the transducer response function and integrated over the region for which we have .
Remark: When two particles are close toghether ( to be close here means to be separated a distance wavelength), the electromagnetic fields form the particles may add coherently and affect the Cherenkov radiation. The Cherenkov radiation for a electron-positron pair at close separation is suppressed compared to two independent leptons!
Remark (II): Coherent radio Cherenkov radiation from electromagnetic showers is significant and it has been applied to the study of cosmic ray air showers. In addition to this, it has been used to search for electron neutrinos induced showers by cosmic rays.
CHERENKOV DETECTOR: MAIN FORMULA AND USES
The applications of Cherenkov detectors for particle identification (generally labelled as PID Cherenkov detectors) are well beyond the own range of high-energy Physics. Its uses includes: A) Fast particle counters. B) Hadronic particle indentifications. C) Tracking detectors performing complete event reconstruction. The PDG gives some examples of each category: a) Polarization detector of SLD, b) the hadronic PID detectors at B factories like BABAR or the aerogel threshold Cherenkov in Belle, c) large water Cherenkov counters liket those in Superkamiokande and other neutrino detector facilities.
Cherenkov detectors contain two main elements: 1) A radiator/material through which the particle passes, and 2) a photodetector. As Cherenkov radiation is a weak source of photons, light collection and detection must be as efficient as possible. The presence of a refractive material specifically designed to detect some special particles is almost vindicated in general.
The number of photoelectrons detected in a given Cherenkov radiation detector device is provided by the following formula (derived from the Tamm-Frank formula simply taking into account the efficiency in a straightforward manner):
where is the path length of the particle in the radiator/material, is the efficiency for the collector of Cherenkov light and transducing it in photoelectrons, and
Remark: The efficiencies and the Cherenkov critical angle are functions of the photon energy, generally speaking. However, since the typical energy dependen variation of the refraction index is modest, a quantity sometimes called Cherenkov detector quality fact can be defined as follows
In this case, we can write
Remark(II): Cherenkov detectors are classified into imaging or threshold types, depending on its ability to make use of Cherenkov angle information. Imaging counters may be used to track particles as well as identify particles.
Other main uses/applications of the Vavilov-Cherenkov effect are:
1st. Detection of labeled biomolecules. Cherenkov radiation is widely used to facilitate the detection of small amounts and low concentrations of biomolecules. For instance, radioactive atoms such as phosphorus-32 are readily introduced into biomolecules by enzymatic and synthetic means and subsequently may be easily detected in small quantities for the purpose of elucidating biological pathways and in characterizing the interaction of biological molecules such as affinity constants and dissociation rates.
2nd. Nuclear reactors. Cherenkov radiation is used to detect high-energy charged particles. In pool-type nuclear reactors, the intensity of Cherenkov radiation is related to the frequency of the fission events that produce high-energy electrons, and hence is a measure of the intensity of the reaction. Similarly, Cherenkov radiation is used to characterize the remaining radioactivityof spent fuel rods.
3rd. Astrophysical experiments. The Cherenkov radiation from these charged particles is used to determine the source and intensity of the cosmic ray,s which is used for example in the different classes of cosmic ray detection experiments. For instance, Ice-Cube, Pierre-Auger, VERITAS, HESS, MAGIC, SNO, and many others. Cherenkov radiation can also be used to determine properties of high-energy astronomical objects that emit gamma rays, such as supernova remnants and blazars. In this last class of experiments we place STACEE, in new Mexico.
4th. High-energy experiments. We have quoted already this, and there many examples in the actual LHC, for instance, in the ALICE experiment.
Vacuum Cherenkov radiation (VCR) is the alledged and conjectured phenomenon which refers to the Cherenkov radiation/effect of a charged particle propagating in the physical vacuum. You can ask: why should it be possible? It is quite straightforward to understand the answer.
The classical (non-quantum) theory of relativity (both special and general) clearly forbids any superluminal phenomena/propagating degrees of freedom for material particles, including this one (the vacuum case) because a particle with non-zero rest mass can reach speed of light only at infinite energy (besides, the nontrivial vacuum itself would create a preferred frame of reference, in violation of one of the relativistic postulates).
However, according to modern views coming from the quantum theory, specially our knowledge of Quantum Field Theory, physical vacuum IS a nontrivial medium which affects the particles propagating through, and the magnitude of the effect increases with the energies of the particles!
Then, a natural consequence follows: an actual speed of a photon becomes energy-dependent and thus can be less than the fundamental constant of speed of light, such that sufficiently fast particles can overcome it and start emitting Cherenkov radiation. In summary, any charged particle surpassing the speed of light in the physical vacuum should emit (Vacuum) Cherenkov radiation. Note that it is an inevitable consequence of the non-trivial nature of the physical vacuum in Quantum Field Theory. Indeed, some crazy people saying that superluminal particles arise in jets from supernovae, or in colliders like the LHC fail to explain why those particles don’t emit Cherenkov radiation. It is not true that real particles become superluminal in space or collider rings. It is also wrong in the case of neutrino propagation because in spite of being chargeless, neutrinos should experiment an analogue effect to the Cherenkov radiation called the Askaryan effect. Other (alternative) possibility or scenario arises in some Lorentz-violating theories ( or even CPT violating theories that can be equivalent or not to such Lorentz violations) when a speed of a propagating particle becomes higher than c which turns this particle into the tachyon. The tachyon with an electric charge would lose energy as Cherenkov radiation just as ordinary charged particles do when they exceed the local speed of light in a medium. A charged tachyon traveling in a vacuum therefore undergoes a constant proper-time acceleration and, by necessity, its worldline would form an hyperbola in space-time. These last type of vacuum Cherenkov effect can arise in theories like the Standard Model Extension, where Lorentz-violating terms do appear.
One of the simplest kinematic frameworks for Lorentz Violating theories is to postulate some modified dispersion relations (MODRE) for particles , while keeping the usual energy-momentum conservation laws. In this way, we can provide and work out an effective field theory for breaking the Lorentz invariance. There are several alternative definitions of MODRE, since there is no general guide yet to discriminate from the different theoretical models. Thus, we could consider a general expansion in integer powers of the momentum, in the next manner (we set units in which ):
However, it is generally used a more soft expansion depending only on positive powers of the momentum in the MODRE. In such a case,
and where . If Lorentz violations are associated to the yet undiscovered quantum theory of gravity, we would get that ordinary deviations of the dispersion relations in the special theory of relativity should appear at the natural scale of the quantum gravity, say the Planck mass/energy. In units where we obtain that Planck mass/energy is:
Lets write and parametrize the Lorentz violations induced by the fundamental scale of quantum gravity (naively this Planck mass scale) by:
Here, is a dimensionless quantity that can differ from one particle (type) to another (type). Considering, for instance , since the seems to be ruled out by previous terrestrial experiments, at higer energies the lowest non-null term will dominate the expansion with . The MODRE reads:
and where the label in the term is specific of the particle type. Such corrections might only become important at the Planck scale, but there are two exclusions:
1st. Particles that propagate over cosmological distances can show differences in their propagation speed.
2nd. Energy thresholds for particle reactions can be shifted or even forbidden processes can be allowed. If the -term is comparable to the -term in the MODRE. Thus, threshold reactions can be significantly altered or shifted, because they are determined by the particle masses. So a threshold shift should appear at scales where:
Imposing/postulating that , the typical scales for the thresholds for some diffent kind of particles can be calculated. Their values for some species are given in the next table:
We can even study some different sources of modified dispersion relationships:
1. Measurements of time of flight.
2. Thresholds creation for: A) Vacuum Cherenkov effect, B) Photon decay in vacuum.
3. Shift in the so-called GZK cut-off.
4. Modified dispersion relationships induced by non-commutative theories of spacetime. Specially, there are time shifts/delays of photon signals induced by non-commutative spacetime theories.
We will analyse this four cases separately, in a very short and clear fashion. I wish!
Case 1. Time of flight. This is similar to the recently controversial OPERA experiment results. The OPERA experiment, and other similar set-ups, measure the neutrino time of flight. I dedicated a post to it early in this blog
http://thespectrumofriemannium.wordpress.com/2012/06/08/
In fact, we can measure the time of flight of any particle, even photons. A modified dispersion relation, like the one we introduced here above, would lead to an energy dependent speed of light. The idea of the time of flight (TOF) approach is to detect a shift in the arrival time of photons (or any other massless/ultra-relativistic particle like neutrinos) with different energies, produced simultaneous in a distant object, where the distance gains the usually Planck suppressed effect. In the following we use the dispersion relation for only, as modifications in higher orders are far below the sensitivity of current or planned experiments. The modified group velocity becomes:
and then, for photons,
The time difference in the photon shift detection time will be:
where D is the distance multiplied (if it were the case) by the redshift to correct the energy with the redshift. In recent years, several measurements on different objects in various energy bands leading to constraints up to the order of 100 for . They can be summarized in the next table ( note that the best constraint comes from a short flare of the Active Galactic Nucleus (AGN) Mrk 421, detected in the TeV band by the Whipple Imaging Air Cherenkov telescope):
There is still room for improvements with current or planned experiments, although the distance for TeV-observations is limited by absorption of TeV photons in low energy metagalactic radiation fields. Depending on the energy density of the target photon field one gets an energy dependent mean free path length, leading to an energy and redshift dependent cut off energy (the cut off energy is defined as the energy where the optical depth is one).
2. Thresholds creation for: A) Vacuum Cherenkov effect, B) Photon decay in vacuum. By the other hand, the interaction vertex in quantum electrodynamics (QED) couples one photon with two leptons. When we assume for photons and leptons the following dispersion relations (for simplicity we adopt all units with M=1). Then:
Let us write the photon tetramomentum like and the lepton tetramomentum and . It can be shown that the transferred tetramomentum will be
where the r.h.s. is always positive. In the Lorentz invariant case the parameters are zero, so that this equation can’t be solved and all processes of the single vertex are forbidden. If these parameters are non-zero, there can exist a solution and so these processes can be allowed. We now consider two of these interactions to derive constraints on the parameters . The vacuum
Cherenkov effect and the spontaneous photon-decay .
A) As we have studied here, the vacuum Cherenkov effect is a spontaneous emission of a photon by a charged particle . These effect occurs if the particle moves faster than the slowest possible radiated photon in vacuum!
In the case of , the maximal attainable speed for the particle is faster than c. This means, that the particle can always be faster than a zero energy photon with
and it is independent of . In the case of , i.e., decreases with energy, you need a photon with . This is only possible if .
Therefore, due to the radiation of photons such an electron loose energy. The observation of high energetic electrons allows to derive constraints on and . In the case of , in the case with , we have the bound
Moreover, from the observation of 50 TeV photons in the Crab Nebula (and its pulsar) one can conclude the existens of 50 TeV electrons due to the inverse Compton scattering of these electrons with those photons. This leads to a constraint on of about
where we have used in this case.
B) The decay of photons into positrons and electrons should be a very rapid spontaneous decay process. Due to the observation of Gamma rays from the Crab Nebula on earth with an energy up to . Thus, we can reason that these rapid decay doesn’t occur on energies below 50 TeV. For the constraints on and these condition means (again we impose n=3):
.
3. Shift in the GZK cut-off. As the energy of a proton increases,the pion production reaction can happen with low energy photons of the Cosmic Microwave Background (CMB).
This leads to an energy dependent mean free path length of the particles, resulting in a cutoff at energies around . This is the the celebrated Greisen-Kuzmin-Zatsepin (GZK) cut off. The resonance for the GZK pion photoproduction with the CMB backgroud can be read from the next condition (I will derive this condition in a future post):
Thus in Lorentz invariant world, the mean free path length of a particle of energy 5.1019 eV is 50 Mpc i.e. particle over this energy are readily absorbed due to pion photoproduction reaction. But most of the sources of particle of ultra high energy are outside 50 Mpc. So, one expects no trace of particles of energy above on Earth. From the experimental point of view AGASA has found a few particles having energy higher than the constraint given by GZK cutoff limit and claimed to be disproving the presence of GZK cutoff or at least for different threshold for GZK cutoff, whereas HiRes is consistent with the GZK effect. So, there are two main questions, not yet completely unsolved:
i) How one can get definite proof of non-existence GZK cut off?
ii) If GZK cutoff doesn’t exist, then find out the reason?
The first question could by answered by observation of a large sample of events at these energies, which is necessary for a final conclusion, since the GZK cutoff is a statistical phenomena. The current AUGER experiment, still under construction, may clarify if the GZK cutoff exists or not. The existence of the GZK cutoff would also yield new limits on Lorentz or CPT violation. For the second question, one explanation can be derived from Lorentz violation. If we do the calculation for GZK cutoff in Lorentz violated world we would get the modified proton dispersion relation as described in our previous equations with MODRE.
4. Modified dispersion relationships induced by non-commutative theories of spacetime. As we said above, there are time shifts/delays of photon signals induced by non-commutative spacetime theories. Noncommutative spacetime theories introduce a new source of MODRE: the fuzzy nature of the discreteness of the fundamental quantum spacetime. Then, the general ansatz of these type of theories comes from:
where
are the components of an antisymmetric Lorentz-like tensor which components are the order one. The fundamental scale of non-commutativity is supposed to be of the Planck length. However, there are models with large extra dimensions that induce non-commutative spacetime models with scale near the TeV scale! This is interesting from the phenomenological aside as well, not only from the theoretical viewpoint. Indeed, we can investigate in the following whether astrophysical observations are able to constrain certain class of models with noncommutative spacetimes which are broken at the TeV scale or higher. However, there due to the antisymmetric character of the noncommutative tensor, we need a magnetic and electric background field in order to study these kind of models (generally speaking, we need some kind of field inducing/producing antisymmetric field backgrounds), and then the dispersion relation for photons remains the same as in a commutative spacetime. Furthermore, there is no photon energy dependence of the dispersion relation. Consequently, the time-of-flight experiments are inappopriate because of their energy-dependent dispersion. Therefore, we suggest the next alternative scenario: suppose, there exists a strong magnetic field (for instance, from a star or a cluster of stars) on the path photons emitted at a light source (e.g. gamma-ray bursts). Then, analogous to gravitational lensing, the photons experience deflection and/or change in time-of-arrival, compared to the same path without a magnetic background field. We can make some estimations for several known objects/examples are shown in this final table:
In summary:
1st. Vacuum Cherenkov and related effects modifying the dispersion relations of special relativity are natural in many scenarios beyond the Standard Relativity (BSR) and beyond the Standard Model (BSM).
2nd. Any theory allowing for superluminal propagation has to explain the null-results from the observation of the vacuum Cherenkov effect. Otherwise, they are doomed.
3rd. There are strong bounds coming from astrophysical processes and even neutrino oscillation experiments that severely imposes and kill many models. However, it is true that current MODRE bound are far from being the most general bounds. We expect to improve these bounds with the next generation of experiments.
4th. Theories that can not pass these tests (SR obviously does) have to be banned.
5th. Superluminality has observable consequences, both in classical and quantum physics, both in standard theories and theories beyond standard theories. So, it you buid a theory allowing superluminal stuff, you must be very careful with what kind of predictions can and can not do. Otherwise, your theory is complentely nonsense.
As a final closing, let me include some nice Cherenkov rings from Superkamiokande and MiniBoone experiments. True experimental physics in action. And a final challenge…
FINAL CHALLENGE: Are you able to identify the kind of particles producing those beautiful figures? Let me know your guesses ( I do know the answer, of course).
Figure 1. Typical SuperKamiokande Ring. I dedicate this picture to my admired Japanase scientists there. I really, really admire that country and their people, specially after disasters like the 2011 Earthquake and the Fukushima accident. If you are a japanase reader/follower, you must know we support your from abroad. You were not, you are not and you shall not be alone.
Figure 2. Typical MiniBooNe ring. History: I used this nice picture in my Master Thesis first page, as the cover/title page main picture!
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#### LOG#046. The Cherenkov effect. — 8 Comments
how can we have Cherenkov radiation in vacuum, since vacuum has no molecular structure?
I mean, the radiation comes from molecules which are polarized and depolarized when a particle with u>c/n passes by.
This way we get the angular distribution etc.
Thank you!
E.Chaniotakis
• The Cherenkov effect IN VACUUM, as I mentioned in this entry, is completely speculative and it has NOT been measured YET. How can we have Cherenkov effect in vacuum? I will be grossy but I hope that “clear”… Well, according to Quantum Field Theory, the vacuum is a “sea of particles-antiparticles” (your “molecules AND antimolecules”), and therefore, it could be possible that this sea of particles/antiparticles polarize/depolarize when a highly energetic particle (enough to polarize the vacuum) passes through it. Please, note that we DO KNOW that QUANTUM vacuum is polarizable, in fact it is inevitable from Quantum Mechanics and Quantum Field Theory… For instance, the vacuum polarization effects are important in the structure of QED calculations and renormalization. The Casimir effect, the dynamical Casimir effect, and some other important effects MATTER. Finally, let me point out some important vacuum effects that have not been measured but are believed to be “real” in the framework of QFT and/or “natural” generalizations of QFT:
1st. Vacuum Cherenkov effect.
2nd. Schwinger effect or particle production in strong gauge (e.g. electromagnetic) fields.
3rd. Unruh effect. Any accelerated particle in vacuum “sees” a thermal spectrum of particles that would be absent in rest. Is the vacuum state relative?
4th. Hawking effect. Particle production in strong GRAVITATIONAL fields close to the Black Hole horizon. Is the information of recicled objects conserved somehow by correlations in the outgoing Hawking radiation?
5th. Casimir effect or (gauge) “repulsion” force of certain configurations and geometries with “vacuum” as “media”. The plates can have many geometries but they are static.
6th. Dynamical Casimir effect. The same case as the previous, but with at least a “moving plate” that allows for a different kind of Casimir effect. Experimental evidence of this new Casimir effect is quite recent…
Please, note that vacuum is polarizable in colliders like the LHC were we produce lots of particles from proton-proton collisions. However, the problem with Cherenkov vacuum effect radiation is that it is likely very weak, but at least, some theoretical physicists believe it has to exist, for consistency, or it doesn’t , it has to be due a good reason!
2. Hello,
From the Heisenberg uncertainty principle, . Thus the higher the mass of the virtual particles created, the shorter the time they will ‘live’.
This favours the scenario of electron-positron pairs created more frequently this way rather than ‘big molecules’.
I assume heuristically that electron positron pairs are stochastically produced in vacuo, which will lead in a stochastic production of vacuum cherenkov radiation.
So therefore there will be no coherence and the signal will be vanishing.
Now, if we want to measure the energy of the emitted photons we can find an order of magnitude this way: Take the positronium and find the energy levels separation.
Energy levels go as ,so a typical transition of n=1 to n=2 gives off ~5eV energy which is in the UV.
Assuming that there exist tachyons:
So to find vacuum cherenkov photons, for a high energy tachyon ;thus low speed , one searches for photons almost collinear with the tachyon track.
As the tachyon will lose energy due to Cherenkov energy loss, the angle of emission is likely to change and get higher. (I believe that it will not be higher than 90 degrees but not really sure ).
In order to investigate the tachyon production experimentally, what would you suggest?
• You have some heuristical ideas right, but you are wrong with respect to tachyons and Cherenkov effect.
Any observed Cherenkov effect is NOT related at all with tachyons, it is related to particles travelling faster than the speed of light in certain medium (not vacuum). It is true that if tachyons were probed to exist, they should leave a track of vacuum polarization signals. Indeed, that is how tachyons could be observed. Tachyons searches were carried out for years with negative results. Today, nobody believes in tachyons, despite the fact the higgs field /mechanism is essentially a tachyon condensation. Moreover, relativistic tachyons are subtle entities, they have to agree with special relativity, so it is really hard to explain it with words how it works (specially without too many mathematics). Maybe I will talk about it in this blog in the future. Cherenkov effect in vacuum is not related with tachyons a priori…Be aware of it…
With respecto to your arguments…Positronium is useful, but it has problems, as you have noted, its mass splitting is about 5eV in the main line transition n=1 to n=2… In fact, experimentally, you can not search for the Cherenkov vacuum effect so easily using positronium as “marker” due to other technical problems (I can not tell you in detail that here). If you go back to your ideas, one should expect that the lighter particles would be more frequently created, and the lighter particles in the Standard Model are not electrons/positros but…Neutrinos! However, these particles interact very weakly with matter, and excepting the neutrino oscillations at relatively low energies, and electroweak interactions at ~100 GeV and higher energies, they are difficult to work with. Hypothetical tachyon/antitachyon production could be handled in very different ways… However, I doubt tachyons exist as real particles. Some people suggested long ago that neutrinos were tachyons, but that idea is completely wrong. Other people suggested that neutrinos were superluminal but no tachyons, …That is nonsense too, since if you have a superluminal particle in vacuum, as I have explained here, it should leave a track of Cherenkov light (vacuum Cherenkov effect indeed) that has never been observed in any current detector in vacuum (and we have many good ones). Of course, it could be the Cherenkov vacuum radiation is weaker than expected and it remains undetected even with Cherenkov detectors…That is completely wrong! Unless you explain why a big quantity is tiny quantity… However, you will always find crackpots defending that neutrinos are tachyons and that tachyons do exist without a solid probe…
3. Hello,
I am aware of the Cherenkov effect !!! (Worked for Km3NeT high energy neutrino telescope in the Greek team-NESTOR- for my BSc thesis).
// Cherenkov effect in vacuum is not related with tachyons a priori…Be aware of it…//
Well, if you have Cherenkov radiation in vacuo could it be created by something else than a superluminal particle?
We know pretty much well from ICARUS and the updated OPERA results that neutrinos are not superluminal. The people that suggested neutrinos were tachyons refered mainly to ambiguous experimental results where they believed they measured E^2-p^2 I had a professor who had designed an experiment to observe vacuum cherenkov radiation (obviously with no result) ) is mainly that IF they exist, none knows how they are produced, how they interact,etc.
Thus, this is an experimentalist’s playground.
You have no clue about anything, so you just conduct easy experiments in your free time instead of going to cinema.
**Just some trivia: The most conclusive results about the non-existence of tachyons come from air shower searches where they tried to find particles preceeding the relativistic shower front.
As you may have concluded from the above posts, I am a permanently excited experimentalist
Do I really believe in tachyons? Not really. Wouldn’t commit my life research in sth like this.
But what excites me most is that there is a universal speed barrier that nothing can surpass.
This only leaves one wonder, why this is so. And what it would imply if there was something superluminal?
• HI. I will answer you step by step:
“I am aware of the Cherenkov effect !!! (Worked for Km3NeT high energy neutrino telescope in the Greek team-NESTOR- for my BSc thesis). ”
I am glad you to say it :).
“Well, if you have Cherenkov radiation in vacuo could it be created by something else than a superluminal particle?” That is a good question. Tachyons in relativistic physics are not generally superluminal at all! In fact, the true superluminal particles, if they exist, would be something more physical, perhaps closer to the idea of Gonzales-Mestres superbradions or Zigaladze’s elvisebrions (I am aiming to talk about them here, but I am busy yet with my current job).
You say: “I had a professor who had designed an experiment to observe vacuum cherenkov radiation (obviously with no result) ) is mainly that IF they exist, none knows how they are produced, how they interact,etc.” That is the point! Vacuum Cherenkov radiation, as I told you, is highly speculative and controversial but some theorists believe it should exist. I am in the middle: I only say that it would interesting to search for it, I am not saying it will arise or not (unlike some stringers or loopers with their own theories I am quite pragmatic with the experimental evidence).
“You have no clue about anything, so you just conduct easy experiments in your free time instead of going to cinema.” I have a non-standard status where I live, and I presently work as a High School professor, and private tutor, …I am humble and it is not that I am going to the cinema. In fact, I am also working in my Ph.D alone at the moment, hopefully I will find supervisors/advisors soon. So, please, no jokes about “cinemas” anymore. I am a theoretical physicist with interests in many fields, my master thesis was about neutrinos and I know about this:
“The most conclusive results about the non-existence of tachyons come from air shower searches where they tried to find particles preceeding the relativistic shower front.” Cosmic rays are between my experimental interests…
“Do I really believe in tachyons? Not really. Wouldn’t commit my life research in sth like this.
But what excites me most is that there is a universal speed barrier that nothing can surpass.
This only leaves one wonder, why this is so. And what it would imply if there was something superluminal?”
Well, I didn’t say you believe in tachyons, did I? The universal speed of light is related to electromagnetism and nothing else,…Were it probed that there are some other (dark) forces with exotic properties or extra timelike dimensions we can not see yet, the speed of light would not be a limit anymore. In fact, I have worked (and yet I work) about theories with different/several “speeds of light” in “vacuum”. A weird idea, usually taken as “crazy” and generally very little is known about theories with several limit velocities/bound velocities/accelerations/jerks/… For instance, in Finsler-like geometries or even in the gravity’s rainbow approach you get modified dispersion relationships that one could interpret as defining “extra speeds of light”. Look at papers about analogue models of gravity, for instance, too. A current oncoming project is to formulate an enhanced special relativity with different invariant velocities AND lower velocities (not only higher velocity limits) limits… I am currently apart a little bit from academia but I have my own research projects, that is why I don’t post so often (beyond my job duties) here.
I am a theoretical physicist but I also know things about experimental HEP … 😉 Best, Amarashiki.
PS: You say at the end
“But what excites me most is that there is a universal speed barrier that nothing can surpass.
This only leaves one wonder, why this is so. And what it would imply if there was something superluminal?”
The speed of light as limit velocity is related to the Lorentz invariance of “vacuum”. Were it probed that we can observe Lorentz invariant phenomena, the speed of light as limit velocity (but likely not as “invariant” velocity) should be dropped out from fundamental postulates of Relativity. Quite an statement! I am sure you agree. During my Master Degree, I learned a lot about Lorentz violating/CPT violating theories, but Astrophysics seems to put stringent bounds to “deformed” versions of relativistic dispersion relationships…Of course, the point is that people have modified the dispersion relationships in a perturbative (power series) approach with momentum. I can safely say you that, e.g., if you take a non-perturbative (non-analytical) modified dispersion relationship, the astrophysical and HEP bounds on the usual Special Relativistic energy-momentum equations are not so tight(according to my calculations they are much softer) as making a perturbative expansion in “p”. This fact is relatively uncovered by literature, as far as I know. But you see, Science is human-based, and well, it seems the ideas I like the most are NOT in the mood yet ! 😉 Finally, about superluminal stuff (not necessarily tachyon-like) the point is that …We have really no idea of what would imply IF superluminal stuff IS NOT what we usually believe it is (tachyon like, and Cherenkov light producing…). Let me point out that if some kind of superluminal “ether” exist, likely we would have not its existence with current technology. After all, we don’t know what would be the physical features of such a thing! Even worst, when I think about superluminal stuff, I tend to question everything, even known physics like the vacuum Cherenkov radiation. Would superluminal particles necessarily polarize the vacuum? If you believe in universality yes, but it is clear that we are not observing vacuum Cherenkov radiation blasts from the outer space, so, I wonder like you what could be the experimental evidence of superluminal particles IF they were “dark” to the vacuum. How could you detect a superluminal particle if you argue it does not polarize the vacuum as it happens with tachyons and vacuum Cherenkov effect? Probably, superluminal particles if they exist, they have very unique features…
PS(II): In order to be able to detect VACUUM Cherenkov effect, there is a basic condition. The particle passing through the vacuum must be CHARGED under the gauge group. If superluminal particles are “uncharged”, I can not guess an easy way to detect them, since vacuum Cherenkov radiation would be useless…Do you agree?
4. Hello!
//You have no clue about anything, so you just conduct easy experiments in your free time instead of going to cinema.” I have a non-standard status where I live, and I presently work as a High School professor, and private tutor, …I am humble and it is not that I am going to the cinema. In fact, I am also working in my Ph.D alone at the moment, hopefully I will find supervisors/advisors soon. So, please, no jokes about “cinemas” anymore. I am a theoretical physicist with interests in many fields, my master thesis was about neutrinos and I know about this://
From what I can tell, I offended you somehow. If so, I really apologise!!
What I was reproducing was some statement of Botaev in his review : from quarks to tachyons
-tachyon experiments cost nothing
-they can be performed during holidays (I tend to go to the movies in the holidays quite often!)
-you can do it with one or two friends ( if you find something you need eye witnesses)
etc.
Now, about the experimental approach on superluminal particle detection.
I believe that one should not exploit the vacuum cherenkov effect . That’s because we don’t know if it exists. Wouldn’t it be simpler to use a mainstream cherenkov detector (WATER!) and search for Cherenkov photons emerging at angles higher than the nominal angle (42 deg)? This would immediately imply tachyons (if your experiment is right).
If you find sth like this then you could search for vacuum Cherenkov as a second step and determine the gauge charges of the tachyon.
(About PS2: I think that there is what they call : Gravitational Cherenkov but don’t quite know about it)
Best
E.X
P.S: Check that out, it might interest you http://elea.inp.demokritos.gr/HEP2009/Nanopoulos-TimeDelays.pdf
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http://aimsciences.org/article/doi/10.3934/cpaa.2018034?viewType=html | American Institute of Mathematical Sciences
March 2018, 17(2): 627-646. doi: 10.3934/cpaa.2018034
A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates
182 Memorial Dr, Cambridge, MA 02142, USA
* Corresponding author:Felipe Hernandez
Received December 2015 Revised September 2017 Published March 2018
A new decomposition for frequency-localized solutions to the Schrodinger equation is given which describes the evolution of the wavefunction using a weighted sum of Lipschitz tubes. As an application of this decomposition, we provide a new proof of the bilinear Strichartz estimate as well as the multilinear restriction theorem for the paraboloid.
Citation: Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034
References:
[1] J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and {K}akeya conjectures, Acta Mathematica, 196 (2006), 261-302. [2] J. Bourgain and C. Demeter, The proof of the $\ell^2$ decoupling conjecture, arXiv preprint arXiv: 1403.5335, 2014. [3] J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geometric and Functional Analysis, 21 (2011), 1239-1295. [4] J. Bourgain, Refinements of {S}trichartz' inequality and applications to 2D-NLS with critical nonlinearity, Intern. Mat. Res. Notices, 5 (1998), 253-283. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservations laws and global rough sol.utions to a nonlinear Schrödinger equation, Math. Res. Letters, 9 (2002), 659-682. [6] L. R. Ford and D. R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics, 8 (1956), 399-404. [7] L. Guth, A short proof of the multilinear Kakeya inequality, In Mathematical Proceedings of the Cambridge Philosophical Society, volume 158, pages 147-153. Cambridge Univ Press, 2015. [8] Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Analysis and PDE, 5 (2012), 339-362. [9] Z. Hani, Global well-posedness of the cubic nonlinear Schrödinger equation on closed manifolds, Communications in Partial Differential Equations, 37 (2012), 1186-1236. [10] S. Joseph, The max-flow min-cut theorem, 2007. [11] S. Klainerman, I. Rodnianski and T. Tao, A physical space approach to wave equation bilinear estimates, Journal d'Analyse Mathématique, 87 (2002), 299-336. [12] A. Staples-Moore, Network flows and the max-flow min-cut theorem, http://www.math.uchicago.edu/may/VIGRE/VIGRE2009/REUPapers/Staples-Moore.pdf. [13] T. Tao, A physical space proof of the bilinear Strichartz and local smoothing estimate for the Schrödinger equation, 2010.
show all references
References:
[1] J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and {K}akeya conjectures, Acta Mathematica, 196 (2006), 261-302. [2] J. Bourgain and C. Demeter, The proof of the $\ell^2$ decoupling conjecture, arXiv preprint arXiv: 1403.5335, 2014. [3] J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geometric and Functional Analysis, 21 (2011), 1239-1295. [4] J. Bourgain, Refinements of {S}trichartz' inequality and applications to 2D-NLS with critical nonlinearity, Intern. Mat. Res. Notices, 5 (1998), 253-283. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservations laws and global rough sol.utions to a nonlinear Schrödinger equation, Math. Res. Letters, 9 (2002), 659-682. [6] L. R. Ford and D. R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics, 8 (1956), 399-404. [7] L. Guth, A short proof of the multilinear Kakeya inequality, In Mathematical Proceedings of the Cambridge Philosophical Society, volume 158, pages 147-153. Cambridge Univ Press, 2015. [8] Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Analysis and PDE, 5 (2012), 339-362. [9] Z. Hani, Global well-posedness of the cubic nonlinear Schrödinger equation on closed manifolds, Communications in Partial Differential Equations, 37 (2012), 1186-1236. [10] S. Joseph, The max-flow min-cut theorem, 2007. [11] S. Klainerman, I. Rodnianski and T. Tao, A physical space approach to wave equation bilinear estimates, Journal d'Analyse Mathématique, 87 (2002), 299-336. [12] A. Staples-Moore, Network flows and the max-flow min-cut theorem, http://www.math.uchicago.edu/may/VIGRE/VIGRE2009/REUPapers/Staples-Moore.pdf. [13] T. Tao, A physical space proof of the bilinear Strichartz and local smoothing estimate for the Schrödinger equation, 2010.
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2016 Impact Factor: 0.801
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http://mathhelpforum.com/advanced-algebra/192013-matrices-linear-maps.html | # Thread: Matrices as Linear Maps
1. ## Matrices as Linear Maps
Hey Team,
Let A be an M by N matrix, B be an N by P matrix
a) Show that AB is the sum of n matrices each of rank at most 1
b) If the rank of A is n, what is the rank of AB?
I'm not sure where to start on part a, but part b I think I have a decent start.
$Rank(AB) = dim(R[AB])$
We know that A maps from Fn to Fm, and B maps from Fp to Fn. So we define the left multiplication of these guys as such,
$L_A: F^N \to F^M$
$L_B: F^P \to F^N$
So to find the range of AB we can let Z be some arbitrary vector in $F^P$ and see for what values is it good for
$dim(R[AB])=dim(L_A L_B (Z))$
We can note that $L_B(Z)$ is a subset of $F^N$ so the following inequality results,
$dim(L_A L_B (Z)) \le dim(L_A (F^N)) = dim(A(F^N)) = Rank(A) = N$
We can further note that if $L_B$ is a surjective map then its range is actually equal to $F^N$ turning the inequality into equality.
Was the above process correct for b, and if so, are there any hints for a?
Thanks!
2. ## Re: Matrices as Linear Maps
Originally Posted by AllanCuz
Hey Team,
I'm not sure where to start on part a, but part b I think I have a decent start.
$Rank(AB) = dim(R[AB])$
We know that A maps from Fn to Fm, and B maps from Fp to Fn. So we define the left multiplication of these guys as such,
$L_A: F^N \to F^M$
$L_B: F^P \to F^N$
So to find the range of AB we can let Z be some arbitrary vector in $F^P$ and see for what values is it good for
$dim(R[AB])=dim(L_A L_B (Z))$
We can note that $L_B(Z)$ is a subset of $F^N$ so the following inequality results,
$dim(L_A L_B (Z)) \le dim(L_A (F^N)) = dim(A(F^N)) = Rank(A) = N$
We can further note that if $L_B$ is a surjective map then its range is actually equal to $F^N$ turning the inequality into equality.
Was the above process correct for b, and if so, are there any hints for a?
Thanks!
That looks right or me. For the second part, consider writing the matrix as a sum of column matrices.
3. ## Re: Matrices as Linear Maps
strictly speaking, such a sum does not exist, perhaps you meant a sum of matrices that are all 0's except for one column?
4. ## Re: Matrices as Linear Maps
Originally Posted by Deveno
strictly speaking, such a sum does not exist, perhaps you meant a sum of matrices that are all 0's except for one column?
Obviously, I was letting the OP fill things in for themselves, you know?
5. ## Re: Matrices as Linear Maps
well you know me, i'm sorta slow-witted.... | {"extraction_info": {"found_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": 20, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9847195744514465, "perplexity": 454.60463930069596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1512948597485.94/warc/CC-MAIN-20171217191117-20171217213117-00197.warc.gz"} |
http://math.stackexchange.com/questions/444144/i-need-help-with-an-assignment-question-please-for-numerical-methods | # I need help with an assignment question please for numerical methods
Consider the function $f (x) = xe^x - 2,$ we want to study the properties of $f (x)$ so that we can apply numerical methods to solve the equation $f (x) = 0$.
Which option is false ?
1. the function, $f (x)$ is well defined and continuous for all $x$ in the interval $(0,2)$
2. the function, $f (x)$ has no discontinuity and no singularities
3. the function, $f '(x)$ is well defined and continuous for all $x$ in the interval $(0,2)$
4. the function, $f '(x)$ has no discontinuity and no singularities
5. all of the above
I do not understand the different options given. How do I know if the function is well defined?
-
I take it assignment means homework, so I'll add that tag. Sorry if I misunderstood. – 1015 Jul 15 '13 at 12:35
"Well-defined", in this context, just means, if you plug in a value of $x$, you get a value of $f$. – Gerry Myerson Jul 15 '13 at 12:39
Given any $x\in (0,2)$, is there a number $f(x)$ defined without ambiguity? If so, the function is well-defined on $(0,2)$. – 1015 Jul 15 '13 at 12:39
Hi, no that is fine. It is homework yes but we have no study guide and I do not understand the material in the textbook – Dee Jul 15 '13 at 12:42
Hint 1: Here is a plot of $f(x)$ and $f'(x)$ over the range (do they look continuous over the range, see any singularities over $\mathbb{R}$). Can you make an analytical argument over the range $(0,2)$ and $\mathbb{R}$, respectively, regarding continuity and singularities? Are both continuous? Do either of them have singularities?
Hint 2: Here is a plot of $f(x) = 0$.
Hint 3: From hint 2, did you try solving $f(x) = 0$ analytically? Can you?
What if you had $1/(x-1)$. Where is the discontinuity? Where is the singularity (en.wikipedia.org/wiki/Mathematical_singularity)? – Amzoti Jul 15 '13 at 13:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8890185356140137, "perplexity": 324.82769550676767}, "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-2014-49/segments/1416931007510.17/warc/CC-MAIN-20141125155647-00237-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/167988-find-t-when-vectors-parallel.html | # Math Help - find t when vectors are parallel
1. ## find t when vectors are parallel
I have this question vector a = (2,5) vector b = (-1,t)
find t when a and b are parallel.
I can't seem to figure this one out. Theres nothing in my text about how to solve this. any help is appreciated
2. The two vectors are parallel when (2,5) = a(-1,t) = (-a, at) where a is some constant.
i.e. parallel vectors are vectors that differ only in magnitude but not in direction.
So now you have:
-a = 2
at = 5
Can you solve for t?
3. Find where the cross product of a and b is 0.
4. could you show me how to solve for t in this problem?
5. Can you solve $\dfrac{-1}{2}=\dfrac{t}{5}$ for $t~?$
6. -5/2 =t?
7. You have it!
8. could you show me a way to solve for t using dot product?
could you show me a way to solve for t using dot product?
Actually the notation of parallel vectors is not usually associated with dot product.
The dot product of two perpendicular vectors is zero.
Rather parallel vectors are scalar multiples of each other.
Therefore, their components are proportional.
That is why we have $\dfrac{-1}{2}=\dfrac{t}{5} .$
10. thanks, yeah that occurred to me as a walked away to take a break. thanks a bunch. this forum has been a life saver already.
11. Plato has the easiest way here.
I think you want to use the following
$\displaystyle \cos \theta = \frac{a\cdot b}{\|a\|\|b\|}$
$\displaystyle \cos 0 = \frac{2\times -1 +5\times t}{\sqrt{29}\sqrt{1+t^2}}$
Solving this you should get
$\displaystyle (2t+5)^2=0\implies t = \frac{-5}{2}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.9268194437026978, "perplexity": 572.5347499935964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1406510264270.11/warc/CC-MAIN-20140728011744-00106-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.ias.ac.in/listing/bibliography/jess/Reid_A._Bryson | • Reid A Bryson
Articles written in Journal of Earth System Science
• A note on the performance of climatic forecasts for Asia based on the periodic portion of the data: 1986–1987
Between 1973 and 1986 a group at the University of Wisconsin worked on the use of the periodic portion of climatic time series with the aim of exploring the potential for year-or-more in advance forecasting. This paper reports on the real time verification of the last sets of forecasts made by the group.
From spectra of temperature and cube-rooted precipitation the dominant frequencies were chosen. These were usually related to tidal frequencies. A Fourier series of these dominant terms was then fitted to the dependent data set and future values calculated. These were analyzed for forecast skill, and the skillful Fourier series retained. Real time forecasts were then made. Verification shows a low probability that the forecast skills were obtained by chance. It is suggested that the periodic term might be a useful addition to more standard approaches to long range forecasting.
• The “susceptibility factor” in the atmospheric response to periodic forcing
Evidence is presented of a periodic component in the inter-annual variability of precipitation and pressure data for India during June, the month of the onset of the Indian southwest monsoon. Two frequencies that explain a statistically significant percent of the variance in these data sets are the same as the two that explain most of the variance of the average monthly lunar tidal potential for June. Not only are the frequencies the same but they are also in phase which strongly suggests that lunar tides in the atmosphere do, in fact, produce an element of climatic variability. The amplitude of the atmospheric response to this periodic forcing was not constant in time but was found to be related to the long term change in northern hemispheric surface temperature. This susceptibility of the atmosphere to an external forcing results in a nonlinear relationship between forcing and response. As a result, nonlinear regression had to be used in order to adequately define the magnitude of the response at a given frequency. The ramifications of this nonlinear response are discussed. The nonlinear interaction of the northern hemisphere temperature and the 18.6 year lunar nodal cycle results in a modulation of the frequency which appears in a linear spectral analysis near 22 years. Thus, the 22-year cycle often found in meteorological data sets may instead be the result of the modulated nodal cycle.
• # Journal of Earth System Science
Current Issue
Volume 128 | Issue 8
December 2019
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | {"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": 0.8455615639686584, "perplexity": 830.4094142662861}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1566027330913.72/warc/CC-MAIN-20190826000512-20190826022512-00532.warc.gz"} |
https://farside.ph.utexas.edu/teaching/celestial/Celestial/node50.html | Next: Coriolis force Up: Rotating reference frames Previous: Rotating reference frames
# Centrifugal acceleration
Let our non-rotating inertial frame be one whose origin lies at the center of the Earth, and let our rotating frame be one whose origin is fixed with respect to some point, of latitude , on the Earth's surface. (See Figure 6.1.) The latter reference frame thus rotates with respect to the former (about an axis passing through the Earth's center) with an angular velocity vector, , which points from the center of the Earth toward its north pole and is of magnitude
(6.10)
Here, is the length of a sidereal day; that is, the Earth's rotation period relative to the distant stars (Yoder 1995).6.1
Consider an object that appears stationary in our rotating reference frame: that is, an object that is stationary with respect to the Earth's surface. According to Equation (6.9), the object's apparent equation of motion in the rotating frame takes the form
(6.11)
Let the non-fictitious force acting on our object be the force of gravity, . Here, the local gravitational acceleration, , points directly toward the center of the Earth. It follows that the apparent gravitational acceleration in the rotating frame is written
(6.12)
where is the displacement vector of the origin of the rotating frame (which lies on the Earth's surface) with respect to the center of the Earth. Here, we are assuming that our object is situated relatively close to the Earth's surface (i.e., ).
It can be seen from Equation (6.12) that the apparent gravitational acceleration of a stationary object close to the Earth's surface has two components: first, the true gravitational acceleration, , of magnitude , which always points directly toward the center of the Earth (Yoder 1995); and, second, the so-called centrifugal acceleration, . The latter acceleration is normal to the Earth's axis of rotation, and always points directly away from this axis. The magnitude of the centrifugal acceleration is , where is the perpendicular distance to the Earth's rotation axis, and is the Earth's radius (Yoder 1995). (See Figure 6.2.)
It is convenient to define Cartesian axes in the rotating reference frame such that the -axis points vertically upward and - and -axes are horizontal, with the -axis pointing directly northward and the -axis pointing directly westward. (See Figure 6.1.) The Cartesian components of the Earth's angular velocity are thus
(6.13)
while the vectors and are written
(6.14) and (6.15)
respectively. It follows, from Equation (6.12), that the Cartesian coordinates of the apparent gravitational acceleration are
(6.16)
The magnitude of this acceleration is approximately
(6.17)
According to the preceding equation, the centrifugal acceleration causes the magnitude of the apparent gravitational acceleration on the Earth's surface to vary by about percent, being largest at the poles, and smallest at the equator. This variation in apparent gravitational acceleration, due (ultimately) to the Earth's rotation, causes the Earth itself to bulge slightly at the equator (see Section 6.5), which has the effect of further intensifying the variation (see Exercise 6), because a point on the surface of the Earth at the equator is slightly further away from the Earth's center than a similar point at one of the poles (and, hence, the true gravitational acceleration is slightly weaker in the former case).
Another consequence of centrifugal acceleration is that the apparent gravitational acceleration on the Earth's surface has a horizontal component aligned in the north/south direction. This horizontal component ensures that the apparent gravitational acceleration does not point directly toward the center of the Earth. In other words, a plumb-line on the surface of the Earth does not point vertically downward (toward the center of the Earth), but is deflected slightly away from a true vertical in the north/south direction. The angular deviation from true vertical can easily be calculated from Equation (6.16):
(6.18)
Here, a positive angle denotes a northward deflection, and vice versa. Thus, the deflection is southward in the northern hemisphere (i.e., ) and northward in the southern hemisphere (i.e., ). The deflection is zero at the poles and at the equator, and it reaches its maximum magnitude (which is very small) at middle latitudes.
Next: Coriolis force Up: Rotating reference frames Previous: Rotating reference frames
Richard Fitzpatrick 2016-03-31 | {"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": 0.9819934368133545, "perplexity": 395.50857482702}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540519149.79/warc/CC-MAIN-20191209145254-20191209173254-00057.warc.gz"} |
https://www.groundai.com/project/decentralized-event-triggered-control-over-wireless-sensoractuator-networks/ | Decentralized Event-triggered Control over WSAN
# Decentralized event-triggered control over wireless sensor/actuator networks
Manuel Mazo Jr and Paulo Tabuada
###### Abstract.
In recent years we have witnessed a move of the major industrial automation providers into the wireless domain. While most of these companies already offer wireless products for measurement and monitoring purposes, the ultimate goal is to be able to close feedback loops over wireless networks interconnecting sensors, computation devices, and actuators. In this paper we present a decentralized event-triggered implementation, over sensor/actuator networks, of centralized nonlinear controllers. Event-triggered control has been recently proposed as an alternative to the more traditional periodic execution of control tasks. In a typical event-triggered implementation, the control signals are kept constant until the violation of a condition on the state of the plant triggers the re-computation of the control signals. The possibility of reducing the number of re-computations, and thus of transmissions, while guaranteeing desired levels of control performance, makes event-triggered control very appealing in the context of sensor/actuator networks. In these systems the communication network is a shared resource and event-triggered implementations of control laws offer a flexible way to reduce network utilization. Moreover reducing the number of times that a feedback control law is executed implies a reduction in transmissions and thus a reduction in energy expenditures of battery powered wireless sensor nodes.
M. Mazo Jr is with INCAS, Assen and the Department of Discrete Technology and Production Automation, University of Groningen, The Netherlands, [email protected]
P. Tabuada is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095-1594,[email protected]
## 1. Introduction
For many years, control engineers have designed their controllers as if there were infinite-bandwidth, noise- and delay-free channels between sensors, controllers, and actuators. The effects of non-idealities in the channels, in practice, could be mitigated by employing better hardware. However, on implementations over Wireless Sensor Actuator Networks (WSAN) these limitations of the communication medium can no longer be neglected. This fact, combined with the recent interest from industry, e.g. the WirelessHART initiative [1], have fueled the study of control under communication constraints in the past decade. Much research has been devoted to the effects of: quantization in the sensors; delay and jitter; limited bandwidth; or even packet losses. Some good overviews of these topics can be found in the report resulting from the RUNES project [2], and the special issue of the IEEE proceedings [3].
One aspect common to most modern control systems, and something assumed in most of the studies mentioned above, is the implementation of control strategies in embedded microprocessors. But in controlling the physical world, which is of continuous nature, the use of microprocessors brings a new question: how often should we sample the physical environment [4]? Many researchers have worked on the analysis of this sole problem. Tools like the delta-transform [5] were developed, and many books discussed this issue [6, 7]. More recently, Nesic and collaborators have proposed techniques to select periods retaining closed-loop stability in networked systems [8, 9]. However, engineers still rely mostly on rules of thumb such as sampling with a frequency 20 times the system bandwidth, and then check if it actually works [4, 6, 7]. A shift in perspective was brought by the notion of event-triggered control [10], [11]. In event-triggered control, instead of periodically updating the control input, the update instants are generated by the violation of a condition on the state of the plant. Many researchers have shown a renewed interest on these techniques [12, 13, 14, 15, 16, 17]. Recently, one of the authors proposed a formalism to generate asymptotically stable event-triggered implementations of nonlinear controllers [18], and in [19] the authors explored the application of event-triggered and self-triggered techniques to distributed implementations of linear controllers. For more details about these event-triggered and self-triggered techniques we refer the reader to [20] and [21]. Following the formalism in [18], Wang and Lemmon proposed a distributed event-triggered implementation for weakly-coupled distributed systems [22]. The present work complements the techniques described in [22] by addressing systems without weak-coupling assumptions.
The main contribution of this paper is a strategy for the construction of decentralized event-triggered implementations over WSAN of centralized controllers. The event-triggered techniques introduced in [18] are based on a criterion that depends on the norm of the vector of measured quantities. This is natural in the setting discussed in [18] since sensors were collocated with the micro-controller. However, in a WSAN the physically distributed sensor nodes do not have access to all the measured quantities. Hence, we cannot use the same criterion to determine when the control signal should be re-computed. Using classical observers or estimators (as the Kalman filter) would require filters of dimension as large as the number of states in each sensor node, which would be unpractical given the low computing capabilities of sensor nodes. Moreover, we do not assume observability from every measured output, thus ruling out observer-based techniques. Approaches based on consensus algorithms are also unpractical as they require large amounts of communication and thus large energy expenditures by the sensor nodes. Instead, we present an approach to decentralize a centralized event-triggered condition that relies only on the locally measured quantities. Our technique also provides a mechanism to enlarge the resulting times between controller re-computations without altering performance guarantees.
We do not address in this paper practical issues such as delays or jitter in the communication and focus solely on the reduction of the actuation frequency (with its associated communication and energy savings). In particular, the issue of communication delays has been shown to be easily addressed in the context of event-triggered control in [18] and similarly in [22]. The approach followed in those papers is applicable to the techniques introduced in this paper. Moreover, our techniques can be implemented over the WirelessHART standard [1], which addresses other communication concerns such as medium access control, power control, and routing.
The present paper is organized as follows: we introduce basic notation in Section 2; Section 3 states the problem, briefly reviews the results of [18] and presents our proposal for decentralization; the paper finalizes with an example in Section 4 and a discussion in Section 5.
## 2. Notation
We denote by the natural numbers, by , by the positive real numbers, and by . The usual Euclidean () vector norm is represented by . When applied to a matrix, denotes the induced matrix norm. A matrix is said to be positive definite, denoted , whenever for all , . By we denote the minimum and maximum eigenvalues of respectively. A function , is of class if it is continuous, strictly increasing, and as . Given an essentially bounded function we denote by its norm, i.e., .
In the following we consider systems defined by differential equations of the form:
(1) ddtξ=f(ξ,υ)
with input an essentially bounded piecewise continuous function of time and a smooth map. We also use the simpler notation to refer to (1). We refer to such systems as control systems. Solutions of (1) with initial condition and input , denoted by , satisfy: and for almost all . The notation will be relaxed by dropping the subindex when it does not contribute to the clarity of exposition. A feedback law for a control system is a smooth map ; we sometimes refer to such a law as a controller for the system.
## 3. Decentralized event-triggered control
Consider a nonlinear control system and a hardware platform consisting of a set of wireless sensors and actuators and a computation node. This last node is in charge of computing the control signal with the measurements obtained from the sensors. We consider scenarios in which none of these sensor nodes has access to the full state of the plant. We model the execution of the control loop in three steps: data retrieval from sensors, controller computation, and provision of the control commands to the actuators. Furthermore, we assume that the computation of the controller happens in just one device which retrieves all the measurement information from the sensors, computes the inputs for all actuators, and disseminates these new commands to the actuator nodes. This scenario is a typical configuration considered in the WirelessHART standard, see [23], which addresses the problem of scheduling links and channels for disseminating the information in WirelessHART networks.
Our goal is to provide a mechanism triggering the execution of the control loop which reduces the frequency of the controller updates. In order to reduce the frequency of controller updates we abandon the periodic transmission paradigm, and instead we propose to close the loop whenever certain events happen. In particular, we consider the event-triggered implementation techniques proposed in [18] which guarantee the asymptotic stability of the closed-loop system. These techniques, however, require the knowledge of the full state to decide when to trigger new updates, but such information is not available at any sensing node under our premises. In the following we discuss a decentralization of the decision process triggering controller updates. We propose the use of conditions depending solely on the information available at each node. Whenever any of these conditions is violated at a node, this node informs the computation device. Upon receipt of such an event, the computation device requests fresh measurements, updates the control signals, and forwards the new commands to the actuation nodes.
### 3.1. Event-triggered control
We begin by revisiting the results from [18], which serve as the basis for the rest of our work. Let us start by considering a nonlinear control system:
(2) ˙ξ=f(ξ,υ)
and assume that a feedback control law , is available, rendering the closed-loop system:
(3) ˙ξ=f(ξ,k(ξ+ε))
input-to-state stable (ISS) [24] with respect to measurement errors . We do not provide the definition of ISS, but rather the following characterization that lies at the heart of our techniques:
###### Definition 3.1.
A smooth function is said to be an ISS Lyapunov function for the closed-loop system (3) if there exists class functions ,, and such that for all and the following is satisfied:
α––(|x|)≤ V(x) ≤¯¯¯¯α(|x|) (4) ∂V∂xf(x,k(x+e)) ≤ −α(|x|)+γ(|e|).
The closed-loop system (3) is said to be ISS with respect to measurement errors , if there exists an ISS Lyapunov function for (3).
In a sample-and-hold implementation of the control law , the input signal is held constant between update times, i.e.:
˙ξ(t) = f(ξ(t),υ(t)) (5) υ(t) = k(ξ(tk)),t∈[tk,tk+1[,
where is a divergent sequence of update times. An event-triggered implementation defines such a sequence of update times for the controller, rendering the closed loop system asymptotically stable.
We now consider the signal defined by for and regard it as a measurement error. By doing so, we can rewrite (3.1) for as:
˙ξ(t) = f(ξ(t),k(ξ(t)+ε(t))), ˙ε(t) = −f(ξ(t),k(ξ(t)+ε(t))),ε(tk)=0.
Hence, as (3) is ISS with respect to measurement errors , from (3.1) we know that by enforcing:
(6) γ(|ε(t)|)≤ρα(|ξ(t)|),∀t>0,ρ∈]0,1[
the following holds:
∂V∂xf(x,k(x+e))≤−(1−ρ)α(|x|),∀x,e∈Rn
and asymptotic stability of the closed-loop follows. Moreover, if one assumes that the system operates in some compact set and and are Lipschitz continuous on , the inequality (6) can be replaced by the simpler inequality , for a suitably chosen . Hence, if the sequence of update times is such that:
(7) |ε(t)|2≤σ|ξ(t)|2,t∈[tk,tk+1[,
the sample-and-hold implementation (3.1) is guaranteed to render the closed loop system asymptotically stable.
Condition (7) defines an event-triggered implementation that consists of continuously checking (7) and triggering the recomputation of the control law as soon as the inequality evaluates to equality. Note that recomputing the controller at time requires a new state measurement and thus resets the error to zero which enforces (7).
### 3.2. Decentralized event-triggering conditions
We consider, for simplicity of presentation, a decentralized scenario in which each state variable is measured by a different sensor. However, the same ideas apply to more general decentralized scenarios as we briefly discuss at the end of Section 3.3. In this setting, no sensor can evaluate condition (7), since (7) requires the knowledge of the full state vector . Our goal is to provide a set of simple conditions that each sensor can check locally to decide when to trigger a controller update, thus triggering also the transmission of fresh measurements from sensors to the controller.
Using a set of parameters such that , we can rewrite inequality (7) as:
n∑i=1(ε2i(t)−σξ2i(t))≤0=n∑i=1θi,
where and denote the -th coordinates of and respectively. Hence, the following implication holds:
(8) n⋀i=1(ε2i(t)−σξ2i(t)≤θi)⇒|ε(t)|2≤σ|ξ(t)|2,
which suggests the use of:
(9) ε2i(t)−σξ2i(t)≤θi
as the local event-triggering conditions.
In this decentralized scheme, whenever any of the local conditions (9) becomes an equality, the controller is recomputed. We denote by the first time at which (9) is violated, when , . If the time elapsed between two events triggering controller updates is smaller than the minimum time between updates of the centralized event-triggered implementation111It was proved in [18] that such a minimum time exists for the centralized condition, and that lower bounds can be explicitly computed., the second event is ignored and the controller update is scheduled units of time after the previous update.
Not having an equivalence in (8) entails that this decentralization approach is in general conservative: times between updates will be shorter than in the centralized case. The vector of parameters can be used to reduce the mentioned conservatism and thus reduce utilization of the communication network. It is important to note that the vector can change every time the control input is updated. From here on we show explicitly this time dependence of by writing to denote its value between the update instants and . Following the presented approach, as long as satisfies , the stability of the closed-loop is guaranteed regardless of the specific value that takes and the rules used to update .
We summarize the previous discussion in the following proposition:
###### Proposition 3.2.
For any choice of satisfying:
n∑i=1θi(k)=0,∀k∈N+0,
the sequence of update times given by:
tk+1 = tk+max{τmin,mini=1,…,nτi(ξ(tk))} τi(ξ(tk)) = min{τ∈R+0|ϵ2i(tk+τ)−σξ2i(tk+τ)=θi(k)}
renders the system (3.1) asymptotically stable.
### 3.3. Decentralized event-triggering with on-line adaptation
We present now a family of heuristics to adjust the vector whenever the control input is updated. We define the decision gap at sensor at time as:
Gi(t)=ε2i(t)−σξ2i(t)−θi(k).
The heuristic aims at equalizing the decision gap at some future time. We propose a family of heuristics parametrized by an equalization time and an approximation order .
For the equalization time we present the following two choices: constant and equal to the minimum time between controller updates ; the previous time between updates .
The approximation order is the order of the Taylor expansion used to estimate the decision gap at the equalization time :
^Gi(tk+te)=^ε2i(tk+te)−σ^ξ2i(tk+te)−θi(k).
where for :
^ξi(t) = ξi(tk)+˙ξi(tk)(t−tk)+12¨ξi(tk)(t−tk)2+… +1q!ξ(q)i(tk)(t−tk)q, ^εi(t) = xi0tk−˙ξi(tk)(t−tk)−12¨ξi(tk)(t−tk)2−… −1q!ξ(q)i(tk)(t−tk)q,
using the fact that and .
Finally, once an equalization time and an approximation order are chosen, the vector is computed so as to satisfy:
^Gi(tk+te)=^Gj(tk+te)∀i,j∈{1,2,…,n}, (10) n∑i=1θi(k)=0.
Note that finding such , after the estimates and have been computed, amounts to solving a system of linear equations. Note also that is computed222The resulting computed in this way could be such that for some sensor , . Such choice of results in an immediate violation of the triggering condition at , i.e., would be zero. In practice, when the unique solution of (10) results in , one resets to some default value such as the zero vector. in the controller node, which has access to .
The choice of and has a great impact on the amount of actuation required. The use of a large leads, in general, to poor estimates of the state of the plant at time and thus degrades the equalization of the gaps. On the other hand, one expects that equalizing at times as close as possible to the next update time (according to the centralized event-triggered implementation) provides larger times between updates. In practice, these two objectives (small , and close to the ideal ) can be contradictory, namely when the time between controller updates is large. The effect of the order of approximation depends heavily on and enlarging does not necessarily improve the estimates. An heuristic providing good results in several case studies performed by the authors is given by Algorithm 1.
While we assumed, for simplicity of presentation, that each node measured a single state of the system, in practice there may be scenarios in which one sensor has access to several (but not all) states of the plant. The same approach applies by considering local triggering rules of the kind , where is now the vector of states sensed at node , is its corresponding error vector, and is a scalar.
### 3.4. Comments on practical implementations
The proposed technique, while clearly reducing the amount of information that needs to be transmitted from sensors to actuators, might suggest that sensor nodes need to be continuously listening for events triggered at other nodes. This poses a practical problem since the energy required to keep the radios on to listen for possible events could potentially be very large. In practice, however, sensor nodes have their radio modules asleep most of the time and are periodically awaken according to a time multiplexing medium access protocol. Time multiplexing is typically used in protocols for control over wireless networks, like WirelessHART, in order to provide non-interference and strict delay guarantees. The use of time multiplexing can be accommodated in the proposed technique by regarding its effect as a bounded and known delay between the generation of an event and the corresponding change in the control signal. As was shown in [18], delays can be accommodated in event-triggered implementations by adequately reducing the value of , therefore making the triggering conditions more conservative.
## 4. Examples and simulation results
We present in what follows an example illustrating the effectiveness of the proposed technique. We select the quadruple-tank model from [25] describing the multi-input multi-output nonlinear system consisting of four water tanks as shown in Figure 1. The water flows from tanks and into tanks and , respectively, and from these two tanks to a reservoir. The state of the plant is composed of the water levels of the tanks: , , and . Two inputs are available: and , the input flows to the tanks. The input flows are split at two valves and into the four tanks. The positions of these valves are given as parameters of the plant. The goal is to stabilize the levels and of the lower tanks at some specified values and .
The system dynamics are given by the equation:
˙ξ(t)=f(ξ(t))+gcυ,
with:
f(x)=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−a1√2gx1A1+a3√2gx3A1−a2√2gx2A2+a4√2gx4A2−a3√2gx3A3−a4√2gx4A4⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,gc=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣γ1A100γ2A201−γ2A31−γ1A40⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,
and denoting gravity’s acceleration and and denoting the cross sections of the tank and outlet hole respectively.
The controller design from [25] requires the extension of the plant with two extra artificial states and . These states are nonlinear integrators used by the controller to achieve zero steady-state offset and evolve according to:
˙ξ5(t)=kI1a1√2g(√ξ1(t)−√x∗1), ˙ξ6(t)=kI2a2√2g(√ξ2(t)−√x∗2),
where and are design parameters of the controller. Note how stabilizing the extended system implies that in steady-state and converge to the desired values and . We assume in our implementation that the sensors measuring and , also compute and locally and sufficiently fast. Hence, we can consider and as regular state variables.
The controller proposed in [25] is given by the following feedback law:
(11) υ(t)=−K(ξ(t)−x∗)+u∗
with
(12) u∗ = [γ11−γ21−γ1γ2]−1[a1√2gx∗1a2√2gx∗2] = [01−γ21−γ10]−1[a1√2gx∗3a2√2gx∗4],
and where is a positive definite matrix and is given by
P=[γ1k1(1−γ1)k20(1−γ1)k4γ1k1(1−γ1)k2(1−γ2)k1γ2k2(1−γ2)k30(1−γ2)k1γ2k2],
where , , and are design parameters of the controller. Note how equation (12) can be used to compute and from the specified and . When computing the control , the remaining entries and of can be set to any arbitrary (fixed) values and . This can be done because the errors: and , between the arbitrary values and the actual states and of the equilibrium, can be reinterpreted as a perturbation on the initial states and .
Using this controller the following function:
Hd(x) =12(x−x∗)TPTQP(x−x∗)−u∗TPx+ 4∑i=123kiaix3/2i√2g+k1a1x5√2gx∗1+k2a2x6√2gx∗2,
which is positive definite and has a global minimum at , is an ISS Lyapunov function with respect to , as evidenced by the following bound 333The expression for the matrix is not included because of space limitations. The value of can be easily deduced from [25].:
ddtHd(ξ)≤−λm(R)|∇Hd(ξ)|2+|∇Hd(ξ)||g′cK||ε|.
This equation suggests the use of the triggering condition:
|∇Hd(ξ)||g′cK||ε|≤ρλm(R)|∇Hd(ξ)|2,ρ∈]0,1[.
Moreover, assuming the operation of the system to be confined to a compact set containing a neighborhood of , can be bounded as and the following triggering rule can be applied to ensure asymptotic stability:
|ε(t)|2≤σ|ξ(t)−x∗|2,σ=(ρmρλm(R)|g′cK|)2>0.
We simulated the decentralized event-triggered implementation of this controller following the techniques in Section 3. The physical parameters of the plant and the parameters of the controller are the same as those in [25]. Assuming that the system operates in the compact set , one can take , and for the choice of a value of was selected. A bound for the minimum time between controller updates, computed as explained in [18], is given by . The decentralized event-triggered controller is implemented adapting as specified by Algorithm 1 with . Furthermore, the pairs of states and are assumed to be measured at the same sensor node, and therefore combined in a single triggering condition at the respective nodes. For comparison purposes, we present in the first row of Figure 2 the time between controller updates, the evolution of the ratio vs and the state trajectories, for a centralized event-triggered implementation, starting from initial condition and setting and . The corresponding results for the proposed decentralized event-triggered implementation are shown in the second row of Figure 2, and the results for a decentralized event-triggered implementation without adaptation, i.e., with for all , are shown in the last row of the same figure.
For completeness, Figure 3 presents the evolution of adaptation vector for the adaptive decentralized event-triggered implementation.
We can observe that, as expected, a centralized event-triggered implementation is far more efficient, in terms of time between updates, than a decentralized event-triggered implementation without adaption. It is also clear that, although Algorithm 1 fails to recover the performance of the centralized event-triggered implementation exactly, it produces very good results. The results are even better if we look at the performance in terms of the number of executions which are presented in the legend of these plots. Finally we would like to remark that, although the times between updates in the three implementations can differ quite drastically, the three systems are stabilized producing almost undistinguishable state trajectories.
## 5. Discussion
In [22] Wang and Lemmon proposed a method for distributed event-triggered control under the assumption that the control system was composed of weakly coupled subsystems. Exploiting this fact, they were able to update inputs independently of each other. Our approach, while not updating inputs independently, does not rely on any internal weak coupling assumptions about the system. Thus, our techniques could be used to complement the techniques in [22] at the local subsystem level.
The proposed techniques have been shown effective in decentralizing an event-triggered implementation of a quadruple water-tank system. The centralized controller of this example is a dynamic controller. In this particular case, by allowing the dynamical part of the controller to be continuously computed by the sensors, we successfully obtained a decentralized event-triggered implementation. However, the implementation of general dynamic controllers in event-triggered form, centralized or not, remains a question for future research. The design of more efficient adaptation rules is another interesting question to investigate further. Finally, we would like to emphasize the low computational requirements of the proposed implementation, which makes it suitable for sensor/actuator networks with limited computation capabilities at the sensor level.
## References
• [1] WirelessHART. [Online]. Available: http://www.hartcomm.org/protocol/wihart/wireless_technology.html
• [2] K.-E. Årzén, A. Bicchi, S. Hailes, K. Johansson, and J. Lygeros, “On the design and control of wireless networked embedded systems,” in 2006 IEEE Computer Aided Control System Design,, Oct. 2006, pp. 440 –445.
• [3] P. Antsaklis and J. Baillieul, “Special issue on technology of networked control systems,” Proceedings of the IEEE, vol. 95, no. 1, pp. 5 –8, Jan. 2007.
• [4] G. Franklin, “Rational rate [ask the experts],” Control Systems Magazine, IEEE, vol. 27, no. 4, pp. 19 –19, Aug. 2007.
• [5] G. Goodwin, R. Middleton, and H. Poor, “High-speed digital signal processing and control,” Proceedings of the IEEE, vol. 80, no. 2, pp. 240 –259, Feb. 1992.
• [6] G. Goodwin, S. Graebe, and M. Salgado, Control System Design. Prentice Hall, 2001.
• [7] C. Houpis and G. B. Lamont, Digital Control Systems. McGraw-Hill Higher Education, 1984.
• [8] D. Nesić and A. Teel, “Sampled-data control of nonlinear systems: An overview of recent results,” in Perspectives in robust control, ser. Lecture Notes in Control and Information Sciences, S. Moheimani, Ed. Springer Berlin / Heidelberg, 2001, vol. 268, pp. 221–239.
• [9] D. Nesić, A. Teel, and D. Carnevale, “Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems,” IEEE Transactions on Automatic Control, vol. 59, pp. 619–624, Mar. 2009.
• [10] K.-E. Årzén, “A simple event based PID controller,” in Proceedings of 14th IFAC World Congress, vol. 18, 1999, pp. 423–428.
• [11] K. Åström and B. Bernhardsson, “Comparison of riemann and lebesgue sampling for first order stochastic systems,” in Proceedings of the 41st IEEE Conference on Decision and Control, vol. 2, Dec. 2002, pp. 2011 – 2016.
• [12] W. Heemels, J. Sandee, and P. van den Bosch, “Analysis of event-driven controllers for linear systems,” International Journal of Control, vol. 81, no. 4, pp. 571–590, 2008.
• [13] A. Cervin and T. Henningsson, “Scheduling of event-triggered controllers on a shared network,” in 47th IEEE Conference on Decision and Control, Dec. 2008, pp. 3601 –3606.
• [14] M. Rabi and K. H. Johansson, “Event-triggered strategies for industrial control over wireless networks,” in WICON, 2008.
• [15] M. Rabi, K. H. Johansson, and M. Johansson, “Optimal stopping for event-triggered sensing and actuation,” in 47th IEEE Conference on Decision and Control., Dec. 2008, pp. 3607 –3612.
• [16] A. Molin and S. Hirche, “Optimal event-triggered control under costly observations,” in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, 2010.
• [17] J. Lunze and D. Lehmann, “A state-feedback approach to event-based control,” Automatica, vol. 46, no. 1, pp. 211 – 215, 2010.
• [18] P. Tabuada, “Event-triggered real-time scheduling of stabilizing control tasks,” IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1680 –1685, Sept. 2007.
• [19] M. Mazo Jr. and P. Tabuada, “On event-triggered and self-triggered control over sensor/actuator networks,” in Proceedings of the 47th IEEE Conference on Decision and Control, 2008., Dec. 2008, pp. 435 –440.
• [20] A. Anta and P. Tabuada, “To sample or not to sample: Self-triggered control for nonlinear systems,” IEEE Transactions on Automatic Control, vol. 55, pp. 2030–2042, Sep. 2010.
• [21] M. Mazo Jr., A. Anta, and P. Tabuada, “An iss self-triggered implementation of linear controller,” Automatica, vol. 46, pp. 1310–1314, Aug. 2010.
• [22] X. Wang and M. D. Lemmon, “Event-triggering in distributed networked systems with data dropouts and delays,” in Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control, Apr. 2009, pp. 366–380.
• [23] H. Zhang, P. Soldati, and M. Johansson, “Optimal link scheduling and channel assignment for convergecast in linear wirelesshart networks,” in Proceedings of the 7th international conference on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, 2009, pp. 82–89.
• [24] E. D. Sontag, “Input to state stability: Basic concepts and results,” in Nonlinear and Optimal Control Theory. Springer, 2006, pp. 163–220.
• [25] J. Johnsen and F. Allgöwer, “Interconnection and damping assignment passivity-based control of a four-tank system,” Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, pp. 111–122, 2007.
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The feedback must be of minimum 40 characters and the title a minimum of 5 characters | {"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": 0.8032421469688416, "perplexity": 942.001536632664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704803308.89/warc/CC-MAIN-20210126170854-20210126200854-00484.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/123793-finite-field.html | 1. ## Finite Field
Prove the following: Every finite field is perfect.
Proof. Suppose $F$ is a finite field of characteristic $p$ and let $E$ be a finite extension of $F$. Also let $\alpha \in E$. We want to show that $\alpha$ is separable over $F$. We know that $f(x) = \text{irr}(\alpha, F)$ factors in $\overline{F}$ as $\prod_{i} (x-\alpha_i)^{v}$ where the $\alpha_i$ are the zeros of $f(x)$. We need to show that $v=1$.
What would be a good way to approach this?
2. Originally Posted by Sampras
Prove the following: Every finite field is perfect.
Proof. Suppose $F$ is a finite field of characteristic $p$ and let $E$ be a finite extension of $F$. Also let $\alpha \in E$. We want to show that $\alpha$ is separable over $F$. We know that $f(x) = \text{irr}(\alpha, F)$ factors in $\overline{F}$ as $\prod_{i} (x-\alpha_i)^{v}$ where the $\alpha_i$ are the zeros of $f(x)$. We need to show that $v=1$.
What would be a good way to approach this?
This question, the one on the primitive element and perhaps others are very basic, standard stuff...I mean, are you trying to cope with this material without the aid of books? Any decent book in algebra/fields/extensions/Galois Theory (with introduction on extensions) deal with this (for example, you could first prove that a field K of characteristic p is perfect iff $K^p=K\Longleftrightarrow$ the p-th power map is an automorphism of K.)
I propose you grab some books, try to get some help from them AND THEN, if you get stuck somewhere, you ask.
I firmly believe that anyone studying mathematics at any level above high school (and also before is highly recommended) must get used to books and must have several by his/her side all the time, either of his property or the library's.
Tonio
3. Originally Posted by tonio
This question, the one on the primitive element and perhaps others are very basic, standard stuff...I mean, are you trying to cope with this material without the aid of books? Any decent book in algebra/fields/extensions/Galois Theory (with introduction on extensions) deal with this (for example, you could first prove that a field K of characteristic p is perfect iff $K^p=K\Longleftrightarrow$ the p-th power map is an automorphism of K.)
I propose you grab some books, try to get some help from them AND THEN, if you get stuck somewhere, you ask.
I firmly believe that anyone studying mathematics at any level above high school (and also before is highly recommended) must get used to books and must have several by his/her side all the time, either of his property or the library's.
Tonio
Do you think using too many books destroys intuition? E.g. what about "discovery-based learning?"
4. Originally Posted by Sampras
Do you think using too many books destroys intuition? E.g. what about "discovery-based learning?"
I'm not sure I fully understand what you mean, but if I get it right the yes: if you ALWAYS manage to find an answer to your questions in books and/or in some maths forum then not precisely your intuition but rather your creativity and/or your working habits can get impaired.
Yet, in the case of standard, non-trivial well-known theorems, I think it is a GOOD habit to have reference books by your side, work by yourself proofs and, in case of getting stuck, then ask someone else.
Tonio | {"extraction_info": {"found_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": 28, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9246229529380798, "perplexity": 282.3334491478516}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1480698541396.20/warc/CC-MAIN-20161202170901-00396-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.officetooltips.com/word_365/tips/how_to_insert_an_equation_with_integral.html | # How to insert an equation with integral
Word
This tip displays how to add an equation with integral, for example, Gauss's law, also known as Gauss's flux theorem.
How to add an equation in your document, see Working with Microsoft Equation.
To add an integral form of the Gauss's law, do the following:
## In the Professional format:
2. On the Equation tab, in the Structures group, click the Integral button:
In the Integral list, choose Contour Integral:
3. In the integral template:
• In the lower box, enter S.
• In the upper box, right-click and choose Hide Upper limit in the popup menu:
• In the right base box, enter E.
4. On the Equation tab, in the Symbols group, in the list of symbols choose .
5. Then enter dA=.
6. On the Equation tab, in the Structures group, click the Fraction button. In the Fraction list choose Stacked Fraction:
7. Enter Q at the top of your fraction.
8. In the bottom of your fraction, do the following:
• On the Equation tab, in the Structures group, click the Script button.
• In the Script list, choose Subscript:
• On the Equation tab, in the Symbols group, in the list of symbols choose ,
• In the lower box of the subscript, enter 0.
## In the Linear format:
2. Do one of the following:
• On the Equation tab, in the Symbols group, click the More button:
In the top of the list of symbols, choose Operators:
In the Operators list, choose .
• Simply enter \oint.
3. Enter _S. Then you enter a space key; this linear formula transformed into a professional format.
4. In the base box of integral, enter E.
5. On the Equation tab, in the Symbols group, choose (or simply enter \bullet).
6. Then enter dA=Q/, choose (or simply enter \varepsilon) and then _0:
Then you enter a space key, your linear formula transformed into the professional format. | {"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": 0.9668456315994263, "perplexity": 4696.6850156042765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323587659.72/warc/CC-MAIN-20211025092203-20211025122203-00340.warc.gz"} |
https://engineeringlibrary.org/reference/fracture-criteria | # Fracture Criteria
This page provides the chapter on fracture criteria from Wang, C. H. "Introduction to Fracture Mechanics," DSTO Aeronautical and Maritime Research Laboratory, DSTO-GD-0103, 1996.
Other related chapters from "Introduction to Fracture Mechanics" can be seen to the right.
Introduction to Fracture Mechanics
## 4. Fracture Criteria
### 4.1 K as a Failure Criterion
From previous analysis, it is clear that when stresses at the crack tip exceed yield (which always happens for engineering materials), plasticity results. However, if the redistribution of stress has a minimal effect on the crack tip elastic stress field, then the K approach to defining the stress field is still of sufficient accuracy for engineering applications. Thus, if plasticity is minimal, then a LEFM approach is justified.
Of importance to practical applications is the critical stress and strain state at the crack tip zone, which, when attained, causes the crack to propagate in a brittle, catastrophic manner. The most dangerous situation occurs when a crack is in a high-energy but constrained field that permits only slight plastic deformation at the crack tip. Expressed another way, the amount of energy absorbed in plastic deformation is reduced to a minimum extent and much more energy is thus available for fracture, i.e. crack propagation. This critical state can be described by a critical stress intensity factor Kc,
$$K = K_C$$
(4.1)
which may imply either a low stress acting on long crack or a small crack suffering a high stress. It is important to note the different meaning of the two sides of the above equation. The left hand side represents the driving force of the crack, which depends on the applied loads and the geometry of the components. The right hand side of equation (4.1) signifies the materials' resistance to fracture, which is an environment and load rate dependent material property.
Laboratory testing indicates that the fracture toughness value depends on the thickness B of the specimen tested. The plane strain fracture toughness of the materials is a material property (denoted as K1C, where subscript I denotes mode I loading). Under plane strain condition, since the crack tip plastic zone is small in relation to the component thickness, plastic contraction in the through thickness direction is suppressed by the surrounding elastic material. Tensile stresses are set up in the thickness direction of the plastic zone so that the stress state is triaxial, giving rise to constrained plastic deformation. Table 4.1 lists some typical values of plane strain fracture toughness. As before, the suffix I refers to the tensile opening mode of crack extension, whilst II and III symbolise shear and anti-plane tear modes, respectively.
When the plastic zone is large compared with the component's thickness, the triaxiality may be relaxed and the through thickness stresses normal to the plane of the component will be negligible. In this case, the fracture toughness may vary with the specimen thickness, B. The form of variation of Kc with specimen thickness is schematically shown in Fig. 4.1. Beyond a certain thickness, a state of plane strain prevails (see Chapter 3) and the toughness reaches asymptotic value. If the thickness of the specimen is reduced, more energy will be dissipated as a result of plastic deformation near the specimen surface which is under plane stress condition. There seems to exist an optimum thickness where the toughness reaches its highest level, see Fig.4.1.
In order to achieve plane strain conditions at the elastic-plastic interface, the plastic zone must be small compared to the specimen thickness, crack length, and width of ligament
$$r_p \le {a \over 50}, ~~{W \over 50}, ~~{B \over 50}$$
(4.2)
According to the ASTM standard, the following requirements must be satisfied
$$a, B, (W - a) \ge 2.5 \left({K_I \over \sigma_{ys}}\right)^2$$
(4.3)
which is equivalent to setting the plasticity constraint factor to be √3.
Table 4.1 Typical values of fracture toughness
Material Young's modulus E (GPa) Yield stress σys (MPa) Toughness KIC (MPa√m) Thickness requirement 2.5(KICys)2 (mm)
Steels 210
medium carbon
260 54 108
pressure vessel
470 208 489.6
high strength alloy
1460 98 11
AFC 77 stainless
1530 83 7.4
Aluminum alloys 72
2024 T8
420 27 10.4
7075 T6
540 30 7.9
7178 T6
560 23 4.2
Titanium alloys 108
Ti-6Al-4V
1060 73 12.6
(high yield)
1100 38 3.1
Comparative data
Concrete
45 80 0.2-1.4
Ice
9.1 85 0.2*
Epoxy
2-3 30-60 0.5-3
Boron fibre
441 3000
Carbon fibre
250-390 2200-2700
Boron/epoxy composite
220-340 725-1730 46
CFRP
70-200 300-1400 32-45
GFRP
38 100-300 20-60
* not at room temperature!
### Looking for Fracture Calculators?
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### 4.2 Residual Strength and Critical Crack Size
Since the severity of a cracked component is characterised by stress intensity factor, K, and failure will occur when K = Kc, the residual strength of a cracked component is,
$$\sigma_c = {K_c \over Y \sqrt{\pi a}}$$
(4.4)
where Y is a geometry correction factor. Note that the stress σ is the gross stress on the section on which the function a is defined, where residual strength implies a net section condition. In the case of plane strain Kc = KIC. It is conservative to assume that Kc = KIC if the detailed stress state is not known. The size of the crack at this stress is called the "critical crack size". This is normally difficult to solve in closed form as Y(a) is normally a complicated function of crack length and component geometry. Nevertheless, it can be solved numerically through iteration or, if the value of Y varies slowly with crack size, e.g. for a relatively small crack in a wide panel, an approximate value may be used. The critical crack size that a component can tolerate for a given load is
$$a_c = {1 \over \pi} \left({K_c \over Y (a_c / W) \sigma}\right)^2$$
(4.5)
The above two equations provide the basis for fracture mechanics based design methodologies.
It should be pointed that equation (4.4) is valid only when linear fracture mechanics is applicable, that is the net section stress level is far below the material's yield stress. Otherwise the component will fail in a different mode: plastic collapse. Consider a centre cracked panel with a finite width W, the absolute highest load carrying capability is bounded by the plastic collapse strength: the stress level over the entire section exceeds the yield or ultimate tensile strength of the material. It is easy to show that the nominal stress at collapse is
$$\sigma_{pc} = {W - 2a \over W} \sigma_{ys}$$
(4.6)
When this happens, the plastic deformation becomes unbounded and fracture will occur, regardless of the fracture toughness.
Therefore there are two possible failure modes: brittle fracture and plastic collapse. Should the fracture stress σc be higher than the stress causing failure by collapse, then collapse will prevail. As a result, the actual residual strength is the lowest of σc and σpc. Considering a centre cracked panel, there are three situations in which a plastic collapse failure would prevail: (1) the toughness is very high; (2) the crack is very small; and (3) the width W is very small. A sketch is shown in Fig.4.2. The intersection of the two curves is given by
$${W - 2a \over W} \sigma_{ys} \gt {K_c \over \sqrt{\pi a} \sqrt{\sec (\pi a / W)}}$$
(4.7)
In the short crack regime, the exact transition from one mechanism to the other is not clear, but a plausible engineering approximation is the "tangent" rule: drawing a tangent line passing through the ultimate tensile strength point. More accurate prediction can be achieved by using elasto-plastic fracture mechanics methods.
Example 4.11 Estimate the failure load under uniaxial tension for a centre-cracked panel of aluminium alloy of width W = 500 mm, and thickness B = 4 mm, for the following values of crack length 2a = 20 mm and 2a = 100 mm. Yield stress σy = 350 MPa and fracture toughness KIC = 70 MPa√m.
Solution There are two possible failure modes: plastic collapse and brittle fracture. We will assess the load level required for each mode to prevail.
(i) 2a = 20 mm.
Plastic collapse load Fpc = σys·(W - 2a)·B = 672 kN
Fracture load Fc = σc·W·B where $$\sigma_c = {K_{IC} \over \sqrt{\pi a \sec (\pi a / W)}} =$$ 394.6 MPa
thus Fc = 790 kN.
The actual failure load is the smaller of the above results, 672 kN.
(ii) 2a = 100 mm.
Plastic collapse load Fpc = σys·(W - 2a)·B = 560 kN
Fracture load Fc = σc·W·B where $$\sigma_c = {K_{IC} \over \sqrt{\pi a \sec (\pi a / W)}} =$$ 172.2 MPa
thus Fc = 334.57 kN.
The actual failure load is the smaller of the above results, 334.6 kN.
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### 4.3 R-curve
Crack extension occurs when the stress intensity factor or the strain energy release rate attains a critical value. In a truly brittle material like glass or ice, the energy for crack growth is the surface energy to form the new surface, i.e
$$G = 2 \gamma_f$$
(4.8)
where the factor "2" is included to represent the two crack surfaces being created. It should be noted that the energy required for a crack to grow in an engineering material is much larger than the surface energy. This is because plastic deformation will inevitably occur near the crack tip region and during crack extension energy is consumed in deforming the material plastically. In general the fracture criterion can be written as
$$G = 2 W_f = 2 (\gamma_f + \gamma_p)$$
(4.9)
Where γp refers the plastic work per unit area of surface created, and is typically much larger than γf.
Normally it is convenient to replace 2Wf with R, the material resistance to crack extension. A plot of R versus crack extension is called a resistance curve or R curve, whereas the plot of G versus crack extension is the driving force curve. It is important to note that the driving force curve is entirely dependent on the structure geometry and loading condition, whilst the R curve is a material property dependent on temperature, environment, and loading rate etc. Most brittle materials exhibit a constant resistance sometimes called "no R-curve" effect, as shown in Fig.4.3(a). Many ductile materials, such as low strength steels, possess a rising R curve: a plastic zone at the tip of crack increases with crack length, hence the energy that would dissipate to overcome plastic deformation would increase. This is illustrated in Fig.4.3(b). The exact shape of the R curve depends on the material and, to a lesser extent, on the configuration of the cracked structure.
If a component, containing a crack or crack-like defect, and experiencing some plasticity in the vicinity of the crack, is loaded by increments the crack will extend and stop after each increase in load. This condition is defined as slow-stable crack growth. In this condition the value of the material resistance KR is equal to the applied value K at any given applied stress. Consequently the fracture toughness (Kc) may be obtained by the use of crack growth resistance curves (commonly called R-curves). These curves are a continuous record of toughness development in terms of crack growth resistance, denoted KR, plotted against crack extension under continuously increasing values of stress intensity factor, K. The R-curves characterise the resistance to fracture of materials during incremental slow-stable crack extension as a result of the growth of the plastic zone as the crack extends.
Consider a plate with a through crack of initial length 2a0. At a fixed remote stress, σ, the energy release rate varies linearly with crack size. If the material has a flat R-curve, as shown in Fig 4.3(a), one can define a critical value of energy release rate, Gc, unambiguously. The crack will grow if the applied G reaches this value. For materials with a rising R curve, such as a crack plate reinforced with a composite patch, however, one cannot uniquely characterise a single value toughness value. In this case, normally we define that crack growth will occur when
$${dG \over da} \gt {dR \over da} ~~\text{and}~~ G \ge R$$
(4.10)
This corresponds to when the driving force curve is tangent with the R curve, as depicted in Fig.4.3(b). This can be interpreted as the critical condition when the energy available in the component for crack growth exceeds the maximum amount that the material can dissipate. This point of tangency depends on the shape of the driving force, which itself depends on the shape of the configuration of the structure. For example, the driving force curve for a through crack configuration is linear, but G in the double cantilever beam specimen varies with a2; these two configurations would have different Gc values for a given R curve.
Example 4.12 The following data were obtained from a series of tests conducted on pre-cracked specimens of thickness 10 mm,
Crack length
a (mm)
P (kN)
Critical displacement
u (mm)
30 4 0.4
40 3.5 0.5
50.5 3.12 0.63
61.6 2.8 0.78
71.7 2.67 0.94
79 2.56 1.09
where P and u are the critical load and displacement at each crack growth. The load displacement record for all crack lengths is linear up to a critical point. Determine the critical value of the strain energy release rate Gc = R from (a) the load displacement records and (b) the compliance-crack length curve.
Solution The load-deflection curve can be constructed from the tabulated data, as shown in Fig.4.4(a). The area for a triangle depicted in Fig.4.4(b) is,
Area = P1u2 − ½P1u1 − ½P2u2 − ½(P1 − P2)(u2 − u1) = ½(P1u2 − P2u1)
and so the energy released during each crack growth can be calculated
$$G = R = { \text{Area} \over 2 \Delta a \cdot B} = { {1 \over 2} (P_i u_j - P_j u_i) \over 2 B (a_j - a_i)}$$
The results for the five crack increments are: 30.0, 30.7, 30.2, 29.1, 30.8. (The unit is kJ/m2). Clearly this material exhibits little R-curve behaviour.
### Looking for Fracture Calculators?
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### 4.4 Mixed Mode Loading: Fracture and Crack Path
Most structures and components are subjected to more than one loading. When two or more modes of loading are present, equation (2.20) indicates that energy release rate contributions from each mode are additive. This equation assumes self-similar crack growth, however. If we consider an angled crack problem as depicted in Fig.4.5, coplanar growth means that the crack would grow at an angle 90° − β degrees from the applied stress. In practice, the crack tends to propagate in a direction orthogonal to the applied normal stress; i.e. the mixed-mode crack becomes a mode I crack. This is because a propagating crack seeks the path of least resistance (or the path of maximum driving force, or the path that the maximum amount of energy can be released) and need not be confined to its initial plane. A number of criteria have been proposed to account for such effects. Among them, the most widely used are (i) crack growth will take place in the direction of maximum energy release rate; (ii) crack growth occurs in a direction perpendicular to the maximum principal stress; (iii) crack growth occurs where the strain energy density is the minimum. It can be shown that criteria (i) and (ii) are identical and the differences between these criteria are generally small.
If a crack is loaded in combined mode I and II, the stresses σθ and τ at the crack tip can be derived from the expressions in Table.2.2, by adding the stresses due to the separate mode I and mode II. The result is as follows:
$$\sigma_{\theta} = {1 \over \sqrt{2 \pi r}} \cos^2 \left({\theta \over 2}\right) \left[ K_I \cos{\theta \over 2} - 3 K_{II} \sin{\theta \over 2} \right]$$
(4.11)
$$\tau_{r \theta} = {1 \over \sqrt{2 \pi r}} \cos{\theta \over 2} \left[ K_I \sin{\theta \over 2} \cos{\theta \over 2} + K_{II} \left( 1 - 3 \sin^2 {\theta \over 2} \right) \right]$$
(4.12)
Suppose that the crack in question forms an infinitesimal kink at an angle α from the plane of crack, as shown in Fig.4.6. The local stress intensity factors at the tip of this kink differ from the nominal K values of the main crack. If we define a local x'-y' coordinate system at the tip of the kink, we can define the local mode I and mode II stress intensity factors,
$$K_I (\alpha) = \lim_{r \to 0} \sigma_{\theta} \sqrt{2 \pi r} = \cos^2 {\alpha \over 2} \left[ K_I \cos{\alpha \over 2} - 3 K_{II} \sin{\alpha \over 2} \right]$$
(4.13)
$$K_{II} (\alpha) = \lim_{r \to 0} \tau_{r \theta} \sqrt{2 \pi r} = \cos {\alpha \over 2} \left[ K_I \sin{\alpha \over 2} \cos{\alpha \over 2} + K_{II} \left( 1 - 3 \sin^2 {\alpha \over 2} \right) \right]$$
(4.14)
The energy release rate for the kinked crack is
$$G(\alpha) = {K_I^2 (\alpha) + K_{II}^2 (\alpha) \over E}$$
(4.15)
According to the energy release rate criterion, crack propagation would occur in a direction along which the above energy release rate attains a maximum value. This is shown in Fig. 4.7, where the energy release rate G(α) is normalised by G(α = 0).
Since
$${d K_I \over d \alpha} = -{3 \over 2} K_I \cos^2 {\alpha \over 2} \sin{\alpha \over 2} - {3 \over 2} K_{II} \cos{\alpha \over 2} \left(1 - 3 \sin^2 {\alpha \over 2} \right) = -{3 \over 2} K_{II}$$
the maximum of the strain energy release rate dG(α)/dα = 0 is equivalent to KII(α) = 0 or dKI/dα = 0, thus the peak in G(α) at each α0 corresponds to the point where KI(α) exhibits a maximum and KII0) = 0. In other words, the energy release rate criterion is identical to maximum hoop stress criterion. Figs.4.8 show the hoop stress distributions for three mixed mode ratios: KII/KI = 0 (mode I), KII/KI = 1, KII/KI = ∞ (mode II).
The arrows in the figures mark the direction of crack propagation, which is given by the following equation
$$K_{II} (\alpha_0) = 0$$
(4.16)
so
$$K_I \sin {\alpha_0 \over 2} \cos {\alpha_0 \over 2} + K_{II} \left( 1 - 3 \sin^2 {\alpha_0 \over 2} \right) = 0$$
(4.17)
which yields,
$$\left( \tan {\alpha_0 \over 2} \right)_{1,2} = {1 \over 4} {K_I \over K_{II}} \pm \sqrt{ \left({ K_I \over 4 K_{II} }\right)^2 + {1 \over 2} }$$
(4.18)
The critical value of KI or KII at which crack propagation occurs can be determined from the following equation,
$$K_I(\alpha) = K_{IC} ~~~\text{i.e.}~~~ K_I \cos^3 {\alpha_0 \over 2} - 3 K_{II} \cos^2 {\alpha_0 \over 2} \sin {\alpha_0 \over 2} = K_{IC}$$
(4.19)
Example 4.13 Determine the propagation angle for an inclined crack subjected to uniaxial tension.
Solution: Assume the crack is inclined at an angle β to the applied load, as depicted in Fig.4.5. The mode I and mode II stress intensity factors can be determined as,
$$K_I = \sigma \sqrt{\pi a} \cos^2 \beta ~~~~\text{and}~~~~ K_{II} = \sigma \sqrt{\pi a} \cos \beta \sin \beta$$
consequently the mode I to mode II ratio is equal to (1/tanβ), hence the kink angle is equal to β + α0,
$$\beta + \alpha_0 = \beta + 2 \tan^{-1} \left( {1 \over 4 \tan \beta} - \sqrt{ \left({1 \over 4 \tan \beta}\right)^2 + {1 \over 2} } \right)$$
which is depicted in Fig.4.9, together with some experimental data. | {"extraction_info": {"found_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": 0.8806573748588562, "perplexity": 1372.2636285670378}, "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-2022-21/segments/1652662531762.30/warc/CC-MAIN-20220520061824-20220520091824-00389.warc.gz"} |
https://de.maplesoft.com/support/help/maple/view.aspx?path=XMLTools/XMLCData | XMLCData - Maple Help
XMLTools
XMLCData
XML CDATA constructor
Calling Sequence XMLCData(s)
Parameters
s - string; text of the CDATA section
Description
• The XMLCData(s) command takes the text of a CDATA section s and returns it as an XML data structure in the form of a Maple XML tree.
By using this function, you can include a CDATA section in any Maple representation of an XML document.
Examples
> $\mathrm{with}\left(\mathrm{XMLTools}\right):$
> $\mathrm{XMLCData}\left("This is some CDATA with embedded markupNot even a close tag!"\right)$
${\mathrm{_XML_CDATA}}{}\left({"This is some CDATA with embedded markupNot even a close tag!"}\right)$ (1) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9970030784606934, "perplexity": 4989.721141145067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662521883.7/warc/CC-MAIN-20220518083841-20220518113841-00723.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=588382 | # Electromagnetic field strength
P: 34 hello world. it is know that electrostatic (coulomb's law) and magnetostatic (biot-savart law) fields lose their strength like 1/r^2. why do they say that electromagnetic field falls like 1/r ? is that true ? if yes how, can you explain please ? after all energy radiated from a point source must fall like 1/r^2, because the area of surface of a sphere increases like r^2.
Mentor
P: 17,318
Quote by roboticmehdi why do they say that electromagnetic field falls like 1/r ?
Can you provide a reference for this? It is hard to say one way or the other without knowing the details.
P: 34
Quote by DaleSpam Can you provide a reference for this? It is hard to say one way or the other without knowing the details.
http://en.wikipedia.org/wiki/Larmor_formula there in the part ''Derivation 2: Using Edward M. Purcell approach'' it says stuff related to this.
Mentor P: 17,318 Electromagnetic field strength Both Coulomb's law and the Biot-Savart law are approximations for 0 velocity and 0 acceleration respectively. The full general field produced by a point charge moving with arbitrary velocity and acceleration is given by the Lienard Wiechert potential: http://en.wikipedia.org/wiki/Li%C3%A...hert_potential If you look at the formula for the LW fields you see that for a stationary charge you get a 0 B field and a 1/r² E field, corresponding with Coulomb's law. If you look at the formula for the LW fields for a moving but not accelerating charge you get a 1/r² B field, corresponding with the Biot-Savart law. However, if you look at the formula for an accelerating charge you also get a 1/r E and a 1/r B field.
PF Gold P: 954 One way to shed light on this is to note that the 1/r fields (unlike the 1/r2 fields) are propagating away from the source, carrying energy with them. In a wave, the intensity (energy per unit time per unit normal area) is proportional to the square of the amplitude, so to 1/r2 for the 1/r propagating field. But this 1/r2 intensity law is just what we get by assuming energy not to be lost from the wave as it propagates outwards through larger and larger spherical surfaces – whose areas are proportional to r2.
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