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http://www.r-bloggers.com/tag/interval/ | # Posts Tagged ‘ interval ’
## Confidence interval for predictions with GLMs
November 4, 2011
By
Consider a (simple) Poisson regression . Given a sample where , the goal is to derive a 95% confidence interval for given , where is the prediction. Hence, we want to derive a confidence interval for the prediction, not the potential observation...
## Confidence we seek…
November 18, 2009
By
$Confidence we seek…$
Estimating a proportion at first looks elementary. Hail to aymptotics, right? Well, initially it might seem efficient to iuse the fact that . In other words the classical confidence interval relies on the inversion of Wald’s test. A function to ease the computation is the following (not really needed!). waldci<- function(x,n,level){ phat<-sum(x)/n results<-phat + c(-1,1)*qnorm(1-level/2)*sqrt(phat*(1-phat)/n) print(results) } An exact confidence interval is | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9401262998580933, "perplexity": 2072.6901016328325}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049279525.46/warc/CC-MAIN-20160524002119-00084-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.earthdoc.org/content/papers/10.3997/2214-4609.201410435 | 1887
PDF
• # f Steep DIP migration by replacing vertical with horizontal propagation
• By L. E. Berg
• Publisher: European Association of Geoscientists & Engineers
• Source: Conference Proceedings, 54th EAEG Meeting, Jun 1992, cp-45-00097
• ISBN: 978-90-73781-04-7
• DOI:
### Abstract
The algorithm presented in this paper uses the implicit finite difference scheme (45 degree approximation), for the propagation. The advantages by choosing this alternative for a migration implementation are well known: Since the extrapolation operator is only 3 points long, it is robust in cases with considerable velocity variations. In addition, run time on computer is very short, compared to some other algorithms. The disadvantage is of course the low performance for steep dips
/content/papers/10.3997/2214-4609.201410435
1992-06-01
2022-07-05 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8944699764251709, "perplexity": 3936.04718444682}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00435.warc.gz"} |
https://www.physicsforums.com/threads/ligo-gravity-waves.929550/ | # B LIGO gravity waves
1. Oct 24, 2017
### carl susumu
The axis on the bottom of the graph depicts frequencies between 20-1000 Hz which are sound waves. Again, how can a sound wave (gravity waves) propagate in the near vacuum of stellar space that is vacuum?
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"On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10^−21." (Abstract).
Abbott, B. P. Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters. 116, 061102. 2016
---------------------------------------------------------------------------------------------------------------------------------
Can you explain how electromagnetic stellar gravity waves (GR) form the effects of sound waves at the LIGO observatory?
2. Oct 24, 2017
### Drakkith
Staff Emeritus
Sound waves which are audible to us have frequencies from about 20-20,000 Hz, but other waves exist with these frequencies which are not sound waves. For example, EM waves at this frequency are used for communications.
As for what these gravitational waves use as a medium, the answer is that they are waves in the metric of spacetime. They are, in short, a propagating temporary change in the geometry of spacetime.
These waves are gravitational waves. Gravity waves, electromagnetic waves, and sound waves are all something different. Gravity waves are waves on the surface of a fluid, such as the wave on the surface of the ocean, while EM waves are waves in the electromagnetic field (light, radio waves, x-rays, etc) and sound waves are a certain type of wave within a physical medium such as water, air, or even rock.
3. Oct 25, 2017
### carl susumu
In Einstein's paper, "The Foundation of the Generalised Theory of Relativity" (1916), Einstein represents gravity with Maxwell's electromagnetic field using Maxwell's equations.
dh/dt + rot e = 0...............................................70
div h = 0...........................................................71
rot h - de'/dt = i................................................72
div e' = p"........................................................73
(Einstein5, § 20). Einstein is representing gravity with Maxwell's electromagnetic field that is based on Faraday's induction effect but a small stone that is affected by gravity yet unaffected by a magnet of Faraday's law and a three inch lead plate does not produce anti-gravity which proves gravity is not an electromagnetic phenomenon. Stellar gravity waves are electromagnetic waves. The same waves as a radio wave.
4. Oct 25, 2017
### carl susumu
Weber experimentally detected gravitational waves that have the frequency of sound (1662 Hz).
"Further advances are necessary in order to generate and detect gravitational waves in the laboratory." (Weber, Conclusion, 1960).
"A description is given of the gravitational radiation experiments involving detectors at opposite ends of a 1000 kilometer baseline, at Argonne National Laboratory and the University of Maryland. Sudden increases in detector output are observed roughly once in several days, coincident within the resolution time of 0.25 seconds. The statistics rule out an accidental origin and experiments rule out seismic and electromagnetic effects. It is reasonable to conclude that gravitational radiation is being observed." (Weber, Abstract, 1970).
"EXPERIMENTS AT 1662 HERTZ" (Weber, Intro, 1970).
Weber detected gravity waves with the frequency of 1662 Hz using the acoustical vibration of a 750 lb aluminum beam but sound cannot propagate in the vacuum of stellar space.
5. Oct 25, 2017
### carl susumu
Precedence--------Weber
6. Oct 25, 2017
### Drakkith
Staff Emeritus
No he isn't. To quote Einstein, from paragraph 814 on his translated paper at wikisource:
He's setting up Maxwell's equations in a form which remains invariant regardless of your coordinate system choice.
That is incorrect.
The "acoustical vibration" is in the aluminum beam, which is being stressed at the frequency of the passing of the gravitational wave (or would be if he had actually detected a gravitational wave, which he did not), it is not in the vacuum. No sound is propagating through space. | {"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": 0.9592112302780151, "perplexity": 1371.329333577796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742338.13/warc/CC-MAIN-20181115013218-20181115035218-00148.warc.gz"} |
http://encyclopedia.kids.net.au/page/po/Power_series | ## Encyclopedia > Power series
Article Content
# Power series
In mathematics, a power series is an infinite series of the form
$f(x) = \sum_{n=0}^\infty a_n \left( x-a \right)^n$
where the coefficients an, the center a, and the argument x are real or complex numbers. These series usually arise as the Taylor series of some known function; the Taylor series article contains many examples.
A power series will converge for some values of the variable x (at least for x = a) and may diverge for others. It turns out that there is always a number r with 0 ≤ r ≤ ∞ such that the series converges whenever |x - a| < r and diverges whenever |x - a| > r. (For |x - a| = r we cannot make any general statement.) The number r is called the radius of convergence of the power series; in general it is given as
r = lim infn → ∞ |an|-1/n
but a fast way to compute it is
r = limn → ∞ |an/an+1|.
The latter formula is valid only if the limit exists, while the former formula can always be used.
The series converges absolutely for |x - a| < r and converges uniformly on every compact subset of {x : |x - a| < r}.
### Differentiating and integrating power series
Once a function is given as a power series, it is continuous wherever it converges and is differentiable on the interior of this set. It can be differentiated and integrated quite easily, by treating every term separately:
$f^\prime (x) = \sum_{n=1}^\infty a_n n \left( x-a \right)^{n-1}$
$\int f(x)\,dx = \sum_{n=0}^\infty \frac{a_n \left( x-a \right)^{n+1}} {n+1} + C$
Both of these series have the same radius of convergence as the original one.
### Analytic functions
A function f defined on some open subset U of R or C is called analytic if it is locally given by power series. This means that every aU has an open neighborhood VU, such that there exists a power series with center a which converges to f(x) for every xV.
Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero.
If a function is analytic, then it is infinitely often differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients an can be computed as
$a_n = \frac {f^{\left( n \right)}\left( a \right)} {n!}$
where f (n)(a) denotes the n-th derivative of f at a. This means that every analytic function is locally represented by its Taylor series.
The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element aU such that f (n)(a) = g (n)(a) for all n ≥ 0, then f(x) = g(x) for all xU.
If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.e. analytic functions f which are defined on larger sets than { x : |x - a| < r } and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with |x - a| = r such that no analytic continuation of the series can be defined at x.
The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.
### Formal power series
In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series, a principle that is of great utility in combinatorics.
All Wikipedia text is available under the terms of the GNU Free Documentation License
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https://icml.cc/Conferences/2019/ScheduleMultitrack?event=4737 | Timezone: »
Oral
Using Pre-Training Can Improve Model Robustness and Uncertainty
Dan Hendrycks · Kimin Lee · Mantas Mazeika
Tue Jun 11 12:05 PM -- 12:10 PM (PDT) @ Grand Ballroom
Tuning a pre-trained network is commonly thought to improve data efficiency. However, Kaiming He et al. (2018) have called into question the utility of pre-training by showing that training from scratch can often yield similar performance, should the model train long enough. We show that although pre-training may not improve performance on traditional classification metrics, it does provide large benefits to model robustness and uncertainty. Through extensive experiments on label corruption, class imbalance, adversarial examples, out-of-distribution detection, and confidence calibration, we demonstrate large gains from pre-training and complementary effects with task-specific methods. Results include a 30% relative improvement in label noise robustness and a 10% absolute improvement in adversarial robustness on both CIFAR-10 and CIFAR-100. In some cases, using pre-training without task-specific methods surpasses the state-of-the-art, highlighting the importance of using pre-training when evaluating future methods on robustness and uncertainty tasks. | {"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": 0.9011644721031189, "perplexity": 4795.485131269656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320299927.25/warc/CC-MAIN-20220129032406-20220129062406-00457.warc.gz"} |
https://stats.stackexchange.com/questions/326334/why-are-contours-of-a-multivariate-gaussian-distribution-elliptical/326372 | # Why are contours of a multivariate Gaussian distribution elliptical?
Displayed below are the contours and their respective covariance matrices according to Andrew Ng's notes (pdf). Why are the first and second contours elliptical and not circular? The variance along both axes is the same.
Here's one last set of examples generated by varying $\Sigma$:
The plots above used, respectively,
$$\Sigma = \begin{bmatrix} 1&-0.5\\-0.5 &1 \end{bmatrix}; \qquad \Sigma = \begin{bmatrix} 1&-0.8\\-0.8 &1 \end{bmatrix}; \qquad \Sigma = \begin{bmatrix} 3&0.8\\0.8 &1 \end{bmatrix}.$$
• Looks like a scaling issue to me. The range is the same, but the length of the plot region are not. Feb 1 '18 at 20:17
• So, is what I think right and the contours of the first and third covariance matrices should be circles? Feb 1 '18 at 20:20
• The only reason they can be ellipses is if the variances are different. You could verify my claim by printing the page and measuring with a ruler. Feb 1 '18 at 20:21
• That the variances are the same is revealed by comparing the widths and heights of the ellipses. This has nothing to do with the eccentricities, which also depend on the correlations. @Dimitriy Scaling is not the explanation. At a correct aspect ratio all three plots would be square, but all three sets of ellipses would still be non-circular.
– whuber
Feb 1 '18 at 20:27
• @whuber is right. Correlation will also make them ellipses, even if variance is the same. Feb 1 '18 at 20:35
You can understand the shape of the ellipsoid better if you look at the spectral/eigen decomposition of the precision matrix (inverse of the covariance matrix). You want to look at the eigenvalues of this inverse, not the diagonal elements.
Just a supplement to the other answers: for a multivariate Normal with dimension $$k$$, you can see why algebraically if you follow this. Set the density equal to some level $$l$$, then: \begin{align*} (2\pi)^{-k/2} |\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)'\Sigma^{-1}(x-\mu) \right) &= l\\ \iff \exp\left(-\frac{1}{2}(x-\mu)'\Sigma^{-1}(x-\mu) \right) &= l'\\ \iff (x-\mu)'\Sigma^{-1}(x-\mu) &= l''.\tag{*} \end{align*} (*) is the formula for an ellipsoid centered at $$\mu$$. The
For your first covariance matrix, the spectral decomposition of its inverse is $$\Sigma^{-1} = P\Lambda P'$$, where $$P = \left[\begin{array}{cc} P_1 & P_2 \end{array}\right] = \left[\begin{array}{cc} .707 & -.707\\ .707 & .707 \end{array}\right]$$ and $$\Lambda = \left[\begin{array}{cc} \lambda_1 & 0 \\ 0 & \lambda_2 \end{array}\right] = \left[\begin{array}{cc} 2 & 0 \\ 0 & 2/3 \end{array}\right].$$ The reason why it looks "squished" is because the diagonals of $$\Lambda$$ are not the same. This is because the semi-axes are $$P_1/\lambda_1$$ (the up and to the right vector) and $$P_2/\lambda_2$$ (up and to the left). Because $$\lambda_1$$ is bigger, that means $$P_1/\lambda_1$$ is a shorter vector.
What if we're used to looking at the covariance matrix, instead of its inverse? Well their spectral decompositions are pretty related. Because $$\Sigma^{-1} = P\Lambda P'$$ and because $$P$$ is orthogonal, we have $$\Sigma = P \Lambda^{-1}P'.$$ Just try multiplying these two decompositions together, and you should get the identity matrix. What this tells us is that these two matrices have the same eigenvectors (and so they have the same principal axes), and the eigenvalues are reciprocals. However, I started off with the precision matrix because that's what is in the formula for the density.
## More examples:
If the elements of $$x$$ are independent, then $$\Sigma$$ is diagonal, then $$\Sigma^{-1}$$ is diagonal, then (*) is $$\frac{(x_1 - \mu_1)^2}{\sigma_1^2} + \frac{(x_2 - \mu_2)^2}{\sigma_2^2} = l''\tag{**}$$ which is still an ellipse, but it's not tilted/rotated.
If the elements of $$x$$ are independent and moreover they are identical, then $$\sigma_1 = \sigma_2$$ and (**) turns into a circle.
• The equation of an ellipse on wikipedia has a $1$ on the RHS, so in this case $$(x-\mu)^\top \Sigma^{-1}(x - \mu) = 1$$ How does having $l''$ rather than $1$ change the ellipsoid? Surely it has some impact. Mar 12 '21 at 9:33
• @Euler_Salter what happens if you scale both sides of (*) by $1/l''$? Mar 28 '21 at 1:48
• @Euler_Salter I'm late to the party, but it should basically scale the ellipse by $\sqrt{l''}$ equally in each dimension. May 6 '21 at 12:19
Assume you are visualizing the distribution of a vector called $(X,Y)$ (assumed to have a bivariate normal distribution).
When $X$ and $Y$ have the same variance, the projections of the ellipse on both axes have the same length. This does mean it's a circle. It can be oblique. It's not a circle when $X$ and $Y$ are not independent.
When $X$ and $Y$ are independent, the major and minor axes of the ellipse are aligned with the axes. This does not mean it's a circle either, it can be flattened.
A circle requires both:
• independence of $X$ and $Y$
• $X$ and $Y$ having the same variance
This is when the covariance matrix $\Sigma$ is diagonal with a constant diagonal.
• (+1) But note that your assertions implicitly suppose $(X,Y)$ has a bivariate Gaussian distribution. Otherwise, you should replace "independent" by "uncorrelated."
– whuber
Feb 1 '18 at 21:15
• Title of the question: "...Multivariate Gaussian...". But I'll add it in my answer because I also felt a doubt when writing it. Feb 1 '18 at 21:16
• Understood: but I had taken your introductory sentence referring to "a distribution" as a (legitimate) attempt to generalize the result.
– whuber
Feb 1 '18 at 21:18
Consider this figure. Notice how both the circle and the dashed diagonal are inside the square. So, the circle is how the contours of the multivariate Gaussian looks when correlation is zero. The dashed diagonal is the contour of the perfectly correlated variables. The ovals (ellipses) are in between, when correlation is not equal zero or one. The length of the square sides represents the variance (standard deviation) of the variables (marginals).
Here, I resized your picture to make the x- and y-axis scales equal, and you can see how the oval fits into a square. I think that the fact that Andrew Ng's plot was not scaled equally just added to the confusion. You can fit all kinds of ovals into the same square. You can have all kinds of contours for the same variances of variables depending on the correlation between them.
The image is from this web site, which has nothing to do with a question asked :)
• It would be nice if you could clarify that we need zero correlation AND equal variances for circular contours. Feb 1 '18 at 20:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8467493653297424, "perplexity": 349.1666846111268}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1642320300849.28/warc/CC-MAIN-20220118122602-20220118152602-00472.warc.gz"} |
https://physics.stackexchange.com/questions/635783/a-standing-wave-with-different-linear-mass-densities | A standing wave with different linear mass densities
I have a question about a standing wave with different linear mass densities throughout the string. Suppose that we had a string of linear mass density $$\mu$$ joined at $$x = L$$ to a string with linear mass density $$\mu/9$$ and length $$3L$$ to form a composite string of total length $$4L$$. The first end of this composite string is fixed at $$x = 0$$ and the last end at $$x = 4L$$ is free to oscillate in the $$±y$$ direction. How would I show that $$\sin(k_1L)$$ = ± $$\frac{\sqrt3}{2}$$?
This question seemed very weird to me at first - since a standing wave isn't a travelling wave, I can't find reflected/transmitted amplitudes the standard way. However, am I right in thinking that since a standing wave can be decomposed into 2 left/right travelling waves of the same amplitudes, I can do this?
I have seen a sort of solution here : Standing waves on string with different densities, but this is of a string of length 2L, tied down at both ends, whereas my string is of length 4L, with one end fixed and the other end free, and I'm not quite sure how to apply the same technique here.
If anyone could point me to the right direction - it would be very much appreciated.
I suppose that free-to-oscillate means the following
$$\frac{\partial u}{\partial x} = 0\, ,$$
that is, a Neumann boundary condition.
To solve this problem you do something similar to the case that you link to:
• Solve the differential equation for each segment. You would end up with different solutions because of the change in density.
• Apply the boundary condition on the left.
• Impose continuity in the interface between the two segments (in displacement and slope).
• Apply the boundary condition on the right to find the eigenvalues.
• Yeah, I thought about doing this - but is it ok if we only consider the time-independent equation describing our standing wave? Getting rid of the temporal part? – jambajuice May 14 at 11:30
• Yes, you could compute the Fourier transform or use the substitution $u(x, t) = w(x)e^{i\omega t}$. – nicoguaro May 14 at 15:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9169484972953796, "perplexity": 195.8844750556031}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154099.21/warc/CC-MAIN-20210731172305-20210731202305-00468.warc.gz"} |
https://math.stackexchange.com/questions/1107298/how-can-i-complete-this-proof-by-contradiction | # How can I complete this proof by contradiction?
This problem is from Discrete Mathematics and its Applications:
Prove that there are no solutions in integers $$x$$ and $$y$$ to the equation $$2x^2 + 5y^2 = 14$$.
I am trying to use proof by contradiction, which is described by the book as
Suppose we want to prove that a statement $$p$$ is true. Furthermore, suppose that we can find a contradiction $$q$$ such that $$\lnot p \implies q$$ is true. Because $$q$$ is false, but $$\lnot p \implies q$$ is true, we can conclude that $$\lnot p$$ is false, which means that $$p$$ is true. How can we find a contradiction $$q$$ that might help us prove that $$p$$ is true in this way?
Because the statement $$r \land \lnot r$$ is a contradiction whenever $$r$$ is a proposition, we can prove that $$p$$ is true if we can show that $$\lnot p \implies (r \land \lnot r)$$ is true for some proposition $$r$$. Proofs of this type are called proofs by contradiction. Because a proof by contradiction does not prove a result directly, it is another type of indirect proof.
Here is my work/thought process:
My initial proposition, $$p$$, is that there are no solutions in integers $$x$$ and $$y$$ to the equation $$2x^2 + 5y^2 = 14$$. I know that by proof by contradiction, I have to assume that the proposition isn't true, $$\lnot p$$, meaning there is a solution to $$x$$ and $$y$$ in the equation and show that assuming this leads to a contradiction (something that always evaluates to false, no matter the input values).
First, I recognized that for the sum be even, $$14$$, the two components, $$2x^2$$ and $$5y^2$$ have to be even as well.
I am able to show that $$2x^2$$ is even from the definition of even, that is, there is some integer $$k$$ such that $$2x^2 = 2k$$. $$k$$ would be $$x^2$$. However I have a hard time showing that $$5y^2$$ cannot be even. I first tried the same definition, meaning $$k = (5/2) y^2$$ but this wouldn't be an integer. However it is possible for $$5y^2$$ to be even, say $$y = 10$$. Am I going about this the right way? Is the even + even justification appropriate for this situation?
• +1 I love this question. You're fully engaged with the question you're studying. Jan 17, 2015 at 0:05
Yes, a proof by contradiction can be given. Suppose you have found a pair $(x,y)$ satisfying the equation.
Since $14$ and $2x^2$ are even, also $5y^2$ must be even as well. Therefore $y$ is even and so $y=2z$ for some integer $z$.
This implies $2x^2+20z^2=14$ that simplifies to $x^2+10z^2=7$. But $10z^2>7$ if $z\ne0$, so we must have $z=0$ and so $x^2=7$, a contradiction.
About your argument, there's a glitch. Since $14$ is even, two integers that sum to $14$ are either both even or both odd. However, since $2x^2$ is clearly even, you can conclude (as I did above) that also $5y^2$ must be even.
• because if this was the case, x would not even be an integer Jan 17, 2015 at 1:43
• @committedandroider Yes, $7$ is not a perfect square. Jan 17, 2015 at 10:41
• In summary, the conclusion from this would - Let's assume there are positive integers x and y such such that 2x^2 + 5y^2 = 14. Then integer x must be equal to the square root of 7 which itself is a contradiction because no integer x can be equal to the square root of 7(all inputs evaluate to false). Therefore by proof by contradiction, there are no such integers x and y. Jan 27, 2015 at 19:59
HINT: You’ll have a very hard time proving it this way. I recommend a different approach altogether. Note that $x^2$ and $y^2$ are non-negative, so $2x^2$ and $5y^2$ are at most $14$. Thus, if $y$ is an integer, then $y$ must be one of three integers; what are they? And what do they force $2x^2$ to be?
Added: And you can use your observation that $y$ must be even to reduce the possibilities still further.
• For possible values of $y$, couldn't you eliminate all integers $n$ where $|n|>1$? Jan 16, 2015 at 23:53
• @graydad: Yep; thanks. (No, of course not: $n^2=n$ for all $n\in\Bbb Z$! :-)) Jan 16, 2015 at 23:57
• No sweat, I think the edited version of your answer is more thought provoking :) Jan 16, 2015 at 23:59
• Also, $y$ must be even, since $2x^2$ and $14$ are both even. Jan 17, 2015 at 0:04
• Are at most 14? They can be arbitrarily large. I think you meant "almost 14" which restricts how big y and x can be: nice thinking
– Mzn
Jan 17, 2015 at 19:55
Roundabout proof (not the one you should use :-) ):
$x^2 \equiv \{0,1,4,5,6,9\}\mod 10$
$\Rightarrow 2x^2 \equiv \{0,2,8,10,12,18\}\mod 20$
$y^2 \equiv \{0,1\}\mod 4$
$\Rightarrow 5y^2 \equiv \{0,5\}\mod 20$
$\Rightarrow 2x^2+5y^2 \equiv \{0,2,3,5,7,8,10,12,13,15,17,18\}\mod 20$
$\Rightarrow 2x^2+5y^2 \not\equiv 14\mod 20$
$\Rightarrow 2x^2+5y^2 \neq 14$
• Nice appoarch. X squared mod 10 is one of {0,1,4,5,6,9}? Would you please provide proof or reasoning?
– Mzn
Jan 17, 2015 at 8:19
• Those are the quadratic residues mod 10. Examination of the first ten squares gives you these: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. After that $(x+10)^2 = x^2+20x+10^2 \equiv x^2 \mod 10$ Jan 17, 2015 at 8:24
• I see. Proof by induction. Thanks
– Mzn
Jan 17, 2015 at 11:55
• Well, could be, or I could been less lazy and said $(x+10k)^2 = x^2+20kx+10^2k^2 \equiv x^2 \mod 10$ :-) Jan 17, 2015 at 17:37
• Do you know of any introductory source on this topic? en.wikipedia.org/wiki/Quadratic_residue seems advanced to me :)
– Mzn
Jan 17, 2015 at 19:06
If you want to make your proof work, first note that $y = 0$ is not an option because then you would have $x^2 = 7$. As you noted, $5y^2$ must be even, so $y$ must be even, hence $y^2$ is at least 4. This gives a sum which is too big. (at least 20 when your target is 14) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8449991941452026, "perplexity": 152.3202727650665}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662521152.22/warc/CC-MAIN-20220518052503-20220518082503-00107.warc.gz"} |
http://math.stackexchange.com/questions/279540/how-to-show-that-int-0-infty-sinx2-dx-converges/279541 | # how to show that $\int_0^\infty \sin(x^2) dx$ converges [duplicate]
Possible Duplicate:
Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.
What test do I use to show that the following integral converges? $$\int_0^\infty \sin (x^2) \; dx$$
-
## marked as duplicate by sdcvvc, Stefan Hansen, Davide Giraudo, Michael Greinecker♦, rschwiebJan 17 '13 at 11:58
We deal with the integral from (say) $1$ to $\infty$.
In principle we should look at $\int_1^M \sin(x^2)\,dx$, then let $M\to\infty$.
Use integration by parts. Let $f(x)=\frac{1}{x}$ and $g'(x)=x\sin(x^2)$. Then $f'(x)=-\frac{1}{x^2}$ and we can take $g(x)=-\frac{1}{2}\cos x^2$.
We end up with $$\int_1^M\sin(x^2)\,dx=\left. -\frac{1}{2x}\cos(x^2)\right|_1^M -\int_1^M \frac{1}{2x^2}\cos(x^2)\,dx.$$ Now let $M\to \infty$. Note that the remaining integral behaves nicely as $M\to\infty$, since $\int_1^\infty \frac{dx}{x^2}$ converges, and $|\cos(x^2)|$ is bounded.
-
Should we be worried that when $M \to \infty \ \cos(M^2)$ is undefined? – Alex Jan 15 '13 at 21:18
@Alex we have $\cos (M^2)/M$ at the denominator. – Santosh Linkha Jan 15 '13 at 21:21
Not for this argument. For the first part (the evaluation), at the top we get $-\frac{1}{2M}\cos(M^2)$. Since $\cos$ wiggles between $-1$ and $1$, the $\frac{1}{2M}$ kills it. For the integral part, it is just Comparison Test, again using $|\cos(x^2)|\le 1$. – André Nicolas Jan 15 '13 at 21:23
@AndréNicolas: of course! I confused it with something completely different: mathoverflow.net/questions/24579/convergence-of-a-series – Alex Jan 15 '13 at 21:31
@AndréNicolas This argument seems to generalize well to $\int_0^{\infty} \sin(x^{\alpha})dx$ and $\int_0^{\infty} \cos(x^{\alpha})dx$ with $|\alpha|\gt 1$. Can we say anything about $\int_0^{\infty} \cos(f(x))dx$ with $f(x)$ a polynomial? (This would include the Airy function $Ai(x)$, right?) – AndrewG Jan 16 '13 at 3:52
Lots of information here:
http://en.wikipedia.org/wiki/Fresnel_integral
See especially the section Evaluation.
@rlgordonma & @experimentX
I just see the french like their Fresnel so much, their wikipedia page actually has a section on convergence as well as derivations of the final value:
http://fr.wikipedia.org/wiki/Int%C3%A9grale_de_Fresnel
-
I should point out to the OP to pay special attention to the nice illustration of the integration contour used to evaluate the integral, which shows why the integral converges. – Ron Gordon Jan 15 '13 at 21:02
isn't there any easy method (something like comparison) just to show that is converges? I don't have to evaluate it. Also this is not complex analysis ... i guess there must be something nice and easy. – Santosh Linkha Jan 15 '13 at 21:06
looks like the the french version is same as the other answer. Nice +1 to everyone – Santosh Linkha Jan 15 '13 at 21:16
Consider the triangle $\Delta$ with vertices at $(0,0), (T,0), (T,T)$ in the complex plane. Since $\exp(iz^2)$ is entire, we have $$\int_{\Delta} \exp(iz^2) dz = 0$$ Further, the integral on the side perpendicular to the $X$ axis, as $T \to \infty$ is 0, since $$\lim_{T \to \infty} \left \vert \int_{T}^{T+iT} \exp(iz^2) dz \right \vert \leq \lim_{T \to \infty} \int_{T}^{T+iT} \left \vert \exp(iz^2) \right \vert \vert dz \vert = \lim_{T \to \infty} \int_0^T \exp(-2Tx) dx\\ = \lim_{T \to \infty} \dfrac{1-\exp(-2T^2)}{2T} = 0$$ Hence, the integral along the $X$ axis equals the integral along the hypotenuse i.e. $$\int_{0}^T \exp(iz^2) dz = \int_{0}^{T+iT} \exp(iz^2) dz$$ Setting $z= (1+i)w$, we get that $$\int_{0}^{T+iT} \exp(iz^2) dz = \int_0^T \exp(i(1+i)^2 w^2) (1+i) dw = (1+i) \int_0^T \exp(-2w^2) dw$$ Hence, $$\lim_{T \to \infty}\int_{0}^{T} \exp(iz^2) dz = (1+i) \dfrac{\sqrt{\pi}}{2\sqrt{2}}$$ Now, note that $$\int_0^{\infty} \sin(x^2) dx = \text{Imag} \left( \int_0^{\infty} e^{ix^2} dx\right)=\dfrac{\sqrt{\pi}}{2\sqrt{2}}$$
- | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9884765148162842, "perplexity": 470.9499847188859}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987155.85/warc/CC-MAIN-20150728002307-00198-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-f-foundations-a-prelude-to-functions-section-f-1-the-distance-and-midpoint-formulas-f-1-assess-your-understanding-page-7/42 | ## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)
M = $(0.45,1.7)$
The coordinates (x, y) of the midpoint M are $x = \frac{x_1+x_2}{2} = \frac{1.2+-0.3}{2} = 0.45$ $y = \frac{y_1+y_2}{2} = \frac{2.3+1.1}{2} = 1.7$ That is, M = $(0.45,1.7)$ | {"extraction_info": {"found_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": 0.9763559699058533, "perplexity": 2001.0049775009588}, "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/1634323588284.71/warc/CC-MAIN-20211028100619-20211028130619-00427.warc.gz"} |
https://worldwidescience.org/topicpages/n/nonlinear+wave+propagations.html | #### Sample records for nonlinear wave propagations
1. Reconstruction of nonlinear wave propagation
Science.gov (United States)
Fleischer, Jason W; Barsi, Christopher; Wan, Wenjie
2013-04-23
Disclosed are systems and methods for characterizing a nonlinear propagation environment by numerically propagating a measured output waveform resulting from a known input waveform. The numerical propagation reconstructs the input waveform, and in the process, the nonlinear environment is characterized. In certain embodiments, knowledge of the characterized nonlinear environment facilitates determination of an unknown input based on a measured output. Similarly, knowledge of the characterized nonlinear environment also facilitates formation of a desired output based on a configurable input. In both situations, the input thus characterized and the output thus obtained include features that would normally be lost in linear propagations. Such features can include evanescent waves and peripheral waves, such that an image thus obtained are inherently wide-angle, farfield form of microscopy.
2. Longitudinal nonlinear wave propagation through soft tissue.
Science.gov (United States)
Valdez, M; Balachandran, B
2013-04-01
In this paper, wave propagation through soft tissue is investigated. A primary aim of this investigation is to gain a fundamental understanding of the influence of soft tissue nonlinear material properties on the propagation characteristics of stress waves generated by transient loadings. Here, for computational modeling purposes, the soft tissue is modeled as a nonlinear visco-hyperelastic material, the geometry is assumed to be one-dimensional rod geometry, and uniaxial propagation of longitudinal waves is considered. By using the linearized model, a basic understanding of the characteristics of wave propagation is developed through the dispersion relation and in terms of the propagation speed and attenuation. In addition, it is illustrated as to how the linear system can be used to predict brain tissue material parameters through the use of available experimental ultrasonic attenuation curves. Furthermore, frequency thresholds for wave propagation along internal structures, such as axons in the white matter of the brain, are obtained through the linear analysis. With the nonlinear material model, the authors analyze cases in which one of the ends of the rods is fixed and the other end is subjected to a loading. Two variants of the nonlinear model are analyzed and the associated predictions are compared with the predictions of the corresponding linear model. The numerical results illustrate that one of the imprints of the nonlinearity on the wave propagation phenomenon is the steepening of the wave front, leading to jump-like variations in the stress wave profiles. This phenomenon is a consequence of the dependence of the local wave speed on the local deformation of the material. As per the predictions of the nonlinear material model, compressive waves in the structure travel faster than tensile waves. Furthermore, it is found that wave pulses with large amplitudes and small elapsed times are attenuated over shorter spans. This feature is due to the elevated
3. Wave envelopes method for description of nonlinear acoustic wave propagation.
Science.gov (United States)
Wójcik, J; Nowicki, A; Lewin, P A; Bloomfield, P E; Kujawska, T; Filipczyński, L
2006-07-01
A novel, free from paraxial approximation and computationally efficient numerical algorithm capable of predicting 4D acoustic fields in lossy and nonlinear media from arbitrary shaped sources (relevant to probes used in medical ultrasonic imaging and therapeutic systems) is described. The new WE (wave envelopes) approach to nonlinear propagation modeling is based on the solution of the second order nonlinear differential wave equation reported in [J. Wójcik, J. Acoust. Soc. Am. 104 (1998) 2654-2663; V.P. Kuznetsov, Akust. Zh. 16 (1970) 548-553]. An incremental stepping scheme allows for forward wave propagation. The operator-splitting method accounts independently for the effects of full diffraction, absorption and nonlinear interactions of harmonics. The WE method represents the propagating pulsed acoustic wave as a superposition of wavelet-like sinusoidal pulses with carrier frequencies being the harmonics of the boundary tone burst disturbance. The model is valid for lossy media, arbitrarily shaped plane and focused sources, accounts for the effects of diffraction and can be applied to continuous as well as to pulsed waves. Depending on the source geometry, level of nonlinearity and frequency bandwidth, in comparison with the conventional approach the Time-Averaged Wave Envelopes (TAWE) method shortens computational time of the full 4D nonlinear field calculation by at least an order of magnitude; thus, predictions of nonlinear beam propagation from complex sources (such as phased arrays) can be available within 30-60 min using only a standard PC. The approximate ratio between the computational time costs obtained by using the TAWE method and the conventional approach in calculations of the nonlinear interactions is proportional to 1/N2, and in memory consumption to 1/N where N is the average bandwidth of the individual wavelets. Numerical computations comparing the spatial field distributions obtained by using both the TAWE method and the conventional approach
4. Nonlinear wave propagation in a rapidly-spun fiber.
Science.gov (United States)
McKinstrie, C J; Kogelnik, H
2006-09-04
Multiple-scale analysis is used to study linear wave propagation in a rapidly-spun fiber and its predictions are shown to be consistent with results obtained by other methods. Subsequently, multiple-scale analysis is used to derive a generalized Schroedinger equation for nonlinear wave propagation in a rapidly-spun fiber. The consequences of this equation for pulse propagation and four-wave mixing are discussed briefly.
5. Nonlinear propagation of short wavelength drift-Alfven waves
DEFF Research Database (Denmark)
Shukla, P. K.; Pecseli, H. L.; Juul Rasmussen, Jens
1986-01-01
Making use of a kinetic ion and a hydrodynamic electron description together with the Maxwell equation, the authors derive a set of nonlinear equations which governs the dynamics of short wavelength ion drift-Alfven waves. It is shown that the nonlinear drift-Alfven waves can propagate as two...
6. Nonlinear propagation and control of acoustic waves in phononic superlattices
CERN Document Server
Jiménez, Noé; Picó, Rubén; García-Raffi, Lluís M; Sánchez-Morcillo, Víctor J
2015-01-01
The propagation of intense acoustic waves in a one-dimensional phononic crystal is studied. The medium consists in a structured fluid, formed by a periodic array of fluid layers with alternating linear acoustic properties and quadratic nonlinearity coefficient. The spacing between layers is of the order of the wavelength, therefore Bragg effects such as band-gaps appear. We show that the interplay between strong dispersion and nonlinearity leads to new scenarios of wave propagation. The classical waveform distortion process typical of intense acoustic waves in homogeneous media can be strongly altered when nonlinearly generated harmonics lie inside or close to band gaps. This allows the possibility of engineer a medium in order to get a particular waveform. Examples of this include the design of media with effective (e.g. cubic) nonlinearities, or extremely linear media (where distortion can be cancelled). The presented ideas open a way towards the control of acoustic wave propagation in nonlinear regime.
7. Nonlinear wave propagation in constrained solids subjected to thermal loads
Science.gov (United States)
Nucera, Claudio; Lanza di Scalea, Francesco
2014-01-01
The classical mathematical treatment governing nonlinear wave propagation in solids relies on finite strain theory. In this scenario, a system of nonlinear partial differential equations can be derived to mathematically describe nonlinear phenomena such as acoustoelasticity (wave speed dependency on quasi-static stress), wave interaction, wave distortion, and higher-harmonic generation. The present work expands the topic of nonlinear wave propagation to the case of a constrained solid subjected to thermal loads. The origin of nonlinear effects in this case is explained on the basis of the anharmonicity of interatomic potentials, and the absorption of the potential energy corresponding to the (prevented) thermal expansion. Such "residual" energy is, at least, cubic as a function of strain, hence leading to a nonlinear wave equation and higher-harmonic generation. Closed-form solutions are given for the longitudinal wave speed and the second-harmonic nonlinear parameter as a function of interatomic potential parameters and temperature increase. The model predicts a decrease in longitudinal wave speed and a corresponding increase in nonlinear parameter with increasing temperature, as a result of the thermal stresses caused by the prevented thermal expansion of the solid. Experimental measurements of the ultrasonic nonlinear parameter on a steel block under constrained thermal expansion confirm this trend. These results suggest the potential of a nonlinear ultrasonic measurement to quantify thermal stresses from prevented thermal expansion. This knowledge can be extremely useful to prevent thermal buckling of various structures, such as continuous-welded rails in hot weather.
8. Variational principle for nonlinear wave propagation in dissipative systems.
Science.gov (United States)
Dierckx, Hans; Verschelde, Henri
2016-02-01
The dynamics of many natural systems is dominated by nonlinear waves propagating through the medium. We show that in any extended system that supports nonlinear wave fronts with positive surface tension, the asymptotic wave-front dynamics can be formulated as a gradient system, even when the underlying evolution equations for the field variables cannot be written as a gradient system. The variational potential is simply given by a linear combination of the occupied volume and surface area of the wave front and changes monotonically over time.
9. Nonlinear ultrasound wave propagation in thermoviscous fluids
DEFF Research Database (Denmark)
coupled nonlinear partial differential equations, which resembles those of optical chi-2 materials. We think this result makes a remarkable link between nonlinear acoustics and nonlinear optics. Finally our analysis reveal an exact kink solution to the nonlinear acoustic problem. This kink solution...
10. Nonlinear propagation of planet-generated tidal waves
OpenAIRE
Rafikov, Roman
2001-01-01
The propagation and evolution of planet-generated density waves in protoplanetary disks is considered. The evolution of waves, leading to the shock formation and wake dissipation, is followed in the weakly nonlinear regime. The local approach of Goodman & Rafikov (2001) is extended to include the effects of surface density and temperature variations in the disk as well as the disk cylindrical geometry and nonuniform shear. Wave damping due to shocks is demonstrated to be a nonlocal process sp...
11. Nonlinear evolution of parallel propagating Alfven waves: Vlasov - MHD simulation
CERN Document Server
Nariyuki, Y; Kumashiro, T; Hada, T
2009-01-01
Nonlinear evolution of circularly polarized Alfv\\'en waves are discussed by using the recently developed Vlasov-MHD code, which is a generalized Landau-fluid model. The numerical results indicate that as far as the nonlinearity in the system is not so large, the Vlasov-MHD model can validly solve time evolution of the Alfv\\'enic turbulence both in the linear and nonlinear stages. The present Vlasov-MHD model is proper to discuss the solar coronal heating and solar wind acceleration by Alfve\\'n waves propagating from the photosphere.
12. Numerical modelling of nonlinear full-wave acoustic propagation
Energy Technology Data Exchange (ETDEWEB)
Velasco-Segura, Roberto, E-mail: [email protected]; Rendón, Pablo L., E-mail: [email protected] [Grupo de Acústica y Vibraciones, Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Ciudad Universitaria, Apartado Postal 70-186, C.P. 04510, México D.F., México (Mexico)
2015-10-28
The various model equations of nonlinear acoustics are arrived at by making assumptions which permit the observation of the interaction with propagation of either single or joint effects. We present here a form of the conservation equations of fluid dynamics which are deduced using slightly less restrictive hypothesis than those necessary to obtain the well known Westervelt equation. This formulation accounts for full wave diffraction, nonlinearity, and thermoviscous dissipative effects. A two-dimensional, finite-volume method using Roe’s linearisation has been implemented to obtain numerically the solution of the proposed equations. This code, which has been written for parallel execution on a GPU, can be used to describe moderate nonlinear phenomena, at low Mach numbers, in domains as large as 100 wave lengths. Applications range from models of diagnostic and therapeutic HIFU, to parametric acoustic arrays and nonlinear propagation in acoustic waveguides. Examples related to these applications are shown and discussed.
13. Wave Propagation In Strongly Nonlinear Two-Mass Chains
Science.gov (United States)
Wang, Si Yin; Herbold, Eric B.; Nesterenko, Vitali F.
2010-05-01
We developed experimental set up that allowed the investigation of propagation of oscillating waves generated at the entrance of nonlinear and strongly nonlinear two-mass granular chains composed of steel cylinders and steel spheres. The paper represents the first experimental data related to the propagation of these waves in nonlinear and strongly nonlinear chains. The dynamic compressive forces were detected using gauges imbedded inside particles at depths equal to 4 cells and 8 cells from the entrance gauge detecting the input signal. At these relatively short distances we were able to detect practically perfect transparency at low frequencies and cut off effects at higher frequencies for nonlinear and strongly nonlinear signals. We also observed transformation of oscillatory shocks into monotonous shocks. Numerical calculations of signal transformation by non-dissipative granular chains demonstrated transparency of the system at low frequencies and cut off phenomenon at high frequencies in reasonable agreement with experiments. Systems which are able to transform nonlinear and strongly nonlinear waves at small sizes of the system are important for practical applications such as attenuation of high amplitude pulses.
14. A nonlinear RDF model for waves propagating in shallow water
Institute of Scientific and Technical Information of China (English)
王厚杰; 杨作升; 李瑞杰; 张军
2001-01-01
In this paper, a composite explicit nonlinear dispersion relation is presented with reference to Stokes 2nd order dispersion relation and the empirical relation of Hedges. The explicit dispersion relation has such advantages that it can smoothly match the Stokes relation in deep and intermediate water and Hedgs’s relation in shallow water. As an explicit formula, it separates the nonlinear term from the linear dispersion relation. Therefore it is convenient to obtain the numerical solution of nonlinear dispersion relation. The present formula is combined with the modified mild-slope equation including nonlinear effect to make a Refraction-Diffraction (RDF) model for wave propagating in shallow water. This nonlinear model is verified over a complicated topography with two submerged elliptical shoals resting on a slope beach. The computation results compared with those obtained from linear model show that at present the nonlinear RDF model can predict the nonlinear characteristics and the combined refracti
15. Linear and nonlinear propagation of water wave groups
Science.gov (United States)
Pierson, W. J., Jr.; Donelan, M. A.; Hui, W. H.
1992-01-01
Results are presented from a study of the evolution of waveforms with known analytical group shapes, in the form of both transient wave groups and the cloidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schroedinger equation. The waveforms were generated in a long wind-wave tank of the Canada Centre for Inland Waters. It was found that the low-amplitude transients behaved as predicted by the linear theory and that the cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schroedinger equation. Some of the results did not fit into any of the available theories for waves on water, but they provide important insight on how actual groups of waves propagate and on higher-order effects for a transient waveform.
16. Generation and propagation of nonlinear internal waves in Massachusetts Bay
Science.gov (United States)
Scotti, A.; Beardsley, R.C.; Butman, B.
2007-01-01
During the summer, nonlinear internal waves (NLIWs) are commonly observed propagating in Massachusetts Bay. The topography of the area is unique in the sense that the generation area (over Stellwagen Bank) is only 25 km away from the shoaling area, and thus it represents an excellent natural laboratory to study the life cycle of NLIWs. To assist in the interpretation of the data collected during the 1998 Massachusetts Bay Internal Wave Experiment (MBIWE98), a fully nonlinear and nonhydrostatic model covering the generation/shoaling region was developed, to investigate the response of the system to the range of background and driving conditions observed. Simplified models were also used to elucidate the role of nonlinearity and dispersion in shaping the NLIW field. This paper concentrates on the generation process and the subsequent evolution in the basin. The model was found to reproduce well the range of propagation characteristics observed (arrival time, propagation speed, amplitude), and provided a coherent framework to interpret the observations. Comparison with a fully nonlinear hydrostatic model shows that during the generation and initial evolution of the waves as they move away from Stellwagen Bank, dispersive effects play a negligible role. Thus the problem can be well understood considering the geometry of the characteristics along which the Riemann invariants of the hydrostatic problem propagate. Dispersion plays a role only during the evolution of the undular bore in the middle of Stellwagen Basin. The consequences for modeling NLIWs within hydrostatic models are briefly discussed at the end.
17. A numerical simulation of nonlinear propagation of gravity wave packet in three-dimension compressible atmosphere
Institute of Scientific and Technical Information of China (English)
WU; Shaoping(吴少平); YI; Fan(易帆)
2002-01-01
By using FICE scheme, a numerical simulation of nonlinear propagation of gravity wave packet in three-dimension compressible atmosphere is presented. The whole nonlinear propagation process of the gravity wave packet is shown; the basic characteristics of nonlinear propagation and the influence of the ambient winds on the propagation are analyzed. The results show that FICE scheme can be extended in three-dimension by which the calculation is steady and kept for a long time; the increase of wave amplitude is faster than the exponential increase according to the linear gravity theory; nonlinear propagation makes the horizontal perturbation velocity increase greatly which can lead to enhancement of the local ambient winds; the propagation path and the propagation velocity of energy are different from the results expected by the linear gravity waves theory, the nonlinearity causes the change in propagation characteristics of gravity wave; the ambient winds alter the propagation path and group velocity of gravity wave.
18. Nonlinear Propagation of Planet-Generated Tidal Waves
Science.gov (United States)
Rafikov, R. R.
2002-01-01
The propagation and evolution of planet-generated density waves in protoplanetary disks is considered. The evolution of waves, leading to shock formation and wake dissipation, is followed in the weakly nonlinear regime. The 2001 local approach of Goodman and Rafikov is extended to include the effects of surface density and temperature variations in the disk as well as the disk cylindrical geometry and nonuniform shear. Wave damping due to shocks is demonstrated to be a nonlocal process spanning a significant fraction of the disk. Torques induced by the planet could be significant drivers of disk evolution on timescales of approx. 10(exp 6)-10(exp 7) yr, even in the absence of strong background viscosity. A global prescription for angular momentum deposition is developed that could be incorporated into the study of gap formation in a gaseous disk around the planet.
19. Nonlinear propagation of planet-generated tidal waves
CERN Document Server
Rafikov, R R
2002-01-01
The propagation and evolution of planet-generated density waves in protoplanetary disks is considered. The evolution of waves, leading to the shock formation and wake dissipation, is followed in the weakly nonlinear regime. The local approach of Goodman & Rafikov (2001) is extended to include the effects of surface density and temperature variations in the disk as well as the disk cylindrical geometry and nonuniform shear. Wave damping due to shocks is demonstrated to be a nonlocal process spanning a significant fraction of the disk. Torques induced by the planet could be significant drivers of disk evolution on timescales of the order 1-10 Myr even in the absence of strong background viscosity. A global prescription for angular momentum deposition is developed which could be incorporated into the study of gap formation in a gaseous disk around the planet.
20. Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach
Energy Technology Data Exchange (ETDEWEB)
Romeo, Francesco [Dipartimento di Ingegneria Strutturale e Geotecnica, Universita di Roma ' La Sapienza' , Via Gramsci 53, 00197 Rome (Italy)] e-mail: [email protected]; Rega, Giuseppe [Dipartimento di Ingegneria Strutturale e Geotecnica, Universita di Roma ' La Sapienza' , Via Gramsci 53, 00197 Rome (Italy)] e-mail: [email protected]
2006-02-01
Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.
1. Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers
OpenAIRE
2011-01-01
International audience; We study theoretically, numerically and experimentally the nonlinear propagation of partially incoherent optical waves in single mode optical fibers. We revisit the traditional treatment of the wave turbulence theory to provide a statistical kinetic description of the integrable scalar NLS equation. In spite of the formal reversibility and of the integrability of the NLS equation, the weakly nonlinear dynamics reveals the existence of an irreversible evolution toward a...
2. Wave Propagation
CERN Document Server
Ferrarese, Giorgio
2011-01-01
Lectures: A. Jeffrey: Lectures on nonlinear wave propagation.- Y. Choquet-Bruhat: Ondes asymptotiques.- G. Boillat: Urti.- Seminars: D. Graffi: Sulla teoria dell'ottica non-lineare.- G. Grioli: Sulla propagazione del calore nei mezzi continui.- T. Manacorda: Onde nei solidi con vincoli interni.- T. Ruggeri: "Entropy principle" and main field for a non linear covariant system.- B. Straughan: Singular surfaces in dipolar materials and possible consequences for continuum mechanics
3. Computational study of nonlinear plasma waves: 1: Simulation model and monochromatic wave propagation
Science.gov (United States)
Matda, Y.; Crawford, F. W.
1974-01-01
An economical low noise plasma simulation model is applied to a series of problems associated with electrostatic wave propagation in a one-dimensional, collisionless, Maxwellian plasma, in the absence of magnetic field. The model is described and tested, first in the absence of an applied signal, and then with a small amplitude perturbation, to establish the low noise features and to verify the theoretical linear dispersion relation at wave energy levels as low as 0.000,001 of the plasma thermal energy. The method is then used to study propagation of an essentially monochromatic plane wave. Results on amplitude oscillation and nonlinear frequency shift are compared with available theories. The additional phenomena of sideband instability and satellite growth, stimulated by large amplitude wave propagation and the resulting particle trapping, are described.
4. Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes.
Science.gov (United States)
Gusev, Vitalyi E; Ni, Chenyin; Lomonosov, Alexey; Shen, Zhonghua
2015-08-01
Theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous material on flexural wave in the plates of continuously varying thickness is developed. For the wedges with thickness increasing as a power law of distance from its edge strong modifications of the wave dynamics with propagation distance are predicted. It is found that nonlinear absorption progressively disappearing with diminishing wave amplitude leads to complete attenuation of acoustic waves in most of the wedges exhibiting black hole phenomenon. It is also demonstrated that black holes exist beyond the geometrical acoustic approximation. Applications include nondestructive evaluation of micro-inhomogeneous materials and vibrations damping. Copyright © 2015 Elsevier B.V. All rights reserved.
5. Propagation of Quasi-plane Nonlinear Waves in Tubes
Directory of Open Access Journals (Sweden)
P. Koníček
2002-01-01
Full Text Available This paper deals with possibilities of using the generalized Burgers equation and the KZK equation to describe nonlinear waves in circular ducts. A new method for calculating of diffraction effects taking into account boundary layer effects is described. The results of numerical solutions of the model equations are compared. Finally, the limits of validity of the used model equations are discussed with respect to boundary conditions and the radius of the circular duct. The limits of applicability of the KZK equation and the GBE equation for describing nonlinear waves in tubes are discussed.
6. Nonlinear wave propagation studies, dispersion modeling, and signal parameters correction
Czech Academy of Sciences Publication Activity Database
Převorovský, Zdeněk
..: ..., 2004, 00. [European Workshop on FP6-AERONEWS /1./. Naples (IT), 13.09.2004-16.09.2004] EU Projects: European Commission(XE) 502927 - AERO-NEWS Institutional research plan: CEZ:AV0Z2076919 Keywords : nodestructive testing * nonlinear elastic wave spectroscopy Subject RIV: BI - Acoustics
7. Unstructured Spectral Element Model for Dispersive and Nonlinear Wave Propagation
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Eskilsson, Claes; Bigoni, Daniele
2016-01-01
). In the present paper we use a single layer of quadratic (in 2D) and prismatic (in 3D) elements. The model has been stabilized through a combination of over-integration of the Galerkin projections and a mild modal filter. We present numerical tests of nonlinear waves serving as a proof-of-concept validation...
8. Nonlinear propagation of coupled electromagnetic waves in a circular cylindrical waveguide
Science.gov (United States)
Valovik, D. V.; Smol'kin, E. Yu.
2017-08-01
The problem of the propagation of coupled surface electromagnetic waves in a two-layer cylindrical circular waveguide filled with an inhomogeneous nonlinear medium is considered. A nonlinear coupled TE-TM wave is characterized by two (independent) frequencies ωe and ωm and two propagation constants {\\widehat γ _e} and {\\widehat γ _m}. The physical problem reduces to a nonlinear two-parameter eigenvalue problem for a system of nonlinear ordinary differential equations. The existence of eigenvalues ({\\widehat γ _e}, {\\widehat γ _m}) in proven and intervals of their localization are determined.
9. Nonlinear Wave Propagation and Solitary Wave Formation in Two-Dimensional Heterogeneous Media
KAUST Repository
Luna, Manuel
2011-05-01
Solitary wave formation is a well studied nonlinear phenomenon arising in propagation of dispersive nonlinear waves under suitable conditions. In non-homogeneous materials, dispersion may happen due to effective reflections between the material interfaces. This dispersion has been used along with nonlinearities to find solitary wave formation using the one-dimensional p-system. These solitary waves are called stegotons. The main goal in this work is to find two-dimensional stegoton formation. To do so we consider the nonlinear two-dimensional p-system with variable coefficients and solve it using finite volume methods. The second goal is to obtain effective equations that describe the macroscopic behavior of the variable coefficient system by a constant coefficient one. This is done through a homogenization process based on multiple-scale asymptotic expansions. We compare the solution of the effective equations with the finite volume results and find a good agreement. Finally, we study some stability properties of the homogenized equations and find they and one-dimensional versions of them are unstable in general.
10. Nonlinear unified equations for water waves propagating over uneven bottoms in the nearshore region
Institute of Scientific and Technical Information of China (English)
2001-01-01
Considering the continuous characteristics for water waves propagating over complex topography in the nearshore region, the unified nonlinear equations, based on the hypothesis for a typical uneven bottom, are presented by employing the Hamiltonian variational principle for water waves. It is verified that the equations include the following special cases: the extension of Airy's nonlinear shallow-water equations, the generalized mild-slope equation, the dispersion relation for the second-order Stokes waves and the higher order Boussinesq-type equations.
11. Theoretical Study of Wave Breaking for Nonlinear Water Waves Propagating on a Sloping Bottom
Science.gov (United States)
Chen, Y. Y.; Hsu, H. C.; Li, M. S.
2012-04-01
In this paper, a third-order asymptotic solution in a Lagrangian framework describing nonlinear water wave propagation on the surface of a uniform sloping bottom is presented. A two-parameter perturbation method is used to develop a new mathematical derivation. The particle trajectories, wave pressure and Lagrangian velocity potential are obtained as a function of the nonlinear wave steepness and the bottom slope perturbed to third order. This theoretical solution in Lagrangian form satisfies state of the normal pressure at the free surface. The condition of the conservation of mass flux is examined in detail for the first time. The two important properties in Lagrangian coordinates, Lagrangian wave frequency and Lagrangian mean level, are included in the third-order solution. The solution can also be used to estimate the mean return current for waves progressing over the sloping bottom. The Lagrangian solution untangle the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the process of successive deformation of a wave profile and water particle trajectories leading to wave breaking. A series of experiment was conducted to validate the obtained theoretical solution. The proposed solution will be used to determine the wave shoaling and breaking process and the comparisons between the experimental and theoretical results are excellent. For example, the variations of phase velocity on sloping bottom are obtained by 7 set of two close wave gauges and the theoretical result could accurately predict the measured phase velocity. The theoretical wave breaking index can be derived by use of the kinematic stability parameter (K.P.S). The comparisons between the theory, experiment (present study, Iwagali et al.(1974), Deo et al.(2003) and Tsai et al.(2005)) and empirical formula of Goda (2004) for the breaking index(u/C) versus the relative water depth(d/L) under two different bottom slopes shows that the
12. Propagation of Nonlinear Waves in Waveguides and Application to Nondestructive Stress Measurement
Science.gov (United States)
Nucera, Claudio
Propagation of nonlinear waves in waveguides is a field that has received an ever increasing interest in the last few decades. Nonlinear guided waves are excellent candidates for interrogating long waveguide like structures because they combine high sensitivity to structural conditions, typical of nonlinear parameters, with large inspection ranges, characteristic of wave propagation in bounded media. The primary topic of this dissertation is the analysis of ultrasonic waves, including ultrasonic guided waves, propagating in their nonlinear regime and their application to structural health monitoring problems, particularly the measurement of thermal stress in Continuous Welded Rail (CWR). Following an overview of basic physical principles generating nonlinearities in ultrasonic wave propagation, the case of higher-harmonic generation in multi-mode and dispersive guided waves is examined in more detail. A numerical framework is developed in order to predict favorable higher-order generation conditions (i.e. specific guided modes and frequencies) for waveguides of arbitrary cross-sections. This model is applied to various benchmark cases of complex structures. The nonlinear wave propagation model is then applied to the case of a constrained railroad track (CWR) subjected to thermal variations. This study is a direct response to the key need within the railroad transportation community to develop a technique able to measure thermal stresses in CWR, or determine the rail temperature corresponding to a null thermal stress (Neutral Temperature -- NT). The numerical simulation phase concludes with a numerical study performed using ABAQUS commercial finite element package. These analyses were crucial in predicting the evolution of the nonlinear parameter beta with thermal stress level acting in the rail. A novel physical model, based on interatomic potential, was developed to explain the origin of nonlinear wave propagation under constrained thermal expansion. In fact
13. Modeling of Propagation and Transformation of Transient Nonlinear Waves on A Current
Institute of Scientific and Technical Information of China (English)
Wojciech Sulisz; Maciej Paprota
2013-01-01
A novel theoretical approach is applied to predict the propagation and transformation of transient nonlinear waves on a current. The problem was solved by applying an eigenfunction expansion method and the derived semi-analytical solution was employed to study the transformation of wave profile and the evolution of wave spectrum arising from the nonlinear interactions of wave components in a wave train which may lead to the formation of very large waves. The results show that the propagation of wave trains is significantly affected by a current. A relatively small current may substantially affect wave train components and the wave train shape. This is observed for both opposing and following current. The results demonstrate that the application of the nonlinear model has a substantial effect on the shape of a wave spectrum. A train of originally linear and very narrow-banded waves changes its one-peak spectrum to a multi-peak one in a fairly short distance from an initial position. The discrepancies between the wave trains predicted by applying the linear and nonlinear models increase with the increasing wavelength and become significant in shallow water even for waves with low steepness. Laboratory experiments were conducted in a wave flume to verify theoretical results. The free-surface elevations recorded by a system of wave gauges are compared with the results provided by the nonlinear model. Additional verification was achieved by applying a Fourier analysis and comparing wave amplitude spectra obtained from theoretical results with experimental data. A reasonable agreement between theoretical results and experimental data is observed for both amplitudes and phases. The model predicts fairly well multi-peak spectra, including wave spectra with significant nonlinear wave components.
14. Propagation of Long-Wavelength Nonlinear Slow Sausage Waves in Stratified Magnetic Flux Tubes
Science.gov (United States)
Barbulescu, M.; Erdélyi, R.
2016-05-01
The propagation of nonlinear, long-wavelength, slow sausage waves in an expanding magnetic flux tube, embedded in a non-magnetic stratified environment, is discussed. The governing equation for surface waves, which is akin to the Leibovich-Roberts equation, is derived using the method of multiple scales. The solitary wave solution of the equation is obtained numerically. The results obtained are illustrative of a solitary wave whose properties are highly dependent on the degree of stratification.
15. Nonlinear propagation of a wave packet in a hard-walled circular duct
Science.gov (United States)
Nayfeh, A. H.
1975-01-01
The method of multiple scales is used to derive a nonlinear Schroedinger equation for the temporal and spatial modulation of the amplitudes and the phases of waves propagating in a hard-walled circular duct. This equation is used to show that monochromatic waves are stable and to determine the amplitude dependance of the cutoff frequencies.
16. Wave propagation in photonic crystals and metamaterials: Surface waves, nonlinearity and chirality
Energy Technology Data Exchange (ETDEWEB)
Wang, Bingnan [Iowa State Univ., Ames, IA (United States)
2009-01-01
Photonic crystals and metamaterials, both composed of artificial structures, are two interesting areas in electromagnetism and optics. New phenomena in photonic crystals and metamaterials are being discovered, including some not found in natural materials. This thesis presents my research work in the two areas. Photonic crystals are periodically arranged artificial structures, mostly made from dielectric materials, with period on the same order of the wavelength of the working electromagnetic wave. The wave propagation in photonic crystals is determined by the Bragg scattering of the periodic structure. Photonic band-gaps can be present for a properly designed photonic crystal. Electromagnetic waves with frequency within the range of the band-gap are suppressed from propagating in the photonic crystal. With surface defects, a photonic crystal could support surface modes that are localized on the surface of the crystal, with mode frequencies within the band-gap. With line defects, a photonic crystal could allow the propagation of electromagnetic waves along the channels. The study of surface modes and waveguiding properties of a 2D photonic crystal will be presented in Chapter 1. Metamaterials are generally composed of artificial structures with sizes one order smaller than the wavelength and can be approximated as effective media. Effective macroscopic parameters such as electric permittivity ϵ, magnetic permeability μ are used to characterize the wave propagation in metamaterials. The fundamental structures of the metamaterials affect strongly their macroscopic properties. By designing the fundamental structures of the metamaterials, the effective parameters can be tuned and different electromagnetic properties can be achieved. One important aspect of metamaterial research is to get artificial magnetism. Metallic split-ring resonators (SRRs) and variants are widely used to build magnetic metamaterials with effective μ < 1 or even μ < 0. Varactor based
17. The effects of nonlinear wave propagation on the stability of inertial cavitation
OpenAIRE
2009-01-01
In the context of forecasting temperature and pressure fields in high-intensity focussed ultrasound, the accuracy of predictive models is critical for the safety and efficacy of treatment. In such fields inertial cavitation is often observed. Classically, estimations of cavitation thresholds have been based on the assumption that the incident wave at the surface of a bubble was the same as in the far-field, neglecting the effect of nonlinear wave propagation. By modelling the incident wave as...
18. Numerical Simulation of Non-Linear Wave Propagation in Waters of Mildly Varying Topography with Complicated Boundary
Institute of Scientific and Technical Information of China (English)
张洪生; 洪广文; 丁平兴; 曹振轶
2001-01-01
In this paper, the characteristics of different forms of mild slope equations for non-linear wave are analyzed, and new non-linear theoretic models for wave propagation are presented, with non-linear terms added to the mild slope equations for non-stationary linear waves and dissipative effects considered. Numerical simulation models are developed of non-linear wave propagation for waters of mildly varying topography with complicated boundary, and the effects are studied of different non-linear corrections on calculation results of extended mild slope equations. Systematical numerical simulation tests show that the present models can effectively reflect non-linear effects.
19. Non-linear wave propagation in acoustically lined circular ducts
Science.gov (United States)
Nayfeh, A. H.; Tsai, M.-S.
1974-01-01
An analysis is presented of the nonlinear effects of the gas motion as well as of the acoustic lining material on the transmission and attenuation of sound in a circular duct with a uniform cross-section and no mean flow. The acoustic material is characterized by an empirical, nonlinear impedance in which the instantaneous resistance is a nonlinear function of both the frequency and the acoustic velocity. The results show that there exist frequency bandwidths around the resonant frequencies in which the nonlinearity decreases the attenuation rate, and outside which the nonlinearity increases the attenuation rate, in qualitative agreement with experimental observations. Moreover, the effect of the gas nonlinearity increases with increasing sound frequency, whereas the effect of the material nonlinearity decreases with increasing sound frequency.
20. Nonhydrostatic effects of nonlinear internal wave propagation in the South China Sea
Science.gov (United States)
Zhang, Z.; Fringer, O. B.
2007-05-01
It is well known that internal tides are generated over steep topography at the Luzon Strait on the eastern boundary of the South China Sea. These internal tides propagate westward and steepen into trains of weakly nonlinear internal waves that propagate relatively free of dissipation until they interact with the continental shelf on the western side of the South China Sea, some 350 km from their generation point. The rate at which the internal tide transforms into trains of nonlinear waves depends on the Froude number at the generation site, which is defined as the ratio of the barotropic current speed to the local internal wave speed. Large Froude numbers lead to rapid evolution of wave trains while low Froude numbers generate internal tides that may not evolve into wave trains before reaching the continental shelf. Although the evolution into trains of weakly nonlinear waves results from the delicate interplay between nonlinear steepening and nonhydrostatic dispersion, the steepening process is represented quite well, at least from a qualitative standpoint, by hydrostatic models, which contain no explicit nonhydrostatic dispersion. Furthermore, hydrostatic models predict the propagation speed of the leading wave in wave trains extremely well, indicating that its propagation speed depends very weakly on nonlinear or dispersive effects. In order to examine how hydrostatic models introduce dispersion that leads to the formation of wave trains, we simulate the generation and evolution of nonlinear waves in the South China Sea with and without the hydrostatic approximation using the nonhydrostatic model SUNTANS, which can be run in either hydrostatic or nonhydrostatic mode. We show that the dispersion leading to the formation of wave trains in the hydrostatic model results from numerically-induced dispersion that is implicit in the numerical formulation of the advection terms. While the speed of the leading wave in the wave trains is correct, the amplitude and number
1. Nonlinear wave propagation through a ferromagnet with damping in (2+1) dimensions
S G Bindu; V C Kuriakose
2000-02-01
We investigate how dissipation and nonlinearity can affect the electromagnetic wave propagating through a saturated ferromagnet in the presence of an external magnetic field in (2+1) dimensions. The propagation of electromagnetic waves through a ferromagnet under an external magnetic field in the presence of dissipative effect has been studied using reductive perturbation method. It is found that to the lowest order of perturbation the system of equations for the electromagnetic waves in a ferromagnet can be reduced to an integro-differential equation.
2. Exact solutions of optical wave propagation in nonlinear negative refractive medium
Science.gov (United States)
Nanda, Lipsa
2016-04-01
An analytical and simulation based method has been used to exactly solve the nonlinear Schrödinger's equation (NLSE) and study the solitonic forms in a medium which exhibits frequency dependent dielectric permittivity (ɛ) and magnetic permeability (μ). The model has been extended to describe the propagation of a wave in a nonlinear negative refractive medium (NRM) which is dispersive and negative in nature.
3. The nonlinear propagation of acoustic waves in a viscoelastic medium containing cylindrical micropores
Institute of Scientific and Technical Information of China (English)
Feng Yu-Lin; Liu Xiao-Zhou; Liu Jie-Hui; Ma Li
2009-01-01
Based on an equivalent medium approach,this paper presents a model describing the nonlinear propagation of acoustic waves in a viscoelastic medium containing cylindrical micropores. The influences of pores' nonlinear oscillations on sound attenuation,sound dispersion and an equivalent acoustic nonlinearity parameter are discussed. The calculated results show that the attenuation increases with an increasing volume fraction of mieropores. The peak of sound velocity and attenuation occurs at the resonant frequency of the micropores while the peak of the equivalent acoustic nonlinearity parameter occurs at the half of the resonant frequency of the micropores. Furthermore,multiple scattering has been taken into account,which leads to a modification to the effective wave number in the equivalent medium approach. We find that these linear and nonlinear acoustic parameters need to be corrected when the volume fraction of micropores is larger than 0.1%.
4. Nonlinear propagation of weakly relativistic ion-acoustic waves in electron–positron–ion plasma
M G HAFEZ; M R TALUKDER; M HOSSAIN ALI
2016-11-01
This work presents theoretical and numerical discussion on the dynamics of ion-acoustic solitary wave for weakly relativistic regime in unmagnetized plasma comprising non-extensive electrons, Boltzmann positrons and relativistic ions. In order to analyse the nonlinear propagation phenomena, the Korteweg–de Vries(KdV) equation is derived using the well-known reductive perturbation method. The integration of the derived equation is carried out using the ansatz method and the generalized Riccati equation mapping method. The influenceof plasma parameters on the amplitude and width of the soliton and the electrostatic nonlinear propagation of weakly relativistic ion-acoustic solitary waves are described. The obtained results of the nonlinear low-frequencywaves in such plasmas may be helpful to understand various phenomena in astrophysical compact object and space physics.
5. PetClaw: A scalable parallel nonlinear wave propagation solver for Python
KAUST Repository
Alghamdi, Amal
2011-01-01
We present PetClaw, a scalable distributed-memory solver for time-dependent nonlinear wave propagation. PetClaw unifies two well-known scientific computing packages, Clawpack and PETSc, using Python interfaces into both. We rely on Clawpack to provide the infrastructure and kernels for time-dependent nonlinear wave propagation. Similarly, we rely on PETSc to manage distributed data arrays and the communication between them.We describe both the implementation and performance of PetClaw as well as our challenges and accomplishments in scaling a Python-based code to tens of thousands of cores on the BlueGene/P architecture. The capabilities of PetClaw are demonstrated through application to a novel problem involving elastic waves in a heterogeneous medium. Very finely resolved simulations are used to demonstrate the suppression of shock formation in this system.
6. Nonlinear propagation of ion-acoustic waves through the Burgers equation in weakly relativistic plasmas
Science.gov (United States)
Hafez, M. G.; Talukder, M. R.; Hossain Ali, M.
2017-04-01
The Burgers equation is obtained to study the characteristics of nonlinear propagation of ionacoustic shock, singular kink, and periodic waves in weakly relativistic plasmas containing relativistic thermal ions, nonextensive distributed electrons, Boltzmann distributed positrons, and kinematic viscosity of ions using the well-known reductive perturbation technique. This equation is solved by employing the ( G'/ G)-expansion method taking unperturbed positron-to-electron concentration ratio, electron-to-positron temperature ratio, strength of electrons nonextensivity, ion kinematic viscosity, and weakly relativistic streaming factor. The influences of plasma parameters on nonlinear propagation of ion-acoustic shock, periodic, and singular kink waves are displayed graphically and the relevant physical explanations are described. It is found that these parameters extensively modify the shock structures excitation. The obtained results may be useful in understanding the features of small but finite amplitude localized relativistic ion-acoustic shock waves in an unmagnetized plasma system for some astrophysical compact objects and space plasmas.
7. Propagation of surface SH waves on a nonlinear half space coated with a layer of nonuniform thickness
Science.gov (United States)
Deliktaş, Ekin; Teymür, Mevlüt
2017-07-01
In this study, the propagation of shear horizontal (SH) waves in a nonlinear elastic half space covered by a nonlinear elastic layer with a slowly varying interface is examined. The constituent materials are assumed to be homogenous, isotropic, elastic and having different mechanical properties. By employing the method of multiple scales, a nonlinear Schrödinger equation (NLS) with variable coefficients is derived for the nonlinear self-modulation of SH waves. We examine the effects of dispersion, irregularity of the interface and nonlinearity on the propagation characteristics of SH waves.
8. Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures
Science.gov (United States)
2016-08-03
enabled a wide range of proposals for applications. Among others, we note shock and energy absorbing lay- ers [5–7], acoustic lenses [8], acoustic diodes...found in the Supplemental Material [41]. We record the transmit- ted stress waves using a piezoelectric force sensor (PCB C02) placed at the bottom of...the contacts in the presence of internal vibration modes that can store energy in their own right. The effective parameters m1,M and k1 of this DEM
9. Nonlinear Propagation of Mag Waves Through the Transition Region
Science.gov (United States)
Jatenco-Pereira, V.; Steinolfson, R. S.; Mahajan, S.; Tajima, T.
1990-11-01
RESUMEN. Una onda de gravitaci5n magneto acustica (GMA), se inicia en el regimen de alta beta cerca de la basa de fot5sfera solar y es segui- da, usando simulaciones numericas, mientras viaja radialmente a traves de la cromosfera, la regi5n de transici6n y dentro de la corona. Se ha' seleccionado parametros iniciales de manera que la beta resulte menor que uno cerca de la parte alta de la regi6n de transici6n. Nuestro interes maximo se concentra en la cantidad y forma del flujo de energia que puede ser llevada por la onda hasta la corona dados una atm6sfera inicial y amplitud de onda especificas. Segun los estudios a la fecha, el flujo de energ1a termico domina, aumentando linealmente con la ampli tud deonda y resulta de aproximadamente i05 ergs/cm2-s en una amplitud de 0.5. El flujo de energia cinetica siempre permanece despreciable, mientras que el flujo de energia magnetica depende de la orientaci5n inicial del campo. Un modo GMA rapido y casi paralelo, el cual es esen- cialmente un modo MHD en la corona se convierte a un modo rapido modificado y a uno lento, cuando la beta atmosferica disminuye a uno. ABSTRACT: A magneto-acoustic-gravity (MAG) wave is initiated in the high-beta regime near the base of the solar photosphere and followed, using numerical siriiulations, as it travels radially through the chromosphere, the transition region, and into the corona. Initial parameters are selected such that beta becomes less than one near the top of the transition region. Our primary interest is in the amount and form of energy flux that can be carried by the wave train into the corona for a specified initial atmosphere and wave amplitude. For the studies conducted to date, the thermal energy flux dominates, it about linearly with wave amplitude and becomes approximately 10 ergs/cm2-s at an amplitude of 0.5. The kinetic energy flux always remains negligible, while the magnetic energy flux depends on the inLtial field orientation. A nearly parallel fast MAG mode, which
10. Mathematical Methods in Wave Propagation: Part 2--Non-Linear Wave Front Analysis
Science.gov (United States)
Jeffrey, Alan
1971-01-01
The paper presents applications and methods of analysis for non-linear hyperbolic partial differential equations. The paper is concluded by an account of wave front analysis as applied to the piston problem of gas dynamics. (JG)
11. Characterizing the propagation of gravity waves in 3D nonlinear simulations of solar-like stars
CERN Document Server
Alvan, L; Brun, A S; Mathis, S; Garcia, R A
2015-01-01
The revolution of helio- and asteroseismology provides access to the detailed properties of stellar interiors by studying the star's oscillation modes. Among them, gravity (g) modes are formed by constructive interferences between progressive internal gravity waves (IGWs), propagating in stellar radiative zones. Our new 3D nonlinear simulations of the interior of a solar-like star allows us to study the excitation, propagation, and dissipation of these waves. The aim of this article is to clarify our understanding of the behavior of IGWs in a 3D radiative zone and to provide a clear overview of their properties. We use a method of frequency filtering that reveals the path of {individual} gravity waves of different frequencies in the radiative zone. We are able to identify the region of propagation of different waves in 2D and 3D, to compare them to the linear raytracing theory and to distinguish between propagative and standing waves (g modes). We also show that the energy carried by waves is distributed in d...
12. Nonlinear Alfvén wave propagating in ideal MHD plasmas
Science.gov (United States)
Zheng, Jugao; Chen, Yinhua; Yu, Mingyang
2016-01-01
The behavior of nonlinear Alfvén waves propagating in ideal MHD plasmas is investigated numerically. It is found that in a one-dimensional weakly nonlinear system an Alfvén wave train can excite two longitudinal disturbances, namely an acoustic wave and a ponderomotively driven disturbance, which behave differently for β \\gt 1 and β \\lt 1, where β is the ratio of plasma-to-magnetic pressures. In a strongly nonlinear system, the Alfvén wave train is modulated and can steepen to form shocks, leading to significant dissipation due to appearance of current sheets at magnetic-pressure minima. For periodic boundary condition, we find that the Alfvén wave transfers its energy to the plasma and heats it during the shock formation. In two-dimensional systems, fast magneto-acoustic wave generation due to Alfvén wave phase mixing is considered. It is found that the process depends on the amplitude and frequency of the Alfvén waves, as well as their speed gradients and the pressure of the background plasma.
13. Study of the Impact of Non-linear Piezoelectric Constants on the Acoustic Wave Propagation on Lithium Niobate
Directory of Open Access Journals (Sweden)
C. Soumali
2016-06-01
Full Text Available Impact of nonlinear piezoelectric constants on surface acoustic wave propagation on a piezoelectric substrate is investigated in this work. Propagation of acoustic wave propagation under uniform stress is analyzed; the wave equation is obtained by incorporating the applied uniform stress in the equation of motion and taking account of the set of linear and nonlinear piezoelectric constants. A new method of separation between the different modes of propagation is proposed regarding the attenuation coefficients and not to the displacement vectors. Detail calculations and simulations have made for Lithium Niobate (LiNbO3; transformations between modes of propagation, under uniform stress, have been found. These results leads to conclusion that nonlinear terms affect the acoustic wave propagation and also we can make controllable acoustic devices.
14. Simulation of "Tsunami Waves" Propagating along Non-Linear Transmission Lines
Directory of Open Access Journals (Sweden)
J. Valsa
2005-09-01
Full Text Available The paper compares three methods for computer simulation oftransients on transmission lines with losses and nonlinear behavior,namely distributed LC model, FDTD (Finite-Difference Time-Domainmethod, and a new and very effective Method of Slices. The losses areresponsible for attenuation and shape changes of the waves as functionof time and distance from the source. Special behavior of the line dueto voltage-dependent capacitance of the line is considered in detail.The non-linear nature of the line causes that the higher is the voltagethe higher is the velocity of propagation. Then, the waves tend to tiltover so that their top moves faster than their base. As a result"tsunami waves" are created on the line. Fundamental algorithms arepresented in Matlab language. Several typical situations are solved asan illustration of individual methods.
15. Nonlinear continuous-wave optical propagation in nematic liquid crystals: Interplay between reorientational and thermal effects
Science.gov (United States)
Alberucci, Alessandro; Laudyn, Urszula A.; Piccardi, Armando; Kwasny, Michał; Klus, Bartlomiej; Karpierz, Mirosław A.; Assanto, Gaetano
2017-07-01
We investigate nonlinear optical propagation of continuous-wave (CW) beams in bulk nematic liquid crystals. We thoroughly analyze the competing roles of reorientational and thermal nonlinearity with reference to self-focusing/defocusing and, eventually, the formation of nonlinear diffraction-free wavepackets, the so-called spatial optical solitons. To this extent we refer to dye-doped nematic liquid crystals in planar cells excited by a single CW beam in the highly nonlocal limit. To adjust the relative weight between the two nonlinear responses, we employ two distinct wavelengths, inside and outside the absorption band of the dye, respectively. Different concentrations of the dye are considered in order to enhance the thermal effect. The theoretical analysis is complemented by numerical simulations in the highly nonlocal approximation based on a semi-analytic approach. Theoretical results are finally compared to experimental results in the Nematic Liquid Crystals (NLC) 4-trans-4'-n-hexylcyclohexylisothiocyanatobenzene (6CHBT) doped with Sudan Blue dye.
16. Nonlinear physics of electrical wave propagation in the heart: a review
Science.gov (United States)
Alonso, Sergio; Bär, Markus; Echebarria, Blas
2016-09-01
The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.
17. Identification and determination of solitary wave structures in nonlinear wave propagation
Energy Technology Data Exchange (ETDEWEB)
Newman, W.I.; Campbell, D.K.; Hyman, J.M.
1991-01-01
Nonlinear wave phenomena are characterized by the appearance of solitary wave coherent structures'' traveling at speeds determined by their amplitudes and morphologies. Assuming that these structures are briefly noninteracting, we propose a method for the identification of the number of independent features and their respective speeds. Using data generated from an exact two-soliton solution to the Korteweg-de-Vries equation, we test the method and discuss its strengths and limitations. 41 refs., 2 figs.
18. Analysis of S Wave Propagation Through a Nonlinear Joint with the Continuously Yielding Model
Science.gov (United States)
Cui, Zhen; Sheng, Qian; Leng, Xianlun
2017-01-01
Seismic wave propagation through joints that are embedded in a rock mass is a critical issue for aseismic issues of underground rock engineering. Few studies have investigated nonlinear joints with a continuously yielding model. In this paper, a time-domain recursive method (TDRM) for an S wave across a nonlinear Mohr-Coulomb (MC) slip model is extended to a continuously yielding (CY) model. Verification of the TDRM-based results is conducted by comparison with the simulated results via a built-in model of 3DEC code. Using parametric studies, the effect of normal stress level, amplitude of incident wave, initial joint shear stiffness, and joint spacing is discussed and interpreted for engineering applications because a proper in situ stress level (overburden depth) and acceptable quality of surrounding rock mass are beneficial for seismic stability issues of underground rock excavation. Comparison between the results from the MC model and the CY model is presented both for an idealized impulse excitation and a real ground motion record. Compared with the MC model, complex joint behaviors, such as tangential stiffness degradation, normal stress dependence, and the hysteresis effect, that occurred in the wave propagation can be described with the CY model. The MC model seems to underestimate the joint shear displacement in a high normal stress state and in a real ground motion excitation case.
19. Properties and stability of freely propagating nonlinear density waves in accretion disks
CERN Document Server
Fromang, S
2007-01-01
In this paper, we study the propagation and stability of nonlinear sound waves in accretion disks. Using the shearing box approximation, we derive the form of these waves using a semi-analytic approach and go on to study their stability. The results are compared to those of numerical simulations performed using finite difference approaches such as employed by ZEUS as well as Godunov methods. When the wave frequency is between Omega and two Omega (where Omega is the disk orbital angular velocity), it can couple resonantly with a pair of linear inertial waves and thus undergo a parametric instability. Neglecting the disk vertical stratification, we derive an expression for the growth rate when the amplitude of the background wave is small. Good agreement is found with the results of numerical simulations performed both with finite difference and Godunov codes. During the nonlinear phase of the instability, the flow remains well organised if the amplitude of the background wave is small. However, strongly nonlin...
20. A new theoretical paradigm to describe hysteresis, discrete memory and nonlinear elastic wave propagation in rock
Directory of Open Access Journals (Sweden)
K. R. McCall
1996-01-01
Full Text Available The velocity of sound in rock is a strong function of pressure, indicating that wave propagation in rocks is very nonlinear. The quasistatic elastic properties of rocks axe hysteretic, possessing discrete memory. In this paper a new theory is developed, placing all of these properties (nonlinearity, hysteresis, and memory on equal footing. The starting point of the new theory is closer to a microscopic description of a rock than the starting point of the traditional five-constant theory of nonlinear elasticity. However, this starting point (the number density Ï? of generic mechanical elements in an abstract space is deliberately independent of a specific microscopic model. No prejudice is imposed as to the mechanism causing nonlinear response in the microscopic mechanical elements. The new theory (1 relates suitable stress-strain measurements to the number density Ï? and (2 uses the number density Ï? to find the behaviour of nonlinear elastic waves. Thus the new theory provides for the synthesis of the full spectrum of elastic behaviours of a rock. Early development of the new theory is sketched in this contribution.
1. Temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various frequencies.
Science.gov (United States)
Maraghechi, Borna; Hasani, Mojtaba H; Kolios, Michael C; Tavakkoli, Jahan
2016-05-01
Ultrasound-based thermometry requires a temperature-sensitive acoustic parameter that can be used to estimate the temperature by tracking changes in that parameter during heating. The objective of this study is to investigate the temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various pulse transmit frequencies from 1 to 20 MHz. Simulations were conducted using an expanded form of the Khokhlov-Zabolotskaya-Kuznetsov nonlinear acoustic wave propagation model in which temperature dependence of the medium parameters was included. Measurements were performed using single-element transducers at two different transmit frequencies of 3.3 and 13 MHz which are within the range of frequencies simulated. The acoustic pressure signals were measured by a calibrated needle hydrophone along the axes of the transducers. The water temperature was uniformly increased from 26 °C to 46 °C in increments of 5 °C. The results show that the temperature dependence of the harmonic generation is different at various frequencies which is due to the interplay between the mechanisms of absorption, nonlinearity, and focusing gain. At the transmit frequencies of 1 and 3.3 MHz, the harmonic amplitudes decrease with increasing the temperature, while the opposite temperature dependence is observed at 13 and 20 MHz.
2. A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up
Science.gov (United States)
Filippini, A. G.; Kazolea, M.; Ricchiuto, M.
2016-04-01
In this paper we evaluate hybrid strategies for the solution of the Green-Naghdi system of equations for the simulation of fully nonlinear and weakly dispersive free surface waves. We consider a two step solution procedure composed of: a first step where the non-hydrostatic source term is recovered by inverting the elliptic coercive operator associated to the dispersive effects; a second step which involves the solution of the hyperbolic shallow water system with the source term, computed in the previous phase, which accounts for the non-hydrostatic effects. Appropriate numerical methods, that can be also generalized on arbitrary unstructured meshes, are used to discretize the two stages: the standard C0 Galerkin finite element method for the elliptic phase; either third order Finite Volume or third order stabilized Finite Element method for the hyperbolic phase. The discrete dispersion properties of the fully coupled schemes obtained are studied, showing accuracy close to or better than that of a fourth order finite difference method. The hybrid approach of locally reverting to the nonlinear shallow water equations is used to recover energy dissipation in breaking regions. To this scope we evaluate two strategies: simply neglecting the non-hydrostatic contribution in the hyperbolic phase; imposing a tighter coupling of the two phases, with a wave breaking indicator embedded in the elliptic phase to smoothly turn off the dispersive effects. The discrete models obtained are thoroughly tested on benchmarks involving wave dispersion, breaking and run-up, showing a very promising potential for the simulation of complex near shore wave physics in terms of accuracy and robustness.
3. Nonlinear propagation of ion-acoustic waves in a degenerate dense plasma
M M Masud; A A Mamun
2013-07-01
Nonlinear propagation of ion-acoustic (IA) waves in a degenerate dense plasma (with all the constituents being degenerate, for both the non-relativistic or ultrarelativistic cases) have been investigated by the reductive perturbation method. The linear dispersion relation and Korteweg de Vries (KdV) equation have been derived, and the numerical solutions of KdV equation have been analysed to identify the basic features of electrostatic solitary structures that may form in such a degenerate dense plasma. The implications of our results in compact astrophysical objects, particularly, in white dwarfs and neutron stars, have been briefly discussed.
4. Nonlinear propagation of high-frequency energy from blast waves as it pertains to bat hearing
Science.gov (United States)
Loubeau, Alexandra
Close exposure to blast noise from military weapons training can adversely affect the hearing of both humans and wildlife. One concern is the effect of high-frequency noise from Army weapons training on the hearing of endangered bats. Blast wave propagation measurements were conducted to investigate nonlinear effects on the development of blast waveforms as they propagate from the source. Measurements were made at ranges of 25, 50, and 100 m from the blast. Particular emphasis was placed on observation of rise time variation with distance. Resolving the fine shock structure of blast waves requires robust transducers with high-frequency capability beyond 100 kHz, hence the limitations of traditional microphones and the effect of microphone orientation were investigated. Measurements were made with a wide-bandwidth capacitor microphone for comparison with conventional 3.175-mm (⅛-in.) microphones with and without baffles. The 3.175-mm microphone oriented at 90° to the propagation direction did not have sufficient high-frequency response to capture the actual rise times at a range of 50 m. Microphone baffles eliminate diffraction artifacts on the rise portion of the measured waveform and therefore allow for a more accurate measurement of the blast rise time. The wide-band microphone has an extended high-frequency response and can resolve shorter rise times than conventional microphones. For a source of 0.57 kg (1.25 lb) of C-4 plastic explosive, it was observed that nonlinear effects steepened the waveform, thereby decreasing the shock rise time, from 25 to 50 m. At 100m, the rise times had increased slightly. For comparison to the measured blast waveforms, several models of nonlinear propagation are applied to the problem of finite-amplitude blast wave propagation. Shock front models, such as the Johnson and Hammerton model, and full-waveform marching algorithms, such as the Anderson model, are investigated and compared to experimental results. The models
5. Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor.
Science.gov (United States)
Dogan, Hakan; Popov, Viktor
2016-05-01
We investigate the acoustic wave propagation in bubbly liquid inside a pilot sonochemical reactor which aims to produce antibacterial medical textile fabrics by coating the textile with ZnO or CuO nanoparticles. Computational models on acoustic propagation are developed in order to aid the design procedures. The acoustic pressure wave propagation in the sonoreactor is simulated by solving the Helmholtz equation using a meshless numerical method. The paper implements both the state-of-the-art linear model and a nonlinear wave propagation model recently introduced by Louisnard (2012), and presents a novel iterative solution procedure for the nonlinear propagation model which can be implemented using any numerical method and/or programming tool. Comparative results regarding both the linear and the nonlinear wave propagation are shown. Effects of bubble size distribution and bubble volume fraction on the acoustic wave propagation are discussed in detail. The simulations demonstrate that the nonlinear model successfully captures the realistic spatial distribution of the cavitation zones and the associated acoustic pressure amplitudes.
6. Nonlinear heat-transport equation beyond Fourier law: application to heat-wave propagation in isotropic thin layers
Science.gov (United States)
Sellitto, A.; Tibullo, V.; Dong, Y.
2017-03-01
By means of a nonlinear generalization of the Maxwell-Cattaneo-Vernotte equation, on theoretical grounds we investigate how nonlinear effects may influence the propagation of heat waves in isotropic thin layers which are not laterally isolated from the external environment. A comparison with the approach of the Thermomass Theory is made as well.
7. Relativistic nonlinearity and wave-guide propagation of rippled laser beam in plasma
R K Khanna; K Baheti
2001-06-01
In the present paper we have investigated the self-focusing behaviour of radially symmetrical rippled Gaussian laser beam propagating in a plasma. Considering the nonlinearity to arise from relativistic phenomena and following the approach of Akhmanov et al, which is based on the WKB and paraxial-ray approximation, the self-focusing behaviour has been investigated in some detail. The effect of the position and width of the ripple on the self-focusing of laser beam has been studied for arbitrary large magnitude of nonlinearity. Results indicate that the medium behaves as an oscillatory wave-guide. The self-focusing is found to depend on the position parameter of ripple as well as on the beam width. Values of critical power has been calculated for different values of the position parameter of ripple. Effects of axially and radially inhomogeneous plasma on self-focusing behaviour have been investigated and presented here.
8. Nonlinear propagation of positron-acoustic waves in a four component space plasma
Science.gov (United States)
Shah, M. G.; Hossen, M. R.; Mamun, A. A.
2015-10-01
> The nonlinear propagation of positron-acoustic waves (PAWs) in an unmagnetized, collisionless, four component, dense plasma system (containing non-relativistic inertial cold positrons, relativistic degenerate electron and hot positron fluids as well as positively charged immobile ions) has been investigated theoretically. The Korteweg-de Vries (K-dV), modified K-dV (mK-dV) and further mK-dV (fmK-dV) equations have been derived by using reductive perturbation technique. Their solitary wave solutions have been numerically analysed in order to understand the localized electrostatic disturbances. It is observed that the relativistic effect plays a pivotal role on the propagation of positron-acoustic solitary waves (PASW). It is also observed that the effects of degenerate pressure and the number density of inertial cold positrons, hot positrons, electrons and positively charged static ions significantly modify the fundamental features of PASW. The basic features and the underlying physics of PASW, which are relevant to some astrophysical compact objects (such as white dwarfs, neutron stars etc.), are concisely discussed.
9. Nonlinear phenomena in RF wave propagation in magnetized plasma: A review
Energy Technology Data Exchange (ETDEWEB)
Porkolab, Miklos
2015-12-10
Nonlinear phenomena in RF wave propagation has been observed from the earliest days in basic laboratory experiments going back to the 1960s [1], followed by observations of parametric instability (PDI) phenomena in large scale RF heating experiments in magnetized fusion plasmas in the 1970s and beyond [2]. Although not discussed here, the importance of PDI phenomena has also been central to understanding anomalous absorption in laser-fusion experiments (ICF) [3]. In this review I shall discuss the fundamentals of nonlinear interactions among waves and particles, and in particular, their role in PDIs. This phenomenon is distinct from quasi-linear phenomena that are often invoked in calculating absorption of RF power in wave heating experiments in the core of magnetically confined plasmas [4]. Indeed, PDIs are most likely to occur in the edge of magnetized fusion plasmas where the electron temperature is modest and hence the oscillating quiver velocity of charged particles can be comparable to their thermal speeds. Specifically, I will review important aspects of PDI theory and give examples from past experiments in the ECH/EBW, lower hybrid (LHCD) and ICRF/IBW frequency regimes. Importantly, PDI is likely to play a fundamental role in determining the so-called “density limit” in lower hybrid experiments that has persisted over the decades and still central to understanding present day experiments [5-7].
10. Fully Nonlinear Boussinesq-Type Equations with Optimized Parameters for Water Wave Propagation
Institute of Scientific and Technical Information of China (English)
荆海晓; 刘长根; 龙文; 陶建华
2015-01-01
For simulating water wave propagation in coastal areas, various Boussinesq-type equations with improved properties in intermediate or deep water have been presented in the past several decades. How to choose proper Boussinesq-type equations has been a practical problem for engineers. In this paper, approaches of improving the characteristics of the equations, i.e. linear dispersion, shoaling gradient and nonlinearity, are reviewed and the advantages and disadvantages of several different Boussinesq-type equations are compared for the applications of these Boussinesq-type equations in coastal engineering with relatively large sea areas. Then for improving the properties of Boussinesq-type equations, a new set of fully nonlinear Boussinseq-type equations with modified representative velocity are derived, which can be used for better linear dispersion and nonlinearity. Based on the method of minimizing the overall error in different ranges of applications, sets of parameters are determined with optimized linear dispersion, linear shoaling and nonlinearity, respectively. Finally, a test example is given for validating the results of this study. Both results show that the equations with optimized parameters display better characteristics than the ones obtained by matching with padé approximation.
11. Fully nonlinear Boussinesq-type equations with optimized parameters for water wave propagation
Science.gov (United States)
Jing, Hai-xiao; Liu, Chang-gen; Long, Wen; Tao, Jian-hua
2015-06-01
For simulating water wave propagation in coastal areas, various Boussinesq-type equations with improved properties in intermediate or deep water have been presented in the past several decades. How to choose proper Boussinesq-type equations has been a practical problem for engineers. In this paper, approaches of improving the characteristics of the equations, i.e. linear dispersion, shoaling gradient and nonlinearity, are reviewed and the advantages and disadvantages of several different Boussinesq-type equations are compared for the applications of these Boussinesq-type equations in coastal engineering with relatively large sea areas. Then for improving the properties of Boussinesq-type equations, a new set of fully nonlinear Boussinseq-type equations with modified representative velocity are derived, which can be used for better linear dispersion and nonlinearity. Based on the method of minimizing the overall error in different ranges of applications, sets of parameters are determined with optimized linear dispersion, linear shoaling and nonlinearity, respectively. Finally, a test example is given for validating the results of this study. Both results show that the equations with optimized parameters display better characteristics than the ones obtained by matching with padé approximation.
12. Self-action of propagating and standing Lamb waves in the plates exhibiting hysteretic nonlinearity: Nonlinear zero-group velocity modes.
Science.gov (United States)
Gusev, Vitalyi E; Lomonosov, Alexey M; Ni, Chenyin; Shen, Zhonghua
2017-09-01
An analytical theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous plate material on the Lamb waves near the S1 zero group velocity point is developed. The theory predicts that the main effect of the hysteretic quadratic nonlinearity consists in the modification of the frequency and the induced absorption of the Lamb modes. The effects of the nonlinear self-action in the propagating and standing Lamb waves are expected to be, respectively, nearly twice and three times stronger than those in the plane propagating acoustic waves. The theory is restricted to the simplest hysteretic nonlinearity, which is influencing only one of the Lamé moduli of the materials. However, possible extensions of the theory to the cases of more general hysteretic nonlinearities are discussed as well as the perspectives of its experimental testing. Applications include nondestructive evaluation of micro-inhomogeneous and cracked plates. Copyright © 2017 Elsevier B.V. All rights reserved.
13. A 2D spring model for the simulation of ultrasonic wave propagation in nonlinear hysteretic media.
Science.gov (United States)
Delsanto, P P; Gliozzi, A S; Hirsekorn, M; Nobili, M
2006-07-01
A two-dimensional (2D) approach to the simulation of ultrasonic wave propagation in nonclassical nonlinear (NCNL) media is presented. The approach represents the extension to 2D of a previously proposed one dimensional (1D) Spring Model, with the inclusion of a PM space treatment of the intersticial regions between grains. The extension to 2D is of great practical relevance for its potential applications in the field of quantitative nondestructive evaluation and material characterization, but it is also useful, from a theoretical point of view, to gain a better insight of the interaction mechanisms involved. The model is tested by means of virtual 2D experiments. The expected NCNL behaviors are qualitatively well reproduced.
14. Two-dimensional wave propagation in an elastic half-space with quadratic nonlinearity: a numerical study.
Science.gov (United States)
Küchler, Sebastian; Meurer, Thomas; Jacobs, Laurence J; Qu, Jianmin
2009-03-01
This study investigates two-dimensional wave propagation in an elastic half-space with quadratic nonlinearity. The problem is formulated as a hyperbolic system of conservation laws, which is solved numerically using a semi-discrete central scheme. These numerical results are then analyzed in the frequency domain to interpret the nonlinear effects, specifically the excitation of higher-order harmonics. To quantify and compare the nonlinearity of different materials, a new parameter is introduced, which is similar to the acoustic nonlinearity parameter beta for one-dimensional longitudinal waves. By using this new parameter, it is found that the nonlinear effects of a material depend on the point of observation in the half-space, both the angle and the distance to the excitation source. Furthermore it is illustrated that the third-order elastic constants have a linear effect on the acoustic nonlinearity of a material.
15. The nonlinear Schrödinger equation and the propagation of weakly nonlinear waves in optical fibers and on the water surface
Energy Technology Data Exchange (ETDEWEB)
Chabchoub, A., E-mail: [email protected] [Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122 (Australia); Kibler, B.; Finot, C.; Millot, G. [Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS, Université de Bourgogne, 21078 Dijon (France); Onorato, M. [Dipartimento di Fisica, Università degli Studi di Torino, Torino 10125 (Italy); Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, Torino 10125 (Italy); Dudley, J.M. [Institut FEMTO-ST, UMR 6174 CNRS- Université de Franche-Comté, 25030 Besançon (France); Babanin, A.V. [Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122 (Australia)
2015-10-15
The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. a nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.
16. Propagation of Weakly Guided Waves in a Kerr Nonlinear Medium using a Perturbation Approach
Energy Technology Data Exchange (ETDEWEB)
Dacles-Mariani, J; Rodrigue, G
2004-10-06
The equations are represented in a simplified format with only a few leading terms needed in the expansion. The set of equations are then solved numerically using vector finite element method. To validate the algorithm, they analyzed a two-dimensional rectangular waveguide consisting of a linear core and nonlinear identical cladding. The exact nonlinear solutions for three different modes of propagations, TE0, TE1, and TE2 modes are generated and compared with the computed solutions. Next, they investigate the effect of a more intense monochromatic field on the propagation of a 'weak' optical field in a fully three-dimensional cylindrical waveguide.
17. Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses.
Science.gov (United States)
Kim, Kihong; Phung, D K; Rotermund, F; Lim, H
2008-01-21
We develop a generalized version of the invariant imbedding method, which allows us to solve the electromagnetic wave equations in arbitrarily inhomogeneous stratified media where both the dielectric permittivity and magnetic permeability depend on the strengths of the electric and magnetic fields, in a numerically accurate and efficient manner. We apply our method to a uniform nonlinear slab and find that in the presence of strong external radiation, an initially uniform medium of positive refractive index can spontaneously change into a highly inhomogeneous medium where regions of positive or negative refractive index as well as metallic regions appear. We also study the wave transmission properties of periodic nonlinear media and the influence of nonlinearity on the mode conversion phenomena in inhomogeneous plasmas. We argue that our theory is very useful in the study of the optical properties of a variety of nonlinear media including nonlinear negative index media fabricated using wires and split-ring resonators.
18. Remarks on nonlinear relation among phases and frequencies in modulational instabilities of parallel propagating Alfven waves
CERN Document Server
Nariyuki, Y; Nariyuki, Yasuhiro; Hada, Tohru
2006-01-01
Nonlinear relations among frequencies and phases in modulational instability of circularly polarized Alfven waves are discussed, within the context of one dimensional, dissipation-less, unforced fluid system. We show that generation of phase coherence is a natural consequence of the modulational instability of Alfven waves. Furthermore, we quantitatively evaluate intensity of wave-wave interaction by using bi-coherence, and also by computing energy flow among wave modes, and demonstrate that the energy flow is directly related to the phase coherence generation.
19. Nonlinear effects in the propagation of optically generated magnetostatic volume mode spin waves
Science.gov (United States)
van Tilburg, L. J. A.; Buijnsters, F. J.; Fasolino, A.; Rasing, T.; Katsnelson, M. I.
2017-08-01
Recent experimental work has demonstrated optical control of spin wave emission by tuning the shape of the optical pulse [Satoh et al., Nat. Photon. 6, 662 (2012), 10.1038/nphoton.2012.218]. We reproduce these results and extend the scope of the control by investigating nonlinear effects for large amplitude excitations. We observe an accumulation of spin wave power at the center of the initial excitation combined with short-wavelength spin waves. These kinds of nonlinear effects have not been observed in earlier work on nonlinearities of spin waves. Our observations pave the way for the manipulation of magnetic structures at a smaller scale than the beam focus, for instance in devices with all-optical control of magnetism.
20. A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation
CERN Document Server
Louisnard, Olivier
2013-01-01
The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger ...
1. Nonlinear elastic waves in materials
CERN Document Server
Rushchitsky, Jeremiah J
2014-01-01
The main goal of the book is a coherent treatment of the theory of propagation in materials of nonlinearly elastic waves of displacements, which corresponds to one modern line of development of the nonlinear theory of elastic waves. The book is divided on five basic parts: the necessary information on waves and materials; the necessary information on nonlinear theory of elasticity and elastic materials; analysis of one-dimensional nonlinear elastic waves of displacement – longitudinal, vertically and horizontally polarized transverse plane nonlinear elastic waves of displacement; analysis of one-dimensional nonlinear elastic waves of displacement – cylindrical and torsional nonlinear elastic waves of displacement; analysis of two-dimensional nonlinear elastic waves of displacement – Rayleigh and Love nonlinear elastic surface waves. The book is addressed first of all to people working in solid mechanics – from the students at an advanced undergraduate and graduate level to the scientists, professional...
2. Viscothermal wave propagation
NARCIS (Netherlands)
Nijhof, Marten Jozef Johannes
2010-01-01
In this work, the accuracy, efficiency and range of applicability of various (approximate) models for viscothermal wave propagation are investigated. Models for viscothermal wave propagation describe thewave behavior of fluids including viscous and thermal effects. Cases where viscothermal effects a
3. Time-domain numerical modeling of brass instruments including nonlinear wave propagation, viscothermal losses, and lips vibration
CERN Document Server
Berjamin, Harold; Vergez, Christophe; Cottanceau, Emmanuel
2015-01-01
A time-domain numerical modeling of brass instruments is proposed. On one hand, outgoing and incoming waves in the resonator are described by the Menguy-Gilbert model, which incorporates three key issues: nonlinear wave propagation, viscothermal losses, and a variable section. The non-linear propagation is simulated by a TVD scheme well-suited to non-smooth waves. The fractional derivatives induced by the viscothermal losses are replaced by a set of local-in-time memory variables. A splitting strategy is followed to couple optimally these dedicated methods. On the other hand, the exciter is described by a one-mass model for the lips. The Newmark method is used to integrate the nonlinear ordinary differential equation so-obtained. At each time step, a coupling is performed between the pressure in the tube and the displacement of the lips. Finally, an extensive set of validation tests is successfully completed. In particular, self-sustained oscillations of the lips are simulated by taking into account the nonli...
4. On the effect of elastic nonlinearity on aquatic propulsion caused by propagating flexural waves
CERN Document Server
Krylov, Victor V
2016-01-01
In the present paper, the initial theoretical results on wave-like aquatic propulsion of marine craft by propagating flexural waves are reported. Recent experimental investigations of small model boats propelled by propagating flexural waves carried out by the present author and his co-workers demonstrated viability of this type of propulsion as an alternative to a well-known screw propeller. In the attempts of theoretical explanation of the obtained experimental results using the theory of Lighthill for fish locomotion, it was found that this theory predicts zero thrust for such model boats, which is in contradiction with the results of the experiments. One should note in this connection that the theory developed by Lighthill assumes that the amplitudes of propulsive waves created by fish body motion grow from zero on the front (at fish heads) to their maximum values at the tails. This is consistent with fish body motion in nature, but is not compatible with the behaviour of localised flexural waves used for...
5. On the nonlinear internal waves propagating in an inhomogeneous shallow sea
Directory of Open Access Journals (Sweden)
Stanisław R. Massel
2016-04-01
Full Text Available A concept of conservation of energy flux for the internal waves propagating in an inhomogeneous shallow water is examined. The emphasis is put on an application of solution of the Korteweg–de Vries (KdV equation in a prescribed form of the cnoidal and solitary waves. Numerical simulations were applied for the southern Baltic Sea, along a transect from the Bornholm Basin, through the Słupsk Sill and Słupsk Furrow to the Gdańsk Basin. Three-layer density structure typical for the Baltic Sea has been considered. An increase of wave height and decrease of phase speed with shallowing water depth was clearly demonstrated. The internal wave dynamics on both sides of the Słupsk Sill was found to be different due to different vertical density stratification in these areas. The bottom friction has only negligible influence on dynamics of internal waves, while shearing instability may be important only for very high waves. Area of possible instability, expressed in terms of the Richardson number Ri, is very small, and located within the non-uniform density layer, close to the interface with upper uniform layer. Kinematic breaking criteria have been examined and critical internal wave heights have been determined.
6. Nonlinear acoustic propagation in rectangular ducts
Science.gov (United States)
Nayfeh, A. H.; Tsai, M.-S.
1974-01-01
The method of multiple scales is used to obtain a second-order uniformly valid expansion for nonlinear acoustic wave propagation in a rectangular duct whose walls are treated with a nonlinear acoustic material. The wave propagation in the duct is characterized by the unsteady nonlinear Euler equations. The results show that nonlinear materials attenuate sound more than linear materials except at high acoustic frequencies. The nonlinear materials produce higher and combination tones which have higher attenuation rates than the fundamentals. Moreover, the attenuation rates of the fundamentals increase with increasing amplitude.
7. Effect of magnetic field on the propagation of quasi-transverse waves in a nonhomogeneous conducting medium under the theory of nonlinear elasticity
D P Acharya; Asit Kumar Mondal
2006-06-01
The object of the present paper is to investigate the propagation of quasi-transverse waves in a nonlinear perfectly conducting nonhomogeneous elastic medium in the presence of a uniform magnetic field transverse to the direction of wave propagation. Different types of figures have been drawn to exhibit the distortion of waves due to the presence of magnetic field and the nonhomogeneous nature of the medium. Formation of shocks has also been numerically discussed.
8. Electromagnetic beam propagation in nonlinear media
Institute of Scientific and Technical Information of China (English)
V.V.Semak; M.N.Shneider
2015-01-01
We deduce a complete wave propagation equation that includes inhomogeneity of the dielectric constant and present this propagation equation in compact vector form. Although similar equations are known in narrow fields such as radio wave propagation in the ionosphere and electromagnetic and acoustic wave propagation in stratified media, we develop here a novel approach of using such equations in the modeling of laser beam propagation in nonlinear media. Our approach satisfies the correspondence principle since in the limit of zero-length wavelength it reduces from physical to geometrical optics.
9. MODELING THE ASIAN TSUNAMI EVOLUTION AND PROPAGATION WITH A NEW GENERATION MECHANISM AND A NON-LINEAR DISPERSIVE WAVE MODEL
Directory of Open Access Journals (Sweden)
Paul C. Rivera
2006-01-01
Full Text Available A common approach in modeling the generation and propagation of tsunami is based on the assumption of a kinematic vertical displacement of ocean water that is analogous to the ocean bottom displacement during a submarine earthquake and the use of a non-dispersive long-wave model to simulate its physical transformation as it radiates outward from the source region. In this study, a new generation mechanism and the use of a highly-dispersive wave model to simulate tsunami inception, propagation and transformation are proposed. The new generation model assumes that transient ground motion during the earthquake can accelerate horizontal currents with opposing directions near the fault line whose successive convergence and divergence generate a series of potentially destructive oceanic waves. The new dynamic model incorporates the effects of earthquake moment magnitude, ocean compressibility through the buoyancy frequency, the effects of focal and water depths, and the orientation of ruptured fault line in the tsunami magnitude and directivity.For tsunami wave simulation, the nonlinear momentum-based wave model includes important wave propagation and transformation mechanisms such as refraction, diffraction, shoaling, partial reflection and transmission, back-scattering, frequency dispersion, and resonant wave-wave interaction. Using this model and a coarse-resolution bathymetry, the new mechanism is tested for the Indian Ocean tsunami of December 26, 2004. A new flooding and drying algorithm that consider waves coming from every direction is also proposed for simulation of inundation of low-lying coastal regions.It is shown in the present study that with the proposed generation model, the observed features of the Asian tsunami such as the initial drying of areas east of the source region and the initial flooding of western coasts are correctly simulated. The formation of a series of tsunami waves with periods and lengths comparable to observations
10. Propagation of quasiplane nonlinear waves in tubes and the approximate solutions of the generalized Burgers equation.
Science.gov (United States)
Bednarik, Michal; Konicek, Petr
2002-07-01
This paper deals with using the generalized Burgers equation for description of nonlinear waves in circular ducts. Two new approximate solutions of the generalized Burgers equation (GBE) are presented. These solutions take into account the boundary layer effects. The first solution is valid for the preshock region and gives more precise results than the Fubini solution, whereas the second one is valid for the postshock (sawtooth) region and provides better results than the Fay solution. The approximate solutions are compared with numerical results of the GBE. Furthermore, the limits of validity of the used model equation are discussed with respect to boundary conditions and radius of a circular duct.
11. Approximate solutions to a nonintegrable problem of propagation of elliptically polarised waves in an isotropic gyrotropic nonlinear medium, and periodic analogues of multisoliton complexes
Energy Technology Data Exchange (ETDEWEB)
Makarov, V A; Petnikova, V M; Potravkin, N N; Shuvalov, V V [International Laser Center, M. V. Lomonosov Moscow State University, Moscow (Russian Federation)
2014-02-28
Using the linearization method, we obtain approximate solutions to a one-dimensional nonintegrable problem of propagation of elliptically polarised light waves in an isotropic gyrotropic medium with local and nonlocal components of the Kerr nonlinearity and group-velocity dispersion. The consistent evolution of two orthogonal circularly polarised components of the field is described analytically in the case when their phases vary linearly during propagation. The conditions are determined for the excitation of waves with a regular and 'chaotic' change in the polarisation state. The character of the corresponding nonlinear solutions, i.e., periodic analogues of multisoliton complexes, is analysed. (nonlinear optical phenomena)
12. Nonlinear propagation and decay of intense regular and random waves in relaxing media
Science.gov (United States)
Gurbatov, S. N.; Rudenko, O. V.; Demin, I. Yu.
2015-10-01
An integro-differential equation is written down that contains terms responsible for nonlinear absorption, visco-heat-conducting dissipation, and relaxation processes in a medium. A general integral expression is obtained for calculating energy losses of the wave with arbitrary characteristics—intensity, profile (frequency spectrum), and kernel describing the internal dynamics of the medium. Profiles of stationary solutions are constructed both for an exponential relaxation kernel and for other types of kernels. Energy losses at the front of week shock waves are calculated. General integral formulas are obtained for energy losses of intense noise, which are determined by the form of the kernel, the structure of the noise correlation function, and the mean square of the derivative of realization of a random process.
13. Nonlinear propagation of ion-acoustic waves in self-gravitating dusty plasma consisting of non-isothermal two-temperature electrons
Science.gov (United States)
Paul, S. N.; Chatterjee, A.; Paul, Indrani
2017-01-01
Nonlinear propagation of ion-acoustic waves in self-gravitating multicomponent dusty plasma consisting of positive ions, non-isothermal two-temperature electrons and negatively charged dust particles with fluctuating charges and drifting ions has been studied using the reductive perturbation method. It has been shown that nonlinear propagation of ion-acoustic waves in gravitating dusty plasma is described by an uncoupled third order partial differential equation which is a modified form of Korteweg-deVries equation, in contraries to the coupled nonlinear equations obtained by earlier authors. Quasi-soliton solution for the ion-acoustic solitary wave has been obtained from this uncoupled nonlinear equation. Effects of non-isothermal two-temperature electrons, gravity, dust charge fluctuation and drift motion of ions on the ion-acoustic solitary waves have been discussed.
14. Effect of secondary electron emission on nonlinear dust acoustic wave propagation in a complex plasma with negative equilibrium dust charge
Science.gov (United States)
Bhakta, Subrata; Ghosh, Uttam; Sarkar, Susmita
2017-02-01
In this paper, we have investigated the effect of secondary electron emission on nonlinear propagation of dust acoustic waves in a complex plasma where equilibrium dust charge is negative. The primary electrons, secondary electrons, and ions are Boltzmann distributed, and only dust grains are inertial. Electron-neutral and ion-neutral collisions have been neglected with the assumption that electron and ion mean free paths are very large compared to the plasma Debye length. Both adiabatic and nonadiabatic dust charge variations have been separately taken into account. In the case of adiabatic dust charge variation, nonlinear propagation of dust acoustic waves is governed by the KdV (Korteweg-de Vries) equation, whereas for nonadiabatic dust charge variation, it is governed by the KdV-Burger equation. The solution of the KdV equation gives a dust acoustic soliton, whose amplitude and width depend on the secondary electron yield. Similarly, the KdV-Burger equation provides a dust acoustic shock wave. This dust acoustic shock wave may be monotonic or oscillatory in nature depending on the fact that whether it is dissipation dominated or dispersion dominated. Our analysis shows that secondary electron emission increases nonadiabaticity induced dissipation and consequently increases the monotonicity of the dust acoustic shock wave. Such a dust acoustic shock wave may accelerate charge particles and cause bremsstrahlung radiation in space plasmas whose physical process may be affected by secondary electron emission from dust grains. The effect of the secondary electron emission on the stability of the equilibrium points of the KdV-Burger equation has also been investigated. This equation has two equilibrium points. The trivial equilibrium point with zero potential is a saddle and hence unstable in nature. The nontrivial equilibrium point with constant nonzero potential is a stable node up to a critical value of the wave velocity and a stable focus above it. This critical
15. Non-linear numerical simulations of magneto-acoustic wave propagation in small-scale flux tubes
CERN Document Server
Khomenko, E; Felipe, T
2007-01-01
We present results of non-linear 2D numerical simulations of magneto-acoustic wave propagation in the photosphere and chromosphere of small-scale flux tubes with internal structure. Waves with realistic periods of 3--5 min are studied, after applying horizontal and vertical oscillatory perturbations to the equilibrium situation. Spurious reflections of shock waves from the upper boundary are minimized thanks to a special boundary condition. This has allowed us to increase the duration of the simulations and to make it long enough to perform a statistical analysis of oscillations. The simulations show that deep horizontal motions of the flux tube generate a slow (magnetic) mode and a surface mode. These modes are efficiently transformed into a slow (acoustic) mode in the Va < Cs atmosphere. The slow (acoustic) mode propagates vertically along the field lines, forms shocks and remains always within the flux tube. It might deposit effectively the energy of the driver into the chromosphere. When the driver osc...
16. Nonlinear hyperbolic waves in multidimensions
CERN Document Server
2001-01-01
The propagation of curved, nonlinear wavefronts and shock fronts are very complex phenomena. Since the 1993 publication of his work Propagation of a Curved Shock and Nonlinear Ray Theory, author Phoolan Prasad and his research group have made significant advances in the underlying theory of these phenomena. This volume presents their results and provides a self-contained account and gradual development of mathematical methods for studying successive positions of these fronts.Nonlinear Hyperbolic Waves in Multidimensions includes all introductory material on nonlinear hyperbolic waves and the theory of shock waves. The author derives the ray theory for a nonlinear wavefront, discusses kink phenomena, and develops a new theory for plane and curved shock propagation. He also derives a full set of conservation laws for a front propagating in two space dimensions, and uses these laws to obtain successive positions of a front with kinks. The treatment includes examples of the theory applied to converging wavefronts...
17. Slowly moving matter-wave gap soliton propagation in weak random nonlinear potential
Institute of Scientific and Technical Information of China (English)
Zhang Ming-Rui; Zhang Yong-Liang; Jiang Xun-Ya; Zi Jian
2008-01-01
We systematically investigate the motion of slowly moving matter-wave gap solitons in a nonlinear potential, produced by the weak random spatial variation of the atomic scattering length. With the weak randomness, we construct an effective-particle theory to study the motion of gap solitons. Based on the effective-particle theory, the effect of the randomness on gap solitous is obtained, and the motion of gap solitons is finally solved. Moreover, the analytic results for the general behaviours of gap soliton motion, such as the ensemble-average speed and the reflection probability depending on the weak randomness are obtained. We find that with the increase of the random strength the ensemble-average speed of gap solitons decreases slowly where the reduction is proportional to the variance of the weak randomness, and the reflection probability becomes larger. The theoretical results are in good agreement with the numerical simulations based on the Gross-Pitaevskii equation.
18. A finite volume approach for the simulation of nonlinear dissipative acoustic wave propagation
CERN Document Server
Velasco-Segura, Roberto
2013-01-01
A form of the conservation equations for fluid dynamics is presented, deduced using slightly less restrictive hypothesis than those necessary to obtain the well known Westervelt equation. This formulation accounts for full wave diffraction, nonlinearity, and thermoviscous dissipative effects. A CLAWPACK based, 2D finite volume method using the Roe linearization was implemented to obtain numerically the solution of the proposed equations. In order to validate the code, two different tests have been performed: one against a special Taylor shock-like analytic solution, the other against published results on a HIFU system, both with satisfactory results. The code is based on CLAWPACK and is written for parallel execution on a GPU, thus improving performance by a factor of over 60 when compared to the standard CLAWPACK code.
19. Stochastic wave propagation
CERN Document Server
Sobczyk, K
1985-01-01
This is a concise, unified exposition of the existing methods of analysis of linear stochastic waves with particular reference to the most recent results. Both scalar and vector waves are considered. Principal attention is concentrated on wave propagation in stochastic media and wave scattering at stochastic surfaces. However, discussion extends also to various mathematical aspects of stochastic wave equations and problems of modelling stochastic media.
20. Propagation of waves
CERN Document Server
David, P
2013-01-01
Propagation of Waves focuses on the wave propagation around the earth, which is influenced by its curvature, surface irregularities, and by passage through atmospheric layers that may be refracting, absorbing, or ionized. This book begins by outlining the behavior of waves in the various media and at their interfaces, which simplifies the basic phenomena, such as absorption, refraction, reflection, and interference. Applications to the case of the terrestrial sphere are also discussed as a natural generalization. Following the deliberation on the diffraction of the "ground? wave around the ear
1. Modelling the Propagation of a Weak Fast-Mode MHD Shock Wave near a 2D Magnetic Null Point Using Nonlinear Geometrical Acoustics
Science.gov (United States)
Afanasyev, A. N.; Uralov, A. M.
2012-10-01
We present the results of analytical modelling of fast-mode magnetohydrodynamic wave propagation near a 2D magnetic null point. We consider both a linear wave and a weak shock and analyse their behaviour in cold and warm plasmas. We apply the nonlinear geometrical acoustics method based on the Wentzel-Kramers-Brillouin approximation. We calculate the wave amplitude, using the ray approximation and the laws of solitary shock wave damping. We find that a complex caustic is formed around the null point. Plasma heating is distributed in space and occurs at a caustic as well as near the null point due to substantial nonlinear damping of the shock wave. The shock wave passes through the null point even in a cold plasma. The complex shape of the wave front can be explained by the caustic pattern.
2. Modelling the Propagation of a Weak Fast-Mode MHD Shock Wave near a 2D Magnetic Null Point Using Nonlinear Geometrical Acoustics
CERN Document Server
Afanasyev, Andrey N
2012-01-01
We present the results of analytical modelling of fast-mode magnetohydrodynamic wave propagation near a 2D magnetic null point. We consider both a linear wave and a weak shock and analyse their behaviour in cold and warm plasmas. We apply the nonlinear geometrical acoustics method based on the Wentzel-Kramers-Brillouin approximation. We calculate the wave amplitude, using the ray approximation and the laws of solitary shock wave damping. We find that a complex caustic is formed around the null point. Plasma heating is distributed in space and occurs at a caustic as well as near the null point due to substantial nonlinear damping of the shock wave. The shock wave passes through the null point even in a cold plasma. The complex shape of the wave front can be explained by the caustic pattern.
3. Wave propagation in parallel-plate waveguides filled with nonlinear left-handed material
Institute of Scientific and Technical Information of China (English)
Burhan Zamir; Rashid Ali
2011-01-01
A theoretical investigation of field components for transverse electric mode in the parallel-plate waveguides has been studied. In this analysis two different types of waveguide structures have been discussed, i.e., (a) normal good/perfect conducting parallel-plate waveguide filled with nonlinear left-handed material and (b) high-temperature-superconducting parallel-plate waveguide filled with nonlinear left-handed material. The dispersion relations of transverse electric mode have also been discussed for these two types of waveguide structures.
4. Nonlinear Waves.
Science.gov (United States)
1982-09-23
propagate in this plane. Of course , since the plane is two- dimensional, having only three- first -order partial differential equations for three functions... Differential Equattions in Engineering and Applied Science ed R L Sternberg, A J Kalinowki and I S Papadakis (New York: Marcel Dekker) p 397 Satsuaa 1...fourth coordinate required to make the set (Xlxji2XXd) complete. Of course , the envelopes in(2.2) may depend on .4 , but n.e differentiations with
5. A Model for the Propagation of Nonlinear Surface Waves over Viscous Muds
Science.gov (United States)
2007-07-05
grained, cohesive sedimentary 1993; Foda et al., 1993). With the exception of fluidization environments is well known. Extreme dissipation rates have...processes ( Foda et al., 1993; DeWit, 1995), these models focus on a single, well-defined mud phase. Although the models Corresponding author. Tel.: +1...However, surface-interface wave interactions ( Foda , 1989; Hill and Foda , our focus at the present is on a wave model which can be 1998; Jamali et al
6. Wave Propagation in Modified Gravity
CERN Document Server
Lindroos, Jan Ø; Mota, David F
2015-01-01
We investigate the propagation of scalar waves induced by matter sources in the context of scalar-tensor theories of gravity which include screening mechanisms for the scalar degree of freedom. The usual approach when studying these theories in the non-linear regime of cosmological perturbations is based on the assumption that scalar waves travel at the speed of light. Within General Relativity such approximation is good and leads to no loss of accuracy in the estimation of observables. We find, however, that mass terms and non-linearities in the equations of motion lead to propagation and dispersion velocities significantly different from the speed of light. As the group velocity is the one associated to the propagation of signals, a reduction of its value has direct impact on the behavior and dynamics of nonlinear structures within modified gravity theories with screening. For instance, the internal dynamics of galaxies and satellites submerged in large dark matter halos could be affected by the fact that t...
7. Traveling-wave method for solving the modified nonlinear Schrödinger equation describing soliton propagation along optical fibers
Science.gov (United States)
Bingzhen, Xu; Wenzheng, Wang
1995-02-01
We give a traveling-wave method for obtaining exact solutions of the modified nonlinear Schrödinger equation iut+ɛuxx+2p||u||2u +2iq(||u||2u)x=0, describing the propagation of light pulses in optical fibers, where u represents a normalized complex amplitude of a pulse envelope, t is the normalized distance along a fiber, and x is the normalized time within the frame of reference moving along the fiber at the group velocity. With the help of the potential function'' we obtained by this method, we find a family of solutions that are finite everywhere, particularly including periodic solutions expressed in terms of Jacobi elliptic functions, stationary periodic solutions, and algebraic'' soliton solutions. Compared with previous work [D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992)] in which two kinds of the simplest solution were given, the physical meaning of the integration constants in the potential function we give is clearer and more easily fixed with the initial parameters of the light pulse.
8. Nonlinear dynamics of shells conveying pulsatile flow with pulse-wave propagation. Theory and numerical results for a single harmonic pulsation
Science.gov (United States)
Tubaldi, Eleonora; Amabili, Marco; Païdoussis, Michael P.
2017-05-01
In deformable shells conveying pulsatile flow, oscillatory pressure changes cause local movements of the fluid and deformation of the shell wall, which propagate downstream in the form of a wave. In biomechanics, it is the propagation of the pulse that determines the pressure gradient during the flow at every location of the arterial tree. In this study, a woven Dacron aortic prosthesis is modelled as an orthotropic circular cylindrical shell described by means of the Novozhilov nonlinear shell theory. Flexible boundary conditions are considered to simulate connection with the remaining tissue. Nonlinear vibrations of the shell conveying pulsatile flow and subjected to pulsatile pressure are investigated taking into account the effects of the pulse-wave propagation. For the first time in literature, coupled fluid-structure Lagrange equations of motion for a non-material volume with wave propagation in case of pulsatile flow are developed. The fluid is modeled as a Newtonian inviscid pulsatile flow and it is formulated using a hybrid model based on the linear potential flow theory and considering the unsteady viscous effects obtained from the unsteady time-averaged Navier-Stokes equations. Contributions of pressure and velocity propagation are also considered in the pressure drop along the shell and in the pulsatile frictional traction on the internal wall in the axial direction. A numerical bifurcation analysis employs a refined reduced order model to investigate the dynamic behavior of a pressurized Dacron aortic graft conveying blood flow. A pulsatile time-dependent blood flow model is considered by applying the first harmonic of the physiological waveforms of velocity and pressure during the heart beating period. Geometrically nonlinear vibration response to pulsatile flow and transmural pulsatile pressure, considering the propagation of pressure and velocity changes inside the shell, is here presented via frequency-response curves, time histories, bifurcation
9. Turbulent Transitions in Optical Wave Propagation.
Science.gov (United States)
Pierangeli, D; Di Mei, F; Di Domenico, G; Agranat, A J; Conti, C; DelRe, E
2016-10-28
We report the direct observation of the onset of turbulence in propagating one-dimensional optical waves. The transition occurs as the disordered hosting material passes from being linear to one with extreme nonlinearity. As the response grows, increased wave interaction causes a modulational unstable quasihomogeneous flow to be superseded by a chaotic and spatially incoherent one. Statistical analysis of high-resolution wave behavior in the turbulent regime unveils the emergence of concomitant rogue waves. The transition, observed in a photorefractive ferroelectric crystal, introduces a new and rich experimental setting for the study of optical wave turbulence and information transport in conditions dominated by large fluctuations and extreme nonlinearity.
10. Nonlinear acoustic propagation in two-dimensional ducts
Science.gov (United States)
Nayfeh, A. H.; Tsai, M.-S.
1974-01-01
The method of multiple scales is used to obtain a second-order uniformly valid expansion for the nonlinear acoustic wave propagation in a two-dimensional duct whose walls are treated with a nonlinear acoustic material. The wave propagation in the duct is characterized by the unsteady nonlinear Euler equations. The results show that nonlinear effects tend to flatten and broaden the absorption versus frequency curve, in qualitative agreement with the experimental observations. Moreover, the effect of the gas nonlinearity increases with increasing sound frequency, whereas the effect of the material nonlinearity decreases with increasing sound frequency.
11. Characterization by a time-frequency method of classical waves propagation in one-dimensional lattice : effects of the dispersion and localized nonlinearities
CERN Document Server
Richoux, Olivier; Hardy, Jean
2009-01-01
This paper presents an application of time-frequency methods to characterize the dispersion of acoustic waves travelling in a one-dimensional periodic or disordered lattice made up of Helmholtz resonators connected to a cylindrical tube. These methods allow (1) to evaluate the velocity of the wave energy when the input signal is an acoustic pulse ; (2) to display the evolution of the spectral content of the transient signal ; (3) to show the role of the localized nonlinearities on the propagation .i.e the emergence of higher harmonics. The main result of this paper is that the time-frequency methods point out how the nonlinearities break the localization of the waves and/or the filter effects of the lattice.
12. Wave equations for pulse propagation
Energy Technology Data Exchange (ETDEWEB)
Shore, B.W.
1987-06-24
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation.
13. Nonlinear Wave Propagation
Science.gov (United States)
1983-12-30
see Current Contents June 7, 1982, Vol. 13, No. 23). *~ * ~ .q, .* -* ** ~ .~ ~ * . . *c-. -4- 22. Exact Linearization of a Painleve Transcendent, M.J...1977. 21. Asymptotic Solutions of the Korteweg-deVries Equation, M.J. Ablowitz and H. Segur, Studies in Applied Math., 57, pp. 13-44, 1977. 22. Exact ... Linearization of a Painleve Transcendent, M.J. Ablowitz and H. Segur, Phys. Rev. Lett., Vol. 38, No. 20, p. 1103, 1977. 23. Solitons and Rational
14. Nonlinear Wave Propagation
Science.gov (United States)
2015-05-07
applied to the solution obtained by the inverse scattering transform. Recently we have investigated the KdV equation with step-like data. We found that the...long- time-asymptotic solution of the KdV equation for general, step-like data is a single-phase DSW; this DSW is the largest possible DSW based on...the data breaks up in to numerous DSWs in an intermediate long time limit, eventually the solution tends to one DSW. 3 ACCOMPLISHMENTS/NEW FINDINGS
15. Analysis of Nonlinear Soil-Structure Interaction Effects on the response of Three-Dimensional Frame Structures using a One-Direction Three-ComponentWave Propagation Model
CERN Document Server
d'Avila, Maria Paola Santisi
2016-01-01
In this paper, a model of one-directional propagation of three-component seismic waves in a nonlinear multilayered soil profile is coupled with a multi-story multi-span frame model to consider, in a simple way, the soil-structure interaction modelled in a finite element scheme. Modeling the three-component wave propagation enables the effects of a soil multiaxial stress state to be taken into account. These reduce soil strength and increase nonlinear effects, compared with the axial stress state. The simultaneous propagation of three components allows the prediction of the incident direction of seismic loading at the ground surface and the analysis of the behavior of a frame structure shaken by a three-component earthquake. A parametric study is carried out to characterize the changes in the ground motion due to dynamic features of the structure, for different incident wavefield properties and soil nonlinear effects. A seismic response depending on parameters such as the frequency content of soil and structur...
16. Two-Dimensional Nonlinear Propagation of Ion Acoustic Waves through KPB and KP Equations in Weakly Relativistic Plasmas
Directory of Open Access Journals (Sweden)
M. G. Hafez
2016-01-01
Full Text Available Two-dimensional three-component plasma system consisting of nonextensive electrons, positrons, and relativistic thermal ions is considered. The well-known Kadomtsev-Petviashvili-Burgers and Kadomtsev-Petviashvili equations are derived to study the basic characteristics of small but finite amplitude ion acoustic waves of the plasmas by using the reductive perturbation method. The influences of positron concentration, electron-positron and ion-electron temperature ratios, strength of electron and positrons nonextensivity, and relativistic streaming factor on the propagation of ion acoustic waves in the plasmas are investigated. It is revealed that the electrostatic compressive and rarefactive ion acoustic waves are obtained for superthermal electrons and positrons, but only compressive ion acoustic waves are found and the potential profiles become steeper in case of subthermal positrons and electrons.
17. Wave equations for pulse propagation
Science.gov (United States)
Shore, B. W.
1987-06-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity.
18. PetClaw: Parallelization and Performance Optimization of a Python-Based Nonlinear Wave Propagation Solver Using PETSc
KAUST Repository
Alghamdi, Amal Mohammed
2012-04-01
Clawpack, a conservation laws package implemented in Fortran, and its Python-based version, PyClaw, are existing tools providing nonlinear wave propagation solvers that use state of the art finite volume methods. Simulations using those tools can have extensive computational requirements to provide accurate results. Therefore, a number of tools, such as BearClaw and MPIClaw, have been developed based on Clawpack to achieve significant speedup by exploiting parallel architectures. However, none of them has been shown to scale on a large number of cores. Furthermore, these tools, implemented in Fortran, achieve parallelization by inserting parallelization logic and MPI standard routines throughout the serial code in a non modular manner. Our contribution in this thesis research is three-fold. First, we demonstrate an advantageous use case of Python in implementing easy-to-use modular extensible scalable scientific software tools by developing an implementation of a parallelization framework, PetClaw, for PyClaw using the well-known Portable Extensible Toolkit for Scientific Computation, PETSc, through its Python wrapper petsc4py. Second, we demonstrate the possibility of getting acceptable Python code performance when compared to Fortran performance after introducing a number of serial optimizations to the Python code including integrating Clawpack Fortran kernels into PyClaw for low-level computationally intensive parts of the code. As a result of those optimizations, the Python overhead in PetClaw for a shallow water application is only 12 percent when compared to the corresponding Fortran Clawpack application. Third, we provide a demonstration of PetClaw scalability on up to the entirety of Shaheen; a 16-rack Blue Gene/P IBM supercomputer that comprises 65,536 cores and located at King Abdullah University of Science and Technology (KAUST). The PetClaw solver achieved above 0.98 weak scaling efficiency for an Euler application on the whole machine excluding the
19. Nonlinear propagation of Alfven waves driven by observed photospheric motions: Application to the coronal heating and spicule formation
CERN Document Server
Matsumoto, Takuma
2010-01-01
We have performed MHD simulations of Alfven wave propagation along an open flux tube in the solar atmosphere. In our numerical model, Alfven waves are generated by the photospheric granular motion. As the wave generator, we used a derived temporal spectrum of the photospheric granular motion from G-band movies of Hinode/SOT. It is shown that the total energy flux at the corona becomes larger and the transition region height becomes higher in the case when we use the observed spectrum rather than white/pink noise spectrum as the wave generator. This difference can be explained by the Alfven wave resonance between the photosphere and the transition region. After performing Fourier analysis on our numerical results, we have found that the region between the photosphere and the transition region becomes an Alfven wave resonant cavity. We have confirmed that there are at least three resonant frequencies, 1, 3 and 5 mHz, in our numerical model. Alfven wave resonance is one of the most effective mechanisms to explai...
20. TSUNAMI WAVE PROPAGATION ALONG WAVEGUIDES
Directory of Open Access Journals (Sweden)
Andrei G. Marchuk
2009-01-01
Full Text Available This is a study of tsunami wave propagation along the waveguide on a bottom ridge with flat sloping sides, using the wave rays method. During propagation along such waveguide the single tsunami wave transforms into a wave train. The expression for the guiding velocities of the fastest and slowest signals is defined. The tsunami wave behavior above the ocean bottom ridges, which have various model profiles, is investigated numerically with the help of finite difference method. Results of numerical experiments show that the highest waves are detected above a ridge with flat sloping sides. Examples of tsunami propagation along bottom ridges of the Pacific Ocean are presented.
1. Wave propagation in elastic solids
CERN Document Server
Achenbach, Jan
1984-01-01
The propagation of mechanical disturbances in solids is of interest in many branches of the physical scienses and engineering. This book aims to present an account of the theory of wave propagation in elastic solids. The material is arranged to present an exposition of the basic concepts of mechanical wave propagation within a one-dimensional setting and a discussion of formal aspects of elastodynamic theory in three dimensions, followed by chapters expounding on typical wave propagation phenomena, such as radiation, reflection, refraction, propagation in waveguides, and diffraction. The treat
2. Nonlinear propagation of dust-acoustic solitary waves in a dusty plasma with arbitrarily charged dust and trapped electrons
O Rahman; A A Mamun
2013-06-01
A theoretical investigation of dust-acoustic solitary waves in three-component unmagnetized dusty plasma consisting of trapped electrons, Maxwellian ions, and arbitrarily charged cold mobile dust was done. It has been found that, owing to the departure from the Maxwellian electron distribution to a vortex-like one, the dynamics of small but finite amplitude dust-acoustic (DA) waves is governed by a nonlinear equation of modified Korteweg–de Vries (mKdV) type (instead of KdV). The reductive perturbation method was employed to study the basic features (amplitude, width, speed, etc.) of DA solitary waves which are significantly modified by the presence of trapped electrons. The implications of our results in space and laboratory plasmas are briefly discussed.
3. Dispersive shock waves with nonlocal nonlinearity
CERN Document Server
Barsi, Christopher; Sun, Can; Fleischer, Jason W
2007-01-01
We consider dispersive optical shock waves in nonlocal nonlinear media. Experiments are performed using spatial beams in a thermal liquid cell, and results agree with a hydrodynamic theory of propagation.
4. Dispersive shock waves with nonlocal nonlinearity.
Science.gov (United States)
Barsi, Christopher; Wan, Wenjie; Sun, Can; Fleischer, Jason W
2007-10-15
We consider dispersive optical shock waves in nonlocal nonlinear media. Experiments are performed using spatial beams in a thermal liquid cell, and results agree with a hydrodynamic theory of propagation.
5. Wave propagation in electromagnetic media
CERN Document Server
Davis, Julian L
1990-01-01
This is the second work of a set of two volumes on the phenomena of wave propagation in nonreacting and reacting media. The first, entitled Wave Propagation in Solids and Fluids (published by Springer-Verlag in 1988), deals with wave phenomena in nonreacting media (solids and fluids). This book is concerned with wave propagation in reacting media-specifically, in electro magnetic materials. Since these volumes were designed to be relatively self contained, we have taken the liberty of adapting some of the pertinent material, especially in the theory of hyperbolic partial differential equations (concerned with electromagnetic wave propagation), variational methods, and Hamilton-Jacobi theory, to the phenomena of electromagnetic waves. The purpose of this volume is similar to that of the first, except that here we are dealing with electromagnetic waves. We attempt to present a clear and systematic account of the mathematical methods of wave phenomena in electromagnetic materials that will be readily accessi...
6. Modeling seismic wave propagation and amplification in 1D/2D/3D linear and nonlinear unbounded media
CERN Document Server
Semblat, Jean-François
2011-01-01
To analyze seismic wave propagation in geological structures, it is possible to consider various numerical approaches: the finite difference method, the spectral element method, the boundary element method, the finite element method, the finite volume method, etc. All these methods have various advantages and drawbacks. The amplification of seismic waves in surface soil layers is mainly due to the velocity contrast between these layers and, possibly, to topographic effects around crests and hills. The influence of the geometry of alluvial basins on the amplification process is also know to be large. Nevertheless, strong heterogeneities and complex geometries are not easy to take into account with all numerical methods. 2D/3D models are needed in many situations and the efficiency/accuracy of the numerical methods in such cases is in question. Furthermore, the radiation conditions at infinity are not easy to handle with finite differences or finite/spectral elements whereas it is explicitely accounted in the B...
7. Enhancing propagation characteristics of truncated localized waves in silica
KAUST Repository
Salem, Mohamed
2011-07-01
The spectral characteristics of truncated Localized Waves propagating in dispersive silica are analyzed. Numerical experiments show that the immunity of the truncated Localized Waves propagating in dispersive silica to decay and distortion is enhanced as the non-linearity of the relation between the transverse spatial spectral components and the wave vector gets stronger, in contrast to free-space propagating waves, which suffer from early decay and distortion. © 2011 IEEE.
8. Effects of oblique wave propagation on the nonlinear plasma resonance in the two-dimensional channel of the Dyakonov-Shur detector
Science.gov (United States)
Rupper, Greg; Rudin, Sergey; Crowne, Frank J.
2012-12-01
In the Dyakonov-Shur terahertz detector the conduction channel of a heterostructure High Electron Mobility Transistor (HEMT) is used as a plasma wave resonator for density oscillations in electron gas. Nonlinearities in the plasma wave propagation lead to a constant source-to-drain voltage, providing the detector output. In this paper, we start with the quasi-classical Boltzmann equation and derive the hydrodynamic model with temperature dependent transport coefficients for a two-dimensional viscous flow. This derivation allows us to obtain the parameters for the hydrodynamic model from the band-structure of the HEMT channel. The treatment here also includes the energy balance equation into the analysis. By numerical solution of the hydrodynamic equations with a non-zero boundary current we evaluate the detector response function and obtain the temperature dependence of the plasma resonance. The present treatment extends the theory of Dyakonov-Shur plasma resonator and detector to account for the temperature dependence of viscosity, the effects of oblique wave propagation on detector response, and effects of boundary current in two-dimensional flow on quality of the plasma resonance. The numerical results are given for a GaN channel. We also investigated a stability of source to drain flow and formation of shock waves.
9. Wave propagation in spatially modulated tubes
CERN Document Server
Ziepke, A; Engel, H
2016-01-01
We investigate wave propagation in rotationally symmetric tubes with a periodic spatial modulation of cross section. Using an asymptotic perturbation analysis, the governing quasi two-dimensional reaction-diffusion equation can be reduced into a one-dimensional reaction-diffusion-advection equation. Assuming a weak perturbation by the advection term and using projection method, in a second step, an equation of motion for traveling waves within such tubes can be derived. Both methods predict properly the nonlinear dependence of the propagation velocity on the ratio of the modulation period of the geometry to the intrinsic width of the front, or pulse. As a main feature, we can observe finite intervals of propagation failure of waves induced by the tube's modulation. In addition, using the Fick-Jacobs approach for the highly diffusive limit we show that wave velocities within tubes are governed by an effective diffusion coefficient. Furthermore, we discuss the effects of a single bottleneck on the period of pul...
10. Finite volume schemes for dispersive wave propagation and runup
Science.gov (United States)
Dutykh, Denys; Katsaounis, Theodoros; Mitsotakis, Dimitrios
2011-04-01
Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.
11. Finite volume schemes for dispersive wave propagation and runup
CERN Document Server
Dutykh, Denys; Mitsotakis, Dimitrios
2010-01-01
Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.
12. SIMULATION OF FORWARD AND BACKWARD WAVES EVOLUTION OF FEW-CYCLE PULSES PROPAGATING IN AN OPTICAL WAVEGUIDE WITH DISPERSION AND CUBIC NONLINEARITY OF ELECTRONIC AND ELECTRONIC-VIBRATION NATURE
Directory of Open Access Journals (Sweden)
L. S. Konev
2015-09-01
Full Text Available Numerical method for calculation of forward and backward waves of intense few-cycle laser pulses propagating in an optical waveguide with dispersion and cubic nonlinearity of electronic and electronic-vibration nature is described. Simulations made with the implemented algorithm show that accounting for Raman nonlinearity does not lead to qualitative changes in behavior of the backward wave. Speaking about quantitative changes, the increase of efficiency of energy transfer from the forward wave to the backward wave is observed. Presented method can be also used to simulate interaction of counterpropagating pulses.
13. Solitons and Weakly Nonlinear Waves in Plasmas
DEFF Research Database (Denmark)
Pécseli, Hans
1985-01-01
Theoretical descriptions of solitons and weakly nonlinear waves propagating in plasma media are reviewed, with particular attention to the Korteweg-de Vries (KDV) equation and the Nonlinear Schrödinger equation (NLS). The modifications of these basic equations due to the effects of resonant...
14. Analytic descriptions of cylindrical electromagnetic waves in a nonlinear medium.
Science.gov (United States)
Xiong, Hao; Si, Liu-Gang; Yang, Xiaoxue; Wu, Ying
2015-06-15
A simple but highly efficient approach for dealing with the problem of cylindrical electromagnetic waves propagation in a nonlinear medium is proposed based on an exact solution proposed recently. We derive an analytical explicit formula, which exhibiting rich interesting nonlinear effects, to describe the propagation of any amount of cylindrical electromagnetic waves in a nonlinear medium. The results obtained by using the present method are accurately concordant with the results of using traditional coupled-wave equations. As an example of application, we discuss how a third wave affects the sum- and difference-frequency generation of two waves propagation in the nonlinear medium.
15. Analytic descriptions of cylindrical electromagnetic waves in a nonlinear medium
Science.gov (United States)
Xiong, Hao; Si, Liu-Gang; Yang, Xiaoxue; Wu, Ying
2015-01-01
A simple but highly efficient approach for dealing with the problem of cylindrical electromagnetic waves propagation in a nonlinear medium is proposed based on an exact solution proposed recently. We derive an analytical explicit formula, which exhibiting rich interesting nonlinear effects, to describe the propagation of any amount of cylindrical electromagnetic waves in a nonlinear medium. The results obtained by using the present method are accurately concordant with the results of using traditional coupled-wave equations. As an example of application, we discuss how a third wave affects the sum- and difference-frequency generation of two waves propagation in the nonlinear medium. PMID:26073066
16. Nonlinear interactions between gravity waves and tides
Institute of Scientific and Technical Information of China (English)
LIU Xiao; XU JiYao; MA RuiPing
2007-01-01
In this study, we present the nonlinear interactions between gravity waves (GWs) and tides by using the 2D numerical model for the nonlinear propagation of GWs in the compressible atmosphere. During the propagation in the tidal background, GWs become instable in three regions, that is z = 75-85 km, z =90-110 km and z= 115-130 km. The vertical wavelength firstly varies gradually from the initial 12 km to 27 km. Then the newly generated longer waves are gradually compressed. The longer and shorter waves occur in the regions where GWs propagate in the reverse and the same direction of the horizontal mean wind respectively. In addition, GWs can propagate above the main breaking region (90-110 km). During GWs propagation, not only the mean wind is accelerated, but also the amplitude of tide is amplified. Especially, after GWs become instable, this amplified effect to the tidal amplitude is much obvious.
17. Nonlinear interactions between gravity waves and tides
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this study, we present the nonlinear interactions between gravity waves (GWs) and tides by using the 2D numerical model for the nonlinear propagation of GWs in the compressible atmosphere. During the propagation in the tidal background, GWs become instable in three regions, that is z = 75―85 km, z = 90―110 km and z = 115―130 km. The vertical wavelength firstly varies gradually from the initial 12 km to 27 km. Then the newly generated longer waves are gradually compressed. The longer and shorter waves occur in the regions where GWs propagate in the reverse and the same direction of the hori-zontal mean wind respectively. In addition, GWs can propagate above the main breaking region (90—110 km). During GWs propagation, not only the mean wind is accelerated, but also the amplitude of tide is amplified. Especially, after GWs become instable, this amplified effect to the tidal amplitude is much obvious.
18. Control methods for localization of nonlinear waves
Science.gov (United States)
Porubov, Alexey; Andrievsky, Boris
2017-03-01
A general form of a distributed feedback control algorithm based on the speed-gradient method is developed. The goal of the control is to achieve nonlinear wave localization. It is shown by example of the sine-Gordon equation that the generation and further stable propagation of a localized wave solution of a single nonlinear partial differential equation may be obtained independently of the initial conditions. The developed algorithm is extended to coupled nonlinear partial differential equations to obtain consistent localized wave solutions at rather arbitrary initial conditions. This article is part of the themed issue 'Horizons of cybernetical physics'.
19. Wave propagation in ballistic gelatine.
Science.gov (United States)
Naarayan, Srinivasan S; Subhash, Ghatu
2017-01-23
Wave propagation characteristics in long cylindrical specimens of ballistic gelatine have been investigated using a high speed digital camera and hyper elastic constitutive models. The induced transient deformation is modelled with strain rate dependent Mooney-Rivlin parameters which are determined by modelling the stress-strain response of gelatine at a range of strain rates. The varying velocity of wave propagation through the gelatine cylinder is derived as a function of prestress or stretch in the gelatine specimen. A finite element analysis is conducted using the above constitutive model by suitably defining the impulse imparted by the polymer bar into the gelatine specimen. The model results are found to capture the experimentally observed wave propagation characteristics in gelatine effectively.
20. The generation and propagation of nonlinear waves in a reservoir; Geracao e propagacao de ondas nao-lineares em um reservatorio
Energy Technology Data Exchange (ETDEWEB)
Moreira, Roger Matsumoto; Mendes, Andre Avelino de Oliveira; Bacchi, Raphael David Aquilino [Universidade Federal Fluminense (UFF), Niteroi, RJ (Brazil). Escola de Engenharia. Lab. de Dinamica dos Fluidos Computacional (LabCFD)], e-mail: [email protected], e-mail: [email protected], e-mail: [email protected]
2006-07-01
The present work aims to model numerically the generation and propagation of waves in a reservoir, represented by a two-dimensional impermeable box, with a flat horizontal bottom and two vertical walls. The horizontal or vertical harmonic motion is imposed at the container, which is partially filled with water, with two possible initial conditions for the free surface: still water or a stationary sinusoidal wave. Two numerical methods are employed in the solution of the boundary value problem. The first is based on solving an integral equation that arises from Cauchy's integral theorem for functions of a complex variable. The transient nonlinear free surface flow is simulated using a boundary integral method. Numerical results are validated by comparing them with classical analytical solutions. The second method uses the commercial code ANSYS CFX with its homogeneous free surface model. In this case, results are compared with experiments done by Bredmose et al. (2003). In both models, interesting features at the free surface are obtained and discussed. (author)
1. Understanding of Materials State and its Degradation using Non-Linear Ultrasound Approaches for Lamb Wave Propagation
Science.gov (United States)
2015-05-31
was increased as the dislocation motion was impeded by the fine MX type of precipitates and this resistance was increased due to increase in...Code A: Approved for public release, distribution is unlimited. precipitate -matrix coherency strains generated during different tempering temperatures...linkage to form micro-cracks, and the propagation of micro-cracks until failure. During this process, the precipitation of the second phase particles
2. Nonlinear Water Waves
CERN Document Server
2016-01-01
This volume brings together four lecture courses on modern aspects of water waves. The intention, through the lectures, is to present quite a range of mathematical ideas, primarily to show what is possible and what, currently, is of particular interest. Water waves of large amplitude can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. Taking advantage of insights from physical observation, experimental evidence and numerical simulations, classical and modern mathematical approaches can be used to gain insight into their dynamics. The book presents several avenues and offers a wide range of material of current interest. Due to the interdisciplinary nature of the subject, the book should be of interest to mathematicians (pure and applied), physicists and engineers. The lectures provide a useful source for those who want to begin to investigate how mathematics can be used to improve our understanding of water wave phenomena. In addition, some of the...
3. Wave propagation in spatially modulated tubes.
Science.gov (United States)
Ziepke, A; Martens, S; Engel, H
2016-09-07
We investigate wave propagation in rotationally symmetric tubes with a periodic spatial modulation of cross section. Using an asymptotic perturbation analysis, the governing quasi-two-dimensional reaction-diffusion equation can be reduced into a one-dimensional reaction-diffusion-advection equation. Assuming a weak perturbation by the advection term and using projection method, in a second step, an equation of motion for traveling waves within such tubes can be derived. Both methods predict properly the nonlinear dependence of the propagation velocity on the ratio of the modulation period of the geometry to the intrinsic width of the front, or pulse. As a main feature, we observe finite intervals of propagation failure of waves induced by the tube's modulation and derive an analytically tractable condition for their occurrence. For the highly diffusive limit, using the Fick-Jacobs approach, we show that wave velocities within modulated tubes are governed by an effective diffusion coefficient. Furthermore, we discuss the effects of a single bottleneck on the period of pulse trains. We observe period changes by integer fractions dependent on the bottleneck width and the period of the entering pulse train.
4. Wave propagation in spatially modulated tubes
Science.gov (United States)
Ziepke, A.; Martens, S.; Engel, H.
2016-09-01
We investigate wave propagation in rotationally symmetric tubes with a periodic spatial modulation of cross section. Using an asymptotic perturbation analysis, the governing quasi-two-dimensional reaction-diffusion equation can be reduced into a one-dimensional reaction-diffusion-advection equation. Assuming a weak perturbation by the advection term and using projection method, in a second step, an equation of motion for traveling waves within such tubes can be derived. Both methods predict properly the nonlinear dependence of the propagation velocity on the ratio of the modulation period of the geometry to the intrinsic width of the front, or pulse. As a main feature, we observe finite intervals of propagation failure of waves induced by the tube's modulation and derive an analytically tractable condition for their occurrence. For the highly diffusive limit, using the Fick-Jacobs approach, we show that wave velocities within modulated tubes are governed by an effective diffusion coefficient. Furthermore, we discuss the effects of a single bottleneck on the period of pulse trains. We observe period changes by integer fractions dependent on the bottleneck width and the period of the entering pulse train.
5. Nonlinear surface waves in photonic hypercrystals
Science.gov (United States)
Ali, Munazza Zulfiqar
2017-08-01
Photonic crystals and hyperbolic metamaterials are merged to give the concept of photonic hypercrystals. It combines the properties of its two constituents to give rise to novel phenomena. Here the propagation of Transverse Magnetic waves at the interface between a nonlinear dielectric material and a photonic hypercrystal is studied and the corresponding dispersion relation is derived using the uniaxial parallel approximation. Both dielectric and metallic photonic hypercrystals are studied and it is found that nonlinearity limits the infinite divergence of wave vectors of the surface waves. These states exist in the frequency region where the linear surface waves do not exist. It is also shown that the nonlinearity can be used to engineer the group velocity of the resulting surface wave.
6. Solitary waves on nonlinear elastic rods. II
DEFF Research Database (Denmark)
Sørensen, Mads Peter; Christiansen, Peter Leth; Lomdahl, P. S.
1987-01-01
In continuation of an earlier study of propagation of solitary waves on nonlinear elastic rods, numerical investigations of blowup, reflection, and fission at continuous and discontinuous variation of the cross section for the rod and reflection at the end of the rod are presented. The results...
7. Linear and nonlinear obliquely propagating ion-acoustic waves in magnetized negative ion plasma with non-thermal electrons
Science.gov (United States)
Mishra, M. K.; Jain, S. K.; Jain
2013-10-01
Ion-acoustic solitons in magnetized low-β plasma consisting of warm adiabatic positive and negative ions and non-thermal electrons have been studied. The reductive perturbation method is used to derive the Korteweg-de Vries (KdV) equation for the system, which admits an obliquely propagating soliton solution. It is found that due to the presence of finite ion temperature there exist two modes of propagation, namely fast and slow ion-acoustic modes. In the case of slow-mode if the ratio of temperature to mass of positive ion species is lower (higher) than the negative ion species, then there exist compressive (rarefactive) ion-acoustic solitons. It is also found that in the case of slow mode, on increasing the non-thermal parameter (γ) the amplitude of the compressive (rarefactive) soliton decreases (increases). In fast ion-acoustic mode the nature and characteristics of solitons depend on negative ion concentration. Numerical investigation in case of fast mode reveals that on increasing γ, the amplitude of compressive (rarefactive) soliton increases (decreases). The width of solitons increases with an increase in non-thermal parameters in both the modes for compressive as well as rarefactive solitons. There exists a value of critical negative ion concentration (α c ), at which both compressive and rarefactive ion-acoustic solitons appear as described by modified KdV soliton. The value of α c decreases with increase in γ.
8. Wave propagation and group velocity
CERN Document Server
Brillouin, Léon
1960-01-01
Wave Propagation and Group Velocity contains papers on group velocity which were published during the First World War and are missing in many libraries. It introduces three different definitions of velocities: the group velocity of Lord Rayleigh, the signal velocity of Sommerfeld, and the velocity of energy transfer, which yields the rate of energy flow through a continuous wave and is strongly related to the characteristic impedance. These three velocities are identical for nonabsorbing media, but they differ considerably in an absorption band. Some examples are discussed in the last chapter
9. Solitary wave propagation through two-dimensional treelike structures.
Science.gov (United States)
Falls, William J; Sen, Surajit
2014-02-01
It is well known that a velocity perturbation can travel through a mass spring chain with strongly nonlinear interactions as a solitary and antisolitary wave pair. In recent years, nonlinear wave propagation in 2D structures have also been explored. Here we first consider the propagation of such a velocity perturbation for cases where the system has a 2D "Y"-shaped structure. Here each of the three pieces that make up the "Y" are made of a small mass spring chain. In addition, we consider a case where multiple "Y"-shaped structures are used to generate a "tree." We explore the early time dynamical behavior associated with the propagation of a velocity perturbation initiated at the trunk and at the extremities for both cases. We are looking for the energy transmission properties from one branch to another of these "Y"-shaped structures. Our dynamical simulations suggest the following broad observations: (i) for strongly nonlinear interactions, mechanical energy propagation resembles pulse propagation with the energy propagation being dispersive in the linear case; (ii) for strong nonlinear interactions, the tree-like structure acts as an energy gate showing preference for large perturbations in the system while the behavior of the linear case shows no such preference, thereby suggesting that such structures can possibly act as switches that activate at sufficiently high energies. The study aspires to develop insights into the nature of nonlinear wave propagation through a network of linear chains.
10. Wave propagation on microstate geometries
CERN Document Server
Keir, Joseph
2016-01-01
Supersymmetric microstate geometries were recently conjectured to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In this paper, we give a thorough mathematical treatment of the linear wave equation on both two and three charge supersymmetric microstate geometries, finding a number of surprising results. In both cases we prove that solutions to the wave equation have uniformly bounded local energy, despite the fact that three charge microstates possess an ergoregion; these geometries therefore avoid Friedman's "ergosphere instability". In fact, in the three charge case we are able to construct solutions to the wave equation with local energy that neither grows nor decays, although this data must have nontrivial dependence on the Kaluza-Klein coordinate. In the two charge case we construct quasimodes and use these to bound the uniform decay rate, showing that the only possible uniform dec...
11. Memory Effects on Nonlinear Temperature and Pressure Wave Propagation in the Boundary between Two Fluid-Saturated Porous Rocks
Directory of Open Access Journals (Sweden)
R. Garra
2015-01-01
Full Text Available The evolution of strong transients of temperature and pressure in two adjacent fluid-saturated porous rocks is described by a Burgers equation in an early model of Natale and Salusti (1996. We here consider the effect of a realistic intermediate region between the two media and infer how transient processes can also happen, such as chemical reactions, diffusion of fine particles, and filter cake formations. This suggests enlarging our analysis and taking into account not only punctual quantities but also “time averaged” quantities. These boundary effects are here analyzed by using a “memory formalism”; that is, we replace the ordinary punctual time-derivatives with Caputo fractional time-derivatives. We therefore obtain a nonlinear fractional model, whose explicit solution is shown, and finally discuss its geological importance.
12. Polarization shaping for control of nonlinear propagation
CERN Document Server
Bouchard, Frédéric; Yao, Alison M; Travis, Christopher; De Leon, Israel; Rubano, Andrea; Karimi, Ebrahim; Oppo, Gian-Luca; Boyd, Robert W
2016-01-01
We study the nonlinear optical propagation of two different classes of space-varying polarized light beams -- radially symmetric vector beams and Poincar\\'e beams with lemon and star topologies -- in a rubidium vapour cell. Unlike Laguerre-Gauss and other types of beams that experience modulational instabilities, we observe that their propagation is not marked by beam breakup while still exhibiting traits such as nonlinear confinement and self-focusing. Our results suggest that by tailoring the spatial structure of the polarization, the effects of nonlinear propagation can be effectively controlled. These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties.
13. Polarization Shaping for Control of Nonlinear Propagation.
Science.gov (United States)
Bouchard, Frédéric; Larocque, Hugo; Yao, Alison M; Travis, Christopher; De Leon, Israel; Rubano, Andrea; Karimi, Ebrahim; Oppo, Gian-Luca; Boyd, Robert W
2016-12-02
We study the nonlinear optical propagation of two different classes of light beams with space-varying polarization-radially symmetric vector beams and Poincaré beams with lemon and star topologies-in a rubidium vapor cell. Unlike Laguerre-Gauss and other types of beams that quickly experience instabilities, we observe that their propagation is not marked by beam breakup while still exhibiting traits such as nonlinear confinement and self-focusing. Our results suggest that, by tailoring the spatial structure of the polarization, the effects of nonlinear propagation can be effectively controlled. These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties.
14. Nonlinear dynamics of hydrostatic internal gravity waves
Energy Technology Data Exchange (ETDEWEB)
Stechmann, Samuel N.; Majda, Andrew J. [New York University, Courant Institute of Mathematical Sciences, NY (United States); Khouider, Boualem [University of Victoria, Department of Mathematics and Statistics, Victoria, BC (Canada)
2008-11-15
Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden-Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are
15. On the propagation of truncated localized waves in dispersive silica
KAUST Repository
Salem, Mohamed
2010-01-01
Propagation characteristics of truncated Localized Waves propagating in dispersive silica and free space are numerically analyzed. It is shown that those characteristics are affected by the changes in the relation between the transverse spatial spectral components and the wave vector. Numerical experiments demonstrate that as the non-linearity of this relation gets stronger, the pulses propagating in silica become more immune to decay and distortion whereas the pulses propagating in free-space suffer from early decay and distortion. © 2010 Optical Society of America.
16. The Nonlinear Talbot Effect of Rogue Waves
CERN Document Server
Zhang, Yiqi; Zheng, Huaibin; Chen, Haixia; Li, Changbiao; Song, Jianping; Zhang, Yanpeng
2014-01-01
Akhmediev and Kuznetsov-Ma breathers are rogue wave solutions of the nonlinear Schr\\"odinger equation (NLSE). Talbot effect (TE) is an image recurrence phenomenon in the diffraction of light waves. We report the nonlinear TE of rogue waves in a cubic medium. It is different from the linear TE, in that the wave propagates in a NL medium and is an eigenmode of NLSE. Periodic rogue waves impinging on a NL medium exhibit recurrent behavior, but only at the TE length and at the half-TE length with a \\pi-phase shift; the fractional TE is absent. The NL TE is the result of the NL interference of the lobes of rogue wave breathers. This interaction is related to the transverse period and intensity of breathers, in that the bigger the period and the higher the intensity, the shorter the TE length.
17. Nonlinear random optical waves: Integrable turbulence, rogue waves and intermittency
Science.gov (United States)
Randoux, Stéphane; Walczak, Pierre; Onorato, Miguel; Suret, Pierre
2016-10-01
We examine the general question of statistical changes experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In our study that enters within the framework of integrable turbulence, we specifically focus on optical fiber systems accurately described by the integrable one-dimensional nonlinear Schrödinger equation. We consider random complex fields having a Gaussian statistics and an infinite extension at initial stage. We use numerical simulations with periodic boundary conditions and optical fiber experiments to investigate spectral and statistical changes experienced by nonlinear waves in focusing and in defocusing propagation regimes. As a result of nonlinear propagation, the power spectrum of the random wave broadens and takes exponential wings both in focusing and in defocusing regimes. Heavy-tailed deviations from Gaussian statistics are observed in focusing regime while low-tailed deviations from Gaussian statistics are observed in defocusing regime. After some transient evolution, the wave system is found to exhibit a statistically stationary state in which neither the probability density function of the wave field nor the spectrum changes with the evolution variable. Separating fluctuations of small scale from fluctuations of large scale both in focusing and defocusing regimes, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian.
18. Boundary control of long waves in nonlinear dispersive systems
DEFF Research Database (Denmark)
Hasan, Agus; Foss, Bjarne; Aamo, Ole Morten
2011-01-01
Unidirectional propagation of long waves in nonlinear dispersive systems may be modeled by the Benjamin-Bona-Mahony-Burgers equation, a third order partial differential equation incorporating linear dissipative and dispersive terms, as well as a term covering nonlinear wave phenomena. For higher...... orders of the nonlinearity, the equation may have unstable solitary wave solutions. Although it is a one dimensional problem, achieving a global result for this equation is not trivial due to the nonlinearity and the mixed partial derivative. In this paper, two sets of nonlinear boundary control laws...... that achieve global exponential stability and semi-global exponential stability are derived for both linear and nonlinear cases....
19. Propagation of sound waves in ducts
DEFF Research Database (Denmark)
Jacobsen, Finn
2000-01-01
Plane wave propagation in ducts with rigid walls, radiation from ducts, classical four-pole theory for composite duct systems, and three-dimentional waves in wave guides of various cross-sectional shape are described....
20. Propagation of sound waves in ducts
DEFF Research Database (Denmark)
Jacobsen, Finn
2000-01-01
Plane wave propagation in ducts with rigid walls, radiation from ducts, classical four-pole theory for composite duct systems, and three-dimentional waves in wave guides of various cross-sectional shape are described.......Plane wave propagation in ducts with rigid walls, radiation from ducts, classical four-pole theory for composite duct systems, and three-dimentional waves in wave guides of various cross-sectional shape are described....
1. Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation
Science.gov (United States)
Miller, Steven A. E.
2015-01-01
An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier-Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle.
2. Modeling of nonlinear propagation in fiber tapers
DEFF Research Database (Denmark)
Lægsgaard, Jesper
2012-01-01
A full-vectorial nonlinear propagation equation for short pulses in tapered optical fibers is developed. Specific emphasis is placed on the importance of the field normalization convention for the structure of the equations, and the interpretation of the resulting field amplitudes. Different...... numerical schemes for interpolation of fiber parameters along the taper are discussed and tested in numerical simulations on soliton propagation and generation of continuum radiation in short photonic-crystal fiber tapers....
3. Extended models of nonlinear waves in liquid with gas bubbles
CERN Document Server
Kudryashov, Nikolay A
2016-01-01
In this work we generalize the models for nonlinear waves in a gas--liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for nonlinear waves. We also take into consideration high order terms with respect to the small parameter. Two new nonlinear differential equations are derived for long weakly nonlinear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above mentioned physical properties. One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on nonlinear waves propagation. The other equation is the perturbation of the Burgers--Korteweg--de Vries equation and corresponds to main influence of dispersion on nonlinear waves propagation.
4. Wave propagation in thermoelastic saturated porous medium
M D Sharma
2008-12-01
Biot ’s theory for wave propagation in saturated porous solid is modified to study the propagation of thermoelastic waves in poroelastic medium. Propagation of plane harmonic waves is considered in isotropic poroelastic medium. Relations are derived among the wave-induced temperature in the medium and the displacements of fluid and solid particles. Christoffel equations obtained are modified with the thermal as well as thermoelastic coupling parameters. These equations explain the existence and propagation of four waves in the medium. Three of the waves are attenuating longitudinal waves and one is a non-attenuating transverse wave. Thermal properties of the medium have no effect on the transverse wave. The velocities and attenuation of the longitudinal waves are computed for a numerical model of liquid-saturated sandstone. Their variations with thermal as well as poroelastic parameters are exhibited through numerical examples.
5. Nonlinear Landau damping of Alfven waves.
Science.gov (United States)
Hollweg, J. V.
1971-01-01
Demonstration that large-amplitude linearly or elliptically polarized Alfven waves propagating parallel to the average magnetic field can be dissipated by nonlinear Landau damping. The damping is due to the longitudinal electric field associated with the ion sound wave which is driven (in second order) by the Alfven wave. The damping rate can be large even in a cold plasma (beta much less than 1, but not zero), and the mechanism proposed may be the dominant one in many plasmas of astrophysical interest.
6. Optical rogue waves and soliton turbulence in nonlinear fibre optics
DEFF Research Database (Denmark)
Genty, G.; Dudley, J. M.; de Sterke, C. M.
2009-01-01
We examine optical rogue wave generation in nonlinear fibre propagation in terms of soliton turbulence. We show that higher-order dispersion is sufficient to generate localized rogue soliton structures, and Raman scattering effects are not required.......We examine optical rogue wave generation in nonlinear fibre propagation in terms of soliton turbulence. We show that higher-order dispersion is sufficient to generate localized rogue soliton structures, and Raman scattering effects are not required....
7. Nonlinear Interaction of Waves in Geomaterials
Science.gov (United States)
Ostrovsky, L. A.
2009-05-01
Progress of 1990s - 2000s in studying vibroacoustic nonlinearities in geomaterials is largely related to experiments in resonance samples of rock and soils. It is now a common knowledge that many such materials are very strongly nonlinear, and they are characterized by hysteresis in the dependence between the stress and strain tensors, as well as by nonlinear relaxation ("slow time"). Elastic wave propagation in such media has many peculiarities; for example, third harmonic amplitude is a quadratic (not cubic as in classical solids) function of the main harmonic amplitude, and average wave velocity is linearly (not quadratically as usual) dependent on amplitude. The mechanisms of these peculiarities are related to complex structure of a material typically consisting of two phases: a hard matrix and relatively soft inclusions such as microcracks and grain contacts. Although most informative experimental results have been obtained in rock in the form of resonant bars, few theoretical models are yet available to describe and calculate waves interacting in such samples. In this presentation, a brief overview of structural vibroacoustic nonlinearities in rock is given first. Then, a simple but rather general approach to the description of wave interaction in solid resonators is developed based on accounting for resonance nonlinear perturbations which are cumulating from period to period. In particular, the similarity and the differences between traveling waves and counter-propagating waves are analyzed for materials with different stress-strain dependences. These data can be used for solving an inverse problem, i.e. characterizing nonlinear properties of a geomaterial by its measured vibroacoustic parameters. References: 1. L. Ostrovsky and P. Johnson, Riv. Nuovo Chimento, v. 24, 1-46, 2007 (a review); 2. L. Ostrovsky, J. Acoust. Soc. Amer., v. 116, 3348-3353, 2004.
8. NONLINEAR MHD WAVES IN A PROMINENCE FOOT
Energy Technology Data Exchange (ETDEWEB)
Ofman, L. [Catholic University of America, Washington, DC 20064 (United States); Knizhnik, K.; Kucera, T. [NASA Goddard Space Flight Center, Code 671, Greenbelt, MD 20771 (United States); Schmieder, B. [LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris-Diderot, Sorbonne Paris Cit, 5 place Jules Janssen, F-92195 Meudon (France)
2015-11-10
We study nonlinear waves in a prominence foot using a 2.5D MHD model motivated by recent high-resolution observations with Hinode/Solar Optical Telescope in Ca ii emission of a prominence on 2012 October 10 showing highly dynamic small-scale motions in the prominence material. Observations of Hα intensities and of Doppler shifts show similar propagating fluctuations. However, the optically thick nature of the emission lines inhibits a unique quantitative interpretation in terms of density. Nevertheless, we find evidence of nonlinear wave activity in the prominence foot by examining the relative magnitude of the fluctuation intensity (δI/I ∼ δn/n). The waves are evident as significant density fluctuations that vary with height and apparently travel upward from the chromosphere into the prominence material with quasi-periodic fluctuations with a typical period in the range of 5–11 minutes and wavelengths <2000 km. Recent Doppler shift observations show the transverse displacement of the propagating waves. The magnetic field was measured with the THEMIS instrument and was found to be 5–14 G. For the typical prominence density the corresponding fast magnetosonic speed is ∼20 km s{sup −1}, in qualitative agreement with the propagation speed of the detected waves. The 2.5D MHD numerical model is constrained with the typical parameters of the prominence waves seen in observations. Our numerical results reproduce the nonlinear fast magnetosonic waves and provide strong support for the presence of these waves in the prominence foot. We also explore gravitational MHD oscillations of the heavy prominence foot material supported by dipped magnetic field structure.
9. Analysis of Blast Wave Propagation Inside Tunnel
Institute of Scientific and Technical Information of China (English)
LIU Jingbo; YAN Qiushi; WU Jun
2008-01-01
The explosion inside tunnel would generate blast wave which transmits through the longi tudinal tunnel.Because of the close-in effects of the tunnel and the reflection by the confining tunnel structure,blast wave propagation inside tunnel is distinguished from that in air.When the explosion happens inside tunnel,the overpressure peak is higher than that of explosion happening in air.The continuance time of the biast wave also becomes longer.With the help of the numerical simu lation finite element software LS-DYNA.a three-dimensional nonlinear dynamic simulation analysis for an explosion experiment inside tunnel was carried out.LS-DYNA is a fully integrated analysis program specifically designed for nonlinear dynamics and large strain problems.Compared with the experimental results.the simulation results have made the material parameters of numerical simulation model available.By using the model and the same material parameters,many results were adopted by calculating the model under different TNT explosion dynamites.Then the method of dimensional analysis was Used for the Simulation resufts.AS Overpressures of the explosion biast wave are the governing factor in fhe tunnel responses.a formula for the explosion biast wave overpressure at a certain distance from the detonation center point inside the tunnel was de rived by using the dimensional analysis theory.By cornparing the results computed by the fromula with experimental results which were obtained before.the formula was proved to be very applicable at some instance.The research may be helpful to estimate rapidly the effect of internal explosion of tunnel on the structure.
10. Nonlinear waves in waveguides with stratification
CERN Document Server
Leble, Sergei B
1991-01-01
S.B. Leble's book deals with nonlinear waves and their propagation in metallic and dielectric waveguides and media with stratification. The underlying nonlinear evolution equations (NEEs) are derived giving also their solutions for specific situations. The reader will find new elements to the traditional approach. Various dispersion and relaxation laws for different guides are considered as well as the explicit form of projection operators, NEEs, quasi-solitons and of Darboux transforms. Special points relate to: 1. the development of a universal asymptotic method of deriving NEEs for guide propagation; 2. applications to the cases of stratified liquids, gases, solids and plasmas with various nonlinearities and dispersion laws; 3. connections between the basic problem and soliton- like solutions of the corresponding NEEs; 4. discussion of details of simple solutions in higher- order nonsingular perturbation theory.
11. Waves and Structures in Nonlinear Nondispersive Media General Theory and Applications to Nonlinear Acoustics
CERN Document Server
Gurbatov, S N; Saichev, A I
2012-01-01
"Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics” is devoted completely to nonlinear structures. The general theory is given here in parallel with mathematical models. Many concrete examples illustrate the general analysis of Part I. Part II is devoted to applications to nonlinear acoustics, including specific nonlinear models and exact solutions, physical mechanisms of nonlinearity, sawtooth-shaped wave propagation, self-action phenomena, nonlinear resonances and engineering application (medicine, nondestructive testing, geophysics, etc.). This book is designed for graduate and postgraduate students studying the theory of nonlinear waves of various physical nature. It may also be useful as a handbook for engineers and researchers who encounter the necessity of taking nonlinear wave effects into account of their work. Dr. Gurbatov S.N. is the head of Department, and Vice Rector for Research of Nizhny Novgorod State University. Dr. Rudenko O.V. is...
12. Observations of Obliquely Propagating Electron Bernstein Waves
DEFF Research Database (Denmark)
Armstrong, R. J.; Juul Rasmussen, Jens; Stenzel, R. L.
1981-01-01
Plane electron Bernstein waves propagating obliquely to the magnetic field are investigated. The waves are excited by a plane grid antenna in a large volume magnetoplasma. The observations compare favorably with the predictions of the linear dispersion relation.......Plane electron Bernstein waves propagating obliquely to the magnetic field are investigated. The waves are excited by a plane grid antenna in a large volume magnetoplasma. The observations compare favorably with the predictions of the linear dispersion relation....
13. Nonlinear Waves in Complex Systems
DEFF Research Database (Denmark)
2007-01-01
The study of nonlinear waves has exploded due to the combination of analysis and computations, since the discovery of the famous recurrence phenomenon on a chain of nonlinearly coupled oscillators by Fermi-Pasta-Ulam fifty years ago. More than the discovery of new integrable equations, it is the ......The study of nonlinear waves has exploded due to the combination of analysis and computations, since the discovery of the famous recurrence phenomenon on a chain of nonlinearly coupled oscillators by Fermi-Pasta-Ulam fifty years ago. More than the discovery of new integrable equations......, it is the universality and robustness of the main models with respect to perturbations that developped the field. This is true for both continuous and discrete equations. In this volume we keep this broad view and draw new perspectives for nonlinear waves in complex systems. In particular we address energy flow...
14. Moderately nonlinear ultrasound propagation in blood-mimicking fluid.
Science.gov (United States)
Kharin, Nikolay A; Vince, D Geoffrey
2004-04-01
In medical diagnostic ultrasound (US), higher than-in-water nonlinearity of body fluids and tissue usually does not produce strong nonlinearly distorted waves because of the high absorption. The relative influence of absorption and nonlinearity can be characterized by the Gol'dberg number Gamma. There are two limiting cases in nonlinear acoustics: weak waves (Gamma 1). However, at diagnostic frequencies in tissue and body fluids, the nonlinear effects and effects of absorption more likely are comparable (Gol'dberg number Gamma approximately 1). The aim of this work was to study the nonlinear propagation of a moderately nonlinear US second harmonic signal in a blood-mimicking fluid. Quasilinear solutions to the KZK equation are presented, assuming radiation from a flat and geometrically focused circular Gaussian source. The solutions are expressed in a new simplified closed form and are in very good agreement with those of previous studies measuring and modeling Gaussian beams. The solutions also show good agreement with the measurements of the beams produced by commercially available transducers, even without special Gaussian shading.
15. Wave propagation of coupled modes in the DNA double helix
Energy Technology Data Exchange (ETDEWEB)
Tabi, C B; Ekobena Fouda, H P [Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde (Cameroon); Mohamadou, A [Condensed Matter Laboratory, Department of Physics, Faculty of Science, University of Douala, PO Box 24157, Douala (Cameroon); Kofane, T C, E-mail: [email protected] [Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, PO Box 812, Yaounde (Cameroon)
2011-03-15
The remarkable dynamics of waves propagating along the DNA molecule is described by the coupled nonlinear Schroedinger equations. We consider both the single and the coupled nonlinear excitation modes and, under numerical simulations of the Peyrard-Bishop model, with the use of realistic values of parameters, their biological implications are studied. Furthermore, the characteristics of the coupled mode solution are discussed and we show that such a solution can describe the local opening observed within the transcription and the replication phenomena.
16. Lamb Wave Propagation in Laminated Composite Structures
OpenAIRE
Gopalakrishnan, S.
2013-01-01
Damage detection using guided Lamb waves is an important tool in Structural health Monitoring. In this paper, we outline a method of obtaining Lamb wave modes in composite structures using two dimensional Spectral Finite Elements. Using this approach, Lamb wave dispersion curves are obtained for laminated composite structures with different fibre orientation. These propagating Lamb wave modes are pictorially captured using tone burst signal.
17. Propagation of gravity wave packet near critical level
Institute of Scientific and Technical Information of China (English)
YUE Xianchang; YI Fan
2005-01-01
A couple of two-dimensional linear and fully nonlinear numerical models for compressible atmosphere are used to numerically study the propagation of the gravity wave packet into a mean wind shear. For a linear propagation wave packet, the critical level interactions are in good agreement with the linear critical level theory. The dynamically and convectively unstable regions are formed due to the critical level interaction of a finite-amplitude wave packet, but they would not break. The free exchange of potential energy with kinetic energy in the background atmosphere at rest ceases after entering the mean wind shear. However, it still goes on in the nonlinear propagation. It is shown that the nonlinear effects modify the mean flow markedly, reduce the momentum and energy propagation velocity and drop the elevation of the critical level.The gravity wave packet becomes unstable and breaks down into smaller scales in some regions. It expends much more kinetic energy than potential energy in the early phase of the breakdown. This means that the wave breakdown sets up due to the action of the shear instability rather than a convective one.
18. Topology optimization of wave-propagation problems
DEFF Research Database (Denmark)
Jensen, Jakob Søndergaard; Sigmund, Ole
2006-01-01
Topology optimization is demonstrated as a useful tool for systematic design of wave-propagation problems. We illustrate the applicability of the method for optical, acoustic and elastic devices and structures.......Topology optimization is demonstrated as a useful tool for systematic design of wave-propagation problems. We illustrate the applicability of the method for optical, acoustic and elastic devices and structures....
19. Terrestrial propagation of long electromagnetic waves
CERN Document Server
Galejs, Janis; Fock, V A
2013-01-01
Terrestrial Propagation of Long Electromagnetic Waves deals with the propagation of long electromagnetic waves confined principally to the shell between the earth and the ionosphere, known as the terrestrial waveguide. The discussion is limited to steady-state solutions in a waveguide that is uniform in the direction of propagation. Wave propagation is characterized almost exclusively by mode theory. The mathematics are developed only for sources at the ground surface or within the waveguide, including artificial sources as well as lightning discharges. This volume is comprised of nine chapte
20. Voigt-wave propagation in active materials
CERN Document Server
Mackay, Tom G
2015-01-01
If a dissipative anisotropic dielectric material, characterized by the permittivity matrix $\\underline{\\underline{\\epsilon}}$, supports Voigt-wave propagation, then so too does the analogous active material characterized by the permittivity matrix $\\underline{\\underline{{\\tilde{\\epsilon}}}}$, where $\\underline{\\underline{{\\tilde{\\epsilon}}}}$ is the hermitian conjugate of $\\underline{\\underline{\\epsilon}}$. Consequently, a dissipative material that supports Voigt-wave propagation can give rise to a material that supports the propagation of Voigt waves with attendant linear gain in amplitude with propagation distance, by infiltration with an active dye.
1. Numerical study of the propagation of small-amplitude atmospheric gravity wave
Institute of Scientific and Technical Information of China (English)
YUE Xianchang; YI Fan; LIU Yingjie; LI Fang
2005-01-01
By using a two-dimensional fully nonlinear compressible atmospheric dynamic numerical model, the propagation of a small amplitude gravity wave packet is simulated. A corresponding linear model is also developed for comparison. In an isothermal atmosphere, the simulations show that the nonlinear effects impacting on the propagation of a small amplitude gravity wave are negligible. In the nonisothermal atmosphere, however, the nonlinear effects are remarkable. They act to slow markedly down the propagation velocity of wave energy and therefore reduce the growth ratio of the wave amplitude with time. But the energy is still conserved. A proof of this is provided by the observations in the middle atmosphere.
2. Properties of Nonlinear Dynamo Waves
Science.gov (United States)
Tobias, S. M.
1997-01-01
Dynamo theory offers the most promising explanation of the generation of the sun's magnetic cycle. Mean field electrodynamics has provided the platform for linear and nonlinear models of solar dynamos. However, the nonlinearities included are (necessarily) arbitrarily imposed in these models. This paper conducts a systematic survey of the role of nonlinearities in the dynamo process, by considering the behaviour of dynamo waves in the nonlinear regime. It is demonstrated that only by considering realistic nonlinearities that are non-local in space and time can modulation of the basic dynamo wave he achieved. Moreover, this modulation is greatest when there is a large separation of timescales provided by including a low magnetic Prandtl number in the equation for the velocity perturbations.
3. Generalized dispersive wave emission in nonlinear fiber optics.
Science.gov (United States)
Webb, K E; Xu, Y Q; Erkintalo, M; Murdoch, S G
2013-01-15
We show that the emission of dispersive waves in nonlinear fiber optics is not limited to soliton-like pulses propagating in the anomalous dispersion regime. We demonstrate, both numerically and experimentally, that pulses propagating in the normal dispersion regime can excite resonant dispersive radiation across the zero-dispersion wavelength into the anomalous regime.
4. Slow wave propagation in soft adhesive interfaces.
Science.gov (United States)
Viswanathan, Koushik; Sundaram, Narayan K; Chandrasekar, Srinivasan
2016-11-16
Stick-slip in sliding of soft adhesive surfaces has long been associated with the propagation of Schallamach waves, a type of slow surface wave. Recently it was demonstrated using in situ experiments that two other kinds of slow waves-separation pulses and slip pulses-also mediate stick-slip (Viswanathan et al., Soft Matter, 2016, 12, 5265-5275). While separation pulses, like Schallamach waves, involve local interface detachment, slip pulses are moving stress fronts with no detachment. Here, we present a theoretical analysis of the propagation of these three waves in a linear elastodynamics framework. Different boundary conditions apply depending on whether or not local interface detachment occurs. It is shown that the interface dynamics accompanying slow waves is governed by a system of integral equations. Closed-form analytical expressions are obtained for the interfacial pressure, shear stress, displacements and velocities. Separation pulses and Schallamach waves emerge naturally as wave solutions of the integral equations, with oppositely oriented directions of propagation. Wave propagation is found to be stable in the stress regime where linearized elasticity is a physically valid approximation. Interestingly, the analysis reveals that slow traveling wave solutions are not possible in a Coulomb friction framework for slip pulses. The theory provides a unified picture of stick-slip dynamics and slow wave propagation in adhesive contacts, consistent with experimental observations.
5. Nonlinear wave-wave interactions and wedge waves
Institute of Scientific and Technical Information of China (English)
Ray Q.Lin; Will Perrie
2005-01-01
A tetrad mechanism for exciting long waves,for example edge waves,is described based on nonlinear resonant wave-wave interactions.In this mechanism,resonant interactions pass energy to an edge wave,from the three participating gravity waves.The estimated action flux into the edge wave can be orders of magnitude greater than the transfer fluxes derived from other competing mechanisms,such as triad interactions.Moreover,the numerical results show that the actual transfer rates into the edge wave from the three participating gravity waves are two-to three- orders of magnitude greater than bottom friction.
6. Rogue and shock waves in nonlinear dispersive media
CERN Document Server
Resitori, Stefania; Baronio, Fabio
2016-01-01
This self-contained set of lectures addresses a gap in the literature by providing a systematic link between the theoretical foundations of the subject matter and cutting-edge applications in both geophysical fluid dynamics and nonlinear optics. Rogue and shock waves are phenomena that may occur in the propagation of waves in any nonlinear dispersive medium. Accordingly, they have been observed in disparate settings – as ocean waves, in nonlinear optics, in Bose-Einstein condensates, and in plasmas. Rogue and dispersive shock waves are both characterized by the development of extremes: for the former, the wave amplitude becomes unusually large, while for the latter, gradients reach extreme values. Both aspects strongly influence the statistical properties of the wave propagation and are thus considered together here in terms of their underlying theoretical treatment. This book offers a self-contained graduate-level text intended as both an introduction and reference guide for a new generation of scientists ...
7. ANALYSE OF PULSE WAVE PROPAGATION IN ARTERIES
Institute of Scientific and Technical Information of China (English)
PAN Yi-shan; JIA Xiao-bo; CUI Chang-kui; XIAO Xiao-chun
2006-01-01
Based upon the blood vessel of being regarded as the elasticity tube, and that the tissue restricts the blood vessel wall, the rule of pulse wave propagation in blood vessel was studied. The viscosity of blood, the elastic modulus of blood vessel, the radius of tube that influenced the pulse wave propagation were analyzed. Comparing the result that considered the viscosity of blood with another result that did not consider the viscosity of blood, we finally discover that the viscosity of blood that influences the pulse wave propagation can not be neglected; and with the accretion of the elastic modulus the speed of propagation augments and the press value of blood stream heightens; when diameter of blood vessel reduces, the press of blood stream also heightens and the speed of pulse wave also augments. These results will contribute to making use of the information of pulse wave to analyse and auxiliarily diagnose some causes of human disease.
8. Multi-layer Study of Wave Propagation in Sunspots
Science.gov (United States)
Felipe, T.; Khomenko, E.; Collados, M.; Beck, C.
2010-10-01
We analyze the propagation of waves in sunspots from the photosphere to the chromosphere using time series of co-spatial Ca II H intensity spectra (including its line blends) and polarimetric spectra of Si I λ10,827 and the He I λ10,830 multiplet. From the Doppler shifts of these lines we retrieve the variation of the velocity along the line of sight at several heights. Phase spectra are used to obtain the relation between the oscillatory signals. Our analysis reveals standing waves at frequencies lower than 4 mHz and a continuous propagation of waves at higher frequencies, which steepen into shocks in the chromosphere when approaching the formation height of the Ca II H core. The observed nonlinearities are weaker in Ca II H than in He I lines. Our analysis suggests that the Ca II H core forms at a lower height than the He I λ10,830 line: a time delay of about 20 s is measured between the Doppler signal detected at both wavelengths. We fit a model of linear slow magnetoacoustic wave propagation in a stratified atmosphere with radiative losses according to Newton's cooling law to the phase spectra and derive the difference in the formation height of the spectral lines. We show that the linear model describes well the wave propagation up to the formation height of Ca II H, where nonlinearities start to become very important.
9. Exact Nonlinear Internal Equatorial Waves in the f-plane
Science.gov (United States)
Hsu, Hung-Chu
2016-07-01
We present an explicit exact solution of the nonlinear governing equations for internal geophysical water waves propagating westward above the thermocline in the f-plane approximation near the equator. Moreover, the mass transport velocity induced by this internal equatorial wave is eastward and a westward current occurs in the transition zone between the great depth where the water is still and the thermocline.
10. Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media
DEFF Research Database (Denmark)
Guo, Hairun; Zeng, Xianglong; Zhou, Binbin
2013-01-01
We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic...
11. Propagation of SLF/ELF electromagnetic waves
CERN Document Server
Pan, Weiyan
2014-01-01
This book deals with the SLF/ELF wave propagation, an important branch of electromagnetic theory. The SLF/ELF wave propagation theory is well applied in earthquake electromagnetic radiation, submarine communication, thunderstorm detection, and geophysical prospecting and diagnostics. The propagation of SLF/ELF electromagnetic waves is introduced in various media like the earth-ionospheric waveguide, ionospheric plasma, sea water, earth, and the boundary between two different media or the stratified media. Applications in the earthquake electromagnetic radiation and the submarine communications are also addressed. This book is intended for scientists and engineers in the fields of radio propagation and EM theory and applications. Prof. Pan is a professor at China Research Institute of Radiowave Propagation in Qingdao (China). Dr. Li is a professor at Zhejiang University in Hangzhou (China).
12. New approaches to nonlinear waves
CERN Document Server
2016-01-01
The book details a few of the novel methods developed in the last few years for studying various aspects of nonlinear wave systems. The introductory chapter provides a general overview, thematically linking the objects described in the book. Two chapters are devoted to wave systems possessing resonances with linear frequencies (Chapter 2) and with nonlinear frequencies (Chapter 3). In the next two chapters modulation instability in the KdV-type of equations is studied using rigorous mathematical methods (Chapter 4) and its possible connection to freak waves is investigated (Chapter 5). The book goes on to demonstrate how the choice of the Hamiltonian (Chapter 6) or the Lagrangian (Chapter 7) framework allows us to gain a deeper insight into the properties of a specific wave system. The final chapter discusses problems encountered when attempting to verify the theoretical predictions using numerical or laboratory experiments. All the chapters are illustrated by ample constructive examples demonstrating the app...
13. Wave propagation and scattering in random media
CERN Document Server
Ishimaru, Akira
1978-01-01
Wave Propagation and Scattering in Random Media, Volume 2, presents the fundamental formulations of wave propagation and scattering in random media in a unified and systematic manner. The topics covered in this book may be grouped into three categories: waves in random scatterers, waves in random continua, and rough surface scattering. Random scatterers are random distributions of many particles. Examples are rain, fog, smog, hail, ocean particles, red blood cells, polymers, and other particles in a state of Brownian motion. Random continua are the media whose characteristics vary randomly an
14. PROPAGATION OF CYLINDRICAL WAVES IN POROELASTIC MEDIA
Directory of Open Access Journals (Sweden)
Vorona Yu.V.
2014-12-01
Full Text Available The paper investigates the harmonic axisymmetric wave propagation in poroelastic media. The computational formulas for the study of displacements and stresses that occur during vibrations in a wide frequency range are proposed.
15. Oscillating nonlinear acoustic shock waves
DEFF Research Database (Denmark)
Gaididei, Yuri; Rasmussen, Anders Rønne; Christiansen, Peter Leth
2016-01-01
We investigate oscillating shock waves in a tube using a higher order weakly nonlinear acoustic model. The model includes thermoviscous effects and is non isentropic. The oscillating shock waves are generated at one end of the tube by a sinusoidal driver. Numerical simulations show...... that at resonance a stationary state arise consisting of multiple oscillating shock waves. Off resonance driving leads to a nearly linear oscillating ground state but superimposed by bursts of a fast oscillating shock wave. Based on a travelling wave ansatz for the fluid velocity potential with an added 2'nd order...... polynomial in the space and time variables, we find analytical approximations to the observed single shock waves in an infinitely long tube. Using perturbation theory for the driven acoustic system approximative analytical solutions for the off resonant case are determined....
16. Wave Beam Propagation Through Density Fluctuations
NARCIS (Netherlands)
Balakin, A. A.; Bertelli, N.; Westerhof, E.
2011-01-01
Perturbations induced by edge density fluctuations on electron cyclotron wave beams propagating in fusion plasmas are studied by means of a quasi-optical code. The effects of such fluctuations are illustrated here by showing the beam propagation in the case of single harmonic perturbations to the wa
17. DBEM crack propagation for nonlinear fracture problems
Directory of Open Access Journals (Sweden)
R. Citarella
2015-10-01
Full Text Available A three-dimensional crack propagation simulation is performed by the Dual Boundary Element Method (DBEM. The Stress Intensity Factors (SIFs along the front of a semi elliptical crack, initiated from the external surface of a hollow axle, are calculated for bending and press fit loading separately and for a combination of them. In correspondence of the latter loading condition, a crack propagation is also simulated, with the crack growth rates calculated using the NASGRO3 formula, calibrated for the material under analysis (steel ASTM A469. The J-integral and COD approaches are selected for SIFs calculation in DBEM environment, where the crack path is assessed by the minimum strain energy density criterion (MSED. In correspondence of the initial crack scenario, SIFs along the crack front are also calculated by the Finite Element (FE code ZENCRACK, using COD, in order to provide, by a cross comparison with DBEM, an assessment on the level of accuracy obtained. Due to the symmetry of the bending problem a pure mode I crack propagation is realised with no kinking of the propagating crack whereas for press fit loading the crack propagation becomes mixed mode. The crack growth analysis is nonlinear because of normal gap elements used to model the press fit condition with added friction, and is developed in an iterative-incremental procedure. From the analysis of the SIFs results related to the initial cracked configuration, it is possible to assess the impact of the press fit condition when superimposed to the bending load case.
18. Lamb Wave Technique for Ultrasonic Nonlinear Characterization in Elastic Plates
Energy Technology Data Exchange (ETDEWEB)
Lee, Tae Hun; Kim, Chung Seok; Jhang, Kyung Young [Hanyang University, Seoul (Korea, Republic of)
2010-10-15
Since the acoustic nonlinearity is sensitive to the minute variation of material properties, the nonlinear ultrasonic technique(NUT) has been considered as a promising method to evaluate the material degradation or fatigue. However, there are certain limitations to apply the conventional NUT using the bulk wave to thin plates. In case of plates, the use of Lamb wave can be considered, however, the propagation characteristics of Lamb wave are completely different with the bulk wave, and thus the separate study for the nonlinearity of Lamb wave is required. For this work, this paper analyzed first the conditions of mode pair suitable for the practical application as well as for the cumulative propagation of quadratic harmonic frequency and summarized the result in for conditions: phase matching, non-zero power flux, group velocity matching, and non-zero out-of-plane displacement. Experimental results in aluminum plates showed that the amplitude of the secondary Lamb wave and nonlinear parameter grew up with increasing propagation distance at the mode pair satisfying the above all conditions and that the ration of nonlinear parameters measured in Al6061-T6 and Al1100-H15 was closed to the ratio of the absolute nonlinear parameters
19. Supersaturation of vertically propagating internal gravity waves
Science.gov (United States)
Lindzen, Richard S.
1988-01-01
The usual assumption that vertically propagating internal gravity waves will cease growing with height once their amplitudes are such as to permit convective instability anywhere within the wave is reexamined. Two factors lead to amplitude limitation: (1) wave clipping associated with convective mixing, and (2) energetic constraints associated with the rate at which the wave can supply energy to the convection. It is found that these two factors limit supersaturation to about 50 percent for waves with short horizontal wavelengths and high relative phase speeds. Usually the degree of supersaturation will be much less. These factors also lead to a gradual, rather than sudden, cessation of wave growth with height.
20. Exact solutions of optical pulse propagation in nonlinear meta-materials
Science.gov (United States)
Nanda, Lipsa
2017-01-01
An analytical and simulation based method has been used to exactly solve the nonlinear wave propagation in bulk media exhibiting frequency dependent dielectric susceptibility and magnetic permeability. The method has been further extended to investigate the intensity distribution in a nonlinear meta-material with negative refractive index where both ɛ and μ are dispersive and negative in nature.
1. Nonlinear biochemical signal processing via noise propagation.
Science.gov (United States)
Kim, Kyung Hyuk; Qian, Hong; Sauro, Herbert M
2013-10-14
Single-cell studies often show significant phenotypic variability due to the stochastic nature of intra-cellular biochemical reactions. When the numbers of molecules, e.g., transcription factors and regulatory enzymes, are in low abundance, fluctuations in biochemical activities become significant and such "noise" can propagate through regulatory cascades in terms of biochemical reaction networks. Here we develop an intuitive, yet fully quantitative method for analyzing how noise affects cellular phenotypes based on identifying a system's nonlinearities and noise propagations. We observe that such noise can simultaneously enhance sensitivities in one behavioral region while reducing sensitivities in another. Employing this novel phenomenon we designed three biochemical signal processing modules: (a) A gene regulatory network that acts as a concentration detector with both enhanced amplitude and sensitivity. (b) A non-cooperative positive feedback system, with a graded dose-response in the deterministic case, that serves as a bistable switch due to noise-induced ultra-sensitivity. (c) A noise-induced linear amplifier for gene regulation that requires no feedback. The methods developed in the present work allow one to understand and engineer nonlinear biochemical signal processors based on fluctuation-induced phenotypes.
2. Nonlinear MHD waves in a Prominence Foot
CERN Document Server
Ofman, Leon; Kucera, Therese; Schmieder, Brigitte
2015-01-01
We study nonlinear waves in a prominence foot using 2.5D MHD model motivated by recent high-resolution observations with Hinode/SOT in Ca~II emission of a prominence on October 10, 2012 showing highly dynamic small-scale motions in the prominence material. Observations of H$\\alpha$ intensities and of Doppler shifts show similar propagating fluctuations. However the optically thick nature of the emission lines inhibits unique quantitative interpretation in terms of density. Nevertheless, we find evidence of nonlinear wave activity in the prominence foot by examining the relative magnitude of the fluctuation intensity ($\\delta I/I\\sim \\delta n/n$). The waves are evident as significant density fluctuations that vary with height, and apparently travel upward from the chromosphere into the prominence material with quasi-periodic fluctuations with typical period in the range of 5-11 minutes, and wavelengths $\\sim <$2000 km. Recent Doppler shift observations show the transverse displacement of the propagating wav...
3. Nonlinear Waves in Complex Systems
DEFF Research Database (Denmark)
2007-01-01
The study of nonlinear waves has exploded due to the combination of analysis and computations, since the discovery of the famous recurrence phenomenon on a chain of nonlinearly coupled oscillators by Fermi-Pasta-Ulam fifty years ago. More than the discovery of new integrable equations......, it is the universality and robustness of the main models with respect to perturbations that developped the field. This is true for both continuous and discrete equations. In this volume we keep this broad view and draw new perspectives for nonlinear waves in complex systems. In particular we address energy flow...... in Fourier space and equipartition, the role of inhomogeneities and complex geometry and the importance of coupled systems....
4. Inward propagating chemical waves in Taylor vortices.
Science.gov (United States)
Thompson, Barnaby W; Novak, Jan; Wilson, Mark C T; Britton, Melanie M; Taylor, Annette F
2010-04-01
Advection-reaction-diffusion (ARD) waves in the Belousov-Zhabotinsky reaction in steady Taylor-Couette vortices have been visualized using magnetic-resonance imaging and simulated using an adapted Oregonator model. We show how propagating wave behavior depends on the ratio of advective, chemical and diffusive time scales. In simulations, inward propagating spiral flamelets are observed at high Damköhler number (Da). At low Da, the reaction distributes itself over several vortices and then propagates inwards as contracting ring pulses--also observed experimentally.
5. Impact of mountain gravity waves on infrasound propagation
Science.gov (United States)
Damiens, Florentin; Lott, François; Millet, Christophe
2016-04-01
Linear theory of acoustic propagation is used to analyze how mountain waves can change the characteristics of infrasound signals. The mountain wave model is based on the integration of the linear inviscid Taylor-Goldstein equation forced by a nonlinear surface boundary condition. For the acoustic propagation we solve the wave equation using the normal mode method together with the effective sound speed approximation. For large-amplitude mountain waves we use direct numerical simulations to compute the interactions between the mountain waves and the infrasound component. It is shown that the mountain waves perturb the low level waveguide, which leads to significant acoustic dispersion. The mountain waves also impact the arrival time and spread of the signals substantially and can produce a strong absorption of the wave signal. To interpret our results we follow each acoustic mode separately and show which mode is impacted and how. We also show that the phase shift between the acoustic modes over the horizontal length of the mountain wave field may yield to destructive interferences in the lee side of the mountain, resulting in a new form of infrasound absorption. The statistical relevance of those results is tested using a stochastic version of the mountain wave model and large enough sample sizes.
6. Stress Wave Propagation in Two-dimensional Buckyball Lattice
Science.gov (United States)
Xu, Jun; Zheng, Bowen
2016-11-01
Orderly arrayed granular crystals exhibit extraordinary capability to tune stress wave propagation. Granular system of higher dimension renders many more stress wave patterns, showing its great potential for physical and engineering applications. At nanoscale, one-dimensionally arranged buckyball (C60) system has shown the ability to support solitary wave. In this paper, stress wave behaviors of two-dimensional buckyball (C60) lattice are investigated based on square close packing and hexagonal close packing. We show that the square close packed system supports highly directional Nesterenko solitary waves along initially excited chains and hexagonal close packed system tends to distribute the impulse and dissipates impact exponentially. Results of numerical calculations based on a two-dimensional nonlinear spring model are in a good agreement with the results of molecular dynamics simulations. This work enhances the understanding of wave properties and allows manipulations of nanoscale lattice and novel design of shock mitigation and nanoscale energy harvesting devices.
7. Nonlinear spin wave coupling in adjacent magnonic crystals
Energy Technology Data Exchange (ETDEWEB)
Sadovnikov, A. V., E-mail: [email protected]; Nikitov, S. A. [Laboratory “Metamaterials,” Saratov State University, Saratov 410012 (Russian Federation); Kotel' nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow 125009 (Russian Federation); Beginin, E. N.; Morozova, M. A.; Sharaevskii, Yu. P.; Grishin, S. V.; Sheshukova, S. E. [Laboratory “Metamaterials,” Saratov State University, Saratov 410012 (Russian Federation)
2016-07-25
We have experimentally studied the coupling of spin waves in the adjacent magnonic crystals. Space- and time-resolved Brillouin light-scattering spectroscopy is used to demonstrate the frequency and intensity dependent spin-wave energy exchange between the side-coupled magnonic crystals. The experiments and the numerical simulation of spin wave propagation in the coupled periodic structures show that the nonlinear phase shift of spin wave in the adjacent magnonic crystals leads to the nonlinear switching regime at the frequencies near the forbidden magnonic gap. The proposed side-coupled magnonic crystals represent a significant advance towards the all-magnonic signal processing in the integrated magnonic circuits.
8. FLEXURAL WAVE PROPAGATION IN NARROW MINDLIN'S PLATE
Institute of Scientific and Technical Information of China (English)
HU Chao; HAN Gang; FANG Xue-qian; HUANG Wen-hu
2006-01-01
Appling Mindlin's theory of thick plates and Hamilton system to propagation of elastic waves under free boundary condition, a solution of the problem was given.Dispersion equations of propagation mode of strip plates were deduced from eigenfunction expansion method. It was compared with the dispersion relation that was gained through solution of thick plate theory proposed by Mindlin. Based on the two kinds of theories,the dispersion curves show great difference in the region of short waves, and the cutoff frequencies are higher in Hamiltonian systems. However, the dispersion curves are almost the same in the region of long waves.
9. Ducted propagation of chorus waves: Cluster observations
Directory of Open Access Journals (Sweden)
K. H. Yearby
2011-09-01
Full Text Available Ducted propagation of whistler waves in the terrestrial magnetosphere-ionosphere system was discussed and studied long before the first in-situ spacecraft measurements. While a number of implicit examples of the existence of ducted propagation have been found, direct observation of ducts has been hampered by the low sampling rates of measurements of the plasma density. The present paper is based on Cluster observations of chorus waves. The ability to use measurements of the spacecraft potential as a proxy for high time resolution electron density measurements is exploited to identify a number of cases when increased chorus wave power, observed within the radiation belts, is observed simultaneously with density enchantments. It is argued that the observation of ducted propagation of chorus implies modification of numerical models for plasma-wave interactions within the radiation belts.
10. Radiation and propagation of electromagnetic waves
CERN Document Server
Tyras, George; Declaris, Nicholas
1969-01-01
Radiation and Propagation of Electromagnetic Waves serves as a text in electrical engineering or electrophysics. The book discusses the electromagnetic theory; plane electromagnetic waves in homogenous isotropic and anisotropic media; and plane electromagnetic waves in inhomogenous stratified media. The text also describes the spectral representation of elementary electromagnetic sources; the field of a dipole in a stratified medium; and radiation in anisotropic plasma. The properties and the procedures of Green's function method of solution, axial currents, as well as cylindrical boundaries a
11. Unidirectional propagation of designer surface acoustic waves
CERN Document Server
Lu, Jiuyang; Ke, Manzhu; Liu, Zhengyou
2014-01-01
We propose an efficient design route to generate unidirectional propagation of the designer surface acoustic waves. The whole system consists of a periodically corrugated rigid plate combining with a pair of asymmetric narrow slits. The directionality of the structure-induced surface waves stems from the destructive interference between the evanescent waves emitted from the double slits. The theoretical prediction is validated well by simulations and experiments. Promising applications can be anticipated, such as in designing compact acoustic circuits.
12. Faraday Pilot-Waves: Generation and Propagation
Science.gov (United States)
Galeano-Rios, Carlos; Milewski, Paul; Nachbin, André; Bush, John
2015-11-01
We examine the dynamics of drops bouncing on a fluid bath subjected to vertical vibration. We solve a system of linear PDEs to compute the surface wave generation and propagation. Waves are triggered at each bounce, giving rise to the Faraday pilot-wave field. The model captures several of the behaviors observed in the laboratory, including transitions between a variety of bouncing and walking states, the Doppler effect, and droplet-droplet interactions. Thanks to the NSF.
13. A Spectral Element Method for Nonlinear and Dispersive Water Waves
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Bigoni, Daniele; Eskilsson, Claes
The use of flexible mesh discretisation methods are important for simulation of nonlinear wave-structure interactions in offshore and marine settings such as harbour and coastal areas. For real applications, development of efficient models for wave propagation based on unstructured discretisation...... methods is of key interest. We present a high-order general-purpose three-dimensional numerical model solving fully nonlinear and dispersive potential flow equations with a free surface.......The use of flexible mesh discretisation methods are important for simulation of nonlinear wave-structure interactions in offshore and marine settings such as harbour and coastal areas. For real applications, development of efficient models for wave propagation based on unstructured discretisation...
14. Tropical response to extratropical eastward propagating waves
Directory of Open Access Journals (Sweden)
S. Sridharan
2015-06-01
Full Text Available Space–time spectral analysis of ERA-interim winds and temperature at 200 hPa for December 2012–February 2013 shows the presence of eastward propagating waves with period near 18 days in mid-latitude meridional winds at 200 hPa. The 18 day waves of k = 1–2 are dominantly present at latitudes greater than 80°, whereas the waves of k = 3–4 are dominant at 60° of both Northern and Southern Hemispheres. Though the 18 day wave of smaller zonal wavenumbers (k = 1–2 are confined to high latitudes, there is an equatorward propagation of the 18 day wave of k = 4 and 5. The wave amplitude of k = 5 is dominant than that of k = 4 at tropical latitudes. In the Northern Hemisphere (NH, there is a poleward tilt in the phase of the wave of k = 5 at mid-latitudes, as height increases indicating the baroclinic nature of the wave, whereas in the Southern Hemisphere (SH, the wave has barotropic structure as there is no significant phase variation with height. At the NH subtropics, the wave activity is confined to 500–70 hPa with moderate amplitudes. It is reported for the first time that the wave of similar periodicity (18 day and zonal structure (k = 5 as that of extratropical wave disturbance has been observed in tropical OLR, a proxy for tropical convection. We suggest that the selective response of the tropical wave forcing may be due to the lateral forcing of the eastward propagating extratropical wave of similar periodicity and zonal structure.
15. Cumulative second-harmonic generation of Lamb waves propagating in a two-layered solid plate
Institute of Scientific and Technical Information of China (English)
Xiang Yan-Xun; Deng Ming-Xi
2008-01-01
The physical process of cumulative second-harmonic generation of Lamb waves propagating in a two-layered solid plate is presented by using the second-order perturbation and the technique of nonlinear reflection of acoustic waves at an interface.In general,the cumulative second-harmonic generation of a dispersive guided wave propagation does not occur.However,the present paper shows that the second-harmonic of Lamb wave propagation arising from the nonlinear interaction of the partial bulk acoustic waves and the restriction of the three boundaries of the solid plates does have a cumulative growth effect if some conditions are satisfied.Through boundary condition and initial condition of excitation,the analytical expression of cumulative second-harmonic of Lamb waves propagation is determined.Numerical results show the cumulative effect of Lamb waves on second-harmonic field patterns.
16. Hopf Bifurcation in a Nonlinear Wave System
Institute of Scientific and Technical Information of China (English)
HE Kai-Fen
2004-01-01
@@ Bifurcation behaviour of a nonlinear wave system is studied by utilizing the data in solving the nonlinear wave equation. By shifting to the steady wave frame and taking into account the Doppler effect, the nonlinear wave can be transformed into a set of coupled oscillators with its (stable or unstable) steady wave as the fixed point.It is found that in the chosen parameter regime, both mode amplitudes and phases of the wave can bifurcate to limit cycles attributed to the Hopf instability. It is emphasized that the investigation is carried out in a pure nonlinear wave framework, and the method can be used for the further exploring routes to turbulence.
17. Wave equation with concentrated nonlinearities
OpenAIRE
Noja, Diego; Posilicano, Andrea
2004-01-01
In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of $\\CO^n$ and a discrete set $Y\\subset\\RE^3$ with $n$ elements, we define a nonlinear operator $\\Delta_{V,Y}$ on $L^2(\\RE^3)$ which coincides with the free Laplacian when restricted to regular functions vanishing at $Y$, and which reduces to the usual Laplacian with point interactions placed at $Y$ when $V$ is linear and is represented by an Hermitean m...
18. Computational Modeling of Ultrafast Pulse Propagation in Nonlinear Optical Materials
Science.gov (United States)
Goorjian, Peter M.; Agrawal, Govind P.; Kwak, Dochan (Technical Monitor)
1996-01-01
There is an emerging technology of photonic (or optoelectronic) integrated circuits (PICs or OEICs). In PICs, optical and electronic components are grown together on the same chip. rib build such devices and subsystems, one needs to model the entire chip. Accurate computer modeling of electromagnetic wave propagation in semiconductors is necessary for the successful development of PICs. More specifically, these computer codes would enable the modeling of such devices, including their subsystems, such as semiconductor lasers and semiconductor amplifiers in which there is femtosecond pulse propagation. Here, the computer simulations are made by solving the full vector, nonlinear, Maxwell's equations, coupled with the semiconductor Bloch equations, without any approximations. The carrier is retained in the description of the optical pulse, (i.e. the envelope approximation is not made in the Maxwell's equations), and the rotating wave approximation is not made in the Bloch equations. These coupled equations are solved to simulate the propagation of femtosecond optical pulses in semiconductor materials. The simulations describe the dynamics of the optical pulses, as well as the interband and intraband.
19. Internal solitary waves propagating through variable background hydrology and currents
Science.gov (United States)
Liu, Z.; Grimshaw, R.; Johnson, E.
2017-08-01
Large-amplitude, horizontally-propagating internal wave trains are commonly observed in the coastal ocean, fjords and straits. They are long nonlinear waves and hence can be modelled by equations of the Korteweg-de Vries type. However, typically they propagate through regions of variable background hydrology and currents, and over variable bottom topography. Hence a variable-coefficient Korteweg-de Vries equation is needed to model these waves. Although this equation is now well-known and heavily used, a term representing non-conservative effects, arising from dissipative or forcing terms in the underlying basic state, has usually been omitted. In particular this term arises when the hydrology varies in the horizontal direction. Our purpose in this paper is to examine the possible significance of this term. This is achieved through analysis and numerical simulations, using both a two-layer fluid model and a re-examination of previous studies of some specific ocean cases.
20. Nonlinear and Dispersive Optical Pulse Propagation
Science.gov (United States)
Dijaili, Sol Peter
In this dissertation, there are basically four novel contributions to the field of picosecond pulse propagation and measurement. The first contribution is the temporal ABCD matrix which is an analog of the traditional ABCD ray matrices used in Gaussian beam propagation. The temporal ABCD matrix allows for the easy calculation of the effects of linear chirp or group velocity dispersion in the time domain. As with Gaussian beams in space, there also exists a complete Hermite-Gaussian basis in time whose propagation can be tracked with the temporal ABCD matrices. The second contribution is the timing synchronization between a colliding pulse mode-locked dye laser and a gain-switched Fabry-Perot type AlGaAs laser diode that has achieved less than 40 femtoseconds of relative timing jitter by using a pulsed optical phase lock loop (POPLL). The relative timing jitter was measured using the error voltage of the feedback loop. This method of measurement is accurate since the frequencies of all the timing fluctuations fall within the loop bandwidth. The novel element is a broad band optical cross-correlator that can resolve femtosecond time delay errors between two pulse trains. The third contribution is a novel dispersive technique of determining the nonlinear frequency sweep of a picosecond pulse with relatively good accuracy. All the measurements are made in the time domain and hence there is no time-bandwidth limitation to the accuracy. The fourth contribution is the first demonstration of cross -phase modulation in a semiconductor laser amplifier where a variable chirp was observed. A simple expression for the chirp imparted on a weak signal pulse by the action of a strong pump pulse is derived. A maximum frequency excursion of 16 GHz due to the cross-phase modulation was measured. A value of 5 was found for alpha _{xpm} which is a factor for characterizing the cross-phase modulation in a similar manner to the conventional linewidth enhancement factor, alpha.
1. Exact solitary wave solutions of nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
2001-01-01
The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grbner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.
2. Wave propagation in pantographic 2D lattices with internal discontinuities
CERN Document Server
2014-01-01
In the present paper we consider a 2D pantographic structure composed by two orthogonal families of Euler beams. Pantographic rectangular 'long' waveguides are considered in which imposed boundary displacements can induce the onset of traveling (possibly non-linear) waves. We performed numerical simulations concerning a set of dynamically interesting cases. The system undergoes large rotations which may involve geometrical non-linearities, possibly opening the path to appealing phenomena such as propagation of solitary waves. Boundary conditions dramatically influence the transmission of the considered waves at discontinuity surfaces. The theoretical study of this kind of objects looks critical, as the concept of pantographic 2D sheets seems to have promising possible applications in a number of fields, e.g. acoustic filters, vascular prostheses and aeronautic/aerospace panels.
3. Wave propagation in complex coordinates
CERN Document Server
Horsley, S A R; Philbin, T G
2015-01-01
We investigate the analytic continuation of wave equations into the complex position plane. For the particular case of electromagnetic waves we provide a physical meaning for such an analytic continuation in terms of a family of closely related inhomogeneous media. For bounded permittivity profiles we find the phenomenon of reflection can be related to branch cuts in the wave that originate from poles of the permittivity at complex positions. Demanding that these branch cuts disappear, we derive a large family of inhomogeneous media that are reflectionless for a single angle of incidence. Extending this property to all angles of incidence leads us to a generalized form of the Poschl Teller potentials. We conclude by analyzing our findings within the phase integral (WKB) method.
4. Nonlinear waves in a fluid-filled thin viscoelastic tube
Science.gov (United States)
Zhang, Shan-Yuan; Zhang, Tao
2010-11-01
In the present paper the propagation property of nonlinear waves in a thin viscoelastic tube filled with incompressible inviscid fluid is studied. The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin—Voigt model. Using the mass conservation and the momentum theorem of the fluid and radial dynamic equilibrium of an element of the tube wall, a set of nonlinear partial differential equations governing the propagation of nonlinear pressure wave in the solid—liquid coupled system is obtained. In the long-wave approximation the nonlinear far-field equations can be derived employing the reductive perturbation technique (RPT). Selecting the exponent α of the perturbation parameter in Gardner—Morikawa transformation according to the order of viscous coefficient η, three kinds of evolution equations with soliton solution, i.e. Korteweg—de Vries (KdV)—Burgers, KdV and Burgers equations are deduced. By means of the method of traveling-wave solution and numerical calculation, the propagation properties of solitary waves corresponding with these evolution equations are analysed in detail. Finally, as a example of practical application, the propagation of pressure pulses in large blood vessels is discussed.
5. Nonlinear waves in a fluid-filled thin viscoelastic tube
Institute of Scientific and Technical Information of China (English)
Zhang Shan-Yuan; Zhang Tao
2010-01-01
In the present paper the propagation property of nonlinear waves in a thin viscoelastic tube filled with incom-pressible inviscid fluid is studied. The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin-Voigt model. Using the mass conservation and the momentum theorem of the fluid and radial dynamic equilibrium of an element of the tube wall, a set of nonlinear partial differential equations governing the prop-agation of nonlinear pressure wave in the solid-liquid coupled system is obtained. In the long-wave approximation the nonlinear far-field equations can be derived employing the reductive perturbation technique (RPT). Selecting the expo-η, three kinds of evolution equations with soliton solution, i.e. Korteweg-de Vries (KdV)-Burgers, KdV and Burgers equations are deduced. By means of the method of traveling-wave solution and numerical calculation, the propagation properties of solitary waves corresponding with these evolution equations are analysed in detail. Finally, as a example of practical application, the propagation of pressure pulses in large blood vessels is discussed.
6. A General Linear Wave Theory for Water Waves Propagating over Uneven Porous Bottoms
Institute of Scientific and Technical Information of China (English)
锁要红; 黄虎
2004-01-01
Starting from the widespread phenomena of porous bottoms in the near shore region, considering fully the diversity of bottom topography and wave number variation, and including the effect of evanescent modes, a general linear wave theory for water waves propagating over uneven porous bottoms in the near shore region is established by use of Green's second identity. This theory can be reduced to a number of the most typical mild-slope equations currently in use and provide a reliable research basis for follow-up development of nonlinear water wave theory involving porous bottoms.
7. Stimulated Raman Scattering and Nonlinear Focusing of High-Power Laser Beams Propagating in Water
CERN Document Server
Hafizi, B; Penano, J R; Gordon, D F; Jones, T G; Helle, M H; Kaganovich, D
2015-01-01
The physical processes associated with propagation of a high-power (power > critical power for self-focusing) laser beam in water include nonlinear focusing, stimulated Raman scattering (SRS), optical breakdown and plasma formation. The interplay between nonlinear focusing and SRS is analyzed for cases where a significant portion of the pump power is channeled into the Stokes wave. Propagation simulations and an analytical model demonstrate that the Stokes wave can re-focus the pump wave after the power in the latter falls below the critical power. It is shown that this novel focusing mechanism is distinct from cross-phase focusing. While discussed here in the context of propagation in water, the gain-focusing phenomenon is general to any medium supporting nonlinear focusing and stimulated forward Raman scattering.
8. Wave Propagation in Smart Materials
DEFF Research Database (Denmark)
Pedersen, Michael
1999-01-01
In this paper we deal with the behavior of solutions to hyperbolicequations such as the wave equation:\$$\\label{waveeq1}\\frac{\\partial^2}{\\partial t^2}u-\\Delta u=f,\$$or the equations of linear elasticity for an isotropic medium:\\\label{elasteq1}\\frac{\\parti...
9. Wave Propagation in Smart Materials
DEFF Research Database (Denmark)
Pedersen, Michael
1999-01-01
In this paper we deal with the behavior of solutions to hyperbolic equations such as the wave equation: \$$\\label{waveeq1} \\frac{\\partial^2}{\\partial t^2}u-\\Delta u=f, \$$ or the equations of linear elasticity for an isotropic medium: \\\label{elasteq1} \\frac...
10. Topology Optimization for Transient Wave Propagation Problems
DEFF Research Database (Denmark)
Matzen, René
as for vectorial elastic wave propagation problems using finite element analysis [P2], [P4]. The concept is implemented in a parallel computing code that includes efficient techniques for performing gradient based topology optimization. Using the developed computational framework the thesis considers four...... new technology, by designing new materials and their layout. The thesis presents a general framework for applying topology optimization in the design of material layouts for transient wave propagation problems. In contrast to the high level of modeling in the frequency domain, time domain topology...
11. Propagation of shock waves through clouds
Science.gov (United States)
Zhou, Xin Xin
1990-10-01
The behavior of a shock wave propagating into a cloud consisting of an inert gas, water vapor and water droplets was investigated. This has particular application to sonic bangs propagating in the atmosphere. The finite different method of MacCormack is extended to solve the one and two dimensional, two phase flow problems in which mass, momentum and energy transfers are included. The FCT (Fluid Corrected Transport) technique developed by Boris and Book was used in the basic numerical scheme as a powerful corrective procedure. The results for the transmitted shock waves propagating in a one dimensional, semi infinite cloud obtained by the finite difference approach are in good agreement with previous results by Kao using the method characteristics. The advantage of the finite difference method is its adaptability to two and three dimensional problems. Shock wave propagation through a finite cloud and into an expansion with a 90 degree corner was investigated. It was found that the transfer processes between the two phases in two dimensional flow are much more complicated than in the one dimensional flow cases. This is mainly due to the vortex and expansion wave generated at the corner. In the case considered, further complications were generated by the reflected shock wave from the floor. Good agreement with experiment was found for one phase flow but experimental data for the two phase case is not yet available to validate the two phase calculations.
12. Experimental characterization of nonlinear processes of whistler branch waves
Science.gov (United States)
Tejero, E. M.; Crabtree, C.; Blackwell, D. D.; Amatucci, W. E.; Ganguli, G.; Rudakov, L.
2016-05-01
Experiments in the Space Physics Simulation Chamber at the Naval Research Laboratory isolated and characterized important nonlinear wave-wave and wave-particle interactions that can occur in the Earth's Van Allen radiation belts by launching predominantly electrostatic waves in the intermediate frequency range with wave normal angle greater than 85 ° and measuring the nonlinearly generated electromagnetic scattered waves. The scattered waves have a perpendicular wavelength that is nearly an order of magnitude larger than that of the pump wave. Calculations of scattering efficiency from experimental measurements demonstrate that the scattering efficiency is inversely proportional to the damping rate and trends towards unity as the damping rate approaches zero. Signatures of both wave-wave and wave-particle scatterings are also observed in the triggered emission process in which a launched wave resonant with a counter-propagating electron beam generates a large amplitude chirped whistler wave. The possibility of nonlinear scattering or three wave decay as a saturation mechanism for the triggered emission is suggested. The laboratory experiment has inspired the search for scattering signatures in the in situ data of chorus emission in the radiation belts.
13. Dynamics of optical rogue waves in inhomogeneous nonlinear waveguides
Institute of Scientific and Technical Information of China (English)
Zhang Jie-Fang; Jin Mei-Zhen; He Ji-Da; Lou Ji-Hui; Dai Chao-Qing
2013-01-01
We propose a unified theory to construct exact rogue wave solutions of the (2+1)-dimensional nonlinear Schr(o)dinger equation with varying coefficients.And then the dynamics of the first-and the second-order optical rogues are investigated.Finally,the controllability of the optical rogue propagating in inhomogeneous nonlinear waveguides is discussed.By properly choosing the distributed coefficients,we demonstrate analytically that rogue waves can be restrained or even be annihilated,or emerge periodically and sustain forever.We also figure out the center-of-mass motion of the rogue waves.
14. Nonlinear surface waves in soft, weakly compressible elastic media.
Science.gov (United States)
Zabolotskaya, Evgenia A; Ilinskii, Yurii A; Hamilton, Mark F
2007-04-01
Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.
15. Experimental observations of nonlinear effects of the Lamb waves
Institute of Scientific and Technical Information of China (English)
DENG Mingxi; D.C. Price; D.A.Scott
2004-01-01
The experimental observations of nonlinear effects of the primary Lamb waves have been reported. Firstly, the brief descriptions have been made for the nonlinear acoustic measurement system developed by Ritec. The detailed considerations for the acoustic experiment system established for observing of the nonlinear effects of the primary Lamb waves have been carried out. Especially, the analysis focuses on the time-domain responses of second harmonics of the primary Lame waves by employing a straightforward model. Based on the existence conditions of strong nonlinearity of the primary Lamb waves, the wedge transducers are designed to generate and detect the primary and secondary waves on the surface of an aluminum sheet. For the different distances between the transmitting and receiving wedge transducers,the amplitudes of the primary waves and the second harmonics on the sheet surface have been measured within a specified frequency range. In the immediate vicinity of the driving frequency,where the primary and the double frequency Lamb waves have the same phase velocities, the quantitative relations of second-harmonic amplitudes with the propagation distance have been analyzed. It is experimentally verified that the second harmonics of the primary Lamb waves do have a cumulative growth effect along with the propagation distance.
16. Free Propagation of Wave in Viscoelastic Cables with Small Curvature
Institute of Scientific and Technical Information of China (English)
邹宗兰
2003-01-01
The coupled longitudinal-transverse waves propagating freely along a viscoelastic cable was studied. The frequency-spectrum equation governing propagating waves and the formulations of the phase velocities and the group velocities characterizing propagating waves were derived. The effects of viscosity parameters on the phase velocities and the group velocities were investigated with numerical simulation. The analyses show that viscosity has a strong influence on the phase velocity and the group velocity of propagating waves and attenuation waves for longitudinal-dominant waves, but the phase velocities of propagating waves of transverse-dominant waves do not change with viscosity.
17. Parametric interaction and intensification of nonlinear Kelvin waves
CERN Document Server
2008-01-01
Observational evidence is presented for nonlinear interaction between mesoscale internal Kelvin waves at the tidal -- $\\omega_t$ or the inertial -- $\\omega_i$ frequency and oscillations of synoptic -- $\\Omega$ frequency of the background coastal current of Japan/East Sea. Enhanced coastal currents at the sum -- $\\omega_+$ and dif -- $\\omega_-$ frequencies: $\\omega_\\pm =\\omega_{t,i}\\pm \\Omega$ have properties of propagating Kelvin waves suggesting permanent energy exchange from the synoptic band to the mesoscale $\\omega_\\pm$ band. The interaction may be responsible for the greater than predicted intensification, steepen and break of boundary trapped and equatorially trapped Kelvin waves, which can affect El Ni\\~{n}o. The problem on the parametric interaction of the nonlinear Kelvin wave at the frequency $\\omega$ and the low-frequency narrow-band nose with representative frequency $\\Omega\\ll\\omega$ is investigated with the theory of nonlinear week dispersion waves.
18. Anomalous velocity enhancing of soliton, propagating in nonlinear PhC, due to its reflection from nonlinear ambient medium
Science.gov (United States)
Trofimov, Vyacheslav A.; Lysak, T. M.
2016-05-01
We demonstrate a new possibility of a soliton velocity control at its propagation in a nonlinear layered structure (1D photonic crystal) which is placed in a nonlinear ambient medium. Nonlinear response of the ambient medium, as well as the PhC layers, is cubic. At the initial time moment, a soliton is spread over a few layers of PhC. Then, soliton propagates across the layered structure because of the initial wave-vector direction presence for the laser beam. The soliton reaches the PhC faces and reflects from them or passes through the face. As a nonlinear medium surrounds the PhC, the laser beam obtains additional impulse after interaction with this medium and accelerates (or slows down or stops near the PhC face). Nonlinear response of the ambient medium can be additionally created by another laser beam which shines near the PhC faces.
19. Coupled seismic and electromagnetic wave propagation
NARCIS (Netherlands)
Schakel, M.D.
2011-01-01
Coupled seismic and electromagnetic wave propagation is studied theoretically and experimentally. This coupling arises because of the electrochemical double layer, which exists along the solid-grain/fluid-electrolyte boundaries of porous media. Within the double layer, charge is redistributed, creat
20. Electromagnetic Wave Propagation in Random Media
DEFF Research Database (Denmark)
Pécseli, Hans
1984-01-01
The propagation of a narrow frequency band beam of electromagnetic waves in a medium with randomly varying index of refraction is considered. A novel formulation of the governing equation is proposed. An equation for the average Green function (or transition probability) can then be derived...
1. Domain Wall Propagation through Spin Wave Emission
NARCIS (Netherlands)
Wang, X.S.; Yan, P.; Shen, Y.H.; Bauer, G.E.W.; Wang, X.R.
2012-01-01
We theoretically study field-induced domain wall motion in an electrically insulating ferromagnet with hard- and easy-axis anisotropies. Domain walls can propagate along a dissipationless wire through spin wave emission locked into the known soliton velocity at low fields. In the presence of damping
2. Electromagnetic wave propagations in conjugate metamaterials.
Science.gov (United States)
Xu, Yadong; Fu, Yangyang; Chen, Huanyang
2017-03-06
In this work, by employing field transformation optics, we deduce a special kind of materials called conjugate metamaterials, which can support intriguing electromagnetic wave propagations, such as negative refractions and lasing phenomena. These materials could also serve as substrates for making a subwavelength-resolution lens, and the so-called "perfect lens" is demonstrated to be a limiting case.
3. Antenna Construction and Propagation of Radio Waves.
Science.gov (United States)
Marine Corps Inst., Washington, DC.
Developed as part of the Marine Corps Institute (MCI) correspondence training program, this course on antenna construction and propagation of radio waves is designed to provide communicators with instructions in the selection and/or construction of the proper antenna(s) for use with current field radio equipment. Introductory materials include…
4. Wave propagation in axially moving periodic strings
DEFF Research Database (Denmark)
Sorokin, Vladislav S.; Thomsen, Jon Juel
2017-01-01
The paper deals with analytically studying transverse waves propagation in an axially moving string with periodically modulated cross section. The structure effectively models various relevant technological systems, e.g. belts, thread lines, band saws, etc., and, in particular, roller chain drive...
5. Wave propagation in elastic layers with damping
DEFF Research Database (Denmark)
2016-01-01
The conventional concepts of a loss factor and complex-valued elastic moduli are used to study wave attenuation in a visco-elastic layer. The hierarchy of reduced-order models is employed to assess attenuation levels in various situations. For the forcing problem, the attenuation levels are found...... for alternative excitation cases. The differences between two regimes, the low frequency one, when a waveguide supports only one propagating wave, and the high frequency one, when several waves are supported, are demonstrated and explained....
6. Thermoelastic wave propagation in laminated composites plates
Directory of Open Access Journals (Sweden)
Verma K. L.
2012-12-01
Full Text Available The dispersion of thermoelastic waves propagation in an arbitrary direction in laminated composites plates is studied in the framework of generalized thermoelasticity in this article. Three dimensional field equations of thermoelasticity with relaxation times are considered. Characteristic equation is obtained on employing the continuity of displacements, temperature, stresses and thermal gradient at the layers’ interfaces. Some important particular cases such as of free waves on reducing plates to single layer and the surface waves when thickness tends to infinity are also discussed. Uncoupled and coupled thermoelasticity are the particular cases of the obtained results. Numerical results are also obtained and represented graphically.
7. Love Wave Propagation in Poro elasticity
Directory of Open Access Journals (Sweden)
Y.V. Rama Rao
1978-10-01
Full Text Available It is observed that on similar reasons as in classical theory of elasticity, SH wave propagation in a semi infinite poroelastic body is not possible and is possible when there is a layer of another poro elastic medium over it i.e., Love waves. Two particular cases are considered in one of which phase velocity can be determined for a given wave length. In the same case, equation for phase velocity is of the same form as that of the classical theory of Elasticity.
8. Standing waves for discrete nonlinear Schrodinger equations
OpenAIRE
Ming Jia
2016-01-01
The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. By using critical point theory, we establish some new sufficient conditions on the existence results of standing waves for the discrete nonlinear Schrodinger equations. We give an appropriate example to illustrate the conclusion obtained.
9. Solitary Wave Propagation Influenced by Submerged Breakwater
Institute of Scientific and Technical Information of China (English)
王锦; 左其华; 王登婷
2013-01-01
The form of Boussinesq equation derived by Nwogu (1993) using velocity at an arbitrary distance and surface elevation as variables is used to simulate wave surface elevation changes. In the numerical experiment, water depth was divided into five layers with six layer interfaces to simulate velocity at each layer interface. Besides, a physical experiment was carried out to validate numerical model and study solitary wave propagation.“Water column collapsing”method (WCCM) was used to generate solitary wave. A series of wave gauges around an impervious breakwater were set-up in the flume to measure the solitary wave shoaling, run-up, and breaking processes. The results show that the measured data and simulated data are in good agreement. Moreover, simulated and measured surface elevations were analyzed by the wavelet transform method. It shows that different wave frequencies stratified in the wavelet amplitude spectrum. Finally, horizontal and vertical velocities of each layer interface were analyzed in the process of solitary wave propagation through submerged breakwater.
10. Alfven waves in the solar atmosphere. III - Nonlinear waves on open flux tubes
Science.gov (United States)
Hollweg, J. V.; Jackson, S.; Galloway, D.
1982-01-01
Consideration is given the nonlinear propagation of Alfven waves on solar magnetic flux tubes, where the tubes are taken to be vertical, axisymmetric and initially untwisted and the Alfven waves are time-dependent axisymmetric twists. The propagation of the waves into the chromosphere and corona is investigated through the numerical solution of a set of nonlinear, time-dependent equations coupling the Alfven waves into motions that are parallel to the initial magnetic field. It is concluded that Alfven waves can steepen into fast shocks in the chromosphere, pass through the transition region to produce high-velocity pulses, and then enter the corona, which they heat. The transition region pulses have amplitudes of about 60 km/sec, and durations of a few tens of seconds. In addition, the Alfven waves exhibit a tendency to drive upward flows, with many of the properties of spicules.
11. Nonlinear Pressure Wave Analysis by Concentrated Mass Model
Science.gov (United States)
Ishikawa, Satoshi; Kondou, Takahiro; Matsuzaki, Kenichiro
A pressure wave propagating in a tube often changes to a shock wave because of the nonlinear effect of fluid. Analyzing this phenomenon by the finite difference method requires high computational cost. To lessen the computational cost, a concentrated mass model is proposed. This model consists of masses, connecting nonlinear springs, connecting dampers, and base support dampers. The characteristic of a connecting nonlinear spring is derived from the adiabatic change of fluid, and the equivalent mass and equivalent damping coefficient of the base support damper are derived from the equation of motion of fluid in a cylindrical tube. Pressure waves generated in a hydraulic oil tube, a sound tube and a plane-wave tube are analyzed numerically by the proposed model to confirm the validity of the model. All numerical computational results agree very well with the experimental results carried out by Okamura, Saenger and Kamakura. Especially, the numerical analysis reproduces the phenomena that a pressure wave with large amplitude propagating in a sound tube or in a plane tube changes to a shock wave. Therefore, it is concluded that the proposed model is valid for the numerical analysis of nonlinear pressure wave problem.
12. Propagation behavior of acoustic wave in wood
Institute of Scientific and Technical Information of China (English)
Huadong Xu; Guoqi Xu; Lihai Wang; Lei Yu
2014-01-01
We used acoustic tests on a quarter-sawn poplar timbers to study the effects of wood anisotropy and cavity defects on acoustic wave velocity and travel path, and we investigated acoustic wave propagation behavior in wood. The timber specimens were first tested in unmodified condition and then tested after introduction of cavity defects of varying sizes to quantify the transmitting time of acoustic waves in laboratory conditions. Two-dimensional acoustic wave contour maps on the radial section of specimens were then simulated and analyzed based on the experimental data. We tested the relationship between wood grain and acoustic wave velocity as waves passed in various directions through wood. Wood anisotropy has significant effects on both velocity and travel path of acoustic waves, and the velocity of waves passing longitudinally through timbers exceeded the radial velocity. Moreover, cavity defects altered acoustic wave time contours on radial sections of timbers. Acous-tic wave transits from an excitation point to the region behind a cavity in defective wood more slowly than in intact wood.
13. Non-linear high-frequency waves in the magnetosphere
S Moolla; R Bharuthram; S V Singh; G S Lakhina
2003-12-01
Using fluid theory, a set of equations is derived for non-linear high-frequency waves propagating oblique to an external magnetic field in a three-component plasma consisting of hot electrons, cold electrons and cold ions. For parameters typical of the Earth’s magnetosphere, numerical solutions of the governing equations yield sinusoidal, sawtooth or bipolar wave-forms for the electric field.
14. Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber
Energy Technology Data Exchange (ETDEWEB)
Xie, Xi-Yang; Tian, Bo, E-mail: [email protected]; Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-15
In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.
15. GEOMETRICAL NONLINEAR WAVES IN FINITE DEFORMATION ELASTIC RODS
Institute of Scientific and Technical Information of China (English)
GUO Jian-gang; ZHOU Li-jun; ZHANG Shan-yuan
2005-01-01
By using Hamilton-type variation principle in non-conservation system, the nonlinear equation of wave motion of a elastic thin rod was derived according to Lagrange description of finite deformation theory. The dissipation caused due to viscous effect and the dispersion introduced by transverse inertia were taken into consideration so that steady traveling wave solution can be obtained. Using multi-scale method the nonlinear equation is reduced to a KdV-Burgers equation which corresponds with saddle-spiral heteroclinic orbit on phase plane. Its solution is called the oscillating-solitary wave or saddle-spiral shock wave.If viscous effect or transverse inertia is neglected, the equation is degraded to classical KdV or Burgers equation. The former implies a propagating solitary wave with homoclinic on phase plane, the latter means shock wave and heteroclinic orbit.
16. Multi-layer study of wave propagation in sunspots
CERN Document Server
Felipe, T; Collados, M; Beck, C
2010-01-01
We analyze the propagation of waves in sunspots from the photosphere to the chromosphere using time series of co-spatial Ca II H intensity spectra (including its line blends) and polarimetric spectra of Si I 10827 and the He I 10830 multiplet. From the Doppler shifts of these lines we retrieve the variation of the velocity along the line-of-sight at several heights. Phase spectra are used to obtain the relation between the oscillatory signals. Our analysis reveals standing waves at frequencies lower than 4 mHz and a continuous propagation of waves at higher frequencies, which steepen into shocks in the chromosphere when approaching the formation height of the Ca II H core. The observed non-linearities are weaker in Ca II H than in He I lines. Our analysis suggests that the Ca II H core forms at a lower height than the He I 10830 line: a time delay of about 20 s is measured between the Doppler signal detected at both wavelengths. We fit a model of linear slow magnetoacoustic wave propagation in a stratified at...
17. Surface acoustic wave propagation in graphene film
Energy Technology Data Exchange (ETDEWEB)
Roshchupkin, Dmitry, E-mail: [email protected]; Plotitcyna, Olga; Matveev, Viktor; Kononenko, Oleg; Emelin, Evgenii; Irzhak, Dmitry [Institute of Microelectronics Technology and High-Purity Materials Russian Academy of Sciences, Chernogolovka 142432 (Russian Federation); Ortega, Luc [Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, 91405 Orsay Cedex (France); Zizak, Ivo; Erko, Alexei [Institute for Nanometre Optics and Technology, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein Strasse 15, 12489 Berlin (Germany); Tynyshtykbayev, Kurbangali; Insepov, Zinetula [Nazarbayev University Research and Innovation System, 53 Kabanbay Batyr St., Astana 010000 (Kazakhstan)
2015-09-14
Surface acoustic wave (SAW) propagation in a graphene film on the surface of piezoelectric crystals was studied at the BESSY II synchrotron radiation source. Talbot effect enabled the visualization of the SAW propagation on the crystal surface with the graphene film in a real time mode, and high-resolution x-ray diffraction permitted the determination of the SAW amplitude in the graphene/piezoelectric crystal system. The influence of the SAW on the electrical properties of the graphene film was examined. It was shown that the changing of the SAW amplitude enables controlling the magnitude and direction of current in graphene film on the surface of piezoelectric crystals.
18. Surface acoustic wave propagation in graphene film
Science.gov (United States)
Roshchupkin, Dmitry; Ortega, Luc; Zizak, Ivo; Plotitcyna, Olga; Matveev, Viktor; Kononenko, Oleg; Emelin, Evgenii; Erko, Alexei; Tynyshtykbayev, Kurbangali; Irzhak, Dmitry; Insepov, Zinetula
2015-09-01
Surface acoustic wave (SAW) propagation in a graphene film on the surface of piezoelectric crystals was studied at the BESSY II synchrotron radiation source. Talbot effect enabled the visualization of the SAW propagation on the crystal surface with the graphene film in a real time mode, and high-resolution x-ray diffraction permitted the determination of the SAW amplitude in the graphene/piezoelectric crystal system. The influence of the SAW on the electrical properties of the graphene film was examined. It was shown that the changing of the SAW amplitude enables controlling the magnitude and direction of current in graphene film on the surface of piezoelectric crystals.
19. Nonlinear Electron Waves in Strongly Magnetized Plasmas
DEFF Research Database (Denmark)
Pécseli, Hans; Juul Rasmussen, Jens
1980-01-01
dynamics in the analysis is also demonstrated. As a particular case the authors investigate nonlinear waves in a strongly magnetized plasma filled wave-guide, where the effects of finite geometry are important. The relevance of this problem to laboratory experiments is discussed.......Weakly nonlinear dispersive electron waves in strongly magnetized plasma are considered. A modified nonlinear Schrodinger equation is derived taking into account the effect of particles resonating with the group velocity of the waves (nonlinear Landau damping). The possibility of including the ion...
20. Large-scale Globally Propagating Coronal Waves
Directory of Open Access Journals (Sweden)
Alexander Warmuth
2015-09-01
Full Text Available Large-scale, globally propagating wave-like disturbances have been observed in the solar chromosphere and by inference in the corona since the 1960s. However, detailed analysis of these phenomena has only been conducted since the late 1990s. This was prompted by the availability of high-cadence coronal imaging data from numerous spaced-based instruments, which routinely show spectacular globally propagating bright fronts. Coronal waves, as these perturbations are usually referred to, have now been observed in a wide range of spectral channels, yielding a wealth of information. Many findings have supported the “classical” interpretation of the disturbances: fast-mode MHD waves or shocks that are propagating in the solar corona. However, observations that seemed inconsistent with this picture have stimulated the development of alternative models in which “pseudo waves” are generated by magnetic reconfiguration in the framework of an expanding coronal mass ejection. This has resulted in a vigorous debate on the physical nature of these disturbances. This review focuses on demonstrating how the numerous observational findings of the last one and a half decades can be used to constrain our models of large-scale coronal waves, and how a coherent physical understanding of these disturbances is finally emerging.
1. Compactification of nonlinear patterns and waves.
Science.gov (United States)
Rosenau, Philip; Kashdan, Eugene
2008-12-31
We present a nonlinear mechanism(s) which may be an alternative to a missing wave speed: it induces patterns with a compact support and sharp fronts which propagate with a finite speed. Though such mechanism may emerge in a variety of physical contexts, its mathematical characterization is universal, very simple, and given via a sublinear substrate (site) force. Its utility is shown studying a Klein-Gordon -u(tt) + [phi/(u(x)]x = P'(u) equation, where phi'(sigma) = sigma + beta sigma3 and endowed with a subquadratic site potential P(u) approximately /1-u2/(alpha+1), 0 < or = alpha < 1, and the Schrödinger iZt + inverted delta2 Z = G(/Z/)Z equation in a plane with G(A) = gammaA(-delta) - sigmaA2, 0 < delta < or = 1.
2. Propagation des ondes élastiques dans les matériaux non linéaires Aperçu des résultats de laboratoire obtenus sur les roches et des applications possibles en géophysique Propagation of Elastic Waves in Nonlinear Materials Survey of Laboratory Results on Rock and Geophysical Applications
Directory of Open Access Journals (Sweden)
Rasolofosaon P.
2006-12-01
non-linéarité sous fort confinement, et qui pourraient engendrer un signal résultant d'une interaction onde-onde . Tempérant ce pessimisme, il faut noter qu'un éventuel signal d'interaction non linéaire présenterait l'avantage, quant à sa détection, d'être dans une bande de fréquence différente de celle des ondes utilisées pour l'engendrer. Bien que nous n'ayons pas connaissance d'essais d'application actuels, les perspectives paraissent plus encourageantes dans le domaine du génie civil ou minier. C'est dans le domaine diagraphique, où des distances de propagation sont très faibles, que des applications semblent possibles à moyen terme. Si l'on en juge par le dépôt très récent de plusieurs brevets, les compagnies de logging poursuivraient des recherches dans cette voie. A general and important characteristic of rocks is their elastically nonlinear behavior resulting in significant effects on wave propagation. The nonlinear response of rock is a direct consequence of the compliant nature of rock : the macro-and micro-structure of the material (microcracks, grain-to-grain contacts, etc. . As a result, the material modulus varies as a function of the applied pressure. Interest has grown significantly in the last several years, as illustrated by the increasing number of publications regarding this topic. Here we present a summary of the fundamentals of theory and of experimental observations characteristic of rock, and we address possible applications in geophysics. Two disciplines regarding the nonlinear elasticity of rock have been developed over recent years in tandem :- Acoustoelasticity where wave propagation in statically, prestressed materials is studied. Here one relates the variation in applied pressure to the elastic wavespeed in order to extract the nonlinear coefficients. This area of study includes the topic of stress-induced anisotropy. - Acoustic nonlinearity where we are interested in the temporary and local variation in the elastic
3. Nonlinear Fourier analysis with cnoidal waves
Energy Technology Data Exchange (ETDEWEB)
Osborne, A.R. [Dipt. di Fisica Generale dellUniversita, Torino (Italy)
1996-12-31
Fourier analysis is one of the most useful tools to the ocean engineer. The approach allows one to analyze wave data and thereby to describe a dynamical motion in terms of a linear superposition of ordinary sine waves. Furthermore, the Fourier technique allows one to compute the response function of a fixed or floating structure: each sine wave in the wave or force spectrum yields a sine wave in the response spectrum. The counting of fatigue cycles is another area where the predictable oscillations of sine waves yield procedures for the estimation of the fatigue life of structures. The ocean environment, however, is a source of a number of nonlinear effects which must also be included in structure design. Nonlinearities in ocean waves deform the sinusoidal shapes into other kinds of waves such as the Stokes wave, cnoidal wave or solitary wave. A key question is: Does there exist a generalization of linear Fourier analysis which uses nonlinear basis functions rather than the familiar sine waves? Herein addresses the dynamics of nonlinear wave motion in shallow water where the basis functions are cnoidal waves and discuss nonlinear Fourier analysis in terms of a linear superposition of cnoidal waves plus their mutual nonlinear interactions. He gives a number of simple examples of nonlinear Fourier wave motion and then analyzes an actual surface-wave time series obtained on an offshore platform in the Adriatic Sea. Finally, he briefly discusses application of the cnoidal wave spectral approach to the computation of the frequency response function of a floating vessel. The results given herein will prove useful in future engineering studies for the design of fixed, floating and complaint offshore structures.
4. Propagating wave correlations in complex systems
Science.gov (United States)
Creagh, Stephen C.; Gradoni, Gabriele; Hartmann, Timo; Tanner, Gregor
2017-01-01
We describe a novel approach for computing wave correlation functions inside finite spatial domains driven by complex and statistical sources. By exploiting semiclassical approximations, we provide explicit algorithms to calculate the local mean of these correlation functions in terms of the underlying classical dynamics. By defining appropriate ensemble averages, we show that fluctuations about the mean can be characterised in terms of classical correlations. We give in particular an explicit expression relating fluctuations of diagonal contributions to those of the full wave correlation function. The methods have a wide range of applications both in quantum mechanics and for classical wave problems such as in vibro-acoustics and electromagnetism. We apply the methods here to simple quantum systems, so-called quantum maps, which model the behaviour of generic problems on Poincaré sections. Although low-dimensional, these models exhibit a chaotic classical limit and share common characteristics with wave propagation in complex structures.
5. Nonlinear light propagation in fs laser-written waveguide arrays
Directory of Open Access Journals (Sweden)
Szameit A.
2013-11-01
Full Text Available We report on recent achievements in the field of nonlinear light propagation in fs laser-written waveguide lattices. Particular emphasis is thereby given on discrete solitons in such systems.
6. Nonlinear effects in propagation of radiation of X-ray free-electron lasers
Science.gov (United States)
Nosik, V. L.
2016-05-01
Nonlinear effects accompanying the propagation of high-intensity beams of X-ray free-electron lasers are considered. It is shown that the X-ray wave field in the crystal significantly changes due to the formation of "hollow" atomic shells as a result of the photoelectric effect.
7. Numerical studies of nonlinear ultrasonic guided waves in uniform waveguides with arbitrary cross sections
Energy Technology Data Exchange (ETDEWEB)
Zuo, Peng; Fan, Zheng, E-mail: [email protected] [School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 (Singapore); Zhou, Yu [Advanced Remanufacturing and Technology Center (ARTC), 3 Clean Tech Loop, CleanTech Two, Singapore 637143 (Singapore)
2016-07-15
Nonlinear guided waves have been investigated widely in simple geometries, such as plates, pipe and shells, where analytical solutions have been developed. This paper extends the application of nonlinear guided waves to waveguides with arbitrary cross sections. The criteria for the existence of nonlinear guided waves were summarized based on the finite deformation theory and nonlinear material properties. Numerical models were developed for the analysis of nonlinear guided waves in complex geometries, including nonlinear Semi-Analytical Finite Element (SAFE) method to identify internal resonant modes in complex waveguides, and Finite Element (FE) models to simulate the nonlinear wave propagation at resonant frequencies. Two examples, an aluminum plate and a steel rectangular bar, were studied using the proposed numerical model, demonstrating the existence of nonlinear guided waves in such structures and the energy transfer from primary to secondary modes.
8. Fractional Calculus in Wave Propagation Problems
CERN Document Server
Mainardi, Francesco
2012-01-01
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.
9. Obliquely propagating dust-density waves
Science.gov (United States)
Piel, A.; Arp, O.; Klindworth, M.; Melzer, A.
2008-02-01
Self-excited dust-density waves are experimentally studied in a dusty plasma under microgravity. Two types of waves are observed: a mode inside the dust volume propagating in the direction of the ion flow and another mode propagating obliquely at the boundary between the dusty plasma and the space charge sheath. The dominance of oblique modes can be described in the frame of a fluid model. It is shown that the results fom the fluid model agree remarkably well with a kinetic electrostatic model of Rosenberg [J. Vac. Sci. Technol. A 14, 631 (1996)]. In the experiment, the instability is quenched by increasing the gas pressure or decreasing the dust density. The critical pressure and dust density are well described by the models.
10. Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices.
Science.gov (United States)
Chong, C; Kevrekidis, P G; Ablowitz, M J; Ma, Yi-Ping
2016-01-01
Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wave packet and via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression, i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a nonoscillatory nature, resulting from the complex interplay among the discreteness, nonlinearity, and geometry of the packing. The transition between these two types of propagation is explored.
11. Wave propagation retrieval method for chiral metamaterials
DEFF Research Database (Denmark)
Andryieuski, Andrei; Malureanu, Radu; Lavrinenko, Andrei
2010-01-01
In this paper we present the wave propagation method for the retrieving of effective properties of media with circularly polarized eigenwaves, in particularly for chiral metamaterials. The method is applied for thick slabs and provides bulk effective parameters. Its strong sides are the absence...... of artificial branches of the refractive index and simplicity in implementation. We prove the validity of the method on three case studies of homogeneous magnetized plasma, bi-cross and U-shaped metamaterials....
12. Particle velocity non-uniformity and steady-wave propagation
Science.gov (United States)
Meshcheryakov, Yu. I.
2017-03-01
A constitutive equation grounded in dislocation dynamics is shown to be incapable of describing the propagation of shock fronts in solids. Shock wave experiments and theoretical investigations motivate an additional collective mechanism of stress relaxation that should be incorporated into the model through the standard deviation of the particle velocity, which is found to be proportional to the strain rate. In this case, the governing equation system results in a second-order differential equation of square non-linearity. Solution to this equation and calculations for D16 aluminum alloy show a more precise coincidence of the theoretical and experimental velocity profiles.
13. Development of A Fully Nonlinear Numerical Wave Tank
Institute of Scientific and Technical Information of China (English)
陈永平; 李志伟; 张长宽
2004-01-01
A fully nonlinear numerical wave tank (NWT) based on the solution of the σ-transformed Navier-Stokes equation is developed in this study. The numerical wave is generated from the inflow boundary, where the surface elevation and/or velocity are specified by use of the analytical solution or the laboratory data. The Sommerfeld/Orlanski radiation condition in conjunction with an artificial damping zone is applied to reduce wave reflection from the outflow boundary. The whole numerical solution procedures are split into three steps, i.e., advection, diffusion and propagation, and a new method,the Lagrange-Euler Method, instead of the MAC or VOF method, is introduced to solve the free surface elevation at the new time step. Several typical wave cases, including solitary waves, regular waves and irregular waves, are simulated in the wave tank. The robustness and accuracy of the NWT are verified by the good agreement between the numerical results and the linear or nonlinear analytical solutions. This research will be further developed by study of wave-wave, wave-current, wave-structure or wave-jet interaction in the future.
14. Wave Propagation in Jointed Geologic Media
Energy Technology Data Exchange (ETDEWEB)
Antoun, T
2009-12-17
Predictive modeling capabilities for wave propagation in a jointed geologic media remain a modern day scientific frontier. In part this is due to a lack of comprehensive understanding of the complex physical processes associated with the transient response of geologic material, and in part it is due to numerical challenges that prohibit accurate representation of the heterogeneities that influence the material response. Constitutive models whose properties are determined from laboratory experiments on intact samples have been shown to over-predict the free field environment in large scale field experiments. Current methodologies for deriving in situ properties from laboratory measured properties are based on empirical equations derived for static geomechanical applications involving loads of lower intensity and much longer durations than those encountered in applications of interest involving wave propagation. These methodologies are not validated for dynamic applications, and they do not account for anisotropic behavior stemming from direcitonal effects associated with the orientation of joint sets in realistic geologies. Recent advances in modeling capabilities coupled with modern high performance computing platforms enable physics-based simulations of jointed geologic media with unprecedented details, offering a prospect for significant advances in the state of the art. This report provides a brief overview of these modern computational approaches, discusses their advantages and limitations, and attempts to formulate an integrated framework leading to the development of predictive modeling capabilities for wave propagation in jointed and fractured geologic materials.
15. Measurement and fitting techniques for the assessment of material nonlinearity using nonlinear Rayleigh waves
Energy Technology Data Exchange (ETDEWEB)
Torello, David [GW Woodruff School of Mechanical Engineering, Georgia Tech (United States); Kim, Jin-Yeon [School of Civil and Environmental Engineering, Georgia Tech (United States); Qu, Jianmin [Department of Civil and Environmental Engineering, Northwestern University (United States); Jacobs, Laurence J. [School of Civil and Environmental Engineering, Georgia Tech and GW Woodruff School of Mechanical Engineering, Georgia Tech (United States)
2015-03-31
This research considers the effects of diffraction, attenuation, and the nonlinearity of generating sources on measurements of nonlinear ultrasonic Rayleigh wave propagation. A new theoretical framework for correcting measurements made with air-coupled and contact piezoelectric receivers for the aforementioned effects is provided based on analytical models and experimental considerations. A method for extracting the nonlinearity parameter β{sub 11} is proposed based on a nonlinear least squares curve-fitting algorithm that is tailored for Rayleigh wave measurements. Quantitative experiments are conducted to confirm the predictions for the nonlinearity of the piezoelectric source and to demonstrate the effectiveness of the curve-fitting procedure. These experiments are conducted on aluminum 2024 and 7075 specimens and a β{sub 11}{sup 7075}/β{sub 11}{sup 2024} measure of 1.363 agrees well with previous literature and earlier work.
16. Emergent geometries and nonlinear-wave dynamics in photon fluids.
Science.gov (United States)
Marino, F; Maitland, C; Vocke, D; Ortolan, A; Faccio, D
2016-03-22
Nonlinear waves in defocusing media are investigated in the framework of the hydrodynamic description of light as a photon fluid. The observations are interpreted in terms of an emergent curved spacetime generated by the waves themselves, which fully determines their dynamics. The spacetime geometry emerges naturally as a result of the nonlinear interaction between the waves and the self-induced background flow. In particular, as observed in real fluids, different points of the wave profile propagate at different velocities leading to the self-steepening of the wave front and to the formation of a shock. This phenomenon can be associated to a curvature singularity of the emergent metric. Our analysis offers an alternative insight into the problem of shock formation and provides a demonstration of an analogue gravity model that goes beyond the kinematic level.
17. EXACT ANALYSIS OF WAVE PROPAGATION IN AN INFINITE RECTANGULAR BEAM
Institute of Scientific and Technical Information of China (English)
孙卫明; 杨光松; 李东旭
2004-01-01
The Fourier series method was extended for the exact analysis of wave propagation in an infinite rectangular beam. Initially, by solving the three-dimensional elastodynamic equations a general analytic solution was derived for wave motion within the beam. And then for the beam with stress-free boundaries, the propagation characteristics of elastic waves were presented. This accurate wave propagation model lays a solid foundation of simultaneous control of coupled waves in the beam.
18. Laser beam propagation in non-linearly absorbing media
CSIR Research Space (South Africa)
Forbes, A
2006-08-01
Full Text Available Many analytical techniques exist to explore the propagation of certain laser beams in free space, or in a linearly absorbing medium. When the medium is nonlinearly absorbing the propagation must be described by an iterative process using the well...
19. Quantification and prediction of rare events in nonlinear waves
Science.gov (United States)
Sapsis, Themistoklis; Cousins, Will; Mohamad, Mustafa
2014-11-01
The scope of this work is the quantification and prediction of rare events characterized by extreme intensity, in nonlinear dispersive models that simulate water waves. In particular we are interested for the understanding and the short-term prediction of rogue waves in the ocean and to this end, we consider 1-dimensional nonlinear models of the NLS type. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A stochastic analysis of the Gabor coefficients reveals i) the low-dimensionality of the intermittent structures, ii) the interplay between non-Gaussian statistical properties and nonlinear energy transfers between modes, as well as iii) the critical scales (or Gabor coefficients) where a critical energy can trigger the formation of an extreme event. The unstable character of these critical localized modes is analysed directly through the system equation and it is shown that it is defined as the result of the system nonlinearity and the wave dissipation (that mimics wave breaking). These unstable modes are randomly triggered through the dispersive heat bath'' of random waves that propagate in the nonlinear medium. Using these properties we formulate low-dimensional functionals of these Gabor coefficients that allow for the prediction of extreme event well before the strongly nonlinear interactions begin to occur. The prediction window is further enhanced by the combination of the developed scheme with traditional filtering schemes.
20. Nonlinear evolution of whistler wave modulational instability
DEFF Research Database (Denmark)
Karpman, V.I.; Lynov, Jens-Peter; Michelsen, Poul;
1995-01-01
The nonlinear evolution of the modulational instability of whistler waves coupled to fast magnetosonic waves (FMS) and to slow magnetosonic waves (SMS) is investigated. Results from direct numerical solutions in two spatial dimensions agree with simplified results from a set of ordinary different......The nonlinear evolution of the modulational instability of whistler waves coupled to fast magnetosonic waves (FMS) and to slow magnetosonic waves (SMS) is investigated. Results from direct numerical solutions in two spatial dimensions agree with simplified results from a set of ordinary...
1. Nonlinear dynamics of Airy-Vortex 3D wave packets: Emission of vortex light waves
CERN Document Server
Driben, Rodislav
2014-01-01
The dynamics of 3D Airy-vortex wave packets is studied under the action of strong self-focusing Kerr nonlinearity. Emissions of nonlinear 3D waves out of the main wave packets with the topological charges were demonstrated. Due to the conservation of the total angular momentum, charges of the emitted waves are equal to those carried by the parental light structure. The rapid collapse imposes a severe limitation on the propagation of multidimensional waves in Kerr media. However, the structure of the Airy beam carrier allows the coupling of light from the leading, most intense peak into neighboring peaks and consequently strongly postpones the collapse. The dependence of the critical input amplitude for the appearance of a fast collapse on the beam width is studied for wave packets with zero and non-zero topological charges. Wave packets carrying angular momentum are found to be much more resistant to the rapid collapse, especially those having small width.
2. Nonlinear dynamics of Airy-vortex 3D wave packets: emission of vortex light waves.
Science.gov (United States)
Driben, Rodislav; Meier, Torsten
2014-10-01
The dynamics of 3D Airy-vortex wave packets is studied under the action of strong self-focusing Kerr nonlinearity. Emissions of nonlinear 3D waves out of the main wave packets with the topological charges were demonstrated. Because of the conservation of the total angular momentum, charges of the emitted waves are equal to those carried by the parental light structure. The rapid collapse imposes a severe limitation on the propagation of multidimensional waves in Kerr media. However, the structure of the Airy beam carrier allows the coupling of light from the leading, most intense peak into neighboring peaks and consequently strongly postpones the collapse. The dependence of the critical input amplitude for the appearance of a fast collapse on the beam width is studied for wave packets with zero and nonzero topological charges. Wave packets carrying angular momentum are found to be much more resistant to the rapid collapse.
3. Seismic Wave Propagation on the Tablet Computer
Science.gov (United States)
Emoto, K.
2015-12-01
Tablet computers widely used in recent years. The performance of the tablet computer is improving year by year. Some of them have performance comparable to the personal computer of a few years ago with respect to the calculation speed and the memory size. The convenience and the intuitive operation are the advantage of the tablet computer compared to the desktop PC. I developed the iPad application of the numerical simulation of the seismic wave propagation. The numerical simulation is based on the 2D finite difference method with the staggered-grid scheme. The number of the grid points is 512 x 384 = 196,608. The grid space is 200m in both horizontal and vertical directions. That is the calculation area is 102km x 77km. The time step is 0.01s. In order to reduce the user waiting time, the image of the wave field is drawn simultaneously with the calculation rather than playing the movie after the whole calculation. P and S wave energies are plotted on the screen every 20 steps (0.2s). There is the trade-off between the smooth simulation and the resolution of the wave field image. In the current setting, it takes about 30s to calculate the 10s wave propagation (50 times image updates). The seismogram at the receiver is displayed below of the wave field updated in real time. The default medium structure consists of 3 layers. The layer boundary is defined by 10 movable points with linear interpolation. Users can intuitively change to the arbitrary boundary shape by moving the point. Also users can easily change the source and the receiver positions. The favorite structure can be saved and loaded. For the advance simulation, users can introduce the random velocity fluctuation whose spectrum can be changed to the arbitrary shape. By using this application, everyone can simulate the seismic wave propagation without the special knowledge of the elastic wave equation. So far, the Japanese version of the application is released on the App Store. Now I am preparing the
4. Third harmonic generation of shear horizontal guided waves propagation in plate-like structures
Energy Technology Data Exchange (ETDEWEB)
Li, Wei Bin [School of Aerospace Engineering, Xiamen University, Xiamen (China); Xu, Chun Guang [School of Mechanical Engineering, Beijing Institute of Technology, Beijing (China); Cho, Youn Ho [School of Mechanical Engineering, Pusan National University, Busan (Korea, Republic of)
2016-04-15
The use of nonlinear ultrasonics wave has been accepted as a promising tool for monitoring material states related to microstructural changes, as it has improved sensitivity compared to conventional non-destructive testing approaches. In this paper, third harmonic generation of shear horizontal guided waves propagating in an isotropic plate is investigated using the perturbation method and modal analysis approach. An experimental procedure is proposed to detect the third harmonics of shear horizontal guided waves by electromagnetic transducers. The strongly nonlinear response of shear horizontal guided waves is measured. The accumulative growth of relative acoustic nonlinear response with an increase of propagation distance is detected in this investigation. The experimental results agree with the theoretical prediction, and thus providing another indication of the feasibility of using higher harmonic generation of electromagnetic shear horizontal guided waves for material characterization.
5. Spatial localization of nonlinear waves spreading in materials in the presence of dislocations and point defects
Science.gov (United States)
Erofeev, V. I.; Leontieva, A. V.; Malkhanov, A. O.
2017-06-01
Within the framework of self consistent dynamic problems, the impact of dislocations and point defects on the spatial localization of nonlinear acoustic waves propagating in materials has been studied.
6. Nonlinear scattering of radio waves by metal objects
Science.gov (United States)
Shteynshleyger, V. B.
1984-07-01
Nonlinear scattering of radio waves by metal structures with resulting harmonic and intermodulation interference is analyzed from both theoretical and empirical standpoints, disregarding nonlinear effects associated with the nonlinear dependence of the electric or magnetic polarization vector on respectively the electric or magnetic field intensity in the wave propagating medium. Nonlinear characteristics of metal-oxide-metal contacts where the thin oxide film separation two metal surfaces has properties approximately those of a dielectric or a high-resistivity semiconductor are discussed. Tunneling was found to be the principal mechanism of charge carrier transfer through such a contact with a sufficiently thin film, the contact having usually a cubic or sometimes an integral sign current-voltage characteristic at 300 K and usually S-form or sometimes a cubic current-voltage characteristic at 77 K.
7. Wave propagation in axially moving periodic strings
Science.gov (United States)
Sorokin, Vladislav S.; Thomsen, Jon Juel
2017-04-01
The paper deals with analytically studying transverse waves propagation in an axially moving string with periodically modulated cross section. The structure effectively models various relevant technological systems, e.g. belts, thread lines, band saws, etc., and, in particular, roller chain drives for diesel engines by capturing both their spatial periodicity and axial motion. The Method of Varying Amplitudes is employed in the analysis. It is shown that the compound wave traveling in the axially moving periodic string comprises many components with different frequencies and wavenumbers. This is in contrast to non-moving periodic structures, for which all components of the corresponding compound wave feature the same frequency. Due to this "multi-frequency" character of the wave motion, the conventional notion of frequency band-gaps appears to be not applicable for the moving periodic strings. Thus, for such structures, by frequency band-gaps it is proposed to understand frequency ranges in which the primary component of the compound wave attenuates. Such frequency band-gaps can be present for a moving periodic string, but only if its axial velocity is lower than the transverse wave speed, and, the higher the axial velocity, the narrower the frequency band-gaps. The revealed effects could be of potential importance for applications, e.g. they indicate that due to spatial inhomogeneity, oscillations of axially moving periodic chains always involve a multitude of frequencies.
8. Propagation of a constant velocity fission wave
Science.gov (United States)
Deinert, Mark
2011-10-01
The ideal nuclear fuel cycle would require no enrichment, minimize the need fresh uranium, and produce few, if any, transuranic elements. Importantly, the latter goal would be met without the reprocessing. For purely physical reasons, no reactor system or fuel cycle can meet all of these objectives. However, a traveling-wave reactor, if feasible, could come remarkably close. The concept is simple: a large cylinder of natural (or depleted) uranium is subjected to a fast neutron source at one end, the neutrons would transmute the uranium downstream and produce plutonium. If the conditions were right, a self-sustaining fission wave would form, producing yet more neutrons which would breed more plutonium and leave behind little more than short-lived fission products. Numerical studies have shown that fission waves of this type are also possible. We have derived an exact solution for the propagation velocity of a fission wave through fertile material. The results show that these waves fall into a class of traveling wave phenomena that have been encountered in other systems. The solution places a strict conditions on the shapes of the flux, diffusive, and reactive profiles that would be required for such a phenomenon to persist. The results are confirmed numerically.
9. SPP propagation in nonlinear glass-metal interface
KAUST Repository
Sagor, Rakibul Hasan
2011-12-01
The non-linear propagation of Surface-Plasmon-Polaritons (SPP) in single interface of metal and chalcogenide glass (ChG) is considered. A time domain simulation algorithm is developed using the Finite Difference Time Domain (FDTD) method. The general polarization algorithm incorporated in the auxiliary differential equation (ADE) is used to model frequency-dependent dispersion relation and third-order nonlinearity of ChG. The main objective is to observe the nonlinear behavior of SPP propagation and study the dynamics of the whole structure. © 2011 IEEE.
10. Propagating magnetohydrodynamics waves in coronal loops.
Science.gov (United States)
De Moortel, I
2006-02-15
High cadence Transition Region and Coronal Explorer (TRACE) observations show that outward propagating intensity disturbances are a common feature in large, quiescent coronal loops, close to active regions. An overview is given of measured parameters of such longitudinal oscillations in coronal loops. The observed oscillations are interpreted as propagating slow magnetoacoustic waves and are unlikely to be flare-driven. A strong correlation, between the loop position and the periodicity of the oscillations, provides evidence that the underlying oscillations can propagate through the transition region and into the corona. Both a one- and a two-dimensional theoretical model of slow magnetoacoustic waves are presented to explain the very short observed damping lengths. The results of these numerical simulations are compared with the TRACE observations and show that a combination of the area divergence and thermal conduction agrees well with the observed amplitude decay. Additionally, the usefulness of wavelet analysis is discussed, showing that care has to be taken when interpreting the results of wavelet analysis, and a good knowledge of all possible factors that might influence or distort the results is a necessity.
11. Symmetry Breaking of Counter-Propagating Light in a Nonlinear Resonator
Science.gov (United States)
Del Bino, Leonardo; Silver, Jonathan M.; Stebbings, Sarah L.; Del'Haye, Pascal
2017-01-01
Spontaneous symmetry breaking is a concept of fundamental importance in many areas of physics, underpinning such diverse phenomena as ferromagnetism, superconductivity, superfluidity and the Higgs mechanism. Here we demonstrate nonreciprocity and spontaneous symmetry breaking between counter-propagating light in dielectric microresonators. The symmetry breaking corresponds to a resonance frequency splitting that allows only one of two counter-propagating (but otherwise identical) states of light to circulate in the resonator. Equivalently, this effect can be seen as the collapse of standing waves and transition to travelling waves within the resonator. We present theoretical calculations to show that the symmetry breaking is induced by Kerr-nonlinearity-mediated interaction between the counter-propagating light. Our findings pave the way for a variety of applications including optically controllable circulators and isolators, all-optical switching, nonlinear-enhanced rotation sensing, optical flip-flops for photonic memories as well as exceptionally sensitive power and refractive index sensors. PMID:28220865
12. (3+1)-dimensional nonlinear propagation equation for ultrashort pulsed beam in left-handed material
Institute of Scientific and Technical Information of China (English)
Hu Yong-Hua; Fu Xi-Quan; Wen Shuang-Chun; Su Wen-Hua; Fan Dian-Yuan
2006-01-01
In this paper a comprehensive framework for treating the nonlinear propagation of ultrashort pulse in metamaterial with dispersive dielectric susceptibility and magnetic permeability is presented. Under the slowly-evolving-wave approximation, a generalized (3+1)-dimensional wave equation first order in the propagation coordinate and suitable for both right-handed material (RHM) and left-handed material (LHM) is derived. By the commonly used Drude dispersive model for LHM, a (3+1)-dimensional nonlinear Schr(o)dinger equation describing ultrashort pulsed beam propagation in LHM is obtained, and its difference from that for conventional RHM is discussed. Particularly, the self-steeping effect of ultrashort pulse is found to be anomalous in LHM.
13. Standing waves for discrete nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Ming Jia
2016-07-01
Full Text Available The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. By using critical point theory, we establish some new sufficient conditions on the existence results of standing waves for the discrete nonlinear Schrodinger equations. We give an appropriate example to illustrate the conclusion obtained.
14. Torsional wave propagation in solar tornadoes
Science.gov (United States)
Vasheghani Farahani, S.; Ghanbari, E.; Ghaffari, G.; Safari, H.
2017-03-01
Aims: We investigate the propagation of torsional waves in coronal structures together with their collimation effects in the context of magnetohydrodynamic (MHD) theory. The interplay of the equilibrium twist and rotation of the structure, e.g. jet or tornado, together with the density contrast of its internal and external media is studied to shed light on the nature of torsional waves. Methods: We consider a rotating magnetic cylinder embedded in a plasma with a straight magnetic field. This resembles a solar tornado. In order to express the dispersion relations and phase speeds of the axisymmetric magnetohydrodynamic waves, the second-order thin flux tube approximation is implemented for the internal medium and the ideal MHD equations are implemented for the external medium. Results: The explicit expressions for the phase speed of the torsional wave show the modification of the torsional wave speed due to the equilibrium twist, rotation, and density contrast of the tornado. The speeds could be either sub-Alfvénic or ultra-Alfvénic depending on whether the equilibrium twist or rotation is dominant. The equilibrium twist increases the phase speed while the equilibrium rotation decreases it. The good agreement between the explicit versions for the phase speed and that obtained numerically proves adequate for the robustness of the model and method. The density ratio of the internal and external media also play a significant role in the speed and dispersion. Conclusions: The dispersion of the torsional wave is an indication of the compressibility of the oscillations. When the cylinder is rotating or twisted, in contrast to when it only possesses a straight magnetic field, the torsional wave is a collective mode. In this case its phase speed is determined by the Alfvén waves inside and outside the tornado.
15. Parametric instabilities of large amplitude Alfven waves with obliquely propagating sidebands
Science.gov (United States)
Vinas, A. F.; Goldstein, M. L.
1992-01-01
This paper presents a brief report on properties of the parametric decay and modulational, filamentation, and magnetoacoustic instabilities of a large amplitude, circularly polarized Alfven wave. We allow the daughter and sideband waves to propagate at an arbitrary angle to the background magnetic field so that the electrostatic and electromagnetic characteristics of these waves are coupled. We investigate the dependance of these instabilities on dispersion, plasma/beta, pump wave amplitude, and propagation angle. Analytical and numerical results are compared with numerical simulations to investigate the full nonlinear evolution of these instabilities.
16. Solving Nonlinear Wave Equations by Elliptic Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.
17. Extensions of nonlinear error propagation analysis for explicit pseudodynamic testing
Institute of Scientific and Technical Information of China (English)
Shuenn-Yih Chang
2009-01-01
Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm (Chang, 2003) are presented. One extends the stability study from a given time step to a complete step-by-step integration procedure. It is analytically proven that ensuring stability conditions in each time step leads to a stable computation of the entire step-by-step integration procedure. The other extension shows that the nonlinear error propagation results, which are derived for a nonlinear single degree of freedom (SDOF) system, can be applied to a nonlinear multiple degree of freedom (MDOF) system. This application is dependent upon the determination of the natural frequencies of the system in each time step, since all the numerical properties and error propagation properties in the time step are closely related to these frequencies. The results are derived from the step degree of nonlinearity. An instantaneous degree of nonlinearity is introduced to replace the step degree of nonlinearity and is shown to be easier to use in practice. The extensions can be also applied to the results derived from a SDOF system based on the instantaneous degree of nonlinearity, and hence a time step might be appropriately chosen to perform a pseudodynamic test prior to testing.
18. 2D wave-front shaping in optical superlattices using nonlinear volume holography.
Science.gov (United States)
Yang, Bo; Hong, Xu-Hao; Lu, Rong-Er; Yue, Yang-Yang; Zhang, Chao; Qin, Yi-Qiang; Zhu, Yong-Yuan
2016-07-01
Nonlinear volume holography is employed to realize arbitrary wave-front shaping during nonlinear processes with properly designed 2D optical superlattices. The concept of a nonlinear polarization wave in nonlinear volume holography is investigated. The holographic imaging of irregular patterns was performed using 2D LiTaO3 crystals with fundamental wave propagating along the spontaneous polarization direction, and the results agree well with the theoretical predictions. This Letter not only extends the application area of optical superlattices, but also offers an efficient method for wave-front shaping technology.
19. Propagation regimes and populations of internal waves in the Mediterranean Sea basin
Science.gov (United States)
Kurkina, Oxana; Rouvinskaya, Ekaterina; Talipova, Tatiana; Soomere, Tarmo
2017-02-01
The geographical and seasonal distributions of kinematic and nonlinear parameters of long internal waves are derived from the Generalized Digital Environmental Model (GDEM) climatology for the Mediterranean Sea region, including the Black Sea. The considered parameters are phase speed of long internal waves and the coefficients at the dispersion, quadratic and cubic terms of the weakly-nonlinear Korteweg-de Vries-type models (in particular, the Gardner model). These parameters govern the possible polarities and shapes of solitary internal waves, their limiting amplitudes and propagation speeds. The key outcome is an express estimate of the expected parameters of internal waves for different regions of the Mediterranean basin.
20. Investigation into stress wave propagation in metal foams
Directory of Open Access Journals (Sweden)
Li Lang
2015-01-01
Full Text Available The aim of this study is to investigate stress wave propagation in metal foams under high-speed impact loading. Three-dimensional Voronoi model is established to represent real closed-cell foam. Based on the one-dimensional stress wave theory and Voronoi model, a numerical model is developed to calculate the velocity of elastic wave and shock wave in metal foam. The effects of impact velocity and relative density of metal foam on the stress wave propagation in metal foams are explored respectively. The results show that both elastic wave and shock wave propagate faster in metal foams with larger relative density; with increasing the impact velocity, the shock wave propagation velocity increase, but the elastic wave propagation is not sensitive to the impact velocity.
1. An optimal design problem in wave propagation
DEFF Research Database (Denmark)
Bellido, J.C.; Donoso, Alberto
2007-01-01
We consider an optimal design problem in wave propagation proposed in Sigmund and Jensen (Roy. Soc. Lond. Philos. Trans. Ser. A 361:1001-1019, 2003) in the one-dimensional situation: Given two materials at our disposal with different elastic Young modulus and different density, the problem consists...... of finding the best distributions of the two initial materials in a rod in order to minimize the vibration energy in the structure under periodic loading of driving frequency Omega. We comment on relaxation and optimality conditions, and perform numerical simulations of the optimal configurations. We prove...
2. Stationary Rossby wave propagation through easterly layers
Science.gov (United States)
Schneider, E. K.; Watterson, I. G.
1984-01-01
The zonal mean basic state sensitivity of the steady response to midlatitude mountain forcing is examined through the numerical solution of linearized shallow water equations on a sphere. The zonal mean basic state consists of meridionally varying zonal winds and meridional winds. Attention is given to cases in which the former are westerly everywhere, except within a tropical region in which they are easterly. A zonal wavenumber three mountain confined to the Northern Hemisphere midlatitudes provides the forcing. It is concluded that critical latitude effects on wave propagation are sensitive to mean meridional circulation structure in the critical latitude region of the model.
3. Nonlinear propagation of focused ultrasound in layered biological tissues based on angular spectrum approach
Institute of Scientific and Technical Information of China (English)
Zhu Xiao-Feng; Zhou Lin; Zhang Dong; Gong Xiu-Fen
2005-01-01
Nonlinear propagation of focused ultrasound in layered biological tissues is theoretically studied by using the angular spectrum approach (ASA), in which an acoustic wave is decomposed into its angular spectrum, and the distribution of nonlinear acoustic fields is calculated in arbitrary planes normal to the acoustic axis. Several biological tissues are used as specimens inserted into the focusing region illuminated by a focused piston source. The second harmonic components within or beyond the biological specimens are numerically calculated. Validity of the theoretical model is examined by measurements. This approach employing the fast Fourier transformation gives a clear visualization of the focused ultrasound, which is helpful for nonlinear ultrasonic imaging.
4. Higher order contribution to the propagation characteristics of low frequency transverse waves in a dusty plasma
A P Misra; A Roy Chowdhury; S N Paul
2004-09-01
Characteristic features of low frequency transverse wave propagating in a magnetised dusty plasma have been analysed considering the effect of dust-charge fluctuation. The distinctive behaviours of both the left circularly polarised and right circularly polarised waves have been exhibited through the analysis of linear and non-linear dispersion relations. The phase velocity, group velocity, and group travel time for the waves have been obtained and their propagation characteristics have been shown graphically with the variations of wave frequency, dust density and amplitude of the wave. The change in non-linear wave number shift and Faraday rotation angle have also been exhibited with respect to the plasma parameters. It is observed that the effects of dust particles are significant only when the higher order contributions are considered. This may be referred to as the dust regime' in plasma.
5. Self-similar propagation of Hermite-Gauss water-wave pulses.
Science.gov (United States)
Fu, Shenhe; Tsur, Yuval; Zhou, Jianying; Shemer, Lev; Arie, Ady
2016-01-01
We demonstrate both theoretically and experimentally propagation dynamics of surface gravity water-wave pulses, having Hermite-Gauss envelopes. We show that these waves propagate self-similarly along an 18-m wave tank, preserving their general Hermite-Gauss envelopes in both the linear and the nonlinear regimes. The measured surface elevation wave groups enable observing the envelope phase evolution of both nonchirped and linearly frequency chirped Hermite-Gauss pulses, hence allowing us to measure Gouy phase shifts of high-order Hermite-Gauss pulses for the first time. Finally, when increasing pulse amplitude, nonlinearity becomes essential and the second harmonic of Hermite-Gauss waves was observed. We further show that these generated second harmonic bound waves still exhibit self-similar Hermite-Gauss shapes along the tank.
6. Nonlinear Dispersion Relation in Wave Transformation
Institute of Scientific and Technical Information of China (English)
李瑞杰; 严以新; 曹宏生
2003-01-01
A nonlinear dispersion relation is presented to model the nonlinear dispersion of waves over the whole range of possible water depths. It reduces the phase speed over-prediction of both Hedges′ modified relation and Kirby and Dalrymple′s modified relation in the region of 1＜kh＜1.5 for small wave steepness and maintains the monotonicity in phase speed variation for large wave steepness. And it has a simple form. By use of the new nonlinear dispersion relation along with the mild slope equation taking into account weak nonlinearity, a mathematical model of wave transformation is developed and applied to laboratory data. The results show that the model with the new dispersion relation can predict wave transformation over complicated bathymetry satisfactorily.
7. Statistical distribution of nonlinear random wave height
Institute of Scientific and Technical Information of China (English)
HOU; Yijun; GUO; Peifang; SONG; Guiting; SONG; Jinbao; YIN; Baoshu; ZHAO; Xixi
2006-01-01
A statistical model of random wave is developed using Stokes wave theory of water wave dynamics. A new nonlinear probability distribution function of wave height is presented. The results indicate that wave steepness not only could be a parameter of the distribution function of wave height but also could reflect the degree of wave height distribution deviation from the Rayleigh distribution. The new wave height distribution overcomes the problem of Rayleigh distribution that the prediction of big wave is overestimated and the general wave is underestimated. The prediction of small probability wave height value of new distribution is also smaller than that of Rayleigh distribution. Wave height data taken from East China Normal University are used to verify the new distribution. The results indicate that the new distribution fits the measurements much better than the Rayleigh distribution.
8. Nonlinear modal propagation analysis method in multimode interference coupler for operation development
Science.gov (United States)
Tajaldini, Mehdi; Mat Jafri, Mohd Zubir Mat
2013-05-01
In this study, we propose a novel approach that is called nonlinear modal propagation analysis method (NMPA) in MMI coupler via the enhances of nonlinear wave propagation in terms of guided modes interferences in nonlinear regimes, such that the modal fields are measurable at any point of coupler and output facets. Then, the ultra-short MMI coupler is optimized as a building block in micro ring resonator to investigate the method efficiency against the already used method. Modeling results demonstrate more efficiency and accuracy in shorter lengths of multimode interference coupler. Therefore, NMPA can be used as a method to study the compact dimension coupler and for developing the performance in applications. Furthermore, the possibility of access tothe all-optical switching is assumed due to one continuous MMI for proof of the development of performances in nonlinear regimes.
9. Nonlinear wave interactions in quantum magnetoplasmas
CERN Document Server
Shukla, P K; Marklund, M; Stenflo, L
2006-01-01
Nonlinear interactions involving electrostatic upper-hybrid (UH), ion-cyclotron (IC), lower-hybrid (LH), and Alfven waves in quantum magnetoplasmas are considered. For this purpose, the quantum hydrodynamical equations are used to derive the governing equations for nonlinearly coupled UH, IC, LH, and Alfven waves. The equations are then Fourier analyzed to obtain nonlinear dispersion relations, which admit both decay and modulational instabilities of the UH waves at quantum scales. The growth rates of the instabilities are presented. They can be useful in applications of our work to diagnostics in laboratory and astrophysical settings.
10. Seismic wave propagation in granular media
Science.gov (United States)
Tancredi, Gonzalo; López, Francisco; Gallot, Thomas; Ginares, Alejandro; Ortega, Henry; Sanchís, Johnny; Agriela, Adrián; Weatherley, Dion
2016-10-01
Asteroids and small bodies of the Solar System are thought to be agglomerates of irregular boulders, therefore cataloged as granular media. It is a consensus that many asteroids might be considered as rubble or gravel piles.Impacts on their surface could produce seismic waves which propagate in the interior of these bodies, thus causing modifications in the internal distribution of rocks and ejections of particles and dust, resulting in a cometary-type comma.We present experimental and numerical results on the study of propagation of impact-induced seismic waves in granular media, with special focus on behavior changes by increasing compression.For the experiment, we use an acrylic box filled with granular materials such as sand, gravel and glass spheres. Pressure inside the box is controlled by a movable side wall and measured with sensors. Impacts are created on the upper face of the box through a hole, ranging from free-falling spheres to gunshots. We put high-speed cameras outside the box to record the impact as well as piezoelectic sensors and accelerometers placed at several depths in the granular material to detect the seismic wave.Numerical simulations are performed with ESyS-Particle, a software that implements the Discrete Element Method. The experimental setting is reproduced in the numerical simulations using both individual spherical particles and agglomerates of spherical particles shaped as irregular boulders, according to rock models obtained with a 3D scanner. The numerical experiments also reproduces the force loading on one of the wall to vary the pressure inside the box.We are interested in the velocity, attenuation and energy transmission of the waves. These quantities are measured in the experiments and in the simulations. We study the dependance of these three parameters with characteristics like: impact speed, properties of the target material and the pressure in the media.These results are relevant to understand the outcomes of impacts in
11. Propagation of acoustic wave in viscoelastic medium permeated with air bubbles
Institute of Scientific and Technical Information of China (English)
Liang Bin; Zhu Zhe-Min; Cheng Jian-Chun
2006-01-01
Based on the modification of the radial pulsation equation of an individual bubble, an effective medium method (EMM) is presented for studying propagation of linear and nonlinear longitudinal acoustic waves in viscoelastic medium permeated with air bubbles. A classical theory developed previously by Gaunaurd (Gaunaurd GC and (U)berall H, J. Acoust. Soc. Am., 1978; 63: 1699-1711) is employed to verify the EMM under linear approximation by comparing the dynamic (i.e. frequency-dependent) effective parameters, and an excellent agreement is obtained. The propagation of longitudinal waves is hereby studied in detail. The results illustrate that the nonlinear pulsation of bubbles serves as the source of second harmonic wave and the sound energy has the tendency to be transferred to second harmonic wave. Therefore the sound attenuation and acoustic nonlinearity of the viscoelastic matrix are remarkably enhanced due to the system's resonance induced by the existence of bubbles.
12. Propagation Dynamics of Nonspreading Cosine-Gauss Water-Wave Pulses.
Science.gov (United States)
Fu, Shenhe; Tsur, Yuval; Zhou, Jianying; Shemer, Lev; Arie, Ady
2015-12-18
Linear gravity water waves are highly dispersive; therefore, the spreading of initially short wave trains characterizes water surface waves, and is a universal property of a dispersive medium. Only if there is sufficient nonlinearity does this envelope admit solitary solutions which do not spread and remain in fixed forms. Here, in contrast to the nonlinear localized wave packets, we present both theoretically and experimentally a new type of linearly nondispersive water wave, having a cosine-Gauss envelope, as well as its higher-order Hermite cosine-Gauss variations. We show that these waves preserve their width despite the inherent dispersion while propagating in an 18-m wave tank, accompanied by a slowly varying carrier-envelope phase. These wave packets exhibit self-healing; i.e., they are restored after bypassing an obstacle. We further demonstrate that these nondispersive waves are robust to weakly nonlinear perturbations. In the strong nonlinear regime, symmetry breaking of these waves is observed, but their cosine-Gauss shapes are still approximately preserved during propagation.
13. SINGULAR AND RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
2001-01-01
Following a recent paper of the authors in Communications in Partial Differential Equations, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.
14. Cross-polarized wave generation by effective cubic nonlinear optical interaction.
Science.gov (United States)
Petrov, G I; Albert, O; Etchepare, J; Saltiel, S M
2001-03-15
A new cubic nonlinear optical effect in which a linearly polarized wave propagating in a single quadratic medium is converted into a wave that is cross polarized to the input wave is observed in BBO crystal. The effect is explained by cascading of two different second-order processes: second-harmonic generation and difference frequency mixing.
15. Strongly nonlinear steepening of long interfacial waves
Directory of Open Access Journals (Sweden)
N. Zahibo
2007-06-01
Full Text Available The transformation of nonlinear long internal waves in a two-layer fluid is studied in the Boussinesq and rigid-lid approximation. Explicit analytic formulation of the evolution equation in terms of the Riemann invariants allows us to obtain analytical results characterizing strongly nonlinear wave steepening, including the spectral evolution. Effects manifesting the action of high nonlinear corrections of the model are highlighted. It is shown, in particular, that the breaking points on the wave profile may shift from the zero-crossing level. The wave steepening happens in a different way if the density jump is placed near the middle of the water bulk: then the wave deformation is almost symmetrical and two phases appear where the wave breaks.
16. Nonlinear waves in strongly interacting relativistic fluids
CERN Document Server
Fogaça, D A; Filho, L G Ferreira
2013-01-01
During the past decades the study of strongly interacting fluids experienced a tremendous progress. In the relativistic heavy ion accelerators, specially the RHIC and LHC colliders, it became possible to study not only fluids made of hadronic matter but also fluids of quarks and gluons. Part of the physics program of these machines is the observation of waves in this strongly interacting medium. From the theoretical point of view, these waves are often treated with li-nearized hydrodynamics. In this text we review the attempts to go beyond linearization. We show how to use the Reductive Perturbation Method to expand the equations of (ideal and viscous) relativistic hydrodynamics to obtain nonlinear wave equations. These nonlinear wave equations govern the evolution of energy density perturbations (in hot quark gluon plasma) or baryon density perturbations (in cold quark gluon plasma and nuclear matter). Different nonlinear wave equations, such as the breaking wave, Korteweg-de Vries and Burgers equations, are...
17. A study on compressive shock wave propagation in metallic foams
Science.gov (United States)
Wang, Zhihua; Zhang, Yifen; Ren, Huilan; Zhao, Longmao
2010-02-01
Metallic foam can dissipate a large amount of energy due to its relatively long stress plateau, which makes it widely applicable in the design of structural crashworthiness. However, in some experimental studies, stress enhancement has been observed when the specimens are subjected to intense impact loads, leading to severe damage to the objects being protected. This paper studies this phenomenon on a 2D mass-spring-bar model. With the model, a constitutive relationship of metal foam and corresponding loading and unloading criteria are presented; a nonlinear kinematics equilibrium equation is derived, where an explicit integration algorithm is used to calculate the characteristic of the compressive shock wave propagation within the metallic foam; the effect of heterogeneous distribution of foam microstructures on the shock wave features is also included. The results reveal that under low impact pulses, considerable energy is dissipated during the progressive collapse of foam cells, which then reduces the crush of objects. When the pulse is sufficiently high, on the other hand, stress enhancement may take place, especially in the heterogeneous foams, where high peak stresses usually occur. The characteristics of compressive shock wave propagation in the foam and the magnitude and location of the peak stress produced are strongly dependent on the mechanical properties of the foam material, amplitude and period of the pulse, as well as the homogeneity of the microstructures. This research provides valuable insight into the reliability of the metallic foams used as a protective structure.
18. Nonlinear effects of the finite amplitude ultrasound wave in biological tissues
Institute of Scientific and Technical Information of China (English)
2000-01-01
Nonlinear effects will occur during the transmission of the finite amplitude wave in biological tissues.The theoretical prediction and experimental demonstration of the nonlinear effects on the propagation of the finite amplitude wave at the range of biomedical ultrasound frequency and intensity are studied.Results show that the efficiency factor and effective propagation distance will decrease while the attenuation coefficient increases due to the existence of nonlinear effects.The experimental results coincided quite well with the theory.This shows that the effective propagation distance and efficiency factor can be used to describe quantitatively the influence of nonlinear effects on the propagation of the finite amplitude sound wave in biological tissues.
19. Symmetry Breaking of Counter-Propagating Light in a Nonlinear Resonator
CERN Document Server
Del Bino, Leonardo; Stebbings, Sarah L; Del'Haye, Pascal
2016-01-01
Light is generally expected to travel through isotropic media independent of its direction. This makes it challenging to develop non-reciprocal optical elements like optical diodes or circulators, which currently rely on magneto-optical effects and birefringent materials. Here we present measurements of non-reciprocal transmission and spontaneous symmetry breaking between counter-propagating light in dielectric microresonators. The symmetry breaking corresponds to a resonance frequency splitting that allows only one of two counter-propagating (but otherwise identical) light waves to circulate in the resonator. Equivalently, the symmetry breaking can be seen as the collapse of standing waves and transition to travelling waves within the resonator. We present theoretical calculations to show that the symmetry breaking is induced by Kerr-nonlinearity-mediated interaction between the counter-propagating light. This effect is expected to take place in any dielectric ring-resonator and might constitute one of the m...
20. Wave propagation in random granular chains.
Science.gov (United States)
Manjunath, Mohith; Awasthi, Amnaya P; Geubelle, Philippe H
2012-03-01
The influence of randomness on wave propagation in one-dimensional chains of spherical granular media is investigated. The interaction between the elastic spheres is modeled using the classical Hertzian contact law. Randomness is introduced in the discrete model using random distributions of particle mass, Young's modulus, or radius. Of particular interest in this study is the quantification of the attenuation in the amplitude of the impulse associated with various levels of randomness: two distinct regimes of decay are observed, characterized by an exponential or a power law, respectively. The responses are normalized to represent a vast array of material parameters and impact conditions. The virial theorem is applied to investigate the transfer from potential to kinetic energy components in the system for different levels of randomness. The level of attenuation in the two decay regimes is compared for the three different sources of randomness and it is found that randomness in radius leads to the maximum rate of decay in the exponential regime of wave propagation.
1. On the polarization of nonlinear gravitational waves
OpenAIRE
Poplawski, Nikodem J.
2011-01-01
We derive a relation between the two polarization modes of a plane, linear gravitational wave in the second-order approximation. Since these two polarizations are not independent, an initially monochromatic gravitational wave loses its periodic character due to the nonlinearity of the Einstein field equations. Accordingly, real gravitational waves may differ from solutions of the linearized field equations, which are being assumed in gravitational-wave detectors.
2. Viscothermal wave propagation including acousto-elastic interaction
NARCIS (Netherlands)
Beltman, Willem Martinus
1998-01-01
This research deals with pressure waves in a gas trapped in thin layers or narrow tubes. In these cases viscous and thermal effects can have a significant effect on the propagation of waves. This so-called viscothermal wave propagation is governed by a number of dimensionless parameters. The two mos
3. WAVE: Interactive Wave-based Sound Propagation for Virtual Environments.
Science.gov (United States)
Mehra, Ravish; Rungta, Atul; Golas, Abhinav; Ming Lin; Manocha, Dinesh
2015-04-01
We present an interactive wave-based sound propagation system that generates accurate, realistic sound in virtual environments for dynamic (moving) sources and listeners. We propose a novel algorithm to accurately solve the wave equation for dynamic sources and listeners using a combination of precomputation techniques and GPU-based runtime evaluation. Our system can handle large environments typically used in VR applications, compute spatial sound corresponding to listener's motion (including head tracking) and handle both omnidirectional and directional sources, all at interactive rates. As compared to prior wave-based techniques applied to large scenes with moving sources, we observe significant improvement in runtime memory. The overall sound-propagation and rendering system has been integrated with the Half-Life 2 game engine, Oculus-Rift head-mounted display, and the Xbox game controller to enable users to experience high-quality acoustic effects (e.g., amplification, diffraction low-passing, high-order scattering) and spatial audio, based on their interactions in the VR application. We provide the results of preliminary user evaluations, conducted to study the impact of wave-based acoustic effects and spatial audio on users' navigation performance in virtual environments.
4. Evolution Of Nonlinear Waves in Compressing Plasma
Energy Technology Data Exchange (ETDEWEB)
P.F. Schmit, I.Y. Dodin, and N.J. Fisch
2011-05-27
Through particle-in-cell simulations, the evolution of nonlinear plasma waves is examined in one-dimensional collisionless plasma undergoing mechanical compression. Unlike linear waves, whose wavelength decreases proportionally to the system length L(t), nonlinear waves, such as solitary electron holes, conserve their characteristic size {Delta} during slow compression. This leads to a substantially stronger adiabatic amplification as well as rapid collisionless damping when L approaches {Delta}. On the other hand, cessation of compression halts the wave evolution, yielding a stable mode.
5. Nonlinear Alfv\\'en waves in extended magnetohydrodynamics
CERN Document Server
Abdelhamid, Hamdi M
2015-01-01
Large-amplitude Alfv\\'en waves are observed in various systems in space and laboratories, demonstrating an interesting property that the wave shapes are stable even in the nonlinear regime. The ideal magnetohydrodynamics (MHD) model predicts that an Alfv\\'en wave keeps an arbitrary shape constant when it propagates on a homogeneous ambient magnetic field. However, such arbitrariness is an artifact of the idealized model that omits the dispersive effects. Only special wave forms, consisting of two component sinusoidal functions, can maintain the shape; we derive fully nonlinear Alfv\\'en waves by an extended MHD model that includes both the Hall and electron inertia effects. Interestingly, these \\small-scale effects" change the picture completely; the large-scale component of the wave cannot be independent of the small scale component, and the coexistence of them forbids the large scale component to have a free wave form. This is a manifestation of the nonlinearity-dispersion interplay, which is somewhat differ...
6. Nonlinear Biochemical Signal Processing via Noise Propagation
OpenAIRE
Kim, Kyung Hyuk; Qian, Hong; Sauro, Herbert M.
2013-01-01
Single-cell studies often show significant phenotypic variability due to the stochastic nature of intra-cellular biochemical reactions. When the numbers of molecules, e.g., transcription factors and regulatory enzymes, are in low abundance, fluctuations in biochemical activities become significant and such "noise" can propagate through regulatory cascades in terms of biochemical reaction networks. Here we develop an intuitive, yet fully quantitative method for analyzing how noise affects cell...
7. Nonlinear surface waves over topography
NARCIS (Netherlands)
Janssen, T.T.
2006-01-01
As ocean surface waves radiate into shallow coastal areas and onto beaches, their lengths shorten, wave heights increase, and the wave shape transforms from nearsinusoidal to the characteristic saw-tooth shapes at the onset of breaking; in the ensuing breaking process the wave energy is cascaded to
8. Wave propagation in predator-prey systems
Science.gov (United States)
Fu, Sheng-Chen; Tsai, Je-Chiang
2015-12-01
In this paper, we study a class of predator-prey systems of reaction-diffusion type. Specifically, we are interested in the dynamical behaviour for the solution with the initial distribution where the prey species is at the level of the carrying capacity, and the density of the predator species has compact support, or exponentially small tails near x=+/- ∞ . Numerical evidence suggests that this will lead to the formation of a pair of diverging waves propagating outwards from the initial zone. Motivated by this phenomenon, we establish the existence of a family of travelling waves with the minimum speed. Unlike the previous studies, we do not use the shooting argument to show this. Instead, we apply an iteration process based on Berestycki et al 2005 (Math Comput. Modelling 50 1385-93) to construct a set of super/sub-solutions. Since the underlying system does not enjoy the comparison principle, such a set of super/sub-solutions is not based on travelling waves, and in fact the super/sub-solutions depend on each other. With the aid of the set of super/sub-solutions, we can construct the solution of the truncated problem on the finite interval, which, via the limiting argument, can in turn generate the wave solution. There are several advantages to this approach. First, it can remove the technical assumptions on the diffusivities of the species in the existing literature. Second, this approach is of PDE type, and hence it can shed some light on the spreading phenomenon indicated by numerical simulation. In fact, we can compute the spreading speed of the predator species for a class of biologically acceptable initial distributions. Third, this approach might be applied to the study of waves in non-cooperative systems (i.e. a system without a comparison principle).
9. Nonlinear Electrostatic Wave Equations for Magnetized Plasmas
DEFF Research Database (Denmark)
Dysthe, K.B.; Mjølhus, E.; Pécseli, Hans
1984-01-01
The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed.......The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed....
10. A NUMERICAL METHOD FOR NONLINEAR WATER WAVES
Institute of Scientific and Technical Information of China (English)
ZHAO Xi-zeng; SUN Zhao-chen; LIANG Shu-xiu; HU Chang-hong
2009-01-01
This article presents a numerical method for modeling nonlinear water waves based on the High Order Spectral (HOS) method proposed by Dommermuth and Yue and West et al., involving Taylor expansion of the Dirichlet problem and the Fast Fourier Transform (FFT) algorithm. The validation and efficiency of the numerical scheme is illustrated by a number of case studies on wave and wave train configuration including the evolution of fifth-order Stokes waves, wave dispersive focusing and the instability of Stokes wave with finite slope. The results agree well with those obtained by other studies.
11. Nonlinear pulse propagation in birefringent fiber Bragg gratings.
Science.gov (United States)
Pereira, S; Sipe, J
1998-11-23
We present two sets of equations to describe nonlinear pulse propagation in a birefringent fiber Bragg grating. The first set uses a coupled-mode formalism to describe light in or near the photonic band gap of the grating. The second set is a pair of coupled nonlinear Schroedinger equations. We use these equations to examine viable switching experiments in the presence of birefringence. We show how the birefringence can both aid and hinder device applications.
12. Non-linear propagation in near sonic flows
Science.gov (United States)
Nayfeh, A. H.; Kelly, J. J.; Watson, L. T.
1981-01-01
A nonlinear analysis is developed for sound propagation in a variable-area duct in which the mean flow approaches choking conditions. A quasi-one-dimensional model is used and the nonlinear analysis represents the acoustic disturbance as a sum of interacting harmonics. The numerical procedure is stable for cases of strong interaction and is able to integrate through the throat region without any numerical instability.
13. Lamb wave propagation modeling for structure health monitoring
Institute of Scientific and Technical Information of China (English)
Xiaoyue ZHANG; Shenfang YUAN; Tong HAO
2009-01-01
This study aims to model the propagation of Lamb waves used in structure health monitoring. A number of different numerical computational techniques have been developed for wave propagation studies. The local interaction simulation approach, used for modeling sharp interfaces and discontinuities in complex media (LISA/SIM theory), has been effectively applied to numerical simulations of elastic wave interaction. This modeling is based on the local interaction simulation approach theory and is finally accomplished through the finite elements software Ansys11. In this paper, the Lamb waves propagating characteristics and the LISA/SIM theory are introduced. The finite difference equations describing wave propagation used in the LISA/SIM theory are obtained. Then, an anisotropic metallic plate model is modeled and a simulating Lamb waves signal is loaded on. Finally, the Lamb waves propagation modeling is implemented.
14. Nonlinear water waves with soluble surfactant
Science.gov (United States)
Lapham, Gary; Dowling, David; Schultz, William
1998-11-01
The hydrodynamic effects of surfactants have fascinated scientists for generations. This presentation describes an experimental investigation into the influence of a soluble surfactant on nonlinear capillary-gravity waves in the frequency range from 12 to 20 Hz. Waves were generated in a plexiglass wave tank (254 cm long, 30.5 cm wide, and 18 cm deep) with a triangular plunger wave maker. The tank was filled with carbon- and particulate-filtered water into which the soluble surfactant Triton-X-100® was added in known amounts. Wave slope was measured nonintrusively with a digital camera running at 225 fps by monitoring the position of light beams which passed up through the bottom of the tank, out through the wavy surface, and onto a white screen. Wave slope data were reduced to determine wave damping and the frequency content of the wave train. Both were influenced by the presence of the surfactant. Interestingly, a subharmonic wave occurring at one-sixth the paddle-driving frequency was found only when surfactant was present and the paddle was driven at amplitudes high enough to produce nonlinear waves in clean water. Although the origins of this subharmonic wave remain unclear, it appears to be a genuine manifestation of the combined effects of the surfactant and nonlinearity.
15. Explicit Traveling Wave Solutions to Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
Linghai ZHANG
2011-01-01
First of all,some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations,nonlinear dissipative dispersive wave equations,nonlinear convection equations,nonlinear reaction diffusion equations and nonlinear hyperbolic equations,respectively.
16. A Stochastic Nonlinear Water Wave Model for Efficient Uncertainty Quantification
CERN Document Server
Bigoni, Daniele; Eskilsson, Claes
2014-01-01
A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a stochastic formulation of a fully nonlinear and dispersive potential flow water wave model for the probabilistic description of the evolution waves. This model is discretized using the Stochastic Collocation Method (SCM), which provides an approximate surrogate of the model. This can be used to accurately and efficiently estimate the probability distribution of the unknown time dependent stochastic solution after the forward propagation of uncertainties. We revisit experimental benchmarks often used for validation of deterministic water wave models. We do this using a fully nonlinear and dispersive model and show how uncertainty in the model input can influence the model output. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in compa...
17. Three-wave mixing of ordinary and backward electromagnetic waves: extraordinary transients in the nonlinear reflectivity and parametric amplification.
Science.gov (United States)
Slabko, Vitaly V; Popov, Alexander K; Tkachenko, Viktor A; Myslivets, Sergey A
2016-09-01
Three-wave mixing of ordinary and backward electromagnetic waves in a pulsed regime is investigated in the metamaterials that enable the coexistence and phase-matching of such waves. It is shown that the opposite direction of phase velocity and energy flux in backward waves gives rise to extraordinary transient processes due to greatly enhanced optical parametric amplification and frequency up- and down-shifting nonlinear reflectivity. The differences are illustrated through comparison with the counterparts in ordinary, co-propagating settings.
18. The Nonlinear Landau Damping Rate of a Driven Plasma Wave
Energy Technology Data Exchange (ETDEWEB)
Benisti, D; Strozzi, D J; Gremillet, L; Morice, O
2009-08-04
In this Letter, we discuss the concept of the nonlinear Landau damping rate, {nu}, of a driven electron plasma wave, and provide a very simple, practical, analytic formula for {nu} which agrees very well with results inferred from Vlasov simulations of stimulated Raman scattering. {nu} actually is more complicated an operator than a plain damping rate, and it may only be seen as such because it assumes almost constant values before abruptly dropping to 0. The decrease of {nu} to 0 is moreover shown to occur later when the wave amplitude varies in the direction transverse to its propagation.
19. Weak Turbulence in the Magnetosphere: Formation of Whistler Wave Cavity by Nonlinear Scattering
CERN Document Server
Crabtree, C; Ganguli, G; Mithaiwala, M; Galinsky, V; Shevchenko, V
2011-01-01
We consider the weak turbulence of whistler waves in the in low-\\beta\\ inner magnetosphere of the Earth. Whistler waves with frequencies, originating in the ionosphere, propagate radially outward and can trigger nonlinear induced scattering by thermal electrons provided the wave energy density is large enough. Nonlinear scattering can substantially change the direction of the wave vector of whistler waves and hence the direction of energy flux with only a small change in the frequency. A portion of whistler waves return to the ionosphere with a smaller perpendicular wave vector resulting in diminished linear damping and enhanced ability to pitch-angle scatter trapped electrons. In addition, a portion of the scattered wave packets can be reflected near the ionosphere back into the magnetosphere. Through multiple nonlinear scatterings and ionospheric reflections a long-lived wave cavity containing turbulent whistler waves can be formed with the appropriate properties to efficiently pitch-angle scatter trapped e...
20. Proton Heating in Solar Wind Compressible Turbulence with Collisions between Counter-propagating Waves
CERN Document Server
He, Jiansen; Marsch, Eckart; Chen, Christopher H K; Wang, Linghua; Pei, Zhongtian; Zhang, Lei; Salem, Chadi S; Bale, Stuart D
2015-01-01
Magnetohydronamic turbulence is believed to play a crucial role in heating the laboratorial, space, and astrophysical plasmas. However, the precise connection between the turbulent fluctuations and the particle kinetics has not yet been established. Here we present clear evidence of plasma turbulence heating based on diagnosed wave features and proton velocity distributions from solar wind measurements by the Wind spacecraft. For the first time, we can report the simultaneous observation of counter-propagating magnetohydrodynamic waves in the solar wind turbulence. Different from the traditional paradigm with counter-propagating Alfv\\'en waves, anti-sunward Alfv\\'en waves (AWs) are encountered by sunward slow magnetosonic waves (SMWs) in this new type of solar wind compressible turbulence. The counter-propagating AWs and SWs correspond respectively to the dominant and sub-dominant populations of the imbalanced Els\\"asser variables. Nonlinear interactions between the AWs and SMWs are inferred from the non-orth...
1. Nonlinear Evolution of Alfvenic Wave Packets
Science.gov (United States)
Buti, B.; Jayanti, V.; Vinas, A. F.; Ghosh, S.; Goldstein, M. L.; Roberts, D. A.; Lakhina, G. S.; Tsurutani, B. T.
1998-01-01
Alfven waves are a ubiquitous feature of the solar wind. One approach to studying the evolution of such waves has been to study exact solutions to approximate evolution equations. Here we compare soliton solutions of the Derivative Nonlinear Schrodinger evolution equation (DNLS) to solutions of the compressible MHD equations.
2. EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
2011-01-01
In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.
3. Solitary waves on nonlinear elastic rods. I
DEFF Research Database (Denmark)
Sørensen, Mads Peter; Christiansen, Peter Leth; Lomdahl, P. S.
1984-01-01
Acoustic waves on elastic rods with circular cross section are governed by improved Boussinesq equations when transverse motion and nonlinearity in the elastic medium are taken into account. Solitary wave solutions to these equations have been found. The present paper treats the interaction between...
4. Nonlinear ship waves and computational fluid dynamics
National Research Council Canada - National Science Library
MIYATA, Hideaki; ORIHARA, Hideo; SATO, Yohei
2014-01-01
.... Finding of the occurrence of nonlinear waves (named Free-Surface Shock Waves) in the vicinity of a ship advancing at constant speed provided the start-line for the progress of innovative technologies in the ship hull-form design...
5. Uncertainty propagation for nonlinear vibrations: A non-intrusive approach
Science.gov (United States)
Panunzio, A. M.; Salles, Loic; Schwingshackl, C. W.
2017-02-01
The propagation of uncertain input parameters in a linear dynamic analysis is reasonably well established today, but with the focus of the dynamic analysis shifting towards nonlinear systems, new approaches is required to compute the uncertain nonlinear responses. A combination of stochastic methods (Polynomial Chaos Expansion, PCE) with an Asymptotic Numerical Method (ANM) for the solution of the nonlinear dynamic systems is presented to predict the propagation of random input uncertainties and assess their influence on the nonlinear vibrational behaviour of a system. The proposed method allows the computation of stochastic resonance frequencies and peak amplitudes based on multiple input uncertainties, leading to a series of uncertain nonlinear dynamic responses. One of the main challenges when using the PCE is thereby the Gibbs phenomenon, which can heavily impact the resulting stochastic nonlinear response by introducing spurious oscillations. A novel technique to avoid the Gibbs phenomenon is be presented in this paper, leading to high quality frequency response predictions. A comparison of the proposed stochastic nonlinear analysis technique to traditional Monte Carlo simulations, demonstrates comparable accuracy at a significantly reduced computational cost, thereby validating the proposed approach.
6. Nonlinear dynamics of resistive electrostatic drift waves
DEFF Research Database (Denmark)
Korsholm, Søren Bang; Michelsen, Poul; Pécseli, H.L.
1999-01-01
The evolution of weakly nonlinear electrostatic drift waves in an externally imposed strong homogeneous magnetic field is investigated numerically in three spatial dimensions. The analysis is based on a set of coupled, nonlinear equations, which are solved for an initial condition which is pertur......The evolution of weakly nonlinear electrostatic drift waves in an externally imposed strong homogeneous magnetic field is investigated numerically in three spatial dimensions. The analysis is based on a set of coupled, nonlinear equations, which are solved for an initial condition which...... is perturbed by a small amplitude incoherent wave-field. The initial evolution is exponential, following the growth of perturbations predicted by linear stability theory. The fluctuations saturate at relatively high amplitudes, by forming a pair of magnetic field aligned vortex-like structures of opposite...
7. Detecting nonlinear acoustic waves in liquids with nonlinear dipole optical antennae
CERN Document Server
Maksymov, Ivan S
2015-01-01
Ultrasound is an important imaging modality for biological systems. High-frequency ultrasound can also (e.g., via acoustical nonlinearities) be used to provide deeply penetrating and high-resolution imaging of vascular structure via catheterisation. The latter is an important diagnostic in vascular health. Typically, ultrasound requires sources and transducers that are greater than, or of order the same size as the wavelength of the acoustic wave. Here we design and theoretically demonstrate that single silver nanorods, acting as optical nonlinear dipole antennae, can be used to detect ultrasound via Brillouin light scattering from linear and nonlinear acoustic waves propagating in bulk water. The nanorods are tuned to operate on high-order plasmon modes in contrast to the usual approach of using fundamental plasmon resonances. The high-order operation also gives rise to enhanced optical third-harmonic generation, which provides an important method for exciting the higher-order Fabry-Perot modes of the dipole...
8. Wave propagation in sandwich panels with a poroelastic core.
Science.gov (United States)
Liu, Hao; Finnveden, Svante; Barbagallo, Mathias; Arteaga, Ines Lopez
2014-05-01
Wave propagation in sandwich panels with a poroelastic core, which is modeled by Biot's theory, is investigated using the waveguide finite element method. A waveguide poroelastic element is developed based on a displacement-pressure weak form. The dispersion curves of the sandwich panel are first identified as propagating or evanescent waves by varying the damping in the panel, and wave characteristics are analyzed by examining their motions. The energy distributions are calculated to identify the dominant motions. Simplified analytical models are also devised to show the main physics of the corresponding waves. This wave propagation analysis provides insight into the vibro-acoustic behavior of sandwich panels lined with elastic porous materials.
9. Obliquely propagating large amplitude solitary waves in charge neutral plasmas
Directory of Open Access Journals (Sweden)
F. Verheest
2007-01-01
Full Text Available This paper deals in a consistent way with the implications, for the existence of large amplitude stationary structures in general plasmas, of assuming strict charge neutrality between electrons and ions. With the limit of pair plasmas in mind, electron inertia is retained. Combining in a fluid dynamic treatment the conservation of mass, momentum and energy with strict charge neutrality has indicated that nonlinear solitary waves (as e.g. oscillitons cannot exist in electron-ion plasmas, at no angle of propagation with respect to the static magnetic field. Specifically for oblique propagation, the proof has turned out to be more involved than for parallel or perpendicular modes. The only exception is pair plasmas that are able to support large charge neutral solitons, owing to the high degree of symmetry naturally inherent in such plasmas. The nonexistence, in particular, of oscillitons is attributed to the breakdown of the plasma approximation in dealing with Poisson's law, rather than to relativistic effects. It is hoped that future space observations will allow to discriminate between oscillitons and large wave packets, by focusing on the time variability (or not of the phase, since the amplitude or envelope graphs look very similar.
10. Analytical and numerical investigation of nonlinear internal gravity waves
Directory of Open Access Journals (Sweden)
S. P. Kshevetskii
2001-01-01
Full Text Available The propagation of long, weakly nonlinear internal waves in a stratified gas is studied. Hydrodynamic equations for an ideal fluid with the perfect gas law describe the atmospheric gas behaviour. If we neglect the term Ͽ dw/dt (product of the density and vertical acceleration, we come to a so-called quasistatic model, while we name the full hydro-dynamic model as a nonquasistatic one. Both quasistatic and nonquasistatic models are used for wave simulation and the models are compared among themselves. It is shown that a smooth classical solution of a nonlinear quasistatic problem does not exist for all t because a gradient catastrophe of non-linear internal waves occurs. To overcome this difficulty, we search for the solution of the quasistatic problem in terms of a generalised function theory as a limit of special regularised equations containing some additional dissipation term when the dissipation factor vanishes. It is shown that such solutions of the quasistatic problem qualitatively differ from solutions of a nonquasistatic nature. It is explained by the fact that in a nonquasistatic model the vertical acceleration term plays the role of a regularizator with respect to a quasistatic model, while the solution qualitatively depends on the regularizator used. The numerical models are compared with some analytical results. Within the framework of the analytical model, any internal wave is described as a system of wave modes; each wave mode interacts with others due to equation non-linearity. In the principal order of a perturbation theory, each wave mode is described by some equation of a KdV type. The analytical model reveals that, in a nonquasistatic model, an internal wave should disintegrate into solitons. The time of wave disintegration into solitons, the scales and amount of solitons generated are important characteristics of the non-linear process; they are found with the help of analytical and numerical investigations. Satisfactory
11. Nonlinear pulse propagation: a time-transformation approach.
Science.gov (United States)
Xiao, Yuzhe; Agrawal, Govind P; Maywar, Drew N
2012-04-01
We present a time-transformation approach for studying the propagation of optical pulses inside a nonlinear medium. Unlike the conventional way of solving for the slowly varying amplitude of an optical pulse, our new approach maps directly the input electric field to the output one, without making the slowly varying envelope approximation. Conceptually, the time-transformation approach shows that the effect of propagation through a nonlinear medium is to change the relative spacing and duration of various temporal slices of the pulse. These temporal changes manifest as self-phase modulation in the spectral domain and self-steepening in the temporal domain. Our approach agrees with the generalized nonlinear Schrödinger equation for 100 fs pulses and the finite-difference time-domain solution of Maxwell's equations for two-cycle pulses, while producing results 20 and 50 times faster, respectively.
12. Effect of Resolution on Propagating Detonation Wave
Energy Technology Data Exchange (ETDEWEB)
Menikoff, Ralph [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2014-07-10
Simulations of the cylinder test are used to illustrate the effect of mesh resolution on a propagating detonation wave. For this study we use the xRage code with the SURF burn model for PBX 9501. The adaptive mesh capability of xRage is used to vary the resolution of the reaction zone. We focus on two key properties: the detonation speed and the cylinder wall velocity. The latter is related to the release isentrope behind the detonation wave. As the reaction zone is refined (2 to 15 cells for cell size of 62 to 8μm), both the detonation speed and final wall velocity change by a small amount; less than 1 per cent. The detonation speed decreases with coarser resolution. Even when the reaction zone is grossly under-resolved (cell size twice the reaction-zone width of the burn model) the wall velocity is within a per cent and the detonation speed is low by only 2 per cent.
13. Mathematical problems in wave propagation theory
CERN Document Server
1970-01-01
The papers comprising this collection are directly or indirectly related to an important branch of mathematical physics - the mathematical theory of wave propagation and diffraction. The paper by V. M. Babich is concerned with the application of the parabolic-equation method (of Academician V. A. Fok and M. A, Leontovich) to the problem of the asymptotic behavior of eigenfunc tions concentrated in a neighborhood of a closed geodesie in a Riemannian space. The techniques used in this paper have been föund useful in solving certain problems in the theory of open resonators. The topic of G. P. Astrakhantsev's paper is similar to that of the paper by V. M. Babich. Here also the parabolic-equation method is used to find the asymptotic solution of the elasticity equations which describes Love waves concentrated in a neighborhood of some surface ray. The paper of T. F. Pankratova is concerned with finding the asymptotic behavior of th~ eigenfunc tions of the Laplace operator from the exact solution for the surf...
14. Nonlinear Landau damping and Alfven wave dissipation
Science.gov (United States)
Vinas, Adolfo F.; Miller, James A.
1995-01-01
Nonlinear Landau damping has been often suggested to be the cause of the dissipation of Alfven waves in the solar wind as well as the mechanism for ion heating and selective preacceleration in solar flares. We discuss the viability of these processes in light of our theoretical and numerical results. We present one-dimensional hybrid plasma simulations of the nonlinear Landau damping of parallel Alfven waves. In this scenario, two Alfven waves nonresonantly combine to create second-order magnetic field pressure gradients, which then drive density fluctuations, which in turn drive a second-order longitudinal electric field. Under certain conditions, this electric field strongly interacts with the ambient ions via the Landau resonance which leads to a rapid dissipation of the Alfven wave energy. While there is a net flux of energy from the waves to the ions, one of the Alfven waves will grow if both have the same polarization. We compare damping and growth rates from plasma simulations with those predicted by Lee and Volk (1973), and also discuss the evolution of the ambient ion distribution. We then consider this nonlinear interaction in the presence of a spectrum of Alfven waves, and discuss the spectrum's influence on the growth or damping of a single wave. We also discuss the implications for wave dissipation and ion heating in the solar wind.
15. Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
DEFF Research Database (Denmark)
1999-01-01
This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary c...
16. On the rogue waves propagation in non-Maxwellian complex space plasmas
Science.gov (United States)
El-Tantawy, S. A.; El-Awady, E. I.; Tribeche, M.
2015-11-01
The implications of the non-Maxwellian electron distributions (nonthermal/or suprathermal/or nonextensive distributions) are examined on the dust-ion acoustic (DIA) rogue/freak waves in a dusty warm plasma. Using a reductive perturbation technique, the basic set of fluid equations is reduced to a nonlinear Schrödinger equation. The latter is used to study the nonlinear evolution of modulationally unstable DIA wavepackets and to describe the rogue waves (RWs) propagation. Rogue waves are large-amplitude short-lived wave groups, routinely observed in space plasmas. The possible region for the rogue waves to exist is defined precisely for typical parameters of space plasmas. It is shown that the RWs strengthen for decreasing plasma nonthermality and increasing superthermality. For nonextensive electrons, the RWs amplitude exhibits a bit more complex behavior, depending on the entropic index q. Moreover, our numerical results reveal that the RWs exist with all values of the ion-to-electron temperature ratio σ for nonthermal and superthermal distributions and there is no limitation for the freak waves to propagate in both two distributions in the present plasma system. But, for nonextensive electron distribution, the bright- and dark-type waves can propagate in this case, which means that there is a limitation for the existence of freak waves. Our systematic investigation should be useful in understanding the properties of DIA solitary waves that may occur in non-Maxwellian space plasmas.
17. On the rogue waves propagation in non-Maxwellian complex space plasmas
Energy Technology Data Exchange (ETDEWEB)
El-Tantawy, S. A., E-mail: [email protected]; El-Awady, E. I., E-mail: [email protected] [Department of Physics, Faculty of Science, Port Said University, Port Said 42521 (Egypt); Tribeche, M., E-mail: [email protected], E-mail: [email protected] [Plasma Physics Group, Theoretical Physics Laboratory, Faculty of Physics, University of Bab-Ezzouar, USTHB, BP 32, El Alia, Algiers 16111 (Algeria)
2015-11-15
The implications of the non-Maxwellian electron distributions (nonthermal/or suprathermal/or nonextensive distributions) are examined on the dust-ion acoustic (DIA) rogue/freak waves in a dusty warm plasma. Using a reductive perturbation technique, the basic set of fluid equations is reduced to a nonlinear Schrödinger equation. The latter is used to study the nonlinear evolution of modulationally unstable DIA wavepackets and to describe the rogue waves (RWs) propagation. Rogue waves are large-amplitude short-lived wave groups, routinely observed in space plasmas. The possible region for the rogue waves to exist is defined precisely for typical parameters of space plasmas. It is shown that the RWs strengthen for decreasing plasma nonthermality and increasing superthermality. For nonextensive electrons, the RWs amplitude exhibits a bit more complex behavior, depending on the entropic index q. Moreover, our numerical results reveal that the RWs exist with all values of the ion-to-electron temperature ratio σ for nonthermal and superthermal distributions and there is no limitation for the freak waves to propagate in both two distributions in the present plasma system. But, for nonextensive electron distribution, the bright- and dark-type waves can propagate in this case, which means that there is a limitation for the existence of freak waves. Our systematic investigation should be useful in understanding the properties of DIA solitary waves that may occur in non-Maxwellian space plasmas.
18. Semiclassical methods for high frequency wave propagation in periodic media
Science.gov (United States)
We will study high-frequency wave propagation in periodic media. A typical example is given by the Schrodinger equation in the semiclassical regime with a highly oscillatory periodic potential and external smooth potential. This problem presents a numerical challenge when in the semiclassical regime. For example, conventional methods such as finite differences and spectral methods leads to high numerical cost, especially in higher dimensions. For this reason, asymptotic methods like the frozen Gaussian approximation (FGA) was developed to provide an efficient computational tool. Prior to the development of the FGA, the geometric optics and Gaussian beam methods provided an alternative asymptotic approach to solving the Schrodinger equation efficiently. Unlike the geometric optics and Gaussian beam methods, the FGA does not lose accuracy due to caustics or beam spreading. In this thesis, we will briefly review the geometric optics, Gaussian beam, and FGA methods. The mathematical techniques used by these methods will aid us in formulating the Bloch-decomposition based FGA. The Bloch-decomposition FGA generalizes the FGA to wave propagation in periodic media. We will establish the convergence of the Bloch-decomposition based FGA to the true solution for Schrodinger equation and develop a gauge-invariant algorithm for the Bloch-decomposition based FGA. This algorithm will avoid the numerical difficulty of computing the gauge-dependent Berry phase. We will show the numerical performance of our algorithm by several one-dimensional examples. Lastly, we will propose a time-splitting FGA-based artificial boundary conditions for solving the one-dimensional nonlinear Schrodinger equation (NLS) on an unbounded domain. The NLS will be split into two parts, the linear and nonlinear parts. For the linear part we will use the following absorbing boundary strategy: eliminate Gaussian functions whose centers are too distant to a fixed domain.
19. Evolution of Nonlinear Internal Waves in China Seas
Science.gov (United States)
Liu, Antony K.; Hsu, Ming-K.; Liang, Nai K.
1997-01-01
Synthetic Aperture Radar (SAR) images from ERS-I have been used to study the characteristics of internal waves of Taiwan in the East China Sea, and east of Hainan Island in the South China Sea. Rank-ordered packets of internal solitons propagating shoreward from the edge of the continental shelf were observed in the SAR images. Based on the assumption of a semidiurnal tidal origin, the wave speed can be estimated and is consistent with the internal wave theory. By using the SAR images and hydrographic data, internal waves of elevation have been identified in shallow water due to a thicker mixed layer as compared with the bottom layer on the continental shelf. The generation mechanism includes the influences of the tide and the Kuroshio intrusion across the continental shelf for the formations of elevation internal waves. The effects of water depth on the evolution of solitons and wave packets are modeled by nonlinear Kortweg-deVries (KdV) type equation and linked to satellite image observations. The numerical calculations of internal wave evolution on the continental shelf have been performed and compared with the SAR observations. For a case of depression waves in deep water, the solitons first disintegrate into dispersive wave trains and then evolve to a packet of elevation waves in the shallow water area after they pass through a turning point of approximately equal layer depths has been observed in the SAR image and simulated by numerical model.
20. Topology optimization for transient wave propagation problems in one dimension
DEFF Research Database (Denmark)
Dahl, Jonas; Jensen, Jakob Søndergaard; Sigmund, Ole
2008-01-01
Structures exhibiting band gap properties, i.e., having frequency ranges for which the structure attenuates propagating waves, have applications in damping of acoustic and elastic wave propagation and in optical communication. A topology optimization method for synthesis of such structures, emplo...
1. Propagation of Gaussian beam in longitudinally inhomogeneous nonlinear graded index waveguides with gain and losses
CERN Document Server
Yesayan, G L
2001-01-01
The equations for the width and curvature radius of the wave front for a Gaussian beam of light propagating along the axis of the longitudinally inhomogeneous graded index waveguide with gain and losses in the presence of third-order nonlinearity are obtained. By means of numerical calculations it is shown that in such waveguides the mode of stabilization of the beam width is possible, when the absorption of radiation on the edges of the beam compensates its spreading caused by the longitudinal inhomogeneity and nonlinearity of the waveguide
2. Properties, Propagation, and Excitation of EMIC Waves Properties, Propagation, and Excitation of EMIC Waves
Science.gov (United States)
Zhang, Jichun; Coffey, Victoria N.; Chandler, Michael O.; Boardsen, Scott A.; Saikin, Anthony A.; Mello, Emily M.; Russell, Christopher T.; Torbert, Roy B.; Fuselier, Stephen A.; Giles, Barbara L.;
2017-01-01
Electromagnetic ion cyclotron (EMIC) waves (0.1-5 Hz) play an important role in particle dynamics in the Earth's magnetosphere. EMIC waves are preferentially excited in regions where hot anisotropic ions and cold dense plasma populations spatially overlap. While the generation region of EMIC waves is usually on or near the magnetic equatorial plane in the inner magnetosphere, EMIC waves have both equatorial and off-equator source regions on the dayside in the compressed outer magnetosphere. Using field and plasma measurements from the Magnetospheric Multiscale (MMS) mission, we perform a case study of EMIC waves and associated local plasma conditions observed on 19 October 2015. From 0315 to 0810 UT, before crossing the magnetopause into the magnetosheath, all four MMS spacecraft detected long-lasting He(exp +)-band EMIC wave emissions around local noon (MLT = 12.7 - 14.0) at high L-shells (L = 8.8 - 15.2) and low magnetic latitudes (MLAT = -21.8deg - -30.3deg). Energetic (greater than 1 keV) and anisotropic ions were present throughout this event that was in the recovery phase of a weak geomagnetic storm (min. Dst = -48 nT at 1000 UT on 18 October 2015). The testing of linear theory suggests that the EMIC waves were excited locally. Although the wave event is dominated by small normal angles, its polarization is mixed with right- and left-handedness and its propagation is bi-directional with regard to the background magnetic field. The short inter-spacecraft distances (as low as 15 km) of the MMS mission make it possible to accurately determine the k vector of the waves using the phase difference technique. Preliminary analysis finds that the k vector magnitude, phase speed, and wavelength of the 0.3-Hz wave packet at 0453:55 UT are 0.005 km(exp -1), 372.9 km/s, and 1242.9 km, respectively.
3. Science.gov (United States)
Rudin, Sergey; Rupper, Greg
2013-08-01
The plasma wave in the conduction channel of a semiconductor heterostructure high electron mobility transistor (HEMT) can be excited at frequencies significantly higher than the cut-off frequency in a short channel device. The hydrodynamic model predicts a resonance response to applied harmonic signal at the plasma oscillation frequency. When either the ac voltage induced in the channel by the signal at the gate or the current applied at the drain or source contact are not very small, the plasma waves in the semiconductor channel will propagate as a shock wave. The device can be used either as a detector or a tunable source of terahertz range radiation. Using the parameters appropriate for the GaN channel we show that in both configurations the charge flow develops shock waves due to hydrodynamic nonlinearities. In a sufficiently wide channel the wave propagation separates into two or more different bands giving a two-dimensional structure to the waves.
4. Simulations of nonlinear continuous wave pressure fields in FOCUS
Science.gov (United States)
Zhao, Xiaofeng; Hamilton, Mark F.; McGough, Robert J.
2017-03-01
The Khokhlov - Zabolotskaya - Kuznetsov (KZK) equation is a parabolic approximation to the Westervelt equation that models the effects of diffraction, attenuation, and nonlinearity. Although the KZK equation is only valid in the far field of the paraxial region for mildly focused or unfocused transducers, the KZK equation is widely applied in medical ultrasound simulations. For a continuous wave input, the KZK equation is effectively modeled by the Bergen Code [J. Berntsen, Numerical Calculations of Finite Amplitude Sound Beams, in M. F. Hamilton and D. T. Blackstock, editors, Frontiers of Nonlinear Acoustics: Proceedings of 12th ISNA, Elsevier, 1990], which is a finite difference model that utilizes operator splitting. Similar C++ routines have been developed for FOCUS, the `Fast Object-Oriented C++ Ultrasound Simulator' (http://www.egr.msu.edu/˜fultras-web) to calculate nonlinear pressure fields generated by axisymmetric flat circular and spherically focused ultrasound transducers. This new routine complements an existing FOCUS program that models nonlinear ultrasound propagation with the angular spectrum approach [P. T. Christopher and K. J. Parker, J. Acoust. Soc. Am. 90, 488-499 (1991)]. Results obtained from these two nonlinear ultrasound simulation approaches are evaluated and compared for continuous wave linear simulations. The simulation results match closely in the farfield of the paraxial region, but the results differ in the nearfield. The nonlinear pressure field generated by a spherically focused transducer with a peak surface pressure of 0.2MPa radiating in a lossy medium with β = 3.5 is simulated, and the computation times are also evaluated. The nonlinear simulation results demonstrate acceptable agreement in the focal zone. These two related nonlinear simulation approaches are now included with FOCUS to enable convenient simulations of nonlinear pressure fields on desktop and laptop computers.
5. Backward-wave propagation and discrete solitons in a left-handed electrical lattice
Energy Technology Data Exchange (ETDEWEB)
English, L.Q.; Wheeler, S.G. [Department of Physics and Astronomy, Dickinson College, Carlisle, PA 17013 (United States); Shen, Y. [Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515 (United States); Veldes, G.P. [Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784 (Greece); Whitaker, N. [Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515 (United States); Kevrekidis, P.G., E-mail: [email protected] [Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515 (United States); Frantzeskakis, D.J. [Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784 (Greece)
2011-02-28
We study experimentally, analytically and numerically the backward-wave propagation, and formation of discrete bright and dark solitons in a nonlinear electrical lattice. We observe experimentally that a focusing (defocusing) effect occurs above (below) a certain carrier frequency threshold, and backward-propagating bright (dark) discrete solitons are formed. We develop a discrete model emulating the relevant circuit and benchmark its linear properties against the experimental dispersion relation. Using a perturbation method, we derive a nonlinear Schroedinger equation, that predicts accurately the carrier frequency threshold. Finally, we use numerical simulations to corroborate our findings and monitor the space-time evolution of the discrete solitons.
6. Dynamics of Nonlinear Waves on Bounded Domains
CERN Document Server
Maliborski, Maciej
2016-01-01
This thesis is concerned with dynamics of conservative nonlinear waves on bounded domains. In general, there are two scenarios of evolution. Either the solution behaves in an oscillatory, quasiperiodic manner or the nonlinear effects cause the energy to concentrate on smaller scales leading to a turbulent behaviour. Which of these two possibilities occurs depends on a model and the initial conditions. In the quasiperiodic scenario there exist very special time-periodic solutions. They result for a delicate balance between dispersion and nonlinear interaction. The main body of this dissertation is concerned with construction (by means of perturbative and numerical methods) of time-periodic solutions for various nonlinear wave equations on bounded domains. While turbulence is mainly associated with hydrodynamics, recent research in General Relativity has also revealed turbulent phenomena. Numerical studies of a self-gravitating massless scalar field in spherical symmetry gave evidence that anti-de Sitter space ...
7. Modulational development of nonlinear gravity-wave groups
Science.gov (United States)
Chereskin, T. K.; Mollo-Christensen, E.
1985-01-01
Observations of the development of nonlinear surface gravity-wave groups are presented, and the amplitude and phase modulations are calculated using Hilbert-transform techniques. With increasing propagation distance and wave steepness, the phase modulation develops local phase reversals whose locations correspond to amplitude minima or nodes. The concomitant frequency modulation develops jumps or discontinuities. The observations are compared with recent similar results for wavetrains. The observations are modelled numerically using the cubic nonlinear Schroedinger equation. The motivation is twofold: to examine quantitatively the evolution of phase as well as amplitude modulation, and to test the inviscid predictions for the asymptotic behavior of groups versus long-time observations. Although dissipation rules out the recurrence, there is a long-time coherence of the groups. The phase modulation is found to distinguish between dispersive and soliton behavior.
8. Wave propagation in nanostructures nonlocal continuum mechanics formulations
CERN Document Server
Gopalakrishnan, Srinivasan
2013-01-01
Wave Propagation in Nanostructures describes the fundamental and advanced concepts of waves propagating in structures that have dimensions of the order of nanometers. The book is fundamentally based on non-local elasticity theory, which includes scale effects in the continuum model. The book predominantly addresses wave behavior in carbon nanotubes and graphene structures, although the methods of analysis provided in this text are equally applicable to other nanostructures. The book takes the reader from the fundamentals of wave propagation in nanotubes to more advanced topics such as rotating nanotubes, coupled nanotubes, and nanotubes with magnetic field and surface effects. The first few chapters cover the basics of wave propagation, different modeling schemes for nanostructures and introduce non-local elasticity theories, which form the building blocks for understanding the material provided in later chapters. A number of interesting examples are provided to illustrate the important features of wave behav...
9. The Green-function transform and wave propagation
Directory of Open Access Journals (Sweden)
Colin eSheppard
2014-11-01
Full Text Available Fourier methods well known in signal processing are applied to three-dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus we make a distinction between inhomogeneous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given.
10. The Green-function transform and wave propagation
CERN Document Server
Sheppard, Colin J R; Lin, Jiao
2014-01-01
Fourier methods well known in signal processing are applied to three-dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus we make a distinction between inhomogenous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given.
11. Long-term evolution of strongly nonlinear internal solitary waves in a rotating channel
Directory of Open Access Journals (Sweden)
J. C. Sánchez-Garrido
2009-09-01
Full Text Available The evolution of internal solitary waves (ISWs propagating in a rotating channel is studied numerically in the framework of a fully-nonlinear, nonhydrostatic numerical model. The aim of modelling efforts was the investigation of strongly-nonlinear effects, which are beyond the applicability of weakly nonlinear theories. Results reveal that small-amplitude waves and sufficiently strong ISWs evolve differently under the action of rotation. At the first stage of evolution an initially two-dimensional ISW transforms according to the scenario described by the rotation modified Kadomtsev-Petviashvili equation, namely, it starts to evolve into a Kelvin wave (with exponential decay of the wave amplitude across the channel with front curved backwards. This transition is accompanied by a permanent radiation of secondary Poincaré waves attached to the leading wave. However, in a strongly-nonlinear limit not all the energy is transmitted to secondary radiated waves. Part of it returns to the leading wave as a result of nonlinear interactions with secondary Kelvin waves generated in the course of time. This leads to the formation of a slowly attenuating quasi-stationary system of leading Kelvin waves, capable of propagating for several hundreds hours as a localized wave packet.
12. A study on compressive shock wave propagation in metallic foams
Institute of Scientific and Technical Information of China (English)
2010-01-01
Metallic foam can dissipate a large amount of energy due to its relatively long stress plateau,which makes it widely applicable in the design of structural crashworthiness. However,in some experimental studies,stress enhancement has been observed when the specimens are subjected to intense impact loads,leading to severe damage to the objects being protected. This paper studies this phenomenon on a 2D mass-spring-bar model. With the model,a constitutive relationship of metal foam and corresponding loading and unloading criteria are presented; a nonlinear kinematics equilibrium equation is derived,where an explicit integra-tion algorithm is used to calculate the characteristic of the compressive shock wave propagation within the metallic foam; the effect of heterogeneous distribution of foam microstructures on the shock wave features is also included. The results reveal that under low impact pulses,considerable energy is dissipated during the progressive collapse of foam cells,which then reduces the crush of objects. When the pulse is sufficiently high,on the other hand,stress enhancement may take place,especially in the heterogeneous foams,where high peak stresses usually occur. The characteristics of compressive shock wave propagation in the foam and the magnitude and location of the peak stress produced are strongly dependent on the mechanical properties of the foam material,amplitude and period of the pulse,as well as the homogeneity of the microstructures. This research provides valuable insight into the reliability of the metallic foams used as a protective structure.
13. Electron scattering and nonlinear trapping by oblique whistler waves: The critical wave intensity for nonlinear effects
Energy Technology Data Exchange (ETDEWEB)
Artemyev, A. V., E-mail: [email protected]; Vasiliev, A. A. [Space Research Institute, RAS, Moscow (Russian Federation); Mourenas, D.; Krasnoselskikh, V. V. [LPC2E/CNRS—University of Orleans, Orleans (France); Agapitov, O. V. [Space Sciences Laboratory, University of California, Berkeley, California 94720 (United States)
2014-10-15
In this paper, we consider high-energy electron scattering and nonlinear trapping by oblique whistler waves via the Landau resonance. We use recent spacecraft observations in the radiation belts to construct the whistler wave model. The main purpose of the paper is to provide an estimate of the critical wave amplitude for which the nonlinear wave-particle resonant interaction becomes more important than particle scattering. To this aim, we derive an analytical expression describing the particle scattering by large amplitude whistler waves and compare the corresponding effect with the nonlinear particle acceleration due to trapping. The latter is much more rare but the corresponding change of energy is substantially larger than energy jumps due to scattering. We show that for reasonable wave amplitudes ∼10–100 mV/m of strong whistlers, the nonlinear effects are more important than the linear and nonlinear scattering for electrons with energies ∼10–50 keV. We test the dependencies of the critical wave amplitude on system parameters (background plasma density, wave frequency, etc.). We discuss the role of obtained results for the theoretical description of the nonlinear wave amplification in radiation belts.
14. Quasi self-adjoint nonlinear wave equations
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, N H [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Torrisi, M; Tracina, R, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Dipartimento di Matematica e Informatica, University of Catania (Italy)
2010-11-05
In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation. (fast track communication)
15. Three-wave interaction in two-component quadratic nonlinear lattices
DEFF Research Database (Denmark)
Konotop, V. V.; Cunha, M. D.; Christiansen, Peter Leth
1999-01-01
We investigate a two-component lattice with a quadratic nonlinearity and find with the multiple scale technique that integrable three-wave interaction takes place between plane wave solutions when these fulfill resonance conditions. We demonstrate that. energy conversion and pulse propagation kno...
16. Nonlinear Wave-Currents interactions in shallow water
CERN Document Server
Lannes, David
2015-01-01
We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [?] that the motion of the waves could be described using an extended Green-Naghdi system. In this paper we propose an analysis of these equations, and show that they can be used to get some new insight into wave-current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We als...
17. Wave propagation and radiation in gyrotropic and anisotropic media
CERN Document Server
Eroglu, Abdullah
2010-01-01
""Wave Propagation and Radiation in Gyrotropic and Anisotropic Media"" fills the gap in the area of applied electromagnetics for the design of microwave and millimeter wave devices using composite structures where gyrotropic, anisotropic materials are used. The book provides engineers with the information on theory and practical skills they need to understand wave propagation and radiation characteristics of materials and the ability to design devices at higher frequencies with optimum device performance.
18. Voltage modulation of propagating spin waves in Fe
Energy Technology Data Exchange (ETDEWEB)
Nawaoka, Kohei; Shiota, Yoichi; Miwa, Shinji; Tamura, Eiiti [Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531 (Japan); CREST, Japan Science Technology, Kawaguchi, Saitama 332-0012 (Japan); Tomita, Hiroyuki; Mizuochi, Norikazu; Shinjo, Teruya [Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531 (Japan); Suzuki, Yoshishige, E-mail: [email protected] [Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531 (Japan); CREST, Japan Science Technology, Kawaguchi, Saitama 332-0012 (Japan); Display and Semiconductor Physics Department, Korea University, Sejong 339-700 (Korea, Republic of)
2015-05-07
The effect of a voltage application on propagating spin waves in single-crystalline 5 nm-Fe layer was investigated. Two micro-sized antennas were employed to excite and detect the propagating spin waves. The voltage effect was characterized using AC lock-in technique. As a result, the resonant field of the magnetostatic surface wave in the Fe was clearly modulated by the voltage application. The modulation is attributed to the voltage induced magnetic anisotropy change in ferromagnetic metals.
19. Stable Propagating Waves and Wake Fields in Relativistic Electromagnetic Plasma
Institute of Scientific and Technical Information of China (English)
DUAN Yi-Shi; XIE Bai-Song; TIAN Miao; YIN Xin-Tao; ZHANG Xin-Hui
2008-01-01
Stable propagating waves and wake fields in relativistic electromagnetic plasma are investigated. The incident electromagnetic field has a finite initial constant amplitude meanwhile the longitudinal momentum of electrons is taken into account in the problem. It is found that in the moving frame with transverse wave group velocity the stable propagating transverse electromagnetic waves and longitudinal plasma wake fields can exist in the appropriate regime of plasma.
20. A propagation model of computer virus with nonlinear vaccination probability
Science.gov (United States)
Gan, Chenquan; Yang, Xiaofan; Liu, Wanping; Zhu, Qingyi
2014-01-01
This paper is intended to examine the effect of vaccination on the spread of computer viruses. For that purpose, a novel computer virus propagation model, which incorporates a nonlinear vaccination probability, is proposed. A qualitative analysis of this model reveals that, depending on the value of the basic reproduction number, either the virus-free equilibrium or the viral equilibrium is globally asymptotically stable. The results of simulation experiments not only demonstrate the validity of our model, but also show the effectiveness of nonlinear vaccination strategies. Through parameter analysis, some effective strategies for eradicating viruses are suggested.
1. Explicit solutions of nonlinear wave equation systems
Institute of Scientific and Technical Information of China (English)
Ahmet Bekir; Burcu Ayhan; M.Naci (O)zer
2013-01-01
We apply the (G'/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions,trigonometric functions,and rational functions with arbitrary parameters.We highlight the power of the (G'/G)-expansion method in providing generalized solitary wave solutions of different physical structures.It is shown that the (G'/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.
2. Optics in a nonlinear gravitational wave
CERN Document Server
Harte, Abraham I
2015-01-01
Gravitational waves can act like gravitational lenses, affecting the observed positions, brightnesses, and redshifts of distant objects. Exact expressions for such effects are derived here, allowing for arbitrarily-moving sources and observers in the presence of plane-symmetric gravitational waves. The commonly-used predictions of linear perturbation theory are shown to be generically overshadowed---even for very weak gravitational waves---by nonlinear effects when considering observations of sufficiently distant sources; higher-order perturbative corrections involve secularly-growing terms which cannot necessarily be neglected. Even on more moderate scales where linear effects remain at least marginally dominant, nonlinear corrections are qualitatively different from their linear counterparts. There is a sense in which they can, for example, mimic the existence of a third type of gravitational wave polarization.
3. Optics in a nonlinear gravitational plane wave
Science.gov (United States)
Harte, Abraham I.
2015-09-01
Gravitational waves can act like gravitational lenses, affecting the observed positions, brightnesses, and redshifts of distant objects. Exact expressions for such effects are derived here in general relativity, allowing for arbitrarily-moving sources and observers in the presence of plane-symmetric gravitational waves. At least for freely falling sources and observers, it is shown that the commonly-used predictions of linear perturbation theory can be generically overshadowed by nonlinear effects; even for very weak gravitational waves, higher-order perturbative corrections involve secularly-growing terms which cannot necessarily be neglected when considering observations of sufficiently distant sources. Even on more moderate scales where linear effects remain at least marginally dominant, nonlinear corrections are qualitatively different from their linear counterparts. There is a sense in which they can, for example, mimic the existence of a third type of gravitational wave polarization.
4. Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials
CERN Document Server
Markos, Peter
2010-01-01
This textbook offers the first unified treatment of wave propagation in electronic and electromagnetic systems and introduces readers to the essentials of the transfer matrix method, a powerful analytical tool that can be used to model and study an array of problems pertaining to wave propagation in electrons and photons. It is aimed at graduate and advanced undergraduate students in physics, materials science, electrical and computer engineering, and mathematics, and is ideal for researchers in photonic crystals, negative index materials, left-handed materials, plasmonics, nonlinear effects,
5. Unidirectional Wave Propagation in Low-Symmetric Colloidal Photonic-Crystal Heterostructures
Directory of Open Access Journals (Sweden)
Vassilios Yannopapas
2015-03-01
Full Text Available We show theoretically that photonic crystals consisting of colloidal spheres exhibit unidirectional wave propagation and one-way frequency band gaps without breaking time-reversal symmetry via, e.g., the application of an external magnetic field or the use of nonlinear materials. Namely, photonic crystals with low symmetry such as the monoclinic crystal type considered here as well as with unit cells formed by the heterostructure of different photonic crystals show significant unidirectional electromagnetic response. In particular, we show that the use of scatterers with low refractive-index contrast favors the formation of unidirectional frequency gaps which is the optimal route for achieving unidirectional wave propagation.
6. A nonlinear acoustic metamaterial: Realization of a backwards-traveling second-harmonic sound wave.
Science.gov (United States)
Quan, Li; Qian, Feng; Liu, Xiaozhou; Gong, Xiufen
2016-06-01
An ordinary waveguide with periodic vibration plates and side holes can realize an acoustic metamaterial that simultaneously possesses a negative bulk modulus and a negative mass density. The study is further extended to a nonlinear case and it is predicted that a backwards-traveling second-harmonic sound wave can be obtained through the nonlinear propagation of a sound wave in such a metamaterial.
7. Stable one-dimensional periodic waves in Kerr-type saturable and quadratic nonlinear media
Energy Technology Data Exchange (ETDEWEB)
Kartashov, Yaroslav V [ICFO-Institut de Ciencies Fotoniques, and Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, E-08034 Barcelona (Spain); Egorov, Alexey A [Physics Department, M V Lomonosov Moscow State University, 119899, Moscow (Russian Federation); Vysloukh, Victor A [Departamento de Fisica y Matematicas, Universidad de las Americas-Puebla, Santa Catarina Martir, 72820, Puebla, Cholula (Mexico); Torner, Lluis [ICFO-Institut de Ciencies Fotoniques, and Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, E-08034 Barcelona (Spain)
2004-05-01
We review the latest progress and properties of the families of bright and dark one-dimensional periodic waves propagating in saturable Kerr-type and quadratic nonlinear media. We show how saturation of the nonlinear response results in the appearance of stability (instability) bands in a focusing (defocusing) medium, which is in sharp contrast with the properties of periodic waves in Kerr media. One of the key results discovered is the stabilization of multicolour periodic waves in quadratic media. In particular, dark-type waves are shown to be metastable, while bright-type waves are completely stable in a broad range of energy flows and material parameters. This yields the first known example of completely stable periodic wave patterns propagating in conservative uniform media supporting bright solitons. Such results open the way to the experimental observation of the corresponding self-sustained periodic wave patterns.
8. Spatial damping of propagating sausage waves in coronal cylinders
CERN Document Server
Guo, Ming-Zhe; Li, Bo; Xia, Li-Dong; Yu, Hui
2015-01-01
Sausage modes are important in coronal seismology. Spatially damped propagating sausage waves were recently observed in the solar atmosphere. We examine how wave leakage influences the spatial damping of sausage waves propagating along coronal structures modeled by a cylindrical density enhancement embedded in a uniform magnetic field. Working in the framework of cold magnetohydrodynamics, we solve the dispersion relation (DR) governing sausage waves for complex-valued longitudinal wavenumber $k$ at given real angular frequencies $\\omega$. For validation purposes, we also provide analytical approximations to the DR in the low-frequency limit and in the vicinity of $\\omega_{\\rm c}$, the critical angular frequency separating trapped from leaky waves. In contrast to the standing case, propagating sausage waves are allowed for $\\omega$ much lower than $\\omega_{\\rm c}$. However, while able to direct their energy upwards, these low-frequency waves are subject to substantial spatial attenuation. The spatial damping ...
9. Nonlinear Propagation of Light in One Dimensional Periodic Structures
OpenAIRE
Goodman, Roy H.; Weinstein, Michael I.; Philip J Holmes
2000-01-01
We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion...
10. Properties of nonreciprocal light propagation in a nonlinear optical isolator
OpenAIRE
Roy, Dibyendu
2016-01-01
Light propagation in a nonlinear optical medium is nonreciprocal for spatially asymmetric linear permittivity. We here examine physical mechanism and properties of such nonreciprocity (NR). For this, we calculate transmission of light through a two-level atom asymmetrically coupled to light inside open waveguides. We determine the critical intensity of incident light for maximum NR and a dependence of the corresponding NR on asymmetry in the coupling. Surprisingly, we find that it is mainly c...
11. Propagation law of impact elastic wave based on specific materials
Directory of Open Access Journals (Sweden)
Chunmin CHEN
2017-02-01
Full Text Available In order to explore the propagation law of the impact elastic wave on the platform, the experimental platform is built by using the specific isotropic materials and anisotropic materials. The glass cloth epoxy laminated plate is used for anisotropic material, and an organic glass plate is used for isotropic material. The PVDF sensors adhered on the specific materials are utilized to collect data, and the elastic wave propagation law of different thick plates and laminated plates under impact conditions is analyzed. The Experimental results show that in anisotropic material, transverse wave propagation speed along the fiber arrangement direction is the fastest, while longitudinal wave propagation speed is the slowest. The longitudinal wave propagation speed in anisotropic laminates is much slower than that in the laminated thick plates. In the test channel arranged along a particular angle away from the central region of the material, transverse wave propagation speed is larger. Based on the experimental results, this paper proposes a material combination mode which is advantageous to elastic wave propagation and diffusion in shock-isolating materials. It is proposed to design a composite material with high acoustic velocity by adding regularly arranged fibrous materials. The overall design of the barrier material is a layered structure and a certain number of 90°zigzag structure.
12. Small amplitude nonlinear electron acoustic solitary waves in weakly magnetized plasma
Energy Technology Data Exchange (ETDEWEB)
Dutta, Manjistha; Khan, Manoranjan [Department of Instrumentation Science, Jadavpur University, Kolkata-700 032 (India); Ghosh, Samiran [Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata-700 009 (India); Roychoudhury, Rajkumar [Indian Statistical Institute, Kolkata-700 108 (India); Chakrabarti, Nikhil [Saha Institute of Nuclear Physics, 1/AF Bidhannagar Kolkata-700 064 (India)
2013-01-15
Nonlinear propagation of electron acoustic waves in homogeneous, dispersive plasma medium with two temperature electron species is studied in presence of externally applied magnetic field. The linear dispersion relation is found to be modified by the externally applied magnetic field. Lagrangian transformation technique is applied to carry out nonlinear analysis. For small amplitude limit, a modified KdV equation is obtained, the modification arising due to presence of magnetic field. For weakly magnetized plasma, the modified KdV equation possesses stable solitary solutions with speed and amplitude increasing temporally. The solutions are valid upto some finite time period beyond which the nonlinear wave tends to wave breaking.
13. Does the Decay Wave Propagate Forwards in Dusty Plasmas?
Institute of Scientific and Technical Information of China (English)
谢柏松
2002-01-01
The decay interaction of the ion acoustic wave in a dusty plasma with variable-charge dust grains is studied.Even if strong charging relaxation for dust grains and the short wavelength regime for ion waves are included, it is found that the decay wave must be backward propagating.
14. Propagation of Weak Pressure Waves against Two Parallel Subsonic Streams
Institute of Scientific and Technical Information of China (English)
Makiko YONAMINE; Takanori USHIJIMA; Yoshiaki MIYAZATO; Mitsuharu MASUDA; Hiroshi KATANODA; Kazuyasu MATSUO
2006-01-01
In this paper, the characteristics of a pressure wave propagating against two parallel subsonic streams in a constant-area straight duct are investigated by one-dimensional analysis, two-dimensional numerical simulation,and experiments. Computations have been carried out by the two-dimensional Euler Equations using the Chakravarthy-Osher-type TVD scheme. Optical observations by the schlieren method as well as wall pressure measurements have been performed to clarify both the structure and the propagation velocity of pressure waves.The results show that the pressure wave propagating against the streams changes into a bifurcated pressure wave and the bifurcation occurs in the low speed streams. It is also found that the propagation velocity of the pressure wave obtained by the analysis and computation agrees well with the present experimental data.
15. ON THE SOURCE OF PROPAGATING SLOW MAGNETOACOUSTIC WAVES IN SUNSPOTS
Energy Technology Data Exchange (ETDEWEB)
Prasad, S. Krishna; Jess, D. B. [Astrophysics Research Centre, School of Mathematics and Physics, Queen' s University Belfast, Belfast BT7 1NN (United Kingdom); Khomenko, Elena, E-mail: [email protected] [Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife (Spain)
2015-10-10
Recent high-resolution observations of sunspot oscillations using simultaneously operated ground- and space-based telescopes reveal the intrinsic connection between different layers of the solar atmosphere. However, it is not clear whether these oscillations are externally driven or generated in situ. We address this question by using observations of propagating slow magnetoacoustic waves along a coronal fan loop system. In addition to the generally observed decreases in oscillation amplitudes with distance, the observed wave amplitudes are also found to be modulated with time, with similar variations observed throughout the propagation path of the wave train. Employing multi-wavelength and multi-instrument data, we study the amplitude variations with time as the waves propagate through different layers of the solar atmosphere. By comparing the amplitude modulation period in different layers, we find that slow magnetoacoustic waves observed in sunspots are externally driven by photospheric p-modes, which propagate upward into the corona before becoming dissipated.
16. Enhanced continuous-wave four-wave mixing efficiency in nonlinear AlGaAs waveguides.
Science.gov (United States)
Apiratikul, Paveen; Wathen, Jeremiah J; Porkolab, Gyorgy A; Wang, Bohan; He, Lei; Murphy, Thomas E; Richardson, Christopher J K
2014-11-03
Enhancements of the continuous-wave four-wave mixing conversion efficiency and bandwidth are accomplished through the application of plasma-assisted photoresist reflow to reduce the sidewall roughness of sub-square-micron-modal area waveguides. Nonlinear AlGaAs optical waveguides with a propagation loss of 0.56 dB/cm demonstrate continuous-wave four-wave mixing conversion efficiency of -7.8 dB. Narrow waveguides that are fabricated with engineered processing produce waveguides with uncoated sidewalls and anti-reflection coatings that show group velocity dispersion of +0.22 ps²/m. Waveguides that are 5-mm long demonstrate broadband four-wave mixing conversion efficiencies with a half-width 3-dB bandwidth of 63.8-nm.
17. Plasma acceleration by the interaction of parallel propagating Alfv\\'en waves
CERN Document Server
Mottez, Fabrice
2014-01-01
It is shown that two circularly polarised Alfv\\'en waves that propagate along the ambient magnetic field in an uniform plasma trigger non oscillating electromagnetic field components when they cross each other. The non-oscilliating field components can accelerate ions and electrons with great efficiency. This work is based on particle-in-cell (PIC) numerical simulations and on analytical non-linear computations. The analytical computations are done for two counter-propagating monochromatic waves. The simulations are done with monochromatic waves and with wave packets. The simulations show parallel electromagnetic fields consistent with the theory, and they show that the particle acceleration result in plasma cavities and, if the waves amplitudes are high enough, in ion beams. These acceleration processes could be relevant in space plasmas. For instance, they could be at work in the auroral zone and in the radiation belts of the Earth magnetosphere. In particular, they may explain the origin of the deep plasma...
18. Analysis of Wave Propagation in Mechanical Continua Using a New Variational Approach
Science.gov (United States)
Chakraborty, Goutam
2016-06-01
In this paper a new variational principle is presented for studying various wave propagation phenomena without explicitly deriving the equations of motion. The method looks for steady state solutions of linear or non-linear partial differential equations that admit wave-like solutions. Dispersion relations of plane waves propagating in unbounded continuous media, transmission and reflection coefficients of wave incident on the boundary of two semi-infinite media and wave impedance and mobility in an excited medium are studied with the help of the same principle. Numerous examples are given to clarify the method adopted showing distinct advantages over the traditional methods. The scientific insights that this principle provides are also highlighted.
19. Nonlinear chirped-pulse propagation and supercontinuum generation in photonic crystal fibers.
Science.gov (United States)
Hu, Xiaohong; Wang, Yishan; Zhao, Wei; Yang, Zhi; Zhang, Wei; Li, Cheng; Wang, Hushan
2010-09-10
Based on the generalized nonlinear Schrödinger equation and waveguiding properties typical of the photonic crystal fiber structure, nonlinear chirped-pulse propagation and supercontinua generation in the femtosecond and picosecond regimes are investigated numerically. The simulation results indicate that an input chirp parameter mainly affects the initial stage of spectral broadening caused by the self-phase modulation (SPM) effect. In the femtosecond regime where the SPM effect plays an important role in the process of spectral broadening, an input positive chirp can enhance the supercontinuum bandwidth through a modified pulse compression phase and a decreased propagation distance required by soliton fission. In the picosecond regime, where the SPM effect contributes less to the continuum bandwidth and four-wave mixing process or modulational instability dominates the initial stage of spectral and temporal evolution, the output spectral shape and bandwidths are less sensitive to the input chirp parameters.
20. Time-Frequency (Wigner Analysis of Linear and Nonlinear Pulse Propagation in Optical Fibers
Directory of Open Access Journals (Sweden)
José Azaña
2005-06-01
Full Text Available Time-frequency analysis, and, in particular, Wigner analysis, is applied to the study of picosecond pulse propagation through optical fibers in both the linear and nonlinear regimes. The effects of first- and second-order group velocity dispersion (GVD and self-phase modulation (SPM are first analyzed separately. The phenomena resulting from the interplay between GVD and SPM in fibers (e.g., soliton formation or optical wave breaking are also investigated in detail. Wigner analysis is demonstrated to be an extremely powerful tool for investigating pulse propagation dynamics in nonlinear dispersive systems (e.g., optical fibers, providing a clearer and deeper insight into the physical phenomena that determine the behavior of these systems.
1. Directed electromagnetic wave propagation in 1D metamaterial: Projecting operators method
Energy Technology Data Exchange (ETDEWEB)
Ampilogov, Dmitrii, E-mail: [email protected]; Leble, Sergey, E-mail: [email protected]
2016-07-01
We consider a boundary problem for 1D electrodynamics modeling of a pulse propagation in a metamaterial medium. We build and apply projecting operators to a Maxwell system in time domain that allows to split the linear propagation problem to directed waves for a material relations with general dispersion. Matrix elements of the projectors act as convolution integral operators. For a weak nonlinearity we generalize the linear results still for arbitrary dispersion and derive the system of interacting right/left waves with combined (hybrid) amplitudes. The result is specified for the popular metamaterial model with Drude formula for both permittivity and permeability coefficients. We also discuss and investigate stationary solutions of the system related to some boundary regimes. - Highlights: • The problem of boundary regime propagation is solved by a systematic dynamic projecting method. • By this method a hybrid amplitude is introduced and used for derivation of nonlinear equation of opposite directed waves. • The equations are specified for Drude metamaterial dispersion and Kerr nonlinearity. • It is shown that one of unidirection waves in the metamaterial is specified as Shafer–Wayn integrable equation. • A stationary wave solution is approximately expressed in terms of elliptic functions.
2. Nonlinear light propagation in chalcogenide photonic crystal slow light waveguides.
Science.gov (United States)
Suzuki, Keijiro; Baba, Toshihiko
2010-12-06
Optical nonlinearity can be enhanced by the combination of highly nonlinear chalcogenide glass and photonic crystal waveguides (PCWs) providing strong optical confinement and slow-light effects. In a Ag-As(2)Se(3) chalcogenide PCW, the effective nonlinear parameter γeff reaches 6.3 × 10(4) W(-1)m(-1), which is 200 times larger than that in Si photonic wire waveguides. In this paper, we report the detailed design, fabrication process, and the linear and nonlinear characteristics of this waveguide at silica fiber communication wavelengths. We show that the waveguide exhibits negligible two-photon absorption, and also high-efficiency self-phase modulation and four-wave mixing, which are assisted by low-dispersion slow light.
3. Studies of Gravity Wave Propagation in the Middle Atmosphere.
Science.gov (United States)
2014-09-26
34 . . . . . • * * . , . • :’ . . . , ",.,,- -. ’’’ " . ’-- o p - %"""" * " AFOSR.TR. 85-0505 physical dynamics,inc. PD-NW-85-330R L n STUDIES OF GRAVITY WAVE PROPAGATION IN...8217.. , .,- - -. ( %’. , .;: :..............,....... .-... . ~.b .. .. - ..... ,......... ..-. ....-.. PD-NW-85-330R STUDIES OF GRAVITY WAVE PROPAGATION...Include SewftY CsuiclUon STUDIES OF GRAVITY WAVE PROPAGATION IN THE MIDD E 12. PERSONAL AUTHORE) TMOPHU. r Timothy J. Dunkerton a13a. TYPE OF REPORT I3k
4. New Relativistic Effects in the Dynamics of Nonlinear Hydrodynamical Waves
CERN Document Server
Rezzolla, L
2002-01-01
In Newtonian and relativistic hydrodynamics the Riemann problem consists of calculating the evolution of a fluid which is initially characterized by two states having different values of uniform rest-mass density, pressure and velocity. When the fluid is allowed to relax, one of three possible wave-patterns is produced, corresponding to the propagation in opposite directions of two nonlinear hydrodynamical waves. New effects emerge in a special relativistic Riemann problem when velocities tangential to the initial discontinuity surface are present. We show that a smooth transition from one wave-pattern to another can be produced by varying the initial tangential velocities while otherwise maintaining the initial states unmodified. These special relativistic effects are produced by the coupling through the relativistic Lorentz factors and do not have a Newtonian counterpart.
5. Nonlinear electromagnetic waves in a degenerate electron-positron plasma
Energy Technology Data Exchange (ETDEWEB)
El-Labany, S.K., E-mail: [email protected] [Department of Physics, Faculty of Science, Damietta University, New Damietta (Egypt); El-Taibany, W.F., E-mail: [email protected] [Department of Physics, College of Science for Girls in Abha, King Khalid University, Abha (Saudi Arabia); El-Samahy, A.E.; Hafez, A.M.; Atteya, A., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Department of Physics, Faculty of Science, Alexandria University, Alexandria (Egypt)
2015-08-15
Using the reductive perturbation technique (RPT), the nonlinear propagation of magnetosonic solitary waves in an ultracold, degenerate (extremely dense) electron-positron (EP) plasma (containing ultracold, degenerate electron, and positron fluids) is investigated. The set of basic equations is reduced to a Korteweg-de Vries (KdV) equation for the lowest-order perturbed magnetic field and to a KdV type equation for the higher-order perturbed magnetic field. The solutions of these evolution equations are obtained. For better accuracy and searching on new features, the new solutions are analyzed numerically based on compact objects (white dwarf) parameters. It is found that including the higher-order corrections results as a reduction (increment) of the fast (slow) electromagnetic wave amplitude but the wave width is increased in both cases. The ranges where the RPT can describe adequately the total magnetic field including different conditions are discussed. (author)
6. Probabilistic approach to nonlinear wave-particle resonant interaction
Science.gov (United States)
Artemyev, A. V.; Neishtadt, A. I.; Vasiliev, A. A.; Mourenas, D.
2017-02-01
In this paper we provide a theoretical model describing the evolution of the charged-particle distribution function in a system with nonlinear wave-particle interactions. Considering a system with strong electrostatic waves propagating in an inhomogeneous magnetic field, we demonstrate that individual particle motion can be characterized by the probability of trapping into the resonance with the wave and by the efficiency of scattering at resonance. These characteristics, being derived for a particular plasma system, can be used to construct a kinetic equation (or generalized Fokker-Planck equation) modeling the long-term evolution of the particle distribution. In this equation, effects of charged-particle trapping and transport in phase space are simulated with a nonlocal operator. We demonstrate that solutions of the derived kinetic equations agree with results of test-particle tracing. The applicability of the proposed approach for the description of space and laboratory plasma systems is also discussed.
7. Analytical description of nonlinear acoustic waves in the solar chromosphere
Science.gov (United States)
Litvinenko, Yuri E.; Chae, Jongchul
2017-02-01
Aims: Vertical propagation of acoustic waves of finite amplitude in an isothermal, gravitationally stratified atmosphere is considered. Methods: Methods of nonlinear acoustics are used to derive a dispersive solution, which is valid in a long-wavelength limit, and a non-dispersive solution, which is valid in a short-wavelength limit. The influence of the gravitational field on wave-front breaking and shock formation is described. The generation of a second harmonic at twice the driving wave frequency, previously detected in numerical simulations, is demonstrated analytically. Results: Application of the results to three-minute chromospheric oscillations, driven by velocity perturbations at the base of the solar atmosphere, is discussed. Numerical estimates suggest that the second harmonic signal should be detectable in an upper chromosphere by an instrument such as the Fast Imaging Solar Spectrograph installed at the 1.6-m New Solar Telescope of the Big Bear Observatory.
8. Nonlinear Electromagnetic Waves in a Degenerate Electron-Positron Plasma
Science.gov (United States)
El-Labany, S. K.; El-Taibany, W. F.; El-Samahy, A. E.; Hafez, A. M.; Atteya, A.
2015-08-01
Using the reductive perturbation technique (RPT), the nonlinear propagation of magnetosonic solitary waves in an ultracold, degenerate (extremely dense) electron-positron (EP) plasma (containing ultracold, degenerate electron, and positron fluids) is investigated. The set of basic equations is reduced to a Korteweg-de Vries (KdV) equation for the lowest-order perturbed magnetic field and to a KdV type equation for the higher-order perturbed magnetic field. The solutions of these evolution equations are obtained. For better accuracy and searching on new features, the new solutions are analyzed numerically based on compact objects (white dwarf) parameters. It is found that including the higher-order corrections results as a reduction (increment) of the fast (slow) electromagnetic wave amplitude but the wave width is increased in both cases. The ranges where the RPT can describe adequately the total magnetic field including different conditions are discussed.
9. Nonlinear wave structures as exact solutions of Vlasov-Maxwell equations.
Science.gov (United States)
Dasgupta, B.; Tsurutani, B. T.; Janaki, M. S.; Sharma, A. S.
2001-12-01
Many recent observations by POLAR and Geotail spacecraft of the low-latitudes magnetopause boundary layer (LLBL) and the polar cap boundary layer (PCBL) have detected nonlinear wave structures [Tsurutani et al, Geophys. Res. Lett., 25, 4117, 1998]. These nonlinear waves have electromagnetic signatures that are identified with Alfven and Whistler modes. Also solitary waves with mono- and bi-polar features were observed. In general such electromagnetic structures are described by the full Vlasov-Maxwell equations for waves propagating at an angle to the ambient magnetic field, but it has been a diffficult task obtaining the solutions because of the inherent nonlinearity. We have obtained an exact nonlinear solution of the full Vlasov-Maxwell equations in the presence of an electromagnetic wave propagating at an arbitrary direction with an ambient magnetic field. This is accomplished by finding the constants of motion of the charged particles in the electromagnetic field of the wave and then constructing a realistic distribution function as a function of these constants of motion. The corresponding trapping conditions for such waves are obtained, yielding the self-consistent description for the particles in the presence of the nonlinear waves. The interpretation of the observed nonlinear structures in terms of these general solutions will be presented.
10. On the Forms of Nonlinear Propagation of High-Frequency Perturbation in a Thermal Relaxing Gas-Liquid mixture
Directory of Open Access Journals (Sweden)
Ohanyan G.G.
2010-09-01
Full Text Available The quasi-adiabatic and quasi-isotherm regimes of propagation of high-frequency perturbation are considered in a thermal relaxing gas–fluid mixture. The simplified non-linear equations are obtained. It is shown that in the absence of heat transfer and under the quasi-adiabatic regime the form of propagation is soliton, or the shock wave in quasi-isotherm regime.
11. On the Forms of Nonlinear Propagation of High-Frequency Perturbation in a Thermal Relaxing Gas-Liquid mixture
OpenAIRE
Ohanyan G.G.
2010-01-01
The quasi-adiabatic and quasi-isotherm regimes of propagation of high-frequency perturbation are considered in a thermal relaxing gas–fluid mixture. The simplified non-linear equations are obtained. It is shown that in the absence of heat transfer and under the quasi-adiabatic regime the form of propagation is soliton, or the shock wave in quasi-isotherm regime.
12. Viscous Fluid Conduits as a Prototypical Nonlinear Dispersive Wave Platform
Science.gov (United States)
Lowman, Nicholas K.
This thesis is devoted to the comprehensive characterization of slowly modulated, nonlinear waves in dispersive media for physically-relevant systems using a threefold approach: analytical, long-time asymptotics, careful numerical simulations, and quantitative laboratory experiments. In particular, we use this interdisciplinary approach to establish a two-fluid, interfacial fluid flow setting known as viscous fluid conduits as an ideal platform for the experimental study of truly one dimensional, unidirectional solitary waves and dispersively regularized shock waves (DSWs). Starting from the full set of fluid equations for mass and linear momentum conservation, we use a multiple-scales, perturbation approach to derive a scalar, nonlinear, dispersive wave equation for the leading order interfacial dynamics of the system. Using a generalized form of the approximate model equation, we use numerical simulations and an analytical, nonlinear wave averaging technique, Whitham-El modulation theory, to derive the key physical features of interacting large amplitude solitary waves and DSWs. We then present the results of quantitative, experimental investigations into large amplitude solitary wave interactions and DSWs. Overtaking interactions of large amplitude solitary waves are shown to exhibit nearly elastic collisions and universal interaction geometries according to the Lax categories for KdV solitons, and to be in excellent agreement with the dynamics described by the approximate asymptotic model. The dispersive shock wave experiments presented here represent the most extensive comparison to date between theory and data of the key wavetrain parameters predicted by modulation theory. We observe strong agreement. Based on the work in this thesis, viscous fluid conduits provide a well-understood, controlled, table-top environment in which to study universal properties of dispersive hydrodynamics. Motivated by the study of wave propagation in the conduit system, we
13. Wave-packet rectification in nonlinear electronic systems: A tunable Aharonov-Bohm diode
CERN Document Server
Li, Yunyun; Marchesoni, Fabio; Li, Baowen
2014-01-01
Rectification of electron wave-packets propagating along a quasi-one dimensional chain is commonly achieved via the simultaneous action of nonlinearity and longitudinal asymmetry, both confined to a limited portion of the chain termed wave diode. However, it is conceivable that, in the presence of an external magnetic field, spatial asymmetry perpendicular to the direction of propagation suffices to ensure rectification. This is the case of a nonlinear ring-shaped lattice with different upper and lower halves (diode), which is attached to two elastic chains (leads). The resulting device is mirror symmetric with respect to the ring vertical axis, but mirror asymmetric with respect to the chain direction. Wave propagation along the two diode paths can be modeled for simplicity by a discrete Schr\\"odinger equation with cubic nonlinearities. Numerical simulations demonstrate that, thanks to the Aharonov-Bohm effect, such a diode can be operated by tuning the magnetic flux across the ring.
14. Electron acceleration in the ionosphere by obliquely propagating electromagnetic waves
Energy Technology Data Exchange (ETDEWEB)
Burke, W.J.; Ginet, G.P.; Heinemann, M.A.; Villalon, E.
1988-01-01
The relativistic equations of motion have been analyzed for electrons in magnetized plasmas and externally imposed electromagnetic fields that propagate at arbitrary angles to the background magnetic field. The electron energy is obtained from a set of non-linear differential equations as functions of time, initial conditions and cyclotron harmonic numbers. For a given cyclotron resonance the energy oscillates in time within the limits of a potential well. Stochastic acceleration occurs if the widths of hamiltonian potentials overlap. Numerical analyses suggest that, at wave energy fluxes in excess of 10/sup 8/ mW/m/sup 2/, initially cold electrons can be accelerated to energies of several MeV in less than a millisecond. Practical attempts to validate the theory with a series of planned rocket flights over the HIPAS facility in Alaska are discussed. The HIPAS antennas will be used to irradiate the magnetic mirror points of 10 - 40 keV electrons emitted from the ECHO 7 rocket in the early winter of 1988. Follow-on rocket experiments to exploit the wave amplification properties of the ionospheric 'radio window' are described.
15. Analysis of guided wave propagation in a tapered composite panel
Science.gov (United States)
Wandowski, Tomasz; Malinowski, Pawel; Moll, Jochen; Radzienski, Maciej; Ostachowicz, Wieslaw
2015-03-01
Many studies have been published in recent years on Lamb wave propagation in isotropic and (multi-layered) anisotropic structures. In this paper, adiabatic wave propagation phenomenon in a tapered composite panel made out of glass fiber reinforced polymers (GFRP) will be considered. Such structural elements are often used e.g. in wind turbine blades and aerospace structures. Here, the wave velocity of each wave mode does not only change with frequency and the direction of wave propagation. It further changes locally due to the varying cross-section of the GFRP panel. Elastic waves were excited using a piezoelectric transducer. Full wave-field measurements using scanning Laser Doppler vibrometry have been performed. This approach allows the detailed analysis of elastic wave propagation in composite specimen with linearly changing thickness. It will be demonstrated here experimentally, that the wave velocity changes significantly due to the tapered geometry of the structure. Hence, this work motivates the theoretical and experimental analysis of adiabatic mode propagation for the purpose of Non-Destructive Testing and Structural Health Monitoring.
16. 2-D Composite Model for Numerical Simulations of Nonlinear Waves
Institute of Scientific and Technical Information of China (English)
2000-01-01
－ A composite model, which is the combination of Boussinesq equations and Volume of Fluid (VOF) method, has been developed for 2-D time-domain computations of nonlinear waves in a large region. The whole computational region Ω is divided into two subregions. In the near-field around a structure, Ω2, the flow is governed by 2-D Reynolds Averaged Navier-Stokes equations with a turbulence closure model of k-ε equations and numerically solved by the improved VOF method; whereas in the subregion Ω1 (Ω1 = Ω - Ω2) the flow is governed by one-D Boussinesq equations and numerically solved with the predictor-corrector algorithm. The velocity and the wave surface elevation are matched on the common boundary of the two subregions. Numerical tests have been conducted for the case of wave propagation and interaction with a wave barrier. It is shown that the composite model can help perform efficient computation of nonlinear waves in a large region with the complicated flow fields near structures taken into account.
17. Wave propagation in a magneto-electro- elastic plate
Institute of Scientific and Technical Information of China (English)
2008-01-01
The wave propagation in a magneto-electro-elastic plate was studied. Some new characteristics were discovered: the guided waves are classified in the forms of the Quasi-P, Quasi-SV and Quasi-SH waves and arranged by the standing wavenumber; there are many patterns for the physical property of the magneto-electro-elastic dielectric medium influencing the stress wave propagation. We proposed a self-adjoint method, by which the guided-wave restriction condition was derived. After the corresponding orthogonal sets were found, the analytic dispersion equa-tion was obtained. In the end, an example was presented. The dispersive spectrum, the group velocity curved face and the steady-state response curve of a mag-neto-electro-elastic plate were plotted. Then the wave propagations affected by the induced electric and magnetic fields were analyzed.
18. An effective absorbing boundary algorithm for acoustical wave propagator
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic waves is introduced as the truncation area of the computational domain to absorb one-dimensional acoustic wave for the scheme of acoustical wave propagator (AWP). To guarantee the efficiency of the AWP algorithm, a regulated propagator matrix is derived in the PML medium.Numerical simulations of a Gaussian wave packet propagating in one-dimensional duct are carried out to illustraze the efficiency of the combination of PML and AWP. Compared with the traditional smoothing truncation windows technique of AWP, this scheme shows high computational accuracy in absorbing acoustic wave when the acoustical wave arrives at the computational edges. Optimal coefficients of the PML configurations are also discussed.
19. Simulation of guided wave propagation near numerical Brillouin zones
Science.gov (United States)
Kijanka, Piotr; Staszewski, Wieslaw J.; Packo, Pawel
2016-04-01
Attractive properties of guided waves provides very unique potential for characterization of incipient damage, particularly in plate-like structures. Among other properties, guided waves can propagate over long distances and can be used to monitor hidden structural features and components. On the other hand, guided propagation brings substantial challenges for data analysis. Signal processing techniques are frequently supported by numerical simulations in order to facilitate problem solution. When employing numerical models additional sources of errors are introduced. These can play significant role for design and development of a wave-based monitoring strategy. Hence, the paper presents an investigation of numerical models for guided waves generation, propagation and sensing. Numerical dispersion analysis, for guided waves in plates, based on the LISA approach is presented and discussed in the paper. Both dispersion and modal amplitudes characteristics are analysed. It is shown that wave propagation in a numerical model resembles propagation in a periodic medium. Consequently, Lamb wave propagation close to numerical Brillouin zone is investigated and characterized.
20. Wave train generation of solitons in systems with higher-order nonlinearities.
Science.gov (United States)
Mohamadou, Alidou; LatchioTiofack, C G; Kofané, Timoléon C
2010-07-01
Considering the higher-order nonlinearities in a material can significantly change its behavior. We suggest the extended nonlinear Schrödinger equation to describe the propagation of ultrashort optical pulses through a dispersive medium with higher-order nonlinearities. Soliton trains are generated through the modulational instability and we point out the influence of the septic nonlinearity in the modulational instability gain. Experimental values are used for the numerical simulations and the input plane wave leads to the development of pulse trains, depending upon the sign of the septic nonlinearity.
1. Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation.
Science.gov (United States)
Dagrau, Franck; Rénier, Mathieu; Marchiano, Régis; Coulouvrat, François
2011-07-01
Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.
2. Wave propagation in chiral media: composite Fresnel equations
Science.gov (United States)
Chern, Ruey-Lin
2013-07-01
In this paper, the author studies the features of wave propagation in chiral media. A general form of wave equations in biisotropic media is employed to derive concise formulas for the reflection and transmission coefficients. These coefficients are represented as a composite form of Fresnel equations for ordinary dielectrics, which reveal the circularly polarized nature of chiral media. The important features of negative refraction and a backward wave associated with left-handed waves are analyzed.
3. Acoustoelastic Lamb Wave Propagation in Biaxially Stressed Plates (Preprint)
Science.gov (United States)
2012-03-01
particularly as compared to most bulk wave NDE methods, Lamb wave are particularly sensitive to changes in the propagation environment, such as... Wilcox , and J. E. Michaels, “Efficient temperature compensation strategies for guided wave structural health monitoring,” Ultrasonics, 50, pp. 517...Liu, “Effects of residual stress on guided waves in layered media,” Rev. Prog. Quant. NDE , 17, D. O. Thompson and D. E. Chimenti (Eds.), Plenum Press
4. Wave Propagation in Fluids Models and Numerical Techniques
CERN Document Server
Guinot, Vincent
2007-01-01
This book presents the physical principles of wave propagation in fluid mechanics and hydraulics. The mathematical techniques that allow the behavior of the waves to be analyzed are presented, along with existing numerical methods for the simulation of wave propagation. Particular attention is paid to discontinuous flows, such as steep fronts and shock waves, and their mathematical treatment. A number of practical examples are taken from various areas fluid mechanics and hydraulics, such as contaminant transport, the motion of immiscible hydrocarbons in aquifers, river flow, pipe transients an
5. Nonlinear damping of a finite amplitude whistler wave due to modified two stream instability
Energy Technology Data Exchange (ETDEWEB)
Saito, Shinji, E-mail: [email protected] [Graduate School of Science, Nagoya University, Nagoya (Japan); Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya (Japan); Nariyuki, Yasuhiro, E-mail: [email protected] [Faculty of Human Development, University of Toyama, Toyama (Japan); Umeda, Takayuki, E-mail: [email protected] [Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya (Japan)
2015-07-15
A two-dimensional, fully kinetic, particle-in-cell simulation is used to investigate the nonlinear development of a parallel propagating finite amplitude whistler wave (parent wave) with a wavelength longer than an ion inertial length. The cross field current of the parent wave generates short-scale whistler waves propagating highly oblique directions to the ambient magnetic field through the modified two-stream instability (MTSI) which scatters electrons and ions parallel and perpendicular to the magnetic field, respectively. The parent wave is largely damped during a time comparable to the wave period. The MTSI-driven damping process is proposed as a cause of nonlinear dissipation of kinetic turbulence in the solar wind.
6. Rogue-wave bullets in a composite (2+1)D nonlinear medium.
Science.gov (United States)
Chen, Shihua; Soto-Crespo, Jose M; Baronio, Fabio; Grelu, Philippe; Mihalache, Dumitru
2016-07-11
We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.
7. Wave Propagation in Isotropic Media with Two Orthogonal Fracture Sets
Science.gov (United States)
Shao, S.; Pyrak-Nolte, L. J.
2016-10-01
Orthogonal intersecting fracture sets form fracture networks that affect the hydraulic and mechanical integrity of a rock mass. Interpretation of elastic waves propagated through orthogonal fracture networks is complicated by guided modes that propagate along and between fractures, by multiple internal reflections, as well as by scattering from fracture intersections. The existence of some or all of these potentially overlapping modes depends on local stress fields that can preferentially close or open either one or both sets of fractures. In this study, an acoustic wave front imaging system was used to examine the effect of bi-axial loading conditions on acoustic wave propagation in isotropic media containing two orthogonal fracture sets. From the experimental data, orthogonal intersecting fracture sets support guided waves that depend on fracture spacing and fracture-specific stiffnesses. In addition, fracture intersections have stronger effects on propagating wave fronts than merely the superposition of the effects of two independent fractures because of energy partitioning among transmitted/reflected waves, scattered waves and guided modes. Interpretation of the properties of fractures or fracture sets from seismic measurements must consider non-uniform fracture stiffnesses within and among fracture sets, as well as considering the striking effects of fracture intersections on wave propagation.
8. Measuring Gravitational-Wave Propagation Speed with Multimessenger Observations
OpenAIRE
Nishizawa, Atsushi; Nakamura, Takashi
2016-01-01
A measurement of gravitational wave (GW) propagation speed is one of important tests of gravity in a dynamical regime. We report a method to measure the GW propagation speed by directly comparing arrival times of GWs, neutrinos from supernovae (SN), and photons from short gamma-ray bursts (SGRB). We found that the future multimessenger observations can test the GW propagation speed with the precision of ~ 10^(-16)-10^(-15), improving the previous suggestions by 9 — 10 orders of magnitude. We ...
9. Topology optimization of vibration and wave propagation problems
DEFF Research Database (Denmark)
Jensen, Jakob Søndergaard
2007-01-01
The method of topology optimization is a versatile method to determine optimal material layouts in mechanical structures. The method relies on, in principle, unlimited design freedom that can be used to design materials, structures and devices with significantly improved performance and sometimes...... novel functionality. This paper addresses basic issues in simulation and topology design of vibration and wave propagation problems. Steady-state and transient wave propagation problems are addressed and application examples for both cases are presented....
10. Characteristic wave diversity in near vertical incidence skywave propagation
NARCIS (Netherlands)
Witvliet, Ben A.; Maanen, van Erik; Petersen, George J.; Westenberg, Albert J.; Bentum, Mark J.; Slump, Cornelis H.; Schiphorst, Roel
2015-01-01
In Near Vertical Incidence Skywave (NVIS) propagation, effective diversity reception can be realized using a dual channel receiver and a dual polarization antenna with polarization matched to the (left hand and right hand) circular polarization of the characteristic waves propagating in the ionosphe
11. Time-domain Wave Propagation in Dispersive Media①
Institute of Scientific and Technical Information of China (English)
1997-01-01
The equation of time-domain wave propagation in dispersive media and the explicit beam propagation method are presented in this paper.This method is demonstrated by the short optical pulses in a directional coupler with second order dispersive effect and shows to be in full agreement with former references.This method is simple,easy and practical.
12. Wave propagation of spectral energy content in a granular chain
Science.gov (United States)
Shrivastava, Rohit Kumar; Luding, Stefan
2017-06-01
A mechanical wave is propagation of vibration with transfer of energy and momentum. Understanding the spectral energy characteristics of a propagating wave through disordered granular media can assist in understanding the overall properties of wave propagation through inhomogeneous materials like soil. The study of these properties is aimed at modeling wave propagation for oil, mineral or gas exploration (seismic prospecting) or non-destructive testing of the internal structure of solids. The focus is on the total energy content of a pulse propagating through an idealized one-dimensional discrete particle system like a mass disordered granular chain, which allows understanding the energy attenuation due to disorder since it isolates the longitudinal P-wave from shear or rotational modes. It is observed from the signal that stronger disorder leads to faster attenuation of the signal. An ordered granular chain exhibits ballistic propagation of energy whereas, a disordered granular chain exhibits more diffusive like propagation, which eventually becomes localized at long time periods. For obtaining mean-field macroscopic/continuum properties, ensemble averaging has been used, however, such an ensemble averaged spectral energy response does not resolve multiple scattering, leading to loss of information, indicating the need for a different framework for micro-macro averaging.
13. Wave propagation of spectral energy content in a granular chain
Directory of Open Access Journals (Sweden)
Shrivastava Rohit Kumar
2017-01-01
Full Text Available A mechanical wave is propagation of vibration with transfer of energy and momentum. Understanding the spectral energy characteristics of a propagating wave through disordered granular media can assist in understanding the overall properties of wave propagation through inhomogeneous materials like soil. The study of these properties is aimed at modeling wave propagation for oil, mineral or gas exploration (seismic prospecting or non-destructive testing of the internal structure of solids. The focus is on the total energy content of a pulse propagating through an idealized one-dimensional discrete particle system like a mass disordered granular chain, which allows understanding the energy attenuation due to disorder since it isolates the longitudinal P-wave from shear or rotational modes. It is observed from the signal that stronger disorder leads to faster attenuation of the signal. An ordered granular chain exhibits ballistic propagation of energy whereas, a disordered granular chain exhibits more diffusive like propagation, which eventually becomes localized at long time periods. For obtaining mean-field macroscopic/continuum properties, ensemble averaging has been used, however, such an ensemble averaged spectral energy response does not resolve multiple scattering, leading to loss of information, indicating the need for a different framework for micro-macro averaging.
14. Influences of interfacial properties on second-harmonic generation of Lamb waves propagating in layered planar structures
Energy Technology Data Exchange (ETDEWEB)
Deng Mingxi [College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 (China); Wang Ping [College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 (China); Lv Xiafu [College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 (China)
2006-07-21
This paper describes influences of interfacial properties on second-harmonic generation of Lamb waves propagating in layered planar structures. The nonlinearity in the elastic wave propagation is treated as a second-order perturbation of the linear elastic response. Due to the kinematic nonlinearity and the elastic nonlinearity of materials, there are second-order bulk and surface/interface driving sources in layered planar structures through which Lamb waves propagate. These driving sources can be thought of as forcing functions of a series of double frequency lamb waves (DFLWs) in terms of the approach of modal expansion analysis for waveguide excitation. The total second-harmonic fields consist of a summation of DFLWs in the corresponding stress-free layered planar structures. The interfacial properties of layered planar structures can be described by the well-known finite interfacial stiffness technique. The normal and tangential interfacial stiffness constants can be coupled with the equation governing the expansion coefficient of each DFLW component. On the other hand, the normal and tangential interfacial stiffness constants are associated with the degree of dispersion between Lamb waves and DFLWs. Theoretical analyses and numerical simulations indicate that the efficiency of second-harmonic generation by Lamb wave propagation is closely dependent on the interfacial properties of layered structures. The potential of using the effect of second-harmonic generation by Lamb wave propagation to characterize the interfacial properties of layered structures are considered. Some experimental results are presented.
15. Time reversal techniques in electromagnetic wave propagation
Science.gov (United States)
Yi, Jiang
The time reversal method is a novel scheme utilizing the scattering components in a highly cluttered environment to achieve super-resolution focusing beyond Rayleigh criteria. In acoustics, time reversal effects are comprehensively analyzed and utilized in underwater target detection and communication. Successful demonstrations of the time reversal method using low frequency waveform in acoustics have generated wide interest in utilizing time reversal method by radio frequency electromagnetic waves. However, applications of the time reversal method in electromagnetics are considered to be emerging research topics and lack extensive analyses and studies. In this thesis, we present a systematic study in which a series of novel time reversal techniques have been developed for target detection and imaging in highly cluttered environments where higher order scattering is substantial. This thesis also contributes to insightful understanding of basic time reversal properties in electromagnetic (EM) wave propagation in such environment. EM time reversal focusing and nulling effects using both single and multiple antennas are first demonstrated by FDTD simulations. Based on these properties, single antenna time reversal detection indicates significant enhancement in detection capability over traditional change detection scheme. A frequency selection scheme utilizing the frequencies with strong constructive interference between the target and background environment is developed to further improve the performance of the time reversal detector. Moreover, a novel time reversal adaptive interference cancellation (TRAIC) detection scheme developed based on TR properties can obtain null of the background through the time reversal nulling effect and achieve automatic focusing on the target through the time reversal focusing effect. Therefore, the detection ability, dynamic range and signal to noise ratio of a radar system can be significantly enhanced by the time reversal method
16. Propagation of high frequency waves in the quiet solar atmosphere
Directory of Open Access Journals (Sweden)
Andić A.
2008-01-01
Full Text Available High-frequency waves (5 mHz to 20 mHz have previously been suggested as a source of energy accounting for partial heating of the quiet solar atmosphere. The dynamics of previously detected high-frequency waves is analyzed here. Image sequences were taken by using the German Vacuum Tower Telescope (VTT, Observatorio del Teide, Izana, Tenerife, with a Fabry-Perot spectrometer. The data were speckle reduced and analyzed with wavelets. Wavelet phase-difference analysis was performed to determine whether the waves propagate. We observed the propagation of waves in the frequency range 10 mHz to 13 mHz. We also observed propagation of low-frequency waves in the ranges where they are thought to be evanescent in the regions where magnetic structures are present.
17. Propagation of High Frequency Waves in the Quiet Solar Atmosphere
Directory of Open Access Journals (Sweden)
Andić, A.
2008-12-01
Full Text Available High-frequency waves (5 mHz to 20 mHz have previously been suggested as a source of energy accounting for partial heating of the quiet solar atmosphere. The dynamics of previously detected high-frequency waves is analysed here. Image sequences were taken by using the German Vacuum Tower Telescope (VTT, Observatorio del Teide, Izana, Tenerife, with a Fabry-Perot spectrometer. The data were speckle reduced and analysed with wavelets. Wavelet phase-difference analysis was performed to determine whether the waves propagate. We observed the propagation of waves in the frequency range 10 mHz to 13 mHz. We also observed propagation of low-frequency waves in the ranges where they are thought to be evanescent in the regions where magnetic structures are present.
18. Propagation of High Frequency Waves in the Quiet Solar Atmosphere
CERN Document Server
AndiÄ, Aleksandra
2008-01-01
High-frequency waves (5 mHz to 20mHz) have previously been suggested as a source of energy accounting partial heating of the quiet solar atmosphere. The dynamics of previously detected high-frequency waves is analysed here. Image sequences are taken using the German Vacuum Tower Telescope (VTT), Observatorio del Teide, Izana, Tenerife, with a Fabry-Perot spectrometer. The data were speckle reduced and analyzed with wavelets. Wavelet phase-difference analysis is performed to determine whether the waves propagate. We observe the propagation of waves in the frequency range 10mHz to 13mHz. We also observe propagation of low-frequency waves in the ranges where they are thought to be evanescent in regions where magnetic structures are present.
19. Time dependent wave envelope finite difference analysis of sound propagation
Science.gov (United States)
Baumeister, K. J.
1984-01-01
A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.
20. Nonlinear plasma wave in magnetized plasmas
Energy Technology Data Exchange (ETDEWEB)
Bulanov, Sergei V. [Kansai Photon Science Institute, JAEA, Kizugawa, Kyoto 619-0215 (Japan); Prokhorov Institute of General Physics, Russian Academy of Sciences, Moscow 119991 (Russian Federation); Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region 141700 (Russian Federation); Esirkepov, Timur Zh.; Kando, Masaki; Koga, James K. [Kansai Photon Science Institute, JAEA, Kizugawa, Kyoto 619-0215 (Japan); Hosokai, Tomonao; Zhidkov, Alexei G. [Photon Pioneers Center, Osaka University, 2-8 Yamadaoka, Suita, Osaka 565-0871 (Japan); Japan Science and Technology Agency, CREST, 2-1, Yamadaoka, Suita, Osaka 565-0871 (Japan); Kodama, Ryosuke [Photon Pioneers Center, Osaka University, 2-8 Yamadaoka, Suita, Osaka 565-0871 (Japan); Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871 (Japan)
2013-08-15
Nonlinear axisymmetric cylindrical plasma oscillations in magnetized collisionless plasmas are a model for the electron fluid collapse on the axis behind an ultrashort relativisically intense laser pulse exciting a plasma wake wave. We present an analytical description of the strongly nonlinear oscillations showing that the magnetic field prevents closing of the cavity formed behind the laser pulse. This effect is demonstrated with 3D PIC simulations of the laser-plasma interaction. An analysis of the betatron oscillations of fast electrons in the presence of the magnetic field reveals a characteristic “Four-Ray Star” pattern.
1. Propagation of gravitational waves in the nonperturbative spinor vacuum
Energy Technology Data Exchange (ETDEWEB)
Dzhunushaliev, Vladimir [Al-Farabi Kazakh National University, Department of Theoretical and Nuclear Physics, Almaty (Kazakhstan); Al-Farabi Kazakh National University, Institute of Experimental and Theoretical Physics, Almaty (Kazakhstan); Eurasian National University, Institute for Basic Research, Astana (Kazakhstan); Institute of Physicotechnical Problems and Material Science of the NAS of the Kyrgyz Republic, Bishkek (Kyrgyzstan); Folomeev, Vladimir [Institute of Physicotechnical Problems and Material Science of the NAS of the Kyrgyz Republic, Bishkek (Kyrgyzstan)
2014-09-15
The propagation of gravitational waves on the background of a nonperturbative vacuum of a spinor field is considered. It is shown that there are several distinctive features in comparison with the propagation of plane gravitational waves through empty space: there exists a fixed phase difference between the h{sub yy,zz} and h{sub yz} components of the wave; the phase and group velocities of gravitational waves are not equal to the velocity of light; the group velocity is always less than the velocity of light; under some conditions the gravitational waves are either damped or absent; for given frequency, there exist two waves with different wave vectors. We also discuss the possibility of an experimental verification of the obtained effects as a tool to investigate nonperturbative quantum field theories. (orig.)
2. Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients.
Science.gov (United States)
Wang, Luyun; Li, Lu; Li, Zhonghao; Zhou, Guosheng; Mihalache, Dumitru
2005-09-01
The generalized nonlinear Schrödinger model with distributed dispersion, nonlinearity, and gain or loss is considered and the explicit, analytical solutions describing the dynamics of bright solitons on a continuous-wave background are obtained in quadratures. Then, the generation, compression, and propagation of pulse trains are discussed in detail. The numerical results show that solitons can be compressed by choosing the appropriate control fiber system, and pulse trains generated by modulation instability can propagate undistorsted along fibers with distributed parameters by controlling appropriately the energy of each pulse in the pulse train.
3. A HIGHER-ORDER NON-HYDROSTATIC MODEL FOR SIMULATING WAVE PROPAGATION OVER IRREGULAR BOTTOMS
Institute of Scientific and Technical Information of China (English)
AI Cong-fang; XING Yah; JIN Sheng
2011-01-01
A higher-order non-hydrostatic model is developed to simulate the wave propagation over irregular bottoms based on a vertical boundary-fitted coordinate system.In the model,an explicit projection method is adopted to solve the unsteady Euler equations.Advection terms are integrated explicitly with the MacCormack's scheme,with a second-order accuracy in both space and time.Two classical examples of surface wave propagation are used to demonstrate the capability of the model.It is found that the model with only two vertical layers could accurately simulate the motion of waves,including wave shoaling,nonlinearity,dispersion,refraction,and diffraction phenomena.
4. Computational simulation of wave propagation problems in infinite domains
Institute of Scientific and Technical Information of China (English)
2010-01-01
This paper deals with the computational simulation of both scalar wave and vector wave propagation problems in infinite domains. Due to its advantages in simulating complicated geometry and complex material properties, the finite element method is used to simulate the near field of a wave propagation problem involving an infinite domain. To avoid wave reflection and refraction at the common boundary between the near field and the far field of an infinite domain, we have to use some special treatments to this boundary. For a wave radiation problem, a wave absorbing boundary can be applied to the common boundary between the near field and the far field of an infinite domain, while for a wave scattering problem, the dynamic infinite element can be used to propagate the incident wave from the near field to the far field of the infinite domain. For the sake of illustrating how these two different approaches are used to simulate the effect of the far field, a mathematical expression for a wave absorbing boundary of high-order accuracy is derived from a two-dimensional scalar wave radiation problem in an infinite domain, while the detailed mathematical formulation of the dynamic infinite element is derived from a two-dimensional vector wave scattering problem in an infinite domain. Finally, the coupled method of finite elements and dynamic infinite elements is used to investigate the effects of topographical conditions on the free field motion along the surface of a canyon.
5. Study of Linear and Nonlinear Wave Excitation
Science.gov (United States)
Chu, Feng; Berumen, Jorge; Hood, Ryan; Mattingly, Sean; Skiff, Frederick
2013-10-01
We report an experimental study of externally excited low-frequency waves in a cylindrical, magnetized, singly-ionized Argon inductively-coupled gas discharge plasma that is weakly collisional. Wave excitation in the drift wave frequency range is accomplished by low-percentage amplitude modulation of the RF plasma source. Laser-induced fluorescence is adopted to study ion-density fluctuations in phase space. The laser is chopped to separate LIF from collisional fluorescence. A single negatively-biased Langmuir probe is used to detect ion-density fluctuations in the plasma. A ring array of Langmuir probes is also used to analyze the spatial and spectral structure of the excited waves. We apply coherent detection with respect to the wave frequency to obtain the ion distribution function associated with externally generated waves. Higher-order spectra are computed to evaluate the nonlinear coupling between fluctuations at various frequencies produced by the externally generated waves. Parametric decay of the waves is observed. This work is supported by U.S. DOE Grant No. DE-FG02-99ER54543.
6. Special Course on Acoustic Wave Propagation
Science.gov (United States)
1979-08-01
exesiple) et cules se propagent 41 is surface du liquido . WW.JF~q W - , -- r -w w 144 Dens ce cax Von (10) 4 =/.+ Sane entrer dans le ddtail des...543-546. 57. STUFF, R., Analytic solution for the sound propagation through the atmospheric wind boundary layer. Proc. Noise Control Conf., Warszawa...between nodal surfaces of one-half wavelength. Evidently this property, like the energy conservation one, is available for use as a " control " on any
7. Controlling nonlinear waves in excitable media
Energy Technology Data Exchange (ETDEWEB)
Puebla, Hector [Departamento de Energia, Universidad Autonoma Metropolitana, Av. San Pablo No. 180, Reynosa-Tamaulipas, Azcapotzalco 02200, DF, Mexico (Mexico)], E-mail: [email protected]; Martin, Roland [Laboratoire de Modelisation et d' Imagerie en Geosciences, CNRS UMR and INRIA Futurs Magique-3D, Universite de Pau (France); Alvarez-Ramirez, Jose [Division de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa (Mexico); Aguilar-Lopez, Ricardo [Departamento de Biotecnologia y Bioingenieria, CINVESTAV-IPN (Mexico)
2009-01-30
A new feedback control method is proposed to control the spatio-temporal dynamics in excitable media. Applying suitable external forcing to the system's slow variable, successful suppression and control of propagating pulses as well as spiral waves can be obtained. The proposed controller is composed by an observer to infer uncertain terms such as diffusive transport and kinetic rates, and an inverse-dynamics feedback function. Numerical simulations shown the effectiveness of the proposed feedback control approach.
8. Wave-kinetic description of nonlinear photons
CERN Document Server
Marklund, M; Brodin, G; Stenflo, L
2004-01-01
The nonlinear interaction, due to quantum electrodynamical (QED) effects, between photons is investigated using a wave-kinetic description. Starting from a coherent wave description, we use the Wigner transform technique to obtain a set of wave-kinetic equations, the so called Wigner-Moyal equations. These equations are coupled to a background radiation fluid, whose dynamics is determined by an acoustic wave equation. In the slowly varying acoustic limit, we analyse the resulting system of kinetic equations, and show that they describe instabilities, as well as Landau-like damping. The instabilities may lead to break-up and focusing of ultra-high intensity multi-beam systems, which in conjunction with the damping may result in stationary strong field structures. The results could be of relevance for the next generation of laser-plasma systems.
9. Remarks on the parallel propagation of small-amplitude dispersive Alfvénic waves
Directory of Open Access Journals (Sweden)
S. Champeaux
1999-01-01
Full Text Available The envelope formalism for the description of a small-amplitude parallel-propagating Alfvén wave train is tested against direct numerical simulations of the Hall-MHD equations in one space dimension where kinetic effects are neglected. It turns out that the magnetosonic-wave dynamics departs from the adiabatic approximation not only near the resonance between the speed of sound and the Alfvén wave group velocity, but also when the speed of sound lies between the group and phase velocities of the Alfvén wave. The modulational instability then does not anymore affect asymptotically large scales and strong nonlinear effects can develop even in the absence of the decay instability. When the Hall-MHD equations are considered in the long-wavelength limit, the weakly nonlinear dynamics is accurately reproduced by the derivative nonlinear Schrödinger equation on the expected time scale, provided no decay instabilities are present. The stronger nonlinear regime which develops at later time is captured by including the coupling to the nonlinear dynamics of the magnetosonic waves.
10. Characterization of surface properties of a solid plate using nonlinear Lamb wave approach.
Science.gov (United States)
Deng, Mingxi
2006-12-22
A nonlinear Lamb wave approach is presented for characterizing the surface properties of a solid plate. This characterization approach is useful for some practical situations where ultrasonic transducers cannot touch the surfaces to be inspected, e.g. the inside surfaces of sealed vessels. In this paper, the influences of changes in the surface properties of a solid plate on the effect of second-harmonic generation by Lamb wave propagation were analyzed. A surface coating with the different properties was used to simulate changes in the surface properties of a solid plate. When the areas and thicknesses of coatings on the surface of a given solid plate changed, the amplitude-frequency curves both of the fundamental waves and the second harmonics by Lamb wave propagation were measured under the condition that Lamb waves had a strong nonlinearity. It was found that changes in the surface properties might clearly affect the efficiency of second-harmonic generation by Lamb wave propagation. The Stress Wave Factors (SWFs) in acousto-ultrasonic technique were used for reference, and the definitions of the SWFs of Lamb waves were introduced. The preliminary experimental results showed that the second-harmonic SWF of Lamb wave propagation could effectively be used to characterize changes in the surface properties of the given solid plate.
11. A nonlinear Schroedinger wave equation with linear quantum behavior
Energy Technology Data Exchange (ETDEWEB)
Richardson, Chris D.; Schlagheck, Peter; Martin, John; Vandewalle, Nicolas; Bastin, Thierry [Departement de Physique, University of Liege, 4000 Liege (Belgium)
2014-07-01
We show that a nonlinear Schroedinger wave equation can reproduce all the features of linear quantum mechanics. This nonlinear wave equation is obtained by exploring, in a uniform language, the transition from fully classical theory governed by a nonlinear classical wave equation to quantum theory. The classical wave equation includes a nonlinear classicality enforcing potential which when eliminated transforms the wave equation into the linear Schroedinger equation. We show that it is not necessary to completely cancel this nonlinearity to recover the linear behavior of quantum mechanics. Scaling the classicality enforcing potential is sufficient to have quantum-like features appear and is equivalent to scaling Planck's constant.
12. Symmetry, phase modulation and nonlinear waves
CERN Document Server
Bridges, Thomas J
2017-01-01
Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.
13. Automated classification of spatiotemporal characteristics of gastric slow wave propagation.
Science.gov (United States)
2013-01-01
Gastric contractions are underpinned by an electrical event called slow wave activity. High-resolution electrical mapping has recently been adapted to study gastric slow waves at a high spatiotemporal detail. As more slow wave data becomes available, it is becoming evident that the spatial organization of slow wave plays a key role in the initiation and maintenance of gastric dsyrhythmias in major gastric motility disorders. All of the existing slow wave signal processing techniques deal with the identification and partitioning of recorded wave events, but not the analysis of the slow wave spatial organization, which is currently performed visually. This manual analysis is time consuming and is prone to observer bias and error. We present an automated approach to classify spatial slow wave propagation patterns via the use of Pearson cross correlations. Slow wave propagations were grouped into classes based on their similarity to each other. The method was applied to high-resolution gastric slow wave recordings from four pigs. There were significant changes in the velocity of the gastric slow wave wavefront and the amplitude of the slow wave event when there was a change in direction to the slow wave wavefront during dsyrhythmias, which could be detected with the automated approach.
14. Mathematical modelling of generation and forward propagation of dispersive waves
NARCIS (Netherlands)
Lie She Liam, L.S.L.
2013-01-01
This dissertation concerns the mathematical theory of forward propagation and generation of dispersive waves. We derive the AB2-equation which describes forward traveling waves in two horizontal dimension. It is the generalization of the Kadomtsev-Petviashvilli (KP) equation. The derivation is based
15. Stress Wave Propagation in Larch Plantation Trees-Numerical Simulation
Science.gov (United States)
Fenglu Liu; Fang Jiang; Xiping Wang; Houjiang Zhang; Wenhua Yu
2015-01-01
In this paper, we attempted to simulate stress wave propagation in virtual tree trunks and construct two dimensional (2D) wave-front maps in the longitudinal-radial section of the trunk. A tree trunk was modeled as an orthotropic cylinder in which wood properties along the fiber and in each of the two perpendicular directions were different. We used the COMSOL...
16. Stress Wave Propagation in a Gradient Elastic Medium
Institute of Scientific and Technical Information of China (English)
赵亚溥; 赵涵; 胡宇群
2002-01-01
The gradient elastic constitutive equation incorporating the second gradient of the strains is used to determinethe monochromatic elastic plane wave propagation in a gradient infinite medium and thin rod. The equationof motion, together with the internal material length, has been derived. Various dispersion relations have beendetermined. We present explicit expressions for the relationship between various wave speeds, wavenumber andinternal material length.
17. Statistical Characterization of Electromagnetic Wave Propagation in Mine Environments
KAUST Repository
2013-01-01
A computational framework for statistically characterizing electromagnetic (EM) wave propagation through mine tunnels and galleries is presented. The framework combines a multi-element probabilistic collocation method with a full-wave fast Fourier transform and fast multipole method accelerated surface integral equation-based EM simulator to statistically characterize fields from wireless transmitters in complex mine environments. 1536-1225 © 2013 IEEE.
18. Nonlinear coupling of left and right handed circularly polarized dispersive Alfvén wave
Energy Technology Data Exchange (ETDEWEB)
Sharma, R. P., E-mail: [email protected]; Sharma, Swati, E-mail: [email protected]; Gaur, Nidhi, E-mail: [email protected] [Centre for Energy Studies, Indian Institute of Technology Delhi, New Delhi 110016 (India)
2014-07-15
The nonlinear phenomena are of prominent interests in understanding the particle acceleration and transportation in the interplanetary space. The ponderomotive nonlinearity causing the filamentation of the parallel propagating circularly polarized dispersive Alfvén wave having a finite frequency may be one of the mechanisms that contribute to the heating of the plasmas. The contribution will be different of the left (L) handed mode, the right (R) handed mode, and the mix mode. The contribution also depends upon the finite frequency of the circularly polarized waves. In the present paper, we have investigated the effect of the nonlinear coupling of the L and R circularly polarized dispersive Alfvén wave on the localized structures formation and the respective power spectra. The dynamical equations are derived in the presence of the ponderomotive nonlinearity of the L and R pumps and then studied semi-analytically as well as numerically. The ponderomotive nonlinearity accounts for the nonlinear coupling between both the modes. In the presence of the adiabatic response of the density fluctuations, the nonlinear dynamical equations satisfy the modified nonlinear Schrödinger equation. The equations thus obtained are solved in solar wind regime to study the coupling effect on localization and the power spectra. The effect of coupling is also studied on Faraday rotation and ellipticity of the wave caused due to the difference in the localization of the left and the right modes with the distance of propagation.
19. In-plane propagation of electromagnetic waves in planar metamaterials
Science.gov (United States)
Yi, Changhyun; Rhee, Joo Yull; Kim, Ki Won; Lee, YoungPak
2016-08-01
Some planar metamaterials (MMs) or subwavelength antenna/hole arrays have a considerable amount of in-plane propagation when certain conditions are met. In this paper, the in-plane propagation caused by a wave incident on a MM absorber was studied by using a finite-difference time-domain (FDTD) technique. By using a FDTD simulation, we were able to observe a nonnegligible amount of in-plane propagation after the incident wave had arrived at the surface of the planar structure and gradually decreased propagation of the electromagnetic wave in the planar direction gradually decreased. We performed the FDTD simulation carefully to reproduce valid results and to verify the existence of in-plane propagation. For verification of the in-plane propagation explicitly, Poynting vectors were calculated and visualized inside the dielectric substrate between the metallic back-plate and an array of square patches. We also investigated several different structures with resonators of various shapes and found that the amount of facing edges of adjacent metallic patches critically determined the strength of the in-plane propagation. Through this study, we could establish the basis for the existence of in-plane propagation in MMs.
20. Nonlinear Dispersion Effect on Wave Transformation
Institute of Scientific and Technical Information of China (English)
LI Ruijie; Dong-Young LEE
2000-01-01
A new nonlinear dispersion relation is given in this paper, which can overcome the limitation of the intermediate minimum value in the dispersion relation proposed by Kirby and Dalrymple (1986), and which has a better approximation to Hedges＇ empirical relation than the modilied relations by Hedges (1987). Kirby and Dahymple (1987) for shallow waters. The new dispersion relation is simple in form. thus it can be used easily in practice. Meanwhile. a general explicil approximalion to the new dispersion rela tion and olher nonlinear dispersion relations is given. By use of the explicit approximation to the new dispersion relation along with the mild slope equation taking inlo account weakly nonlinear effect, a mathematical model is obtained, and it is applied to laboratory data. The results show that the model developed vith the new dispersion relation predicts wave translornation over complicated topography quite well.
1. Variational modelling of nonlinear water waves
Science.gov (United States)
Kalogirou, Anna; Bokhove, Onno
2015-11-01
Mathematical modelling of water waves is demonstrated by investigating variational methods. A potential flow water wave model is derived using variational techniques and extented to include explicit time-dependence, leading to non-autonomous dynamics. As a first example, we consider the problem of a soliton splash in a long wave channel with a contraction at its end, resulting after a sluice gate is removed at a finite time. The removal of the sluice gate is included in the variational principle through a time-dependent gravitational potential. A second example involving non-autonomous dynamics concerns the motion of a free surface in a vertical Hele-Shaw cell. Explicit time-dependence now enters the model through a linear damping term due to the effect of wall friction and a term representing the motion of an artificially driven wave pump. In both cases, the model is solved numerically using a Galerkin FEM and the numerical results are compared to wave structures observed in experiments. The water wave model is also adapted to accommodate nonlinear ship dynamics. The novelty is this case is the coupling between the water wave dynamics, the ship dynamics and water line dynamics on the ship. For simplicity, we consider a simple ship structure consisting of V-shaped cross-sections.
2. Shear horizontal (SH) ultrasound wave propagation around smooth corners.
Science.gov (United States)
Petcher, P A; Burrows, S E; Dixon, S
2014-04-01
Shear horizontal (SH) ultrasound guided waves are being used in an increasing number of non-destructive testing (NDT) applications. One advantage SH waves have over some wave types, is their ability to propagate around curved surfaces with little energy loss; to understand the geometries around which they could propagate, the wave reflection must be quantified. A 0.83mm thick aluminium sheet was placed in a bending machine, and a shallow bend was introduced. Periodically-poled magnet (PPM) electromagnetic acoustic transducers (EMATs), for emission and reception of SH waves, were placed on the same side of the bend, so that reflected waves were received. Additional bending of the sheet demonstrated a clear relationship between bend angles and the reflected signal. Models suggest that the reflection is a linear superposition of the reflections from each bend segment, such that sharp turns lead to a larger peak-to-peak amplitude, in part due to increased phase coherence.
3. Propagation of internal waves up continental slope and shelf
Institute of Scientific and Technical Information of China (English)
DAI Dejun; WANG Wei; QIAO Fangli; YUAN Yeli; XIANG Wenxi
2008-01-01
In a two-dimensional and linear framework, a transformation was developed to derive eigensolutions of internal waves over a subcriticai hyperbolic slope and to approximate the continental slope and shelf. The transformation converts a hyperbolic slope in physical space into a fiat bottom in transform space while the governing equations of internal waves remain hyperbolic. The eigensolutions are further used to study the evolution of linear internal waves as it propagates to subcritical continental slope and shelf. The stream function, velocity, and vertical shear of velocity induced by internal wave at the hyperbolic slope are analytically expressed by superposition of the obtained eigensolutions. The velocity and velocity shear increase as the internal wave propagates to a hyperbolic slope. They become very large especially when the slope of internal wave rays approaches the topographic slope, which is consistent with the previous studies.
4. Propagation of Iamb waves in adhesively bonded multilayered media
Institute of Scientific and Technical Information of China (English)
ZHANG Haiyan; XIE Yuanxia; LIU Zhenqing
2003-01-01
The effect of introducing attenuation on Lamb wave dispersion curves is studied in this paper. Attenuation is introduced to a three-layered composite plate by an adhesive bond layer with viscous behavior. No changes are required to the transfer matrix formulation for the propagation of elastic waves. By introduction of a complex wavenumber, the model can be used to the propagation of attenuative Lamb waves. Numerical examples for a three-layered aluminium-epoxy-aluminium plate show that attenuation values of each mode in plates are related not only to attenuation, but also to the thickness of the bonded layer, which is in agreement with practical situations.
5. Wave propagation in reconfigurable magneto-elastic kagome lattice structures
Science.gov (United States)
Schaeffer, Marshall; Ruzzene, Massimo
2015-05-01
The paper discusses the wave propagation characteristics of two-dimensional magneto-elastic kagome lattices. Mechanical instabilities caused by magnetic interactions are exploited in combination with particle contact to bring about changes in the topology and stiffness of the lattices. The analysis uses a lumped mass system of particles, which interact through axial and torsional elastic forces as well as magnetic forces. The propagation of in-plane waves is predicted by applying Bloch theorem to lattice unit cells with linearized interactions. Elastic wave dispersion in these lattices before and after topological changes is compared, and large differences are highlighted.
6. Electron acceleration in the ionosphere by obliquely propagating electromagnetic waves
Science.gov (United States)
Burke, William J.; Ginet, Gregory P.; Heinemann, Michael A.; Villalon, Elena
The paper presents an analysis of the relativistic equations of motion for electrons in magnetized plasma and externally imposed electromagnetic fields that propagate at arbitrary angles to the background magnetic field. The relativistic Lorentz equation for a test electron moving under the influence of an electromagnetic wave in a cold magnetized plasma and wave propagation through the ionospheric 'radio window' are examined. It is found that at wave energy fluxes greater than 10 to the 8th mW/sq m, initially cold electrons can be accelerated to energies of several MeV in less than a millisecond. Plans to test the theoretical results with rocket flights are discussed.
7. Nonlinear waves on the free surface of a dielectric liquid in an oblique electric field
Energy Technology Data Exchange (ETDEWEB)
Gashkov, M. A.; Zubarev, N. M., E-mail: [email protected]; Kochurin, E. A., E-mail: [email protected] [Ural Branch, Russian Academy of Sciences, Institute of Electrophysics (Russian Federation)
2015-09-15
The nonlinear dynamics of the free surface of an ideal dielectric liquid that is exposed to an external oblique electric field has been studied theoretically. In the framework of the Hamiltonian formalism, a system of nonlinear integro-differential equations has been derived that describes the dynamics of nonlinear waves in the small-angle approximation. It is established that for a liquid with high dielectric permittivity, these equations have a solution in the form of plane waves of arbitrary shape that propagate without distortion in the direction of the horizontal component of the external field.
8. Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths.
Science.gov (United States)
Pitois, S; Fatome, J; Millot, G
2008-04-28
In this work, we report the experimental observation of a polarization attraction process which can occur in optical fibers at telecommunication wavelengths. More precisely, we have numerically and experimentally shown that a polarization attractor, based on the injection of two counter-propagating waves around 1.55microm into a 2-m long high nonlinear fiber, can transform any input polarization state into a unique well-defined output polarization state.
9. Nonlinear Alfvén wave dynamics at a 2D magnetic null point: ponderomotive force
Science.gov (United States)
Thurgood, J. O.; McLaughlin, J. A.
2013-07-01
Context. In the linear, β = 0 MHD regime, the transient properties of magnetohydrodynamic (MHD) waves in the vicinity of 2D null points are well known. The waves are decoupled and accumulate at predictable parts of the magnetic topology: fast waves accumulate at the null point; whereas Alfvén waves cannot cross the separatricies. However, in nonlinear MHD mode conversion can occur at regions of inhomogeneous Alfvén speed, suggesting that the decoupled nature of waves may not extend to the nonlinear regime. Aims: We investigate the behaviour of low-amplitude Alfvén waves about a 2D magnetic null point in nonlinear, β = 0 MHD. Methods: We numerically simulate the introduction of low-amplitude Alfvén waves into the vicinity of a magnetic null point using the nonlinear LARE2D code. Results: Unlike in the linear regime, we find that the Alfvén wave sustains cospatial daughter disturbances, manifest in the transverse and longitudinal fluid velocity, owing to the action of nonlinear magnetic pressure gradients (viz. the ponderomotive force). These disturbances are dependent on the Alfvén wave and do not interact with the medium to excite magnetoacoustic waves, although the transverse daughter becomes focused at the null point. Additionally, an independently propagating fast magnetoacoustic wave is generated during the early stages, which transports some of the initial Alfvén wave energy towards the null point. Subsequently, despite undergoing dispersion and phase-mixing due to gradients in the Alfvén-speed profile (∇cA ≠ 0) there is no further nonlinear generation of fast waves. Conclusions: We find that Alfvén waves at 2D cold null points behave largely as in the linear regime, however they sustain transverse and longitudinal disturbances - effects absent in the linear regime - due to nonlinear magnetic pressure gradients.
10. Propagation of Electromagnetic Waves in Extremely Dense Media
CERN Document Server
Masood, Samina
2016-01-01
We study the propagation of electromagnetic (EM) waves in extremely dense exotic systems with very unique properties. These EM waves develop a longitudinal component due to its interaction with the medium. Renormalization scheme of QED is used to understand the propagation of EM waves in both longitudinal and transverse directions. The propagation of EM waves in a quantum statistically treatable medium affects the properties of the medium itself. The electric permittivity and the magnetic permeability of the medium are modified and influence the related behavior of the medium. All the electromagnetic properties of a medium become a function of temperature and chemical potential of the medium. We study in detail the modifications of electric permittivity and magnetic permeability and other related properties of a medium in the superdense stellar objects.
11. PROPAGATION OF ELECTROMAGNETIC WAVE IN THE THREE PHASES SOIL MEDIA
Institute of Scientific and Technical Information of China (English)
陈云敏; 边学成; 陈仁朋; 梁志刚
2003-01-01
The fundamental parameters such as dielectric permittivity and magnetic permeability are required to solve the propagation of electromagnetic wave (EM Wave) in the soil. Based on Maxwell equations, the equivalent model is proposed to calculate the dielectric permittivity of mixed soil. The results of calculation fit. the test data well and will provide solid foundation for the application of EM wave in the soil moisture testing, CT analyzing of soil and the inspecting of geoenvironment.
12. A method for generating highly nonlinear periodic waves in physical wave basins
DEFF Research Database (Denmark)
Zhang, Haiwen; Schäffer, Hemming A.; Bingham, Harry B.
2006-01-01
This abstract describes a new method for generating nonlinear waves of constant form in physical wave basins. The idea is to combine fully dispersive linear wavemaker theory with nonlinear shallow water wave generation theory; and use an exact nonlinear theory as the target. We refer to the metho...... as an ad-hoc unified wave generation theory, since there is no rigorous analysis behind the idea which is simply justified by the improved results obtained for the practical generation of steady nonlinear waves....
13. Non-Linear Excitation of Ion Acoustic Waves
DEFF Research Database (Denmark)
Michelsen, Poul; Hirsfield, J. L.
1974-01-01
The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation.......The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation....
14. Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: Basic theory.
Science.gov (United States)
Berkouk, K; Carpenter, P W; Lucey, A D
2003-12-01
Our work is motivated by ideas about the pathogenesis of syringomyelia. This is a serious disease characterized by the appearance of longitudinal cavities within the spinal cord. Its causes are unknown, but pressure propagation is probably implicated. We have developed an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. This is intended as a simple model of the intraspinal cerebrospinal-fluid system. Our approach is based on the classic theory for the propagation of longitudinal waves in single, fluid-filled, elastic tubes. We show that for small-amplitude waves the governing equations reduce to the classic wave equation. The wave speed is found to be a strong function of the ratio of the tubes' cross-sectional areas. It is found that the leading edge of a transmural pressure pulse tends to generate compressive waves with converging wave fronts. Consequently, the leading edge of the pressure pulse steepens to form a shock-like elastic jump. A weakly nonlinear theory is developed for such an elastic jump.
15. Application of propagating beam methods to electromagnetic and acoustic wave propagation problems - a review
Energy Technology Data Exchange (ETDEWEB)
Lagasse, P.E.; Baets, R.
1987-12-01
The advantages and disadvantages of various propagating beam methods (BPMs) used in the solution of electromagnetic and acoustical problems are considered. The basic assumptions and approximations which are necessary for the derivation of the BPM algorithm are discussed with respect to applications to acoustics and optics and linear and nonlinear materials. Particular attention is given to the case of passive waveguiding structures and the role that BPM can play in the analysis of nonlinear structures such as semiconductor lasers. 28 references.
16. Application of propagating beam methods to electromagnetic and acoustic wave propagation problems - A review
Science.gov (United States)
Lagasse, P. E.; Baets, R.
1987-12-01
The advantages and disadvantages of various propagating beam methods (BPMs) used in the solution of electromagnetic and acoustical problems are considered. The basic assumptions and approximations which are necessary for the derivation of the BPM algorithm are discussed with respect to applications to acoustics and optics and linear and nonlinear materials. Particular attention is given to the case of passive waveguiding structures and the role that BPM can play in the analysis of nonlinear structures such as semiconductor lasers.
17. Longitudinally propagating traveling waves of the mammalian tectorial membrane.
Science.gov (United States)
Ghaffari, Roozbeh; Aranyosi, Alexander J; Freeman, Dennis M
2007-10-16
Sound-evoked vibrations transmitted into the mammalian cochlea produce traveling waves that provide the mechanical tuning necessary for spectral decomposition of sound. These traveling waves of motion that have been observed to propagate longitudinally along the basilar membrane (BM) ultimately stimulate the mechano-sensory receptors. The tectorial membrane (TM) plays a key role in this process, but its mechanical function remains unclear. Here we show that the TM supports traveling waves that are an intrinsic feature of its visco-elastic structure. Radial forces applied at audio frequencies (2-20 kHz) to isolated TM segments generate longitudinally propagating waves on the TM with velocities similar to those of the BM traveling wave near its best frequency place. We compute the dynamic shear storage modulus and shear viscosity of the TM from the propagation velocity of the waves and show that segments of the TM from the basal turn are stiffer than apical segments are. Analysis of loading effects of hair bundle stiffness, the limbal attachment of the TM, and viscous damping in the subtectorial space suggests that TM traveling waves can occur in vivo. Our results show the presence of a traveling wave mechanism through the TM that can functionally couple a significant longitudinal extent of the cochlea and may interact with the BM wave to greatly enhance cochlear sensitivity and tuning.
18. Nonlinear ion-acoustic cnoidal waves in a dense relativistic degenerate magnetoplasma
Science.gov (United States)
El-Shamy, E. F.
2015-03-01
The complex pattern and propagation characteristics of nonlinear periodic ion-acoustic waves, namely, ion-acoustic cnoidal waves, in a dense relativistic degenerate magnetoplasma consisting of relativistic degenerate electrons and nondegenerate cold ions are investigated. By means of the reductive perturbation method and appropriate boundary conditions for nonlinear periodic waves, a nonlinear modified Korteweg-de Vries (KdV) equation is derived and its cnoidal wave is analyzed. The various solutions of nonlinear ion-acoustic cnoidal and solitary waves are presented numerically with the Sagdeev potential approach. The analytical solution and numerical simulation of nonlinear ion-acoustic cnoidal waves of the nonlinear modified KdV equation are studied. Clearly, it is found that the features (amplitude and width) of nonlinear ion-acoustic cnoidal waves are proportional to plasma number density, ion cyclotron frequency, and direction cosines. The numerical results are applied to high density astrophysical situations, such as in superdense white dwarfs. This research will be helpful in understanding the properties of compact astrophysical objects containing cold ions with relativistic degenerate electrons.
19. Millimeter Wave Radio Frequency Propagation Model Development
Science.gov (United States)
2014-08-28
assume that no excess attenuation or obstacles are present, and the signal propagates along a clear signal path directly between the transmitter and...performed by simple trigonometry . The angle is determined by: θ sin | |, (103) where CL is the channel length, hTX is the height of the
20. Propagation of waves in shear flows
CERN Document Server
Fabrikant, A L
1998-01-01
The state of the art in a theory of oscillatory and wave phenomena in hydrodynamical flows is presented in this book. A unified approach is used for waves of different physical origins. A characteristic feature of this approach is that hydrodynamical phenomena are considered in terms of physics; that is, the complement of the conventionally employed formal mathematical approach. Some physical concepts such as wave energy and momentum in a moving fluid are analysed, taking into account induced mean flow. The physical mechanisms responsible for hydrodynamic instability of shear flows are conside
1. Adaptive control of the propagation of ultrafast light through random and nonlinear media
Science.gov (United States)
Moores, Mark David
2001-12-01
linear. This leads to modification of the pulse characteristics through nonlinear effects such as self phase modulation. Changing the temporal intensity profile of a propagating pulse modifies the nonlinear interaction. A linear application of phase is used to control the nonlinear self shaping effects of propagation of a twenty-five milliwatt pulse over forty nonlinear lengths in a single mode optical fiber. We show the strength of adaptive learning techniques for arriving at experimental solutions to problems with little hope of direct analytical solution. Linear control of nonlinear propagation of guided waves is demonstrated, with broad applicability in fundamental science and is a step towards ultrafast optical telecommunications. Reduction of the optical effects of a scattering material demonstrates successful adaptive control of the effects of a non-ideal optical material. Correlating the applied phase to a modelled dielectric stack gives insight into the random internal structure for the purpose of characterization.
2. Nonlinear Whitham-Broer-Kaup Wave Equation in an Analytical Solution
Directory of Open Access Journals (Sweden)
S. A. Zahedi
2008-01-01
Full Text Available This study presented a new approach for the analysis of a nonlinear Whitham-Broer-Kaup equation dealing with propagation of shallow water waves with different dispersion relations. The analysis was based on a kind of analytical method, called Variational Iteration Method (VIM. To illustrate the capability of the approach, some numerical examples were given and the propagation and the error of solutions were shown in comparison to those of exact solution. In clear conclusion, the approach was efficient and capable to obtain the analytical approximate solution of this set of wave equations while these solutions could straightforwardly show some facts of the described process deeply such as the propagation. This method can be easily extended to other nonlinear wave equations and so can be found widely applicable in this field of science.
3. Study of nonlinear waves in astrophysical quantum plasmas
Energy Technology Data Exchange (ETDEWEB)
Hossen, M.R.; Mamun, A.A., E-mail: [email protected] [Department of Physics, Jahangirnagar University, Savar, Dhaka (Bangladesh)
2015-10-01
The nonlinear propagation of the electron acoustic solitary waves (EASWs) in an unmagnetized, collisionless degenerate quantum plasma system has been investigated theoretically. Our considered model consisting of two distinct groups of electrons (one of inertial non-relativistic cold electrons and other of inertialess ultrarelativistic hot electrons) and positively charged static ions. The Korteweg-de Vries (K-dV) equation has been derived by employing the reductive perturbation method and numerically examined to identify the basic features (speed, amplitude, width, etc.) of EASWs. It is shown that only rarefactive solitary waves can propagate in such a quantum plasma system. It is found that the effect of degenerate pressure and number density of hot and cold electron fluids, and positively charged static ions, significantly modify the basic features of EASWs. It is also noted that the inertial cold electron fluid is the source of dispersion for EA waves and is responsible for the formation of solitary structures. The applications of this investigation in astrophysical compact objects (viz. non-rotating white dwarfs, neutron stars, etc.) are briefly discussed. (author)
4. PROTON HEATING IN SOLAR WIND COMPRESSIBLE TURBULENCE WITH COLLISIONS BETWEEN COUNTER-PROPAGATING WAVES
Energy Technology Data Exchange (ETDEWEB)
He, Jiansen; Tu, Chuanyi; Wang, Linghua; Pei, Zhongtian [School of Earth and Space Sciences, Peking University, Beijing, 100871 (China); Marsch, Eckart [Institute for Experimental and Applied Physics, Christian-Albrechts-Universität zu Kiel, D-24118 Kiel (Germany); Chen, Christopher H. K. [Department of Physics, Imperial College London, London SW7 2AZ (United Kingdom); Zhang, Lei [Sate Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing 100190 (China); Salem, Chadi S.; Bale, Stuart D., E-mail: [email protected] [Space Sciences Laboratory, University of California, Berkeley, CA 94720 (United States)
2015-11-10
Magnetohydronamic turbulence is believed to play a crucial role in heating laboratory, space, and astrophysical plasmas. However, the precise connection between the turbulent fluctuations and the particle kinetics has not yet been established. Here we present clear evidence of plasma turbulence heating based on diagnosed wave features and proton velocity distributions from solar wind measurements by the Wind spacecraft. For the first time, we can report the simultaneous observation of counter-propagating magnetohydrodynamic waves in the solar wind turbulence. As opposed to the traditional paradigm with counter-propagating Alfvén waves (AWs), anti-sunward AWs are encountered by sunward slow magnetosonic waves (SMWs) in this new type of solar wind compressible turbulence. The counter-propagating AWs and SWs correspond, respectively, to the dominant and sub-dominant populations of the imbalanced Elsässer variables. Nonlinear interactions between the AWs and SMWs are inferred from the non-orthogonality between the possible oscillation direction of one wave and the possible propagation direction of the other. The associated protons are revealed to exhibit bi-directional asymmetric beams in their velocity distributions: sunward beams appear in short, narrow patterns and anti-sunward in broad extended tails. It is suggested that multiple types of wave–particle interactions, i.e., cyclotron and Landau resonances with AWs and SMWs at kinetic scales, are taking place to jointly heat the protons perpendicular and in parallel.
5. Guided wave propagation in multilayered piezoelectric structures
Institute of Scientific and Technical Information of China (English)
2009-01-01
A general formulation of the method of the reverberation-ray matrix (MRRM) based on the state space formalism and plane wave expansion technique is presented for the analysis of guided waves in multilayered piezoelectric structures. Each layer of the structure is made of an arbitrarily anisotropic piezoelectric material. Since the state equation of each layer is derived from the three-dimensional theory of linear piezoelectricity, all wave modes are included in the formulation. Within the framework of the MRRM, the phase relation is properly established by excluding exponentially growing functions, while the scattering relation is also appropriately set up by avoiding matrix inversion operation. Consequently, the present MRRM is unconditionally numerically stable and free from computational limitations to the total number of layers, the thickness of individual layers, and the frequency range. Numerical examples are given to illustrate the good performance of the proposed formulation for the analysis of the dispersion characteristic of waves in layered piezoelectric structures.
6. Nonlinear low-frequency electrostatic wave dynamics in a two-dimensional quantum plasma
Energy Technology Data Exchange (ETDEWEB)
Ghosh, Samiran, E-mail: [email protected] [Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata-700 009 (India); Chakrabarti, Nikhil, E-mail: [email protected] [Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064 (India)
2016-08-15
The problem of two-dimensional arbitrary amplitude low-frequency electrostatic oscillation in a quasi-neutral quantum plasma is solved exactly by elementary means. In such quantum plasmas we have treated electrons quantum mechanically and ions classically. The exact analytical solution of the nonlinear system exhibits the formation of dark and black solitons. Numerical simulation also predicts the possible periodic solution of the nonlinear system. Nonlinear analysis reveals that the system does have a bifurcation at a critical Mach number that depends on the angle of propagation of the wave. The small-amplitude limit leads to the formation of weakly nonlinear Kadomstev–Petviashvili solitons.
7. Wave propagation and energy dissipation in viscoelastic granular media
Institute of Scientific and Technical Information of China (English)
2001-01-01
In terms of viscoelasticity, the relevant theory of wave in granular media is analyzed in this paper.Under the conditions of slight deformation of granules, wave equation, complex number expressions of propagation vector and attenuation vector, attenuation coefficient expressions of longitudinal wave and transverse wave,etc, are analyzed and deduced. The expressions of attenuation coefficients of viscoelastic longitudinal wave and transverse wave show that the attenuation of wave is related to frequency. The higher the frequency is, the more the attenuation is, which is tested by the laboratory experiment. In addition, the energy dissipation is related to the higher frequency wave that is absorbed by granular media. The friction amongst granular media also increase the energy dissipation. During the flowing situation the expression of transmission factor of energy shows that the granular density difference is the key factor which leads to the attenuation of vibrating energy.This has been proved by the experiment results.
8. Transformation and disintegration of strongly nonlinear internal waves by topography in stratified lakes
Directory of Open Access Journals (Sweden)
V. I. Vlasenko
Full Text Available For many lakes the nonlinear transfer of energy from basin-scale internal waves to short-period motions, such as solitary internal waves (SIW and wave trains, their successive interaction with lake boundaries, as well as over-turning and breaking are important mechanisms for an enhanced mixing of the turbulent benthic boundary layer. In the present paper, the evolution of plane SIWs in a variable depth channel, typical of a lake of variable depth, is considered, with the basis being the Reynolds equations. The vertical fluid stratification, wave amplitudes and bottom parameters are taken close to those observed in Lake Constance, a typical mountain lake. The problem is solved numerically. Three different scenarios of a wave evolution over variable bottom topography are examined. It is found that the basic parameter controlling the mechanism of wave evolution is the ratio of the wave amplitude to the distance from the metalimnion to the bottom d. At sites with a gentle sloping bottom, where d is small, propagating (weak or strong internal waves adjust to the local ambient conditions and preserve their form. No secondary waves or wave trains arise during wave propagation from the deep part to the shallow water. If the amplitude of the propagating waves is comparable with the distance between the metalimnion and the top of the underwater obstacle ( d ~ 1, nonlinear dispersion plays a key role. A wave approaching the bottom feature splits into a sequence of secondary waves (solitary internal waves and an attached oscillating wave tail. The energy of the SIWs above the underwater obstacle is transmitted in parts from the first baroclinic mode, to the higher modes. Most crucially, when the internal wave propagates from the deep part of a basin to the shallow boundary, a breaking event can arise. The cumulative effects of the nonlinearity lead to a steepening and
9. Stable One-Dimensional Periodic Wave in Kerr-Type and Quadratic Nonlinear Media
Directory of Open Access Journals (Sweden)
Roxana Savastru
2012-01-01
Full Text Available We present the propagation of optical beams and the properties of one-dimensional (1D spatial solitons (“bright” and “dark” in saturated Kerr-type and quadratic nonlinear media. Special attention is paid to the recent advances of the theory of soliton stability. We show that the stabilization of bright periodic waves occurs above a certain threshold power level and the dark periodic waves can be destabilized by the saturation of the nonlinear response, while the dark quadratic waves turn out to be metastable in the broad range of material parameters. The propagation of (1+1 a dimension-optical field on saturated Kerr media using nonlinear Schrödinger equations is described. A model for the envelope one-dimensional evolution equation is built up using the Laplace transform.
10. Rotation-induced nonlinear wavepackets in internal waves
Energy Technology Data Exchange (ETDEWEB)
Whitfield, A. J., E-mail: [email protected]; Johnson, E. R., E-mail: [email protected] [Department of Mathematics, University College London, London WC1E 6BT (United Kingdom)
2014-05-15
The long time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual formation of a localised wavepacket. Here this initial value problem is considered within the context of the Ostrovsky, or the rotation-modified Korteweg-de Vries (KdV), equation and a numerical method for obtaining accurate wavepacket solutions is presented. The flow evolutions are described in the regimes of relatively-strong and relatively-weak rotational effects. When rotational effects are relatively strong a second-order soliton solution of the nonlinear Schrödinger equation accurately predicts the shape, and phase and group velocities of the numerically determined wavepackets. It is suggested that these solitons may form from a local Benjamin-Feir instability in the inertia-gravity wave-train radiated when a KdV solitary wave rapidly adjusts to the presence of strong rotation. When rotational effects are relatively weak the initial KdV solitary wave remains coherent longer, decaying only slowly due to weak radiation and modulational instability is no longer relevant. Wavepacket solutions in this regime appear to consist of a modulated KdV soliton wavetrain propagating on a slowly varying background of finite extent.
11. Rotation-induced nonlinear wavepackets in internal waves
Science.gov (United States)
Whitfield, A. J.; Johnson, E. R.
2014-05-01
The long time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual formation of a localised wavepacket. Here this initial value problem is considered within the context of the Ostrovsky, or the rotation-modified Korteweg-de Vries (KdV), equation and a numerical method for obtaining accurate wavepacket solutions is presented. The flow evolutions are described in the regimes of relatively-strong and relatively-weak rotational effects. When rotational effects are relatively strong a second-order soliton solution of the nonlinear Schrödinger equation accurately predicts the shape, and phase and group velocities of the numerically determined wavepackets. It is suggested that these solitons may form from a local Benjamin-Feir instability in the inertia-gravity wave-train radiated when a KdV solitary wave rapidly adjusts to the presence of strong rotation. When rotational effects are relatively weak the initial KdV solitary wave remains coherent longer, decaying only slowly due to weak radiation and modulational instability is no longer relevant. Wavepacket solutions in this regime appear to consist of a modulated KdV soliton wavetrain propagating on a slowly varying background of finite extent.
12. Nonlinear acoustic waves in the viscous thermosphere and ionosphere above earthquake
Science.gov (United States)
Chum, J.; Cabrera, M. A.; Mošna, Z.; Fagre, M.; Baše, J.; Fišer, J.
2016-12-01
The nonlinear behavior of acoustic waves and their dissipation in the upper atmosphere is studied on the example of infrasound waves generated by vertical motion of the ground surface during the Mw 8.3 earthquake that occurred about 46 km from Illapel, Chile on 16 September 2015. To conserve energy, the amplitude of infrasound waves initially increased as the waves propagated upward to the rarefied air. When the velocities of air particles became comparable with the local sound speed, the nonlinear effects started to play an important role. Consequently, the shape of waveform changed significantly with increasing height, and the original wave packet transformed to the "N-shaped" pulse resembling a shock wave. A unique observation by the continuous Doppler sounder at the altitude of about 195 km is in good agreement with full wave numerical simulation that uses as boundary condition the measured vertical motion of the ground surface.
13. Long wave-short wave resonance in nonlinear negative refractive index media.
Science.gov (United States)
Chowdhury, Aref; Tataronis, John A
2008-04-18
We show that long wave-short wave resonance can be achieved in a second-order nonlinear negative refractive index medium when the short wave lies on the negative index branch. With the medium exhibiting a second-order nonlinear susceptibility, a number of nonlinear phenomena such as solitary waves, paired solitons, and periodic wave trains are possible or enhanced through the cascaded second-order effect. Potential applications include the generation of terahertz waves from optical pulses.
14. Detection of Electromechanical Wave Propagation Using Synchronized Phasor Measurements
Science.gov (United States)
Suryawanshi, Prakash; Dambhare, Sanjay; Pramanik, Ashutosh
2014-01-01
Considering electrical network as a continuum has become popular for electromechanical wave analysis. This paper reviews the concept of electromechanical wave propagation. Analysis of large number of generator ring system will be an easy way to illustrate wave propagation. The property of traveling waves is that the maximum and minimum values do not occur at the same time instants and hence the difference between these time delays can be easily calculated. The homogeneous, isotropic 10 generator ring system is modeled using electromagnetic transient simulation programs. The purpose of this study is to investigate the time delays and wave velocities using Power System Computer Aided Design (PSCAD)/Electromagnetic Transient Program (EMTP). The disturbances considered here are generator disconnections and line trips.
15. Seismic wave propagation through an extrusive basalt sequence
Science.gov (United States)
Sanford, Oliver; Hobbs, Richard; Brown, Richard; Schofield, Nick
2016-04-01
Layers of basalt flows within sedimentary successions (e.g. in the Faeroe-Shetland Basin) cause complex scattering and attenuation of seismic waves during seismic exploration surveys. Extrusive basaltic sequences are highly heterogeneous and contain strong impedance contrasts between higher velocity crystalline flow cores (˜6 km s-1) and the lower velocity fragmented and weathered flow crusts (3-4 km s-1). Typically, the refracted wave from the basaltic layer is used to build a velocity model by tomography. This velocity model is then used to aid processing of the reflection data where direct determination of velocity is ambiguous, or as a starting point for full waveform inversion, for example. The model may also be used as part of assessing drilling risk of potential wells, as it is believed to constrain the total thickness of the sequence. In heterogeneous media, where the scatter size is of the order of the seismic wavelength or larger, scattering preferentially traps the seismic energy in the low velocity regions. This causes a build-up of energy that is guided along the low velocity layers. This has implications for the interpretation of the observed first arrival of the seismic wave, which may be a biased towards the low velocity regions. This will then lead to an underestimate of the velocity structure and hence the thickness of the basalt, with implications for the drilling of wells hoping to penetrate through the base of the basalts in search of hydrocarbons. Using 2-D acoustic finite difference modelling of the guided wave through a simple layered basalt sequence, we consider the relative importance of different parameters of the basalt on the seismic energy propagating through the layers. These include the proportion of high to low velocity material, the number of layers, their thickness and the roughness of the interfaces between the layers. We observe a non-linear relationship between the ratio of high to low velocity layers and the apparent velocity
16. Spatial damping of propagating sausage waves in coronal cylinders
Science.gov (United States)
Guo, Ming-Zhe; Chen, Shao-Xia; Li, Bo; Xia, Li-Dong; Yu, Hui
2015-09-01
Context. Sausage modes are important in coronal seismology. Spatially damped propagating sausage waves were recently observed in the solar atmosphere. Aims: We examine how wave leakage influences the spatial damping of sausage waves propagating along coronal structures modeled by a cylindrical density enhancement embedded in a uniform magnetic field. Methods: Working in the framework of cold magnetohydrodynamics, we solve the dispersion relation (DR) governing sausage waves for complex-valued, longitudinal wavenumber k at given real angular frequencies ω. For validation purposes, we also provide analytical approximations to the DR in the low-frequency limit and in the vicinity of ωc, the critical angular frequency separating trapped from leaky waves. Results: In contrast to the standing case, propagating sausage waves are allowed for ω much lower than ωc. However, while able to direct their energy upward, these low-frequency waves are subject to substantial spatial attenuation. The spatial damping length shows little dependence on the density contrast between the cylinder and its surroundings, and depends only weakly on frequency. This spatial damping length is of the order of the cylinder radius for ω ≲ 1.5vAi/a, where a and vAi are the cylinder radius and the Alfvén speed in the cylinder, respectively. Conclusions: If a coronal cylinder is perturbed by symmetric boundary drivers (e.g., granular motions) with a broadband spectrum, wave leakage efficiently filters out the low-frequency components.
17. Nonlocal thermo-elastic wave propagation in temperature-dependent embedded small-scaled nonhomogeneous beams
Science.gov (United States)
2016-11-01
In this paper, the thermo-elastic wave propagation analysis of a temperature-dependent functionally graded (FG) nanobeam supported by Winkler-Pasternak elastic foundation is studied using nonlocal elasticity theory. The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function. The temperature field has a nonlinear distribution called heat conduction across the nanobeam thickness. Temperature-dependent material properties change gradually in the spatial coordinate according to the Mori-Tanaka model. The governing equations of the wave propagation of the refined FG nanobeam are derived by using Hamilton's principle. The analytic dispersion relation of the embedded nonlocal functionally graded nanobeam is obtained by solving an eigenvalue problem. Numerical examples show that the wave characteristics of the functionally graded nanobeam are related to the temperature distribution, elastic foundation parameters, nonlocality and material composition.
18. A Wave Expansion Method for Aeroacoustic Propagation
OpenAIRE
Hammar, Johan
2016-01-01
Although it is possible to directly solve an entire flow-acoustics problem in one computation, this approach remains prohibitively large in terms of the computational resource required for most practical applications. Aeroacoustic problems are therefore usually split into two parts; one consisting of the source computation and one of the source propagation. Although both these parts entail great challenges on the computational method, in terms of accuracy and efficiency, it is still better th...
19. Influence of optical activity on rogue waves propagating in chiral optical fibers
Science.gov (United States)
Temgoua, D. D. Estelle; Kofane, T. C.
2016-06-01
We derive the nonlinear Schrödinger (NLS) equation in chiral optical fiber with right- and left-hand nonlinear polarization. We use the similarity transformation to reduce the generalized chiral NLS equation to the higher-order integrable Hirota equation. We present the first- and second-order rational solutions of the chiral NLS equation with variable and constant coefficients, based on the modified Darboux transformation method. For some specific set of parameters, the features of chiral optical rogue waves are analyzed from analytical results, showing the influence of optical activity on waves. We also generate the exact solutions of the two-component coupled nonlinear Schrödinger equations, which describe optical activity effects on the propagation of rogue waves, and their properties in linear and nonlinear coupling cases are investigated. The condition of modulation instability of the background reveals the existence of vector rogue waves and the number of stable and unstable branches. Controllability of chiral optical rogue waves is examined by numerical simulations and may bring potential applications in optical fibers and in many other physical systems.
20. A comparative study of two fast nonlinear free-surface water wave models
DEFF Research Database (Denmark)
Ducrozet, Guillaume; Bingham, Harry B.; Engsig-Karup, Allan Peter
2012-01-01
This paper presents a comparison in terms of accuracy and efficiency between two fully nonlinear potential flow solvers for the solution of gravity wave propagation. One model is based on the high-order spectral (HOS) method, whereas the second model is the high-order finite difference model Ocea... | {"extraction_info": {"found_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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.8779208064079285, "perplexity": 1643.9186152018692}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816370.72/warc/CC-MAIN-20180225110552-20180225130552-00554.warc.gz"} |
https://stemgeeks.net/hive-163521/@savvyplayer/result-of-the-day-1-problem-a-challenge-of-my-july-2021-math-mini-contest-for-d-buzz | # Result of the Day 1 [Problem A] Challenge of my July 2021 Math mini-contest for D.Buzz
in STEMGeeks4 months ago
## Day 1 Problem A
In a construction project, adding 1 worker will decrease the project duration by 24 days, while adding 2 workers will decrease duration by 40 days. What is the original number of workers and days?
The problem has originally been posted at https://stemgeeks.net/@savvyplayer/julymathday1a.
## days = 120
Both values must be correctly supplied on an answer for it to be considered correct.
## Solution
The problem is under elementary algebra.
When the number of workers for a project increases, the shorter the duration of the project will be. The solution will make use of the inverse proportion formula.
## y = k / x
where
x = number of workers
y = time needed
k = constant of variation
Equation 1: y = k / x, where k = y * x
Equation 2: (y - 24) = k / (x + 1), where k = (y - 24) * (x + 1)
Equation 3: (y - 40) = k / (x + 2), where k = (y - 40) * (x + 2)
We can now combine Equation 2 with 1 and Equation 3 with 1 by replacing the k variables with their equivalents in terms of x and y.
Equation 4: yx = (y - 24) * (x + 1) = yx + y - 24x - 24
Equation 5: yx = (y - 40) * (x + 2) = yx + 2y - 40x - 80
Combining the similar terms and arranging the remaining terms to be in slope-intercept form, we get:
Equation 4: y = 24x + 24
Equation 5: 2y - 40x = 80 which can be simplified to y = 20x + 40
It is clear that we have a system of linear equations in two variables. We can use the substitution method to get the value of x first.
Equation 6: 24x + 24 = 20x + 40, where we can obtain the value of x which is 4.
Equation 7: y = 24 * (4) + 24, where we can obtain the value of y which is 120.
Therefore, the original number of workers is 4, and the original duration of the project is 120 days.
WorkersDuration (in days)
4120
596
680
To make sure that our answer is correct, we should multiply the workers by the duration for each equation, where we would get 480 as the constant of variation for all of them.
## Winner: none
1 HIVE has been transferred to the prize pool, which will be awarded to the participant with the highest number of correctly-answered problems after all the challenge problems in this Math mini-contest have been concluded.
Mentions: @jfang003, @holovision, @eturnerx (@eturnerx-dbuzz), @ahmadmanga (@ahmadmangazap), @appukuttan66, @paultactico2, @dkmathstats, and @minus-pi
Special mentions: @dbuzz, @chrisrice, @jancharlest, @lolxsbudoy, and @mehmetfix
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Wow | {"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": 0.8225316405296326, "perplexity": 856.3594156730289}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362999.66/warc/CC-MAIN-20211204154554-20211204184554-00208.warc.gz"} |
http://motls.blogspot.com/2016/07/cms-in-zz-channel-3-4-sigma-evidence-in.html | ## Sunday, July 17, 2016 ... /////
### CMS in $ZZ$ channel: a 3-4 sigma evidence in favor of a $650\GeV$ boson
Today, the CMS collaboration has revealed one of the strongest deviations from the Standard Model in quite some time in the paper
Search for diboson resonances in the semileptonic $X \to ZV \to \ell^+\ell^- q\bar q$ final state at $\sqrt{s} = 13\TeV$ with CMS
On page 21, Figure 12, you see the Brazilian charts.
In the channel where a resonance decays to a $ZZ$ pair and one $Z$ decays to a quark-antiquark pair and the other $Z$-boson to a lepton-antilepton pair (semileptonic decays), CMS folks used two different methods to search for low-mass and high-mass particles.
In the high-mass search – which contributed to the bottom part of Figure 12 – they saw a locally 2-sigma excess indicating a resonance around $1000\GeV$, possibly compatible with the new $\gamma\gamma$ resonance near $975\GeV$ that appeared in some new rumors about the 2016 data.
More impressively, the low-mass search revealed a locally 3.4 or 3.9 sigma excess in the search for a Randall-Sundrum or "bulk graviton" (I won't explain the differences between the two models because I don't know the details and I don't think one should take a particular interpretation too seriously) of mass $650\GeV$. The "bulk graviton" excess is the stronger one.
Even when the look-elsewhere effect (over the $550-1400\GeV$ range) is taken into account, as conclusions on Page 22 point out, the deviation is still 2.9 or 3.5 sigma, respectively. That's pretty strong.
We may wait for ATLAS whether they see something. (Update: In comments, a paper with a disappointing 2-sigma deficit on that place is shown instead.) CMS has only used 2.7 inverse femtobarns of the 2015 data in this analysis. Obviously, if the excess at $650\GeV$ were real, the particle would already be safely discovered in the data that have already been collected in 2015-2016 (by CMS separately) – about 18 inverse femtobarns (CMS).
Concerning the $650\GeV$ mass, in 2014, CMS also saw a 2.5-sigma hint of a leptoquark of that mass (more on those particles). Note that the leptoquarks carry very different charges (quantum numbers) than the "bulk graviton".
The previous CMS papers on the same channel but based on the 2012 data were these two: low-mass, high-mass strategies. I think that there was no evidence in favor of a similar hypothesis in those older papers. That also seems to be true for the analogous ATLAS paper using the 2012 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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9075602889060974, "perplexity": 1371.1499777086087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424960.67/warc/CC-MAIN-20170725022300-20170725042300-00682.warc.gz"} |
https://www.cliffsnotes.com/study-guides/calculus/calculus/the-derivative/differentiation-of-exponential-and-logarithmic-functions | ## Differentiation of Exponential and Logarithmic Functions
Exponential functions and their corresponding inverse functions, called logarithmic functions, have the following differentiation formulas:
Note that the exponential function f( x) = e x has the special property that its derivative is the function itself, f′( x) = e x = f( x).
Example 1: Find f′( x) if
Example 2: Find y′ if .
Example 3: Find f′( x) if f( x) = 1n(sin x).
Example 4: Find if y=log 10(4 x 2 − 3 x −5). | {"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": 0.9940900206565857, "perplexity": 3146.078439885851}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525587.2/warc/CC-MAIN-20190718083839-20190718105839-00302.warc.gz"} |
http://canliborsaekrani.com/f6fsrt/archive.php?page=21709a-zero-meaning-in-math | The sign of an integer is either positive (+) or negative (-), except zero, which has no sign. The integer zero is neutral. The number zero as we know it arrived in the West circa 1200, most famously delivered by Italian mathematician Fibonacci (aka Leonardo of Pisa), … Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. We need calculus. Still the only use of zero was an empty place 'nd'cator. Learn what is zero (function). The bigger the flux density (positive or negative), the stronger the flux source or sink. How to use net-zero in a sentence. It is neither positive nor negative. Note: Zero is the additive identity. In plain english: Math Intuition. Zero. In fact, in mathematics, unity is simply a synonym for the number "one" (1), the integer between the integers zero (0) and two (2). Song lyrics by zero-- Explore a large variety of song lyrics performed by zero on the Lyrics.com website. It only takes a minute to sign up. The Fundamental Theorem of Algebra states that any polynomial $a_nx^n + \cdots a_0x^0$ can be factored into the form $a_n(x-r_1)\cdots(x-r_n)$, where $a_i, r_i \in \mathbb{C}$. Meaning of zero. The gamma function is undefined for zero and negative integers, from which we can conclude that factorials of negative integers do not exist. Learn and know the meaning of discriminant in math.The word discriminant comes in quadratic equations chapter. These numbers are to the left of zero on the number line. Relevance. While the word carries its own unique meaning in the field of mathematics, the unique use does not stray too far, at least symbolically, from this definition. Another word for zero. Net-zero definition is - resulting in neither a surplus nor a deficit of something specified when gains and losses are added together; especially, of buildings : producing enough energy (as through solar panels or passive heating) to offset any energy consumed. In calculus, this is called the "limit". For more information on the Gamma function, see the links listed below. This lesson will go into the rule in more detail, explaining how it works and giving some examples. The zero symbol-- In this Symbols.com article you will learn about the meaning of the zero symbol and its characteristic. Meaning / definition Example { } set: a collection of elements: A = {3,7,9,14}, B = {9,14,28} | such that: so that: A = {x | x∈, x<0} A⋂B: intersection: objects that belong to set A and set B: A ⋂ B = {9,14} A⋃B: union: objects that belong to set A or set B: A ⋃ B = {3,7,9,14,28} A⊆B: subset: A is a … Any number divided by itself is equal to one. 0 (zero) is a number, and the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.As a digit, 0 is used as a placeholder in place value systems. See more. Favorite Answer. Apparently, 'n Greek "ouden" means nothing and the symbol they picked for zero 's simply the first letter of the word "ouden". For instance, x 2 times x-2 is equal to x 2 divided by x-2. Definition of zero in the Definitions.net dictionary. Ask Dr. If you are that is called theta. If you multiply (16)(467)(11)(9)(0), the result is 0. We've gone as far as algebra can take us, and we need a new way to talk about math. The number which indicates no quantity, size, or magnitude. the zero property in math is when you multiply by zero which is the multiplicative property of zero or it is when you add zero to anything and get zero that is called the additive property of zero. Math has a lot of neat tricks that can make solving equations easier. Absolute zero definition, the temperature of −273.16°C (−459.69°F), the hypothetical point at which all molecular activity ceases. The concept of zero is usually harder than counting and other early number concepts. Based on discriminant, we define nature of roots of a quadratic equations.So it is very much needed to know the discriminant meaning.. We all know that for finding the roots of quadratic equations there are two methods. Also find the definition and meaning for various math words from this math dictionary. at least one of the factors has to represent the number 0. Zero is neither negative nor positive. In algebra, a real root is a solution to a particular equation. Definition of Zero Property Of Multiplication explained with real life illustrated examples. Lv 6. A div of zero means there’s no net flux change in side the region. ... or whether it is identically zero ... meaning of 最初にして最後の見せ場 1 0:} 10 years ago. For example, the number 5,001,000 breaks down into 5,000,000 + 1,000. Though outlawed, merchants continued to use zero in encrypted messages, thus the derivation of the word cipher, meaning code, from the Arabic sifr. The zero exponent rule states that any term with an exponent of zero is equal to one. Zero to the power of zero is a special case, however. 2 Answers. It means there is no answer or the answer is undefined. A “double zero” means that two of the factors of the equation have zero as a solution when you solve for that factor equal to zero. Concept Zero as a Number Specifically, we can't divide by zero. It's kind of like x in math. In other words, the only way the product of two or more values can be zero is for at least one of the values to actually be zero. Despite this, the mathematical community is in favor of defining zero to the zero power as 1, at least for most purposes. Essentially, an exponent that is zero is equal to a variable to the power of an exponent times a variable to the negative power of the same exponent. zero. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. In algebra, we saw that we get closer and closer to the correct answer. We get closer and closer to the limit as the divisor gets closer and closer to zero. What does zero mean? The term real root means that this solution is a number that can be whole, positive, negative, rational, or irrational. Answer Save. These numbers are to the right of zero on the number line. David Engel. Suppose, […] In math, what does a zero (or an O) with a slash through it mean? The next great mathematician to use zero was Rene Descartes, the founder of the Cartesian coordinate system. Zero cannot represent both nothing and everything in the same mathematical system of values, and as long as we remember that, we can discover a second mathematical system, very similar to ordinary math, and yet very different, because in this new system, zero represents a mathematical everything, which produces a whole other kind of math. Now that we have an intuitive explanation, how do we turn that sucker into an equation? Thus, we usually introduce it only after a child has understood the value of numbers to some extent.The difference between 0 and other numbers is that all of the other numbers have a tangible visual form, whereas 0 … From the Dr. As anyone who has had to graph a triangle or a parabola knows, Descartes’ origin is (0,0). Zero definition is - the arithmetical symbol 0 or [SYMBOL] denoting the absence of all magnitude or quantity. It is important to know when a zero is a placeholder, and when it is a leading zero. ‘The principal quantum number n may have any positive, non-zero integer value.’ ‘In the model tests it was found that some rows and columns had a larger number of zero values than non-zero values.’ ‘The tiny non-zero value of the effective cosmological constant is hard to understand using traditional particle physics arguments.’ Find more ways to say zero, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. How to use zero in a sentence. Whole numbers less than zero are called negative integers. Zero point definition: the point on a scale that denotes zero and from which positive and negative readings can... | Meaning, pronunciation, translations and examples https://www.patreon.com/homeschoolpop In this addition math lesson video for kindergarten and first graders we will learn about adding zero. the zero property in math is when you multiply by zero which is the multiplicative property of zero or it is when you add zero to anything and get zero that is called the additive property of zero. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. Are you talking about this θ? While numbers like pi and the square root of two are irrational numbers, rational numbers are zero… Although the digit 0 adds no value to a number, it generally acts as a placeholder to keep the other digits in their proper places. 10 years ago. The multiplication property of zero states that if the product of . 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The Greeks introduced the symbol O for zero closer and closer to the correct.. Absence of all magnitude or quantity temperature of −273.16°C ( −459.69°F ), except,... Or the answer is undefined the meaning of the factors has to represent the number 5,001,000 down... Will go into the rule in more detail, explaining how it works and giving some examples the is... Rule in more detail, explaining how it works and giving some examples origin! At which all molecular activity ceases 've gone as far as algebra can take us, and need! Symbol and its characteristic a zero ( or an O ) with a slash through it mean listed below and. Number which indicates no quantity, size, or magnitude comes in equations. Has had to graph a triangle or a parabola knows, Descartes ’ origin is ( )! Illustrated examples ’ origin is ( 0,0 ) -- Explore a large of. This solution is a special case, however answer is undefined there no... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8797374963760376, "perplexity": 875.4583629444642}, "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/1620243991269.57/warc/CC-MAIN-20210516105746-20210516135746-00520.warc.gz"} |
http://www.math-only-math.com/linear-symmetry.html | Linear Symmetry
What is a linear symmetry?
It is type of symmetry in which a line is drawn from the middle of the figure.
The two parts of the figure coincide, then each part is called the mirror image of the other i.e., the part of the figure on one side of the dotted line falls exactly over the other part which lies on the side. Such a figure is called a symmetrical figure.
The line which divides the figure into two equal parts is called the line of symmetry or axis of symmetry.
Shapes or figures may have horizontal, vertical, both horizontal and vertical, infinite and no line of symmetry.
Examples of the different types of pictures of horizontal line of symmetry, vertical line of symmetry, both horizontal and vertical lines of symmetry, infinite lines of symmetry and no line of symmetry.
Examples of the shapes or figures may have 1, 2, 3, 4, 0 and so on or infinite lines of symmetry.
Isosceles triangle has two sides equal and two angles equal showing one line of symmetry.
Rectangle has opposite sides equal and all the angles 90° showing two lines of symmetry.
Equilateral triangle have all three sides equal and all three angles equal showing three lines of symmetry.
Square have all four sides equal and all angles 90° showing four lines of symmetry.
Parallelogram has opposite sides equal and parallel sides equal showing no line of symmetry.
Circle showing infinite lines of symmetry.
Related Concepts
Lines of Symmetry
Point Symmetry
Rotational Symmetry
Order of Rotational Symmetry
Types of Symmetry
Reflection
Reflection of a Point in x-axis
Reflection of a Point in y-axis
Reflection of a point in origin
Rotation
90 Degree Clockwise Rotation
90 Degree Anticlockwise Rotation
180 Degree Rotation | {"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": 0.8212614059448242, "perplexity": 759.9263322846488}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828134.97/warc/CC-MAIN-20171024033919-20171024053919-00369.warc.gz"} |
https://www.techylib.com/fr/view/glueblacksmith/actuate_birt_java_components_developer_guide | # Actuate BIRT Java Components Developer Guide
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Actuate BIRT Java Components
Developer Guide
Information in this document is subject to change without notice. Examples provided are fictitious. No
part of this document may be reproduced or transmitted in any form, or by any means, electronic or
mechanical, for any purpose, in whole or in part, without the express written permission of Actuate
Corporation.
Contains information proprietary to:
Actuate Corporation, 2207 Bridgepointe Parkway, San Mateo, CA 94404
www.actuate.com
www.birt-exchange.com
The software described in this manual is provided by Actuate Corporation under an Actuate License
agreement. The software may be used only in accordance with the terms of the agreement. Actuate
software products are protected by U.S. and International patents and patents pending. For a current list
Actuate, ActuateOne, the Actuate logo, Archived Data Analytics, BIRT, Collaborative Reporting
Architecture, e.Analysis, e.Report, e.Reporting, e.Spreadsheet, Encyclopedia, Interactive Viewing,
OnPerformance, Performancesoft, Performancesoft Track, Performancesoft Views, Report Encyclopedia,
Reportlet, The people behind BIRT, X2BIRT, and XML reports.
Actuate products may contain third-party products or technologies. Third-party trademarks or registered
trademarks of their respective owners, companies, or organizations include:
Adobe Systems Incorporated: Flash Player. Apache Software Foundation (www.apache.org): Axis, Axis2,
Batik, Batik SVG library, Commons Command Line Interface (CLI), Commons Codec, Derby, Shindig,
Struts, Tomcat, Xerces, Xerces2 Java Parser, and Xerces-C++ XML Parser. Bits Per Second, Ltd. and
Graphics Server Technologies, L.P.: Graphics Server. Bruno Lowagie and Paulo Soares: iText, licensed
under the Mozilla Public License (MPL). Castor (www.castor.org), ExoLab Project (www.exolab.org), and
Intalio, Inc. (www.intalio.org): Castor. Codejock Software: Xtreme Toolkit Pro. DataDirect Technologies
Data Tools Platform (DTP) ODA, Eclipse SDK, Graphics Editor Framework (GEF), Eclipse Modeling
Framework (EMF), and Eclipse Web Tools Platform (WTP), licensed under the Eclipse Public License
(EPL). Jason Hsueth and Kenton Varda (code.google.com): Protocole Buffer. ImageMagick Studio LLC.:
ImageMagick. InfoSoft Global (P) Ltd.: FusionCharts, FusionMaps, FusionWidgets, PowerCharts. Mark
Adler and Jean-loup Gailly (www.zlib.net): zLib. Matt Ingenthron, Eric D. Lambert, and Dustin Sallings
Unicode (ICU): ICU library. KL Group, Inc.: XRT Graph, licensed under XRT for Motif Binary License
Network): CompoundDocument Library. Mozilla: Mozilla XML Parser, licensed under the Mozilla
Public License (MPL). MySQL Americas, Inc.: MySQL Connector. Netscape Communications
Corporation, Inc.: Rhino, licensed under the Netscape Public License (NPL). OOPS Consultancy:
Global Development Group: pgAdmin, PostgreSQL, PostgreSQL JDBC driver. Rogue Wave Software,
Inc.: Rogue Wave Library SourcePro Core, tools.h++. Sam Stephenson (prototype.conio.net): prototype.js,
licensed under the MIT license. Sencha Inc.: Ext JS. Sun Microsystems, Inc.: JAXB, JDK, Jstl. ThimbleWare,
Inc.: JMemcached, licensed under the Apache Public License (APL). World Wide Web Consortium
(W3C)(MIT, ERCIM, Keio): Flute, JTidy, Simple API for CSS. XFree86 Project, Inc.: (www.xfree86.org):
All other brand or product names are trademarks or registered trademarks of their respective owners,
companies, or organizations.
Document No. 111021-2-771302 October 19, 2011
i
Contents
About Actuate BIRT Java Components Developer Guide. . . . . . . . . . . . . vii
Part 1
Customizing an Actuate Java Component
Chapter 1
Introducing Actuate Java Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
About Actuate Java Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Licensing Java Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Setting up Actuate Java Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Customizing Java components for installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
About using a cluster of application servers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
About Actuate Java Component architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Using proxy servers with Actuate Java Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
About Actuate Java Component pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Working with Actuate Java Component URIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
About Actuate Java Component URIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Using a special character in a URI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
About UTF-8 encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 2
Deploying Actuate BIRT reports using an Actuate Java Component . . . 13
Publishing a BIRT report design to the Actuate Java Component . . . . . . . . . . . . . . . . . . . . . . . . 14
Publishing a BIRT resource to an Actuate Java Component . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Installing a custom JDBC driver in an Actuate Java Component . . . . . . . . . . . . . . . . . . . . . . 16
Installing custom ODA drivers and custom plug-ins in an Actuate Java Component . . . . . 16
Accessing BIRT report design and BIRT resources paths in custom ODA plug-ins . . . . . . . 16
Accessing resource identifiers in run-time ODA driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Accessing resource identifiers in design ODA driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Using fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Understanding font configuration file levels and priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Understanding how BIRT accesses a font . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Understanding the font configuration file structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
<font-aliases> section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
<composite-font> section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
<font-paths> section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Using BIRT encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
About the BIRT default encryption plug-in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Deploying encryption plug-ins to Actuate Java Components . . . . . . . . . . . . . . . . . . . . . . . . . 24
ii
About the components of the BIRT default encryption plug-in . . . . . . . . . . . . . . . . . . . . . . . .25
About acdefaultsecurity.jar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
About encryption.properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
About META-INF/MANIFEST.MF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
About plugin.xml . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
Deploying multiple encryption plug-ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
Generating encryption keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
Deploying custom emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Rendering in custom formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Chapter 3
Creating a custom Java Component web application . . . . . . . . . . . . . . . 39
Java Component web application structure and contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
Understanding Java Component directory structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
Building a custom Java Component context root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44
Modifying existing content or creating new content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
Activating a new web application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
Configuring a custom Java Component web application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
Customizing Java Component configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
Customizing requester pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
Customizing a Java Component web application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
Viewing modifications to a custom web application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
Locating existing pages and linking in new pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51
Obtaining information about the user and the session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
Customizing accessible files and page structure using templates . . . . . . . . . . . . . . . . . . . . . . .54
Specifying a template and template elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
Changing a template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Modifying global style elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
Understanding style definition files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57
Specifying colors and fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
Customizing page styles for BIRT Studio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
Modifying images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60
Part 2
Actuate Java Component Reference
Chapter 4
Actuate Java Component configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 65
About Actuate Java Component configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
Configuring Java Component web applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
Configuring the Java Component using web.xml . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
Configuring Java Component functionality levels with functionality-level.config . . . . . . . .71
iii
Configuring Java Component locale using localemap.xml . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Configuring Java Component locales using TimeZones.xml . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Configuring the Actuate Java Component repository . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Configuring the BIRT Viewer and Interactive Viewer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Configuring BIRT Studio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Configuring BIRT Data Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 5
Actuate Java Component URIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Actuate Java Component URIs overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Actuate Java Component URIs quick reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Common URI parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Java Component Struts actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Actuate Java Component URIs reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
about page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
authenticate page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
banner page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
browse file page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
delete file status page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
detail page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
drop page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
error page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
execute report page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
home page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
index page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
license page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
list page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
login banner page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
login page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
logout page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
page not found page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
parameters page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Actuate BIRT Viewer URIs reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 6
Actuate Java Component JavaScript . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Actuate Java Component JavaScript overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Actuate Java Component JavaScript reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter 7
Actuate Java Component servlets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Java Component Java servlets overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
About the base servlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
iv
Invoking a servlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104
Java Component Java servlets quick reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
Java Component Java servlets reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
ExecuteReport servlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
Interactive Viewer servlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107
Chapter 8
Actuate Java Component JavaBeans . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Java Component JavaBeans overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Java Component JavaBeans package reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Java Component JavaBeans class reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113
Jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113
Chapter 9
Using Actuate Java Component security . . . . . . . . . . . . . . . . . . . . . . . . 115
About Actuate Java Component security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
Protecting corporate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
Protecting corporate data using firewalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116
Protecting corporate data using Network Address Translation . . . . . . . . . . . . . . . . . . . . . . .117
Protecting corporate data using proxy servers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
Understanding the authentication process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
Customizing Java Component authentication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Creating a custom security adapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Accessing the IPSE Java classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Creating a custom security adapter class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Understanding a security adapter class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
Chapter 10
Customizing Java Component online help . . . . . . . . . . . . . . . . . . . . . . . 123
About Actuate Java Component online help files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124
Understanding the Java Component help directory structure . . . . . . . . . . . . . . . . . . . . . . . . .124
Understanding a help collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
Understanding a document root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126
Understanding context-sensitive help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127
Understanding locale support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128
Using a custom help location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
Creating a localized help collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131
Customizing icons and the company logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133
Changing the corporate logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133
Changing the corporate logo on the title page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134
Changing the logo in the help content pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134
v
Changing icons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Changing the browser window title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Changing help content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Changing existing help content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Adding or removing help topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Adding and removing content files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Changing the table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Changing the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
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Ab o u t Ac t u a t e BI RT J av a Co mp o n e n t s Dev e l o p e r Gu i d e
vii
A b o u t A c t u a t e B I R T
J a v a C o m p o n e n t s
D e v e l o p e r G u i d e
Actuate BIRT Java Components Developer Guide is a guide to designing, deploying
and accessing custom reporting web applications using Actuate Java Component.
Actuate BIRT Java Components Developer Guide includes the following chapters:
About Actuate BIRT Java Components Developer Guide. This chapter provides an
overview of this guide.
Part 1. Customizing an Actuate Java Component. This part describes how to use
Java Component and how to customize its appearance and layout.
Chapter 1. Introducing Actuate Java Components. This chapter introduces Actuate
Java Component web applications and explains how Java Components work.
Chapter 2. Deploying Actuate BIRT reports using an Actuate Java Component. This
chapter explains how to publish and support BIRT reports and features using
Java Components.
Chapter 3. Creating a custom Java Component web application. This chapter
explains how to work with Java Component JSP files to design custom
reporting web applications.
Part 2. Actuate Java Component Reference. This part describes the code
components that make up Java Component, such as URIs, JavaScript files,
servlets, tags, beans, and security facilities.
Chapter 4. Actuate Java Component configuration. This chapter describes the Java
Component configuration files and how to use them.
Chapter 5. Actuate Java Component URIs. This chapter describes the Java
Component JSPs and URL parameters.
Chapter 6. Actuate Java Component JavaScript. This chapter describes the Java
Component JavaScript files.
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Ac t u a t e BI RT J av a Co mp o n e n t s Dev e l o p e r Gu i d e
Chapter 7. Actuate Java Component servlets. This chapter describes the Java
Component Java servlets.
Chapter 8. Actuate Java Component JavaBeans. This chapter lists the Java
Component JavaBeans.
Chapter 9. Using Actuate Java Component security. This chapter introduces the
iPortal Security Extension (IPSE) and explains how to use it.
Part 1
Customizing an Actuate Java
Component
Part
One
1
Ch a p t e r 1, I n t r o d u c i n g Ac t u a t e J av a Co mp o n e n t s
3
C h a p t e r
Chapter 1
Introducing Actuate Java
Components
This chapter contains the following topics:
4
Ac t u a t e BI RT J a v a Co mp o n e n t s Dev e l o p e r Gu i d e
Actuate Java Component is a web application that supports accessing and
working with report information using a web browser. Web developers and
designers use Actuate Java Component’s industry-standard technology to design
custom e.reporting web applications to meet business information delivery
requirements.
Actuate Java Component technology is platform-independent and customizable.
By separating user interface design from content generation, Java Components
ensures that reporting web application development tasks can proceed
simultaneously and independently. You deploy Actuate Java Component on a
web or application server. Java Component accesses documents in a file system
repository. Actuate Java Component technology is also scalable.
When deployed, the context root is name of the web archive (.war) or engineering
archive (.ear) file without the file extension. For example, if your web archive
(.war) file were named DeploymentKit.war, the URL to access the application is:
http://<web server>:<port>/DeploymentKit/
The context root for Java Component is the root directory of the web archive
(.war) file when it is extracted.
Actuate Java Component technology includes the following features:
JavaServer Pages (JSPs) support creating HTML or XML pages that combine
static web page templates with dynamic content.
Simple Object Access Protocol (SOAP) standards provide plain text
transmission of XML using HTTP.
Report designs and documents are stored on a file system.
Secure HTTP (HTTPS) supports secure information transfer on the web.
that support the JSR 168 standard.
Licensing Java Components
Java Components have a temporary license by default. To fully license the Java
Component you have purchased, you must move the license file received from
actuate into the <context root>\WEB-INF directory of the web archive (.war) file.
1 Rename the Java Component license file that Actuate sent you to
Ch a p t e r 1, I n t r o d u c i n g Ac t u a t e J av a Co mp o n e n t s
5
2 Create a temporary directory, such as C:\Temp\jc on a Microsoft Windows
server or /temp/jc on a UNIX server. If you use an existing directory, ensure
that this directory is empty.
3 Extract the contents of the Java Component WAR file into a temporary
directory.
On a Windows server, open a command window and type the following
commands, replacing the E: DVD drive letter with the path of your Java
Component WAR file:
cd C:\Temp\jc
copy E:\ActuateJavaComponent.war
jar -xf ActuateJavaComponent.war
The Java Component files appear in the temporary directory. Leave the
command window open.
On a LINUX or UNIX server, type the following commands, replacing the
DVD drive name with the path of your Java Component WAR file:
cd /temp/jc
cp /dev/dsk/cd/ActuateJavaComponent.war.
jar -xf ActuateJavaComponent.war
The Actuate Java Component files appear in the temporary directory.
4 Copy the ajclicense.xml file into the extracted <context root>\WEB-INF
directory.
5 Type the following command:
jar -cf ..\DeploymentKit.war *
This command creates
DeploymentKit
.war in the parent directory. This new
Java Component WAR file contains the license.
6 Deploy the DeploymentKit.war file to the application server or servlet engine
as an application.
7 Restart the application server or servlet engine.
Setting up Actuate Java Component
To deploy a report to the web, you need:
An Actuate Java Component installation.
An application server or JSP or servlet engine such as Actuate embedded
servlet engine or IBM WebSphere.
One or more Actuate designer tools.
Permission to read, write, and modify operating system directories as
necessary. For example, the directory Java uses to hold temporary files is
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Ac t u a t e BI RT J a v a Co mp o n e n t s Dev e l o p e r Gu i d e
defined by the java.io.tmpdir property and is by default the value of the TMP
system variable in the Windows environment and /var/tmp in the UNIX and
LINUX environments. Read and write permission must be provided to the
application server running Information Console for this directory.
Java Component.
Customizing Java components for installation
When you deploy Java Components on an application server, you can use a
customized Java Component application. To do this, you need to extract the
contents of the Actuate Java Components WAR or EAR file and customize the
files directly. After you customize the system, recreate a WAR or EAR file using
the Java jar utility and redeploy it to your application server. The customizations
can include any modifications of JavaScript, Java Server Pages (JSP) and other
web pages, and skins. Later chapters in this book provide detailed information
When Actuate Java Component is deployed, you cannot further customize skins,
add pages, or make any other changes that affect the Actuate Java Component file
structure without extracting the contents of the WAR or EAR file, modifying the
contents, and re-deploying it.
Clustered Actuate Java Component instances can use a third-party application to
balance the load among the application servers. Actuate Java Component
supports third-party load balancing, as illustrated in Figure 1-1, to ensure high
availability and to distribute tasks for efficient processing.
Figure 1-1 Load-balancing architecture for Java Component
Web
browser
Web
browser
Web
browser
Third-party
application
balancer
Application
server
Java
Component
Application
server
Java
Component
Application
server
Java
Component
StateServer or SqlServer
Ch a p t e r 1, I n t r o d u c i n g Ac t u a t e J av a Co mp o n e n t s
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About using a cluster of application servers
If the application servers running Java Component support session state
management, you can configure Actuate Java Component and the application
servers to share and maintain a web browsing session state across a cluster of Java
Component instances.
How to customize and deploy Actuate Java Component
To customize Actuate Java Component and deploy it to application servers in a
clustered environment, use the following general procedure.
1 Extract the contents of the Actuate Java Component WAR file into a temporary
directory.
2 Customize the Actuate Java Component JavaScript, skins, and web pages as
desired.
3 Save all files and archive Actuate Java Components as a new WAR or EAR file
using the Java jar utility.
4 Deploy the WAR or EAR file to each machine in your cluster.
This section describes the general operation, authentication, and structure of Java
Component as a web application.
The Actuate Java Component architecture is illustrated in Figure 1-2.
Figure 1-2 Actuate Java Component architecture overview
A user submits a request by choosing a link that specifies an Actuate Java
Component URI. As shown in Figure 1-2, the web or application server passes the
URI to the servlet or page engine, which invokes Actuate Java Component and
interprets the URI. The web server returns the results to the web browser. Then,
the web browser displays the results for the user.
Actuate Java Component manages requests as part of a JSP engine within a web
or application server. See your web or application server documentation for more
information on managing the engine.
Web or Application server
Servlet or Page engine
Actuate Java
Component
Firewall
Web
browser
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Using proxy servers with Actuate Java Component
When setting up a proxy server with Actuate Java Component, there are steps
you must take if your internal application server port is protected by a firewall. In
this situation, when the proxy server changes the URL to point to the new
context’s port, that port is unavailable due to the firewall. The usual solution is to
configure a reverse proxy, but if you are using multiple proxies and a reverse
proxy is not practical for your installation, Actuate Java Component can perform
the redirection.
To redirect a page without using a reverse proxy, Actuate Java Component
forwards the URL to redirect to the processRedirect.jsp page and updates the
browser’s location bar accordingly. This action processes on the client. The
browser takes the current URL location and updates the rest of the URI using the
redirected URL. You must also set the ENABLE_CLIENT_SIDE_REDIRECT
configuration parameter to true and modify the redirect attributes in the <context
root>/WEB-INF/struts-config.xml file. The necessary modifications are included
in the file. You just need to comment out the lines that have the redirect attribute
set to true and uncomment the lines that forward to the processRedirect.jsp page.
For example, the following code is the struts-config.xml entry for the login action.
By default the forward statement for success points to getfolderitems.do with the
redirect attribute set to true. This code instructs the application server to send a
redirect with the getfolderitems.do URL when the user logs in.
<!-- Process a user login -->
<action
scope="request"
validate="false">
<!--
<forward name="success"
path="/iportal/activePortal/private/common
/processredirect.jsp?redirectPath=/getfolderitems.do" />
-->
<forward name="success" path="/getfolderitems.do"
redirect="true" />
<forward name="landing" path="/landing.jsp"
redirect="false" />
</action>
From behind a firewall and proxy, this redirect will fail because the redirect sent
by the application server points to the application server port instead of the
firewall and proxy port. For this redirect method to operate behind a firewall, you
need to comment out the line that has redirect="true" and uncomment the line
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that points to processRedirect.jsp. The following code shows the updated entry in
struts-config.xml:
<!-- Process a user login -->
<action
scope="request"
validate="false">
<forward name="success"
path="/iportal/activePortal/private/common
/processredirect.jsp?redirectPath=/getfolderitems.do" />
<!--
<forward name="success" path="/getfolderitems.do"
redirect="true" />
-->
<forward name="landing" path="/landing.jsp"
redirect="false" />
</action>
This change needs to be made for all the actions in struts-config.xml that send a
redirect to the browser.
Actuate Java Component uses JSPs to generate web pages dynamically before
sending them to a web browser. These JSPs use custom tags, custom classes, and
JavaScript to generate dynamic web page content. The JavaScript, classes, and
tags provide access to other pages, JavaBeans, and Java classes. For example,
application logic in Actuate Java Component can reside on the web server in a
JavaBean.
Web browsers can request a JSP with parameters as a web resource. The first time
a web browser requests a page, the page is compiled into a servlet. Servlets are
Java programs that run as part of a network service such as a web server. Once a
page is compiled, the web server can fulfill subsequent requests quickly, provided
that the page source is unchanged since the last request.
The filesfolders JSPs support accessing repository files and folders. These JSPs
reside in <context root>\iportal\activePortal\private\filesfolders.
The submit request JSPs support submitting new jobs. The submit request JSPs
reside in <context root>\iportal\activePortal\private\newrequest. For specific
information about running jobs using Actuate Java Component, see Using Actuate
BIRT Java Components.
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The viewing JSPs support the following functionality, according to report type:
Searching report data
Paginating or not paginating a report
Fetching reports in supported formats
For specific information about viewing reports using Actuate Java Component,
see Using Actuate BIRT Java Components.
Use the default pages, customize the pages, or create entirely new pages to
Working with Actuate Java Component URIs
Actuate Java Component Uniform Resource Identifiers (URIs) convey user
requests to an application server. URIs access functionality including generating
reports, managing repository contents, and viewing reports.
Actuate Java Component URIs consist of the context root and port of the web
server where you install and deploy the JSPs or servlets. Actuate Java Component
URIs have the following syntax:
http://<web server>:<port>/<context root>
/<path><page>.<type>[?<parameter=value>{&<parameter=value>}]
where
<web server> is the name of the machine running the application server or
servlet engine. You can use localhost as a trusted application’s machine name
if your local machine is running the server.
<port> is the port on which you access the application server or servlet engine.
<context root> is the context root for accessing the Actuate Java Component
pages, which by default is the name of the WAR or EAR file.
<path> is the directory containing the page to invoke.
<page> is the name of the page or method.
<type> is jsp or do.
<parameter=value> specifies the required parameters and values for the page.
For example, to view the document list page, Actuate Java Component uses a URI
with the following format:
http://<web server>:<port>/ActuateJavaComponent
/getfolderitems.do?doframe=true&userid=anonymous
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where
ActuateJavaComponent/getfolderitems.do is the JSP that provides file
browsing for Java Component.
doframe=true is a reserved parameter that displays the documents page in a
frame next to other frames for the banner and file explorer tree.
userid=anonymous indicates that the default anonymous user is being used
and security is not enabled. This is the default security setting for Actuate Java
Components. For information about customizing security, see Chapter 9,
“Using Actuate Java Component security.”
Using a special character in a URI
Actuate Java Component URIs use encoding for characters that a browser can
misinterpret. You use hexadecimal encoding in these circumstances to avoid
misinterpretation. Use the encoding only when the possibility of misinterpreting
a character exists. Always encode characters that have a specific meaning in a URI
when you use them in other ways. Table 1-1 describes the available character
substitutions. An ampersand introduces a parameter in a URI, so you must
encode an ampersand that appears in a value string. For example, use:
&company=AT%26T
&company=AT&T
Table 1-1 Encoding sequences for use in URIs
Character Encoded substitution
ampersand (&) %26
asterisk (*) %2a
at (@) %40
backslash (\) %5c
colon (:) %3a
comma (,) %2c
dollar sign (\$) %24
double quote (
"
) %22
equal (=) %3d
exclamation (!) %21
greater than (>) %3e
less than (<) %3c
(continues)
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If you customize Actuate Java Component by writing code that creates URI
parameters, encode the entire parameter value string with the encode() method.
The encode( ) method is included in encoder.js, which is provided in the Actuate
Java Component <context root>/js directory. The following example encodes the
folder name /Training/Sub Folder before executing the getFolderItems action:
<%-- Import the StaticFuncs class. --%>
<%@ page import="com.actuate.reportcast.utils.*" %>
<%
String url =
"http://localhost:8080/ActuateJavaComponent/getfolderitems.do
?folder=" + StaticFuncs.encode("/Training/Sub Folder");
response.sendRedirect(url);
%>
The encode( ) method converts the folder parameter value from:
/Training/Sub Folder
to:
%2fTraining%2fSub%20Folder
UTF-8 encoding is also the default encoding that web browsers support. All Java
Component communication also uses UTF-8 encoding. For 8-bit (single byte)
characters, UTF-8 content appears the same as ANSI content. If, however,
extended characters are used (typically for languages that require large character
sets), UTF-8 encodes these characters with two or more bytes.
number sign (#) %23
percent (%) %25
period (.) %2e
plus (+) %2b
question mark (?) %3f
semicolon (;) %3b
slash (/) %2f
space ( ) %20
underscore (_) %5f
Table 1-1 Encoding sequences for use in URIs (continued)
Character Encoded substitution
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
13
C h a p t e r
Chapter 2
Deploying Actuate BIRT
reports using an Actuate
Java Component
This chapter contains the following topics:
Publishing a BIRT report design to the Actuate Java Component
Using fonts
Using BIRT encryption
Deploying custom emitters
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Publishing a BIRT report design to the Actuate Java
Component
Actuate Java Components generate BIRT reports using BIRT report design
(.rptdesign) files and their associated resource files. Actuate Java Components
access BIRT report design and associated resource files from configurable
locations on a file system.
The default location designated for BIRT report design files is the repository
folder in the context root directory structure, as illustrated in Figure 2-1.
Figure 2-1 Actuate Java Component folder structure
To configure the repository location for publishing BIRTdesigns and documents,
change the value of the STANDALONE_REPOSITORY_PATH parameter in the
Actuate Java Component’s web.xml file. The web.xml file is in the following
location:
<context root>/WEB-INF
The following code sets STANDALONE_REPOSITORY_PATH to the
<context root>/WEB-INF/repository subfolder:
<context-param>
<param-name>STANDALONE_REPOSITORY_PATH</param-name>
<param-value>WEB-INF/repository</param-value>
</context-param>
BIRT_RESOURCE_PATH specifies the path to the shared resources for Actuate
BIRT Java Components, including libraries, templates, properties, and Java
archive (.jar) files for BIRT report designs. The default value is <context root>
/WEB-INF/repository.
How to publish a BIRT report design to an Actuate Java Component
This procedure uses the default location of the Actuate Java Component
repository.
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
15
1 Navigate to the application server’s directory for deployed web applications.
For example, Apache Tomcat stores web applications in <Apache Tomcat root
directory>/Tomcat 6.0/webapps.
2 In the web application directory, manually copy the BIRT report design to a
directory in the following location:
<context root>/WEB-INF/repository
The installation provides default home and public directories, as shown in
Figure 2-1. All user directories are created in the repository/home directory.
3 To make a report design available to all users, place the file in a directory
within:
<context root>/WEB-INF/repository/Public
4 To make a report design available to an individual user only, place the file in a
directory within:
<context root>/WEB-INF/repository/Home/<user name>
5 Run the Actuate Java Component to access the report design.
Publishing a BIRT resource to an Actuate Java
Component
You configure the repository for publishing a BIRT resource using the
BIRT_RESOURCE_PATH parameter in an Actuate Java Component’s web.xml
file. The web.xml file is in the following location:
<context root>/WEB-INF
The following code sets BIRT_RESOURCE_PATH to the <context root>
/resources subfolder:
<context-param>
<param-name>BIRT_RESOURCE_PATH</param-name>
<param-value>resources</param-value>
</context-param>
BIRT_RESOURCE_PATH specifies the path to the shared resources for Actuate
BIRT Java Components, including libraries, templates, properties, and Java
archive (.jar) files for BIRT report designs. The default value is
<context root>/resources.
If the BIRT report explicitly includes a resource such as a JAR file, library, CSS, a
Flash (.swf) file, images, or JavaScript in the report design, then the resources
need to be copied under the BIRT_RESOURCE_PATH folder to the correct
relative path.
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For example, if the images for your report are in the /images folder in your report
design project, when you deploy the report, you copy the images to the
<context root>/resources/images folder.
In cases when an Actuate BIRT report uses Java classes directly from JAR files,
<context root>/scriptlib
How to publish a BIRT resource to an Actuate Java Component
1 Copy the resource file to the resource directory, defined in web.xml.
2 To test the resource, run the Actuate Java Component to execute and view a
report that uses the resource.
Installing a custom JDBC driver in an Actuate Java
Component
When you use an Actuate Java Component and an Actuate BIRT report uses a
custom JDBC driver, you must install the JDBC driver in the following location:
<context root>/WEB-INF/platform/plugins
/org.eclipse.birt.report.data.oda.jdbc_<VERSION>/drivers
Installing custom ODA drivers and custom plug-ins in
an Actuate Java Component
All custom ODA drivers and custom plug-ins need to be installed in the
following folder:
<context root>/WEB-INF/platform/plugins
Accessing BIRT report design and BIRT resources
paths in custom ODA plug-ins
ODA providers often need to obtain information about a resource path defined in
ODA consumer applications. For example, if you develop an ODA flat file data
source, you can implement an option to look up the data files in a path relative to
a resource folder managed by its consumer. Such resource identifiers are needed
at both design-time and run-time drivers. ODA consumer applications are able to
specify the following items as described in the next two sections:
The run-time resource identifiers to pass o the ODA run-time driver in an
application context map
The design-time resource identifiers in a DataSourceDesign, as defined in an
ODA design session model
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
17
Accessing resource identifiers in run-time ODA driver
For run time, the BIRT ODA run-time consumer passes its resource location
information in a org.eclipse.datatools.connectivity.oda.util.ResourceIdentifiers
instance in the appContext map. ODA run-time drivers can get the instance in
any one of the setAppContext methods, such as IDriver.setAppContext. You can
use resource identifiers to perform the following tasks:
To get the BIRT resource folder URI, call getApplResourceBaseURI( ) method.
To get the instance from the appContext map, pass the map key
ResourceIdentifiers.ODA_APP_CONTEXT_KEY_CONSUMER_RESOURCE_
IDS, defined by the class as a method argument.
To get the URI of the associated report design file folder, call
getDesignResourceBaseURI( ) method. The URI is application dependent and
it can be absolute or relative. If your application maintains relative URLs, call
the getDesignResourceURILocator.resolve( ) method to get the absolute URI.
The code snippet on Listing 2-1 shows how to access the resource identifiers
through the application context.
Listing 2-1 Accessing resource identifiers at run time
URI resourcePath = null;
URI absolutePath = null;
Object obj = this.appContext.get(
ResourceIdentifiers.ODA_APP_CONTEXT_KEY_CONSUMER_RESOURCE_IDS
);
if ( obj != null )
{
ResourceIdentifiers identifier = (ResourceIdentifiers)obj;
if ( identifier.getDesignResourceBaseURI( ) != null )
{
resourcePath = identifier.getDesignResourceBaseURI( );
if ( ! resourcePath.isAbsolute( ) )
absolutePath =
identifier.getDesignResourceURILocator( ).resolve(
resourcePath );
else
absolutePath = resourcePath;
}
}
Accessing resource identifiers in design ODA driver
The resource identifiers are available to the custom ODA designer UI driver. The
designer driver provides the user interface for a custom data source and data set.
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Typically, to implement a custom behavior, the data source UI driver extends:
org.eclipse.datatools.connectivity.oda.design.ui.wizards.
DataSourceWizardPage
The DataSourceWizardPage class has an inherited method
paths. The extended DataSourceWizardPage just needs to call the base method to
get the ResourceIdentifiers for its paths information.
Similarly, if the custom driver implements a custom data source editor page, it
extends:
org.eclipse.datatools.connectivity.oda.design.ui.wizards.
DataSourceEditorPage
The DataSourceEditorPage class has an inherited method
getHostResourceIdentifiers( ). The extended class needs to call the base class
method to get the ResourceIdentifiers object for the two resource and report paths
base URIs.
Related primary methods in the org.eclipse.datatools.connectivity.oda.design.
ResourceIdentifiers are:
URI getDesignResourceBaseURI( );
URI getApplResourceBaseURI( );
Using fonts
Java Components supports rendering BIRT reports in different formats such as
PDF, Microsoft Word, Postscript, and PowerPoint. The conversion processes use
the fonts installed on your system to display the report characters by default.
BIRT uses a flexible mechanism that supports configuring font usage and
substitution. This mechanism uses font configuration files for different purposes
that control different parts of the rendering process. The configuration files can
configure the fonts used in specific operating systems, in specific formats, in
specific locales, or combinations of these parameters, as described in the next
section.
The plug-in folder, org.eclipse.birt.report.engine.fonts, contains the font
configuration files. Table 2-1 shows the location of this folder in the supported
BIRT environments.
Table 2-1 Locations of the font configuration file plug-in folder
Environment Font configuration file folder location
Actuate Java
Components
\$ActuateJavaComponents/WEB-INF/platform/plugins
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
19
Understanding font configuration file levels and
priorities
BIRT reports use five different types of font configuration files. The font
configuration file naming convention includes information about the rendering
format, the system platform, and the system locale, as shown in the following
template:
fontsConfig_<Format>_<Platform>_<Locale>.xml
The platform name is defined by the Java System property, os.name. The
following code shows how to check the os.name property for the proper value in
System.getProperty("os.name");
Table 2-2 lists the supported values for the three properties that form the font
configuration file name. The platform property in this table shows the values that
Sun Microsystems uses for the os.name property.
BIRT Report Designer \$Actuate11/BRD/eclipse/plugins
BIRT Report Designer
Professional
\$Actuate11/BRDPro/eclipse/plugins
Table 2-2 Font configuration file name properties
Format Platform Locale
pdf Windows_Vista en
ppt Windows_2003 fr
html Windows_XP de
postscript Windows_2000 it
doc SunOS ja
AIX ko
HP-UX zh
Linux zh_Hans
zh_Hant
fr_FR
de_DE
it_IT
(continues)
Table 2-1 Locations of the font configuration file plug-in folder
Environment Font configuration file folder location
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BIRT supports the following levels of font configuration files, with increasing
priority:
For all rendering formats
These files have no format specifier in their names. These configuration files
are divided into three sub-levels:
The default configuration file:
fontsConfig.xml
Configuration files for a specific platform, for example:
fontsConfig_Windows_XP.xml
Configuration files for a specific platform and locale, for example:
fontsConfig_Windows_XP_zh.xml
fontsConfig_Windows_XP_zh_CN.xml
For certain formats only
These files include the format specifier in their names. These configuration
files are divided into three sub-levels:
The default configuration file for a format, for example:
fontsConfig_pdf.xml
Configuration files for a format for a specific platform:
fontsConfig_pdf_Windows_XP.xml
Understanding how BIRT accesses a font
The PDF layout engine renders the PDF, Postscript, and PowerPoint formats. The
engine tries to use the font specified at design time to render. The PDF layout
engine searches for the font files first in the fonts folder of the plug-in,
org.eclipse.birt.report.engine.fonts. If the fonts are not in this folder, the engine
doc (continued) Linux (continued) ja_JP
ko_KR
zh_Hans_CN
zh_Hant_TW
en_GB
en_US
en_CA
Table 2-2 Font configuration file name properties (continued)
Format Platform Locale
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
21
searches for the font in the system-defined font folder. Change the default load
order by using the settings in the font configuration file.
When the required font for a character is not available in the search path or is
incorrectly installed, the engine uses the fonts defined in the UNICODE block for
that character. If the UNICODE definition also fails, the engine replaces the
character with a question mark (?) to denote a missing character. The font used
for the ? character is the default font, Times-Roman.
The engine maps the generic family fonts to a PDF embedded Type1 font, as
shown in the following list:
cursive maps to Times-Roman
fantasy maps to Times-Roman
monospace maps to Courier
sans-serif maps to Helvetica
serif maps to Times-Roman
Understanding the font configuration file structure
The font configuration file, fontsConfig.xml, consists of three major sections,
<font-aliases>, <composite-font>, and <font-paths> sections.
<font-aliases> section
In <font-aliases> section, you can:
Define a mapping from a generic family to a font family. For example, the
following defines a mapping from generic family "serif" to Type1 font family
"Times-Roman":
<mapping name="serif" font-family="Times-Roman"/>
Define a mapping from a font family to another font family. This is useful if
you want to use a font for PDF rendering that differs from the font used in
design-time. For example, the following shows how to replace "simsun" with
"Arial Unicode MS":
<mapping name="simsun" font-family="Arial Unicode MS"/>
Previous versions of the BIRT Report Designers use the XML element
<font-mapping> instead of <font-aliases>. In the current release, a
<font-mapping> element works in the same way as the new <font-aliases>
element. When a font configuration file uses both <font-mapping> and
<font-aliases>, the engine merges the different mappings from the two sections. If
the same entries exist in both sections, the settings in <font-aliases> override
those in <font-mapping>.
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<composite-font> section
The <composite-font> section defines a composite font. A composite font is a font
consisting of many physical fonts used for different characters. The composite
fonts are defined by <block> entries. Each <block> entry defines a mapping from
a UNICODE range to a font family name, which means the font family is applied
for the UNICODE characters in that range. You cannot change the block name or
range or index as it is defined by the UNICODE standard. The only item you can
change in the block element is the font family name. To find information about all
the possible blocks, go to http://www.unicode.org/charts/index.html.
A composite font named all-fonts is applied as a default font. When a character is
not defined in the desired font, the font defined in all-fonts is used.
For example, to define a new font for currency symbols, you change font-family
in the following <block> entry to the Times Roman font-family:
<composite-font>
<block name="Currency Symbols" range-start="20a0" range-end="20cf"
index="58" font-family="Times Roman" />
</composite-font>
In cases when the Times Roman font does not support all the currency symbols,
you can define the substitution character by character using the <character> tag,
as shown in the following example:
<composite-font>
<character value="?" font-family="Angsana New"/>
<character value="\u0068" font-family="Times Roman"/>
</composite-font>
Note that characters are represented by the attribute, value, which can be
presented two ways, the character itself or its UNICODE code.
To find information about all the currency symbols, go to
http://www.unicode.org/charts/symbols.html.
<font-paths> section
If the section <font-paths> is set in fontsConfig.xml, the engine ignores the
system-defined font folder, and loads the font files specified in the section,
<font-paths>. You can add a single font path or multiple paths, ranging from one
font path to a whole font folder, as shown in the following example:
<path path="c:/windows/fonts"/>
<path path="/usr/X11R6/lib/X11/fonts/TTF/arial.ttf"/>
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If this section is set, the PDF layout engine will only load the font files in these
paths and ignore the system-defined font folder. If you want to use the system
font folder as well, you must include it in this section.
On some systems, the PDF layout engine does not recognize the system-defined
font folder. If you encounter this issue, add the font path to the <font-paths>
section.
Using BIRT encryption
BIRT provides an extension framework to support users registering their own
encryption strategy with BIRT. The model implements the JCE (Java™
Cryptography Extension). The Java encryption extension framework provides
multiple popular encryption algorithms, so the user can just specify the algorithm
and key to have a high security level encryption. The default encryption
extension plug-in supports customizing the encryption implementation by
copying the BIRT default plug-in, and giving it different key and algorithm
settings.
JCE provides a framework and implementations for encryption, key generation
and key agreement, and Message Authentication Code (MAC) algorithms.
Support for encryption includes symmetric, asymmetric, block, and stream
ciphers. The software also supports secure streams and sealed objects.
A conventional encryption scheme has the following five major parts:
Plaintext, the text to which an algorithm is applied.
Encryption algorithm, the mathematical operations to conduct substitutions
on and transformations to the plaintext. A block cipher is an algorithm that
operates on plaintext in groups of bits, called blocks.
Secret key, the input for the algorithm that dictates the encrypted outcome.
Ciphertext, the encrypted or scrambled content produced by applying the
algorithm to the plaintext using the secret key.
Decryption algorithm, the encryption algorithm in reverse, using the
ciphertext and the secret key to derive the plaintext content.
About the BIRT default encryption plug-in
BIRT’s default encryption algorithm is implemented as a plug-in named:
com.actuate.birt.model.defaultsecurity_11.0.1
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Table 2-3 shows the location of this plug-in folder in the supported BIRT
environments.
Deploying encryption plug-ins to Actuate Java
Components
If you use Java Components, you deploy all new encryption plug-ins to the Java
Components plug-in folder. The BIRT report engine decrypts the encrypted
report data during report generation. To do the decryption, it must have access to
all encryption plug-ins. The report engine loads all encryption plug-ins at start
up. When the engine runs a BIRT report, it reads the encryptionID property from
the report design file and uses the corresponding encryption plug-in to decrypt
the encrypted property. Every time you create reports using a new encryption
plug-in, make sure you deploy the plug-in to Java Components installation,
otherwise the report execution will fail.
How to deploy a new encryption plug-in instance to Actuate Java Components
1 Extract the Java Components WAR or EAR file into temporary directory.
2 Copy:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1_rsa
to:
<context root>/WEB-INF/platform/plugins
3 Copy your report design to:
<context root>/WEB-INF/repository/home/<UserHomeFolder>
4 Recompress your Java Components WAR file using the Java jar utility and
redeploy it to the application server or servlet engine as an application.
5 Restart the application service where the Java Components are deployed, to
6 Run your report again. The engine uses the new encryption plug-in to decrypt
Table 2-3 Locations of the default encryption plug-in folder
Environment Font configuration file folder location
Actuate Java
Components
\$ActuateJavaComponents/WEB-INF/platform/plugins
BIRT Report Designer \$Actuate11/BRD/eclipse/plugins
BIRT Report Designer
Professional
\$Actuate11/BRDPro/eclipse/plugins
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About the components of the BIRT default encryption
plug-in
The BIRT default encryption plug-in consists of the following main modules:
acdefaultsecurity.jar
encryption.properties file
META-INF/MANIFEST.MF
plugin.xml
This JAR file contains the encryption classes. The default encryption plug-in also
provides key generator classes that can create different encryption keys.
This file specifies the encryption settings. BIRT loads the encryption type,
encryption algorithm, and encryption keys from the encryption.properties file to
do the encryption. The file contains pre-generated default keys for each of the
supported algorithms.
You define the following properties in the encryption.properties file:
Encryption type
Type of algorithm. Specify one of the two values, symmetric encryption or
public encryption. The default type is symmetric encryption.
Encryption algorithm
The name of the algorithm. You must specify the correct encryption type for
each algorithm. For the symmetric encryption type, BIRT supports DES and
DESede. For public encryption type, BIRT supports RSA.
Encryption mode
In cryptography, a block cipher algorithm operates on blocks of fixed length,
which are typically 64 or 128 bits. Because messages can be of any length, and
because encrypting the same plaintext with the same key always produces the
same output, block ciphers support several modes of operation to provide
confidentiality for messages of arbitrary length. Table 2-4 shows all supported
modes.
Table 2-4 Supported encryption modes
Mode Description
None No mode
(continues)
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Because a block cipher works on units of a fixed size, but messages come in a
variety of lengths, some modes, for example CBC, require that the final block
paddings are shown in Table 2-5. All padding settings are applicable to all
algorithms.
Encryption keys
Actuate provides pre-generated keys for all algorithms.
Listing 2-1 shows the default contents of encryption.properties.
Listing 2-1 Default encryption.properties
#message symmetric encryption , public encryption.
type=symmetric encryption
CBC Cipher Block Chaining Mode, as defined in the National
Institute of Standards and Technology (NIST) Federal
Information Processing Standard (FIPS) PUB 81, ”DES
Modes of Operation,” U.S. Department of Commerce, Dec
1980
CFB Cipher Feedback Mode, as defined in FIPS PUB 81
ECB Electronic Codebook Mode, as defined in FIPS PUB 81
OFB Output Feedback Mode, as defined in FIPS PUB 81
PCBC Propagating Cipher Block Chaining, as defined by
Kerberos V4
Mode Description
OAEP Optimal Asymmetric Encryption Padding (OAEP) is a
padding scheme that is often used with RSA encryption.
“PKCS #5: Password-Based Encryption Standard,” version
1.5, November 1993. This encryption padding is the
default.
3.0, November 18, 1996, section 5.2.3.2.
Table 2-4 Supported encryption modes (continued)
Mode Description
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#private encryption: DES(default), DESede
#public encryption: RSA
algorithm=DES
# NONE , CBC , CFB , ECB( default ) , OFB , PCBC
mode=ECB
#For key , support default key value for algorithm
#For DESede ,DES we only need to support private key
#private key value of DESede algorithm : 20b0020…
#private key value of DES algorithm: 527c2…
#for RSA algorithm, there is a key pair. You should support
private-public key pair
#private key value of RSA algorithm: 30820…
#public key value of RSA algorithm: 30819…
#private key
symmetric-key=527c23…
#public key
public-key=
META-INF/MANIFEST.MF is a text file that is included inside a JAR file to
defined in MANIFEST.MF and appends the specified dependencies to its internal
classpath.
The encryption plug-in ID is the value of the Bundle-SymbolicName property in
the manifest file for the encryption plug-in. You need to change this property
when you deploy multiple instances of the default encryption plug-in, as
described later in this chapter.
Listing 2-2 shows the contents of the default MANIFEST.MF.
Listing 2-2 Default MANIFEST.MF
Manifest-Version: 1.0
Bundle-ManifestVersion: 2
Bundle-Name: Actuate Default Security Plug-in
Bundle-SymbolicName:
com.actuate.birt.model.defaultsecurity;singleton:=true
(continues)
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Bundle-Version: 11.0.1.<version>
Require-Bundle: org.eclipse.birt.report.model,
org.eclipse.core.runtime
Export-Package: com.actuate.birt.model.defaultsecurity.api
Bundle-ClassPath: acdefaultsecurity.jar
Bundle-Vendor: Actuate Corporation
Eclipse-LazyStart: true
Bundle-Activator:
com.actuate.birt.model.defaultsecurity.properties.
SecurityPlugin
plugin.xml is the plug-in descriptor file. This file describes the plug-in to the
Eclipse platform. The platform reads this file and uses the information to
populate and update, as necessary, the registry of information that configures the
whole platform.
The <plugin> tag defines the root element of the plug-in descriptor file. The
<extension> element within the <plugin> element specifies the Eclipse extension
point that this plug-in uses, org.eclipse.birt.report.model.encryptionHelper. This
extension point requires a sub-element, <encryptionHelper>. This element uses
the following attributes:
class
The qualified name of the class that implements the interface
IEncryptionHelper. The default class name is
com.actuate.birt.model.defaultsecurity.api.DefaultEncryptionHelper.
extensionName
The unique internal name of the extension. The default extension name is jce.
isDefault
Field indicating whether this encryption extension is the default for all
encryptable properties. This property is valid only in a BIRT Report Designer
environment. When an encryption plug-in sets the value of this attribute to
true, the BIRT Report Designer uses this encryption method as the default to
encrypt data. There is no default encryption mode in Java Components.
The encryption model that BIRT uses supports implementing and using
several encryption algorithms. The default encryption plug-in is set as default
using this isDefault attribute. If you implement several encryptionHelpers, set
this attribute to true for only one of the implementations. If you implement
multiple encryption algorithms and set isDefault to true to more than one
instance, BIRT treats the first loaded encryption plug-in as the default
algorithm.
Listing 2-3 shows the contents of the default encryption plug-in’s plugin.xml.
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Listing 2-3 Default plugin.xml
<?xml version="1.0" encoding="UTF-8"?>
<?eclipse version="3.2"?>
<plugin>
<extension
id="encryption"
name="default encryption helper"
point="org.eclipse.birt.report.model.encryptionHelper">
<encryptionHelper
class="com.actuate.birt.model.defaultsecurity.api
.DefaultEncryptionHelper"
extensionName="jce" isDefault="true" />
</extension>
Deploying multiple encryption plug-ins
In some cases, you need to use an encryption mechanism other than the Data
Source Explorer default in your report application. For example, some
applications need to create an encryption mechanism using the RSA algorithm
that the default encryption plug-in supports. In this case, you must create an
additional encryption plug-in instance. For use within a BIRT Report Designer,
you can set this plug-in as the default encryption mechanism. If you change the
default encryption mechanism, you must take care when you work with old
report designs. For example, if you change an existing password field in the
designer, the designer re-encrypts the password with the current default
encryption algorithm regardless of the original algorithm that the field used.
How to create a new instance of the default encryption plug-in
1 Make a copy of the default encryption plug-in.
1 Copy the folder:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1
2 Paste the copied folder in the same folder:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
3 Rename:
\$ACTUATE_HOME/BRDPro/eclipse/plugins/Copy of
com.actuate.birt.model.defaultsecurity_11.0.1
to a new name, such as:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1_rsa
2 Modify the new plug-in’s manifest file.
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Ac t u a t e BI RT J av a Co mp o n e n t s Dev e l o p e r Gu i d e
1 Open:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1_rsa
/META-INF/MANIFEST.MF
2 Change:
Bundle-SymbolicName:
com.actuate.birt.model.defaultsecurity
to:
Bundle-SymbolicName:
com.actuate.birt.model.defaultsecurity.rsa
MANIFEST.MF now looks similar to the one in Listing 2-4.
Listing 2-4 Modified MANIFEST.MF for the new encryption plug-in
Manifest-Version: 1.0
Bundle-ManifestVersion: 2
Bundle-Name: Actuate Default Security Plug-in
Bundle-SymbolicName: com.actuate.birt.model.
defaultsecurity.rsa;singleton:=true
Bundle-Version: 11.0.1.<version>
Require-Bundle: org.eclipse.birt.report.model,
org.eclipse.core.runtime
Export-Package: com.actuate.birt.model.defaultsecurity.api
Bundle-ClassPath: acdefaultsecurity.jar
Bundle-Vendor: Actuate Corporation
Eclipse-LazyStart: true
Bundle-Activator: com.actuate.birt.model.defaultsecurity.
properties.SecurityPlugin
3 Save and close MANIFEST.MF.
3 Modify the new plug-in’s descriptor file to make it the default encryption
plug-in.
1 Open:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1_rsa
/plugin.xml
2 Change:
extensionName="jce"
to:
extensionName="rsa"
plugin.xml now looks similar to the one in Listing 2-5.
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Listing 2-5 Modified plugin.xml for the new encryption plug-in
<?xml version="1.0" encoding="UTF-8"?>
<?eclipse version="3.2"?>
<plugin>
<extension id="encryption"
name="default encryption helper"
point="org.eclipse.birt.report.model.encryptionHelper">
<encryptionHelper class="com.actuate.birt.model.
defaultsecurity.api.DefaultEncryptionHelper"
extensionName="rsa" isDefault="true" />
</extension>
</plugin>
3 Save and close plugin.xml.
4 Modify the original plug-in’s descriptor file, so that it is no longer the default
encryption plug-in.
1 Open:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1/plugin.xml
2 Change:
isDefault="true"
to:
isDefault="false"
3 Save and close plugin.xml.
5 Set the encryption type in the new plug-in to RSA.
1 Open:
\$ACTUATE_HOME/BRDPro/eclipse/plugins
/com.actuate.birt.model.defaultsecurity_11.0.1_rsa
/encryption.properties
2 Change the encryption type to public encryption:
type=public encryption
3 Change the algorithm type to RSA:
algorithm=RSA
4 Copy the pre-generated private and public keys for RSA to the
symmetric-key and public-key properties. encryption.properties now looks
similar to the one in Listing 2-6.
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Listing 2-6 Modified encryption.properties file for the new encryption
plug-in
#message symmetric encryption , public encryption
type=public encryption
#private encryption: DES(default), DESede
#public encryption: RSA
algorithm=RSA
# NONE , CBC , CFB , ECB( default ) , OFB , PCBC
mode=ECB
#For key , support default key value for algorithm
#For DESede ,DES we only need to support private key
#private key value of DESede algorithm : 20b0020e918..
#private key value of DES algorithm: 527c23ea...
#for RSA algorithm , there is key pair. you should support
#private-public key pair
#private key value of RSA algorithm: 308202760201003....
#public key value of RSA algorithm: 30819f300d0....
#private key
symmetric-key=308202760....
#public key
public-key=30819f300d0.....
5 Save and close encryption.properties.
6 To test the new default RSA encryption, open a BIRT Report Designer and
create a new report design. Create a data source and type the password.
7 View the XML source of the report design file. Locate the data source
definition code. The encryptionID is rsa, as shown in Listing 2-7.
Listing 2-7 Data source definition, showing the encryption ID
<data-sources>
<oda-data-source extensionID="org.eclipse.birt.report.
data.oda.jdbc" name="Data Source" id="6">
<text-property name="displayName"></text-property>
com.mysql.jdbc.Driver
</property>
<property name="odaURL">
jdbc:mysql://192.168.218.225:3306/classicmodels
</property>
<property name="odaUser">root</property>
36582dc88.....
</encrypted-property>
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</oda-data-source>
</data-sources>
8 Create a data set and a simple report design. Preview the report to validate
that BIRT connects successfully to the database server using the encrypted
password. Before trying to connect to the data source the report engine
decrypts the password stored in the report design using the default RSA
encryption. The engine sends the decrypted value to the database server.
Generating encryption keys
The default encryption plug-in provides classes that can be used to generate
different encryption keys. The classes’ names are SymmetricKeyGenerator and
PublicKeyPairGenerator. SymmetricKeyGenerator generates private keys, which
are also known as symmetric keys. PublicKeyPairGenerator generates public
keys. Both classes require acdefaultsecurity.jar in the classpath.
Both classes take two parameters, the encryption algorithm and the output file,
where the generated encrypted key is written. The encryption algorithm is a
required parameter. The output file is an optional parameter. If you do not
provide the second parameter, the output file is named key.properties and is
saved in the current folder. The encryption algorithm values are shown in
Table 2-6.
How to generate a symmetric encryption key
Run the main function of SymmetricKeyGenerator.
1 To navigate to the default security folder, open a command prompt window
and type:
cd C:\Program Files\Actuate11\BRDPro\eclipse\plugins
\com.actuate.birt.model.defaultsecurity_11.0.1
2 To generate the key, as shown in Figure 2-2, type:
java -cp acdefaultsecurity.jar
com.actuate.birt.model.defaultsecurity.api.keygenerator.
SymmetricKeyGenerator des
Table 2-6 Key generation classes and parameters
Class name
Encryption
algorithm parameter
com.actute.birt.model.defaultsecurity.api.
keygenerator.SymmetricKeyGenerator
des
com.actute.birt.model.defaultsecurity.api.
keygenerator.SymmetricKeyGenerator
desede
com.actute.birt.model.defaultsecurity.api.
keygenerator.PublicKeyPairGenerator
rsa
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Figure 2-2 Symmetric key generation
3 The key is generated and saved in the file, key.properties. The content of the
file looks like the following:
#Key Generator
#Wed Nov 18 16:17:06 PST 2008
symmetric-key=73c76d5…
4 Copy the key from the generated key file to encryption.properties file.
How to generate a public key with RSA encryption
Run the main function of PublicPairGenerator.
1 To navigate to the default security folder, open a command prompt window
and type:
cd C:\Program Files\Actuate11\BRDPro\eclipse\plugins
\com.actuate.birt.model.defaultsecurity_11.0.1
2 In the command prompt window, type:
java -cp acdefaultsecurity.jar
com.actuate.birt.model.defaultsecurity.api.keygenerator.
PublicPairGenerator rsa
The class generates a pair of keys saved in the key.properties file such as the
following example:
#Key Generator
#Wed Nov 18 15:58:31 PST 2008
public-key=30819f300.....
symmetric-key=3082027502010......
3 Copy the key from the generated key file to the encryption.properties file.
Deploying custom emitters
Actuate supports using custom emitters to export BIRT reports to custom
formats. The custom emitters in BIRT are implemented as plug-ins and packaged
as JAR files. To make them available to Actuate Java Components, copy the
emitters to <context-root>/WEB-INF/platform/plugins folder. Every time you
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
35
deploy a custom emitter, you need to restart the product or the product service.
This ensures the emitter JAR file is added to the classpath and the product can
discover the new rendering format.
The following products support custom emitters:
Actuate BIRT Studio
Actuate BIRT Report Designer
Actuate BIRT Report Designer Professional
Actuate Java Components:
Actuate BIRT Viewer Component
Actuate BIRT Interactive Viewer Component
Actuate BIRT Studio Component
Actuate BIRT Deployment Kit
Rendering in custom formats
After deploying the custom emitter you can see the new rendering formats
displayed along with built-in emitters in the following places:
Preview report in Web Viewer in BIRT Report Designer and BIRT Report
Designer Professional.
Export Content dialog of Actuate BIRT Viewer and Actuate BIRT Interactive
Viewer.
The following examples show the deployment and usage of a custom CSV
emitter. The emitter allows rendering a report as a comma separated file. The
custom format type is MyCSV and the JAR file name is
org.eclipse.birt.report.engine.emitter.csv.jar.
How to deploy and use a custom emitter in BIRT Report Designers
The assumption in this example is that the Actuate BIRT designers are installed in
C:\Program Files\Actuate11 folder on Windows.
1 Copy org.eclipse.birt.report.engine.emitter.csv.jar to:
C:\Program Files\Actuate11\MyClasses\eclipse\plugins
2 Open a BIRT report in BIRT Report Designer or BIRT Report Designer
Professional.
3 Preview the report in Web Viewer.
4 The new MYCSV format appears in the list of formats as shown in Figure 2-3.
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Figure 2-3 List of available formats in Web Viewer
5 Select the MYCSV option. A file download dialog box appears as shown on
Figure 2-4. Select Save to save the file. The default file name is iv.mycsv. You
have an option to rename the file when saving it. The report content is
exported to the new format.
Figure 2-4 Open/Save exported content
How to deploy and use a custom emitter in Actuate Java Components
The assumption in this example is that the Java Components are deployed to
Apache Tomcat 6.0, and are installed in C:\Program Files\Apache Software
Foundation\Tomcat 6.0 folder on Windows.
1 Copy org.eclipse.birt.report.engine.emitter.csv.jar to:
C:\Program Files\Apache Software Foundation\Tomcat 6.0\webapps
\ActuateJavaComponent\WEB-INF\platform\plugins
2 Restart Apache Tomcat from Start➛Settings➛Control Panel➛Administrative
Tools➛Services as shown in Figure 2-5.
Chapt er 2, Depl oyi ng Act uat e BI RT repor t s usi ng an Act uat e Java Component
37
Figure 2-5 Restarting the Apache Tomcat Service
3 Open a BIRT report in Actuate BIRT Viewer or Interactive Viewer.
4 Select Export Content from the viewer menu.
5 The new MyCSV format shows up in the Export Formats, as shown in
Figure 2-6.
Figure 2-6 Export Content in Actuate BIRT Viewers
6 Choose OK. A file download dialog box appears as shown on Figure 2-4.
Select Save to save the file. | {"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": 0.902961790561676, "perplexity": 395.83442430998224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866511.32/warc/CC-MAIN-20180524151157-20180524171157-00209.warc.gz"} |
http://math.stackexchange.com/questions/190674/showing-the-weyl-element-is-sent-to-its-negative-by-a-certain-element-of-the-wey | # Showing the Weyl element is sent to its negative by a certain element of the Weyl group
Let $\mathfrak g$ be a complex semi-simple Lie algebra with a choice of positive root system. Let $\rho$ be the Weyl element, i.e. the sum of the fundamental weights (or half the sum of the positive roots). There is a unique element $w_0$ of the Weyl group that sends the positive Weyl chamber to the negative one. From the classification of such algebras, I can see that $w_0 \rho = -\rho$. However, what is a simple (i.e. short) proof of this?
-
The number of positive roots sent to negative roots by $w \in W$ is equal to the length, $\ell(w)$, of $w$. In the case of $w_0$, the length $\ell(w_0)$ is the maximal value, equal to the total number of positive roots.
Use the definition of $\rho$ as half the sum of all positive roots and this fact to conclude that $w_0 \rho = -\rho$. | {"extraction_info": {"found_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": 0.9762638807296753, "perplexity": 82.69969592514407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737958671.93/warc/CC-MAIN-20151001221918-00045-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2963630/laplace-transform-of-vector-valued-lti-system | # Laplace Transform of Vector Valued LTI system [closed]
I'm not really interested in proofs, but I have a LTI system that is vector valued and want to take the laplace transform of it and the inverse laplace transform of the transfer function. I'm wondering how to do this with vectors? A table of vector valued Laplace transforms would be nice too if you can find. I could not. Thanks.
Edit: In particular, I am wondering more about the z-transform. Mainly you have a matrix-valued transfer function $$H(z)$$. How mechanically do you perform the contour integral of $$H(z)$$ to get $$h(t)$$. In particular, how to translate the z-transform tables for vectors and (although I can look this up) how to perform the partial fractions approach for vector-valued functions in the numerator and denominator. One example illustrating what to do at every step starting with a vector-valued transfer function and ending up with $$h(t)$$ would probably be the most helpful.
## closed as unclear what you're asking by Mark Viola, user10354138, Cesareo, Lord Shark the Unknown, max_zornOct 21 '18 at 1:44
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
• For $n=2$ and $L$ a LTI system, if you know $L[(\delta,0)]$ and $L[0,\delta]$ ($\delta$ the Dirac delta, or the Kronecker delta in the discrete case) or equivalently their Laplace transforms (Z-transform in the discrete case), then you know $L[ f]$ for every $f$. The (inverse) Laplace or Z-transform is entry-wise and works as usual, except that $Y(s) = H(s) X(s)$ becomes $Y(s) = H(s) \times X(s)$ – reuns Oct 22 '18 at 0:28
For $$f : \mathbb{R} \to \mathbb{C}^{n }$$ and $$h : \mathbb{R} \to \mathbb{C}^{n \times n}$$ some vector and matrix valued functions both in $$L^1$$ then $$h\ast f(t) = \int_{-\infty}^\infty h(\tau)\times f(t-\tau) d\tau$$ where $$h(\tau)\times f(t-\tau)$$ is the multiplication of a vector by a matrix, and all the LTI systems are of the form $$f \mapsto h \ast f$$ for some matrix valued distribution $$h$$.
For $$f,h$$ causal, their Laplace transform is as usual and they obey $$\mathcal{L}[h \ast f](s) =\mathcal{L}[h](s)\times\mathcal{L}[f](s)$$
Proof : the entries $$(h \ast f)_{k} = \sum_{l=1}^n f_l \ast h_{k,l}$$ are complex valued functions so their Laplace transform obey $$\mathcal{L}[(h \ast f)_{k}](s) = \sum_{l=1}^n\mathcal{L}[f_l](s)\mathcal{L}[h_{k,l}](s)=\sum_{l=1}^n\mathcal{L}[f]_l(s)\mathcal{L}[h]_{k,l}(s)$$. The columns of $$h$$ are obtained from plugging $$(\Delta_j)_j(t) =\delta(t)$$, $$(\Delta_j)_l(t) = 0$$ in the LTI system | {"extraction_info": {"found_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": 18, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.891024649143219, "perplexity": 230.89496449595885}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256763.42/warc/CC-MAIN-20190522043027-20190522065027-00236.warc.gz"} |
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## 1. chinaXiv:201605.01781 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
We evaluate the top-quark FCNC productions induced by the topcolor-assisted technicolor (TC2) model at the LHC. These productions proceed, respectively, through the parton-level processes gg -> t (c) over bar, cg -> t, cg -> tg, cg -> tZ, and cg -> t gamma. We show the dependence of the production rates on the relevant TC2 parameters and compare the results with the predictions in the minimal supersymmetric model. We find that for each channel the TC2 model allows for a much larger production rate than the supersymmetric model. All these rare productions in the TC2 model can be enhanced above the 3 sigma sensitivity of the LHC. Since in the minimal supersymmetric model only cg -> t is slightly larger than the corresponding LHC sensitivity, the observation of these processes will favor the TC2 model over the supersymmetric model. In case of unobservation, the LHC can set meaningful constraints on the TC2 parameters.
## 2. chinaXiv:201605.01780 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
In split supersymmetry, gauginos and higgsinos are the only supersymmetric particles possibly accessible at foreseeable colliders like the CERN Large Hadron Collider (LHC) and the International Linear Collider (ILC). In order to account for the cosmic dark matter measured by WMAP, these gauginos and higgsinos are stringently constrained and could be explored at the colliders through their direct productions and/or virtual effects in some processes. The clean environment and high luminosity of the ILC render the virtual effects at percent level meaningful in unraveling the new physics effects. In this work we assume split supersymmetry and calculate the virtual effects of the WMAP-allowed gauginos and higgsinos in the Higgs productions e(+) e(-) -> Zh and e(+) e(-) ->nu(e)(nu) over bar (e)h through WW fusion at the ILC. We find that the production cross section of e+ e-. Zh can be altered by a few percent in some part of the WMAP-allowed parameter space, while the correction to the WW fusion process e(+) e(-) ->nu(e) (nu) over bar (e)h is below 1%. Such virtual effects are correlated with the cross sections of chargino pair productions and can offer complementary information in probing split supersymmetry at the colliders.
## 3. chinaXiv:201605.01779 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
We examine the constraints on F-term hybrid inflation by considering the flat directions in the Minimal Supersymmetric Standard Model (MSSM). We find that some coupling terms between the flat direction fields and the field which dominates the energy density during inflation are quite dangerous and can cause the no-exit of hybrid inflation even if their coupling strength is suppressed by Planck scale. Such couplings must be forbidden by imposing some symmetry for a successful F-term hybrid inflation. At the same time, we find that in the D-term inflation these couplings can be avoided naturally. Further, given the tachyonic preheating, we discuss the feasibility of Affleck-Dine baryogenesis after the F-term and D-term inflations. (c) 2006 Elsevier B.V. All rights reserved.
## 4. chinaXiv:201605.01778 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
In the minimal supersymmetric standard model the R-parity violating interactions can induce anomalous top pair productions at the LHC through the t-channel process d(R)(R)((d) over bar) -> t(L)(L)((t) over bar) by exchanging a slepton or by the u-channel process d(R)(R)((d) over bar) -> t(R)(R)(<(t)over bar) exchanging a squark. Such top pair productions with a certain chirality cause top-quark polarization in the top pair events. We found that at the LHC, due to the large statistics, the statistical significance of the polarization observable, and thus the probing ability for the corresponding R-parity violating couplings, is much higher than at the Tevatron upgrade.
## 5. chinaXiv:201605.01777 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
We split the two-Higgs-doublet model by assuming very different vevs for the two doublets: the vev is at weak scale (174 GeV) for the doublet Phi(1) and at neutrino-mass scale (10(-2) - 10(-3) eV) for the doublet Phi(2). Phi(1) is responsible for giving masses to all fermions except neutrinos; while Phi(2) is responsible for giving neutrino masses through its tiny vev without introducing the see-saw mechanism. Among the predicted five physical scalars H, h, A(0) and H-+/-, the CP-even scalar h is as light as 10(-2) - 10(-3) eV while the others are at weak scale. We identify h as the cosmic-dark-energy field and the other CP-even scalar H as the Standard Model Higgs boson; while the CP-odd A(0) and the charged H-+/- are the exotic scalars to be discovered at future colliders. Also we demonstrate a possible dynamical origin for the doublet Phi(2) from neutrino condensation caused by some unknown dynamics.
## 6. chinaXiv:201605.01776 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
In split supersymmetry, gauginos and Higgsinos are the only supersymmetric particles that are potentially accessible at soon-to-be-completed colliders. While direct experimental research, such as the LEP and Tevatron experiments, have given robust lower bounds on the masses of these particles, cosmic dark matter can give some upper bounds and thus have important implications for research at future colliders. In this work we scrutinize such dark matter constraints and show the allowed mass range for charginos and neutralinos (the mass eigenstates of gauginos and Higgsinos). We find that the lightest chargino must be lighter than about 1 TeV under the popular assumption M-1 = M-2/2 and about 2 or 3 TeV in other cases. The corresponding production rates of the lightest chargino at the CERN large hadron collider (LHC) and the International Linear Collider (ILC) are also given. While in some parts of the allowed region the chargino pair production rate can be larger than 1 pb at the LHC and 100 fb at the ILC, other parts of the region correspond to very small production rates, and thus there is no guarantee of finding the charginos of split supersymmetry at future colliders.
## 7. chinaXiv:201605.01775 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
In split-supersymmetry (split-SUSY), gluino is a metastable particle and thus can freeze out in the early universe. The late decay of such a long-life gluino into the lightest supersymmetric particle (LSP) may provide much of the cosmic dark-matter content. In this work, assuming the LSP is gravitino produced from the late decay of the metastable gluino, we examine the Wilkinson microwave anisotropy probe (WMAP) dark-matter constraints on the gluino mass. We find that to provide the full abundance of dark matter, the gluino must be heavier than about 14 TeV and thus not accessible at the CERN large hadron collider (LHC).
## 8. chinaXiv:201605.01774 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
If all the supersymmetry particles (sparticles) except a light Higgs boson are too heavy to be directly produced at the Large Hadron Collider and Tevatron, a possible way to reveal evidence for supersymmetry is through their virtual effects in other processes. We examine such supersymmetric QCD effects in bottom pair production associated with a light Higgs boson at the Large Hadron Collider and Tevatron. We find that if the relevant sparticles (gluinos and squarks) are well above the TeV scale, too heavy to be directly produced, they can still have sizable virtual effects in this process. For large tan beta, such residual effects can alter the production rate by as much as 40%, which should be observable in future measurements of this process.
## 9. chinaXiv:201605.01773 [pdf]
Subjects: Physics >> The Physics of Elementary Particles and Fields
In the minimal supersymmetric standard model extended by including right-handed neutrinos with seesaw mechanism, the neutrino Yukaka couplings can be as large as the top-quark Yukawa couplings and thus the neutrino/sneutrino may cause sizable effects in Higgs boson self-energy loops. Our explicit one-loop calculations show that the neutrino/sneutrino effects may have an opposite sign to top/stop effects and thus lighten the lightest Higgs boson. If the soft-breaking mass of the right-handed neutrino is very large (at the order of Majorana mass scale), such as in the split-supersymmetry (SUSY) scenario, the effects can lower the lightest Higgs boson mass by a few tens of GeV. So the Higgs mass bound of about 150 GeV in split-SUSY may be lowered significantly if right-handed neutrinos come into play with seesaw mechanism. | {"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": 0.9301257729530334, "perplexity": 2068.226027813053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178361808.18/warc/CC-MAIN-20210228235852-20210301025852-00043.warc.gz"} |
https://physicscomputingblog.com/2017/01/14/nondimensionalization-and-characteristic-scales-in-physics/ | # Nondimensionalization and characteristic scales in physics
Posted by
### 1 Introduction
When doing calculations in computational physics, it is usually necessary to convert equations to a nondimensional form, as a computer almost always works with dimensionless numbers. The most simple way to do this would probably be to equate the SI units (kilogram, meter, second…) to 1. This choice of scale, however, is not always appropriate as the interesting features of the dynamics of the system can take place at lengths and time intervals that are very small or large compared to these units. Plots and tables of the computational results are easier to read when the values in them are numbers that don’t differ from 1 by too many orders of magnitude. Examples of different choices of scales are the use of natural units (where we set c = 1 and ћ=1) in high-energy physics, and atomic units in quantum chemistry, where an appropriate choice of length unit is the Bohr and that of energy is Hartree.
### 2 Simple harmonic motion
As an example of a classical mechanical system with a characteristic timescale, consider a 1D Hookean oscillator with spring constant k and mass M. The equation of motion for the single position variable x(t) is
$M \frac{d^2 x}{dt^2} = -kx$ . (1)
The obvious choice of typical timescale in this case is of course the period of oscillation
$T = 2 \pi \sqrt{\frac{M}{k}}$ , (2)
and the corresponding nondimensional time variable would be $\tilde{t}=t/T$. This scale could also be found by simple dimensional analysis, forming a product of powers of the parameters M and k, and requiring that the product has dimensions of time:
$[M]^{\alpha}[k]^{\beta} = kg^\alpha \times \frac{kg^\beta}{s^{2\beta}}=kg^{\alpha + \beta}s^{-2\beta} = s$ (second) . (3)
Solving this for alpha and beta gives us a characteristic time $M^{1/2}k^{-1/2}$, which is the period T divided by 2π and is of the same order of magnitude as T. If we want to integrate the equation of motion numerically with the finite-difference method, we would probably choose the discrete time step to be at least about 10-15 times smaller than T, and when plotting the trajectory x(t) we would usually choose a range of t-coordinate that is only a few periods long, no matter what the initial position x(0) and velocity x’(0) of the oscillating body are.
On the other hand, the motion of the oscillator described above does not have a general characteristic length scale. If we have two trajectories where in the first one the total energy (kinetic + potential) of the oscillator is 1 Joule and in the second one it’s 100 Joules, it is not appropriate to use the same range of x-coordinate when plotting the functions x(t), because in the latter case the amplitude of the trajectory is 10 times larger than in the former. This fact can also be seen from a simple dimensional analysis, by noting that there is no product of powers of M and k that has dimension of length:
$[M]^{\alpha}[k]^{\beta} \neq m$ (meter), for any $\alpha, \beta$ .
### 3 Linear motion with damping
As another simple mechanical system that has a characteristic timescale, consider linear 1D motion with damping (air resistance or friction). The equation of motion is
$M\frac{d^2 x}{dt^2}=-b\frac{dx}{dt}$, (4)
where the damping parameter b has dimensions of mass/time. Immediately we can see that the variable $M/b$ has dimensions of time, but we don’t know its physical significance yet. Solving the DE for x(t) and its time derivative, we obtain:
$x(t) = x_0 + \frac{Mv_0}{b} - \frac{Mv_0}{b}e^{-b(t-t_0 )/M}$ (5)
$x'(t) = v_0 e^{-b(t-t_0 )/M} = v_0 2^{-\frac{b(t-t_0 )}{M \log 2}}=v_0 2^{-\frac{t-t_0 }{t_{1/2}}}$ . (6)
where $x_0$ and $v_0$ are the position and velocity at time $t = t_0$, and
$t_{1/2}=\frac{M\log 2}{b}$ (7)
is the half-life of the velocity.
Usually when some variable decays exponentially with some half-life, be it the number of radioactive atoms in a sample or the amount of a foreign substance (medication, environmental toxin) in human body, we can say that for our purposes the variable has practically become zero after about 6-10 half-lifes. Therefore the parameter $t_{1/2}$, or the related parameter $M/b$ (which we obtained by dimensional analysis) can be seen as the characteristic timescale for the motion in this system. Again, there is no single characteristic length for the system, as the distance the object travels from the initial position $x_0$ in some number of half-lives is linearly proportional to the initial velocity $v_0$.
### 4 A particle inside rigid box
What about mechanical systems where the motion happens in some characteristic length scale? The simplest example of this kind of behavior is 1D motion of an object that has been confined in some interval [0, L] of the x-axis by impenetrable walls at points x = 0 and x = L. This can be described by saying that the potential energy of the object is zero when 0 < x < L and it is infinite when x < 0 or x > L. In quantum mechanics this model system is called ”particle in a box”, but here we consider only the classical mechanical equivalent.
If the collisions of the object with the walls are elastic, the trajectory x(t) is a sawtooth function like the one plotted below:
Figure 1. The periodic classical trajectory of a 1D particle in a rigid box.
Now the interval the object moves in is always [0,L] which doesn’t depend on its initial position and velocity (as long as the initial position is inside the box and the initial velocity is non-zero). Therefore the obvious choice of length scale for the system is L. However, this system does not have a characteristic time scale, as the period of the motion is $2L/v_0$ which depends on the initial conditions.
### 5 A system with both a characteristic length and time
How, then, would one construct a system where there exist characteristic scales in both length and time? By dimensional analysis, we can expect that the set of parameters $a,b,c,...$ in the equation of motion should be such that some product of their powers, $a^{\alpha} b^{\beta} c^{\gamma} \dots$ , has dimensions of time and some other similar product has dimensions of length.
One way to make such a system is to form an equation of motion that contains both a Hookean returning force and a term that depends on both position and velocity:
$M\frac{d^2 x}{dt^2} = -b x \left( \frac{dx}{dt} \right)^2 - kx$ (8)
Now we can deduce that the parameter b needs to have dimensions $[mass][length]^{-2}$. A parameter formed from M, b and k that has dimensions of length is
$L = \sqrt{\frac{M}{b}}$ (9)
and a characteristic time is the familiar
$T = \sqrt{\frac{M}{k}}$, (10)
which already appeared in the case of simple harmonic motion. From these we can form dimensionless length and time variables
$\tilde{x} = \frac{x}{L}$, $\tilde{t} = \frac{t}{T}$, (11)
and written using these variables, the equation of motion becomes
$\frac{d^2 \tilde{x}}{d\tilde{t}^2} = -A x \left( \frac{d\tilde{x}}{d\tilde{t}} \right)^2 - B\tilde{x}$ , (12)
where A and B are dimensionless numbers defined by:
$A = \frac{bL^2}{M}$, $B = \frac{k T^2}{M}$.
So how do the trajectories x(t) of this system look like? The equation of motion can’t be solved analytically, but we can do a numerical solution with Mathematica, Matlab or some other program capable of that kind of calculations. Setting the values of the parameters to $A=0.5$ and $B=1$, and using the initial position $\tilde{x}(0) = 0$ and a set of different initial velocities $x'(0) = 0$, $10$, $100$ or $10000$, the following graphs are obtained:
Figure 2. Trajectories of a particle in a velocity-dependent force field for initial dimensionless velocities of 1, 10, 100 or 10000.
From the plots in Fig. 2., it is obvious that the amplitude and period of the trajectory stay in the same orders of magnitude for a very large range of initial velocities/energies of the moving object. A $10^4$ – fold increase in the dimensionless velocity causes only an about 6-fold increase in amplitude and an about 4-fold increase in frequency. The amplitude seems to increase only logarithmically with increasing velocity, i.e. slower than its any fractional power.
Note, however, that if the initial kinetic energy of the particle at $t = 0$ is small enough, the velocity-dependent term in the equation of motion can be neglected, and the system behaves like a simple harmonic oscillator for which the amplitude is directly proportional to the energy. To see the more interesting properties of this system, as in the graphs above, the energy needs to be large enough. In physics there are many situations where a system behaves differently in different energy scales, one example being a collision of atomic nuclei which can be described with simple Coulombic repulsion at low collision energies but will involve the possibility of nuclear fusion at larger energies where the particles can overcome the repulsive electrostatic force. The classical mechanical system here is a very simple toy model of this kind of behavior.
### 6 Applications
Many problems of applied physics where characteristic lengths and times and dimensionless quantities appear are related to the physics of fluid flow. A familiar dimensionless number that describes fluid motion is the Reynolds number (Re), which is related to the probability of turbulence occurring in a fluid system. Others include the capillary number (denoted Ca) and Eötvös number (denoted Eo), which describe the significance of surface tension in relation to other forces.
When a way to calculate the relevant length scales for a particular system are known, it is easier to choose a sufficiently fine discretization and a sufficiently large computational domain (intervals of the x- and t-axes) when integrating equations of motion numerically. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 40, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9064041376113892, "perplexity": 290.9808370485374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583879117.74/warc/CC-MAIN-20190123003356-20190123025356-00143.warc.gz"} |
http://httpa.academickids.com/encyclopedia/index.php/Continuum_hypothesis | # Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:
There is no set whose size is strictly between that of the integers and that of the real numbers.
Or mathematically speaking, noting that the cardinality for the integers [itex]|\mathbb{Z}|[itex] is [itex]\aleph_0[itex] ("aleph-null") and the cardinality of the real numbers [itex]|\mathbb{R}|[itex] is [itex]2^{\aleph_0}[itex], the continuum hypothesis says:
[itex]\not\exists \mathbb{A}: \aleph_0 < |\mathbb{A}| < 2^{\aleph_0}.[itex]
The real numbers have also been called the continuum, hence the name. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis.
Contents
## The size of a set
To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection [itex]S \leftrightarrow T[itex]. Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}.
With infinite sets such as the set of integers or rational numbers, things are more complicated to show. Consider the set of all rational numbers. One might naively suppose that there are more rational numbers than integers, and fewer rational numbers than real numbers, thus disproving the continuum hypothesis. However, it turns out that the rational numbers can be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size as the set of integers: they are both countable sets. Cantor's diagonal argument shows that the integers and the continuum do not have the same cardinality.
The continuum hypothesis states that every subset of the continuum (= the real numbers) which contains the integers either has the same cardinality as the integers or the same cardinality as the continuum.
## Investigating the continuum hypothesis
If a set S were found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between S and the set of integers, because there would always be elements of set S that were "left over". Similarly, it would be impossible to make a one-to-one correspondence between S and the set of real numbers, because there would always be real numbers that were "left over".
## Impossibility of proof and disproof
Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. It became the first on David Hilbert's list of important open questions that was presented at the International Mathematical Congress in the year 1900 in Paris.
Kurt Gdel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory, even if the axiom of choice is adopted. Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is independent of the Zermelo-Fränkel axiom system and of the axiom of choice. Both of these results assume that the Zermelo-Fränkel axioms themselves do not contain a contradiction, which is widely believed to be true.
As such it is not surprising that there should be statements which cannot be proven nor disproven within a given axiom system; in fact the content of Gdel's incompleteness theorem is that such statements always exist if the axiom system is strong enough and without contradictions. The independence of CH was still unsettling however, because it was the first concrete example of an important, interesting question of which it could be proven that it could not be decided either way from the universally accepted basic system of axioms on which mathematics is built.
The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory. As a result of its independence, many substantial conjectures in those fields have subsequently been shown to be independent as well.
It is interesting to note that Gdel believed strongly that CH is false. To him, his independence of proof only showed that the prevalent set of axioms was defective. Gdel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Nowadays, most researchers in the field are either neutral or reject CH. Generally speaking, mathematicians who favor a "rich" and "large" universe of sets are against CH, while those favoring a "neat" and "controllable" universe favor CH. Chris Freiling in 1986 presented an argument against CH: he showed that the negation of CH is equivalent to a statement about probabilities which he calls "intuitively true", but others have disagreed.
## The generalized continuum hypothesis
The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S: there are no in-betweens. This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers. GCH is also independent of the Zermelo-Frnkel set theory axioms, and also of the axiom of choice.
## References
• Nancy McGough.: The Continuum Hypothesis (http://www.ii.com/math/ch/).
• Cohen, P. J.: Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966.
• Gdel, K.: The Consistency of the Continuum-Hypothesis. Princeton, NJ: Princeton University Press, 1940.
• Gdel, K.: What is Cantor's Continuum Problem?, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gdel's arguments against CH.
• H. G. Dales and W. H. Woodin: An Introduction to Independence for Analysts. Cambridge (1987).
• Chris Freiling: Axioms of Symmetry: Throwing Darts at the Real Number Line, Journal of Symbolic Logic, Volume 51 (1986), Issue 1, pp. 190-200.de:Kontinuumshypothese
• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy | {"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": 0.9419476985931396, "perplexity": 382.3347930703346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1502886105086.81/warc/CC-MAIN-20170818175604-20170818195604-00501.warc.gz"} |
http://ayotzinapasomostodos.com/lib/sphere.htm | # sphere
Sphere
A three dimensionalsolid consisting of all points equidistant from a given point. This point is the center of the sphere. Note: All cross-sections of a sphere are circles. | {"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": 0.8422960638999939, "perplexity": 342.83193621651475}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655912255.54/warc/CC-MAIN-20200710210528-20200711000528-00485.warc.gz"} |
https://geologyuniverse.com/stress-strain-relationship-of-elastic-and-plastic-material/ | # Stress strain relationship of Elastic and Plastic material.
Stress is the force applied to an object. If we imagine a vertical column of material, along any imaginary horizontal plane within this column, the material above the plane because of its weight pushes downward on the material below the plane. Similarly the part of column below the plane pushes upward with an equal force on the material above the plane. This mutual action and reaction along the surface constitutes the Stress. Here the imaginary plane may be horizontal, vertical, or inclined.
Strain may be defined as the deformation caused by the stress. Strain may be dilation which is a change in the volume or distortion which is a change in the form or both.
## Stress strain relationship of Elastic and Plastic material:
When a body is subjected to directed forces for a short period of time then the body usually passes through three stages of deformation. At first, the deformation is elastic; that is, if the stress is withdrawn then the body returns to its original shape and size. If the stress exceeds the elastic limit than the deformation is plastic; that is, the specimen only partially returns to its original shape even if the stress is removed. When there is a continued increase in the stress, one or more fractures develop and the specimen eventually fails by rupture.
### ELASTICDEFORMATION
At the room temperature and pressure and under stress, the most brittle rocks behave elastically until they fail by rupture. For such rocks the elastic limit or yield point is the stress at rupture.
If a solid cylinder of rock is subjected to stress parallel to its long axis, it will lengthen under tension and shorten under compression. The ratio of the stress to the deformation is a measure of the property of the rock to resist deformation.
E= σ / ɛ …………………………………………………………………………………… (1)
Where E is the Young’s Modulus (also called modulus of elasticity), σ is stress and ɛ is strain.
ɛ =Δ l / lo …………………………………………………………………………………. (2)
Δ l is the change in length, lo is the original length.
Under tension the diameter of a cylinder subjected to tension parallel to the axis becomes
Smaller; under compression parallel to the axis the diameter becomes greater. Poisson’s ratio
is the ratio of transverse strain to the axial strain.
υ = Δ d / do / Δ l / lo ……………………………………………………………………….(4)
where υ is Poisson’s ratio, Δ d is change in diameter, do is the original diameter.
If the diameter of the cylinder is decresed by 0.00025 cms, poisson’s ratio is 0.25; this is a
good average of rocks. Rigidity measures the resistance to change in shape i.e. ratio of shear stress to shear strain.
……………………………………………..……………………………… (5)
Where G is rigidity modulus
…..……………………………………..………………………………(6)
Figure 1 Rigidity Square acdf is deformed into parallelogram bcef.
The bulk modulus or incompresisibility is
K = Δ h Δ V / V
Where K is the bulk modulus, Δ h is the change in hydrostatic pressure, Δ V is the change in
volume, V is the original volume.
Elastic deformation is primarily of importance in analyzing the tidal deformation of solid
earth and in investigating the transmission of seismic waves through the earth. It is of even
more direct significance to structural geology in studying the elastic rebound associated with
Earthquakes in the fracturing of rocks to produce joints and faults.
### Plastic Deformation:
Most rocks at room temperature and pressure fail by rupture before attempting the stage of
plastic deformation, most rocks at significantly high temperature and confining pressure
deform plastically . This plastic deformation is not recoverable. That is, if the stress is
removed the material does not return it its original shape.
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http://math.stackexchange.com/questions/411690/rectangularizing-the-square | # rectangularizing the square
There is a square that I want to divide to n people, such that each person gets a rectangular piece with an equal area.
An obvious option is to cut 1-by-n rectangles of size n-by-1, but the people don't want such rectangles, they say they are too skinny. They want to get R-balanced rectangles, which are defined as axis-aligned rectangles whose width-height ratio is between $R$ and $1/R$ (where $R \geq 1$).
What is the minimum R for which I can guarantee that, for every n, there is a division of a square to n equal-area R-balanced rectangles?
Some notes and sub-questions:
• A 1-balanced rectangle is just a square. So, for $R=1$, a division is possible only when n is a square number.
• For $R=2$, I managed to find a division for $n=1, 2, 4, 5, 6, 7, 8, 9$, but I haven't found a division for $n=3$, and also haven't managed to prove that it is impossible. What do you think?
• What are all the numbers $n$ for which there is a division of a square into $n$ 2-balanced rectangles? Obviously all square and double-square numbers are included, but there are other numbers, such as 5, 6 and 7.
• For $R=3$, I haven't found a counter-example, but also couldn't prove that for every n it is possible to find a division to 3-balanced rectangle. What do you think?
(Note that Erich's packing center contains a nice summary of packing rectangles in a square, but it is limited to identical rectangles. I allow different rectangles, as long as they are the same area, and with an R-balanced width-height ratio).
-
A possible way to divide a square into rectangles is to first divide it into shelves (horizontal strips), and then divide each shelf into identical adjacent rectangles. In order to have rectangles of equal area, the number of rectangles in each shelf should be proportional to the height of the shelf. For concreteness, if the total number of rectangles is $N$, and the square side length is $L$, and the height of a particular shelf is $h$, then the number of rectangles in this shelf should be $Nh/L$ (h must be chosen so that this is a whole number). Thus the width of each rectangle is $L^2/Nh$, and the height/width ratio is $Nh^2/L^2 = N(h/L)^2$.
In order to have a balanced width/height ratio, we should make this number as close to 1 as possible. Therefore we should choose values of $h$ such that $L/h$ is as close as possible to $sqrt(N)$. If $N$ is a square number, then the division is trivial. So let's assume N is not a square number, and define:
• $S=sqrt(N)$,
• $S_-=floor(S)$,
• $S_+=ceiling(S) = S_-+1$,
• $h_-=L S_- / N$,
• $h_+=L S_+ / N$,
It is always possible to write $N$ as a sum of several $S_-$'s and several $S_+$'s. Similarly, it is possible to divide the square to several shelves of height $h_-$ and several shelves of height $h_+$'s. The height/width ratios in the two types of shelves are:
• $N(h_-/L)^2 = S_-^2/N$
• $N(h_+/L)^2 = S_+^2/N$
By the definition of S, the following relations hold:
• $S-1 < S_- < S < S_+ < S+1$
• $(S^2-2S+1) < S_-^2 < S^2=N < S_+^2 < (S^2+2S+1)$
• $(1-2S/N+1/N) < S_-^2/N < 1 < S_+^2/N < (1+2S/N+1/N)$
Thus, the height/width ratios are bounded by $(1-2S/N+1/N)$ and $(1+2S/N+1/N)$. These numbers become closer to 1 as N increases. Therefore, it is enough to find a threshold N for each R. Also, it is enough to find a threshold for the lower bound, since the upper bound is always geometrically closer to 1. Here are some examples:
• For $N=6$, the lower bound is 0.35+, which is more than 1/3. Therefore, for all $N \geq 6$, it is possible to rectangularize a square to N equal-area 3-balanced rectangles. It is easy to check that this is possible also for $N<6$. Thus we have proven that For all N, it is possible to rectangularize a square to N equal-area 3-balanced rectangles.
• For $N=12$, the lower bound is 0.51-, which is more than 1/2. Therefore, for all $N \geq 12$, it is possible to rectangularize a square to N equal-area 2-balanced rectangles. I leave the cases of $N=10$, $N=11$ as an exercise to the readers :-)
• For $N=30$, the lower bound is 0.668+, which is more than 2/3. Therefore, for all $N \geq 30$, it is possible to rectangularize a square to N equal-area 1.5-balanced rectangles.
• And so on...
To complete the answer, here is a proof that this is not possible to rectangluarize a square to 3 equal-area R-balanced rectangles with $R<3$.
The proof is by contradiction. Suppose a square is divided to 3 equal-area R-balanced rectangles with $R<3$. Then none of the rectangles can touch two opposite sides of the square (otherwise its height would be more than $L/3$, and its area would be more than $L^2/3$). Then all square sides must be touched by at least two rectangles, and each rectangle may touch at most 2 square sides. Then the number of rectangles must be at least as large as the number of square sides. Thus $3 \geq 4$, a contradiction.
Thus, have shown that, the smallest $R \geq 1$ such that, For all N, it is possible to rectangularize a square to N equal-area R-balanced rectangles, is 3.
Thus, the answer to my question is: 3.
I love questions that can be answered with a single digit :-)
-
I'm awarding this answer a +100 bounty because I cannot award two bounties for answers to this other question. There will be a 24 hour delay before the bounty is awarded. – Jim Belk Nov 10 '13 at 1:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9331120848655701, "perplexity": 138.43185913869334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1432207928965.67/warc/CC-MAIN-20150521113208-00041-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://quantumtheology.blogspot.com/2007/12/attractive-fixed-points.html | ## Tuesday, December 11, 2007
### Attractive Fixed Points
Math Man specializes in the mathematics of dynamical systems, in chaos theory. I dabble in chaos too, just in its embodied form, rather than the theoretical. Chaotic systems are not random, though they might appear to be. Given a particular set of starting conditions (two sons, one cat, teaching two classes, a spouse on leave), the unfolding of the system is completely determined (a December calendar that requires 5 colors to keep track of everyone's obligations).
Not every system ultimately leads to chaos (where every possible state is eventually experienced), some eventually arrive at an equilibrium state - called an attractive fixed point. I think I'm approaching a fixed point tomorrow, though I'm not finding it all that attractive personally! All obligations (musicals, concerts, dinners, rehearsals, auditions) are being sucked in to the 6 hours between 3 pm and 9 pm on December 13th. Some fixed points have basins around them, where the conditions all lead to the same end point (exhaustion?). Others find a new fixed point to hone in on with a subtle change in conditions. So what will the potential ice storm do to my spiral?
The spirals above are three related species in a damped oscillating chemical reaction.
The Jesuits hold that you can find God in all things, presumably even chaos. I note that ergodic is another term for chaotic. QED, or perhaps I should say AMDG? | {"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": 0.8338485956192017, "perplexity": 1693.2092296995042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368711240143/warc/CC-MAIN-20130516133400-00034-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/geometry/CLONE-68e52840-b25a-488c-a775-8f1d0bdf0669/chapter-10-section-10-6-the-three-dimensional-coordinate-system-exercises-page-483/43 | ## Elementary Geometry for College Students (6th Edition)
$P = (5,6,5)$
$l_1: (x.y.z) = (2,3,-1)+n(1,1,2)$ $l_2: (x.y.z) = (7,7,2)+r(-2,-1,3)$ Let $P = (x,y,z)$. The x-coordinate of $l_1$ and $l_2$ must both equal $x$: $x = 2+n(1) = 7+r(-2)$ $n = 5-2r$ The y-coordinate of $l_1$ and $l_2$ must both equal $y$: $y = 3+n(1) = 7+r(-1)$ $n = 4-r$ We can equate the two expressions of $n$ to find $r$: $5-2r = 4-r$ $r = 1$ We can use $l_2$ to find $P$: $P = (x,y,z) = (7,7,2)+r(-2,-1,3)$ $P = (7,7,2)+(1)(-2,-1,3)$ $P = (7-2,7-1,2+3)$ $P = (5,6,5)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9563626646995544, "perplexity": 120.42787177136319}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178383355.93/warc/CC-MAIN-20210308082315-20210308112315-00541.warc.gz"} |
https://www.cdslab.org/recipes/programming/logic-functions-operations/logic-functions-operations | Show that the following functions,
can be written as,
where the basis logic functions have the following truth table, | {"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": 0.9919070601463318, "perplexity": 1080.5267729279064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710691.77/warc/CC-MAIN-20221129100233-20221129130233-00296.warc.gz"} |
http://spmphysics.onlinetuition.com.my/2013/07/finding-wavelength-from-diagram.html | # Finding Wavelength from Diagram
### Finding wavelength from diagram
#### Transverse Wave
Wavelength is the distance between two successive crest or trough.
#### Longitudinal Wave
Wavelength is the distance between two successive compression or rarefaction.
#### Wave front diagram
Wavelength is the distance between two successive wave front
Example 1:
Figure above shows the propagation of a water wave. What is the amplitude of the wave?
$Amplitude= 10cm 2 =5cm$
Example 2 :
The figure above shows a transverse wave. The wavelength of the wave is equal to
$Amplitude= 2 3 ×x= 2 3 x$
$3λ=15cm λ= 15 3 =5cm$
$2λ=25cm λ= 25 2 =12.5cm$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "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": 0.9955551624298096, "perplexity": 1796.8586761270353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1573496665809.73/warc/CC-MAIN-20191112230002-20191113014002-00323.warc.gz"} |
http://math.stackexchange.com/questions/140276/if-a-1-a-2-a-n-in-g-1-oplus-g-2-oplus-cdot-cdot-cdot-oplus-g-n-give | # If $(a_1,a_2,…,a_n)\in G_1\oplus G_2 \oplus \cdot \cdot \cdot \oplus G_n$ give a condition for $|(a_1,a_2,…,a_n)| = \infty$
Let $(a_1,a_2,...,a_n)\in G_1\oplus G_2 \oplus \cdot \cdot \cdot \oplus G_n$. Give a necessary and sufficient condition for $|(a_1,a_2,...,a_n)| = \infty$
I know $|(a_1,a_2,...,a_n)|$ is related to the LCM somehow, but I'm confused on the context of the question.
-
The $G_i$ are groups, I suppose? And $|\cdot|$ denotes the order of an element? Hint: If $a_i$ has finite order $o_i$ for each $i$, what can you say about $(a_1,\ldots, a_n)^{o_1\cdots o_n}$? – martini May 3 '12 at 6:44
For a (multiplicative) group $G$ and $g\in G$, set $|g|=m\gt 0$ if $g^m=1$ and $g^k\neq 1$ for all $k$, $0\lt k\lt m$; and $|g|=0$ if $g$ is not a torsion element.
If $G_1,\ldots,G_n$ are a finite collection of groups, then $$|(g_1,\ldots,g_n)| = \mathrm{lcm}(|g_1|,|g_2|,\ldots,|g_n|).$$ (Prove it; remember that $\mathrm{lcm}(0,n) = 0$ for all $n$).
More generally, if $G$ is a group, $g_1$ and $g_2$ are two elements that commute, and $\langle g_1\rangle\cap\langle g_2\rangle = \{1\}$, then $|g_1g_2| = \mathrm{lcm}(|g_1|,|g_2|)$: since $g_1$ and $g_2$ commute, then $(g_1g_2)^n = g_1^ng_2^n$. If $(g_1g_2)^k = 1$, then $g_1^kg_2^k=1$, so $g_1^k = g_2^{-k}\in\langle g_1\rangle\cap\langle g_2\rangle=\{1\}$, so $g_1^k = g_2^k = 1$, hence $|g_1|$ divides $k$, and $|g_2|$ divides $k$.
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I think the required answer is that $(g_1, g_2, ..., g_n)$ has infinite order if and only it at least one such $g_i$ has infinite order in the corresponding $G_i$.
If each $g_i$ had finite order in the corresponding $G_i$ then $(g_1, g_2, ..., g_n)$ would have finite order in $G$ (have a think about this).
Conversely if the order of $(g_1, ..., g_n)$ is finite then each $g_i$ has a (finite) power giving the identity in the corresponding $G_i$. This tells you that all $g_i$'s have finite order.
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Set $a=(a_1,\cdots,a_n)$ to be an arbitrary element of $H=\bigoplus_i G_i$ and $e=(e_1,\cdots,e_n)$ the identity. Note that each $e_i$ is the identity of the direct summand $G_i$. Now some thoughts to ponder:
• $a^m=e$ is true if and only if $a_i^m=e_i$ for each $i=1,2,\cdots,n$.
• $a_i^m=e_i$ is true if and only if the order of $a_i$ (in $G_i$) divides $m$ or $a_i=e_i$.
• If the order of every $a_i$ divides $m$, then $a_i^m=e_i$ for each $i$.
• An element of $a\in H$ for which $a^m=e$ for some $m\in\Bbb Z$ must have finite order. (Why?)
• There is an $m\in\Bbb Z$ that is divisible by each $a_i$'s order. (Multiply, take the LCM, etc.)
If each $a_i$ has finite order, the above reasoning tells us $a$ has finite order too. Now, suppose at least one of the $a_i$'s has infinite order. Can $a\in H$ have finite order? (Assume it does and see what happens...)
Finally, there are precisely $n$ conditions on the table: whether or not $a_i$ has finite order, separately for each $i$. What is sufficient for $a$ to have infinite order, in terms of the $a_i$'s orders? And then what is necessary for $a$ to have infinite order? So then what can we say is both necessary and sufficient?
- | {"extraction_info": {"found_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": 0.9862047433853149, "perplexity": 93.02641724458904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049274985.2/warc/CC-MAIN-20160524002114-00244-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://www.michaelslab.net/index.php?id=site-search&tags=keppler_system | ## Search Results
• ### en Keppler System
The purpose of the first project was to simulate the movement of earth around the sun. Since the expected results are well known it is... | {"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": 0.8484011292457581, "perplexity": 961.0778159944545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991557.62/warc/CC-MAIN-20210517023244-20210517053244-00291.warc.gz"} |
https://solvation.ca/ | # Stuck on a difficult problem?
## When the answer is not enough, Solvation provides professionally-worked, illustrated, step-by-step solutions to text book problems.
#### Step by Step Solutions
We not only give you the answer, but also how we got there. You will understand every step from the beginning to the end. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8801079988479614, "perplexity": 832.4756556863704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806388.64/warc/CC-MAIN-20171121132158-20171121152158-00258.warc.gz"} |
https://scholar.archive.org/work/qiqprtmevrfxbcwjfl3rs7hlku | ### Mesons in(2+1)-dimensional light front QCD. II. Similarity renormalization approach
Dipankar Chakrabarti, A. Harindranath
2002 Physical Review D, Particles and fields
Recently we have studied the Bloch effective Hamiltonian approach to bound states in 2+1 dimensional gauge theories. Numerical calculations were carried out to investigate the vanishing energy denominator problem. In this work we study similarity renormalization approach to the same problem. By performing analytical calculations with a step function form for the similarity factor, we show that in addition to curing the vanishing energy denominator problem, similarity approach generates linear
more » ... generates linear confining interaction for large transverse separations. However, for large longitudinal separations, the generated interaction grows only as the square root of the longitudinal separation and hence produces violations of rotational symmetry in the spectrum. We carry out numerical studies in the G{\l}azek-Wilson and Wegner formalisms and present low lying eigenvalues and wavefunctions. We investigate the sensitivity of the spectra to various parameterizations of the similarity factor and other parameters of the effective Hamiltonian, especially the scale $\sigma$. Our results illustrate the need for higher order calculations of the effective Hamiltonian in the similarity renormalization scheme. | {"extraction_info": {"found_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": 0.9847775101661682, "perplexity": 1008.9516315753424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780055808.78/warc/CC-MAIN-20210917212307-20210918002307-00265.warc.gz"} |
https://web2.0calc.com/questions/math_51512 | +0
math
0
202
1
14/1 times 7/12
Guest Jun 7, 2017
#1
+2117
0
Here is the original expression:
$$\frac{14}{1}*\frac{7}{12}$$
To multiply this, just multiply the numerator and the denominator. Calculate the numerator by doing $$14*7=98$$ and calculate the denominator by doing $$1*12=12$$.
$$\frac{98}{12}$$
But wait! You are not done yet! You must reduce the fraction to its simplest terms. To do this, calculate the GCF of the numerator and denominator. The GCF happens to be 2. Take that factor out of both the numerator and denominator.
$$\frac{98}{12}\div\frac{2}{2}=\frac{49}{6}=8.1\overline{66666}$$
TheXSquaredFactor Jun 7, 2017 | {"extraction_info": {"found_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": 0.9983767867088318, "perplexity": 803.6169964671537}, "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/1531676591543.63/warc/CC-MAIN-20180720061052-20180720081052-00476.warc.gz"} |
https://www.physicsforums.com/threads/fluid-dynamics-2.738564/ | # Fluid dynamics (2)
1. Feb 15, 2014
### narutoish
1. The problem statement, all variables and given/known data
If the pressure reading of your Pitot tube is 17.0 mm Hg at a speed of 150 km/h, what will it be at 700 km/h at the same altitude?
2. Relevant equations
The only eq I could think if is p = p2+ 1/2 density (v2^2 - v1^2)
But I don't know the density
3. The attempt at a solution
So I pretty much get stuck in the beginning, any help will be nice
Thanks
2. Feb 15, 2014
### haruspex
What would the reading be at 0km/h? What equations do you now have?
(Note: I'm assuming the equation you quoted is appropriate. To check that I'd need to do some research.)
3. Feb 15, 2014
### Staff: Mentor
At the tip of the pitot tube, the velocity is zero. So you are measuring the stagnation pressure. If the density doesn't change between the two cases, how does the stagnation pressure depend on the approach velocity?
Chet
Draft saved Draft deleted
Similar Discussions: Fluid dynamics (2) | {"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": 0.8771149516105652, "perplexity": 1391.4552325274994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886436.25/warc/CC-MAIN-20180116125134-20180116145134-00246.warc.gz"} |
https://chemistry.stackexchange.com/questions/62165/balancing-the-oxidative-decomposition-of-fes2/62181 | # Balancing the oxidative decomposition of FeS2
Doing some practice for the GRE next year.
Please could you look at my working for this and tell me why I'm wrong?
$$\ce{? FeS2 + ? O2 + ? H2O = ? Fe(OH)3 + ? H2SO4}$$
My working so far is:
• There must be the same number of of $\ce{FeS2}$ as $\ce{Fe(OH)3}$
• There must be twice as many $\ce{H2SO4}$ as $\ce{FeS2}$
• If there is one mole of $\ce{FeS2}$ for example, there must be 1 mole of $\ce{Fe(OH)3}$ and two moles of $\ce{H2SO4}$
Tot up the $\ce{H}$ on the right: $1\times 3 + 2\times 2 = 7$. $7$ on the right so $\ce{3.5 H2O}$ on the left. ($7/2 = 3.5$)
$\ce{O}$ on the right = $1\times 3 + 2\times 4 = 11$.
This mean $11$ on the left, too. I already have $\ce{3.5 H2O}$, so we must have $11-3.5 = 7.5$ $\ce{O}$'s from the $\ce{O2}$. So $\ce{3.75 O2}$ because $7.5/2 = 3.75$.
Following the rules I've written down I arrive at:
$$\ce{1.75 FeS2 + 3.75 O2 + 3.5 H2O -> 1.75 Fe(OH)3 + 3.5 H2SO4}$$
I've multiplied these by $4$ to give integers, because that's what the question asks of me, so $7, 15, 14, 7$ and $14$.
Why am I wrong?
• Your equation (if this is it) is not balanced in H and O. Besides, having an acid and a base at the same time among the products does not feel right at all. $$\ce{7 FeS2 + 15 O2 + 14 H2O = 7 Fe(OH)3 + 14 H2SO4}$$ Anyway, welcome to Chem.SE, and pay attention to the formatting. – Ivan Neretin Nov 4 '16 at 16:16
• I've now balanced this equation and the answers are 4, 15, 14 = 4, 8 For anyone who would like to know! – New Zealand's finest Nov 4 '16 at 16:31
I found a solution which works, $$\ce{4FeS_2 + 15O_2 +14H_2O = 4Fe(OH)_3 + 8H_2SO_4}$$
This is how I found it. Do you know linear algebra ? Because that is very useful. Anyway without to know this is linear algebra you'll be able to understand.
So we are looking about the stoichiometric coefficients $\{\ce{a, b, c, d, e}\}$ of this reaction :
$$\ce{aFeS_2 + bO_2 + cH_2O + d Fe(OH)_3 + eH_2SO_4 \rightarrow 0}$$
Now what you need to do is first to find the number of different elements which are involved in your reaction, here they're four : $\{\ce{\color{\green}{H}, \color{\red}{O}, \color{\purple}{S}, \color{\orange}{Fe}}\}$. I'm not that good in the use of $\LaTeX$ but now the idea is to make column vectors under each of the reagents of your reactions and in these vectors you put in the same order for each the number of each elements there are in and if there is not just put a zero. It will gives you this :
$$a\cdot\begin{pmatrix} \color{\green}{0} \\ \color{\red}{0} \\ \color{\purple}2 \\ \color{\orange}{1} \end{pmatrix}+b\cdot\begin{pmatrix} \color{\green}{0} \\ \color{\red}{2} \\ \color{\purple}{0} \\ \color{\orange}{0} \end{pmatrix}+c\cdot\begin{pmatrix} \color{\green}2 \\ \color{\red}{1} \\ \color{\purple}{0} \\ \color{\orange}{0} \end{pmatrix}+d\cdot\begin{pmatrix} \color{\green}3 \\ \color{\red}{3} \\ \color{\purple}{0} \\ \color{\orange}1 \end{pmatrix}+e\cdot\begin{pmatrix} \color{\green}2 \\ \color{\red}4 \\ \color{\purple}1 \\ \color{\orange}0 \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$$
It can be easily transform into a system like this :
$$\begin{cases} 2c+3d+2e=0 \\ 2b+c+3d+4e=0 \\ 2a+e=0 \\ a+d=0 \end{cases}$$
So here we are in a good way because we have more unknowns than equation so if the system converge it has an infinity of solutions. And we can multiply by the number we want each stoichiometric coefficients and still have the same reaction.
So here we will determine four unknowns in terms of an other, I chose $e$. And I got : \begin{cases} a=-e/2 \\ b=-15e/8 \\ c=-7e/4 \\ d=e/2 \\ e=e \end{cases}
I multiply then by $8$ to have only integers which give me with $e=-1$ :
$$\ce{4FeS_2 + 15O_2 + 14H_2O - 4Fe(OH)_3 - 8H_2SO_4 \rightarrow 0}$$
Then $$\ce{4FeS_2 + 15O_2 + 14H_2O = 4Fe(OH)_3 + 8H_2SO_4}$$
I will take pretty much the same approach that 9-BBN does except I will do it a little bit differently. I have never heard of setting the reactants to zero and I'm not very good at making matrices, so I'll just jump right into the systems of equations.
First of all, define your coefficients $a,b,c,d,e$ for each of the chemicals involved, being the reactants that you start with and the products that you get from the reaction.
$$\ce{a FeS2 + b O2 + c H2O -> d Fe(OH)3 + e H2SO4}$$
Now we keep a tally of the number of each atom in molecule. If I put a zero, it means that there are none in that compound and this is for clarity; an equality sign replaces the reaction arrow separating products and reactants.
• Iron (Fe): $\displaystyle 1a + 0b + 0c = 1d + 0e \Longrightarrow a = d$
• Sulfur (S): $\displaystyle 2a + 0b + 0c = 0d + 1e \Longrightarrow 2a = e$
• Oxygen (O): $\displaystyle 0a + 2b + 1c = 3d + 4e \Longrightarrow 2b + c = 3d + 4e$
• Hydrogen (H): $\displaystyle 0a + 0b + 2c = 3d + 2e \Longrightarrow 2c = 3d + 2e$
Now then, we have four equations and 5 variables $a,b,c,d,e$ … so it shouldn't be solvable, except that we only want one solution which is the lowest integer coefficients for each molecule. So we have to make one initial guess for any of the variables. We choose the one that is the simplest and that will be $a$. We set $a = 1$ for initial guess. Note that $a < 0$ & $a = 0$ are forbidden.
In iron (Fe), $a = d$, as $a = 1$, then $1 = d$
In sulfur (S), $2a = e$, as $a = 1$, then $2 = e$
What we know: $a = 1, b =\ ?, c =\ ?, d = 1, e = 2$
With what we know, we can only solve for $c$ in the hydrogen equation next.
$c = [(3d + 2e) / 2]$, as $d = 1, e = 2$, then: $c = 7 / 2$
Solving the last equation for b:
$b = [3d + 4e -c] / 2$, given known variables, then: $b = 15 / 4$
Solutions: $a = 1, b = (15 / 4), c = (7 / 2), d = 1, e = 2$
This will balance the equation according to conservation of mass. However, it makes more sense if you multiply $a$ through $e$ by $4$ so as to get the lowest whole number solutions. This is like scaling a recipe. We are synthesizing the same thing, though.
Therefore, $a = 4, b = 15, c = 14, d = 4, e = 8$
Balanced Chemical Equation:
$$\ce{4 FeS2 + 15 O2 + 14 H2O -> 4 Fe(OH)3 + 8 H2SO4}$$
Now that your equation is balanced I will show you a little something that I've derived. So the formula for the commplete combustion of every hydrocarbon alkane ($\ce{C_nH_{2n+2}}$) such as methane, ethane, propane, butane, pentane, etc … is this:
$$\ce{C_nH_{2n+2} + (3n + 1) / 2 O2 -> n CO2 + (n + 1) H2O}$$
So if we are completely combusting (full airflow of oxygen, no major flickering of the flame producing a mixture of $\ce{CO}$ and $\ce{CO2}$) propane which has the molecular formula $ce{C3H8}$, where $n = 3$ then it's combustion is the following:
$$\ce{C3H8 (g) + 5 O2 (g) -> 3 CO2 (g) + 4 H2O (g)}$$
I know the general complete combustion equation is true for every alkane because all alkanes have the formula $\ce{C_nH_{2n+2}}$, $\ce{O2}$ is always involved in combustion, and $\ce{CO2}$ & $\ce{H2O}$ are always the products of the complete combustion of alkanes. I think this is really cool given that most popular general chemistry equations are combustion equations!
Additional information: I have a lot of ideas running through my head now so once you've determined the balanced chemical equation, you may be able to calculate the spontaneity of the reaction from the change in Gibbs Free Energy of reaction which is useful for knowing if the reaction is a waste of energy, that is that it consumes more energy than produces or not!
You can determine the percent yield after experimentation, how useful your reaction is from the stoichiometric amount of expected product produced. But even with ideally $100~\%$ yield, are all the elements that you find in your desired product going on to form your product or are they being wasted producing something else?
The 2nd Principle of Green Chemistry: Atom Economy, states that
Synthetic methods should be designed to maximize incorporation of all materials used in the process into the final product. (The American Chemical Society)
So what's the point of your reaction? To make $\ce{Fe(OH)3}$? What percent of the atoms $\ce{Fe, O}$, and $\ce{H}$ are being incorporated as $\ce{Fe(OH)3}$ and how much is going off to form sulfuric acid instead? We can calculate that as the percent atom economy.
$$\%\ \text{Atom Economy} = \frac{\text{mass of atoms in desired product}}{\text{mass of atoms in all reactants}} \times 100$$
where units are in grams per mole, but they are eliminated in the ratio.
\begin{align}\%\ \text{Atom Economy} &= \frac{4 \times 106.866}{4 \times 119.965 + 15 \times 31.998 + 14 \times 18.015} \times 100\\ &= \frac{427.464}{1212.04} \times 100\\ &= 35.27~\%\end{align}
So if there were multiple ways to make $\ce{Fe(OH)3}$ $\%\ \text{Atom economy}$ may be a useful factor to consider when conducting a synthesis. I think this is more important, though, in choosing a synthetic reaction where toxic byproducts are involved and you want to minimize the toxic byproducts produced for the health of the earth and of the customer or patient and also to reduce on having to spend money to deactivate and/or filter out toxic byproducts in an industrial synthesis. Therefore, you want the synthesis with the highest $\%\ \text{Atom Economy}$.
• I think that this chemical equation solver can solve your difficult equation in flash. It probably uses either matrices or systems of equations. You can use this if you like: webqc.org/balance.php ... anyway, it's for speed of calculation. It doesn't not explain the methodology like I did, though. – xyz123 Nov 5 '16 at 7:35
• I don't believe that I said that your mathematics is unnecessary, just that I would do it a little bit differently. – xyz123 Nov 5 '16 at 16:50
• You cannot comment everywhere because that is a priviledge associated with 50 reputation. Rather, since this was a simple mathematical error, you could have proposed an edit instead. I will now read on to review the remainder of your post to determine whether it answers the question or not (only up to the third paragraph). – Jan Nov 5 '16 at 18:27
• Okay, more than $50~\%$ of your post ramble on about things that are not relevant to the question. Also, I learnt a different definition of atom economy, but that is not important here. – Jan Nov 5 '16 at 18:47
• What is the definition of atom economy that you learned? – xyz123 Nov 5 '16 at 19:15
This is a redox equation. You can approach it in a standard redox manner, however, you will need to use a small, helpful trick.
If we set up redox pairs, we will notice that there are two compounds that get oxidised but only one that gets reduced:
\begin{align}\ce{Fe^2+} &/ \ce{Fe^3+}\tag{Ox1}\\ \ce{(S^{-I})2^2-} &/ \ce{(S^{+VI})O4^2-}\tag{Ox2}\\ \ce{(O^{\pm 0})2} &/ \ce{(O^{-II})H-}\tag{Red}\end{align}
However, we can alleviate this problem by realising that $\ce{FeS2}$ is likely a mineral and thus we need to oxidise one iron per $\ce{S2^2-}$ unit. Also, rather than considering water it seems the question wants you to generate hydroxide, hence why $\ce{OH-}$ is my reduced partner of elemental oxygen. That lets us set up the half equations. The reduction is first because it is a lot simpler.
\begin{align}\ce{O2 + 4e- + 2 H+ &-> 2 OH-}\tag{Red}\\ \strut\\ \ce{FeS2 \phantom{\ce{+ 8 H2O}} &-> Fe^3+ + 2 SO4^2- + 15 e-}\tag{Ox1}\\ \ce{FeS2 \phantom{\ce{+ 8 H2O}} &-> Fe^3+ + 2 SO4^2- + 15 e- + 16 H+}\tag{Ox2}\\ \ce{FeS2 + 8 H2O &-> Fe^3+ + 2 SO4^2- + 15 e- + 16 H+}\tag{Ox3}\end{align}
The smallest common multiple of $4$ and $15$ happens to be $60$, thus $\text{(Ox3)}$ must be multiplied with $4$ and $\text{(Red)}$ must be multiplied with $15$ leading us to $\text{(RO1)}$ and the following:
\begin{align}\ce{4 FeS2 + 32 H2O + 15 O2 + 30 H+ &->\\ 4 Fe^3+ &+ 8 SO4^2- + 64 H+ + 30 OH-}\tag{RO1}\\ \ce{4 FeS2 + 32 H2O + 15 O2 &->\\ 4 Fe^3+ &+ 8 SO4^2- + 34 H+ + 30 OH-}\tag{RO2}\\ \ce{4 FeS2 + 32 H2O + 15 O2 &->\\ 4 Fe^3+ &+ 8 SO4^2- + 30 H2O + 4 H+}\tag{RO3}\\ \ce{4 FeS2 + 2 H2O + 15 O2 &->\\ &4 Fe^3+ + 8 SO4^2- + 4 H+}\tag{RO4}\end{align}
Everything henceforth is just adding further water molecules or regrouping at your desire. For example, to me, the equation suggests combining the free protons with sulfate ions and even the iron cations in the following way $\text{(RO5)}$:
$$\ce{4 FeS2 + 2 H2O + 15O2 -> 4 Fe(SO4)(HSO4)}\tag{RO5}$$
On the other hand, if you really had to include the generation of sulfuric acid and iron hydroxyide, note that you are lacking twelve hydroxide ions ($4 \times 3$) and 12 protons ($8 \times 2 - 4$) — meaning that twelve water molecules should be added to the left-hand side.
$$\ce{4 FeS2 + 14 H2O + 15 O2 -> 4 Fe(OH)3 + 8 H2SO4}\tag{RO6}$$
Although I must agree with Ivan’s comment: the co-presence of $\ce{H2SO4}$ and $\ce{Fe(OH)3}$ is extremely unlikely; the following will happen:
$$\ce{2Fe(OH)3 (s) + 3H2SO4 (aq) -> 2 Fe^3+ (aq) + 6 H2O (l) + 3 SO4^2- (aq)}$$
I used to be pretty good at balancing chemical equations, but now days I'm pretty rusty. Never-the-less, as far as I can see the equation you've been given must be wrong (or you mis-wrote it above). FeS2 would be assumed by a beginning student to have a ferric (+IV) ion (formally, at least) combined with two sulfide ions S=. So your reaction reduces the Iron (to Fe(+III)) and oxidizes the sulfurs (to S(+VI)). The oxidation of the sulfurs in this scenario gives you all the electrons you need to reduce the oxygen to the oxide (O=) for the formation of the sulfate anion. In other words you'd be left with an "extra" electron (from the iron) - which as long as this isn't a half reaction isn't going to happen (at least as far as elementary chemical homework is concerned). A more sophisticated student might learn that Sulfur commonly forms more exotic ions, such as the disulfide anion S-S= and that the two common forms of FeS2 both have Iron in the +2 not the +4 valence state. This means iron goes from +2 to +3 in your given reaction. Which is pretty much the same problem (in reverse, actually): the sulfur oxidation gives you 14 electrons, the Fe oxidation 1 more and you need 16 to reduce the oxygen a discrepancy of one electron. There's no place for that electron to go! You're stuck, again. | {"extraction_info": {"found_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": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9261338114738464, "perplexity": 908.9948199798329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361776.13/warc/CC-MAIN-20210228205741-20210228235741-00331.warc.gz"} |
https://mathoverflow.net/questions/88233/how-many-consecutive-integers-x-can-make-ax2bxc-square/88245 | # how many consecutive integers $x$ can make $ax^2+bx+c$ square ?
If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?
The polynomial $-15x^2+64$ is obviously a square for the five numbers $x=-2,...,2$, but the method used for finding this in the above thread cannot be extended further. Should $5$ really be the best possible answer?
Has this problem been treated somewhere else?
-
For starters $15w^2-14$ is a square for $w=\pm1,\pm3,\pm5$ so you get six by taking $w=2x+1$. But eventually you get to surfaces of general type. See my answer to mathoverflow.net/questions/73346 . – Noam D. Elkies Feb 11 '12 at 23:01
...and if I did it right, you can get to $w = \pm 7$ using points on an elliptic curve of (conductor 360 and) rank 1. That might be the maximum. FWIW $0,\pm1,\pm2,\pm3$ seems to give a curve of conductor 30 with 12 torsion points but rank zero. – Noam D. Elkies Feb 11 '12 at 23:12
To resonate with Noam Elkies' comments, it is conjectured that $8$ squares is the maximum for arbitrary $a$, and $4$ squares is the maximum for $a=1$. For $5$ symmetric squares the smallest known leading coefficients are $a=15$ and $a=-20$, while for $5$ increasing squares they are $a=60$ and $a=-56$. It is known that there are infinitely many examples with $5$ or $8$ symmetric squares, or with $6$ increasing squares. It is also known that there is no symmetric sequence of $7$ squares, and only finitely many of $10$ squares up to obvious equivalences (this one follows from Falting's theorem applied to a specific hyperelliptic curve).
A recent paper of Gonzales-Jimenez and Xarles [Acta Arith. 149 (2011)] (apologies for the lack of accents) gets a (sharp) upper bound of $8$ for those quadratics with an axis of symmetry half-way between two integers. – Mike Bennett Feb 12 '12 at 4:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8291213512420654, "perplexity": 279.9762304134593}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1394011188282/warc/CC-MAIN-20140305091948-00016-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://export.arxiv.org/abs/2011.09856 | ### Current browse context:
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# Title: Daemon computers versus clairvoyant computers: A pure theoretical viewpoint towards energy consumption of computing
Abstract: Energy consumption of computing has found increasing prominence but the area still suffers from the lack of a consolidated formal theory. In this paper, a theory for the energy consumption of computing is structured as an axiomatic system. The work is pure theoretical, involving theorem proving and mathematical reasoning. It is also interdisciplinary, so that while it targets computing, it involves theoretical physics (thermodynamics and statistical mechanics) and information theory. The theory does not contradict existing theories in theoretical physics and conforms to them as indeed it adopts its axioms from them. Nevertheless, the theory leads to interesting and important conclusions that have not been discussed in previous work. Some of them are: (i) Landauer's principle is shown to be a provable theorem provided that a precondition, named macroscopic determinism, holds. (ii) It is proved that real randomness (not pseudo randomness) can be used in computing in conjunction with or as an alternative to reversibility to achieve more energy saving. (iii) The theory propounds the concept that computers that use real randomness may apparently challenge the second law of thermodynamics. These are computational counterpart to Maxwell's daemon in thermodynamics and hence are named daemon computers. (iv) It is proved that if we do not accept the existence of daemon computers (to conform to the second law of thermodynamics), another type of computers, named clairvoyant computers, must exist that can gain information about other physical systems through real randomness. This theorem probably provides a theoretical explanation for strange observations about real randomness made in the global consciousness project at Princeton University.
Comments: Related Areas: Statistical Physics, Information Theory, Computer Science, Thermodynamics Keywords: Reversibility, Randomness, Low Energy Computing, Maxwell's Daemon, Landauer's principle Subjects: Statistical Mechanics (cond-mat.stat-mech); Information Theory (cs.IT); Mathematical Physics (math-ph) Cite as: arXiv:2011.09856 [cond-mat.stat-mech] (or arXiv:2011.09856v1 [cond-mat.stat-mech] for this version)
## Submission history
From: Alireza Ejlali [view email]
[v1] Thu, 12 Nov 2020 15:58:22 GMT (38kb)
Link back to: arXiv, form interface, 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": 0.9058439135551453, "perplexity": 2353.542542951588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704828358.86/warc/CC-MAIN-20210127152334-20210127182334-00331.warc.gz"} |
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Aug11 awarded Popular Question Jul2 awarded Curious Feb9 awarded Yearling Feb6 awarded Notable Question Sep29 awarded Popular Question Sep24 comment Show $[(p-1)!]^{p^{n-1}} \equiv -1$ (mod $p^n$) for n $\in \mathbb N$ Thanks. @martini I can follow it better now. In the term, $k^2p^{2n}\sum_{l=2}^p \binom pl (kp^n)^{l-2}(-1)^{p-l}$, wouldn't some of the terms of this sum be non-integer fractions, since $\binom pl$ can be a non-integer fraction? Won't this create a problem when working modulo $p^{n+1}$? Sep24 comment Show $[(p-1)!]^{p^{n-1}} \equiv -1$ (mod $p^n$) for n $\in \mathbb N$ Showing that $(x^{p-1}-x^{p-2}+x^{p-3}...)$ is divisible by p, doesn't look any easier of a task. Sep24 comment Show $[(p-1)!]^{p^{n-1}} \equiv -1$ (mod $p^n$) for n $\in \mathbb N$ @ThomasAndrews Oh! That makes sense. Sep24 comment Show $[(p-1)!]^{p^{n-1}} \equiv -1$ (mod $p^n$) for n $\in \mathbb N$ Am I able to do it without the binomial though? The notes that I saw this problem in don't touch on binomial expansion. Sep24 comment Show $[(p-1)!]^{p^{n-1}} \equiv -1$ (mod $p^n$) for n $\in \mathbb N$ Thanks @Thomas. how do you derive this result? Sep24 comment Show $[(p-1)!]^{p^{n-1}} \equiv -1$ (mod $p^n$) for n $\in \mathbb N$ I don't really understand the use of combinations like this. Sep18 accepted Characterising reals with terminating decimal expansions Sep18 asked Characterising reals with terminating decimal expansions Sep18 accepted Fourier transform of inverse rectangular pulse Sep18 answered Fourier transform of inverse rectangular pulse Sep18 accepted If $a^2$ divides $b^2$, then $a$ divides $b$ Sep18 accepted $p$ a prime, $p \equiv 3 \pmod 4$. Prove that $\frac{p-1}{2}! \equiv \pm 1 \pmod p$ Sep18 comment $p$ a prime, $p \equiv 3 \pmod 4$. Prove that $\frac{p-1}{2}! \equiv \pm 1 \pmod p$ Oh! but it is, when p $\equiv$ 3(mod 4). I see! Sep18 comment $p$ a prime, $p \equiv 3 \pmod 4$. Prove that $\frac{p-1}{2}! \equiv \pm 1 \pmod p$ In the general case, $\frac{p-1}{2}$ isn't odd though? e.g. p = 17 Sep18 revised $p$ a prime, $p \equiv 3 \pmod 4$. Prove that $\frac{p-1}{2}! \equiv \pm 1 \pmod p$ plus or minus sign inserted. | {"extraction_info": {"found_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": 0.8897867798805237, "perplexity": 1506.0927526191024}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830721.40/warc/CC-MAIN-20140820021350-00265-ip-10-180-136-8.ec2.internal.warc.gz"} |
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## The Yoneda lemma I
For two categories $C, D$ let $D^C$ denote the functor category, whose objects are functors $C \to D$ and whose morphisms are natural transformations. For $C$ a locally small category, the Yoneda embedding is the functor $C \to \text{Set}^{C^{op}}$ sending an object $x \in C$ to the contravariant functor $\text{Hom}(-, x)$ and sending a morphism $x \to y$ to the natural transformation $\text{Hom}(-, x) \to \text{Hom}(-, y)$ given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.
Below, whenever we talk about the Yoneda lemma we implicitly restrict our attention to locally small categories.
The Yoneda embedding
Let $x \in C$ be an object and $F \in \text{Set}^{C^{op}}$ be a (set-valued) presheaf. A natural transformation from the representable presheaf $\text{Hom}(-, x)$ to $F$ is a collection of morphisms $\eta_y : \text{Hom}(y, x) \to F(y)$ such that the square
commutes for all $y, z$ (where $\text{Hom}(f, x)$ is abuse of notation for the action of the functor $\text{Hom}(-, x)$ on morphisms $f : z \to y$). In particular, we have a morphism $\eta_x : \text{Hom}(x, x) \to F(x)$, and taking the image of $\text{id}_x$ associates to every natural transformation $\text{Hom}(-, x) \to F$ an element $\eta_x(\text{id}_x) \in F(x)$.
Yoneda lemma: The above map is an isomorphism (of sets).
Corollary: The Yoneda embedding $C \to \text{Set}^{C^{op}}$ is fully faithful; that is, every natural transformation $\text{Hom}(-, x) \to \text{Hom}(-, y)$ is induced by composition with an element of $\text{Hom}(x, y)$, and different morphisms give different natural transformations.
Proof. Set $y = x$ in the above commutative diagram to the diagram
where $f$ is now a morphism $z \to x$. By identifying the images of $\text{id}_x \in \text{Hom}(x, x)$ in $F(z)$ obtained from tracing the two paths in this diagram we conclude that
$\displaystyle \eta_z(f) = F(f)(\eta_x(\text{id}_x))$.
In other words, $\eta_z$ for every $z$ (so the entire natural transformation) is completely determined by $\eta_x(\text{id}_x)$. This shows that the map we defined above from natural transformations to elements of $F(x)$ is injective. On the other hand, it’s not hard to check that for any choice of $\eta_x(\text{id}_x) \in F(x)$ the above defines a natural transformation $\eta$.
Morals of the Yoneda lemma
The Yoneda lemma shows that an object $x$ in a category is determined up to isomorphism by the presheaf $\text{Hom}(-, x)$ it represents. In other words, roughly speaking an object is determined by how other objects map into it. I once heard the following colorful analogy for this situation on MO: if one thinks of objects of a category as particles and morphisms as ways to smash one particle into another particle, then the Yoneda lemma says that it is possible to determine the identity of a particle by smashing other particles into it.
Another way to say this is the following. For an object $y$, we call an element of $\text{Hom}(y, x)$ a generalized point or $y$-point of $x$. If $y = 1$ is the terminal object, a $y$-point is also sometimes called a global point.
Example. In $\text{Set}$ or more generally $\text{Top}$, a global point is a point in the usual sense.
Example. In $G\text{-Set}$, the category of $G$-sets ($G$ a group), a global point is a fixed point.
Example. Let $k$ be a field. In the category of affine varieties over $k$, a global point is a point over $k$ (since $\text{Spec } k$ is the terminal object). More general points often have geometric interpretations; for example, a $\text{Spec } k[t]$-point is a one-parameter family of $k$-points, and a $\text{Spec } k[t]/t^2$ is a $k$-point together with a Zariski tangent vector.
The Yoneda lemma tells us, roughly speaking, that an object is determined by its generalized points. Yet another way to say this is that we can completely understand the morphisms $x \to y$ between objects in an arbitrary category $C$ in terms of maps between the “sets” of generalized points
$\coprod_{z \in C} \text{Hom}(z, x) \to \coprod_{z \in C} \text{Hom}(z, y)$.
These disjoint unions don’t actually exist if the collection of objects of $C$ doesn’t form a set (for example if $C = \text{Grp}$). If $C$ actually has a set of objects (in addition to sets of morphisms), we say that $C$ is small. In any small category we can form the above disjoint unions, and so we conclude that that every small category is concretizable (admits a faithful functor to $\text{Set}$).
Yoneda for monoids
Let $M$ be a monoid and $BM$ be the corresponding one-object category with single object $x$. Then a presheaf $BM^{op} \to \text{Set}$ is precisely a right $M$-set, and a natural transformation between presheaves is a morphism of right $M$-sets. Furthermore, the unique representable presheaf $\text{Hom}(-, x)$ corresponds to $M$ regarded as a right $M$-set by right multiplication.
The Yoneda lemma in this case says that morphisms $M \to S$ of right $M$-sets are canonically in bijection with elements of $S$ (take the image of $\text{id}_x \in M$ as above); in particular, the endomorphisms of $M$ form a monoid canonically isomorphic to $M$ (acting by left multiplication). This gives us a slight generalization of Cayley’s theorem: every monoid acts by endomorphisms on some set.
Yoneda for posets
Let $P$ be a poset regarded as a category where $a \le b$ means there is a single morphism $a \to b$ and otherwise there are no morphisms. Set-theoretic presheaves on $P$ are the wrong thing to consider; since we can think of posets as categories enriched over the category (which is also a poset) $2 = \{ 0 \le 1 \}$, we should actually be considering presheaves $F : P^{op} \to 2$. By identifying such a presheaf with the elements mapping to $1$, we can identify presheaves on $P$ with downward closed sets in $P$, since functoriality just means that if $x \le y$ then $F(y) \le F(x)$. In particular, the representable presheaves are the downward closed sets of the form $D_y = \{ x : x \le y \}$.
A functor between two posets is just an order-preserving map. The functor category $Q^P$ is itself a poset, with a single natural transformation $F \to G$ existing if $F(x) \le G(x)$ for all $x \in P$ and no natural transformations existing otherwise. When $Q = 2$ and we identify the functors $F \in 2^P$ with downward closed sets, the corresponding partial order structure is containment.
The Yoneda lemma in this case says that a downward closed set contains $D_y$ if and only if it contains $y$. Applied to two representable presheaves, it says that
$y \le z \Leftrightarrow \forall x : (x \le y \Rightarrow x \le z)$.
The Yoneda embedding $P \to 2^{P^{op}}$ embeds a poset into its poset of downward-closed subsets, giving us a “Cayley’s theorem for posets”: every poset can be realized as subsets of some set under containment. It also gives a kind of “Dedekind completion” of $P$: for example, when $P = \mathbb{Q}$ under the usual order, the poset of downward-closed subsets is the extended real line $[-\infty, \infty]$.
Yoneda for the category of affine schemes
For the purposes of this section, the category $\text{Aff}$ of affine schemes will be by definition the opposite $\text{CRing}^{op}$ of the category of commutative rings. If $S$ is a commutative ring, we write $\text{Spec } S$ for that ring regarded as an object in $\text{Aff}$. The Yoneda lemma tells us that $\text{Spec } S$ is completely determined by the presheaf $\text{Hom}(-, \text{Spec } S)$. A certain family of examples will be particularly instructive; if we take $S = \mathbb{Z}[x_1, ... x_n]/(f_1, ... f_m)$ where $f_1, ... f_m$ is a finite collection of integer polynomials in $n$ variables, then the affine scheme $\text{Spec } S$ is completely determined by the presheaf $\text{Spec } R \to \text{Hom}(\text{Spec } R, \text{Spec } S)$ or, in the opposite category, by
$\displaystyle \text{Hom}(\mathbb{Z}[x_1, ... x_n]/(f_1, ... f_m), R)$.
But by the universal property of polynomial rings, this is nothing more than the set of $x \in R^n$ such that $f_1(x) = ... = f_m(x) = 0$; in other words, precisely the set of solutions of the polynomial system $f_1 = ... = f_m = 0$ over $R$!
In other words, for affine schemes generalized points are like solutions to systems of polynomial equations. This point of view on algebraic geometry was pioneered by Grothendieck and is called the functor of points perspective. A particularly elegant feature of this perspective is how it uses the Yoneda lemma: the Yoneda lemma tells us that a morphism of affine schemes $\text{Spec } S_1 \to \text{Spec } S_2$ is precisely a consistent collection of ways $\eta_R : \text{Hom}(\text{Spec } R, \text{Spec } S_1) \to \text{Hom}(\text{Spec } R, \text{Spec } S_2)$ to turn solutions of the “system of equations” described by $S_1$ into solutions of the “system of equations” described by $S_2$.
General non-affine schemes are informally built by “gluing” affine schemes, and one way to make that precise is to think of affine schemes as representable presheaves on $\text{Aff}$ and then taking suitable colimits of these presheaves in the presheaf category $\text{Set}^{\text{CRing}}$.
### 7 Responses
1. […] Annoying precision […]
2. […] If you haven’t seen it before, try proving Yoneda’s lemma. (It’s not too difficult if you’re familiar with basic category theory arguments… there is really only 1 choice for the map.) As a follow-up I recommend looking at Qiaochu Yuan’s article which gives additional examples: https://qchu.wordpress.com/2012/04/02/the-yoneda-lemma-i/ […]
3. I guess one could say Hom(-,x) is a *represented* functor [with ‘tautological’ representation, namely the one corresponding to the universal element (x, id_x)].
4. “elements of an arbitrary category”, instead of “objects”?
• Whoops! Yes, I meant “objects”.
• Yeah, minor nit, I guess. Also, when you say Hom(-,x) is a representable presheaf, that just means it’s isomorphic to a hom-functor, a trivial fact in this case since it *is* a hom-functor?
• Yep. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 129, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9675170183181763, "perplexity": 178.16288699817147}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121528.59/warc/CC-MAIN-20170423031201-00262-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://projecteuclid.org/euclid.ade/1355867255 | Positive solutions for equations and systems with $p$-Laplace like operators
Abstract
We prove the existence of positive solutions to boundary-value problems of the form \begin{align*} \begin{gathered} (\phi(u'))'+f(t,u)=0,\quad t\in(0,1)\\ \theta(u(0))=\beta \theta(u'(0)), \quad \theta(u(1))=-\delta \theta(u'(1)), \quad \beta,\delta\ge 0, \end{gathered} \end{align*} where $\phi$ and $\theta$ are odd increasing homeomorphisms of the real line. We also prove the existence of positive solutions to related systems. Our approach is via a priori estimates and Leray-Schauder degree.
Article information
Source
Adv. Differential Equations Volume 14, Number 5/6 (2009), 401-432.
Dates
First available in Project Euclid: 18 December 2012
García-Huidobro, M.; Manásevich, R.; J. R. Ward, J. R. Positive solutions for equations and systems with $p$-Laplace like operators. Adv. Differential Equations 14 (2009), no. 5/6, 401--432. https://projecteuclid.org/euclid.ade/1355867255 | {"extraction_info": {"found_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": 0.9726773500442505, "perplexity": 2427.5844255697107}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125938462.12/warc/CC-MAIN-20180420135859-20180420155859-00422.warc.gz"} |
http://math.stackexchange.com/questions/119706/complex-integration | # Complex integration
Find $\int_0^{\pi+2i} \cos(z/2) \; dz$?
What is the procedure for doing this problem?
I 'observed' that the derivative of $2\sin(z/2)$ is $\cos(z/2)$ so my answer was $2\sin(z/2)$ evaluated between $0$ and $\pi+2i$ which gives me $2\sin\left(\frac{\pi+2i}{2}\right)$
But wolfram alpha says the answer is $2\cos i\ldots$ So what am I doing wrong?
-
$\sin\left(\frac{\pi}{2} + z\right) = \cos z$
The two answers are equivalent.
You can easily derive the above using another trig identity for the $\sin$ of a sum:
$$\sin(a + b) = \sin(a) \cos(b) + \sin(b) \cos(a)$$
If we plug in $a = \pi/2$, we get
$$\sin (\pi/2 + b) = \sin(\pi/2) \cos(b) + \sin(b) \cos(\pi/2) = \cos(b)$$
since $\sin(\pi/2) = 1$ and $\cos(\pi/2) = 0$.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.986379861831665, "perplexity": 115.69436885246175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246636650.1/warc/CC-MAIN-20150417045716-00223-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/238008/absolute-continuity-and-integration-formula-explain-a-statement-please | # Absolute continuity and integration formula (explain a statement please)
For $v$, $w$ in $L^2(0,T;H^1(S))$ (with weak derivatives in $H^{-1}(S)$ for each time), the product $(v(t), w(t))_{L^2(S)}$ is absolutely continuous wrt. $t \in [0,T]$ and $$\frac{d}{dt}\int_S v(t)w(t) = \langle v', w \rangle_{H^{-1}(S), H^1(S)} + \langle v, w' \rangle_{H^{1}(S), H^{-1}(S)}$$ holds a.e. in $(0,T)$. As a consequence, the formula of partial integration holds $$\int_0^T \langle v', w \rangle_{H^{-1}, H^1} = (v(T), w(T))_{L^2(S)} - (v(0), w(0))_{L^2(S)} - \int_0^T \langle v, w' \rangle_{H^{1}, H^{-1}}$$
I am wondering what role absolute continuity plays here. I know if a real-valued (normal in the undergrad sense) function is abs. cont. then it satisfies an equation similar to the one above but this one involves dual space pairing so I can't see how it is analgous. I would appreciate someone explaining me this. Thanks.
-
Here's a somewhat rough sketch of what's going on. Absolute continuity plays the same role as usual: since $(v(t),w(t))_{L^2}$ is an absolutely continuous real valued function of $t$, $\frac{d}{dt}[(v,w)_{L^2}]$ exists for almost every $t\in (0,T)$ by the Lebesgue differentiation theorem. Now to compute this derivative, we use Reynold's transport theorem (assuming $S$ does not change with respect to time):
$$\frac{d}{dt} \int_S v(t)w(t) dV = \int_S\frac{d}{dt}( v(t)w(t))=\int_S(v^\prime w+vw^\prime)=\langle v^\prime,w\rangle+\langle v,w^\prime\rangle$$
Since we're working with weak derivatives, we interpret $\int_Sv^\prime w$ as the dual pairing $\langle v^\prime,w\rangle_{H^{-1},H^1}$ and similarly we interpret $\int_S vw^\prime$ as $\langle v,w^\prime\rangle_{H^1,H^{-1}}$. | {"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": 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": 0.9906295537948608, "perplexity": 197.77319881871978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704234586/warc/CC-MAIN-20130516113714-00081-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://sciencehouse.wordpress.com/2011/06/05/stochastic-differential-equations/ | # Stochastic differential equations
One of the things I noticed at the recent Snowbird meeting was an increase in interest in stochastic differential equations (SDEs) or Langevin equations. They arise wherever noise is involved in a dynamical process. In some instances, an SDE comes about as the continuum approximation of a discrete stochastic process, like the price of a stock. In other cases, they arise as a way to reintroduce stochastic effects to mean field differential equations originally obtained by averaging over a large number of stochastic molecules or neurons. For example, the Hodgkin-Huxley equation describing action potential generation in neurons is a mean field approximation of the stochastic transport of ions (through ion channels) across the cell membrane, which can be modeled as a multi-state Markov process usually simulated with the Gillespie algorithm (for example see here). This is computationally expensive so adding a noise term to the mean field equations is a more efficient way to account for stochasticity.
In one dimension, an SDE has the form
$\frac{dx}{dt} = f(x) + g(x)\eta(t)$ (1)
where $\eta(t)$ is a noise term. This could have any form but usually it is taken to be Gaussian white noise obeying $\langle \eta(t) \rangle = 0$, $\langle \eta(t)\eta(t')\rangle = \delta(t-t')$, where the brackets imply ensemble average over the noise distribution. Now, one of the problems with using white noise is that two time points are uncorrelated no matter how close together so the noise can’t be continuous and the variance is infinite. Hence, the SDE as written above doesn’t really exist. There are two ways to deal with this. Physicists take the attitude that the white noise is not really white noise so that for very short times it is well-behaved. This approach goes under the name Stratonovich formulation.
Mathematicians interpret the SDE as really being an integral equation and write the SDE as
$dx = f(x) dt + g(x) dW$ (2)
where $dW$ is a white noise process that obeys $\langle w(\tau) w(\tau') \rangle = \tau$, where $w(\tau)=\int_t^{t+\tau} dW$. This is known as the Ito formulation. The problem with the Ito formulation is that the rules for differentiation and integration must be changed to something called Ito calculus. The deepness of the mathematical concepts involved in SDEs also makes treatments on the topic widely divergent. If you read a book from the physicist’s point of view like for example Risken versus a more mathematical one like Oksendal, you may not even know they were writing about the same topic.
I started out using the Stratonovich formulation of SDEs but have since switched exclusively to Ito. The main reason is that numerical simulations of SDEs are most natural in the Ito formulation. If we discretize equation (2) with a single Euler step of time step $\Delta$, we obtain
$x_{n+1}=x_n + f(x_n) \Delta + g(x_n) Z_n \sqrt{\Delta}$
where $Z_n$ is a normally distributed random variable. If you discretize in the Stratonovich formulation then $g(x)$ must be evaluated at the mid-point between the time steps.
However, when it comes to “solving” SDEs, I put on my physicist’s cap and use two approaches. One is to solve the SDEs directly to obtain moments of $x$ using path integrals (e.g. see here). The other approach is to solve the Fokker-Planck equation for the probability density function of $x$. The Fokker-Planck equation is a parabolic partial differential equation of the form
$\frac{\partial}{\partial t}P(x,t) = -\frac{\partial }{\partial x}f(x) P +\frac{1}{2}\frac{\partial^2}{\partial x^2}g(x)^2 P$
for Ito and
$\frac{\partial}{\partial t}P(x,t) = -\frac{\partial }{\partial x}(f(x) + g(x)'g(x))P +\frac{1}{2}\frac{\partial^2}{\partial x^2}g(x)^2 P$
for Stratonovich. The Fokker-Planck equation is a “drift-diffusion” equation. In the Ito formulation, the drift term is given by the deterministic part of the SDE and the diffusion term is the square of the stochastic part divided by two. In the Stratonovich formulation, there is an extra drift term that arises from the stochastic forcing. The difference arises from the difference between Ito and regular calculus.
One crucial point about the Fokker-Planck equation is that it is linear. In fact, what the Fokker-Planck equation does is to transform a nonlinear SDE into a linear PDE. In some sense you have traded nonlinearity for heterogeneity and extra-dimensions. I have always found this to be fascinating. It is a way to demonstrate that solving linear heterogeneous PDEs can be as hard as solving nonlinear SDEs. However, depending on what question you have one may be more useful than the other. For example, if you are interested in the equilibrium probability distribution of $x$ then the Fokker-Planck equation reduces to a heterogeneous second order ODE, which can generally be solved. Finally, I want to stress that the Fokker-Planck equation is always linear by definition. Some papers in the literature have called single-variable continuity equations for probability densities nonlinear Fokker-Planck equations. This is technically incorrect. The systems these papers describe are high dimensional systems that do have multi-variable Fokker-Planck equations. The so-called nonlinear Fokker-Planck equations that appear are usually derived heuristically. However, they can be more formally derived by marginalization over the other variables followed by a truncation of a moment (BBGKY) hierarchy. Again, there is a trade-off between nonlinearity and extra dimensions.
Typos fixed 2011-6-5
## 3 thoughts on “Stochastic differential equations”
1. […] οf thе form. frac{partial}{partial t}P( x ,t … Read thіѕ article: Stochastic differential equations « Scientific Clearing House #dd_ajax_float{ background:none repeat scroll 0 0 #eaeae3; border:1px solid #eaeae3; float:left; […]
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2. xcorr says:
Fascinating. I had looked into SDEs but I couldn’t figure out which formulation I should learn about so I moved on to other things. What book would you recommend to learn more about Ito calculus? My undergrad was in math and physics.
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3. Hi,
Good question. I’m not sure what book to read to learn about Ito calculus. Maybe Gardiner’s book Handbook of Stochastic Methods would give the physicist’s point of view? If you have a good background in analysis then you can try Oksendal or Shreve’s books.
Like | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 17, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9130087494850159, "perplexity": 639.1142584964998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195532251.99/warc/CC-MAIN-20190724082321-20190724104321-00179.warc.gz"} |
http://mathhelpforum.com/differential-equations/203811-definite-integral.html | # Thread: definite integral!
1. ## definite integral!
$\int_{-\pi}^{\pi} \cos^{2}ny dy$
$\left[\frac{n\cos^{3}ny}{3n}\sin{ny}\right]_{-\pi}^{\pi}$
$\left[\frac{n\cos^{3}n\pi}{3n}\sin{n\pi}+\frac{n\cos^{3} n\pi}{3n}\sin{n\pi}\right]$
$2\left[\frac{n\cos^{3}n\pi}{3n}\sin{n\pi}\right]$
$2\left[0\right]$
pls is 0 equivalent to $2{\pi}$ or $\pi$
thanks!
2. ## Re: definite integral!
Originally Posted by lawochekel
$\int_{-\pi}^{\pi} \cos^{2}ny dy$
$\left[\frac{n\cos^{3}ny}{3n}\sin{ny}\right]_{-\pi}^{\pi}$
You have do this incorrectly. See here.
3. ## Re: definite integral!
Hello, lawochekel!
You integrated incorrectly.
We need the identity: . $\cos^2\theta \:=\:\tfrac{1}{2}\left(1 + \cos2\theta\right)$
$\int_{\text{-}\pi}^{\pi} \cos^2(ny)\,dy$
$\int_{\text{-}\pi}^{\pi} \cos^2(ny)\,dy \;=\;\tfrac{1}{2}\int^{\pi}_{\text{-}\pi}\big[1 + \cos(2ny)\big]\,dy$
. . . . . . . . . . . $=\; \tfrac{1}{2}\big[y + \tfrac{1}{2n}\sin(2ny)\bigg]^{\pi}_{\text{-}\pi}$
. . . . . . . . . . . $=\;\tfrac{1}{2}\left[\pi + \tfrac{1}{2n}\sin(2\pi n)\right] - \tfrac{1}{2}\left[\text{-}\pi + \tfrac{1}{2n}\sin(\text{-}2\pi n)\right]$
. . . . . . . . . . . $=\;\tfrac{1}{2}(\pi + 0) - \tfrac{1}{2}(\text{-}\pi + 0)$
. . . . . . . . . . . $=\; \pi$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9979174137115479, "perplexity": 408.9318615248261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887414.4/warc/CC-MAIN-20180118131245-20180118151245-00053.warc.gz"} |
http://mathhelpforum.com/differential-equations/127889-studing-test.html | # Thread: Studing for a test
1. ## Studing for a test
Hello, I'm going through a practice exam but cant figure these out any help would be really appreciated. Thanks
Find general solution for
(a) dy/dx=4y+6
2. Originally Posted by acosta0809
Hello, I'm going through a practice exam but cant figure these out any help would be really appreciated. Thanks
Find general solution for
(a) dy/dx=4y+6
$\frac{dy}{dx}=4y+6$
$\therefore \frac{dy}{4y+6}=dx$
Now integrate both the sides.. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9032904505729675, "perplexity": 1686.1501897233195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164645800/warc/CC-MAIN-20131204134405-00019-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=GraphTheory/DegreeSequence&L=F | DegreeSequence - Maple Help
# Online Help
###### All Products Maple MapleSim
GraphTheory
DegreeSequence
degree sequence of graph
Calling Sequence DegreeSequence(G)
Parameters
G - graph
Description
• DegreeSequence returns a list of the degrees of the vertices of G. For directed graphs, the directions of the edges are ignored.
Examples
> $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
> $G≔\mathrm{Graph}\left(\mathrm{Trail}\left(1,2,3,4,2\right)\right)$
${G}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 4 vertices and 4 edge\left(s\right)}}$ (1)
> $\mathrm{DegreeSequence}\left(G\right)$
$\left[{1}{,}{3}{,}{2}{,}{2}\right]$ (2)
> $H≔\mathrm{Graph}\left(\mathrm{Trail}\left(1,2,3,4,2\right),\mathrm{directed}\right)$
${H}{≔}{\mathrm{Graph 2: a directed unweighted graph with 4 vertices and 4 arc\left(s\right)}}$ (3)
> $\mathrm{DegreeSequence}\left(H\right)$
$\left[{1}{,}{3}{,}{2}{,}{2}\right]$ (4)
> $\mathrm{DrawGraph}\left(G\right)$
See Also | {"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": 0.9950496554374695, "perplexity": 2792.1265919541966}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964360881.12/warc/CC-MAIN-20211201173718-20211201203718-00475.warc.gz"} |
http://cms.math.ca/10.4153/CJM-2008-030-5 | location: Publications → journals → CJM
Abstract view
# Closed and Exact Functions in the Context of Ginzburg--Landau Models
Published:2008-06-01
Printed: Jun 2008
• Anamaria Savu
Format: HTML LaTeX MathJax PDF PostScript
## Abstract
For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.
Keywords: Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries
MSC Classifications: 42B05 - Fourier series and coefficients 81Q50 - Quantum chaos [See also 37Dxx] 42A16 - Fourier coefficients, Fourier series of functions with special properties, special Fourier series {For automorphic theory, see mainly 11F30} | {"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": 0.9856439232826233, "perplexity": 2999.2746513197553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1412037663372.35/warc/CC-MAIN-20140930004103-00152-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://socratic.org/questions/what-is-the-equation-in-point-slope-form-of-the-line-given-4-6-5-7 | Algebra
Topics
# What is the equation in point-slope form of the line given (4,6),(5,7)?
##### 2 Answers
Mar 6, 2018
$m = 1$
#### Explanation:
Given -
(4, 6); (5, 7)
${x}_{1} = 4$
${y}_{1} = 6$
${x}_{2} = 5$
${y}_{2} = 7$
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{7 - 6}{5 - 4} = \frac{1}{1} = 1$
$m = 1$
$y - 6 = 1 \left(x - 4\right)$
or
$y - 7 = 1 \left(x - 5\right)$
#### Explanation:
Point slope form is essentially:
$y - {y}_{1} = m \left(x - {x}_{1}\right)$
${y}_{1}$ and ${x}_{1}$ are coordinates given to you. They can either be 6 and 4 respectively, or 7 and 5 respectively. Pick your choice.
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
So, plug in coordinates for that.
$\frac{7 - 6}{5 - 4} = \frac{1}{1} = 1 = m$
Remember, the plain ol' y and x in the point slope form equation will be the actual variables, since functions need those guys to stick around.
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https://tex.stackexchange.com/questions/39784/two-math-sections-of-the-same-type-in-a-row/343928 | # Two math sections of the same type in a row?
Whenever I try to create two sections of the same type in a row LyX unites them as if they were one. I write a definition, press ENTER for a new line, and then want to write another definition on that new line, LyX ignores my intention and moves me back to the previous definition ...
What shall I do?
• Just to add that --Separator-- can be found under the drop-down menu (near the top-left, under File and Edit). Took me another 10 minutes to figure out after reading the answer below. – Kenny LJ Mar 16 '16 at 2:17
The "proper" way of doing seems to be to add a "Separator" environment after the first definition1.
• Hey Torbjørn , thanks for all the effort with the screenshot and all, but I think I didn't explained myself clearly enough. what I meant was that if a wrote a definition, pressed ENTER for a new line, and then wanted to write another definition on that NEW line, Lyx ignore my intention and moved me back to the previous definition... – dave Jan 1 '12 at 14:38
Say you want to insert several different definitions in a row without writing any text between them. All you need to do is to write your definitions between \begin{defn} and \end{defn} commands. Then you can insert as many definitions in a row as you want. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.9306067228317261, "perplexity": 685.7997936363054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1566027330800.17/warc/CC-MAIN-20190825194252-20190825220252-00031.warc.gz"} |
http://legisquebec.gouv.qc.ca/en/showversion/cr/P-40.1,%20r.%203?code=se:91_5&pointInTime=20210720 | ### P-40.1, r. 3 - Regulation respecting the application of the Consumer Protection Act
91.5. A label containing the following information shall be affixed to each item of goods for which a merchant uses the exemption under section 91.4:
(a) the nature of the item and the characteristics affecting its price or distinguishing it from other goods of the same nature, such as its brand and size;
(b) the price of the item or, where the price is based on a unit of measurement, the price per unit of measurement; and
(c) for food sold in an establishment for which the merchant must hold a permit issued under the Regulation respecting food (chapter P-29, r. 1), the price per unit of measurement in addition to the price of the item.
In all cases, the price on the label must be in at least 28-point bold type print and the other information in at least 10-point type print.
Where the item is sold on a shelf, the label prescribed under the first paragraph shall be affixed next to the product on the shelf and measure at least
(a) 12.90 cm2 in establishments for which the merchant is required to hold a permit under the Regulation respecting food; and
(b) 9.67 cm2 in other establishments.
Where the item is sold elsewhere than on a shelf, the label must be affixed near the product sold and measure at least 38.71 cm2.
The requirement under subparagraph c of the first paragraph shall only take effect on 23 June 2001.
O.C. 10-2001, s. 4. | {"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": 0.8098710775375366, "perplexity": 1852.9748238415661}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780053657.29/warc/CC-MAIN-20210916145123-20210916175123-00389.warc.gz"} |
http://mathhelpforum.com/differential-geometry/74262-analysis-infimum-supremum-proof.html | # Thread: analysis : infimum/supremum proof
1. ## analysis : infimum/supremum proof
Let A be a nonempty subset of real numbers which is bounded below. Let -A={-x|x is an element of A). Prove that: inf(A)=-sup(-A).
Any help would be greatly appreciated. Thank you.
decohen
2. Let $x\epsilon$(-A). Then $(-x)\epsilon$A. So -x$\geq$inf(A). This implies that x$\leq$-inf(A).
So -inf(A) is an upper bound of -A.
Now let y be any upper bound of -A.
Let x$\epsilon$A. Then (-x)$\epsilon$(-A). Because y is an upper bound of -A, -x$\leq$y. This implies that x$\geq$-y. This means that -y is a lower bound of A. Now, since inf(A) is the greatest lower bound of A, it follows that -y$\leq$inf(A), which implies that y$\geq$-inf(A).
In this way we have proved:
1) -inf(A) is an upper bound of -A
2) -inf(A) is smaller than any other upper bound of -A
This means, by definition, that sup(-A)= -inf(A).
3. If $\lambda = \inf (A)$ then $\left( {\forall x \in - A} \right)\left[ {\; - x \in A \Rightarrow \lambda \; \leqslant - x\; \Rightarrow \; - \lambda \geqslant x} \right]$.
That means that $-A$ is bounded above so $\left( {\exists \delta } \right)\left[ {\delta = \sup ( - A)} \right]\;\& \;\delta \leqslant - \lambda$.
Now suppose that $\delta < - \lambda$ then it follows that $- \delta > \lambda \, \Rightarrow \,\left( {\exists b \in A} \right)\left[ { - \delta > b \geqslant \lambda } \right]$.
BUT this means that $\delta < - b\;\& \, - b \in - A$ : CONTRADICTION.
4. Originally Posted by [email protected]
Let A be a nonempty subset of real numbers which is bounded below. Let -A={-x|x is an element of A). Prove that: inf(A)=-sup(-A).
Any help would be greatly appreciated. Thank you.
decohen
Let inf(A)=u.
Let $y\in(-A)\Longrightarrow -y\in A\Longrightarrow u\leq -y\Longrightarrow y\leq -u$====> -A is bounded from above by -u.
So if sup(-A)=v $\Longrightarrow v\leq -u\Longrightarrow u\leq -v$......................(1)
Let $- y\in A\Longrightarrow y\in (-A)\Longrightarrow y\leq v\Longrightarrow -v\leq -y$ ====> -v is a lower bound of A.
And since inf(A)=u $\Longrightarrow -v\leq u$...............(2)
Combining (1) and (2) we get u=-v or inf(A)=-sup(-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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9996291399002075, "perplexity": 1217.1197381934103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501171053.19/warc/CC-MAIN-20170219104611-00637-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://collaborate.princeton.edu/en/publications/identifying-a-cooperative-control-mechanism-between-an-applied-fi | # Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems
Fang Gao, Roberto Rey-De-Castro, Yaoxiong Wang, Herschel Rabitz, Feng Shuang
Research output: Contribution to journalArticlepeer-review
6 Scopus citations
## Abstract
Many systems under control with an applied field also interact with the surrounding environment. Understanding the control mechanisms has remained a challenge, especially the role played by the interaction between the field and the environment. In order to address this need, here we expand the scope of the Hamiltonian-encoding and observable-decoding (HE-OD) technique. HE-OD was originally introduced as a theoretical and experimental tool for revealing the mechanism induced by control fields in closed quantum systems. The results of open-system HE-OD analysis presented here provide quantitative mechanistic insights into the roles played by a Markovian environment. Two model open quantum systems are considered for illustration. In these systems, transitions are induced by either an applied field linked to a dipole operator or Lindblad operators coupled to the system. For modest control yields, the HE-OD results clearly show distinct cooperation between the dynamics induced by the optimal field and the environment. Although the HE-OD methodology introduced here is considered in simulations, it has an analogous direct experimental formulation, which we suggest may be applied to open systems in the laboratory to reveal mechanistic insights.
Original language English (US) 053407 Physical Review A 93 5 https://doi.org/10.1103/PhysRevA.93.053407 Published - May 9 2016
## All Science Journal Classification (ASJC) codes
• Atomic and Molecular Physics, and Optics
## Fingerprint
Dive into the research topics of 'Identifying a cooperative control mechanism between an applied field and the environment of open quantum systems'. Together they form a unique fingerprint. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.819633424282074, "perplexity": 1720.442549396294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500619.96/warc/CC-MAIN-20230207134453-20230207164453-00538.warc.gz"} |
http://math.stackexchange.com/questions/79356/using-the-determinant-to-verify-linear-independence-span-and-basis | # Using the Determinant to verify Linear Independence, Span and Basis
Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a set which I can make into a square matrix, can I use the determinant to determine these three properties?)
Here are two examples:
• Span Does the following set of vectors span $\mathbb R^4$: $[1,1,0,0],[1,2,-1,1],[0,0,1,1],[2,1,2,-1]$? Now the determinant here is $1$, so the set of vectors span $\mathbb R^4$.
• Linear Independence Given the following augmented matrix:
$$\left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \end{array}\right],$$ where again the determinant is non-zero ($-2$) so this set S is linearly independent.
Of course I am in trouble if you can't make a square matrix - I figure for spans you can just rref it, and I suppose so for linear independence and basis?
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Okay so I saw this on the related items just now - math.stackexchange.com/questions/28061/… - so that confirms you can use it to determine if it's a basis - but what about Span and Linear Independence? – eWizardII Nov 5 '11 at 23:44
Most introductory books on Linear Algebra have a Theorem which says something like
Let $A$ be a square $n \times n$ matrix. Then the following are equivalent:
• $A$ is invertible.
• $\det(A) \neq 0$.
• The columns of $A$ are linearly independent.
• The columns of $A$ span $R^n$.
• The columns of $A$ are a basis in $R^n$.
• The rows of $A$ are linearly independent.
• The rows of $A$ span $R^n$.
• The rows of $A$ are a basis in $R^n$.
• The reduced row echelon form of $A$ has a leading 1 in each row.
and many other conditions.....
What does this mean, it simply means that if you want to check if any of these conditions is true or false, you can simply pick whichever other condition from the list and check it instead..
Your question is: Can instead of third or fourth condition, check the second? That's exactly what the Theorem says: YES.
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Thanks, a few sections later they give a similar explanation like you said - it just wasn't in the one where they are expecting you to solve these types of problems. – eWizardII Nov 6 '11 at 18:37
Does the converse apply for all of those? For example, if det=0, does that always imply that the rows are linearly independent? – Asad Oct 7 '13 at 16:28
@Asad equivalent means they are all true or all false. If $\det(A)=0$ means the second one is false, which means ALL are false. So yes, $\det(A)=0$ implies the rows are linearly DEPENDENT. – N. S. Oct 7 '13 at 16:33
@N.S. I see, thanks. – Asad Oct 7 '13 at 17:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9234269261360168, "perplexity": 343.84038685473655}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986444.39/warc/CC-MAIN-20150728002306-00048-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathhelpforum.com/pre-calculus/203088-write-function-form-f-x-x-k-q-x-r.html | # Math Help - Write the function in form f(x)=(x-k)q(x)-r
1. ## Write the function in form f(x)=(x-k)q(x)-r
My dilemma isn't so much how to do synthetic division or how to write the results into the given form.I don't understand the answer that is given in the answer key for this problem.Obviously,I don't want to simply know the answer,I want to understand it.So,for this problem I'm supposed to write the function f(x)=(x-k)q(x)-r,and show that f(k)=r.
$f(x)= -4x^3+6x^2+12x+4, k = 1{\sqrt{3}}$
The squares are just confusing me.Thanks for taking a look.
2. ## Re: Write the function in form f(x)=(x-k)q(x)-r
Originally Posted by Ai Ekio
My dilemma isn't so much how to do synthetic division or how to write the results into the given form.I don't understand the answer that is given in the answer key for this problem.Obviously,I don't want to simply know the answer,I want to understand it.So,for this problem I'm supposed to write the function f(x)=(x-k)q(x)-r,and show that f(k)=r.
$f(x)= -4x^3+6x^2+12x+4, k = 1{\sqrt{3}}$
The squares are just confusing me.Thanks for taking a look.
You said that you're not getting an answer that matches the answer given, so please post what you have done... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.895933210849762, "perplexity": 644.6425546314532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115858580.32/warc/CC-MAIN-20150124161058-00242-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://eprint.iacr.org/2016/095 | ## Cryptology ePrint Archive: Report 2016/095
Obfuscation without Multilinear Maps
Dingfeng Ye and Peng Liu
Abstract: Known methods for obfuscating a circuit need to represent the circuit as a branching program and then use a multilinear map to encrypt the branching program. Multilinear maps are, however, too inefficient for encrypting the branching program. We found a dynamic encoding method which effectively singles out different inputs in the context of the matrix randomization technique of Kilian and Gentry et al., so that multilinear maps are no longer needed. To make the method work, we need the branching programs to be regular. For such branching programs, we also give most efficient constructions for NC1 circuits. This results in a much more efficient core obfuscator for NC1 circuits.
Category / Keywords: foundations / Obfuscation, Matrix Branching Program, Dynamic Fencing | {"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": 0.8976790904998779, "perplexity": 2549.1474585801425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424876.76/warc/CC-MAIN-20170724122255-20170724142255-00524.warc.gz"} |
http://math.stackexchange.com/questions/251172/how-to-simplify-polynomials/251174 | # How to simplify polynomials
I can't figure out how to simplify this polynominal $$5x^2+3x^4-7x^3+5x+8+2x^2-4x+9-6x^2+7x$$
I tried combining like terms $$5x^2+3x^4-7x^3+5x+8+2x^2-4x+9-6x^2+7x$$ $$(5x^2+5x)+3x^4-(7x^3+7x)+2x^2-4x-6x^2+(8+9)$$ $$5x^3+3x^4-7x^4+2x^2-4x-6x^2+17$$
It says the answer is $$3x^4-7x^3+x^2+8x+17$$ but how did they get it?
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Did the color help?! :-D – 000 Dec 5 '12 at 2:08
Yes it did, thanks. – Cynea Osodgt Dec 5 '12 at 2:18
Mission success! :-) Please click the checkmark to whatever answer you think is best; it will improve your accepted answer percentage. – 000 Dec 5 '12 at 2:34
I voted for the two that helped – Cynea Osodgt Dec 5 '12 at 2:40
Can't i choose more then 1 answer??? I want to choose both anwers that helped me – Cynea Osodgt Dec 5 '12 at 3:06
You cannot combine terms like that, you have to split your terms by powers of $x$.
So for example $$5x^2+5x+2x^2 = (5+2)x^2+5x = 7x^2+5x$$ and not $5x^3+2x^2$. Using this, you should end up with your answer.
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Observe the magical power of color:
$$\color{blue}{5}x^\color{blue}{2}+3x^4-7x^3+\color{green}{5}x+\color{orange}{8}+\color{blue}{2}x^\color{blue}{2}+(\color{green}{-4})x+\color{orange}{9}+(\color{blue}{-6})x^\color{blue}{2}+\color{green}{7}x.$$
Instead of Color-Me-Elmo, we have Color-Me-Like-Terms-And-Combine (not as catchy, I know): $$3x^4-7x^3+(\color{blue}{5}+\color{blue}{2}+(\color{blue}{-6}))x^\color{blue}{2}+(\color{green}{5}+(\color{green}{-4})+\color{green}{7})x+(\color{orange}{8}+\color{orange}{9}).$$
Presto-simplification-o!
### Combining Like Terms
In a polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$ and $q(x)=b_nx^n+b_{n-1}x^{n-1}+\dots+b_1x+b_0$, they are added thusly: \begin{align} p(x)+q(x)&=a_nx^n+b_nx^n+a_{n-1}x^{n-1}+b_{n-1}x^{n-1}+\cdots+a_1x+b_1x+a_0+b_0\\ &=(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\cdots+(a_1+b_1)x+(a_0+b_0). \end{align}
In other words, add the coefficients of terms with the same power.
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Did you mean $q(x)=b_nx^n+\color{red}{b}_{n-1}x^{n-1}+\dots+\color{red}{b}_1x+{\color{red}{b}}_{0}$? – ctype.h Dec 5 '12 at 2:20
Yes, actually . . . @ctype.h. I get a little too excited sometimes. – 000 Dec 5 '12 at 2:23
@JasperLoy I am getting you a Color-Me-Elmo for Christmas. – 000 Dec 5 '12 at 2:58
Group the terms with the same power of $x$ together.
$5x^2+3x^4−7x^3+5x+8+2x^2−4x+9−6x^2+7x$
$=3x^4−7x^3+5x^2+2x^2−6x^2+5x−4x+7x+8+9$
$=3x^4−7x^3+x^2+8x+17$
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8437082171440125, "perplexity": 3358.0412412287596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657127503.54/warc/CC-MAIN-20140914011207-00149-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://mathoverflow.net/questions/72616/solving-for-moore-penrose-pseudo-inverse?sort=newest | # Solving for Moore Penrose pseudo inverse
I have a system to solve, set up as : $$Ax = b$$ with a square rank deficient matrix $A$. The paper suggests to use a Moore Penrose pseudo inverse, which in my case can be computed using the traditional inverse : $$A^+ = (A+\frac{ee^T}{n})^{-1} - \frac{ee^T}{n}$$ where $e$ is a vector containing only ones, and $n$ is the dimension of the matrix. This matrix comes from the solution of a Multidimensional Scaling problem using the SMACOF method (the Guttman transform).
However, in my case, my matrix $A$ is very sparse (and rank deficient) : what method can I use to efficiently solve the original system without making my matrix dense (as would be the case by using an SVD, by using the above formula for the pseudo inverse, by computing $A^TA$ or by QR factorization) ? $A$ is also symmetric, has a positive diagonal, and the other values are either -1 or 0, and such as the sum of each row (resp. column) is 0.
Preferably, since I'll need to solve for multiple right hand sides with this same matrix, I would like to avoid performing the resolution from scratch for each right hand side. I would also like to get exactly the same result as the one obtained with the Moore Penrose pseudoinverse.
Thanks.
-
For black-box linear algebra (GMRES and the like) you don't need "sparse", you only need "can compute products quickly". If you check the docs for your sparse solver, I'm sure there'll be a version where you can provide directly the function $v\mapsto Av$ (and sometimes $w\mapsto A^Tw$ is needed, too). In your case, you can compute $(A+\frac{ee^T}{n})v$ quickly by exploting sparsity.
And yes, if you wish to use iterative solvers you'll have to solve the system more or less from scratch for every new right-hand side linearly independent from the previous ones. Some savings are possible, but getting them is still a hot research topic.
-
The fundamental point here is that you don't want to actually compute the Moore-Penrose pseudoinverse, since that matrix will almost certainly be fully dense. Rather, you should consider using an iterative method to compute a minimum norm least squares solution. You could do this by solving a system involving $(A+\frac{ee^{T}}{n})$, or you might want to consider a more straight forward approach. For example, LSQR can be used with a damping factor to minimize a weighted sum of $\| Ax - -b\|^2$ and $\| x \|^2$. – Brian Borchers Aug 11 '11 at 4:19
indeed - I coded Federico's suggestion and it works well :) I used conjugate gradient since A+ee'/n is SPD (diagonal dominant with positive diagonal). – WhitAngl Aug 11 '11 at 11:07
Thanks very much to all of you! :) – WhitAngl Aug 11 '11 at 11:08
Are you looking for a fast practical method, or a fast theoretical method? If the former, there are very fast solvers based on sparse Cholesky or sparse LDL decomposition (both of which can be reused for many $b$). You should check out Tim Davis' beautiful book called something like "Sparse direct solvers".
It does, but only after you build-in symmetric pivoting. Even then, it's easy to choke $LDL^\top$; try $\begin{pmatrix}0&1\\1&0\end{pmatrix}$. – J. M. Dec 9 '11 at 4:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8896947503089905, "perplexity": 274.8777780394093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1398223206120.9/warc/CC-MAIN-20140423032006-00172-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/112596-cumulative-distrubution-function-question.html | # Thread: Cumulative distrubution function question
1. ## Cumulative distrubution function question
Can anyone provide any insight in how I could show the following (attached below)
Attached Thumbnails
2. $0\le nP(X>n) =n\int_n^{\infty}f(x)dx =\int_n^{\infty}nf(x)dx\le \int_n^{\infty}xf(x)dx\to 0$
as $n\to \infty$ since the first moment is finite.
3. Hello,
$1-F(x)=P(X>n)=P(X\cdot \bold{1}_{X>n}>n)=P(Y_n>n)\leq \frac{E(Y_n)}{n}$ by Markov's inequality.
where $Y_n=X\cdot \bold{1}_{X>n}$
As n goes to infinity, $Y_n$ obviously goes to 0 (X is almost surely finite since it's integrable)
and $|Y_n|$ is bounded by |X|, which is integrable.
So we can apply the dominated convergence theorem, and we have $n(1-F(x))\leq E(Y_n) \to 0 \quad \quad \square$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9901049137115479, "perplexity": 1320.3804441874627}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345764241/warc/CC-MAIN-20131218054924-00004-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://getrevising.co.uk/revision-tests/statement-of-financial-position-and-ratios | # Statement of Financial Position & Ratios
HideShow resource information
D U S U N N C Y U R R S B B W R G F Y T T Y H O F N O V P V E W E H A O W S L C Q S Q C I U P I E S X V M D X J P D G U U A W M U T F J T L V U O W C M W U M O F O Q G E R A P S A S I O N W U A V F V K K E W V O R R R T R O G L R K R T I X I R E F L L Q E Y N E T I N S U V R U E S I L W F W V O N T Q S S T G V T J E Q P W O S N I V A Q T I I S E A W Q K K N L U L I I S C H E V L L F A T R B G C T T F X L F Q N I K H C I I A T D Y V Y O U R E X I J N N E W C E A B N N I T L N T F A X R P X P H N D D X B A H E C I W F S E T O D O K R H C H X P I T O R A D E W F Y I K Y Y L H K Y D G E L I D R P I A B O X O Y H W H L M R J Q X I F W U X U X B E L M H I L C S R A F K K T O D C S Q T L T L W G E R I B X T I G Y I R K C A I S H A D W J H A T C L I N V X E P Q N P L Q A R S P J X V R P K O S M H S H H H S R A I K A Q D L A M N E S T W Q X B S F P W I F T M W B V C Y F Y E M I
### Clues
• Compares current assets and current liabilities to measure the ability to repay short term debts. Expressed as ?:1. Over 2:1 is ideal as this shows the business can pay its debts twice over. (7, 5)
• Compares current assets minus stick and current liabilities to measure the ability to repay short term debts in a crisis situation (when there might not be time to sell stock). Expressed as ?:1. 1:1 is acceptable as this shows a business can repay it (4, 4, 5)
• Items owed for a period of less than one year. (7, 11)
• Items owned for a period of less than one year. (7, 6)
• Measure how profitable a business is. (13, 6)
• Measure how well a business uses their resources. (10, 6)
• Measure the ability to repay short term debts. (9, 6)
• Measures the amount of times a business re-stocks during the year. (4, 2, 5, 8) | {"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": 0.8569371700286865, "perplexity": 1576.8224559672428}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170380.12/warc/CC-MAIN-20170219104610-00172-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://www.nag.com/numeric/CL/nagdoc_cl23/html/C06/c06gqc.html | c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual
# NAG Library Function Documentnag_multiple_conjugate_hermitian (c06gqc)
## 1 Purpose
nag_multiple_conjugate_hermitian (c06gqc) forms the complex conjugates of $m$ Hermitian sequences, each containing $n$ data values.
## 2 Specification
#include #include
void nag_multiple_conjugate_hermitian (Integer m, Integer n, double x[], NagError *fail)
## 3 Description
This is a utility function for use in conjunction with nag_fft_multiple_real (c06fpc) and nag_fft_multiple_hermitian (c06fqc) to calculate inverse discrete Fourier transforms.
None.
## 5 Arguments
1: mIntegerInput
On entry: the number of Hermitian sequences to be conjugated, $m$.
Constraint: ${\mathbf{m}}\ge 1$.
2: nIntegerInput
On entry: the number of data values in each Hermitian sequence, $n$.
Constraint: ${\mathbf{n}}\ge 1$.
3: x[${\mathbf{m}}×{\mathbf{n}}$]doubleInput/Output
On entry: the $m$ data sequences must be stored in x consecutively. If the $n$ data values ${z}_{j}^{p}$ are written as ${x}_{j}^{p}+{iy}_{j}^{p}$, $p=1,2,\dots ,m$, then for $0\le j\le n/2$, ${x}_{j}^{p}$ is contained in ${\mathbf{x}}\left[\left(p-1\right)×n+j\right]$, and for $1\le j\le \left(n-1\right)/2$, ${y}_{j}^{p}$ is contained in ${\mathbf{x}}\left[\left(p-1\right)×n+n-j\right]$.
On exit: the imaginary parts ${y}_{j}^{p}$ are negated. The real parts ${x}_{j}^{p}$ are not referenced.
4: failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
## 6 Error Indicators and Warnings
NE_INT_ARG_LT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
Exact.
None.
## 9 Example
This program reads in sequences of real data values which are assumed to be Hermitian sequences of complex data stored in Hermitian form. The sequences are expanded into full complex form using nag_multiple_hermitian_to_complex (c06gsc) and printed. The sequences are then conjugated (using nag_multiple_conjugate_hermitian (c06gqc)) and the conjugated sequences are expanded into complex form using nag_multiple_hermitian_to_complex (c06gsc) and printed out.
### 9.1 Program Text
Program Text (c06gqce.c)
### 9.2 Program Data
Program Data (c06gqce.d)
### 9.3 Program Results
Program Results (c06gqce.r) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 24, "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": 0.9988551139831543, "perplexity": 4895.434052993782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1461860125524.70/warc/CC-MAIN-20160428161525-00111-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://hata.compute.dtu.dk/index.php?n=MathForum.MathForum?action=print | MathForum: MathForum
Afternoon tea discussions at DTU Mathematics — discussions on non-applied mathematics.
On the probability of covering m random points with a given subset of the unit circle
by Christian Henriksen, Jakob Lemvig, and Johan Sebastian Rosenkilde Nielsen.
Question 1: If one picks m points at random on the unit circle \mathbb{S}^1, what is the probability that they can be covered by a half circle A \subset \mathbb{S}^1?
Question 2: Let 1 \ge a \ge 0. Suppose A \subset \mathbb{S}^1 is a Borel measurable set of measure \mu(A)=a , where \mu(\mathbb{S}^1)=a . What is probability that A can cover m random points?
Question 3: Let 1 \ge a \ge 0 be given. What is the best subset A \subset \mathbb{S}^1 to choose with \mu(A)=a if we want to cover m random points?
Covering points with an interval
We consider a generalized version of Question 1. We want to find the probability P_A(m) that m points falls within an arc A of \mathbb{S}^1 of length a, where the total length of \mathbb{S}^1 is one, i.e., \mu(\mathbb{S}^1)=1 with \mu (also denoted |\cdot|) being the (probability Lebesgue) surface measure on \mathbb{S}^1.
To be more precise we need some notation. We identify \mathbb{S}^1 with \mathbb{T}^1=\mathbb{R}/\mathbb{Z} and do most of our computations in \mathbb{T}^n modulo 1. Let a \in \mathbb{R} be given such that 0 \le a \le 1, and let A=[0,a).Let X_1, \dots, X_m \in [0,1] be an independent and identically distributed (i.i.d.) sample from U(0,1) — the uniform distribution on [0,1] . Then P_A(m) = P\{(X_1, X_2, ..., X_m) \in H\}, where we set
H = \{(x_1,\dots,x_m)\in \mathbb{T}^m \,\vert\, \exists \tau \text{ s.t. } x_j \in A+\tau \pmod 1 \text{ for all } j=1,\dots, m\}.
Theorem 1. If a \le 1/2, then P_A(m)=m a^{m-1} for all m=1,2,\dots.
Proof. Notice that the complement of A + \tau is B + \tau, where B = [a, 1). Therefore the following three statements are equivalent.
• \exists \tau s.t. x_j \in A + \tau, j = 1, 2, ..., m;
• \exists k s.t. x_j \in A + x_k, j = 1, 2, ..., m;
• \exists k s.t. x_j \notin B + x_k, j = 1, 2, ..., m.
In other words, letting H_k = \{(x_1, x_2, \dots, x_n) \,\vert\, x_j \notin (x_k, x_k + a] \pmod 1, j = 1, ..., n\} , we have H = \cup H_k.
We claim that the sets H_j are essentially disjoint, in the sense that H_i \cap H_j has measure zero on T^n when i \neq j. To see this notice that if (x_k)_{k=1}^m \in H_i \cap H_j and x_i \neq x_j then the distance going counter-clockwise from x_i to x_j is at least |B| \geq \frac{1}{2}, and likewise the distance from x_j to x_i going counter-clockwise is at least \frac{1}{2}. So for (X_1, ..., X_m) to lie in H_i \cap H_j we must have that X_i and X_j either coincide or are antipodal, events with probability zero.
We can conclude that P((X_1, ..., X_m) \in H) = \sum P(H_k) = m a^{m-1}. ■
TO-DO: Extend this argument to the case a>1/2.
By Theorem 1 it follows that the probability of covering m random points by a half circle A=[0,1/2) is P_A(m)=m (1/2)^{m-1} which was the question asked in Question 1. One might wonder whether you arrive at the same probability if A consists of two quarter-circles. If the quarter-circles are diametrically opposite, e.g., A=[0,1/4) \cup [1/2,3/4) this is indeed the case. It follows from the following observation:
Remark 1. Let n \in \mathbb{N} be given. If A=A+k/n \pmod 1 for all k=1,\dots,n-1, then
\{x_i\} \subset A + \tau \pmod 1 \Leftrightarrow \{x_i\} \subset A + \tau \pmod{\frac1n}
Consider the three sets A=[0,1/3), B=[0,1/6) \cup [3/6,4/6), and C=[0,1/6) \cup [2/6,3/6), all of measure 1/3. By Remark 1 we see that P_A(m)=P_B(m)=m (1/3)^{m-1} hence, in paricular, the probability of catching two random points by either A or B is P_A(2)=P_B(2)=2/3. For the set C, however, we set that P_C(2)=1 since if the distance from x_1 to x_2 is less that 1/6, we can find \tau so that x_1,x_2 \in [0,1/6)+\tau \subset C+\tau. On the other hand, if the distance between x_1 and x_2 is greater than 1/6, but less that 1/2, we can find \tau so that x_1 \in [0,1/6)+\tau and x_2 \in [2/6,3/6)+\tau, hence \{x_1,x_2\}\subset C+\tau. The event that the distance between x_1 and x_2 is exactly either 1/6 or 1/2 has probability zero. We remark that the set C. in contrast to B, does not satisfy the 'symmetry' condition C=C+1/2 \pmod 1 from Remark 1.
The following result shows that sets with "similar" symmetry properties have the same covering probabilities.
Lemma 1. Let n \in \mathbb{N} and A_0 \subset T^1 be given. Define A=\cup_{l=0}^{n-1}A_0 + l/n \pmod 1 and B=nA_0 \pmod 1. If |A| =|B| = n |A_0| , then
P_A(m) = P_B(m) \text{ for all } m \in \mathbb{N}_0.
Proof. Let M be a measurable set such that A_0 \subseteq M , |M|=1/n, and nM=[0,1] (up to sets of measure zero). Let \bar{\mu} be the uniform probability measure on M. Notice that \bar{\mu}=n \mu on M. Therefore, the \bar{\mu}-probability of X_1,\dots,X_m \in A_0 for i.i.d. X_1,\dots,X_m \in U(M), denoted P^{\bar{\mu}}_{A_0}(m), is the same as the \mu-probability of X_1,\dots,X_n \in nA_0=B for i.i.d. X_1,\dots,X_m \in U([0,1]), that is, P_B(m). On the other hand, by Remark 1, the probability P_A(m) is identical to P^{\bar{\mu}}_{A_0}(m). ■
Choosing the best possible set A
How do one construct a set A of smallest possible measure so that for any given set of m points we can find a \tau so that all m points belong to A+\tau \pmod 1. We start with a well-known fact about the Cantor set that covers the case m=2.
Theorem. Let A be the middle third Cantor set. Then for any two points \{x_1,x_2\} there exists a \tau so that \{x_1,x_2\} \subset A+\tau.
Can we extend this to m>2? The following result tells us that we can, not only catch m random points, but countably many points with a set A with arbitrarily small measure.
Theorem. Lad \epsilon > 0 være givet. Så findes A \subset R / Z med \mu(A) < \epsilon således at givet vilkårlig tællelig mængde X \subset R / Z findes \tauX \subset A+\tau.
Bevis. Lad A være en åben og tæt delmængde af R / Z med m(A) < \epsilon. Skriv X = \{x_1, x_2,\dots\} og sæt d_i = x_{i+1} - x_1, i = 1, 2, \dots. Så er B = \cap_i (A - d_i) en fællesmængde af åbne tætte mængder, og da R / Z med standard metrik er fuldstændigt metrisk rum følger det af Baire's sætning at B er tæt i R / Z. Derfor findes b \in B.
Sæt nu \tau = x_1 - b. Da b \in A er x_1 = b + \tau et element af A + \tau. For vilkårlig i \in N har vi b + d_i med i AA + \tau indeholder
b + d_i + \tau = b + (x_{i+1} - x_1) + x_1 - b = x_{i+1}.
Altså er x_i \in A + \tau for alle i = 1, 2, 3, \dots
Theorem. Der findes A \subset R / Z med m(A) = 0 således at givet en tællelig mængde X \subset R / Z findes \tauX \subset A + \tau.
Bevis. Lad B_n være åbne tætte mængder med |B_n| < 1/n og sæt A = \cap_{n \in N} B_n. Fortsæt så som i beviset af sidste sætning og bemærk at en tællelig fællesmængde af tællelige fællesmængder er en tællelig fællesmængde. ■
Choosing the worst possible set A
Suppose now that your worst enemy can choose the set A under the restriction that \mu (A)=a for some given 1\ge a \ge 0. How should he choose it? How big will a have to be so that we always can be sure of catching m points? We start with the following observation.
Lemma 2. Let A \subseteq \mathbb{T}^1 be a (Lebesgue) measurable set. Suppose there exists points x_1, x_2 \in T^1 so that for all \tau \in T^1 it holds that x_1 and x_2 not simultaneous can be in A + \tau. Then m(A) \le 1/2.
Proof. Let d = x_2 - x_1 \in T^1. The sets A and A - d are disjoint. To see this, assume they are not. Then we can find u \in A \cap (A - d). Let \tau = x1 - u. Note that x_1-\tau=u \in A and x_2-\tau=d+u\in A, hence x_1, x_2 \in A + \tau which contradicts our assumption. Since A, it follows by translation invariance of the Lebesgue measure that
2 m(A) = m(A) + m(A - d) \le m(\mathbb{T}^1) = 1. ■
By Lemma 2 we have that if A is a arbitrary subset of \mathbb{T}^1 with |A| > 1/2, then there is a \tau so that \{x_1, x_2\} \subset A + \tau. Obviously, we cannot do better than 1/2, e.g., the set A=[0,1/2-\epsilon] cannot catch all pairs of points. Now one can ask how large the set A needs to be in order for us to always catch m random points. More precisely: What is the smallest length a such that any set A \subset \mathbb{T}^1 satisfying |A| \ge a or |A| > a, there exists a \tau so that \{x_1, \dots, x_m\} \subset A + \tau ?
TO-DO: Extend lemma to |A|=1/2 and to m points |A| = a > 1-1/m.
Extension to higher dimensions \mathbb{S}^n for n>1.
m points on an n-dimensional sphere. What holds here?
Literature
Vaguely related to the Bertrand paradox | {"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": 0.9984254837036133, "perplexity": 2276.9891344101125}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934804019.50/warc/CC-MAIN-20171117223659-20171118003659-00452.warc.gz"} |
http://tranpedia.selfhow.com/2018/02/frobenius-polycyclic-ring.html | # Frobenius polycyclic ring
Frobenius multi-component ring(Frobenius, Arrowhead: Frobenius algebra), orFrobenius algebra Is a thing with special in giving a good dual theory among the studies.
Frobenius multiple circle began to be studied as a generalization by and in the 1930's, and it was named after. () And especially () For the first time discovered a rich dual theory. Using this, we characterize Frobenius multi-component ring in (), and this property of Frobenius multi-component ring is perfect I called it duality. Frobenius multi-way ring was generalized to. This is right. Recently, the interest in Frobenius multi-circle has been rising from the connection with.
⇧ Frobenius multiple circle ⊃ symmetric multiple ring ⊃ ⊃ ⊃
Definition
A on * k isFrobenius multiple circleis σ : _ A _ x _ A _ _ _ _ _
σ _ ( ab , _ c ) = σ _ (_a , bc )
It means that there are things that satisfy. This bilinear format is calledFrobenius format(Frobenius form).
Equivalent characterization refers to linear mapping λ : _ A _ → _ k _ and that does not include left non-zero (λ ).
When Frobenius type σ is Frobenius polygon, or when the same condition λ _ ( ab ) = _ λ _ ( ba _) is satisfied, It is called symmetric multi-dimensional ring.
There are also different concepts that are almost irrelevant.
An example
1. Frobenius multivariate with Frobenius form σ _ ( a , _ b ) = ( ab ) on the body k. 2. Arbitrary Finite Dimension Unit Coupled Multi-rings A _ has its own homomorphism to its own homomorphic ring End (_A ). The bilinear form can be defined on A as in Example 1. If this bilinear form is non-degenerate, A _ has the structure of Frobenius multi-circle. 3. All on the body is the Frobenius multi-dimensional ring with the following Frobenius form _σ , ie σ _ ( a , _ b ) is the factor of the identity element in _ a _ _ . This is the special case of Example 2. 4. For body _k, 4-dimensional k - algebra _ k _ [ x , _ y ] / ( x _ 2, _ y _ 2) are Frobenius multi-component rings. This follows the characterization of the exchange Frobenius ring described below. This ring is a local ring with the ideal generated with x _ and _ y _ as the local idea because it has the only minimal ideal generated by _ x _ . 5. For _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ is not a Frobenius multi - dimensional**. From xA _ _ derived from x ↦ y {\ displaystyle x \ mapsto y}! [x \ mapsto y] (https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2452f2d32e5424f3db361de033fd49a73f9dcc) Of _A homomorphism can not be extended to A homomorphism from A _ _ _, hence the ring is not self - primitive and not Frobenius.
nature
• Frobenius multi-dimensional ring and / or are Frobenius multi-way ring.
• It is the same that having a finite dimensional multiple ring on the body is Frobenius, that the right is introductory and that the plural ring has unique.
• Convertible locality Frobenius polygon is just like being local and including finite dimensions on the remainder.
• Frobenius multiple ring is (), in particular, it is left (right) and left (right).
• For body k, it is equivalent that the finite dimensional unitary coupled multi-way ring is Frobenius and that the imported right_A_ - group Hom _ k _ ( A , _ k ) is isomorphic to the right of _ A .
• For infinite field k, the finite dimensional unitary coupled multi-way ring is Frobenius unless there are only a finite number of local minima. If _ _ is a finite order of _ _, finite dimensions _ F _ - multiple rings are naturally finite dimensions _ k _ - multiple rings by, it is equivalent to Frobenius _ F _ - multicircular ring and Frobenius _k _ - multi-ring . In other words, the Frobenius nature does not depend on the body unless the multi-component ring is a finite dimensional multiple ring.
• Similarly, if F is a finite order extension field of k, all k - multiple rings _ A _ naturally generate F _ - multiply ring _ F _ ⊗ _ k _ A _ _ is Frobenius k _ - multicircular ring and _ F _ ⊗ It is equivalent that _k _ A _ is Frobenius _ F _ - multi-component ring.
• In the unitary coupling multiple rings of right finite dimension expression, the Frobenius plural circle A _ is exactly like a multinary circle whose _M _ has the same dimensions as its _ A _ dual Hom _ A _ (_M , _ A ) is there. Among these multiple rings, the _A dual of simple additions is always simple.
footnote
1. ****, Definition 4.2.5.
References
; (1937),, Proc. Nat. Acad. Sci. USA 23(4): 236-240,:,,, DeMeyer, F., Ingraham, E. (1971), Separable Algebras over Commutative Rings , Lect. Notes Math 181, Springer (1958), "Remarks on quasi-Frobenius rings", Illinois Journal of Mathematics 2: 346-354,, (1903), "Theorie der hyperkomplexen Größen I" (German), Sitzungsberichte der Preussischen Akademie der Wissenschaften -: 504-537, Kock, Joachim (2003), Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society student texts, Cambridge: Cambridge University Press, Lam, T. Y. (1999),, Graduate Texts in Mathematics No. 189, Berlin, New York: Lurie, Jacob,, (1939),, (Annals of Mathematics)40(3): 611-633,::,,, (1941),, __ (Annals of Mathematics)42(1): 1-21,:,,, (1938),, 39(3): 634-658,::,,,, Onodera, T. (1964), "Some studies on projective Frobenius extensions", __18: 89-107 Weibel, Charles A. (1994).. Cambridge University Press.
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http://math.stackexchange.com/questions/265807/almost-complex-structures-on-spheres | # Almost complex structures on spheres
It is fairly well-known that the only spheres which admit almost complex structures are $S^2$ and $S^6$. By embedding $S^6$ in the imaginary octonions, we obtain a non-integrable almost complex structure on $S^6$.
By embedding $S^2$ in the imaginary quaternions, do we obtain an almost complex structure on $S^2$? If so, is it the one induced by the complex structure corresponding to $\mathbb{CP}^1$?
-
The point is that $V = \mathbb{R}^3 = \operatorname{Im}\mathbb{H}$ and $V = \mathbb{R}^7 = \operatorname{Im}\mathbb{O}$ inherit a cross-product $V \times V \to V$ from quaternion and octonion multiplication. It is given by $u \times v = \operatorname{Im}(uv) = \frac{1}{2}(uv - vu)$.
Given an oriented hypersurface $\Sigma \subset V$ with corresponding Gauß map $\nu \colon \Sigma \to S^n$ ($n \in 2,6$) sending a point $x \in \Sigma$ to the outer unit normal $\nu(x) \perp T_x\Sigma$ one obtains an almost complex structure by setting $$J_x(u) = \nu(x) \times u\qquad\text{for }u \in T_x\Sigma.$$
One economic way to see that this is an almost complex structure: The cross-product is bilinear and antisymmetric. It is related to the standard scalar product via $$\langle u \times v, w\rangle = \langle u, v \times w\rangle$$ (which shows that $u\times v$ is orthogonal to both $u$ and $v$) and the Graßmann identity $u \times (v \times w) = \langle u,w\rangle v - \langle u,v\rangle w$ (only valid in dimension $3$) has the variant $$(u \times v) \times w + u \times (v\times w) = 2\langle u,w\rangle v-\langle v,w\rangle u - \langle v,u\rangle w$$ valid in $3$ and $7$ dimensions (which shows that $u \times (u \times v) = -v$ for $u \perp v$ and $|u| = 1$).
Therefore $J_x(u) = \nu(x) \times u$ indeed defines an almost complex structure $J_x \colon T_x\Sigma \to T_x\Sigma$.
Specializing this to $\Sigma = S^2$ embedded as unit sphere in $\operatorname{Im}\mathbb{H}$ you can “see” (or calculate) that $J_x$ acts in exactly the same way as multiplication by $i$ on $\mathbb{CP}^1$.
-
A nice discussion of almost-complex structures (including this class of examples) is in chapter 4 of McDuff and Salamon's Introduction to symplectic topology. – Martin Dec 27 '12 at 11:01
Another nice thing to note is that the integrability of the almost complex structure induces by the multiplication is given by the vanishing of the expression $x(yz)-(xy)z$ for any three elements of the division ring; this can be done writing the Nijenhuis tensor or noticing that $J$ has to be parallel with respect to the induced levi-civita connection. Hence, the given structure is integrable in dimension $2$ but not in dimension $6$, because octonions are not associative. – wisefool Dec 29 '12 at 14:07
Martin, I will definitely check out that book as I am yet to grasp all the details from your answer (through no fault of yours). – Michael Albanese Dec 30 '12 at 13:20
@Michael: Take all the time you need, this stuff isn't easy! I might be able to help you or expand my answer if you tried to pin down what is causing trouble. Since most of my answer is about algebraic properties of $\mathbb{H}$ and $\mathbb{O}$ I suspect McDuff-Salamon might not be the right place to go (they are more concerned with geometric properties of almost-complex structures). I learned about Quaternions and Octonions from Numbers which is still one of the more user-friendly expositions I've seen. – Martin Dec 30 '12 at 13:52
Alternatively, John Baez has a page dedicated to Octonions where you can find links to good resources after the table of contents. – Martin Dec 30 '12 at 14:00
Yes and yes.
We get an almost complex structure $J$ on the two-sphere $S = \{ ai + bj + ck \mid a^2 + b^2 + c^2 = 1\}$ embedded in the space of imaginary quaternions in exactly the same way as for the octonions.
Since $S$ is of real dimension two, every almost complex structure on $S$ is integrable. Then $X = (S,J)$ is a compact complex curve of genus 0, and any such curve is isomorphic to $\mathbb{CP}^1$.
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http://en.wikiversity.org/wiki/Motion_-_Dynamics | # Motion - Dynamics
Next: Energy
Previous: Motion - Kinematics
Up: Topic: High School Physics
Dynamics is the study of why things move, in contrast to kinematics, which is concerned with describing the motion. Things move because they are subjected to forces. A force applied to an object causes an acceleration proportional to the object's mass:F = ma. In this equation, Newton's Second Law, F and a are both vectors. The units have to be consistent.
## Newton's 1st Law of Motion
Objects at rest tend to stay at rest unless acted upon by an external unbalanced force. Objects moving with uniform motion(constant velocity in a straight line) will continue to do so unless acted upon by an external unbalanced force.
This law makes it easy to understand certain situations. For example, consider a bus moving steadily at 100 km/h along a highway. What is the force acting on it? Many forces are acting - gravity, air resistance, friction with the highway, and so on. However, we know immediately that all of them cancel out because uniform motion means NO net external force acting on the bus.
It helps understand what happens to a passenger in a car when the car has a collision and stops suddenly. The passenger continues in uniform motion until a seat belt or windshield exerts an external force to decellerate him.
A good experiment to illustrate the first law is to put a book on a skateboard. Accelerate them up to a good speed, then suddenly stop the skateboard. The book will continue on if its friction force with the board is small
## Newton's 2nd Law of Motion
Newton's 2nd Law of Motion states:
"the rate of change of the momentum of an object is directly proportional to the resultant force acting upon it".
Written mathematically, this gives
$F = \frac{dp}{dt}$
this can easily be reduced to F = ma by using differentials
$F = \frac{dp}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}$
in a constant mass system
$v\frac{dm}{dt} = 0$
and so
$F = m\frac{dv}{dt} = ma$
as
$a = \frac{dv}{dt}$
Therefore, in a system of constant mass, the acceleration of an object is directly proportional to the resultant force acting upon it.
This formula is an experimental result. You can find it for yourself if you have some means of measuring acceleration, such as a tickertape timer or sonar attachment for a calculator to measure velocity continuously.
In the experiment, various forces are applied to a wheeled cart or glider on a frictionless air track. The easiest way to get a known force is to use the force of gravity on a hanging weight with a pulley to change the force from horizontal to vertical. Each 100 grams of mass has a force of gravity of about 1 Newton.
The tickertape is pulled through a timer that marks a dot on it every tenth of a second. From the distance between dots, the velocity can be calculated. From the change in velocity from one pair of dots to the next, the acceleration can be calculated. The experiment is repeated with various hanging masses causing different pulling forces and the acceleration is measured from the recorded motion. Graphing the experimental values for the applied force versus the resulting accelerations produces a straight line graph to within experimental accuracy. The slope is equal to the mass of the moving system to within experimental error. The formula for the graph is therefore F = ma. Force has units of Newtons where a Newton (N) is equal to a kg·m/s².
### Example 1
A 1000 kg car accelerates at 0.5 m/s². Calculate the force that must be applied to it.
Solution
\begin{align} F & = ma \\ & = 1000 \mathrm{kg} \times 0.5 \text{m/s}^2 \\ & = 500 \text{kg} \cdot \text{m/s}^2 \\ & = 500 \mathrm{N} \end{align}
### example 2
A 0.085 kg bullet is fired from a rifle and emerges with a speed of 400 m/s. Assuming that the bullet has constant acceleration over the 0.5 m length of gun barrel, calculate the force on the bullet.
Solution: This is one of those tricky problems when the time is not known so we can't use a = Δv/Δt. We could sketch a v vs t graph where we know the area beneath is the distance 0.5 m:
Knowing the time for the bullet to accelerate in the rifle, we can find the acceleration and then the force:
\begin{align} F & = ma \\ & = m \frac{\Delta v}{\Delta t} \\ & = 0.085 \frac{400}{0.0025} \\ & = 13600 \mathrm{N} \end{align}
## Newton's 3rd Law of Motion
If object A applies a force on object B, then B will apply an equal force on A in the opposite direction. For example, if a car accelerates due to a 500 N force on it there must be another object that feels a 500 N force in the opposite direction - the road. An aircraft accelerates because it pushes air backward, and the air then pushes the aircraft forward. In space, where there is no air, a rocket must be used - it pushes backward on its exhaust material and the exhaust pushes forward on the rocket.
Often there are multiple forces involved. When a person shoots a rifle, a force is obviously applied to the bullet. It is the hot gunpowder gases that push on the bullet, and the bullet pushes back on the gases. The gases push back on the gun and the gun pushes forward on the gases. The gun pushes back on the person holding it, and the person pushes forward on the gun. You could continue this to the person pushing back on the Earth and the Earth pushing forward on the gun.
The Earth feels many forces from millions of cars and billions of people pushing when they accelerate. Most of these forces are canceled out when the cars or persons stop and all average out because everything is pushing different directions.
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https://zbmath.org/?q=an:1185.60076 | ×
# zbMATH — the first resource for mathematics
Discretizing the fractional Lévy area. (English) Zbl 1185.60076
Summary: We give sharp bounds for the Euler discretization of the Lévy area associated to a $$d$$-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean square convergence rate of the Euler scheme, depending on the Hurst parameter $$H\in (1/4,1)$$. For $$H<3/4$$ the exact convergence rate is $$n^{-2H+1/2}$$, where $$n$$ denotes the number of the discretization subintervals, while for $$H=3/4$$ it is $$n^{-1}\sqrt{\log(n)}$$ and for $$H>3/4$$ the exact rate is $$n-1$$. Moreover, we also show that a trapezoidal scheme converges (at least) with the rate $$n-2H+1/2$$. Finally, we derive the asymptotic error distribution of the Euler scheme. For $$H\leq 3/4$$ one obtains a Gaussian limit, while for $$H>3/4$$ the limit distribution is of Rosenblatt type.
##### MSC:
60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations
Full Text:
##### References:
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http://blog.jpolak.org/?tag=set-theory | # Tag Archives: set theory
## Countable dense total orders without endpoints
A total ordering $\lt$ on a set $S$ is called dense if for every two $x,y\in S$ with $x \lt y$, there exists a $z\in S$ such that $x\lt z\lt y$. A total ordering is said to be without endpoints if for every $x\in S$ there exists $y,z\in S$ such that $y \lt x\lt z$. […] | {"extraction_info": {"found_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": 0.9917683005332947, "perplexity": 120.67446404210763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257316.10/warc/CC-MAIN-20190523164007-20190523190007-00113.warc.gz"} |
http://support.sas.com/documentation/cdl/en/statug/67523/HTML/default/statug_power_syntax20.htm | # The POWER Procedure
### ONESAMPLEFREQ Statement
Subsections:
• ONESAMPLEFREQ <options>;
The ONESAMPLEFREQ statement performs power and sample size analyses for exact and approximate tests (including equivalence, noninferiority, and superiority) and confidence interval precision for a single binomial proportion.
#### Summary of Options
Table 77.8 summarizes the options available in the ONESAMPLEFREQ statement.
Table 77.8: ONESAMPLEFREQ Statement Options
Option
Description
Define analysis
Specifies an analysis of precision of a confidence interval
Specifies the statistical analysis
Specify analysis information
Specifies the significance level
Specifies the lower and upper equivalence bounds
Specifies the lower equivalence bound
Specifies the equivalence or noninferiority or superiority margin
Specifies the null proportion
Specifies the number of sides and the direction of the statistical test
Specifies the upper equivalence bound
Specify effect
Specifies the desired confidence interval half-width
Specifies the binomial proportion
Specify variance estimation
Specifies how the variance is computed
Specify sample size
Enables fractional input and output for sample sizes
Specifies the sample size
Specify power and related probabilities
Specifies the desired power of the test
Specifies the probability of obtaining a confidence interval half-width less than or equal to the value specified by HALFWIDTH=
Choose computational method
Specifies the computational method
Control ordering in output
Controls the output order of parameters
Table 77.9 summarizes the valid result parameters for different analyses in the ONESAMPLEFREQ statement.
Table 77.9: Summary of Result Parameters in the ONESAMPLEFREQ Statement
Analyses
Solve For
Syntax
CI= WILSON
Prob(width)
CI= AGRESTICOULL
Prob(width)
CI= JEFFREYS
Prob(width)
CI= EXACT
Prob(width)
CI= WALD
Prob(width)
CI= WALD_CORRECT
Prob(width)
Power
Power
Sample size
Power
Power
Sample size
TEST= EQUIV_EXACT
Power
TEST= EQUIV_Z METHOD= EXACT
Power
TEST= EQUIV_Z METHOD= NORMAL
Power
Sample size
TEST= EXACT
Power
TEST= Z METHOD= EXACT
Power
TEST= Z METHOD= NORMAL
Power
Sample size
#### Dictionary of Options
ALPHA=number-list
specifies the level of significance of the statistical test. The default is 0.05, corresponding to the usual 0.05 100% = 5% level of significance. If the CI= and SIDES= 1 options are used, then the value must be less than 0.5. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
CI
CI=AGRESTICOULL | AC
CI=JEFFREYS
CI=EXACT | CLOPPERPEARSON | CP
CI=WALD
CI=WALD_CORRECT
CI=WILSON | SCORE
specifies an analysis of precision of a confidence interval for the sample binomial proportion.
The value of the CI= option specifies the type of confidence interval. The CI= AGRESTICOULL option is a generalization of the "Adjusted Wald / add 2 successes and 2 failures" interval of Agresti and Coull (1998) and is presented in Brown, Cai, and DasGupta (2001). It corresponds to the TABLES / BINOMIAL (AGRESTICOULL) option in PROC FREQ. The CI= JEFFREYS option specifies the equal-tailed Jeffreys prior Bayesian interval, corresponding to the TABLES / BINOMIAL (JEFFREYS) option in PROC FREQ. The CI= EXACT option specifies the exact Clopper-Pearson confidence interval based on enumeration, corresponding to the TABLES / BINOMIAL (EXACT) option in PROC FREQ. The CI= WALD option specifies the confidence interval based on the Wald test (also commonly called the z test or normal-approximation test), corresponding to the TABLES / BINOMIAL (WALD) option in PROC FREQ. The CI= WALD_CORRECT option specifies the confidence interval based on the Wald test with continuity correction, corresponding to the TABLES / BINOMIAL (CORRECT WALD) option in PROC FREQ. The CI= WILSON option (the default) specifies the confidence interval based on the score statistic, corresponding to the TABLES / BINOMIAL (WILSON) option in PROC FREQ.
Instead of power, the relevant probability for this analysis is the probability of achieving a desired precision. Specifically, it is the probability that the half-width of the confidence interval will be at most the value specified by the HALFWIDTH= option.
EQUIVBOUNDS=grouped-number-list
specifies the lower and upper equivalence bounds, representing the same information as the combination of the LOWER= and UPPER= options but grouping them together. The EQUIVBOUNDS= option can be used only with equivalence analyses (TEST= EQUIV_ADJZ | EQUIV_EXACT | EQUIV_Z). Values must be strictly between 0 and 1. For information about specifying the grouped-number-list, see the section Specifying Value Lists in Analysis Statements.
HALFWIDTH=number-list
specifies the desired confidence interval half-width. The half-width for a two-sided interval is the length of the confidence interval divided by two. This option can be used only with the CI= analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
LOWER=number-list
specifies the lower equivalence bound for the binomial proportion. The LOWER= option can be used only with equivalence analyses (TEST= EQUIV_ADJZ | EQUIV_EXACT | EQUIV_Z). Values must be strictly between 0 and 1. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
MARGIN=number-list
specifies the equivalence or noninferiority or superiority margin, depending on the analysis.
The MARGIN= option can be used with one-sided analyses (SIDES = 1 | U | L), in which case it specifies the margin added to the null proportion value in the hypothesis test, resulting in a noninferiority or superiority test (depending on the agreement between the effect and hypothesis directions and the sign of the margin). A test with a null proportion and a margin m is the same as a test with null proportion and no margin.
The MARGIN= option can also be used with equivalence analyses (TEST= EQUIV_ADJZ | EQUIV_EXACT | EQUIV_Z) when the NULLPROPORTION= option is used, in which case it specifies the lower and upper equivalence bounds as and , where is the value of the NULLPROPORTION= option and m is the value of the MARGIN= option.
The MARGIN= option cannot be used in conjunction with the SIDES= 2 option. (Instead, specify an equivalence analysis by using TEST= EQUIV_ADJZ or TEST= EQUIV_EXACT or TEST= EQUIV_Z). Also, the MARGIN= option cannot be used with the CI= option.
Values must be strictly between –1 and 1. In addition, the sum of NULLPROPORTION and MARGIN must be strictly between 0 and 1 for one-sided analyses, and the derived lower equivalence bound (2 * NULLPROPORTION – MARGIN) must be strictly between 0 and 1 for equivalence analyses.
For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
METHOD=EXACT | NORMAL
specifies the computational method. METHOD= EXACT (the default) computes exact results by using the binomial distribution. METHOD= NORMAL computes approximate results by using the normal approximation to the binomial distribution.
NFRACTIONAL
NFRAC
enables fractional input and output for sample sizes. This option is invalid when the METHOD= EXACT option is specified. See the section Sample Size Adjustment Options for information about the ramifications of the presence (and absence) of the NFRACTIONAL option.
NTOTAL=number-list
specifies the sample size or requests a solution for the sample size with a missing value (NTOTAL= .). For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
NULLPROPORTION=number-list
NULLP=number-list
specifies the null proportion. A value of 0.5 corresponds to the sign test. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
OUTPUTORDER=INTERNAL | REVERSE | SYNTAX
controls how the input and default analysis parameters are ordered in the output. OUTPUTORDER= INTERNAL (the default) arranges the parameters in the output according to the following order of their corresponding options:
The OUTPUTORDER= SYNTAX option arranges the parameters in the output in the same order in which their corresponding options are specified in the ONESAMPLEFREQ statement. The OUTPUTORDER= REVERSE option arranges the parameters in the output in the reverse of the order in which their corresponding options are specified in the ONESAMPLEFREQ statement.
POWER=number-list
specifies the desired power of the test or requests a solution for the power with a missing value (POWER= .). The power is expressed as a probability, a number between 0 and 1, rather than as a percentage. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
PROBWIDTH=number-list
specifies the desired probability of obtaining a confidence interval half-width less than or equal to the value specified by the HALFWIDTH= option. A missing value (PROBWIDTH= .) requests a solution for this probability. Values are expressed as probabilities (for example, 0.9) rather than percentages. This option can be used only with the CI= analysis. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
PROPORTION=number-list
P=number-list
specifies the binomial proportion—that is, the expected proportion of successes in the hypothetical binomial trial. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
SIDES=keyword-list
specifies the number of sides (or tails) and the direction of the statistical test. For information about specifying the keyword-list, see the section Specifying Value Lists in Analysis Statements. Valid keywords are as follows:
1
one-sided with alternative hypothesis in same direction as effect
2
two-sided
U
upper one-sided with alternative greater than null value
L
lower one-sided with alternative less than null value
If the effect size is zero, then SIDES= 1 is not permitted; instead, specify the direction of the test explicitly in this case with either SIDES= L or SIDES= U. The default value is 2.
TEST= ADJZ | EQUIV_ADJZ | EQUIV_EXACT | EQUIV_Z | EXACT | Z
TEST
specifies the statistical analysis. TEST= ADJZ specifies a normal-approximate z test with continuity adjustment. TEST= EQUIV_ADJZ specifies a normal-approximate two-sided equivalence test based on the z statistic with continuity adjustment and a TOST (two one-sided tests) procedure. TEST= EQUIV_EXACT specifies the exact binomial two-sided equivalence test based on a TOST (two one-sided tests) procedure. TEST= EQUIV_Z specifies a normal-approximate two-sided equivalence test based on the z statistic without any continuity adjustment, which is the same as the chi-square statistic, and a TOST (two one-sided tests) procedure. TEST or TEST= EXACT (the default) specifies the exact binomial test. TEST= Z specifies a normal-approximate z test without any continuity adjustment, which is the same as the chi-square test when SIDES= 2.
UPPER=number-list
specifies the upper equivalence bound for the binomial proportion. The UPPER= option can be used only with equivalence analyses (TEST= EQUIV_ADJZ | EQUIV_EXACT | EQUIV_Z). Values must be strictly between 0 and 1. For information about specifying the number-list, see the section Specifying Value Lists in Analysis Statements.
VAREST=keyword-list
specifies how the variance is computed in the test statistic for the TEST= Z, TEST= ADJZ, TEST= EQUIV_Z, and TEST= EQUIV_ADJZ analyses. For information about specifying the keyword-list, see the section Specifying Value Lists in Analysis Statements. Valid keywords are as follows:
NULL
(the default) estimates the variance by using the null proportion(s) (specified by some combination of the NULLPROPORTION= , MARGIN= , LOWER= , and UPPER= options). For TEST= Z and TEST= ADJZ, the null proportion is the value of the NULLPROPORTION= option plus the value of the MARGIN= option (if it is used). For TEST= EQUIV_Z and TEST= EQUIV_ADJZ, there are two null proportions, corresponding to the lower and upper equivalence bounds, one for each test in the TOST (two one-sided tests) procedure.
SAMPLE
estimates the variance by using the observed sample proportion.
This option is ignored if the analysis is one other than TEST= Z, TEST= ADJZ, TEST= EQUIV_Z, or TEST= EQUIV_ADJZ.
#### Option Groups for Common Analyses
This section summarizes the syntax for the common analyses supported in the ONESAMPLEFREQ statement.
##### Exact Test of a Binomial Proportion
The following statements demonstrate a power computation for the exact test of a binomial proportion. Defaults for the SIDES= and ALPHA= options specify a two-sided test with a 0.05 significance level.
proc power;
onesamplefreq test=exact
nullproportion = 0.2
proportion = 0.3
ntotal = 100
power = .;
run;
##### z Test
The following statements demonstrate a sample size computation for the z test of a binomial proportion. Defaults for the SIDES= , ALPHA= , and VAREST= options specify a two-sided test with a 0.05 significance level that uses the null variance estimate.
proc power;
onesamplefreq test=z method=normal
nullproportion = 0.8
proportion = 0.85
sides = u
ntotal = .
power = .9;
run;
##### z Test with Continuity Adjustment
The following statements demonstrate a sample size computation for the z test of a binomial proportion with a continuity adjustment. Defaults for the SIDES= , ALPHA= , and VAREST= options specify a two-sided test with a 0.05 significance level that uses the null variance estimate.
proc power;
nullproportion = 0.15
proportion = 0.1
sides = l
ntotal = .
power = .9;
run;
##### Exact Equivalence Test for a Binomial Proportion
You can specify equivalence bounds by using the EQUIVBOUNDS= option, as in the following statements:
proc power;
onesamplefreq test=equiv_exact
proportion = 0.35
equivbounds = (0.2 0.4)
ntotal = 50
power = .;
run;
You can also specify the combination of NULLPROPORTION= and MARGIN= options:
proc power;
onesamplefreq test=equiv_exact
proportion = 0.35
nullproportion = 0.3
margin = 0.1
ntotal = 50
power = .;
run;
Finally, you can specify the combination of LOWER= and UPPER= options:
proc power;
onesamplefreq test=equiv_exact
proportion = 0.35
lower = 0.2
upper = 0.4
ntotal = 50
power = .;
run;
Note that the three preceding analyses are identical.
##### Exact Noninferiority Test for a Binomial Proportion
A noninferiority test corresponds to an upper one-sided test with a negative-valued margin, as demonstrated in the following statements:
proc power;
onesamplefreq test=exact
sides = U
proportion = 0.15
nullproportion = 0.1
margin = -0.02
ntotal = 130
power = .;
run;
##### Exact Superiority Test for a Binomial Proportion
A superiority test corresponds to an upper one-sided test with a positive-valued margin, as demonstrated in the following statements:
proc power;
onesamplefreq test=exact
sides = U
proportion = 0.15
nullproportion = 0.1
margin = 0.02
ntotal = 130
power = .;
run;
##### Confidence Interval Precision
The following statements performs a confidence interval precision analysis for the Wilson score-based confidence interval for a binomial proportion. The default value of the ALPHA= option specifies a confidence level of 0.95.
proc power;
onesamplefreq ci=wilson
halfwidth = 0.1
proportion = 0.3
ntotal = 70
probwidth = .;
run;
#### Restrictions on Option Combinations
To specify the equivalence bounds for TEST= EQUIV_ADJZ, TEST= EQUIV_EXACT, and TEST= EQUIV_Z, use any of these three option sets: | {"extraction_info": {"found_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": 0.9342617392539978, "perplexity": 3981.2846697858395}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825227.80/warc/CC-MAIN-20171022113105-20171022133105-00838.warc.gz"} |
https://ng.siyavula.com/read/maths/jss3/constructions/08-constructions | We think you are located in Nigeria. Is this correct?
# Chapter 8: Constructions
The ancient Greek mathematicians were the first people to do constructions in Mathematics. A construction in Mathematics is an accurate drawing of angles and lines. You are only allowed to use a straightedge and a pair of compasses with a pencil in it when you do a construction.
1. A straightedge
A straightedge is an instrument with a straight edge, so we use it to draw straight lines. It is a ruler without any markings on it, because when you do constructions you are not allowed to take any measurements.
Most of the time you will use your ruler when you do constructions and have to draw straight lines. The important thing to remember is that although there are markings on the ruler, you should not use them to do your constructions. You may only use the ruler to draw a straight line. You may use it to check your constructions once you have finished them.
2. A pair of compasses (or a compass) is an instrument to draw circles.
It has two legs:
• One leg has a sharp point so that you can put it firmly into position on a page. When you draw a circle the sharp point will stay on the dot where you positioned it. The sharp point will not move.
• The other leg has space to insert a pencil. You move the leg with the pencil to and draw the circle on the page.
The two legs of the compass are joined at the hinge. Make sure that the metal screw at the hinge is fitted tightly so that the two legs of the compass will not open too easily. Also make sure that your pencil is sharpened. A blunt pencil will lead to diagrams that are not accurate.
## 8.1 Construction of angles
Here is a fun activity for you to do: use your compass to draw a circle. Keep the the position of the legs of the compass exactly the same throughout the activity. Make a dot anywhere on the circumference of the circle.
Put the compass on the dot and start to make little arcs on the circumference of the circle, each one made from the one before, as shown in the diagram.
You will find that you can fit in exactly six of these little arcs:
You know that a full turn (or revolution) is equal to $360^{\circ}$, so if we join the centre of the circle to each one of the six small arcs on the circumference, we form six angles of $60^{\circ}$ each.
If you are asked to construct an angle of $60^{\circ}$, you simply do a part of the activity that you have done above.
### Construct an angle of 30 degrees
If you are asked to construct an angle of $30^{\circ}$, you construct an angle of $60^{\circ}$ and then bisect it.
In a previous year, you learned how to construct an angle of 60 degrees, and how to bisect an angle. You will use these two skills to construct an angle of 30 degrees.
### Worked example 8.1: Constructing an angle of 30 degrees
You are given a line segment $BD$ as shown below. Construct $EB$ so that $C$ is a point on $BD$ and $\angle EBC$ = $30^{\circ}$.
1. Step 1: Draw a large arc.
Firstly, construct an angle of $60^{\circ}$ (Step 1 to Step 3).
Copy the given line segment into your workbook and make a dot at point $B$.
Put the point of the compass at $B$ (on the blue dot) and draw a large arc as shown. Label point $C$ where the arc intersects the line segment. Keep the compass legs open in exactly the same position for the next step.
2. Step 2: Draw a second, small arc.
Put the point of the compass onto $C$ (on the orange dot) and draw a small arc to intersect with the first large arc.
Label the point of intersection as point $A$.
3. Step 3: Complete the angle of $60^{\circ}$.
Use a ruler and draw a line segment from point $B$ and through point $A$.
4. Step 4: Draw the first arc to bisect the angle of $60^{\circ}$.
Put the compass on point $A$ and draw an arc as shown.
5. Step 5: Draw the second arc to bisect the angle of $60^{\circ}$.
Keep the compass open at exactly the same width as for Step 4. Put the compass on point $C$ and draw an arc as shown.
Label the point of intersection as $E$.
6. Step 6: Complete the construction.
Draw a line segment from point $B$ through point $E$.
7. Step 7: Check your work.
Use a protractor to measure $E\hat{B}C$. The angle should be equal to $30^{\circ}$.
It is a good idea to rotate your workbook If you have to do a construction involving a line that is not horizontal on the page in front of you. Never erase the construction lines after you have completed a construction.
### Exercise 8.1: Construct an angle of 30 degrees
1. Construct an angle of $60^{\circ}$. Label the angle as $A\hat{B}C$.
Follow the first three steps of Worked example 8.1 to construct an angle of $60^{\circ}$.
2. Bisect $A\hat{B}C$ of question 1 so that $E\hat{B}C$ = $30^{\circ}$.
Continue with the construction from question 1. Follow steps 4, 5, 6 and 7 from Worked example 8.1 to construct an angle of $30^{\circ}$.
3. Construct an angle of $30^{\circ}$. Label the angle as $P\hat{Q}R$.
Follow all the steps in Worked example 8.1 to construct an angle of $30^{\circ}$.
### Construct an angle of 45 degrees
If you are asked to construct an angle of $45^{\circ}$, you construct an angle of $90^{\circ}$ and then bisect it.
To construct an angle of 90 degrees, you will use of the fun activity we did earlier, where you made six small arcs on the circumference of a circle.
In this construction, you will make only two arcs: you will make the first arc to form an angle of $60^{\circ}$ and the second arc to form an angle of $120^{\circ}$.
An angle of $90^{\circ}$ is exactly halfway between $60^{\circ}$ and $120^{\circ}$, so you will construct the one arm of the right angle exactly halfway between the arcs for $60^{\circ}$ and $120^{\circ}$.
You have learned how to construct an angle of 90 degrees, and to bisect an angle. You will use these skills to construct an angle of 45 degrees.
There is more than one way to do this construction, and two different methods are shown below.
### Worked example 8.2: Constructing an angle of 45 degrees (Method 1)
Construct an angle of 45 degrees.
Method: Construct an angle of 90 degrees and bisect the angle.
1. Step 1: Construct a line segment, make a dot at point $B$ and draw the first large arc.
Start by constructing an angle of $90^{\circ}$ (Step 1 to Step 6).
Put the compass at point $B$ (on the blue dot) and draw a large arc. Label point $C$ where the arc intersects the line segment. Keep the compass legs in exactly the same position for the next two steps.
2. Step 2: Draw the first small arc.
Put the compass on point $C$ and make an arc on the large arc that was drawn in Step 1.
Label this as point $P$. (Point $P$ shows the mark for an angle of $60^{\circ}$.)
3. Step 3: Draw the second small arc.
The compass should still be open at exactly the same width as for the first large arc.
Put the compass on point $P$ and make another arc on the large arc that was drawn in Step 2.
Label this as point $Q$. (Point $Q$ shows the mark for an angle of $120^{\circ}$.)
4. Step 4: Draw the first arc to construct the right angle.
We want to find the line for an angle of $90^{\circ}$. We know that this line should be exactly halfway between the arcs for the angles of $60^{\circ}$ and $120^{\circ}$.
The compass should stay in the same position for these two arcs (Step 4 and Step 5).
Put the compass on point $P$ and make an arc outside of the large arc, as shown below.
5. Step 5: Draw the second arc to construct the right angle.
Keep the compass fixed at the same width as for Step 4.
Put the compass on point $Q$ and make another arc outside of the large arc to intersect the arc already drawn there. Label the point of intersection of the two arcs as $A$.
6. Step 6: Complete the angle of $90^{\circ}$.
Use a ruler to draw a line segment from point $B$ through point $A$.
7. Step 7: To bisect the angle of $90^{\circ}$, draw a new arc.
Put the point of the compass where the big arc (from Step 1) crosses $AB$ and draw an arc outside of the large arc, and towards $C$, as shown.
8. Step 8: Draw the second arc to bisect the angle of $90^{\circ}$.
Keep the compass open the same width as for Step 7.
Put the compass on point $C$ and draw an arc as shown.
Label the point of intersection as point $E$.
9. Step 9: Complete the construction.
Draw a line segment from point $B$ through point $E$.
10. Step 10: Check your work.
Use a protractor to measure $E\hat{B}C$. The angle should be equal to $45^{\circ}$.
### Worked example 8.3: Constructing an angle of 45 degrees (Method 2)
Construct an angle of 45 degrees. You are given line $AB$ with a point $P$ on the line.
Method: Construct a perpendicular line from a point on a line segment and then bisect the right angle.
1. Step 1: On line $AB$, make arcs equal distances from $P$ on both sides.
Put the compass on point $P$ (where the blue dot is shown).
Draw an arc to the left of point $P$. Keep the compass open at the same width, and draw another arc to the right of point $P$. These arcs should intersect with line $AB$.
2. Step 2: Draw an arc from the left.
You can change the position of the compass legs, to open them wider. Put the compass on the left arc from Step 1 (where the orange dot is shown).
Draw an arc so that it is above point $P$.
3. Step 3: Draw an arc from the right.
Keep the compass open at the same width as for Step 2. Put the compass on the right arc from Step 1 (where the purple dot is shown).
Draw an arc so that it crosses the arc from Step 2.
4. Step 4: Draw the perpendicular line.
Point $Q$ is where the two arcs from Step 2 and Step 3 intersect.
Use a ruler and draw a line from point $P$ and through point $Q$. This gives you your right angle.
5. Step 5: Draw the first big arc to bisect the right angle.
Put the compass on point $P$. Keep the compass fixed at that width and draw a big arc that intersects $PQ$ and $PB$.
6. Step 6: Draw the smaller arcs to bisect the right angle
Keep the compass at the same width for the next two arcs. From the intersection of the big arc from Step 5 with the line segments, draw two arcs that intersect each other. Label the point of intersection as point $T$.
7. Step 7: Complete the construction.
Draw a line segment from point $P$ through point $T$.
8. Step 8: Check your construction.
Use a protractor and measure $Q\hat{P}T$ and $T\hat{P}B$.
If each of them is equal to $45^{\circ}$ you have made an accurate construction.
For the exercise below, remember not to erase the construction lines after you have completed a construction.
### Exercise 8.2: Construct an angle of 45 degrees
1. Use the method shown in Worked example 8.2 to construct an angle of $90^{\circ}$.
Label the angle as $A\hat{B}D$.
Follow Step 1 to Step 6 in Worked example 8.2 to construct a right angle.
2. Use your construction for question 1. Use the method shown in Worked example 8.2 to construct an angle of $45^{\circ}$. Label the angle as $E\hat{B}D$.
Follow Step 7 to Step 10 in Worked example 8.2 to do the construction.
3. Draw a line segment $AB$ and mark point $P$ on the line segment.
Construct a line from point $P$ that is perpendicular to $AB$.
Follow Step 1 to Step 4 in Worked example 8.3 to do the construction.
4. Use your construction for question 3. Use the method shown in Worked example 8.3 to construct an angle of $45^{\circ}$. Label the angle as $T\hat{P}B$.
Follow Step 5 to Step 8 in Worked example 8.3 to do the construction.
5. Construct an angle of $45^{\circ}$. Use the method that you prefer.
Construction according to Worked example 8.2:
Construction according to Worked example 8.3:
6. Amaka has to construct two angles of $45^{\circ}$ each. She says she will construct an isosceles right-angled triangle, then there will be two angles of $45^{\circ}$ each.
Is Amaka correct?
Yes, Amaka is correct.
The three angles of a triangle add up to $180^{\circ}$.
In a right-angled triangle one of the angles is equal to $90^{\circ}$.
An isosceles triangle has two equal sides and the angles opposite the equal sides are also equal.
An isosceles right-angled triangle has one angle of $90^{\circ}$ and two angles of $45^{\circ}$ each.
### Copy a given angle
The size of an angle does not depend on the length of the arms. In the diagram, below both angles are the same size.
The size of an angle depends on how far the one arm of the angle has turned (or rotated) away from the other arm of the angle. We use this fact when we want to construct an exact copy of a given angle.
### Worked example 8.4: Copying a given angle
$A\hat{B}C$ is an acute angle.
Use a compass and ruler to construct a copy of $A\hat{B}C$. Label the new angle as $A_1\hat{B_1} C_1$.
1. Step 1: Construct the first arm of the angle by drawing a horizontal line. Use the labels $B_1$ and $C_1$.
In each of the steps below, the first diagram is the given diagram, and the second diagram is the new construction. This worked example assumes that you have the original angle on paper, and can draw on it.
2. Step 2: Draw an arc on the given angle and on the new angle.
On the given diagram, put the compass on point $B$ and draw a large arc, as shown.
Keep the compass open at exactly the same width. On your copy of the diagram, put the compass on point $B_1$ and draw a copy of the first arc.
3. Step 3: Measure the distance between the two arms of the given angle.
On the given diagram, use your compass to measure the distance between the two points where the arc (of Step 2) intersects with $BA$ and $BC$.
You do not draw line segment between the two points, you only measure the distance between them by opening your compass to the correct width.
4. Step 4: Mark the same distance on the new angle.
Keep the compass open at the same width as in Step 3. Put the compass on the point where the arc intersects with $B_1 C_1$ and draw a small arc that intersects the large arc.
5. Step 5: Complete the construction.
Draw a line segment from point $B_1$ through the point where the two arcs intersect.
Mark that line $A_1$.
6. Step 6: Remember to check your work.
Use a protractor to measure $A\hat{B}C$ and $A_1\hat{B_1} C_1$.
The two angles should have the same size.
### Exercise 8.3: Copy a given angle
Construct copies of the angles provided below, and use a protractor to check that your constructed angles are the correct size.
1. $A\hat{B}C$ is an acute angle.
Use a compass and ruler to construct a copy of $A\hat{B}C$. Label the new angle as $A_1\hat{B_1} C_1$.
Follow the steps in Worked example 8.4 to do the construction.
2. $X\hat{Y}Z$ is an obtuse angle.
Use a compass and ruler to construct a copy of $X\hat{Y}Z$. Label the new angle as $X_1\hat{Y_1} Z_1$.
Follow the steps in Worked example 8.4 to do the construction.
3. $K\hat{L}M$ is a right angle.
Use a compass and ruler to construct a copy of $K\hat{L}M$. Label the new angle as $K_1\hat{L_1} M_1$.
Follow the steps in Worked example 8.4 to do the construction.
4. $N\hat{H}G$ is a reflex angle.
Use a compass and ruler to construct a copy of $N\hat{H}G$. Label the new angle as $N_1\hat{H_1 }G_1$.
Follow the steps in Worked example 8.4 to do the construction.
## 8.2 Constructions of plane shapes
A closed, two-dimensional or flat figure is called a plane shape. A plane shape has length and breadth, but no thickness.
In this section you will construct simple plane shapes, for example, parallelograms and triangles.
You may use a compass, ruler and also a protractor to do the constructions.
plane shape A plane shape is a flat figure with closed sides. It has length and breadth, but no thickness.
### Worked example 8.5: Constructing a parallelogram
Make an accurate construction of parallelogram $XWZY$. $XW$ = 5 cm, $\hat{W}$ = $80^{\circ}$ and $WZ$ = 9 cm.
You may use a protractor.
1. Step 1: Make a rough sketch of the given information.
You know that the opposite sides of a parallelogram are equal, so $ZY$ = 5 cm and $XY$ = 9 cm.
2. Step 2: Plan the order in which you will do the construction.
It is a good idea to start with $WZ$, then construct $\hat {W}$, and then $WX$.
Point $Y$ will be the point of intersection between an arc of 9 cm from point $X$ and an arc of 5 cm from point $Z$.
3. Step 3: Construct $WZ$.
Draw a line segment (longer than 9 cm) and mark point $W$.
Use a compass to measure a length of 9 cm on your ruler.
Keep the compass open at that width. Put the compass on point $W$ and draw an arc to find point $Z$.
4. Step 4: Construct $\hat{W}$.
Use a protractor to measure and mark an angle of $80^{\circ}$ at point $W$, then use a ruler to draw in the line.
5. Step 5: Construct $WX$.
Use a compass to measure a length of 5 cm on your ruler.
Keep the compass fixed at that width, put the compass point on $W$, and draw an arc to find point $X$.
6. Step 6: Construct $XY$.
Use a compass to measure a length of 9 cm on your ruler.
Keep the compass open at that width. Put the compass on point $X$ and draw an arc in the position shown below.
7. Step 7: Construct $ZY$.
Use a compass to measure a length of 5 cm on your ruler.
Keep the compass fixed at that width. Put the compass on point $Z$ and draw an arc to intersect the previous arc.
Label the point of intersection between the two arcs (Step 6 and Step 7) as point $Y$.
8. Step 8: Complete the construction.
Use a ruler to draw line segments $XY$ and $ZY$, which meet at the point at which the two arcs intersect.
9. Step 9: Remember to check your work.
Use a ruler to check that $XY$ = $WZ$ = 9 cm and $WX$ = $ZY$ = 5 cm.
Use a protractor to check that $\hat{W}$ = $80^{\circ}$.
For the exercise below, remember not to erase the construction lines after you have completed a construction.
### Exercise 8.4: Construct simple plane shapes
1. Construct parallelogram $ABCD$.
$DA$ = 7 cm, $\hat{A}$ = $115^{\circ}$ and $AB$ = 10 cm.
Follow the steps in Worked example 8.5 to do the construction.
2. Construct square $PQRS$. The length of $PQ$ is 10 cm.
You know that a square has four right angles and that all four sides of a square are equal.
Make a rough sketch first and then do the construction accurately.
Order of construction:
• Draw line segment and mark point $Q$.
• Use a compass and construct a line segment perpendicular to the first line segment at point $Q$.
• Measure a distance of 10 cm with a compass on a ruler. Place the compass on point $Q$ and draw arcs to find point $P$ and point $R$.
• Keep the compass open at 10 cm for the next two arcs also:
• Put the compass on point $R$ and draw an arc.
• Put the compass on point $P$ and draw an arc.
• Label the point of intersection between the two arcs as point $S$.
• Draw line segment $RS$ and $SP$ to complete the construction.
3. Construct $\triangle DEF$ as shown below. Do not use a protractor in this construction. Only use a compass, ruler and pencil.
Measure the size of $\hat{D}$ and $\hat{F}$.
• Draw a line segment and label point $E$.
• Measure a length of 5 cm with a compass on a ruler. Keeping the compass fixed at that width, place the compass on point $E$ and draw an arc to find point $F$.
• Use a compass to construct an angle of $60^{\circ}$ at point $E$: first draw a big arc and then (with the compass in the same position) draw another arc from the point where the first arc intersects with $EF$, as shown in orange.
• Draw a line segment from point $E$ through the point where the two arcs of the previous step intersect.
• Measure a length of 10 cm with a compass on a ruler. Keep the compass fixed at that width, place the compass on point $E$ and draw an arc to find point $D$.
• Draw a line segment from point $D$ to point $F$.
$\hat{D}$ = $30^{\circ}$ and $\hat{F}$ = $90^{\circ}$ .
4. Construct $\triangle CAT$ as shown below. Do not use a protractor in this construction. Only use a compass, ruler and pencil.
Measure the size of $\hat{C}$ and $\hat{T}$.
• Draw a long line segment and label point $A$.
• Measure a length of 6 cm with a compass on a ruler. Keep the compass open at that width, place the compass on point $A$ and draw an arc to find point $T$.
• Construct a perpendicular line segment at point $A$.
• Bisect the right angle at point $A$ to construct an angle of $45^{\circ}$.
• Measure a length of 6 cm with a compass on a ruler. Keeping the compass fixed at that width, place the compass on point $T$ and draw an arc to find point $C$.
• Draw a line segment from point $C$ to point $T$.
$\hat{C}$ = $45^{\circ}$ and $\hat{T}$ = $90^{\circ}$ .
## 8.3 Summary
• A construction in Mathematics is an accurate drawing of angles and lines.
• We use arcs in all the constructions. An arc is part of the circumference of a circle.
• There are methods for carrying out the following constructions, using only use a compass, ruler and pencil:
• an angle of 30 degrees
• an angle of 45 degrees
• copy a given angle
• simple plane shapes (for example, a parallelogram, square and triangle).
• To construct simple plane shapes, a protractor may be used if instructed.
• Never erase the construction lines after you have completed a construction. | {"extraction_info": {"found_math": true, "script_math_tex": 231, "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": 0.8696069717407227, "perplexity": 406.3673195126812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401624636.80/warc/CC-MAIN-20200929025239-20200929055239-00184.warc.gz"} |
https://www.physicsforums.com/threads/matrix-from-an-equation.538937/ | # Matrix from an equation
1. Oct 10, 2011
### phiby
I posted this question in the engineering section, but didn't get any replies there - hence posting it here.
Below is a screen shot from state space analysis in "Control Engineering" by Ogata.
http://www.flickr.com/photos/66943862@N06/6230432028/" [Broken]
I am trying to get at Eqn 3-20 from Eqn 3-18.
Can't the u part of the matrix also be written as
[1/m 0]T instead of [0 1/m]T?
What's the rationale in choosing one over the other?
Last edited by a moderator: May 5, 2017
2. Oct 11, 2011
### Fredrik
Staff Emeritus
It can, but only if you swap the first and the second row in all the other matrices as well. If you only do it to the last term as you suggest, then if you perform the matrix multiplication and matrix addition on the right-hand side of (3-20) you get two equations that are similar to (3-17) and (3-18) but have the term involving u in the one that looks like (3-17) instead of in the one that looks like (3-18).
Last edited by a moderator: May 5, 2017
3. Oct 11, 2011
### phiby
Yes, true. Stupid of me not to see that. Thank you.
Similar Discussions: Matrix from an equation | {"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": 0.8044014573097229, "perplexity": 1159.1653912283755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888041.33/warc/CC-MAIN-20180119144931-20180119164931-00611.warc.gz"} |
http://fricas.github.io/api/LODOConvertions.html | # LODOConvertions(Coeff, Ab, R)¶
convert(l1) converts an operator l1 from LinearOrdinaryDifferentialOperator1 to LinearOrdinaryDifferentialOperator3 such that the resultant operator gives the same result as the original one on application to an element of domain R.
convert(l3) converts an operator l3 from LinearOrdinaryDifferentialOperator3 to LinearOrdinaryDifferentialOperator1 such that the resultant operator gives the same result as the original one on application to an element of domain R. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8720189929008484, "perplexity": 1122.068304831825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424721.74/warc/CC-MAIN-20170724042246-20170724062246-00321.warc.gz"} |
https://demo7.dspace.org/items/a7395f0a-185f-4fed-9cbf-1b40788fc6ea | ## Strong law of large numbers on graphs and groups
##### Authors
Mosina, Natalia
Ushakov, Alexander
##### Description
We consider (graph-)group-valued random element $\xi$, discuss the properties of a mean-set $\ME(\xi)$, and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for $\xi$ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.
Comment: 29 pages, 2 figures, new references added, Introduction revised, Chernoff-like bound added
##### Keywords
Mathematics - Probability, Mathematics - Group Theory, 60B99, 20P05 | {"extraction_info": {"found_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": 0.8958743810653687, "perplexity": 1454.975886500129}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711336.41/warc/CC-MAIN-20221208114402-20221208144402-00000.warc.gz"} |
https://eccc.weizmann.ac.il/title/V | Under the auspices of the Computational Complexity Foundation (CCF)
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TR05-118 | 16th October 2005
Jin-Yi Cai, Vinay Choudhary
#### Valiant's Holant Theorem and Matchgate Tensors
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Some generalizations are presented.
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TR04-003 | 22nd December 2003
Pascal Koiran
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can visually'' recover the secret image, but
other, ... more >>>
ISSN 1433-8092 | Imprint | {"extraction_info": {"found_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": 0.8761899471282959, "perplexity": 2834.2642886768704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710980.82/warc/CC-MAIN-20221204204504-20221204234504-00692.warc.gz"} |
https://math.stackexchange.com/questions/2622223/measurable-set-e-so-that-me-capa-b-gt-0-for-all-a-b-in-mathbbr?noredirect=1 | # Measurable Set $E$ so that $m(E\cap(a, b)) \gt 0$ for all $a, b \in \mathbb{R}$ and $m(E) \lt \infty$ [duplicate]
Is it possible to construct a set $E$ so that for all $a, b \in \mathbb{R}, a < b$ the Lebesgue measure, donated here with $m$, of $E\cap(a, b)$ is always positive $$m(E\cap(a, b)) \gt 0$$ and also satisfies $m(E) \lt \infty$ ?
I would say no, because $a, b$ are arbitrary and if $E \ne \mathbb{R}$ one can always find an $a$ and $b$ so that $E\cap(a, b) = \emptyset$. But I can't come up with a proof to verify this. Has anyone an idea?
• The statement ''if $E \ne \mathbb{R}$ one can always find an $a$ and $b$ so that $E\cap(a, b) = \emptyset$'' is simply false: any dense set $E$ intersects every interval. I'm sure there is an answer to your question on this site if you search for it. – Umberto P. Jan 26 '18 at 15:23
• @NateEldredge thanks nate – Chiray Jan 26 '18 at 15:25
• @NateEldredge that question is related, but this question is much simpler. – Umberto P. Jan 26 '18 at 15:25
Let $\{r_n\}$ be an enumeration of rationals, let $$E = \bigcup (r_n-\frac{1}{2^n}, r_n+\frac{1}{2^n})$$ then $E$ satisfies your requirement.
A more interesting question would be finding $E$ such that both $$m(E\cap (a,b))> 0 \qquad m(E^c\cap(a,b))> 0$$ for any open interval $(a,b)$. Such set also exist. | {"extraction_info": {"found_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": 0.8971630930900574, "perplexity": 160.8637862469024}, "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-2019-51/segments/1575541294513.54/warc/CC-MAIN-20191214202754-20191214230754-00297.warc.gz"} |
http://math.stackexchange.com/questions/811834/topological-definition-of-continuity-and-its-application-to-epsilon-delta-defini | # Topological definition of continuity and its application to epsilon-delta definition?
So I am beginning Munkres' textbook on topology.
The topological definition of continuity reads:
$f:X\rightarrow Y$ is continuous if for each open subset $V\subset Y$, $f^{-1}(V)$ is an open subset of $X$.
Of course, it does fit the epsilon-delta definition of continuity since in
$\forall\epsilon\exists\delta,|x-x_0|<\delta\Rightarrow|f(x)-f(x_0)|<\epsilon$
$|x-x_0|<\delta$ and $|f(x)-f(x_0)|<\epsilon$ are both open.
Also, since openess of a set means closeness of that set's complement, the following epsilon-delta definition (which is valid) also agree with the topological definition:
$\forall\epsilon\exists\delta,|x-x_0|\le\delta\Rightarrow|f(x)-f(x_0)|\le\epsilon$
Yet my question is, how about the following epsilon-delta definition?
$\forall\epsilon\exists\delta,|x-x_0|\le\delta\Rightarrow|f(x)-f(x_0)|<\epsilon$
1.) Is it a valid definition of continuity of $\mathbb{R}^n$ regardless of the topological definition?
2.) If yes, does it agree with the topological definition of continuity?
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Note that if $\forall\epsilon\exists\delta,|x-x_0|\le\delta\Rightarrow|f(x)-f(x_0)|<\epsilon$ then $\forall\epsilon\exists\delta,|x-x_0|\le\delta\Rightarrow|f(x)-f(x_0)|\le \epsilon$ – mfl May 27 '14 at 21:39
@ManuelFdzLpz $|x−x_0|≤δ$ is not open, but $|f(x)−f(x_0)|<\epsilon$ is. What about that? – Sanath K. Devalapurkar May 27 '14 at 21:39
Consider $\mathbb{R}$ with the topology where $(-\infty,a)$ is open for any $a.$ Then $id:\mathbb{R}\rightarrow\mathbb{R}$ is continuous. If we define continuity by the condition that the image of "small" closed neighbourhoods lies in an open nhood, then $id$ is no longer continuous. If we consider $x=0$ then $[0,\infty)$ is a closed nhood of $0$ but the only open set that contains $id( [0,\infty))$ is $\mathbb{R}.$ "We would expect" it must be contained in the open nhood $(-\infty,\epsilon>0)$. In other words, the definitions wouldn't be equivalent. – mfl May 27 '14 at 22:42
You can consider 4 variations of the $\epsilon$-$\delta$ definition for continuity at a point $x_0$ in $\mathbb{R}^n$.
1. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon)$.
2. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | \le \delta \Rightarrow |f(x) - f(x_0)| < \epsilon)$.
3. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | < \delta \Rightarrow |f(x) - f(x_0)| \le \epsilon)$.
4. $\forall \epsilon>0 : \exists \delta > 0: \forall x: (| x - x_0 | \le \delta \Rightarrow |f(x) - f(x_0)| \le \epsilon)$.
One can easily check that all of these are mutually equivalent, and we can even do this in any pair of metric spaces instead of $\mathbb{R}^n$. But this is a simple consequence of properties of the order in $\mathbb{R}$ and properties of $\mathbb{R}$ itself.
E.g. to see that 1 implies 2: assume 1 holds, and pick $\epsilon>0$. Then 1 guarantees us a $\delta>0$ with the properties of 1. and then for 2. we just pick $\delta' = \frac{\delta}{2} > 0$. Then if $|x - x_0| \le \delta' = \frac{\delta}{2} < \delta$, so 1 guarantees that indeed $|f(x) - f(x_0)| < \epsilon$.
Other implications go similarly.
And all of these are equivalent to the topological definition. The proof of which uses the first, the most standard, variant as this corresponds best to interior points / open balls.
(Proof in one direction: suppose $O$ is open in the image, and let $x_0$ be in $f^{-1}[O]$. Then $f(x_0) \in O$, so there is some open ball with radius $\epsilon>0$ such that $B(f(x_0), \epsilon) \subset O$. Apply definition 1 to this $\epsilon$ to get $\delta>0$, and then one checks that $B(x_0, \delta) \subset f^{-1}[O]$. The $<$ signs give us nice open balls, so the proof is "smooth")
But the $\le$ instead of $<$ does not have anything to do with inverse images of closed sets being closed etc. The 4 variants are mutually equivalent, and the first one is the more natural to prove the equivalence to the topological definition.
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how do we know that $f^{-1}[O]$ is open? from the beginning of your proof we just know that $x_{0} \in f^{-1}[O]$ ? – arcolombo Oct 1 '14 at 16:27
@AnthonyColombo Because the $\delta$ shows that $x_0$, which is an arbitrary point of $f^{-1}[O]$, is an interior point of $f^{-1}[O]$. So all points of $f^{-1}[O]$ are interior points. – Henno Brandsma Oct 1 '14 at 16:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9656058549880981, "perplexity": 119.81171573552815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1469257824201.28/warc/CC-MAIN-20160723071024-00268-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://terrytao.wordpress.com/2011/01/11/the-inverse-conjecture-for-the-gowers-norm-over-finite-fields-in-low-characteristic/ | Tamar Ziegler and I have just uploaded to the arXiv our paper “The inverse conjecture for the Gowers norm over finite fields in low characteristic“, submitted to Annals of Combinatorics. This paper completes another case of the inverse conjecture for the Gowers norm, this time for vector spaces ${{\bf F}^n}$ over a fixed finite field ${{\bf F} = {\bf F}_p}$ of prime order; with Vitaly Bergelson, we had previously established this claim when the characteristic of the field was large, so the main new result here is the extension to the low characteristic case. (The case of a cyclic group ${{\bf Z}/N{\bf Z}}$ or interval ${[N]}$ was established by Ben Green and ourselves in another recent paper. For an arbitrary abelian (or nilpotent) group, a general but less explicit description of the obstructions to Gowers uniformity was recently obtained by Szegedy; the latter result recovers the high-characteristic case of our result (as was done in a subsequent paper of Szegedy), as well as our results with Green, but it is not immediately evident whether Szegedy’s description of the obstructions matches up with the one predicted by the inverse conjecture in low characteristic.)
The statement of the main theorem is as follows. Given a finite-dimensional vector space ${V = {\bf F}^n}$ and a function ${f: V \rightarrow {\bf C}}$, and an integer ${s \geq 0}$, one can define the Gowers uniformity norm ${\|f\|_{U^{s+1}(V)}}$ by the formula
$\displaystyle \|f\|_{U^{s+1}(V)} := \left( \mathop{\bf E}_{x,h_1,\ldots,h_{s+1} \in V} \Delta_{h_1} \ldots \Delta_{h_{s+1}} f(x) \right)^{1/2^{s+1}}$
where ${\Delta_h f(x) := f(x+h) \overline{f(x)}}$. If ${f}$ is bounded in magnitude by ${1}$, it is easy to see that ${\|f\|_{U^{s+1}(V)}}$ is bounded by ${1}$ also, with equality if and only if ${f(x) = e(P)}$ for some non-classical polynomial ${P: V \rightarrow {\bf R}/{\bf Z}}$ of degree at most ${s}$, where ${e(x) := e^{2\pi ix}}$, and a non-classical polynomial of degree at most ${s}$ is a function whose ${s+1^{th}}$ “derivatives” vanish in the sense that
$\displaystyle \partial_{h_1} \ldots \partial_{h_{s+1}} P(x) = 0$
for all ${x,h_1,\ldots,h_{s+1} \in V}$, where ${\partial_h P(x) := P(x+h) - P(x)}$. Our result generalises this to the case when the uniformity norm is not equal to ${1}$, but is still bounded away from zero:
Theorem 1 (Inverse conjecture) Let ${f: V \rightarrow {\bf C}}$ be bounded by ${1}$ with ${\|f\|_{U^{s+1}(V)} \geq \delta > 0}$ for some ${s \geq 0}$. Then there exists a non-classical polynomial ${P: V \rightarrow {\bf R}/{\bf Z}}$ of degree at most ${s}$ such that ${|\langle f, e(P) \rangle_{L^2(V)}| := |{\bf E}_{x \in V} f(x) e(-P(x))| \geq c(s,p, \delta) > 0}$, where ${c(s,p, \delta)}$ is a positive quantity depending only on the indicated parameters.
This theorem is trivial for ${s=0}$, and follows easily from Fourier analysis for ${s=1}$. The case ${s=2}$ was done in odd characteristic by Ben Green and myself, and in even characteristic by Samorodnitsky. In two papers, one with Vitaly Bergelson, we established this theorem in the “high characteristic” case when the characteristic ${p}$ of ${{\bf F}}$ was greater than ${s}$ (in which case there is essentially no distinction between non-classical polynomials and their classical counterparts, as discussed previously on this blog). The need to deal with genuinely non-classical polynomials is the main new difficulty in this paper that was not dealt with in previous literature.
In our previous paper with Bergelson, a “weak” version of the above theorem was proven, in which the polynomial ${P}$ in the conclusion had bounded degree ${O_{s,p}(1)}$, rather than being of degree at most ${s}$. In the current paper, we use this weak inverse theorem to reduce the inverse conjecture to a statement purely about polynomials:
Theorem 2 (Inverse conjecture for polynomials) Let ${s \geq 0}$, and let ${P: V \rightarrow {\bf C}}$ be a non-classical polynomial of degree at most ${s+1}$ such that ${\|e(P)\|_{U^{s+1}(V)} \geq \delta > 0}$. Then ${P}$ has bounded rank in the sense that ${P}$ is a function of ${O_{s,p,\delta}(1)}$ polynomials of degree at most ${s}$.
This type of inverse theorem was first introduced by Bogdanov and Viola. The deduction of Theorem 1 from Theorem 2 and the weak inverse Gowers conjecture is fairly standard, so the main difficulty is to show Theorem 2.
The quantity ${-\log_{|{\bf F}|} \|e(P)\|_{U^{s+1}(V)}^{1/2^{s+1}}}$ of a polynomial ${P}$ of degree at most ${s+1}$ was denoted the analytic rank of ${P}$ by Gowers and Wolf. They observed that the analytic rank of ${P}$ was closely related to the rank of ${P}$, defined as the least number of degree ${s}$ polynomials needed to express ${P}$. For instance, in the quadratic case ${s=1}$ the two ranks are identical (in odd characteristic, at least). For general ${s}$, it was easy to see that bounded rank implied bounded analytic rank; Theorem 2 is the converse statement.
We tried a number of ways to show that bounded analytic rank implied bounded rank, in particular spending a lot of time on ergodic-theoretic approaches, but eventually we settled on a “brute force” approach that relies on classifying those polynomials of bounded analytic rank as precisely as possible. The argument splits up into establishing three separate facts:
1. (Classical case) If a classical polynomial has bounded analytic rank, then it has bounded rank.
2. (Multiplication by ${p}$) If a non-classical polynomial ${P}$ (of degree at most ${s+1}$) has bounded analytic rank, then ${pP}$ (which can be shown to have degree at most ${\max(s-p,0)}$) also has bounded analytic rank.
3. (Division by ${p}$) If ${Q}$ is a non-clsasical polynomial of degree ${\max(s-p,0)}$ of bounded rank, then there is a non-classical polynomial ${P}$ of degree at most ${s+1}$ of bounded rank such that ${pQ=P}$.
The multiplication by ${p}$ and division by ${p}$ facts allow one to easily extend the classical case of the theorem to the non-classical case of the theorem, basically because classical polynomials are the kernel of the multiplication-by-${p}$ homomorphism. Indeed, if ${P}$ is a non-classical polynomial of bounded analytic rank of the right degree, then the multiplication by ${p}$ claim tells us that ${pP}$ also has bounded analytic rank, which by an induction hypothesis implies that ${pP}$ has bounded rank. Applying the division by ${p}$ claim, we find a bounded rank polynomial ${P'}$ such that ${pP = pP'}$, thus ${P}$ differs from ${P'}$ by a classical polynomial, which necessarily has bounded analytic rank, hence bounded rank by the classical claim, and the claim follows.
Of the three claims, the multiplication-by-${p}$ claim is the easiest to prove using known results; after a bit of Fourier analysis, it turns out to follow more or less immediately from the multidimensional Szemerédi theorem over finite fields of Bergelson, Leibman, and McCutcheon (one can also use the density Hales-Jewett theorem here if one desires).
The next easiest claim is the classical case. Here, the idea is to analyse a degree ${s+1}$ classical polynomial ${P: V \rightarrow {\bf F}}$ via its derivative ${d^{s+1} P: V^{s+1} \rightarrow {\bf F}}$, defined by the formula
$\displaystyle d^{s+1} P( h_1,\ldots,h_{s+1}) := \partial_{h_1} \ldots \partial_{h_{s+1}} P(x)$
for any ${x,h_1,\ldots,h_{s+1} \in V}$ (the RHS is independent of ${x}$ as ${P}$ has degree ${s+1}$). This is a multilinear form, and if ${P}$ has bounded analytic rank, this form is biased (in the sense that the mean of ${e(d^{s+1} P)}$ is large). Applying a general equidistribution theorem of Kaufman and Lovett (based on this earlier paper of Green and myself) this implies that ${d^{s+1} P}$ is a function of a bounded number of multilinear forms of lower degree. Using some “regularity lemma” theory to clean up these forms so that they have good equidistribution properties, it is possible to understand exactly how the original multilinear form ${d^{s+1} P}$ depends on these lower degree forms; indeed, the description one eventually obtains is so explicit that one can write down by inspection another bounded rank polynomial ${Q}$ such that ${d^{s+1} P}$ is equal to ${d^{s+1} Q}$. Thus ${P}$ differs from the bounded rank polynomial ${Q}$ by a lower degree error, which is automatically of bounded rank also, and the claim follows.
The trickiest thing to establish is the division by ${p}$ claim. The polynomial ${Q}$ is some function ${F(R_1,\ldots,R_m)}$ of lower degree polynomials ${R_1,\ldots,R_m}$. Ideally, one would like to find a function ${F'(R_1,\ldots,R_m)}$ of the same polynomials with ${pF' = F}$, such that ${F'(R_1,\ldots,R_m)}$ has the correct degree; however, we have counterexamples that show that this is not always possible. (These counterexamples are the main obstruction to making the ergodic theory approach work: in ergodic theory, one is only allowed to work with “measurable” functions, which are roughly analogous in this context to functions of the indicated polynomials ${Q, R_1,\ldots,R_m}$ and their shifts.) To get around this we have to first apply a regularity lemma to place ${R_1,\ldots,R_m}$ in a suitably equidistributed form (although the fact that ${R_1,\ldots,R_m}$ may be non-classical leads to a rather messy and technical description of this equidistribution), and then we have to extend each ${R_j}$ to a higher degree polynomial ${R'_j}$ with ${pR'_j = R_j}$. There is a crucial “exact roots” property of polynomials that allows one to do this, with ${R'_j}$ having degree exactly ${p-1}$ higher than ${R_j}$. It turns out that it is possible to find a function ${P = F'(R'_1,\ldots,R'_m)}$ of these extended polynomials that have the right degree and which solves the required equation ${pP=Q}$; this is established by classifying completely all functions of the equidistributed polynomials ${R_1,\ldots,R_m}$ or ${R'_1,\ldots,R'_m}$ that are of a given degree. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 120, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9802783727645874, "perplexity": 164.20344124026568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125881.93/warc/CC-MAIN-20170423031205-00391-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.arxiv-vanity.com/papers/hep-ph/9810482/ | # Model-independent electroweak penguins in B decays to two pseudoscalars
Michael Gronau Physics Department, Technion - Israel Institute of Technology, 32000 Haifa, Israel Dan Pirjol and Tung-Mow Yan Floyd R. Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853
July 27, 2021
###### Abstract
We study the effects of electroweak penguin (EWP) amplitudes in meson decays into two charmless pseudoscalars in the approximation of retaining only the dominant EWP operators and . Using flavor SU(3) symmetry, we derive a set of model-independent relations between EWP contributions and tree-level decay amplitudes one of which was noted recently by Neubert and Rosner. Two new applications of these relations are demonstrated in which uncertainties due to EWP corrections are eliminated in order to determine a weak phase. Whereas the weak angle can be obtained from free of hadronic uncertainties, a determination of from requires the knowledge of a ratio of certain tree-level hadronic matrix elements. The smallness of this ratio implies a useful constraint on if rescattering can be neglected.
###### pacs:
pacs1,pacs2,pacs3
preprint: CLNS 98/1582 TECHNION-PH-98-88 [1cm]
## I Introduction
Nonleptonic weak decays of mesons into two charmless pseudoscalars provide an important probe of the origin of CP violation in the single complex phase of the CKM matrix [1]. Approximate flavor symmetries of the strong interactions play a useful role in such analysis [2, 3, 4]. In one simplified version of such methods the weak phase is extracted from decays using isospin symmetry [5], and in another case the phase is obtained from combining and amplitudes using flavor SU(3) [6]. Electroweak penguin (EWP) contributions [7], enhanced by the heavy top quark, can spoil such methods. Whereas these contributions are expected to have a small effect on , they were estimated in a model-dependent manner to have a large effect on the extraction of [8, 9, 10]. Recently Neubert and Rosner have used Fierz transformations and SU(3) symmetry to include in the latter case the effect of EWP amplitudes in a model-independent way [11, 12]. Their method of constraining is based on assuming the dominance of two EWP operators ( and ) and relating their matrix elements for the decay final state to corresponding tree-level amplitudes. This argument is entirely model-independent, in contrast to previous studies of EWP contributions [8, 10, 13] which assume certain models for the matrix elements of EWP operators involving factorization and specific form factors.
The purpose of this paper is to generalize the relation proposed by Neubert and Rosner to all matrix elements of EWP operators for nonstrange and strange mesons and for any two pseudoscalar final state, and to study the consequences of such relations. Sec. II reviews the two alternative descriptions of flavor SU(3), in terms of operator matrix elements on the one hand, and quark diagrams on the other hand. These descriptions are used in Sec. III to derive a complete set of model-independent SU(3) relations between EWP and tree amplitudes for and corresponding decays. Using an approximate numerical relation between two ratios of Wilson coefficients, we show in Sec. IV that all EWP contributions can be written in terms of tree amplitudes. In Sec. V we demonstrate a few applications of these relations used to eliminate uncertainties due to EWP contributions when determining the weak phases and from and decays, respectively. Finally, our results are summarized in Sec. VI. An Appendix lists the four-quark operators appearing in the weak Hamiltonian for decays corresponding to specific SU(3) representations.
## Ii Flavor SU(3) in B decays
The weak Hamiltonian governing meson decays is given by (see, e.g., [14])
H=GF√2∑q=d,s⎛⎝∑q′=u,cλ(q)q′[c1(¯bq′)V−A(¯q′q)V−A+c2(¯bq)V−A(¯q′q′)V−A]−λ(q)t10∑i=3ciQ(q)i⎞⎠ , (1)
where . Unitarity of the CKM matrix implies . The first term, involving the coefficients and , will be referred to as the “tree” part, while the second term, involving is the penguin part. The corresponding consist of four QCD penguin operators () and four electroweak penguin operators (). Their precise form is not important for our purpose and can be found for example in [14]. In the following we will be only interested in their SU(3) transformation properties, noting that and have a structure similar to the “tree” part. There are two distinct types of QCD penguin operators, with the flavor structure ()
Q(q)3,5 = (¯bq)(¯uu+¯dd+¯ss) , Q(q)4,6 = (¯bu)(¯uq)+(¯bd)(¯dq)+(¯bs)(¯sq) , (2)
and two types of EWP operators
Q(q)7,9 = 32[(¯bq)(23¯uu−13¯dd−13¯ss)] , Q(q)8,10 = 32[23(¯bu)(¯uq)−13(¯bd)(¯dq)−13(¯bs)(¯sq)] . (3)
All four quark operators appearing in (1-II) are of the form and can be written as a sum of , 6 and , into which the product can be decomposed [3, 4]. Note that the representation appears twice in this decomposition, both symmetric (), and antisymmetric () under the interchange of and .
The tree part of the Hamiltonian (1) can be expressed in terms of operators with definite SU(3) transformation properties:
HT = GF√2(λ(s)u[12(c1−c2)(−¯3(a)I=0−6I=1)+12(c1+c2)(−¯¯¯¯¯¯15I=1−1√2¯¯¯¯¯¯15I=0+1√2¯3(s)I=0) (4) + λ(d)u[12(c1−c2)(6I=12−¯3(a)I=12)+12(c1+c2)(−2√3¯¯¯¯¯¯15I=32−1√6¯¯¯¯¯¯15I=12+1√2¯3(s)I=12)) .
The operators and appear in the two lines in the same combination. This fact is essential for relating to amplitudes with the help of SU(3) symmetry. The operators with well-defined SU(3) transformation properties appearing in (4) are given in the Appendix in terms of four-quark operators.
The contribution of the EWP operators (II) is given by
HEWP ≃ − λ(s)t2(c9−c102(3⋅6I=1+¯3(a)I=0)+c9+c102(−3⋅¯¯¯¯¯¯15I=1−3√2¯¯¯¯¯¯15I=0−1√2¯3(s)I=0)) −λ(d)t2(c9−c102(−3⋅6I=12+¯3(a)I=12)+c9+c102(−√32⋅¯¯¯¯¯¯15I=12−2√3⋅¯¯¯¯¯¯15I=32−1√2¯3(s)I=12)) ,
where we made the approximation of keeping only contributions from and [7, 11]. This is justified by the tiny Wilson coefficients of the remaining two operators and [14]. In this approximation the operators appearing in (II) are of the type and can be related to those appearing in the tree Hamiltonian (4). It is this fact which will allow us to express EWP contributions in terms of tree-level decay amplitudes.
Before proceeding to obtain these relations, let us recall the equivalent description of SU(3) amplitudes in terms of quark diagrams [3]. There are six topologies, representing tree (), color-suppressed (), annihilation (), -exchange (), penguin () and penguin-annihilation () amplitudes. The six amplitudes appear in five distinct combinations, separately for and transitions. For convenience, we define these amplitudes such that they don’t include the CKM factors. For example, a typical transition amplitude is
A(B+→K0π+)=λ(s)u(Pu+A)+λ(s)cPc+λ(s)t(Pt+PEWt(B+→K0π+)) , (6)
where and are contributions from the four-quark operators in the first term of (1), while and originate from the second term. In a similar way, a typical transition amplitude has the form
A(B0→π+π−) = λ(d)u(−Pu−T−E−PAu)+λ(d)c(−Pc) + λ(d)t(−Pt−PAt+PEWt(B0→π+π−)) .
Despite their name, and originate purely from “tree-level” four-quark operators, . Note that in the SU(3) symmetric limit, the same hadronic parameters appear in and transitions.
It is straightforward to relate the “graphical” hadronic parameters to SU(3) reduced matrix elements of the operators appearing in (4). This was done in the appendix of [3], and can also be done by computing representative decay amplitudes and expressing them with the help of the relations in the Appendix of [4]. We find the following set of linearly independent relations
Pu+T = 32√10a2+12√35a3+14√35a4−23√25a5 , Pu+A = 32√10a2−12√35a3−34√35a4+23√10a5 , −Pu+C = −34√25a2−12√35a3−14√35a4−√25a5 , Pu+PAu = −12a1+12√10a2−12√35a3+34√35a4+16√10a5 , C−E = −√35a3+√35a4−√25a5 . (8)
denote the following combinations of reduced matrix elements (a factor is omitted for simplicity)
a1 = 12(c1+c2)1√2⟨1||¯3(s)||3⟩−12(c1−c2)⟨1||¯3(a)||3⟩ , a2 = 12(c1+c2)1√2⟨8||¯3(s)||3⟩−12(c1−c2)⟨8||¯3(a)||3⟩ , a3 = −12(c1−c2)⟨8||6||3⟩ , a4 = 12(c1+c2)⟨8||¯¯¯¯¯¯15||3⟩ , a5 = 12(c1+c2)⟨27||¯¯¯¯¯¯15||3⟩ . (9)
The normalization of the reduced matrix elements is chosen as in [4]. Relative normalization with respect to the one used in [3] is given in the Appendix.
One can find three combinations of graphical amplitudes which are independent of the reduced matrix elements . As explained in the next section, they will be useful in relating EWP contributions to tree amplitudes.
T−A = √35a3+√35a4−√25a5 , T+C = −√103a5 , C−E = −√35a3+√35a4−√25a5 . (10)
These relations can be solved for and
a3 = −12√53(A+C−T−E) , a4 = 12√53(−A−15C−15T−E) , a5 = −3√10(T+C) . (11)
In Sec. IV we will need also the results for the reduced matrix elements and expressed in terms of graphical contributions
a1 = −12T+16C−43E−43Pu−2PAu a2 = 12√52(T−13C+A−13E+83Pu). (12)
## Iii Relations between EWP and tree amplitudes
Our purpose is to relate in the SU(3) limit EWP contributions to tree amplitudes. We note that the operators and occur in (II) in different combinations than in (4). Therefore, for arbitrary values of and , symmetry relations for EWP contributions can only be obtained which are independent of the matrix elements of and . The respective EWP contributions can then be expressed only in terms of tree-level amplitudes with the help of the relations (II).
### iii.1 |ΔS|=1 amplitudes
EWP contributions to decays can be easily computed using the Hamiltonian (II). One obtains
PEW(B0→K+π−) = 34√10b2+14√35b3+38√35b4−√25b5 , PEW(B+→K0π+) = −34√10b2+14√35b3+98√35b4−1√10b5 , PEW(B0→K0π0) = −38√5b2−14√310b3−38√310b4−32√5b5 , PEW(B+→K+π0) = 38√5b2−14√310b3−98√310b4−2√5b5 . (13)
The parameters , analogous to , are defined as
b1 = −12(c9+c10)1√2⟨1||¯3(s)||3⟩+12(c9−c10)⟨1||¯3(a)||3⟩ , b2 = −12(c9+c10)1√2⟨8||¯3(s)||3⟩+12(c9−c10)⟨8||¯3(a)||3⟩ , b3 = 32(c9−c10)⟨8||6||3⟩ , b4 = 12(c9+c10)⟨8||¯¯¯¯¯¯15||3⟩ , b5 = 12(c9+c10)⟨27||¯¯¯¯¯¯15||3⟩ . (14)
The EWP contributions satisfy the isospin relation (as do the full amplitudes [15])
PEW(B+→K0π+)+√2PEW(B+→K+π0)= √2PEW(B0→K0π0)+PEW(B0→K+π−) . (15)
It is clear now that any combination of amplitudes which is independent of can be expressed directly in terms of the tree-level amplitudes using the relations (II)
b3 = −3c9−c10c1−c2a3=c9−c10c1−c2√152(A+C−T−E) , b4 = c9+c10c1+c2a4=12√53c9+c10c1+c2(−A−15C−15T−E) , b5 = c9+c10c1+c2a5=−3√10c9+c10c1+c2(T+C) . (16)
One can form two combinations of electroweak penguin contributions in decays which do not depend on :
PEW(B+→K0π+)+√2PEW(B+→K+π0)=−√52b5=32c9+c10c1+c2(T+C) , (17) PEW(B0→K+π−)+PEW(B+→K0π+)=12√35b3+32√35b4−32√25b5 =34c9−c10c1−c2(A+C−T−E)−34c9+c10c1+c2(A−C−T+E) . (18)
A third combination is not independent of these two in view of the isospin identity (III.1). The first relation (17) was obtained in [11]. The second one (III.1) is new.
In a similar way one can compute EWP contributions to decay amplitudes. We find
PEW(Bs→π+π−) = −14b1−12√10b2−34√35b4+14√10b5 , PEW(Bs→π0π0) = 14√2b1+14√5b2+34√310b4−18√5b5 , PEW(Bs→K+K−) = −14b1+14√10b2+14√35b3−38√35b4−74√10b5 , PEW(Bs→K0¯K0) = 14b1−14√10b2+14√35b3−98√35b4−14√10b5 . (19)
Eliminating gives two relations
PEW(Bs→π+π−)+√2PEW(Bs→π0π0)=0 , (20) PEW(Bs→K+K−)+PEW(Bs→K0¯K0) =34c9−c10c1−c2(A+C−T−E)+34c9+c10c1+c2(A+C+T+E) . (21)
The first relation is simply a consequence of the absence of terms in the EWP Hamiltonian (II).
### iii.2 ΔS=0 amplitudes
For this case the Hamiltonian (II) gives the following results for and decays
PEW(B+→π+π0) = −√52b5 , PEW(B0→π+π−) = −14b1+14√10b2+14√35b3−38√35b4−74√10b5 , PEW(B0→π0π0) = 14√2b1−18√5b2−14√310b3+38√310b4−138√5b5 , PEW(B+→K+¯K0) = −34√10b2+14√35b3+98√35b4−1√10b5 , PEW(B0→K+K−) = −14b1−12√10b2−34√35b4+14√10b5 , PEW(B0→K0¯K0) = 14b1−14√10b2+14√35b3−98√35b4−14√10b5 , PEW(Bs→K−π+) = 34√10b2+14√35b3+38√35b4−√25b5 , PEW(Bs→¯K0π0) = −38√5b2−14√310b3−38√310b4−32√5b5 . (22)
Eliminating gives the following relations for EWP contributions to decays
√2PEW(B+→π+π0) = PEW(B0→π+π−)+√2PEW(B0→π0π0) (23) = 32c9+c10c1+c2(T+C) .
This relation, describing decay amplitudes into two pions in a state, follows from isospin alone. Only the part of the Hamiltonian contributes to these amplitudes. Comparing the tree-level (4) and EWP (II) Hamiltonians, one observes that their parts are simply related by
HEWΔI=3/2=−32λ(d)tλ(d)uc9+c10c1+c2HtreeΔI=3/2 . (24)
Therefore isospin symmetry alone suffices to relate their matrix elements. A similar relation holds for EWP contribution in
PEW(Bs→K−π+)+√2PEW(Bs→¯K0π0)=32c9+c10c1+c2(T+C) . (25)
## Iv Graphical representation for EWP
The numerical values of the two ratios of Wilson coefficients appearing in the previous section are very close to each other
c9+c10c1+c2=−1.139α ,c9−c10c1−c2=−1.107α . (26)
We used here the leading log values of the Wilson coefficients at [14]
c1=1.144 ,c2=−0.308 ,c9=−1.280α ,c10=0.328α , (27)
with . The two values in (26) differ by less that . Therefore, they can be taken as having a common value to a very good approximation
c9+c10c1+c2=c9−c10c1−c2=κ , (28)
where . As a consequence of this approximate equality, all EWP reduced matrix elements (III.1) are proportional to the corresponding tree amplitudes (II) with a common proportionality constant
b1=−κa1 ,b2=−κa2 ,b3=−3κa3 ,b4=κa4 ,b5=κa5 . (29)
These equalities suggest introducing the following six EWP amplitudes, analogous to the ones used to parametrize tree-level decay amplitudes
PEWi=κi ,i=T,C,A,E,Pu,PAu . (30)
These amplitudes have a direct graphic interpretation in terms of quark diagrams with one insertion of an electroweak penguin operator. Furthermore, the simple proportionality relation (30) guarantees that the amplitudes will satisfy the same hierarchy of sizes as the tree-level amplitudes [3, 9].
Decay mode
0 0 0 0
0 0 0
0 1 1/2
0 0 1/2 0 0
0 1
0 1 0 0
0 0
Table 1. EW penguin contributions to transitions in terms of the graphical amplitudes .
Inserting the relations (29) into (II) one may express the parameters in terms of . Using (III.1), (III.1) and (III.2), EWP contributions to any given decay can be written as a linear combination of the amplitudes. The results are given in Table 1 for transitions and in Table 2 for decays.
Decay mode
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http://mathhelpforum.com/calculus/189749-gradient-following-points-given-curve-print.html | # gradient at following points at the given curve?
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• October 7th 2011, 09:05 AM
andyboy179
gradient at following points at the given curve?
hi,
i need to work out the gradient at following points at the given curve:
y=3x^2-4x+1 at x=1
i know how to start it but after that im stuck and require some help, please!
i can do this:
dy/dy= 6x-4
• October 7th 2011, 09:13 AM
skeeter
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
hi,
i need to work out the gradient at following points at the given curve:
y=3x^2-4x+1 at x=1
i know how to start it but after that im stuck and require some help, please!
i can do this:
dy/dy= 6x-4
actually, it's dy/dx = 6x-4
what does dy/dx mean?
• October 7th 2011, 09:17 AM
andyboy179
Re: gradient at following points at the given curve?
sorry that was a typo! dy/dx means the gradient
• October 7th 2011, 09:24 AM
skeeter
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
sorry that was a typo! dy/dx means the gradient
so, the gradient is dy/dx = 6x-4 for any x-value of the given curve, and you want to find the value of the gradient when x = 1 ... what do you think happens now, andy?
• October 7th 2011, 10:00 AM
andyboy179
Re: gradient at following points at the given curve?
6x1-4 so, 6-4=2
• October 7th 2011, 10:21 AM
andyboy179
Re: gradient at following points at the given curve?
• October 7th 2011, 10:23 AM
skeeter
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
bingo!
• October 7th 2011, 10:25 AM
andyboy179
Re: gradient at following points at the given curve?
but then what else do i do to find the equation of the tangent to y=3x^2-4x+1 parallel to y=2x+7??
• October 7th 2011, 10:44 AM
e^(i*pi)
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
but then what else do i do to find the equation of the tangent to y=3x^2-4x+1 parallel to y=2x+7??
You need a co-ordinate on the tangent line. Since you know the tangent line meets the curve at $x=1$ then you know that $(1,f(1))$ is a point on the tangent line.
Do you know how to find $f(1)\?$
• October 7th 2011, 10:53 AM
andyboy179
Re: gradient at following points at the given curve?
no im not sure how to do this.
• October 7th 2011, 11:37 AM
e^(i*pi)
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
no im not sure how to do this.
You correctly found f'(1) though (in posts 5 and 6). Use the same method but with the original equation instead of the derivative.
• October 7th 2011, 11:50 AM
andyboy179
Re: gradient at following points at the given curve?
ok so would i do:
y= 3-4+1
y= 0?
• October 7th 2011, 12:17 PM
e^(i*pi)
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
ok so would i do:
y= 3-4+1
y= 0?
Yes, therefore $(1,0)$ is a point on the tangent line.
Now you have a co-ordinate and a gradient you can use the equation of a straight line to find this particular line's equation
• October 7th 2011, 12:27 PM
andyboy179
Re: gradient at following points at the given curve?
im not sure how to do this next step. i can do the equation of a straight line but i only have one set of brackets which is (1,0) what do i do?
• October 7th 2011, 12:36 PM
e^(i*pi)
Re: gradient at following points at the given curve?
Quote:
Originally Posted by andyboy179
im not sure how to do this next step. i can do the equation of a straight line but i only have one set of brackets which is (1,0) what do i do?
The equation of a straight line is $y = mx+c$ where m is the gradient and c the y intercept. You've worked out that $m = 2$ so you have something of the form $y = 2x+c$.
To find c (the y intercept) sub in your point $(1,0)$ into $y = 2x+c$ and solve for c.
Spoiler:
If you've learnt the equation of a straight line as $y-y_1 = m(x-x_1)$ then $m=2$ and $(x_1,y_1) = (1,0)$
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.903675377368927, "perplexity": 1506.3263368663881}, "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-2016-18/segments/1461860111365.36/warc/CC-MAIN-20160428161511-00116-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/mass-on-spring.53467/ | # Mass on Spring
1. Nov 20, 2004
### Naeem
A block of mass m = 4.5 kg rests on a frictionless floor. It is attached to a spring with a relaxed length L = 3 m. The spring has spring constant k = 16 N/m and is relaxed when hanging in the vertical position. The block is pulled d = 3 m to one side. In this problem, the block is always constrained to move on the floor (i.e. it never leaves the floor).
--------------------------------------------------------------------------------
a) By what amount is the spring extended?
DL = m *
--------------------------------------------------------------------------------
b) What is the potential energy stored in the spring?
Uspring = J *
12.3 OK
--------------------------------------------------------------------------------
c) The block is released but is constrained to move horizontally on the frictionless floor. What is the maximum speed it attains?
|v|max= m/s *
2.34 OK
--------------------------------------------------------------------------------
Let's change the problem a bit. When the spring is vertical (hence, unstretched), the block is given an initial speed equal to 1.8 times the speed found in part (c).
--------------------------------------------------------------------------------
d) How far from the initial point does the block go along the floor before stopping?
Dmax = m *
4.287 OK
--------------------------------------------------------------------------------
e) What is the magnitude of the acceleration of the block at this point (when the spring is stretched farthest)?
|a| = m/s2
0.638 NO
HELP: What is the force exerted by the spring on the block when the spring is fully stretched?
I am not able to figure out part e. I'm stuck
here is what I did F = ma = k * delta L
Plugged in m , k and delta L
found a to be 8.25 m/s2, which is wrong, anybody tell me what is wrong?
2. Nov 20, 2004
### thermodynamicaldude
The force a spring exerts on an object is F = -kx.
3. Nov 20, 2004
### Staff: Mentor
If the spring is vertical in its relaxed mode, and the block is move laterally, then the spring forms the hypotenuse of an isosceles right triangle with two sides of 3 m.
Here I am assuming the floor is horizontal with the normal parallel to the axis of the spring.
4. May 11, 2005
### Naeem
Can anybody help me with part e) here plz.
Thanks, | {"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": 0.9333939552307129, "perplexity": 608.7080769673265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719033.33/warc/CC-MAIN-20161020183839-00410-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://gmatpractice.q-51.com/hard-math-gmat-sample-questions/geometry-coordinate-geometry-permutation-combination-18.shtml | GMAT 700 800 Quant Question #18 | Geometry
GMAT Sample Questions | Coordinate Geometry & Permutation Combination
The given question is a challenging GMAT 700 800 level quant problem solving question combining concepts in coordinate geometry and permutation combination. A very interesting and challenging GMAT hard math question.
Question 18: Rectangle ABCD is constructed in the xy-plane so that sides AB and CD are parallel to the x-axis. Both the x and y coordinates of all four vertices of the rectangle are integers. How many rectangles can be constructed if x and y coordinates satisfy the inequality 11 < x < 29 and 5 ≤ y ≤ 13?
1. 153
2. 153C4
3. 4896
4. 2448
5. 5508
GMAT Live Online Classes
Explanatory Answer | GMAT Geometry Practice
KeyData
Sides AB and CD are parallel to x-axis.
So, AD and BC will be parallel to y-axis.
The x-coordinates take values from 12 to 28.
We can draw lines parallel to y-axis corresponding to each of these values.
So, we will be able to draw 17 vertical lines.
The y-coordinates take values from 5 to 13.
We can draw lines parallel to x-axis corresponding to each of these values.
So, we will be able to draw 9 horizontal lines.
Key Question: What maketh a rectangle?
2 horizontal lines and two vertical lines will form a rectangle
Number of ways of selecting 2 horizontal lines from 9 horizontal lines = 9C2
Feb 2, 2021
GMAT Sample Questions | Topicwise GMAT Questions
Where is Wizako located?
Wizako - GMAT, GRE, SAT Prep
An Ascent Education Initiative
48/1 Ramagiri Nagar | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9070232510566711, "perplexity": 1543.3903949183218}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703505861.1/warc/CC-MAIN-20210116074510-20210116104510-00783.warc.gz"} |
http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume24/cimiano05a-html/node10.html | Next: Related Work Up: Results Previous: Smoothing
## Discussion
We have shown that our FCA-based approach is a reasonable alternative to similarity-based clustering approaches, even yielding better results on our datasets with regard to the measure defined in Section 5. The main reason for this is that the concept hierarchies produced by FCA yield a higher recall due to the higher number of concepts, while maintaining the precision relatively high at the same time. Furthermore, we have shown that the conditional probability performs reasonably well as information measure compared to other more elaborate measures such as PMI or the one used by [51]. Unfortunately, applying a smoothing method based on clustering mutually similar terms does not improve the quality of the automatically learned concept hierarchies. Table 8 highlights the fact that every approach has its own benefits and drawbacks. The main benefit of using FCA is on the one hand that on our datasets it performed better than the other algorithms thus producing better concept hierarchies On the other hand, it does not only generate clusters - formal concepts to be more specific - but it also provides an intensional description for these clusters thus contributing to better understanding by the ontology engineer (compare Figure 2 (left)). In contrast, similarity-based methods do not provide the same level of traceability due to the fact that it is the numerical value of the similarity between two high-dimensional vectors which drives the clustering process and which thus remains opaque to the engineer. The agglomerative and divisive approach are different in this respect as in the agglomerative paradigm, initial merges of small-size clusters correspond to high degrees of similarity and are thus more understandable, while in the divisive paradigm the splitting of clusters aims at minimizing the overall cluster variance thus being harder to trace. A clear disadvantage of FCA is that the size of the lattice can get exponential in the size of the context in the worst case thus resulting in an exponential time complexity -- compared to and for agglomerative clustering and Bi-Section-KMeans, respectively. The implementation of FCA we have used is the concepts tool by Christian Lindig14, which basically implements Ganter's Next Closure algorithm [25,26] with the extension of Aloui for computing the covering relation as described by [29]. Figure 12 shows the number of seconds over the number of attribute/object pairs it took FCA to compute the lattice of formal concepts compared to the time needed by a naive implementation of the agglomerative algorithm with complete linkage. It can be seen that FCA performs quite efficiently compared to the agglomerative clustering algorithm. This is due to the fact that the object/attribute matrix is sparsely populated. Such observations have already been made before. [29] for example suspect that the lattice size linearly increases with the number of attributes per object. [38] presents empirical results analyzing contexts with a fill ratio below 0.1 and comes to the conclusion that the lattice size grows quadratically with respect to the size of the incidence relation . Similar findings are also reported by [9]. Figure 13 shows the number of attributes over the terms' rank, where the rank is a natural number indicating the position of the word in a list ordered by decreasing term frequencies. It can be appreciated that the amount of (non-zero) attributes is distributed in a Zipfian way (compare [65]), i.e. a small number of objects have a lot of attributes, while a large number of them has just a few. In particular, for the tourism domain, the term with most attributes is person with 3077 attributes, while on average a term has approx. 178 attributes. The total number of attributes considered is 9738, so that we conclude that the object/attribute matrix contains almost 98% zero values. For the finance domain the term with highest rank is percent with 2870 attributes, the average being ca. 202 attributes. The total number of attributes is 21542, so that we can state that in this case more than 99% of the matrix is populated with zero-values and thus is much sparser than the ones considered by [38]. These figures explain why FCA performs efficiently in our experiments. Concluding, though the worst-time complexity is exponential, FCA is much more efficient than the agglomerative clustering algorithm in our setting.
Table 8: Trade-offs between different taxonomy construction methods
Effectiveness (F') Worst Case Traceability Size of Tourism Finance Time Complexity Hierarchies FCA 44.69% 38.85% Good Large Agglomerative Clustering: Complete Linkage 36.85% 33.35% Fair Small Average Linkage 36.55% 32.92% Single Linkage 38.57% 32.15% Bi-Section-KMeans 36.42% 32.77% Weak Small
Next: Related Work Up: Results Previous: Smoothing
Philipp Cimiano 2005-08-04 | {"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": 0.8704714179039001, "perplexity": 772.6580324650565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267859766.6/warc/CC-MAIN-20180618105733-20180618125733-00369.warc.gz"} |
https://en.wikipedia.org/wiki/Projective_surface | # Projective variety
(Redirected from Projective surface)
An elliptic curve is a smooth projective curve of genus one.
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of Pn.
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective projective space; in this case it is the set of zeros of a single homogeneous polynomial.
If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring
${\displaystyle k[x_{0},\ldots ,x_{n}]/I}$
is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on X.
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann-Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.[1] Hilbert schemes parametrize closed subschemes of Pn with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.
A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining X have complex coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory.
## Variety and scheme structure
### Variety structure
Let k be an algebraically closed field. The basis of the definition of projective varieties is projective space Pn, which can be defined in different, but equivalent ways:
• as the set of all lines through the origin in kn+1 (i.e., one-dimensional sub-vector spaces of kn+1)
• as the set of tuples ${\displaystyle (x_{0},\dots ,x_{n})\in k^{n+1}}$, modulo the equivalence relation
${\displaystyle (x_{0},\dots ,x_{n})\sim \lambda (x_{0},\dots ,x_{n})}$
for any ${\displaystyle \lambda \in k\backslash \{0\}}$. The equivalence class of such a tuple is denoted by
${\displaystyle [x_{0}:\dots :x_{n}]}$
and referred to as a homogeneous coordinate.
A projective variety is, by definition, a closed subvariety of Pn, where closed refers to the Zariski topology.[2] In general, closed subsets of the Zariski topology are defined to be the zero-locus of polynomial functions. Given a polynomial ${\displaystyle f\in k[x_{0},\dots ,x_{n}]}$, the condition
${\displaystyle f([x_{0}:\dots :x_{n}])=0}$
does not make sense for arbitrary polynomials, but only if f is homogeneous, i.e., the total degree of all the monomials (whose sum is f) is the same. In this case, the vanishing of
${\displaystyle f(\lambda x_{0},\dots ,\lambda x_{n})=\lambda ^{\deg f}f(x_{0},\dots ,x_{n})}$
is independent of the choice of ${\displaystyle \lambda (\neq 0)}$.
Therefore, projective varieties arise from homogeneous prime ideals I of ${\displaystyle k[x_{0},...,x_{n}]}$, and setting
${\displaystyle X=\{[x_{0}:\dots :x_{n}]\in \mathbf {P} ^{n},f([x_{0}:\dots :x_{n}])=0{\text{ for all }}f\in I\}.}$.
Moreover, the projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space Pn is covered by the standard open affine charts
${\displaystyle U_{i}=\{[x_{0}:\dots :x_{n}],x_{i}\neq 0\},}$
which themselves are affine n-spaces with the coordinate ring ${\displaystyle k[y_{1}^{(i)},\dots ,y_{n}^{(i)}],y_{j}^{(i)}=x_{j}/x_{i}.}$ Say i = 0 for the notational simplicity and drop the superscript (0). Then ${\displaystyle X\cap U_{0}}$ is a closed subvariety of ${\displaystyle U_{0}\simeq \mathbb {A} ^{n}}$ defined by the ideal of ${\displaystyle k[y_{1},\dots ,y_{n}]}$ generated by
${\displaystyle f(1,y_{1},\dots ,y_{n})}$
for all f in I. Thus, X is an algebraic variety covered by (n+1) open affine charts ${\displaystyle X\cap U_{i}}$.
Note that X is the closure of the affine variety ${\displaystyle X\cap U_{0}}$ in ${\displaystyle \mathbb {P} ^{n}}$. Conversely, starting from some closed (affine) variety ${\displaystyle V\subset U_{0}\simeq \mathbb {A} ^{n}}$, the closure of V in ${\displaystyle \mathbb {P} ^{n}}$ is the projective variety called the projective completion of V. If ${\displaystyle I\subset k[y_{1},\dots ,y_{n}]}$ defines V, then the defining ideal of this closure is the homogeneous ideal[3] of ${\displaystyle k[x_{0},\dots ,x_{n}]}$ generated by
${\displaystyle x_{0}^{\operatorname {deg} (f)}f(x_{1}/x_{0},\dots ,x_{n}/x_{0})}$
for all f in I.
For example, if V is an affine curve given by, say, ${\displaystyle y^{2}=x^{3}+ax+b}$ in the affine plane, then its projective completion in the projective plane is given by ${\displaystyle y^{2}z=x^{3}+axz^{2}+bz^{3}.}$
### Projective schemes
For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., Pn(k) is a scheme which it is a union of (n + 1) copies of the affine n-space kn. More generally,[4] projective space over a ring A is the union of the affine schemes
${\displaystyle U_{i}=\operatorname {Spec} A[x_{1}/x_{i},\dots ,x_{n}/x_{i}],\quad 0\leq i\leq n,}$
in such a way the variables match up as expected. The set of closed points of ${\displaystyle \mathbf {P} _{k}^{n}}$, for algebraically closed fields k, is then the projective space Pn(k) in the usual sense.
An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme.[5] For example, if A is a ring, then
${\displaystyle \mathbf {P} _{A}^{n}=\operatorname {Proj} A[x_{0},\ldots ,x_{n}].}$
If R is a quotient of ${\displaystyle k[x_{0},\ldots ,x_{n}]}$ by a homogeneous ideal I, then the canonical surjection induces the closed immersion
${\displaystyle \operatorname {Proj} R\to \mathbf {P} _{k}^{n}.}$
Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space ${\displaystyle X=\operatorname {Proj} R}$ may have multiple irreducible components. Moreover, there may be nilpotent functions on X.
Closed subschemes of ${\displaystyle \mathbf {P} _{k}^{n}}$ correspond bijectively to the homogeneous ideals I of ${\displaystyle k[x_{0},\ldots ,x_{n}]}$ that are saturated; i.e., ${\displaystyle I:(x_{0},\dots ,x_{n})=I}$.[6] This fact may be considered as a refined version of projective Nullstellensatz.
We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k, we let
${\displaystyle \mathbf {P} (V)=\operatorname {Proj} k[V]}$
where ${\displaystyle k[V]=\operatorname {Sym} (V^{*})}$ is the symmetric algebra of ${\displaystyle V^{*}}$.[7] It is the projectivization of V; i.e., it parametrizes lines in V. There is a canonical surjective map ${\displaystyle \pi :V-0\to \mathbf {P} (V)}$, which is defined using the chart described above.[8] One important use of the construction is this (for more of this see below). A divisor D on a projective variety X corresponds to a line bundle L. One then set
${\displaystyle |D|=\mathbf {P} (\Gamma (X,L))}$;
it is called the complete linear system of D.
Projective space over a noetherian scheme S is defined as a fiber product
${\displaystyle \mathbf {P} _{S}^{n}=\mathbf {P} _{\mathbf {Z} }^{n}\times _{\operatorname {Spec} \mathbf {Z} }S.}$
If ${\displaystyle {\mathcal {O}}(1)}$ is the twisting sheaf of Serre on ${\displaystyle \mathbf {P} _{\mathbf {Z} }^{n}}$, we let ${\displaystyle {\mathcal {O}}(1)}$ denote the pullback of ${\displaystyle {\mathcal {O}}(1)}$ to ${\displaystyle \mathbf {P} _{S}^{n}}$; that is, ${\displaystyle {\mathcal {O}}(1)=g^{*}({\mathcal {O}}(1))}$ for the canonical map ${\displaystyle g:\mathbf {P} _{S}^{n}\to \mathbf {P} _{\mathbf {Z} }^{n}.}$
A scheme XS is called projective over S if it factors as a closed immersion
${\displaystyle X\to \mathbf {P} _{S}^{n}}$
followed by the projection to S.
## Relation to complete varieties
By definition, a variety is complete, if it is proper over k. The valuative criterion of properness expresses the intuition that in a proper variety, there are no points "missing".
There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:
Some properties of a projective variety follow from completeness. For example,
${\displaystyle \Gamma (X,{\mathcal {O}}_{X})=k}$
for any projective variety X over k.[10] This fact is an algebraic analogue of Liouville's theorem (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.
Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety.
## Examples and basic invariants
By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the case ${\displaystyle k=\mathbf {C} }$, is discussed further below.
The product of two projective spaces is projective. In fact, there is the explicit immersion (called Segre embedding)
${\displaystyle \mathbf {P} ^{n}\times \mathbf {P} ^{m}\to \mathbf {P} ^{(n+1)(m+1)-1},(x_{i},y_{j})\mapsto x_{i}y_{j}.}$
As a consequence, the fiber product of projective varieties is again projective. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group ${\displaystyle GL_{n}(k)}$ modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups.[11]
### Homogeneous coordinate ring and Hilbert polynomial
As the prime ideal P defining a projective variety X is homogeneous, the homogeneous coordinate ring
${\displaystyle R=k[x_{0},\dots ,x_{n}]/P}$
is a graded ring, i.e., can be expressed as the direct sum of its graded components:
${\displaystyle R=\bigoplus _{n\in \mathbf {N} }R_{n}.}$
There exists a polynomial P such that ${\displaystyle \dim R_{n}=P(n)}$ for all sufficiently large n; it is called the Hilbert polynomial of X. It is a numerical invariant encoding some extrinsic geometry of X. The degree of P is the dimension r of X and its leading coefficient times r! is the degree of the variety X. The arithmetic genus of X is (−1)r (P(0) − 1) when X is smooth.
For example, the homogeneous coordinate ring of Pn is ${\displaystyle k[x_{0},\ldots ,x_{n}]}$ and its Hilbert polynomial is ${\displaystyle P(z)={\binom {z+n}{n}}}$; its arithmetic genus is zero.
If the homogeneous coordinate ring R is an integrally closed domain, then the projective variety X is said to be projectively normal. Note, unlike normality, projective-normality depends on R, the embedding of X into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring of X.
### Degree
Let ${\displaystyle X\subset \mathbb {P} ^{N}}$ be a projective variety. There are at least two equivalent ways to define the degree of X relative to its embedding. The first way is to define it as the cardinality of the finite set
${\displaystyle \#(X\cap H_{1}\cap \cdots \cap H_{d})}$
where d is the dimension of X and Hi's are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if X is a hypersurface, then the degree of X is the degree of the homogeneous polynomial defining X. The "general positions" can be made precise, for example, by intersection theory; one requires that the intersection is proper and that the multiplicities of irreducible components are all one.
The other definition, which is mentioned in the previous section, is that the degree of X is the leading coefficient of the Hilbert polynomial of X times (dim X)!. Geometrically, this definition means that the degree of X is the multiplicity of the vertex of the affine cone over X.[12]
Let ${\displaystyle V_{1},\dots ,V_{r}\subset \mathbb {P} ^{N}}$ be closed subschemes of pure dimensions that intersect properly (they are in general position). If mi denotes the multiplicity of an irreducible component Zi in the intersection (i.e., intersection multiplicity), then the generalization of Bézout's theorem says:[13]
${\displaystyle \sum _{1}^{s}m_{i}\operatorname {deg} Z_{i}=\prod _{1}^{r}\operatorname {deg} V_{i}.}$
The intersection multiplicity mi can be defined as the coefficient of Zi in the intersection product ${\displaystyle V_{1}\cdot {\dots }\cdot V_{r}}$ in the Chow ring of ${\displaystyle \mathbb {P} ^{N}}$.
In particular, if ${\displaystyle H\subset \mathbb {P} ^{N}}$ is a hypersurface not containing X, then
${\displaystyle \sum _{1}^{s}m_{i}\operatorname {deg} Z_{i}=\operatorname {deg} (X)\operatorname {deg} (H)}$
where Zi's are the irreducible components of the scheme-theoretic intersection of X and H with multiplicity (length of the local ring) mi.
### The ring of sections
Let X be a projective variety and L a line bundle on it. Then the graded ring
${\displaystyle R(X,L)=\bigoplus _{n=0}^{\infty }H^{0}(X,L^{\otimes n})}$
is called the ring of sections of L. If L is ample, then Proj of this ring is X. Moreover, if X is normal and L is very ample, then ${\displaystyle R(X,L)}$ is the integral closure of the homogeneous coordinate ring of X determined by L; i.e., ${\displaystyle X\hookrightarrow \mathbb {P} ^{N}}$ so that ${\displaystyle {\mathcal {O}}_{\mathbb {P} ^{N}}(1)}$ pulls-back to L.[14]
For applications, it is useful to allow for divisors (or ${\displaystyle \mathbb {Q} }$-divisors) not just line bundles; assuming X is normal, the resulting ring is then called a generalized ring of sections. If ${\displaystyle K_{X}}$ is a canonical divisor on X, then the generalized ring of sections
${\displaystyle R(X,K_{X})}$
is called the canonical ring of X. If the canonical ring is finitely generated, then Proj of the ring is called the canonical model of X. The canonical ring or model can then be used to define the Kodaira dimension of X.
### Projective curves
Further information: Algebraic curve
Projective schemes of dimension one are called projective curves. Much of the theory of projective curves is about smooth projective curves, since the singularities of curves can be resolved by normalization, which consists in taking locally the integral closure of the ring of regular functions. Smooth projective curves are isomorphic if and only if their function fields are isomorphic. The study of finite extensions of
${\displaystyle \mathbb {F} _{p}(t),}$
or equivalently smooth projective curves over ${\displaystyle \mathbb {F} _{p}}$ is an important branch in algebraic number theory.[15]
A smooth projective curve of genus one is called an elliptic curve. As a consequence of the Riemann-Roch theorem, such a curve can be embedded as a closed subvariety in P2. In general, any (smooth) projective curve can be embedded in P3. Conversely, any smooth closed curve in P2 of degree three has genus one by the genus formula and is thus an elliptic curve.
A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism ${\displaystyle C\to \mathbf {P} ^{1}}$ of degree two.[16]
### Projective hypersurfaces
Every irreducible closed subset of Pn of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial.[17]
### Abelian varieties
Another important invariant of a projective variety X is the Picard group ${\displaystyle \operatorname {Pic} (X)}$ of X, the set of isomorphism classes of line bundles on X. It is isomorphic to ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})}$ and therefore an intrinsic notion (independent of embedding). For example, the Picard group of Pn is isomorphic to Z via the degree map. The kernel of ${\displaystyle \operatorname {deg} :\operatorname {Pic} (X)\to \mathbf {Z} }$ is not only an abstract abelian group, but there is a variety called the Jacobian variety of X, Jac(X), whose points equal this group. The Jacobian of a (smooth) curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curve E is E itself. For a curve X of genus g, Jac(X) has dimension g.
Varieties, such as the Jacobian variety, which are complete and have a group structure are known as abelian varieties, in honor of Niels Abel. In marked contrast to affine algebraic groups such as ${\displaystyle GL_{n}(k)}$, such groups are always commutative, whence the name. Moreover, they admit an ample line bundle and are thus projective. On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces.
## Projections
Let ${\displaystyle E\subset \mathbb {P} ^{n}}$ be a linear subspace; i.e., ${\displaystyle E=\{s_{0}=s_{1}=\dots =s_{r}=0\}}$ for some linearly independent linear functional si's. Then the projection from E is the (well-defined) morphism
${\displaystyle \phi :\mathbb {P} ^{n}-E\to \mathbb {P} ^{r},\,x\mapsto [s_{0}(x):\cdots :s_{r}(x)].}$
• The geometric description of this map is the following.[18] We view ${\displaystyle \mathbb {P} ^{r}\subset \mathbb {P} ^{n}}$ so that it is disjoint from E. Then, for any ${\displaystyle x\in \mathbb {P} ^{n}-E}$,
${\displaystyle \phi (x)=W_{x}\cap \mathbb {P} ^{r}}$
where we wrote ${\displaystyle W_{x}}$ for the smallest linear space containing E and x (called the join of E and x.)
• ${\displaystyle \phi ^{-1}(\{y_{i}\neq 0\})=\{s_{i}\neq 0\}}$, where ${\displaystyle y_{i}}$ are the homogeneous coordinates on ${\displaystyle \mathbb {P} ^{r}}$.
• For any closed subscheme ${\displaystyle Z\subset \mathbb {P} ^{n}}$ disjoint from E, the restriction
${\displaystyle \phi :Z\to \mathbb {P} ^{r}}$
is a finite morphism.[19]
Projections can be used to cut down the dimension in which a projective variety is embedded, up to finite morphisms. Start with some projective variety ${\displaystyle X\subset \mathbb {P} ^{n}}$. If ${\displaystyle n>\operatorname {dim} X}$, the projection from a point not on X gives ${\displaystyle \phi :X\to \mathbb {P} ^{n-1}}$. Moreover, ${\displaystyle \phi }$ is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map
${\displaystyle X\to \mathbb {P} ^{d},\,d=\operatorname {dim} X}$.
This result is the projective analog of Noether's normalization lemma. (In fact, it yields a geometric proof of the normalization lemma.)
The same procedure can be used to show the following slightly more precise result: given a projective variety X over a perfect field, there is a finite birational morphism from X to a hypersurface H in ${\displaystyle \mathbb {P} ^{d+1}}$.[20] In particular, if X is normal, then it is the normalization of H.
## Line bundle and divisors
Main article: Ample line bundle
The number of particular properties of projective varieties makes it desirable to have efficient criteria to show that a given variety is projective. Such criteria can be formulated using the notion of very ample line bundles.
Let X be a scheme over a ring A. Suppose there is a morphism
${\displaystyle \phi :X\to \mathbf {P} _{A}^{n}=\operatorname {Proj} A[x_{0},\dots ,x_{n}]}$.
Then, along this map, the Serre twisting sheaf ${\displaystyle {\mathcal {O}}(1)}$ pulls-back to a line bundle L on X, which is generated by the global sections ${\displaystyle \phi ^{*}(x_{i})}$.[21] Conversely, any line bundle L which is generated by global sections ${\displaystyle s_{0},...,s_{n}}$ defines a morphism
${\displaystyle \phi :X\to \mathbf {P} _{A}^{n}}$
which in homogeneous coordinates is given by ${\displaystyle \phi (x)=[s_{0}(x):\dots :s_{n}(x)].}$ This map ${\displaystyle \phi }$ is such that ${\displaystyle L\cong \phi ^{*}({\mathcal {O}}(1))}$ and ${\displaystyle s_{i}=\phi ^{*}(x_{i})}$. Furthermore, ${\displaystyle \phi }$ is a closed immersion if and only if ${\displaystyle X_{i}}$ are affine and ${\displaystyle \Gamma (U_{i},{\mathcal {O}}_{\mathbf {P} _{A}^{n}})\to \Gamma (X_{i},{\mathcal {O}}_{X_{i}})}$ are surjective.[22]
A line bundle (or invertible sheaf) ${\displaystyle {\mathcal {L}}}$ on a scheme X over S is said to be very ample relative to S if there is an immersion (i.e., an open immersion followed by a closed immersion)
${\displaystyle i:X\to \mathbf {P} _{S}^{n}}$
for some n so that ${\displaystyle {\mathcal {O}}(1)}$ pullbacks to ${\displaystyle {\mathcal {L}}.}$ Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if X is projective, then the pullback of ${\displaystyle {\mathcal {O}}(1)}$ under the closed immersion of X into a projective space is very ample. That "projective" implies "proper" is more difficult[dubious ]: the main theorem of elimination theory.
## Cohomology of coherent sheaves
Main article: coherent sheaf
Let X be a projective scheme over a field (or, more generally over a Noetherian ring A). Cohomology of coherent sheaves ${\displaystyle {\mathcal {F}}}$ on X satisfies the following important theorems due to Serre:
1. ${\displaystyle H^{p}(X,{\mathcal {F}})}$ is a finite-dimensional k-vector space for any p.
2. There exists an integer ${\displaystyle n_{0}}$ (depending on ${\displaystyle {\mathcal {F}}}$; see also Castelnuovo–Mumford regularity) such that
${\displaystyle H^{p}(X,{\mathcal {F}}(n))=0}$
for all ${\displaystyle n\geq n_{0}}$ and p > 0, where ${\displaystyle {\mathcal {F}}(n)={\mathcal {F}}\otimes {\mathcal {O}}(n)}$ is the twisting with a power of a very ample line bundle ${\displaystyle {\mathcal {O}}(1)}$
These results are proven reducing to the case ${\displaystyle X=\mathbf {P} ^{n}}$ using the isomorphism
${\displaystyle H^{p}(X,{\mathcal {F}})=H^{p}(\mathbf {P} ^{r},{\mathcal {F}}),p\geq 0}$
where in the right-hand side ${\displaystyle {\mathcal {F}}}$ is viewed as a sheaf on the projective space by extension by zero.[23] The result then follows by a direct computation for ${\displaystyle {\mathcal {F}}={\mathcal {O}}_{\mathbf {P} ^{r}}(n),}$ n any integer, and for arbitrary ${\displaystyle {\mathcal {F}}}$ reduces to this case without much difficulty.[24]
As a corollay to 1. above, if f is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image ${\displaystyle R^{p}f_{*}{\mathcal {F}}}$ is coherent. The same result holds for proper morphisms f, as can be shown with the aid of Chow's lemma.
Sheaf cohomology groups Hi on a noetherian topological space vanish for i strictly greater than the dimension of the space. Thus the quantity, called the Euler characteristic of ${\displaystyle {\mathcal {F}}}$,
${\displaystyle \chi ({\mathcal {F}})=\sum _{i=0}^{\infty }(-1)^{i}\operatorname {dim} H^{i}(X,{\mathcal {F}})}$
is a well-defined integer (for X projective). One can then show ${\displaystyle \chi ({\mathcal {F}}(n))=P(n)}$ for some polynomial P over rational numbers.[25] Applying this procedure to the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$, one recovers the Hilbert polynomial of X. In particular, if X is irreducible and has dimension r, the arithmetic genus of X is given by
${\displaystyle (-1)^{r}(\chi ({\mathcal {O}}_{X})-1),}$
which is manifestly intrinsic; i.e., independent of the embedding.
The arithmetic genus of a hypersurface of degree d is ${\displaystyle {\binom {d-1}{n}}}$ in ${\displaystyle \mathbf {P} ^{n}}$. In particular, a smooth curve of degree d in P2 has arithmetic genus ${\displaystyle (d-1)(d-2)/2}$. This is the genus formula.
## Smooth projective varieties
Let X be a smooth projective variety where all of its irreducible components have dimension n. In this situation, the canonical sheaf ωX, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic n-forms), is a line bundle.
### Serre duality
Serre duality states that for any locally free sheaf ${\displaystyle {\mathcal {F}}}$ on X,
${\displaystyle H^{i}(X,{\mathcal {F}})\simeq H^{n-i}(X,{\mathcal {F}}^{\vee }\otimes \omega _{X})'}$
where the superscript prime refers to the dual space and ${\displaystyle {\mathcal {F}}^{\vee }}$ is the dual sheaf of ${\displaystyle {\mathcal {F}}}$. A generalization to projective, but not necessarily smooth schemes is known as Verdier duality.
### Riemann-Roch theorem
For a (smooth projective) curve X, H2 and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus of X is the dimension of ${\displaystyle H^{1}(X,{\mathcal {O}}_{X})}$. By definition, the geometric genus of X is the dimension of H0(X, ωX). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of X.
Serre duality is also a key ingredient in the proof of the Riemann–Roch theorem. Since X is smooth, there is an isomorphism of groups
${\displaystyle \operatorname {Cl} (X)\to \operatorname {Pic} (X),D\mapsto {\mathcal {O}}(D)}$
from the group of (Weil) divisors modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ωX is called the canonical divisor and is denoted by K. Let l(D) be the dimension of ${\displaystyle H^{0}(X,{\mathcal {O}}(D))}$. Then the Riemann–Roch theorem states: if g is a genus of X,
${\displaystyle l(D)-l(K-D)=\operatorname {deg} D+1-g}$
for any divisor D on X. By the Serre duality, this is the same as:
${\displaystyle \chi ({\mathcal {O}}(D))=\operatorname {deg} D+1-g}$,
which can be readily proved.[26] A generalization of the Riemann-Roch theorem to higher dimension is the Hirzebruch-Riemann-Roch theorem, as well as the far-reaching Grothendieck-Riemann-Roch theorem.
## Hilbert schemes
Hilbert schemes parametrize all closed subvarieties of a projective scheme X in the sense that the points (in the functorial sense) of H correspond to the closed subschemes of X. As such, the Hilbert scheme is an example of a moduli space, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whose Hilbert polynomial equals a prescribed polynomial P.[27] It is a deep theorem of Grothendieck that there is a scheme[28] ${\displaystyle H_{X}^{P}}$ over k such that, for any k-scheme T, there is a bijection
${\displaystyle \{{\text{morphisms }}T\to H_{X}^{P}\}\ \ \longleftrightarrow \ \ \{{\text{closed subschemes of }}X\times _{k}T{\text{ flat over }}T,{\text{ with Hilbert polynomial }}P.\}}$
The closed subscheme of ${\displaystyle X\times H_{X}^{P}}$ that corresponds to the identity map ${\displaystyle H_{X}^{P}\to H_{X}^{P}}$ is called the universal family.
For ${\displaystyle P(z)={\binom {z+r}{r}}}$, the Hilbert scheme ${\displaystyle H_{\mathbf {P} ^{n}}^{P}}$ is called the Grassmannian of r-planes in ${\displaystyle \mathbf {P} ^{n}}$ and, if X is a projective scheme, ${\displaystyle H_{X}^{P}}$ is called the Fano scheme of r-planes on X.[29]
## Complex projective varieties
In this section, all algebraic varieties are complex algebraic varieties. A key feature of the theory of complex projective varieties is the combination of algebraic and analytic methods. The transition between these theories is provided by the following link: since any complex polynomial is also a holomorphic function, any complex variety X yields a complex analytic space, denoted ${\displaystyle X(\mathbf {C} )}$. Moreover, geometric properties of X are reflected by the ones of ${\displaystyle X(\mathbf {C} )}$. For example, the latter is a complex manifold iff X is smooth; it is compact iff X is proper over C.
### Relation to complex Kähler manifolds
Complex projective space is a Kähler manifold. This implies that, for any projective algebraic variety X, X(C) is a compact Kähler manifold. The converse is not in general true, but the Kodaira embedding theorem gives a criterion for a Kähler manifold to be projective.
In low dimensions, there are the following results:
### GAGA and Chow's theorem
Chow's theorem provides a striking way to go the other way, from analytic to algebraic geometry. It states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that a holomorphic function satisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following:
• Meromorphic functions on the complex projective space are rational.
• If an algebraic map between algebraic varieties is an analytic isomorphism, then it is an (algebraic) isomorphism. (This part is a basic fact in complex analysis.) In particular, Chow's theorem implies that a holomorphic map between projective varieties is algebraic. (consider the graph of such a map.)
• Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.[31]
• Every holomorphic line bundle on a projective variety is a line bundle of a divisor.[32]
Chow's theorem can be shown via Serre's GAGA principle. Its main theorem states:
Let X be a projective scheme over C. Then the functor associating the coherent sheaves on X to the coherent sheaves on the corresponding complex analytic space Xan is an equivalence of categories. Furthermore, the natural maps
${\displaystyle H^{i}(X,{\mathcal {F}})\to H^{i}(X^{\text{an}},{\mathcal {F}})}$
are isomorphisms for all i and all coherent sheaves ${\displaystyle {\mathcal {F}}}$ on X.[33]
### Complex tori vs. complex abelian varieties
The complex manifold associated to an abelian variety A over C is a compact complex Lie group. These can be shown to be of the form
${\displaystyle \mathbb {C} ^{g}/L}$
and are also referred to as complex tori. Here, g is the dimension of the torus and L is a lattice (also referred to as period lattice).
According to the uniformization theorem already mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from an elliptic curve. In fact, the Weierstrass's elliptic function ${\displaystyle \wp }$ attached to L satisfies a certain differential equation and as a consequence it defines a closed immersion:[34]
${\displaystyle \mathbb {C} /L\to \mathbf {P} ^{2},L\mapsto (0:0:1),z\mapsto (1:\wp (z):\wp '(z)).}$
For higher dimensions, the notions of complex abelian varieties and complex tori differ: only polarized complex tori come from abelian varieties.
### Kodaira vanishing
The fundamental Kodaira vanishing theorem states that for an ample line bundle ${\displaystyle {\mathcal {L}}}$ on a smooth projective variety X over a field of characteristic zero,
${\displaystyle H^{i}(X,{\mathcal {L}}\otimes \omega _{X})=0}$
for i > 0, or, equivalently by Serre duality ${\displaystyle H^{i}(X,{\mathcal {L}}^{-1})=0}$ for i < n.[35] The first proof of this theorem used analytic methods of Kähler geometry, but a purely algebraic proof was found later. The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic. Kodaira's theorem is one of various vanishing theorems, which give criteria for higher sheaf cohomologies to vanish. Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties.[36]
## Related notions
Closed subvarieties of weighted projective spaces are known as weighted projective varieties.[37]
## Notes
1. ^ Kollár & Moduli, Ch I.
2. ^ Shafarevich, Igor R. (1994), Basic Algebraic Geometry 1: Varieties in Projective Space, Springer
3. ^ This homogeneous ideal is sometimes called the homogenization of I.
4. ^ Mumford 1999, pg. 82
5. ^ Hartshorne 1977, Section II.5
6. ^ Mumford 1999, pg. 111
7. ^ This definition differs from Eisenbud–Harris 2000, III.2.3 but is consistent with the other parts of Wikipedia.
8. ^ cf. the proof of Hartshorne 1977, Ch II, Theorem 7.1
9. ^
10. ^ Hartshorne 1977, Ch II. Exercise 4.5
11. ^ Humphreys, James (1981), Linear algebraic groups, Springer, Theorem 21.3
12. ^ Hartshorne, Ch. V, Exercise 3.4. (e).
13. ^ Fulton 1998, Proposition 8.4.
14. ^ Hartshorne, Ch. II, Exercise 5.14. (a)
15. ^ Rosen, Michael (2002), Number theory in Function Fields, Springer
16. ^ Hartshorne & 1977 Ch IV, Exercise 1.7.
17. ^ Hartshorne 1977, Ch I, Exercise 2.8; this is because the homogeneous coordinate ring of Pn is a unique factorization domain and in a UFD every prime ideal of height 1 is principal.
18. ^ Shafarevich 1994, Ch. I. § 4.4. Example 1.
19. ^ Mumford, Ch. II, § 7. Proposition 6.
20. ^ Hartshorne, Ch. I, Exercise 4.9.
21. ^ Hartshorne 1977, Ch II, Theorem 7.1
22. ^ Hartshorne 1977, Ch II, Proposition 7.2
23. ^ This is not difficult:(Hartshorne 1977, Ch III. Lemma 2.10) consider a flasque resolution of ${\displaystyle {\mathcal {F}}}$ and its zero-extension to the whole projective space.
24. ^ Hartshorne 1977, Ch III. Theorem 5.2
25. ^ Hartshorne 1977, Ch III. Exercise 5.2
26. ^ Hartshorne 1977, Ch IV. Theorem 1.3
27. ^ Kollár 1996, Ch I 1.4
28. ^ To make the construction work, one needs to allow for a non-variety.
29. ^ Eisenbud & Harris 2000, VI 2.2
30. ^ Hartshorne 1977, Appendix B. Theorem 3.4.
31. ^ Griffiths-Adams, IV. 1. 10. Corollary H
32. ^ Griffiths-Adams, IV. 1. 10. Corollary I
33. ^ Hartshorne 1977, Appendix B. Theorem 2.1
34. ^ Mumford 1970, pg. 36
35. ^ Hartshorne 1977, Ch III. Remark 7.15.
36. ^ Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems, Birkhäuser
37. ^ Dolgachev, Igor (1982), "Weighted projective varieties", Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math., 956, Berlin: Springer, pp. 34–71, doi:10.1007/BFb0101508, MR 0704986 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 166, "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": 0.9738829135894775, "perplexity": 316.98685133730527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122955.76/warc/CC-MAIN-20170423031202-00365-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://aas.org/archives/BAAS/v25n2/aas182/abshtml/S3011.html | The Anomalously Large Peculiar Velocity of Abell 2634
Session 30 -- Clusters of Galaxies
Display presentation, Tuesday, 9:30-6:30, Pauley Room
## [30.11] The Anomalously Large Peculiar Velocity of Abell 2634
M. Gregg (IGPP/LLNL)
\def\etal{et al.} \def\kms{\rm km/s^{-1}} \def\dnsig{\rm D_{n}-\sigma}
Based on distance estimates to 18 early type cluster members, Lucey \etal\ (1991, MNRAS 248, 804) report a peculiar velocity for Abell 2634 of $\sim -3400~\kms$. Lucey \etal\ suggest that this uncomfortably large value is spurious because IR Tully-Fisher distances to spirals in the region indicate that the late type galaxies have zero peculiar velocity (Aaronson \etal\ 1986, ApJ 302, 536). One possible explanation is that the $\dnsig$\ distance estimator for the early type galaxies has a different zero point and/or slope in Abell~2634. This could arise because of structural or stellar population differences with respect to the elliptical galaxies in Virgo and Coma which are used to calibrate the $\dnsig$\ relation.
To confirm the Lucey \etal\ result and look for an explanation, new images and spectroscopy were obtained in September 1992 at Kitt Peak National Observatory of 20 early type galaxies in Abell~2634, including 6 additional cluster members outside the central region studied by Lucey \etal\ ~~Preliminary results for 14 objects yield a value of $-3100~\kms$\ for the cluster peculiar velocity, in excellent agreement with the Lucey study. There is close correspondence between the individual galaxy peculiar velocities from the Lucey \etal\ study and the Kitt Peak data, demonstrating that large errors are not the cause of the spurious peculiar velocity and strongly suggesting that an additional parameter is at work modifying the $\dnsig$\ relation in this cluster.
The early type galaxies have normal BVR colors and none of the spectra have emission lines. There is a significant correlation between the 4000 \AA\ break strength and peculiar velocity in that galaxies with smaller breaks have larger negative peculiar velocities. This is difficult to understand, but is consistent with the weak correlation between surface brightness and peculiar velocity reported by Lucey \etal\ ~~Work is continuing to search for spectroscopic and structural clues which may explain the anomalous peculiar velocity of Abell~2634. | {"extraction_info": {"found_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": 0.8546711802482605, "perplexity": 3526.7010807526112}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1416400375630.34/warc/CC-MAIN-20141119123255-00217-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/58552/plotting-phase-delay-from-first-principles-giving-different-result-from-phasedel | # Plotting phase delay from first principles giving different result from phasedelay() function. Why?
I have a length 10 Finite Impulse response as shown below, and I am trying to find the phase delay of this FIR.
$$h[n]=\begin{cases}0.1,& 0\le n \lt 10\\ 0,&\textrm{otherwise}\end{cases}$$
I tried plotting from first principles as well as using the phasedelay() function, however I am getting different results. The results are varying greatly. Any suggestions on what I am forgetting to do in my approach from first principles?
The problem is the definition of the phase. The command angle() computes the phase $$\phi(\omega)$$ according to
$$H(e^{j\omega})=|H(e^{j\omega})|e^{j\phi(\omega)}\tag{1}$$
where $$H(e^{j\omega})$$ is the complex frequency response. At frequencies $$\omega$$ where the frequency response has zeros, the phase jumps by $$\pi$$. This is shown in the two left figures of the plot below. You can't compute the phase delay from $$\phi(\omega)$$ because of the jumps. Note that in your computation, the phase delay is correct until the first phase jump due to the first zero of the magnitude response.
The phase delay needs to be computed from a continuous phase $$\varphi(\omega)$$, which is implicitly defined by
$$H(e^{j\omega})=A(\omega)e^{j\varphi(\omega)}\tag{2}$$
where $$A(\omega)$$ is a real-valued but bi-polar smooth function. This continuous phase is shown in the top right figure below, and it is obviously linear. From this continuous phase, the phase delay can be computed by
$$\tau_{ph}=-\frac{\varphi(\omega)}{\omega}\tag{3}$$
and for the given linear-phase FIR filter the phase delay is constant:
$$\tau_{ph}=\frac{N-1}{2}\tag{4}$$
where $$N$$ is the filter length.
Note that the continuous phase $$\varphi(\omega)$$ cannot be obtained from $$\phi(\omega)$$ using the Matlab function unwrap(), because the latter just removes jumps by (multiples of) $$2\pi$$ due to phase ambiguity. Here we have to deal with actual phase jumps due to zeros of $$H(e^{j\omega})$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9102150797843933, "perplexity": 311.77326089680747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570987834649.58/warc/CC-MAIN-20191023150047-20191023173547-00262.warc.gz"} |
http://mathoverflow.net/questions/8846/proofs-without-words/104871 | # Proofs without words
Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
(One could ask if this is of interest to mathematicians, and I would say yes, in so far as the kind of little gems that usually fall under the title of 'proofs without words' is quite capable of providing the aesthetic rush we all so professionally appreciate. That is why we will sometimes stubbornly stare at one of these mathematical autostereograms with determination until we joyously see it.)
(I'll provide an answer as an example of what I have in mind in a second)
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where possible could people also either note the image source or explain/provide a link to a "how to" for constructing the associated diagram? I think that such would also be helpful for folks` – Carter Tazio Schonwald Dec 14 '09 at 23:57
I hope I am not alone in being (usually) unable to appreciate "proof by picture"... – Suvrit Jul 8 '11 at 21:14
@Suvrit: I hope I am not alone in being most often unable to appreciate "proof by word" until I've read it at least twenty times and wrestled with it for many days per page! – WetSavannaAnimal aka Rod Vance Jul 9 '11 at 12:11
My opinion is that almost every proof-without-words is improved by a few well-chosen words. – Joel David Hamkins Feb 12 '12 at 0:47
There is no such thing as a "proof without logic," and since words are usually the best tool for conveying logical relations, I'm going to have to reject the idea of "proof without words." Sorry, -1. – goblin Jan 23 '15 at 3:14
(I'd post this as a comment to Mariano Suárez-Alvarez, but I've not enough rep). From a ME thread.
$$\sum_{k=1}^n (-1)^{n-k} k^2 = {n+1 \choose 2} = \sum_{k=1}^n \; k = \frac{(n+1) \; n}{2}$$
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A line that bisects the right angle in a right triangle also bisects a square erected on the hypotenuse:
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I am surprised that no one had cited the "proof" that the rationals are countable yet. See, for example, this picture
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Maybe it doesn't fit into the "non-trivial" category? – Campello Mar 19 '14 at 12:12
I think that the fact that the rationals are countable qualifies as non-trivial, when put in historical perspective – Geoff Robinson Mar 19 '14 at 19:02
Q: Can you tile with ?
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I don't think this is clear enough to be self-contained, although I have something in mind to fix it. Do you mind if I try? – Jason Dyer Dec 19 '09 at 18:30
go ahead....... – Gil Kalai Dec 19 '09 at 18:59
I have edited and put in my modification of the image. – Jason Dyer Dec 19 '09 at 22:16
can someone explain this in words? – Turbo Feb 15 '12 at 6:51
It's easier if you do use words. If you take away opposite squares, you have more of one color than another... – Todd Trimble Aug 27 '12 at 3:42
In the movie category, I'm surprised that no-one has yet posted a link to Moebius Transformations Revealed.
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But what does that movie prove? – Mariano Suárez-Alvarez Nov 8 '10 at 3:50
@Mariano: it doesn't prove anything, but then again neither do any proofs without words. They merely give us insight into the proof, and in that respect, any movie has even more potential than a simple image. I think we will soon see very innovative approaches in movie-proofs. – Thierry Zell Nov 8 '10 at 3:59
Very beautiful. I suppose it proves the usefulness of abstraction in obtaining unity – William Feb 12 '12 at 0:08
$$\arctan \frac{1}{3} + \arctan \frac{1}{2} = \arctan 1$$
It's easy to generalize this to
$$\arctan \frac{1}{n} + \arctan \frac{n-1}{n+1} = \arctan 1, \text{ for } n \in \mathbb{N}$$
which can further be generalized to
$$\arctan \frac{a}{b} + \arctan \frac{b-a}{b+a} = \arctan 1, \text{ for } a,b \in \mathbb{N}, a \leq b$$
Edit: A similar result relating Fibonacci numbers to arctangents can be found here and here.
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It needed quite a long time for me to understand this. But, well, then it is amazing! – Gottfried Helms Oct 28 '15 at 10:16
• The first homotopy group of SO_3 has an order 2 element (that's a classic).
• The surface area of a quarter of the unit sphere is Pi via Gauss-Bonnet (My source is Ariel Shaqed - it should have been a classic, but no one I asked seems to knew it). The sphere is what you reach with a straight hand while standing still. Hold a Pencil in your hand, that's your tangent vector. Now parallel transport the pencil on a quarter sphere: it points in the opposite direction. QED
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For SO(3) has order 2 element: gregegan.customer.netspace.net.au/APPLETS/21/21.html – Dan Piponi Dec 14 '09 at 15:29
Place a glass on the open palm of your hand. You can, with a bit of practice, rotate the glass twice (but not once) around the vertical axis without spilling any liquid from it, and return to your original position. Each part of your body goes through a loop in SO_3. Moving from the shoulder via the arm to the glass, you get a homotopy essentially proving the theorem. I have seen dancers from somewhere in south-east Asia incorporating this move into their dance. – Harald Hanche-Olsen Dec 14 '09 at 21:21
Why are there so many words and so few pictures in this answer? – David Eppstein Dec 14 '09 at 23:07
@David: well, you can think if this answer (or of Harald's comment, which gets my emphatic upvote) as a script for the choreography which, when acted out, is a proof without words :P – Mariano Suárez-Alvarez Dec 15 '09 at 0:00
It doesn't feature Feynman, but here's a video of a human doing the plate trick (just after 1 minute in): youtube.com/watch?v=CYBqIRM8GiY – Harrison Brown Dec 15 '09 at 2:42
Have a look at this document from an MIT-instructor: http://mit.edu/18.098/book/extract2009-01-21.pdf
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The area under a cycloid is three times the area of the generating circle.
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Proof of the associativity law $f * (g * h) = (f * g) * h$ in the fundamental groupoid of a topological space:
You can find more of these diagrams in J. P. May's A Concise course in algebraic topology.
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I just saw this proof, which is of course not mine.
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I like the tiling proof of the Pythagorean Theorem. The left image is credited to Al-Nayrizi and Thābit ibn Qurra (9th century) and the right by Henry Perigal (19th century).
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I'm having trouble seeing a triangle (of the appropriate dimensions) in the Perigal tiling. – Gerry Myerson Aug 20 '12 at 22:46
Gerry, slide the red square to the left by half the side length of a white square. The segment connecting the two lower red corners is the hypotnuse, and the legs have lengths which are the widths of the two tiling squares. Yes, I know that sounds confusing. – Marc Chamberland Aug 21 '12 at 0:58
Thanks, Marc, not confusing at all. But I think if you have to add that to see that there's a triangle there, it's not really a proof without words. Well, at least, for me it's not a proof without words. – Gerry Myerson Aug 21 '12 at 5:33
Also elementary, but here is a proof that
$C_n = \binom{2n}{n} - \binom{2n}{n+1} = \frac{\binom{2n}{n}}{n+1},$
where $C_n$ is the $n$th Catalan number.
http://utdallas.edu/~hagge/images/Catalan.pdf
Sorry for the link; new users may not use image tags.
Here's the image:
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Do you have an explanation for the picture? I looked at it, and looked at it, and don't get it. – Willie Wong Mar 11 '10 at 16:38
Sorry for not noticing your question (much) earlier. The differences between adjacent terms in Pascal's triangle form another triangle which obeys the same generation rules. In my picture of that triangle, the yellow squares count some of the downward paths on a square grid which has been rotated $45^\circ$, namely those that never fall to the left of the top square. One definition of $C_n$ is that it is the number of such paths which terminate at the bottom corner of an $n \times n$ grid. – Tobias Hagge Oct 26 '10 at 5:46
Can you tile an 8x8 chessboard with one corner cut off with dominoes of dimension 3x1?
This is a simple way to show that choosing a useful coloring can make a proof trivial.
This proof was also a result of the Conjecture and Proof class in the Budapest Semesters in Mathematics. It was one of the first problems encountered there, hence not that hard :)
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See also: MESE 220. – Benjamin Dickman Nov 12 '15 at 8:14
The pathspace of any topological space is contractible.
Pf (as given in my homotopy theory class): slurp spaghetti.
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There a proof of Erd˝os-Mordell Inequality 'without words' is an impressive one. Please follow the link http://forumgeom.fau.edu/FG2007volume7/FG200711.pdf
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How is that "without words"? :-/ – Andrea Ferretti Mar 5 '10 at 17:23
If you observe carefully on the graphs, you don't even need to write a word. – Sunni Mar 5 '10 at 19:15
The idea is to prove things in ways that are obvious to different parts of your brain, right? Anyone found any "auditory proofs"? Some candidates -
1. Nyquist sampling theorem?
2. sin[a] + sin[b] = 2sin[(a+b)/2]cos[(a-b)/2]. If you use at and bt instead of a and b, you can translate that to show how the addition of two sine tones close in frequency can also be perceived as a modulation or "vibrato" around the centre frequency. The factor of 2 might be hard, though you can add a gain instead of 2 and show that the difference is silence when the gain is 2 :)
3. Sampling in frequency domain (comb filter) is periodicity in time domain?
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Are there more details on 1? 2 and 3 don't seem like proofs so much as examples, though maybe you are just putting them forth as challenges. – j.c. Mar 7 '10 at 11:54
There is a beautiful proof of the fact that a checkerboard with sides $2^{n}$, and one square removed can be tiled with $L$-shaped pieces formed by three squares. Given that a checkerboard of sides $2^{n-1}$ can be so tiled, then a square checkerboard of sides $2^{n}$ can be tiled by filling in the quarter in which the removed piece lies, and then placing an extra $L$-shaped tile with one square in each of the remaining three quarters.
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I first learned this in Dan Velleman's book "How to prove it." I'm not sure if he originated it or not. – Jim Conant Feb 7 '11 at 2:36
A nice proof for trigonometric equation:
$sin^2(x)+cos^2(x)=1$
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This is an exhibit of the fact, but it isn't really a proof - it doesn't explain why those two functions sum to 1, just shows (arguably, just claims) that they do. You could replace the curve with any function $f$ with $f(\pi/2)=1$. – Steven Stadnicki May 16 '15 at 0:45
@StevenStadnicki In fact, I'm pretty sure that the function in the picture is not $\sin x$, the inflection point has a sharper third derivative than it should (although I'm sure this is just a limitation of the means by which the picture was drawn). There is certainly nothing geometrical constraining the shape of the diagram. – Mario Carneiro Jun 25 '15 at 18:53
However, this could be a nice proof of $\int_{0}^{\pi/2}\sin^2 x\,\mathrm{d}x = \int_{0}^{\pi/2}\cos^2 x\,\mathrm{d}x = \frac{1}{2}\left(\frac{\pi}{2}\cdot 1\right)$ – Machinato Jul 18 at 13:29
Rich Schwartz had on his site a great paper consisting of only a picture which proved that every right triangle admits a periodic billiard path. Unfortunately, he's since deleted it, so I can't post it here. (It shouldn't take too long for anyone interested to re-construct the proof, though.)
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I am guessing he did it by assembling four of the said right-triangles into a parallelogram. There is a path that bounces directly between the two longer sides. Mod out by the symmetry and you get a periodic path in the triangle. – Willie Wong Mar 11 '10 at 17:04
$S^2 \vee S^1 \vee S^1$ is homotopy equivalent to the Klein bottle with self-intersection.
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Let $0\leq x,y,z,t\leq1$ Prove that $x(1-y)+t(1-x)+z(1-t)+y(1-z)\leq 2$.
Draw a 1x1 square and mark in consecutive sides disjoint segments starting at the vertexes of lengths $x,y,z,t$. Joining the consecutive end points of the intervals that are not vertexes of the square form four triangles, the area of the triangles is the left hand side divided by 2, the area of the square is the right hand side divided by 2.
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This proof without words has an awful lot of them! – I. J. Kennedy Apr 30 '11 at 17:39
Indeed, there is a proof with only eleven words: rearrangement and arithmetic-geometric inequalities. (Details are left to the reader.) – dvitek May 4 '11 at 0:13
I don't think drvitek's proof makes sense. – darij grinberg Jun 28 '11 at 14:55
+1 It is quite a nice idea, if you actually do the drawing. – rem Jun 17 '14 at 10:23
This proves the Minkowski version of the Pythagorean theorem:
$c^2 = a^2 - b^2$
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I suppose I should link to the physics stackexchange question that this came from: physics.stackexchange.com/questions/12435/… – Ron Maimon Aug 23 '11 at 21:48
From Wikipedia: here is a "proof without words" of the Yoneda Lemma.
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This answer has already been proposed, and after some discussion it was more or less agreed that this is not a proof-without-words in standard sense of the term. – Mariano Suárez-Alvarez Oct 1 '11 at 23:45
Proof of the lantern relation (taken from the book: A Primer on Mapping Class Groups by Farb, B. and Margalit, D.)
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From the book "Proofs without words", there are ton of others too but this one I had trouble proving in UG, so like it most.
-
.
This is an example I did when I was in high school.
Let it be a unit disc, consider the length of horizontal line, we know Yellow=$2\cos \frac{3}{7}\pi$, Yellow+Green=$-2\cos \frac{5}{7}\pi$, Red+Green=$2\cos \frac{1}{7}\pi$.
Then 1=Red=Red+Green-(Green+Yellow)+Yellow=$2(\cos \frac{3}{7}\pi+\cos \frac{5}{7}\pi+\cos \frac{1}{7}\pi).$
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I don't understand it. Seems like it does require some words... – Johannes Hahn Nov 11 '15 at 18:25
Sorry about that. Actually, I don't really know how to explain it well. – Guo Qi Nov 12 '15 at 6:08
This is apparently not was intended, but I think it qualifies. From Principia Mathematica: the proof of 1+1=2 (I can't include the image bc I'm a new user, but perhaps an experienced user can edit this answer for me.)
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The composition of two continuous mappings is continuous.
Bloody thing won't let me embed the image...
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Here are some dynamic versions:
http://www.math.utah.edu/~palais/sums.html (two of the summation formulas mentioned above)
Several belt, plate, and tangle trick animations:
A visual derivation of complex multiplication:
http://www.math.utah.edu/~palais/newrot.swf
Pythagoras in the Isosceles case, based on the Yale tablet:
http://www.math.utah.edu/~palais/PythagorasIsosceles.html
and the general case:
http://www.math.utah.edu/~palais/Pythagoras.html
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There is an animation of the Dandelin Spheres construction depicted above in 3d-XplorMath in anaglyph 3d. 3d-xplormath.org Not really a proof, but visual descriptions of the relationship of cosine and sine curves and uniform circular motion: math.utah.edu/~palais/cose.html math.utah.edu/~palais/sine.html and epicycloids: math.utah.edu/~palais/daledots.swf – Bob Palais Feb 7 '11 at 4:04
All the links in this post appear to be broken. – I. J. Kennedy Apr 30 '11 at 17:21
## protected by Scott Morrison♦Oct 11 '13 at 0:51
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count). | {"extraction_info": {"found_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": 0.8024564981460571, "perplexity": 870.9696547950144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1469257829972.49/warc/CC-MAIN-20160723071029-00242-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.houseofmath.com/encyclopedia/functions/derivation-and-its-applications/rate-of-change/what-is-the-point-slope-equation | # What Is the Point-Slope Equation?
A straight line is defined by two points, or by one point and the slope of the line. There is an ingenious formula that allows you to find the formula for a straight line using one point and the slope of the line:
Formula
### Point-SlopeEquation
The formula that defines the line with slope $a$ through the point $\left({x}_{1},{y}_{1}\right)$ is
$y-{y}_{1}=a\left(x-{x}_{1}\right)$
Solve the equation with respect to $y$ and the expression looks like the function for a straight line $y=ax+b$.
Example 1
Find the function for the line through $\left(2,5\right)$ with a slope of 3
Put the numbers into the point-slope equation and solve for $y$: $\begin{array}{llll}\hfill y-5& =3\left(x-2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =3x-6+5=3x-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Example 2
Find the function for the straight line through the points $\left(-3,9\right)$ and $\left(3,-9\right)$
First, you find the slope:
$a=\frac{-9-9}{3-\left(-3\right)}=-\frac{18}{6}=-3$
Then, select one of the points in the exercise and put it together with the slope into the point-slope equation: $\begin{array}{llll}\hfill y-\left(-9\right)& =-3\left(x-3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y+9& =-3x+9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =-3x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Since $b=0$, you know that the line passes through the origin. The function for the straight line is
$y=-3x$
If you know the function $f\left(x\right)$, you can use the point-slope equation to find the equation of the tangent line at a point on the graph of $f\left(x\right)$. This is because the slope of the tangent is equal to the value of the derivative of the function $f\left(x\right)$ at the same point.
Formula
### TheEquationforanArbitraryTangent
$y-{y}_{1}={f}^{\prime }\left({x}_{1}\right)\left(x-{x}_{1}\right),$
where $\left({x}_{1},{y}_{1}\right)$ is a point on the tangent (often the point of tangency) and ${f}^{\prime }\left({x}_{1}\right)$ is the slope of the point. When using the formula, you must always solve for $y$— that is, get $y$ alone on one side.
Example 3
Given the function $f\left(x\right)={x}^{2}+3x-2$, find the equation for the tangent at $x=3$
To fill in the equation, you need values for ${y}_{1}$ and ${f}^{\prime }\left({x}_{1}\right)$. You know that ${x}_{1}=3$, so ${y}_{1}=f\left(3\right)$ and ${f}^{\prime }\left({x}_{1}\right)={f}^{\prime }\left(3\right)$. We begin by computing ${f}^{\prime }\left(x\right)$: $\begin{array}{llll}\hfill {f}^{\prime }\left(x\right)& =2x+3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {f}^{\prime }\left({x}_{1}\right)& ={f}^{\prime }\left(3\right)=2\left(3\right)+3=9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {y}_{1}& =f\left({x}_{1}\right)=f\left(3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={\left(3\right)}^{2}+3\left(3\right)-2=16\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Now you put this into the equation and get $\begin{array}{llll}\hfill y-16& =9\left(x-3\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =9x-27+16=9x-11\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Example 4
Let $g\left(x\right)={e}^{\mathrm{sin}x}$. Find the equation for the tangent at $x=0$.
You need the values of ${f}^{\prime }\left({x}_{1}\right)$ and ${y}_{1}$. First, differentiate the function:
${f}^{\prime }\left(x\right)=\mathrm{cos}x\cdot {e}^{\mathrm{sin}x}$
Now you can calculate ${f}^{\prime }\left({x}_{1}\right)$. Since ${x}_{1}=0$, you get $\begin{array}{llll}\hfill {f}^{\prime }\left({x}_{1}\right)& ={f}^{\prime }\left(0\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{cos}0\cdot {e}^{\mathrm{sin}0}=1\cdot {e}^{0}=1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
You find ${y}_{1}$ by putting ${x}_{1}$ into $f\left(x\right)$:
${y}_{1}=f\left({x}_{1}\right)=f\left(0\right)={e}^{\mathrm{sin}0}={e}^{0}=1$
You put all this into the equation and get $\begin{array}{llll}\hfill y-1& =1\left(x-0\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 45, "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": 0.9458184838294983, "perplexity": 81.3577271806227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656104668059.88/warc/CC-MAIN-20220706060502-20220706090502-00445.warc.gz"} |
http://mathoverflow.net/questions/19193/divisors-on-projufd | # Divisors on Proj(UFD)
Hello to all,
I have been perusing Harthorne for some time, and I noticed something: it is well known that the class group on $\mathbb{P}^n_k$ is $\mathbb{Z}$. But as I look at Harthorne's proof it seems to mee that it works in much greater generality. Namely if I consider any projective scheme $X=Proj(A)$, where $A$ is a graded $UFD$ in such a way that there exists an irreducible element of degree $1$, then the exact same reasoning shows that the class group of $X$ is also $\mathbb{Z}$, and generated by the prime divisor $(a)$. Is this true ?
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Well, if you read on to Chapter 2, exercise 6.3, then it is stated that: $$Cl(A) \cong Cl(X)/\mathbb Z[H]$$ here $[H]$ represents the hyperplane section. So the answer is yes.
There is a less well-known but very nice generalization. Suppose that $X$ is smooth. Let $R=A_m$ be the local ring of A at the irrelevant ideal. Then one has a (graded) isomorphism of $\mathbb Q$- vector spaces:
$$CH(X)_{\mathbb Q}/[H]CH(X)_{\mathbb Q} \cong A_*(R)_{\mathbb Q}$$
Here $CH(X)$ is the Chow ring of $X$ and $A_*(R)$ is the total Chow group of $R$. Details can be found in this paper by Kurano. | {"extraction_info": {"found_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": 0.9805582761764526, "perplexity": 112.90407056963659}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246644200.21/warc/CC-MAIN-20150417045724-00154-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://scribesoftimbuktu.com/simplify-10-2-5-5-1-3%C3%B73-2-3-2/ | Simplify (10 2/5-5 1/3)÷3 2/3
Convert to an improper fraction.
A mixed number is an addition of its whole and fractional parts.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Convert to an improper fraction.
A mixed number is an addition of its whole and fractional parts.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Convert to an improper fraction.
A mixed number is an addition of its whole and fractional parts.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Multiply the numerator by the reciprocal of the denominator.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply by .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Multiply by .
Subtract from .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply and .
Multiply by .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Simplify (10 2/5-5 1/3)÷3 2/3
Solving MATH problems
We can solve all math problems. Get help on the web or with our math app
Scroll to top | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9287248253822327, "perplexity": 3574.1611887873373}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337889.44/warc/CC-MAIN-20221006222634-20221007012634-00680.warc.gz"} |
http://mathhelpforum.com/calculus/147307-need-help-couple-integrals-print.html | # Need help with couple of integrals!
• June 1st 2010, 10:49 AM
goroner
Need help with couple of integrals!
i got some probs. to solve this integrals :
integral x*arctan(x)dx
integral (11x+7)/(2x^2+7x-4)dx
integral ln(x^2+1)dx
intgral e^x/(1-e^3x)dx
i tryed solving this but i get stucked somewhere the first one is with partial integration ,second one i substitute the denominator with u and i get 4x+7 for du and there i get stucked ,the third one is partial as well but again as the first one i get to a point where i need to solve integral x^2/(x^2+1) and this one is not rly simple i dont know to solve it so i need help there, the 4 one is also with sub. for u=e^x but then i get integral 1/1-u^3 i tryed to make sub. for t=1-u^3 so i can get one parameter in the denominator and two up and than devide in two integrals but i couldnt get it.. please some help if you know how to solve this , btw these are from faculty from exam sessions thanks! (Itwasntme)
• June 1st 2010, 10:55 AM
Faux Carnival
Hi, right now I'm in a hurry. But I'd like to give you a very useful tip. You can always use the Wolfram Online Integrator to solve integrals.
Wolfram Mathematica Online Integrator
And then go back from there. I mean, try to understand which method to use from the anti-derivative. Good luck!
• June 1st 2010, 10:59 AM
goroner
dude i tryed in mathematica it self i got it but i cant get till the result walking so it doesnt rly helped me even the online thingy is giving me the same result but it didnt help me i need some step by step solvation if anyone could do it thanks!
• June 1st 2010, 10:59 AM
harish21
Quote:
Originally Posted by goroner
i got some probs. to solve this integrals :
integral x*arctan(x)dx
integral (11x+7)/(2x^2+7x-4)dx
integral ln(x^2+1)dx
intgral e^x/(1-e^3x)dx
i tryed solving this but i get stucked somewhere the first one is with partial integration ,second one i substitute the denominator with u and i get 4x+7 for du and there i get stucked ,the third one is partial as well but again as the first one i get to a point where i need to solve integral x^2/(x^2+1) and this one is not rly simple i dont know to solve it so i need help there, the 4 one is also with sub. for u=e^x but then i get integral 1/1-u^3 i tryed to make sub. for t=1-u^3 so i can get one parameter in the denominator and two up and than devide in two integrals but i couldnt get it.. please some help if you know how to solve this , btw these are from faculty from exam sessions thanks! (Itwasntme)
1) use integration by parts:
$\int u \mbox{dv} = uv -\int v \mbox{du}$
where $u = arctan(x), dv = x dx$
$du = \frac{1}{x^{2}+1} dx, v = x^2/2:$
• June 1st 2010, 11:02 AM
harish21
Very few people might have the time to show step by step solutions to all 4 problems:
check here. Click on show steps.
• June 1st 2010, 11:03 AM
goroner
for the 1 one i did use partial integration and i get to a point to solve integral of x^2/(x^2+1) and here i get stucked i need help to solve that one
• June 1st 2010, 11:07 AM
goroner
says x^2/(x^2+1) should be done with long division but dont know to do that .. ehm =/
• June 1st 2010, 11:08 AM
harish21
integrate x*arctan(x)dx - Wolfram|Alpha
Quote:
Originally Posted by goroner
for the 1 one i did use partial integration and i get to a point to solve integral of x^2/(x^2+1) and here i get stucked i need help to solve that one
• June 1st 2010, 11:18 AM
harish21
Quote:
Originally Posted by goroner
says x^2/(x^2+1) should be done with long division but dont know to do that .. ehm =/
You need to review your books/ notes:
$\frac{x^2}{x^{2}+1} = 1 - \frac{1}{x^2+1}$
Heard of partial fractions?
$\frac{x^2}{x^2+1} = A + \frac{B}{x^2+1}$
$x^2 = A(x^{2}+1) + B$
solving for A and B gives the required result.
• June 1st 2010, 11:32 AM
goroner
i did heard of partial fractions but i never solve one like this , mostly i was solving partial fractions with factors in the denominator ,anyways thanks for this a lot! | {"extraction_info": {"found_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": 0.915408194065094, "perplexity": 1195.7697916100306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1418802772281.50/warc/CC-MAIN-20141217075252-00019-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/126509-proof-interior-points-closed-sets.html | # Thread: Proof: Interior points and closed sets.
1. ## Proof: Interior points and closed sets.
I need help with another complex problem in a general topological space:
Show that a set S is open if and only if each point in S is an interior point.
2. Originally Posted by Porter1
I need help with another complex problem in a general topological space:
Show that a set S is open if and only if each point in S is an interior point.
If each point of a set $\mathcal{O}$ is an interior point then for all $x \in \mathcal{O}$ there is an open set $Q_x\subseteq \mathcal{O}$.
Prove that $\bigcup\limits_{x \in \mathcal{O}} {Q_x } = \mathcal{O}$
3. Thats the thing, I dont know how to prove it.
4. Originally Posted by Porter1
Thats the thing, I dont know how to prove it.
Please believe me, I mean you no disrespect.
Have you ever considered that you may not have the necessary preparation to tackle this material?
Is a retired chair of a mathematical sciences department, I have had this conversation with a countless number of students.
5. Originally Posted by Porter1
I need help with another complex problem in a general topological space:
Show that a set S is open if and only if each point in S is an interior point.
In the other direction to Plato's without all the details. Let $O$ be an open set in a topological space $X$. Suppose that $x\in O$ was not an interior point, then every neighborhood $N$ of $x$ intersects $X-O$. Therefore, we see that $x$ is a limit point of $O'$. But, we have that $x$ is a limit point of $O'$ that is not in $O'$, so $O'$ is not closed and thus $O$ is not open. This, of course, is a contradiction.
This is an indirect way. Of course it assumes you've defined closed set and limit point. If not, you should be able to see a more direct way.
6. Originally Posted by Porter1
I need help with another complex problem in a general topological space:
Show that a set S is open if and only if each point in S is an interior point.
By writing down two definitions the above statement follows quite easily:
Def 1 x is interior point of S <=> there exists ε>0 and $B(x,\epsilon)\subset S$
Def 2 S is open <=> for all ,x: xεS => there exists ε>0 and $B(x,\epsilon)\subset S$.
And using the two definitions we have:
S is open <=> (for all ,x : xεS => there exists ε>0 and $B(x,\epsilon)\subset S$) <=> (for all ,x : xεS => x is interior of S) <=> (each point of S is interior)
7. Originally Posted by xalk
By writing down two definitions the above statement follows quite easily:
Def 1 x is interior point of S <=> there exists ε>0 and $B(x,\epsilon)\subset S$
Def 2 S is open <=> for all ,x: xεS => there exists ε>0 and $B(x,\epsilon)\subset S$.
And using the two definitions we have:
S is open <=> (for all ,x : xεS => there exists ε>0 and $B(x,\epsilon)\subset S$) <=> (for all ,x : xεS => x is interior of S) <=> (each point of S is interior)
This is not a metric space necessarily. So, while there are still neighborhoods the idea of an open ball is not present.
,
,
,
,
# proofs on interior points
Click on a term to search for related topics. | {"extraction_info": {"found_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": 23, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8720279932022095, "perplexity": 415.17880416197295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1512948568283.66/warc/CC-MAIN-20171215095015-20171215115015-00321.warc.gz"} |
https://researchoutput.ncku.edu.tw/en/publications/reheating-neutron-stars-with-the-annihilation-of-self-interacting | # Reheating neutron stars with the annihilation of self-interacting dark matter
Chian Shu Chen, Yen Hsun Lin
Research output: Contribution to journalArticlepeer-review
17 Citations (Scopus)
## Abstract
Compact stellar objects such as neutron stars (NS) are ideal places for capturing dark matter (DM) particles. We study the effect of self-interacting DM (SIDM) captured by nearby NS that can reheat it to an appreciated surface temperature through absorbing the energy released due to DM annihilation. When DM-nucleon cross section σχn is small enough, DM self-interaction will take over the capture process and make the number of captured DM particles increased as well as the DM annihilation rate. The corresponding NS surface temperature resulted from DM self-interaction is about hundreds of Kelvin and is potentially detectable by the future infrared telescopes. Such observations could act as the complementary probe on DM properties to the current DM direct searches.
Original language English 69 Journal of High Energy Physics 2018 8 https://doi.org/10.1007/JHEP08(2018)069 Published - 2018 Aug 1
## All Science Journal Classification (ASJC) codes
• Nuclear and High Energy Physics
## Fingerprint
Dive into the research topics of 'Reheating neutron stars with the annihilation of self-interacting dark matter'. Together they form a unique fingerprint. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9138770699501038, "perplexity": 4910.900621031824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363134.25/warc/CC-MAIN-20211205005314-20211205035314-00532.warc.gz"} |
http://www.computer.org/csdl/trans/tp/2004/07/i0923-abs.html | Publication 2004 Issue No. 7 - July Abstract - A New Convexity Measure for Polygons
A New Convexity Measure for Polygons
July 2004 (vol. 26 no. 7)
pp. 923-934
ASCII Text x Jovisa Zunic, Paul L. Rosin, "A New Convexity Measure for Polygons," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 7, pp. 923-934, July, 2004.
BibTex x @article{ 10.1109/TPAMI.2004.19,author = {Jovisa Zunic and Paul L. Rosin},title = {A New Convexity Measure for Polygons},journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence},volume = {26},number = {7},issn = {0162-8828},year = {2004},pages = {923-934},doi = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2004.19},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Pattern Analysis and Machine IntelligenceTI - A New Convexity Measure for PolygonsIS - 7SN - 0162-8828SP923EP934EPD - 923-934A1 - Jovisa Zunic, A1 - Paul L. Rosin, PY - 2004KW - ShapeKW - polygonsKW - convexityKW - measurement.VL - 26JA - IEEE Transactions on Pattern Analysis and Machine IntelligenceER -
Abstract—Convexity estimators are commonly used in the analysis of shape. In this paper, we define and evaluate a new convexity measure for planar regions bounded by polygons. The new convexity measure can be understood as a "boundary-based” measure and in accordance with this it is more sensitive to measured boundary defects than the so called "area-based” convexity measures. When compared with the convexity measure defined as the ratio between the Euclidean perimeter of the convex hull of the measured shape and the Euclidean perimeter of the measured shape then the new convexity measure also shows some advantages—particularly for shapes with holes. The new convexity measure has the following desirable properties: 1) the estimated convexity is always a number from (0, 1], 2) the estimated convexity is 1 if and only if the measured shape is convex, 3) there are shapes whose estimated convexity is arbitrarily close to 0, 4) the new convexity measure is invariant under similarity transformations, and 5) there is a simple and fast procedure for computing the new convexity measure.
[1] D.M. Acketa and J. Zunic, On the Maximal Number of Edges of Digital Convex Polygons Included into an$(m,m)\hbox{-}{\rm{grid}}$ J. Combinatorial Theory Series A, vol. 69, no. 2, pp. 358-368, 1995.
[2] F. Bookstein, Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge Univ. Press, 1991.
[3] L. Boxer, Computing Deviations from Convexity in Polygons Pattern Recognition Letters, vol. 14, pp. 163-167, 1993.
[4] L. da F. Costa and R.M. CesarJr., Shape Analysis and Classification. CRC Press, 2001.
[5] Automatic Diatom Identification, J.M.H. du Buf and M.M. Bayer, eds., World Scientific, 2002.
[6] S. Hyde et al., The Language of Shape. Elsevier, 1997.
[7] S. Fischer and H. Bunke, Identification Using Classical and New Features in Combination with Decision Tree Ensembles Automatic Diatom Identification, J.M.H. du Buf and M.M. Bayer, eds., pp. 109-140, World Scientific, 2002.
[8] J. Flusser and T. Suk, Pattern Recognition by Affine Moment Invariants Pattern Recognition, vol. 26, pp. 167-174, 1993.
[9] H. Freeman and R. Shapira, Detemining the Minimum-Area Encasing Rectangle for an Arbitrary Closed Curve Comm. ACM, vol. 18, no. 7, pp. 409-413, 1975.
[10] A.E. Hawkins, The Shape of Powder-Particle Outlines. J. Wiley and Sons, 1993.
[11] L.J. Latecki and R. Lakämper, Convexity Rule for Shape Decomposition Based on Discrete Contour Evolution Computer Vision and Image Understanding, vol. 73, no. 3, pp. 441-454, 1999.
[12] L.J. Latecki and R. Lakamper, Shape Similarity Measure Based on Correspondence of Visual Parts IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 10, pp. 1185-1190, Oct. 2000.
[13] R.R. Martin and P.C. Stephenson, Putting Objects into Boxes Computer Aided Design, vol. 20, pp. 506-514, 1988.
[14] S.K. Murthy, S. Kasif, and S. Salzberg, System for Induction of Oblique Decision Trees J. Artificial Intelligence Research, vol. 2, pp. 1-33, 1994.
[15] A.F. Pitty, Geomorphology. Blackwell, 1984.
[16] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[17] U. Ramer, An Iterative Procedure for the Polygonal Approximation of Plane Curves Computer Graphics and Image Processing, vol. 1, pp. 244-256, 1972.
[18] P.L. Rosin, Shape Partitioning by Convexity IEEE Trans. Systems, Man, and Cybernetics, vol. 30, no. 2, pp. 202-210, 2000.
[19] P.L. Rosin, Measuring Shape: Ellipticity, Rectangularity, and Triangularity Machine Vision and Applications, vol. 14, pp. 172-184, 2003.
[20] M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision. Chapman and Hall, 1993.
[21] H.I. Stern, Polygonal Entropy: A Convexity Measure Pattern Recognition Letters, vol. 10, pp. 229-235, 1989.
[22] G.T. Toussaint, Solving Geometric Problems with the Rotating Calipers Proc. IEEE MELECON '83, pp. A10. 02/1-4, 1983.
[23] W. West and G.S. West, A Monograph of the British Desmidiaceae. London: The Ray Soc., 1904-1923.
Index Terms:
Shape, polygons, convexity, measurement.
Citation:
Jovisa Zunic, Paul L. Rosin, "A New Convexity Measure for Polygons," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 7, pp. 923-934, July 2004, doi:10.1109/TPAMI.2004.19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8427563905715942, "perplexity": 4222.298448161203}, "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-2014-35/segments/1408500826016.5/warc/CC-MAIN-20140820021346-00431-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/geometry/CLONE-df935a18-ac27-40be-bc9b-9bee017916c2/chapter-10-review-exercises-page-495/32 | ## Elementary Geometry for College Students (7th Edition) Clone
We draw the base of the parallelogram on the x-axis, with a vertex at (0,0). We define the midpoints as follows: $B: (b,c) \\ C: (a+2b,2c) \\ D: (2a+b,c) \\E: (a,0)$ We use the equation for slope: $m = \frac{y_2-y_1}{x_2-x_1}$ We find: $m_1 = \frac{2c-c}{2b+a-b}=\frac{c}{b+a}$ $m_2 = \frac{2c-c}{a+2b-2a-b}=\frac{b-a}{c}$ $m_3 = \frac{c-0}{2a+b-a}=\frac{c}{b+a}$ $m_4 =\frac{c}{b-a}$ We see that the slopes of opposite sides are equal, making opposite sides parallel and making the shape a parallelogram. | {"extraction_info": {"found_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": 0.9174028635025024, "perplexity": 334.03340867437726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986669057.0/warc/CC-MAIN-20191016163146-20191016190646-00000.warc.gz"} |
https://physics.stackexchange.com/questions/256797/why-do-i-spill-lesser-water-if-its-hotter | # Why do I spill lesser water if it's hotter?
If I pour water in a glass to make a cup of tea, I noticed that if the water that comes out of the kettle is very hot, almost no water is spilled. If the water is cold though, much more water is spilled. The water streams over the surface of the kettle. Why does the higher velocity of the water molecules cause them to stick less to the kettle?
• Before jumping to "why does the higher velocity of the water molecules cause them to stick less to the kettle?" I'd suggest that what's needed is a controlled experiment demonstrating that the effect is real, rather than just anecdotal (perhaps it's a consequence of you being subconsciously more careful with hot water because of the burn risk?). – Kyle Oman May 19 '16 at 11:39
• The viscosity at 100 C is approximately 4 times smaller than at 20 C. Smaller viscosity is lower resistance in a flow. – nluigi May 19 '16 at 11:41
• @nluigi Huh, that suggests that my hunch that this is just anecdotal may be completely wrong. Maybe expand it into an answer? – Kyle Oman May 19 '16 at 11:45
• @KyleOman - i still agree with your comment that the experiment should be controlled. I wonder how pronounced the effect actually is then. If i should believe valerio's answer, it's an actual measurable effect. – nluigi May 19 '16 at 11:49
• I did the experiment. With the kettle. The less water you pour, the more visible the effect. – descheleschilder May 19 '16 at 12:09
We're talking about the Coanda Effect, right? Then I think this article could provide some useful insights.
Quoting from the article:
When the fluid flows over the heated curved surface in proximity of the curved surface as the temperature of the curved surface increases, dynamic viscosity of the fluid at the vicinity of the wall is increasing with respect to the fluid which is far from the curved surface. Then the thermal heat capacity of the fluid near to surface is increasing, and then consequently, the Prandtl number of the fluid near the curved surface is increasing. In this way the momentum boundary layer would be increasing resulted in the decreased adhesion angle.
The second mechanism can also be given by assuming the constant Prandtl number. Increasing the jet velocity, the thermal gradient near the surface is increasing and the thermal boundary layer would be decreased and consequently the momentum boundary layer would be decreased. In this way the adhesion angle would be increased. Consequently, the observed earlier detachment happens by the effect of the complex equilibrium of the above-mentioned effects.
The conclusion is that
Thermal effect on the flow has been analyzed as shown in table 1. Increment of the temperature of curved surface induced the earlier detachment of the jet.
• Strange. It´s so obvious now! The hot water that leaves the kettle consists of faster-moving water molecules wich causes the water to bend less around the outlet because the faster the molecules move the lesser time they have to exert a force on the kettle. I think the same will happen with a water droplet that you gently push out of a syringe. A hot droplet will fall sooner than a cold one. – descheleschilder May 20 '16 at 11:03 | {"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": 0.810075581073761, "perplexity": 623.8897418832894}, "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/1590347410745.37/warc/CC-MAIN-20200531023023-20200531053023-00579.warc.gz"} |
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# M15-27
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16 Sep 2014, 00:56
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Set $$S$$ consists of consecutive multiples of 3 and set $$T$$ consists of consecutive multiples of 6. Is the median of set $$S$$ larger than the median of set $$T$$?
(1) The least element in both sets is 6.
(2) Set $$T$$ contains twice as many elements as set $$S$$.
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16 Sep 2014, 00:56
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Official Solution:
(1) The least element in both sets is 6. If $$S=\{6\}$$ and $$T=\{6\}$$ then the answer is NO but if $$S=\{6, 9\}$$ and $$T=\{6\}$$ then the answer is YES. Not sufficient.
(2) Set T contains twice as many elements as set $$S$$. If $$S=\{6\}$$ and $$T=\{6, 12\}$$ then the answer is NO but if $$S=\{6\}$$ and $$T=\{0, 6\}$$ then the answer is YES. Not sufficient.
(1)+(2) Since $$T$$ contains twice as many elements as set $$S$$ and the least element in both sets is 6 then the median of $$T$$ will always be more than the median of $$S$$, so the answer to the question is NO. For example we can have that $$S=\{6\}$$ and $$T=\{6, 12\}$$ or $$S=\{6, 9\}$$ and $$T=\{6, 12, 18, 24\}$$ or $$S=\{6, 9, 12\}$$ and $$T=\{6, 12, 18, 24, 30, 36\}$$ ... As you can see in each case the median of $$T$$ is more than the median of $$S$$. Sufficient.
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07 Apr 2015, 10:49
Can we consider only one element in a set and evaluate further if it is specifically mentioned in the question that the set has consecutive multiples/numbers.???? Because for the question to be justified there should be at least two elements in the set.
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13 Sep 2016, 15:25
i think answer should be E, negative numbers are also the multiple, example -9 is multiple of 3
set T contains 13 elements all negatives
example ( -13*3, -12*3, -11*3 ........-1*3 ) so median is -7*3= -21
set S contains 06 elements- all positive integers, multiple of 3
example (3*1, 3*2, .......3*6) so median will be 10.5 so now median of S greater than median of T
if we take T set starting from 21 ( 21, 24,27,30..........57) set T will have much higher MEDIAN than S
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14 Sep 2016, 01:10
vijaisingh2001 wrote:
i think answer should be E, negative numbers are also the multiple, example -9 is multiple of 3
set T contains 13 elements all negatives
example ( -13*3, -12*3, -11*3 ........-1*3 ) so median is -7*3= -21
set S contains 06 elements- all positive integers, multiple of 3
example (3*1, 3*2, .......3*6) so median will be 10.5 so now median of S greater than median of T
if we take T set starting from 21 ( 21, 24,27,30..........57) set T will have much higher MEDIAN than S
Notice that (1) says that the smallest element in either set is 6. So, when considering statements together we cannot take negative values.
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21 Sep 2016, 15:39
language is playing trick here,
if sentence 1 is as mentioned at below, E will be the right answer, one S makes the the difference
(1) The least elementS in either set is 6.
whenever we read least, it is always the numbers of elements in the set
smallest would have been more appropriate THAN LEAST,
looks like this is a question of sentence correction more than the question of DS
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29 Jun 2017, 22:24
1
Bunuel wrote:
Set $$S$$ consists of consecutive multiples of 3 and set $$T$$ consists of consecutive multiples of 6. Is the median of set $$S$$ larger than the median of set $$T$$?
(1) The least element in either set is 6.
(2) Set $$T$$ contains twice as many elements as set $$S$$.
Shouldn't we use 'both' instead of 'either' in statement in (1). Either gives an impression that least element in either of the set is 6. Or am i over thinking this?
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29 Jun 2017, 23:56
warriorguy wrote:
Bunuel wrote:
Set $$S$$ consists of consecutive multiples of 3 and set $$T$$ consists of consecutive multiples of 6. Is the median of set $$S$$ larger than the median of set $$T$$?
(1) The least element in either set is 6.
(2) Set $$T$$ contains twice as many elements as set $$S$$.
Shouldn't we use 'both' instead of 'either' in statement in (1). Either gives an impression that least element in either of the set is 6. Or am i over thinking this?
Edited as suggested. Thank you. Hope all is fine now.
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30 Jun 2017, 00:14
Bunuel wrote:
warriorguy wrote:
Bunuel wrote:
Set $$S$$ consists of consecutive multiples of 3 and set $$T$$ consists of consecutive multiples of 6. Is the median of set $$S$$ larger than the median of set $$T$$?
(1) The least element in either set is 6.
(2) Set $$T$$ contains twice as many elements as set $$S$$.
Shouldn't we use 'both' instead of 'either' in statement in (1). Either gives an impression that least element in either of the set is 6. Or am i over thinking this?
Edited as suggested. Thank you. Hope all is fine now.
FYI. The way it was written was also correct. Here is an official guide question with the same usage of "either of the two": https://gmatclub.com/forum/if-r-and-t-a ... 43303.html
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30 Jun 2017, 02:56
Bunuel wrote:
FYI. The way it was written was also correct. Here is an official guide question with the same usage of "either of the two": https://gmatclub.com/forum/if-r-and-t-a ... 43303.html
Thank you Bunuel. I had a look at the question.
I feel the way it is worded makes it unambiguous (at least to my eyes) since it uses keyword "two" --> Phrase: The tens digit of r is less than either of the other two digits of r.
Not that I wanted to contest the wording of this question, just wanted to clarify my doubt in case I encounter similar wording in future.
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30 Jun 2017, 03:01
warriorguy wrote:
Bunuel wrote:
FYI. The way it was written was also correct. Here is an official guide question with the same usage of "either of the two": https://gmatclub.com/forum/if-r-and-t-a ... 43303.html
Thank you Bunuel. I had a look at the question.
I feel the way it is worded makes it unambiguous (at least to my eyes) since it uses keyword "two" --> Phrase: The tens digit of r is less than either of the other two digits of r.
Not that I wanted to contest the wording of this question, just wanted to clarify my doubt in case I encounter similar wording in future.
Understood. Anyway, now the wording is clear as it is. Thank you fro the suggestion.
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Re: M15-27 &nbs [#permalink] 30 Jun 2017, 03:01
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# M15-27
Moderators: chetan2u, Bunuel
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | {"extraction_info": {"found_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": 0.8090415000915527, "perplexity": 1859.8789241297989}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510866.52/warc/CC-MAIN-20181016180631-20181016202131-00075.warc.gz"} |
http://infamousheelfilcher.blogspot.com/2013/12/a-left-for-reader-in-stability-theory.html | ## Friday, December 6, 2013
### A Left-for-the-reader in stability theory
Yes, yes, I know I haven't posted in forever. I've been busy proving shit.
Completely out of left field, my research has indicated relevance to stability theory, an area of model theory that I've never had the urge to learn. Well, now I feel the urge. So much for purity of intention.
Ralph and I had a little fun today thinking through a left-for-the-reader in a paper that I hope to raid for its methods. So much fun that I want to share it with you, O gentle reader.
Definition 1:
• Let $$\lambda$$ be an infinite cardinal. A theory $$T$$ is $$\lambda$$-stable if for every model $$\mathbf{M} \models T$$ and every $$C \subset M$$ with $$|C| \leq \lambda$$, the space $$S_1(T/C)$$ of 1-types with parameters from $$C$$ has cardinality no greater than $$\lambda$$.
• $$T$$ is stable if it is $$\lambda$$-stable for some $$\lambda$$, and superstable if it is $$\lambda$$-stable for all cardinals $$\lambda$$ greater than some fixed cardinal.
Let $$L$$ be the first-order language with a constant symbol $$0$$, a binary function $$+$$, and a unary function $$F$$, and let $$T$$ be the theory axiomatized by the following sentences:
• $$+$$ is associative
• $$0 = x + x$$
• $$x + 0 = 0 + x = F(x) = F(F(x))$$
• $$x + y = F(x) + F(y) = F(x+y)$$
Note that these axioms imply that $$F$$ is an endomorphism of every model of $$T$$. It is less visually obvious, but still pretty quick, to show that the kernel of $$F$$ is always a strongly abelian congruence.
Exercise: this theory is unstable. That is, it fails to be $$\lambda$$-stable for any infinite cardinal $$\lambda$$.
Proof: For every model $$\mathbf{M} \models T$$, $$\mathbf{M}/ \ker F$$ is an abelian group $$\mathbf{A}$$ of exponent 2. Conversely, for every such group and every function $$j$$ from $$A$$ into cardinals, we can produce a split extension $$\mathbf{A} \leq \mathbf{M} \models T$$ where $$| F^{-1}(a) | = j(a)$$. (Note that $$\mathbf{M}$$ will not in general be a group, since addition will not typically be surjective.)
Let $$\lambda$$ be an infinite cardinal. We must produce a model of $$T$$ and a set $$C = \{ c_i \colon i < \lambda \}$$ such that $$T$$ has more than $$\lambda$$ types with parameters in $$C$$. Let $$\mathbf{A}$$ be the free $$\mathbb{Z}_2$$-module on generators $\{ c_i \colon i < \lambda \} \cup \{ x_g \colon g \in \omega^\lambda \}$and construct the split extension $$\mathbf{M}$$ described above with $$| F^{-1}(x_g + c_i) | = g(i) + 1$$, for each $$g \in \omega^\lambda$$.
Observe that if $$g_1 \neq g_2$$, then $$\mathrm{typ}(x_{g_1}/C) \neq \mathrm{typ}(x_{g_2}/C)$$; indeed, if $$g_2(i) > g_1(i)$$, then $\mathbf{M} \models \exists^{=g_1(i)} y \quad y \neq F(y) = x_{g_1} + c_i$while $\mathbf{M} \models \exists^{>g_1(i)} y \quad y \neq F(y) = x_{g_2} + c_i$This suffices to complete the proof, since $$\omega^\lambda = 2^\lambda > \lambda$$, showing that $$T$$ has more than $$\lambda$$ types with parameters from $$C$$. | {"extraction_info": {"found_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": 0.969752848148346, "perplexity": 177.21613580815637}, "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/1490218189088.29/warc/CC-MAIN-20170322212949-00111-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://math.stackexchange.com/tags/banach-spaces/new | # Tag Info
0
The conclusion of the theorem you want to prove holds if and only if $X^*$ has the Radon–Nikodym property with respect to the Lebesgue measure on $(0,1)$. This is precisely what you need to recover Rademacher's theorem about differentiation of Lipschitz maps from $(0,1)$ to $X^*$. Separability in your proof is redundant but it makes it easier as you may ...
3
One need not to know the whole dual space of $C[0,1]$ to see the lack of reflexivity. Simply note that $C[0,1]^*$, opposed to $C[0,1]$, is non-separable because for each $t\in [0,1]$ the map $$\langle f, \delta_t\rangle = f(t)\quad (f\in C[0,1])$$ is a norm-one linear functional on $C[0,1]$ and $\|\delta_t - \delta_s\| =2$ for distinct $s,t$. To see this, ...
2
No it is not. The dual of $C[0,1]$ is the space $\mathfrak{M}([0,1])$ of complex (or signed) regular Borel measures on $[0,1]$. The dual of $\mathfrak{M}([0,1])$ is the set $\mathcal L^\infty([0,1])$ of bounded Borel functions on $[0,1]$. For example, $\chi_{\{0\}}$, the function which vanishes everywhere, except at $x=0$, where it is equal to $1$, ...
1
Consider the subspace $V_\infty$ of $\mathscr l^2(\mathbb N)$ where only a finite amount of terms in a series is non-zero. This is an infinite dimensional normed vector space. Define also the subspace $V_n$ where only the first $n$ terms of a series are non-zero. $V_n \cong \mathbb R ^n$ with the standard norm. As such there is a sequence of compacta ...
2
No, the last assumption does not follows from the first two one. To see this consider operators $T_n f = f\left(x^n \right).$
0
By definition weak* convergence of $f_k$ to $f$ in $X=L^{\infty}(\mathbb R)$ means: $$lim_{k \to \infty}\langle f^*,f_k\rangle=\langle f^*,f \rangle \space (\forall f^*\in X^*)$$ This is a convergent sequence of reals, hence we have $$sup_{k \in \mathbb N}\Vert \langle f^*,f_k \rangle \Vert = sup_{k \in \mathbb N} \Vert \iota(f_k)(f^*)\Vert < \infty ... 2 Of course this is quite simple, as Jonas showed. Here's a fun way to look at it. Let K=\{x_n'\}\cup\{x'\}. Then K is a compact metric space. Regard x_n and x as functions from K to \Bbb C. Uniform Boundedness shows that ||x_n|| is bounded, and this shows that our family of functions from K to \Bbb C is equicontinuous. And x_n\to x ... 1 Note that$$ |x_n'(x_n)-x'(x)|\le|x_n'(x_n)-x'(x_n)+x'(x_n-x)|\le\|x_n'-x'\|\cdot\|x_n\|+|x'(x_n-x)| $$and the sequence \|x_n\| is bounded because it converges. 1 The proof follows from the following theorem from basic calculus: Let \{a_n\}_{n\in\mathbb N} be a sequence of nonnegative numbers with the property that a_{n+m}\leq a_n \cdot a_m for all n,m\in\mathbb N. Then the limit \lim\limits_{n\to\infty}\sqrt[n]{a_n} exists and it is equal to \inf\limits_{n\in\mathbb N}\sqrt[n]{a_n}. Theorem about ... 1 A(\lambda x_1,\ldots,\lambda x_m) = \lambda^{m-j}\overline{\lambda}^jA(x_1,\ldots,x_m)= \lambda^{m-k}\overline{\lambda}^k A(x_1,\ldots,x_m) From the above, since \lambda \neq 0, we have:$$\left[ \left(\frac{\lambda}{\overline{\lambda}}\right)^{k-j} - 1\right] A(x_1,\ldots,x_m) = 0$$Thus: \left[ \left(\frac{\lambda}{|\lambda|}\right)^{2(k-j)} - ... 0 I find it easier to prove (b) first; using the hint that was given for (a) appears more relevant to (b). (b) If no such C exists, there is a sequence (x_n,y_n) such that$$|B(x_n,y_n)| = 1\quad\text{ and }\quad \|x_n\|\|y_n\|\to 0 \tag{1}$$By replacing (x_n,y_n) with (sx_n, s^{-1}y_n) we can arrange \|x_n\|=\|y_n\| without changing (1). ... 0 EDIT (see below) Following Robert Israel's idea here's another example, which is even Hilbertian: on the space L^2(0,1) consider, again, the polynomials and the linear operator T defined as$$T(x^{2k})=0\quad > T(x^{2k+1})=x^{2k+1}, $$which agrees with the identity on the dense subspace consisting of odd-degree polynomials and is ... 3 Yes, f(B(x_{0}; \epsilon)) is open. It does not follow that B(x_{0}; \epsilon) = f^{-1}((B_{\mathbb{F}}(f(x_{0}); \epsilon'). It doesn't follow even in a finite-dimensional space. To get an idea what's happening here, say X=\Bbb R^2 with the euclidean norm. Say B is the unit ball of X and define f(x,y)=x. Then f(B) is the open interval ... 3 You are correct up until B(x;\epsilon) = f^{-1}(B_\mathbb{F}(f(x_0);\epsilon')). All you can conclude from the statement before that is B(x;\epsilon) \subseteq f^{-1}(B_\mathbb{F}(f(x_0);\epsilon')), which does not imply that B(x;\epsilon) is weakly open. In general, an open ball for the norm topology in an infinite-dimensional Banach space has empty ... 1 It implies that C\|x\|\leqslant \|Tx\| holds true which means that T is injective and has closed range. Note that it is enough to check this only on a dense subspace and {\rm span}\{\delta_x\colon x\in X\} is such a subspace. 1 Let M=\|u\|_{\infty}. Since G' is continuous, there exists a constant K such that |G'(s)|\leq K for all s\in [-M,M]. Thus |G'\circ u|\leq K; it follows that |G'\circ u|^p is bounded by K^p. The desired conclusion follows from Hölder's inequality as you explained with a little correction (you have to change parenthesis by norms): ... 1 Consider the Banach space X = C[0,1] of continuous functions on [0,1], with the operator A of multiplication by x (i.e. Af(x) = x f(x)). This has spectrum [0,1]. Let E be the subspace of X consisting of polynomials. This is invariant under A. However, A - \lambda I is never surjective as an operator from E to E, e.g. there is no ... -1 X=R^2=Vect\{e_1,e_2\}, A(e_1)=e_1, A(e_2)=2e_2, E=R^2-Re_2. The spectrum of A restricted to E is 1 and the spectrum of A is \{1,2\}. 0 The answer can be found in "Calculus without derivatives" of Prof. Jean-Paul Penot. I refer to Lemma 3.94 p.251. The answer is the following : an equivalent norm on the dual X^* is the dual norm of an equivalent norm on X if and only if it is weak-* lower semicontinuous. The major idea is the following. Let us consider the notations introduced in the ... 1 Y is the kernel of the surjective (and continuous) map$$C^1([0,1]^n) \to \mathbb R, f \mapsto f(0).$$In particular, Y is closed (as a kernel) and of codimension 1 (since C^1([0,1]^n)/Y \cong \mathbb R is one-dimensional). 1 The spectrum of a is a compact set that does not contain 0. So there is a disk D around 0 with D\cap\sigma(a)=\emptyset. Thus, on \sigma(a), f:t\longmapsto 1/t is continuous, so f\in C(\sigma(a)). Then f(a)\in C^*(a) via the Gelfand transform. Or, even easier, you could check that f is analytic on \sigma(a), and so a^{-1} belongs ... 2 That \Gamma(y) should be closed doesn't follow from the fact that \Gamma is a closed subalgebra. Example. Let \mathbb N^+ = \mathbb N \setminus \{0\} be the set of positive integers. Consider X = \ell^2(\mathbb N^+) and let R : X \to X be the following weighted right shift:$$ (Rx)(n) = \begin{cases} \quad 0, & \quad\text{if $n = 1$}; ...
2
Lemma (Exercise 3.29, Brezis). Let $E$ be a normed vector space with a uniformly convex norm and fix $p > 1$. If $x$, $y \in \overline{N(0, M)} =: B$ are at least $\epsilon > 0$ apart, then there is some $\delta$ such that$$\left\|{{x + y}\over2}\right\|^p \le {{\|x\|^p + \|y\|^p}\over2} - \delta.$$ Suppose not for the sake of contradiction. Then ...
1
Reflexivity is superfluous. You can use the simple fact that if $T:X\rightarrow X$ is a compact map on a Banach space $X$, with $\dim(X)=\infty$, then $0\in \sigma(T)$.
1
For every normed space $X$, the dual $X^*$ is $1$-complemented in $X^{***}$. Indeed, let $i:X\to X^{**}$ be the canonical embedding; then its adjoint $i^*$ is a projection of norm $1$ of $X^{***}$ to $X^*$. Simply put, it takes a functional $\phi:X^{**}\to \mathbb{C}$ and composes it with $i$. In particular, the above applies to $\mathbb{B}(\mathbb{H})$, ...
2
I don't see the point of reflexivity here, $inf\{\|T(x)\|, \|x\|=1\}>0$ implies that the spectrum of $T$ does not contain zero. Since $T$ is compact, $X$ must be finite dimensional so $S$ is compact.
1
Yes. Say $f_t\in L^2[0,1]$ for $t\in[1,2]$ and $f_t$ depends continuously on $t$. Then $K=\{f_t:t\in[1,2]\}$ is a compact subset of $L^2[0,1]$. Define $T_n:L^2[0,1]\to L^2[0,1]$ by, say, letting $T_n=f*\phi_n$, where $\phi_n$ is an approximate identity. It's easy to see that for each $n$ the function $g_n(s,t)=T_nf_t(s)$ is continuous on $[0,1]\times[1,2]$. ...
1
$\newcommand{\ip}[2]{\left\langle #1,#2\right\rangle}$ $\newcommand{\norm}[1]{\left|\left| #1\right|\right|}$ I think your confusion is due to notation used in Pythagorean theorem: we don't actually require any norm-property of $||\cdot||$ induced by the inner product in proving it. It's only after we recognise $||\cdot||$ as a norm that it has its usual ...
2
Only a partial answer: If you take the norm $\|x\|'=\|x\|+\alpha |x|$ as gerw said in the comments, it is easy to see that since $\|.\|\sim |.|$, i.e $\exists \underline c,\overline c>0: \underline c\|x\|\leq |x|\leq \overline c \|x\|,\forall x\in E\,\,$ you will have $$\|x\|\leq \|x\|+\alpha|x|\leq (1+\alpha \overline c)\|x\|,\forall x\in E$$. It is ...
2
By assumption $p\neq q$. Case 1: $p,q\in(1,+\infty)$. Assume we have a surjection $T:\ell_p\to\ell_q$, then $T^*:\ell_{q'}\to \ell_{p'}$ is an embedding. Since $p',q'\in(1,+\infty)$, we get a contradiction because by corollary of Pitt's theorem theses spaces are totally incomparable. Thus for this case a desired surjection doesn't exists. Case 2: $q=1, ... 0 In the case$V_i = \Bbb R$for all$i$, and the codomain being$\Bbb R^n$, for some$n$, i.e.,$f: \Bbb R^d \to \Bbb R^n$,$x = (x_1,...,x_d) \mapsto f(x) = (f_1(x),...,f_n(x))$, if$a \in \Bbb R^d$is a point at which$f$is differentiable, then$Df(a)$is simply the Jacobian matrix of$f$at$a$. That's, $$Df(a) = \left( \frac{\partial f_i}{\partial ... 1 You are correct. We can note that our space is isometric to the subspace of l^2(\mathbb{N}) comprised of sequences with finitely many non-zero entries, by the isometry \sum_{n=0}^N a_n z^n\mapsto (a_0, a_1, \ldots, a_N, 0, 0, \ldots). (Note that, indeed, \left\langle \sum_{n=0}^N a_n z^n, \sum_{n=0}^M b_n z^n\right\rangle = ... 0 No, this seems not to be possible. Take E = L^2(0,1) and consider$$x_i = \chi_{((i-1)/n,n)}.$$Then,$$\sum_{i=1}^n \|x_i\|_{L^2(0,1)} = \sqrt{n}$$while$$\|\sum_{i=1}^n x_i\|_{L^2(0,1)} = 1.$$It is even worse with L^p(0,1), p > 2. And with p = \infty, you need the constant n. 1 You are definitely on the right track and nearing a complete proof! I'm going to write \mathcal{A}_p for what you're denoting by \sum_p A_n (since I find the placement of the p rather unsettling in your notation ;)). If you're comfortable with the idea of a direct product of vector spaces, you can think of \mathcal{A}_p (before assigning it a ... 1 Let (a_n)_j be a Cauchy sequence of sequences. Now, the "obvious" limit choice would be the sequence (a_n) where a_n = \lim_{j \to \infty} (a_n)_j. To show this, you have to prove that all those (a_n) exist for any j, and then that that is the actual limit. For the first part, it's enough to observe that$$ \|(a_n)_j - (a_n)_k\|_n^p \leq ... 1 You are confused: Let$H$be a Hilbert Space, let$B=\{u_j\}_{j=1}^\infty$be a countable orthonormal basis. So we know that if a set is a complete orthonormal basis, the set of all finite linear combinations is dense in$H$. It is true. Now, since the set of all finite linear combinations is dense let$x\in H$, we have ... 2 A very elegant proof can be based on tensor product representation $$C([0,1],E)= C([0,1]) \tilde \otimes_\varepsilon E.$$ For every dense subspace$L$of$C([0,1])$the tensor product$L\otimes E$is dense. 1 I already know that the unit ball in$X$(denoted$B$) is compact in the$\|.\|$topology. So I just need to have the estimate $$\| . \|_{\alpha} \leq C \|.\|$$ for some$C$to conclude that$B$is compact in the$\|.\|_{\alpha}$topology. Now$X$is closed in$C[0,1]$, the inclusion$i : C^\alpha[0,1] \to C[0,1]$is continuous so that$i^{-1}(X) = ...
3
Seems like the answer is yes. Take $f\in C([0,1],E)$. First extend $f$ to a function in $C(\Bbb R, E)$, say by making $f$ constant on $[1,\infty)$ and constant on $(-\infty,0]$. Now say $\phi_n$ is a smooth (real-valued) approximate identity; then the convolution $f*\phi_n$ should be differentiable and it should be that $f*\phi_n\to f$ uniformly on $[0,1]$. ...
3
You can just take the coefficients of the polynomials in $E$ to have $E$-valued polynomials. Trying to mimic the Stone-Weierstraß proof for $E$-valued functions may however be tricky. But from uniform continuity of continuous functions on $[0,1]$ you immediately get that you can uniformly approximate all functions in $C([0,1],E)$ by piecewise affine ...
1
Hint Consider a Cauchy sequence $\{x_n\}$. See what happens with $$\left\{\frac1{\|x_n\|}x_n\right\}$$ (Note that you also have to consider sequences with zeros).
1
Let $Y$ be the closure of the range of $A$. We define $B : X \to Y$ by $B x = A x$ for all $x \in X$. Let us show that $B' : \tilde Y \to \tilde X$ is invertible. Since the range of $B$ is dense, $B'$ is injective. It remains to show that $B$ is surjective. For any $\tilde x \in \tilde X$, there is $\tilde r \in \tilde X$, such that $A'\tilde r = \tilde x$. ...
1
I think you can find the spectrum directly. First, $$T((a_j))_i=\sum_{j=2}^{\infty}a_j e_1+\sum_{i=2}^{\infty}a_{i-1} e_i=\sum_{j=2}^{\infty}a_j e_1+a_{i-1}$$ Then $$(T-\lambda I)((a_j))_1=\sum_{j=2}^{\infty}a_j -\lambda a_1,$$ $$(T-\lambda I)((a_j))_i=a_{i-1}-\lambda a_i$$ So if $\lambda$ is an eigenvalue, the second equation implies ...
2
You don't need such heavy machinery as Hahn-Banach. This is a completely elementary fact: Let $J\colon E\times F\to F\times E,\,J(x,y)=(-y,x)$. It is a (more or less) immediate consequence of the definiton of $A^\ast$ that $G(A^\ast)=(J G(A))^\perp$, where $G(T)$ denotes the graph of the operator $T$. Then we have $x\in N(A)$ iff $(x,0)\in G(A)$ iff ...
1
If the projection is continuous, then the range of $P$ is closed, as it is the kernel of the continuous projection $I-P$. However not all projections on a Banach space are continuous (as opposed to what happens in the case of a finite dimensional space). So, in general, this is not true. Take for instance the subspace generated by $\{(1,0,\dots), (0,1,0, ... 4 Convexity: To show$\psi$is convex we only need to show$\varphi^*$is convex. Let$f,g\in E^*, \lambda\in [0,1]$. We want to show $$\varphi^*(\lambda f+(1-\lambda)g)\leq \lambda \varphi^*(f)+(1-\lambda)\varphi^*(g)$$ To prove this let$x\in E$with$\varphi(x)<+\infty$. Then \begin{eqnarray} \langle \lambda f+(1-\lambda)g,x\rangle-\varphi(x) & ... 2 You can bound$(f\ast g)_n$below by $$\sum_{m = 1}^{n-1} m^{-\phi} (n-m)^{-\psi} \geqslant \sum_{m = 1}^{n-1} (n-1)^{-\phi}(n-1)^{-\psi} = (n-1)^{1 - \phi - \psi}.$$ So a necessary condition for$f\ast g \in \ell_{\infty}$is$\phi + \psi \geqslant 1$. That is also sufficient, as can be seen by splitting the sum at$n/2\$: \begin{align} \sum_{m = 1}^{n-1} ...
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# Standard Error Two Sample
JSTOR2340569. (Equation 1) of 10/4 = 2.5 standard deviations above the mean of -10. We have $$n_1 However, this method needs additional requirements to be satisfied (at least approximately): Requirement R1: Agreement. But first, a Standard Clicking Here a normal population or large samples? Error Standard Error Of Sample Mean Calculator For the age at first marriage, the population mean Are these independent samples? Bence (1995) Analysis of shortverify the difference Reply Prashant Thanks, the article was very informative and helpful. Remember the Pythagorean from the output. Interpret the above result: We are 99% confident Perspect Clin Res. Sample 5 is shaded blue. By using this site, you agree to standard deviation of the Student t-distribution. Standard Error Of Difference Between Two Means Calculator JSTOR2682923. ^ Sokal and Rohlf (1981) Biometry: Principles andclusters more closely around the population mean and the standard error decreases. The sample standard deviation s = 10.23 is greater The sample standard deviation s = 10.23 is greater Compute the t-statistic: \[s_p= \sqrt{\frac{9\cdot (0.683)^2+9\cdot the Wikimedia Foundation, Inc., a non-profit organization.Reply gloria please can you explain to me5 is shaded blue. We use another theoretical sampling distribution—the sampling distribution of the differencethat the mean GPAs of sophomores and juniors at the university are different. Standard Error Of Difference Calculator that takes into account that spread of possible σ's.Sampling distribution of the "Healthy People 2010 criteria for data suppression" (PDF). Figureare already familiar with the sampling distribution of the mean. When the true underlying distribution is known to be Gaussian, althoughin the average cycle time to deliver a pizza from Pizza Company A vs.Therefore, .08 is not the true difference,assumption of equal variances is violated?Since the p-value is larger than \(\alphathe absolute basics of the two-sample t-test.With unequal sample size, the larger page here to cancel reply. The 95% confidence interval for the average effect of the standard deviation of the sampling distribution of M1 - M2.That is Let n2 be the sample size from population 2, as an average so i can conclude this there is alot of difference after improvement.The question being answered is whether there is a significant (or only random) differenceconclusion in words. ISBN 0-8493-2479-3 p. 626 ^ a b Dietz, Davidl; Barr, walk through of 'Conducting a Pooled t-test in Minitab'. Frankfort-Nachmias and Leon-Guerrero note that the properties of the sampling distribution of the differencetime series: Correcting for autocorrelation.Because these 16 runners are a sample from the population of 9,732 runners,true population mean is the standard deviation of the distribution of the sample means.State the conclusion in words. Finally, check the box Error these are population values. is really significant or if it is due instead to random chance. However, we are usually using sample data Standard Error Of The Difference Between Means Definition between two sample means are determined by a corollary of the Central Limit Theorem.However, since these are samples and therefore involve error, known, we have to estimate the SE mean. The 95% confidence interval contains zero (the null hypothesis, no difference http://typo3master.com/standard-error/guide-standard-error-vs-sample-standard-deviation.php a a sample mean is significantly different from a hypothetical or known population mean.Student approximation when σ value is unknown Further information: Student's t-distribution §Confidence Company B. Two is used to compare two means from two different populations.Example: Comparing Packing Machines In a packing Error Theorem in geometry? Sampling Distribution of the Differences Between the Two Sample Means for Independent Let \(\mu_1$$ denote the mean for the new machine Standard Error Of Difference Definition not related to the students selected from juniors.When we are reasonably sure that the two populations haveworked on a project to reduce the number of shipping errors. population of mean = 90 grams?
See unbiased estimation of Two of green beings on Mars.Notice that it is normally distributed with adistribution of the difference between means.the age was 4.72 years.
The graph below shows the distribution of the sample means read this post here Royal Statistical Society.Reply Pinchy http://www.r-project.org/the test anyhow.When thinking of "population" versus "sample", runners from the population of 9,732 runners. Standard Error Of Difference Between Two Proportions the age was 9.27 years.
The following dialog boxes the separate variances test. Reply k Galloway Is there any freeand asked if they will vote for candidate A or candidate B.The correct z critical value for to estimate the standard error. Note: The Student's probability distribution is a good approximation2.
The Sampling Distribution of the Difference between the Means You conclude this? Similarly, the sample standard deviation will very Two each machine to pack ten cartons are recorded. Of course, T / n {\displaystyle T/n} is Sample Mean Difference Formula Two They may be used
Using Minitab to Perform a Using Minitab Click on this link to follow along with The mean of all possible sample Standard Error Of Sample Mean Formula and the null hypothesis of mean1-mean2=0 will be satisfied.It is rare that thewill focus on the standard error of the mean.
The distribution of the differences between means is 165 and the variance is 64. Standard error of the mean (SEM) This section Error signed in Forgot your password? an application of this formula. The graphs below show the sampling distribution of the will then be displayed.
The rest of the article, however, discusses understanding the two-sample t-test, which is .25081 Reply Tim Erickson Some potentially misleading language here. primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. Actually, if one subtracts the means from two in a population is 34 and the mean of 10-year-olds is 25.
and do not know the population variances.
Using either a Z table or the normal confused between this two. Step of occurrence is less or equal to 5 percent. 2.
The confidence interval
The researchers report that candidate A is expected to receive 52% 1. The ages in that sample were 23, 27, 28, 29, 31, this case each error is occuring for every ten loads as an average. Pooled t-procedure (Assuming Equal Variances) 1.
The mean age for the 16 is 165 - 175 = -10.
For girls, the mean is Lesson 2 - Summarizing Data Software - Describing Data with Minitab II. In this scenario, the 400 patients are a sample How do I show new drug lowers cholesterol by an average of 20 units (mg/dL).
experiment depend on the sampling distribution of the difference between means.
As shown below, the formula for the standard error of the difference between while the mean height of Species 2 is 22. Now let's look at confusion about their interchangeability. | {"extraction_info": {"found_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": 0.9506887793540955, "perplexity": 1102.601657314474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376832559.95/warc/CC-MAIN-20181219151124-20181219173124-00185.warc.gz"} |
https://jasonmaths.wordpress.com/category/physics/ | HAiKQDM:MCfGTtTQCaCQM postmortem
Last week I attended a conference at the Perimeter Institute for Theoretical Physics, titled “Hopf Algebras in Kitaev’s Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics”. My flight from LAX (Los Angeles) to YYZ (Pearson-Toronto) was delayed due to a maintenance issue, leaving me sitting in the plane on the tarmac for about an hour and a half before takeoff, right close to midnight. My flight to Oxford for the QPL conference (Quantum Physics and Logic) two years ago had a similarly long delay… Hmmmn. Despite this conference being fairly small—40 people on the list of participants—there were several people present that I met at that QPL conference (Ross Duncan, Stefano Gogioso, and Pawel Sobocinski), and a couple more that I met while I was a grad student at Riverside (Tobias Fritz and Derek Wise). It turns out Ross and Stefano are sharing an apartment on the floor above my room. Continue reading
QPL 2015
On July 17 I gave a talk at Oxford in the Quantum Physics and Logic 2015 conference. It was recently released on the Oxford Quantum Group youtube channel. There were a bunch of other really cool talks that week. In my talk I refer to Pawel Sobocinski’s Graphical Linear Algebra tutorial on Monday and Tuesday, as well as Sean Tull’s talk on Categories of relations as models of quantum theory. There are benefits to speaking near the end of the conference.
Classical Mechanics formulations (Part 2)
Last time I mentioned looking at two functions, $M(p,\dot p)$ and $J(\dot q,\dot p)$ that would ideally carry all the data needed to solve certain types of problems in classical mechanics. In the case of $M$, I conjectured that equations analogous to the Lagrange equations ought to exist and hold, namely,
$\displaystyle \frac{d}{dt}\frac{\partial M}{\partial \dot p_i} = \frac{\partial M}{\partial p_i}$.
If we got lucky, we could recover $q$ and $\dot q$ from those partials. While my approach on that front (prior to writing this post) has been… heuristic at best, I have found evidence that indicates $M = L$ with the coordinates transformed. At least in some situations. In my attempt to present this evidence yesterday afternoon without complete notes, I was unable to recapture this evidence on the fly. There is probably a good lesson in that, even if nothing else is gained from this exercise.
In the comments for Part 1, Dr. Baez suggested looking at the one-dimensional Lagrangian $L(q,\dot q) = \frac{1}{2}m\dot q\cdot\dot q - V(q)$, for which the equations of motion are well known. Acting on this suggestion over the weekend, during a bout of insomnia while running a fever, I wove together the evidence I found so elusive yesterday with a skein of half-baked yarn. I probably could mix more metaphors in there if I tried, but the point is, the following paragraph may need to be taken with a liberal grain of salt.
At first I was uncertain how to approach an ‘easy’ $V(q)$, like a gravitational potential, so I took a simple spring potential, $V(q) = \frac{1}{2}kq\cdot q$. So that’s already a point in favor of feverish-insomnia me over at-the-whiteboard-without-notes me, who couldn’t even write down the potential correctly. The equation of motion is $m\ddot q = -kq$, from which we can find $q(t) = A\sin(\sqrt{\frac{k}{m}}t+B)$. That can be simplified by choosing units where $A=1$ and a suitable time translation for which $B=0$, but I did not originally make those simplifications, so I won’t here. $\dot q(t) = A\sqrt{\frac{k}{m}}\cos(\sqrt{\frac{k}{m}}t+B) = \frac{p}{m}$, and $\ddot q(t) = -A\frac{k}{m}\sin(\sqrt{\frac{k}{m}}t+B) = \frac{\dot p}{m}$. Noticing those two outside equalities, $\dot q = \frac{p}{m}$, and $\ddot q = \frac{\dot p}{m}$, along with the equation of motion itself, $m\ddot q = -kq$, right off the bat would have saved me from having to do any of these contortions in $t$. There goes that point I gave to feverish-insomnia me.
In particular, $L(q,\dot q) = \frac{1}{2}(m\dot q\cdot\dot q - kq\cdot q)$ can be rewritten by converting $q$ to $\ddot q$ via the equation of motion, and from there $\dot q$ can be written in terms of $p$, so $\ddot q$ can be written in terms of $\dot p$. Presto:
$\displaystyle L = \frac{p\cdot p}{2m} - \frac{\dot p\cdot\dot p}{2k}$
without any reference to $t$. But what happens when we treat this as $M$ and write the Lagrange-analogy equations?
$\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot p} = \frac{d}{dt}\left(-\frac{\dot p}{k}\right) = \frac{d}{dt}(q)$, because $-kq = m\ddot q = \dot p$. Similarly,
$\displaystyle \frac{\partial L}{\partial p} = \frac{p}{m} = \dot q$,
and indeed, $\frac{dq}{dt} = \dot q$.
So that worked out for a simple spring. And it wasn’t just tenuous dreamstuff it was made of, after all. Bolstered by that success, I continued to the ‘simpler’ case of constant gravity. Here, $V(q) = mgq$, where $g$ is the acceleration due to gravity, and is constant. Again, I took the cheap route of writing things in terms of $t$, but this time I was unable to fold all the $t$s back into an expression with only $p,\dot p$ as variables. So that’s the bad news for this one. The good news is that if you take $L(p,\dot p,q)$ and apply the partials, you do get $\frac{dq}{dt} = \dot q$ again. Unfortunately, this requires more data than the original formulations, as the Hamiltonian approach gets by with a proper subset of those coordinate variables. I had thought I had gotten away from requiring $q$, but looking at what I wrote in my delirious state, that really was a phantasm of night.
It would appear this pseudo Lagrangian approach can work for certain situations, most likely ones whose equations of motion are second order homogeneous differential equations such that $q$ can be solved for in terms of $\dot q, \ddot q$, which can be converted to $p,\dot p$. This seems likely at least for the given kinetic energy that was used here. A DHO (damped harmonic oscillator) might be worth looking at in the future, as those provide a generic linear second order homogeneous differential equation of motion.
Classical Mechanics formulations (Part 1)
Given fixed starting position and time, $q_0$ and $t_0$, and fixed ending position and time, $q_1$ and $t_1$, when a particle travels along a path of ‘least’ action, it obeys the Euler-Lagrange equations for some Lagrangian, $L$:
$\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{q}^i} = \frac{\partial L}{\partial q^i}$
which is a fancy way of saying the time derivative of $p_i$ (momentum) is equal to $F_i$ (force). The natural variables for this Lagrangian approach are $q$ (position) and $\dot{q}$ (velocity), from which $p$ and $F$ are built.
Given the right conditions, we can recast this formulation into the Hamiltonian approach, where $q^i$ and $p_i$ are the natural variables, and several nice things happen. By ‘right conditions’ I mean conditions that would allow us to return to the Lagrangian approach. One of the nice things that happen is position and momentum can be seen to be conjugate to each other in a way that is impossible for position and velocity. Indeed, Hamilton’s equations illustrate this conjugacy nicely:
$\displaystyle \frac{dp_i}{dt} = -\frac{\partial H}{\partial q^i}\\ \frac{dq^i}{dt} = \frac{\partial H}{\partial p_i}$
where $H(q^i,p_i)$ is the total energy function.
If $p$ and $q$ are so similar, could we not recast this yet again, but in terms of $p$ and $\dot p = F$, much like the Lagrangian approach was in terms of $q$ and $\dot q$? Or if we are feeling zealous with reformulations, why not consider mechanics recast in terms of $\dot q$ and $F$, like a derived Hamiltonian approach? These are some questions that occurred to me near the end of last quarter in Dr. Baez’ Classical Mechanics (Math 241) course, when he went over how to transition from the Lagrangian approach to the Hamiltonian approach.
At first blush, these ‘new’ approaches seem to give us less information than the old ones. To wit, in Newtonian mechanics, $p = m \dot q$, so any information about the absolute position appears to be lost to a constant of integration, especially for the $\dot q, F$ approach. But it is worse than this. For each particle being considered, the constant of integration may be different. So not only do we lose absolute position, we also lose relative position. This limits our considerations to situations where potential energy is zero, unless something very nice happens that would allow us to recover $q$.
The nice thing about mathematics versus physics, is that I can cast aside the difficulties of whether or not things are physically relevant, as long as they make sense mathematically. So I will set aside any complaints about possible non-utility and forge ahead. I have not actually looked that far ahead yet, but I suspect the first ‘new’ approach will be somewhat similar to the Lagrangian approach, via the analogy between $q$ and $p$. That is, I suspect there will be some function, $M$, analogous to the Lagrangian, $L$, such that:
$\displaystyle \frac{d}{dt}\frac{\partial M}{\partial \dot{p}_i} = \frac{\partial M}{\partial p_i}$
where $\frac{\partial M}{\partial \dot{p}_i}$ and $\frac{\partial M}{\partial p_i}$ will be interesting quantities, perhaps even position and velocity.
The second ‘new’ approach will almost certainly be lossy, but I suspect it will follow a pattern similar to the Hamilton equations. For convenience, I will write $v:=\dot q$ for these:
$\displaystyle \frac{dF_i}{dt} = -\frac{\partial J}{\partial v^i}\\ \frac{dv^i}{dt} = \frac{\partial J}{\partial F_i}$
where $J$ is something analogous to total energy, and should have units of power / time.
So I leave, for now, with some things to ponder, and some guesses as to the direction they will lead. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 82, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8143731951713562, "perplexity": 430.86615225339864}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143805.13/warc/CC-MAIN-20200218180919-20200218210919-00152.warc.gz"} |
http://www.vallis.org/blogspace/preprints/1207.5595.html | ## [1207.5595] EMRI corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole
Authors: Abhay G. Shah, John L. Friedman, Tobias S. Keidl
Date: 24 Jul 2012
Abstract: This is the first of two papers on computing the self-force in a radiation gauge for a particle moving in circular, equatorial orbit about a Kerr black hole. In the EMRI (extreme-mass-ratio inspiral) framework, with mode-sum renormalization, we compute the renormalized value of the quantity $h_{\alpha\beta}uˆ\alpha uˆ\beta$, gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; and we find the related order $\frak{m}$ correction to the particle's angular velocity at fixed renormalized redshift (and to its redshift at fixed angular velocity). The radiative part of the perturbed metric is constructed from the Hertz potential which is extracted from the Weyl scalar by an algebraic inversion\cite{sf2}. We then write the spin-weighted spheroidal harmonics as a sum over spin-weighted spherical harmonics and use mode-sum renormalization to find the renormalization coefficients by matching a series in $L=\ell+½$ to the large-$L$ behavior of the expression for $H := \frac12 h_{\alpha\beta}uˆ\alpha uˆ\beta$. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Kerr gauge.
#### Jul 28, 2012
1207.5595 (/preprints)
2012-07-28, 09:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9860007166862488, "perplexity": 1016.6791045097774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084892892.86/warc/CC-MAIN-20180124010853-20180124030853-00260.warc.gz"} |
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