url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
http://mathhelpforum.com/number-theory/212160-fitting-sine-wave-between-two-lines.html | # Thread: Fitting a sine wave between two lines
1. ## Fitting a sine wave between two lines
I have a data set that appears to be made up of two linear regions, separated by what best looks like a sine wave.
I know the x and y co-ordinates of the two points at the end of each line segment, x1, x2, y1, y2, and the respective gradient of the lines, m1 and m2.
I believe I can parameterise a sine wave of the form "y = Asin(Bx+C)+D" so that it passes through the given points, with the same gradient, but can't work out how to solve for the parameters.
Does anybody have any ideas?
Thanks!
2. ## Re: Fitting a sine wave between two lines
Hi !
The shape of the curve looks like an hyperbola (non orthogonal).
I suggest to fit the curve (least squares fitting method, involving 5 parameters) to the general quadratic equation and then, compute the characteristic parameters of the hyperbola.
It might be much simpler if you already know the gradient of the asymtotes m1 and m2 . In this case, the equation of the hyperbola is :
(y-m1*x+C1)*(y-m2*x+C2)+C3=0
C1, C2 and C3 can be computed thanks to a linear regression involving 3 parameters (to be defined in relation with C1, C2, C3).
3. ## Re: Fitting a sine wave between two lines
Hi,
This works, thanks for the reply!
If anyone has any ideas, I'd still be interested to know whether it's possible to fit a sine wave under these conditions as a matter of interest.
4. ## Re: Fitting a sine wave between two lines
Originally Posted by charlieahill
If anyone has any ideas, I'd still be interested to know whether it's possible to fit a sine wave under these conditions as a matter of interest.
Fitting a sine function y = Asin(Bx+C)+D to experimental data is a difficult problem. Of course only the points on the area of the sine wave have to be considered and the other points excluded. Since they are 4 parameters A, B, C, D to be optimized, the number of experimental points must be large enough.
A method is discribed in the paper "Régressions et équations intégrales" : JJacquelin's Documents | Scribd
(in French. No translation available today) which includes a chapter "Régression sinusoidale" : theory and application pp.21-34 and usable equations pp.35-36 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9314497709274292, "perplexity": 408.67130642927424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218186841.66/warc/CC-MAIN-20170322212946-00049-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/66173/upsampling-with-time-offsets/66185 | # Upsampling with time offsets
Suppose I have done 4x oversampling for a continuous time signal, but the successive sampling times have a linearly increasing offset. Specifically, the samples with indices {4k; k=0, 1, 2,...} are correctly sampled at times {4kTs}, but the samples at indices {4k+1; k= 0, 1, 2,..} are sampled at {(4k+1)Ts+e}, the samples at indices {4k+2; k= 0, 1, 2,..} are sampled at {(4k+2)Ts+2e)}, and the samples with indices{4k+3; k= 0, 1, 2,..} are sampled at {(4k+3)Ts+3e)}. How does the spectrum of this oversampled signal look ?
• Can you choose one of the answers if satisfactory? Or ask if any clarifications needed. – DSP Rookie Apr 19 at 6:12
• how do I choose ? And what's the relevance of choosing ? – voy82 Apr 20 at 7:57
• You choose by accepting whichever answer satisfied your doubt, by clicking on the tick mark. If none of the answers satisfied your doubt then you comment on the answers what is lacking. – DSP Rookie Apr 20 at 8:12
Answer: You will see residual images of $$X(f)$$ at multiples $$f_s$$, $$2f_s$$ and $$3f_s$$, and distorted image of $$X(f)$$ at non-zero multiples of $$4f_s$$, when sampling in the manner you explained. Depending on value $$e$$, the size of residual will change. I have explained how in detail below.
Ideally, sampling at $$4f_s$$ would have completely cancelled those images of $$X(f)$$ at multiples of $$f_s$$, $$2f_s$$ and $$3f_s$$ and you would've seen images of $$X(f)$$ only at multiples of $$4f_s$$ and the magnitude scaled by $$4f_s$$.
(Explanation seems long only because I have included lots of pictures to show. Please follow through.)
I would like to give you an intuition on how to visualize sampling at any rate $$f_s$$. You probably have a pretty good idea about that. But then I would like to show pictorially what happens when you sample at $$2f_s$$ and then you can extend the idea to $$4f_s$$.
1. Sampling a bandlimited signal $$x(t)$$ at sampling rate $$f_s$$ :
When you sample $$x(t)$$ at sampling rate $$f_s$$, you are basically, multiplying $$x(t)$$ with a periodic pulse train with period $$T_s = \frac{1}{f_s}$$ in time domain. Hence, in frequency domain you see a convolution of $$X(f)$$ with Fourier representation of that periodic pulse train.
Since the pulse train is periodic, it's Fourier representation will be obtained by computing Fourier Series. The pulse train can be represented by: $$\sum^{\infty}_{k=-\infty} \delta(t - kT_s)$$ and it's Fourier transform as: $$\frac{1}{T_s}\sum^{\infty}_{k=-\infty} \delta(f - kf_s)$$
Notice that the magnitude of Fourier domain representation of the sampling pulse train has scaled up magnitude of $$\frac{1}{T_s} = f_s$$.
Assume, frequency representation of $$x(t)$$ as below :
Now, as described above, Sampling is nothing but multiplication of $$x(t)$$ with a $$T_s$$ periodic pulse train in time domain and hence a convolution of $$X(f)$$ with Fourier transform of the pulse train in frequency domain. Mathematically, $$x(kT_s) = x(t).\sum^{\infty}_{k=-\infty}\delta(t-kT_s)$$ $$X_{sampled}(f) = X(f) * f_s. \sum^{\infty}_{k=-\infty}\delta(f-kf_s)$$
1. Now, lets explore what happens when we sample at $$2f_s$$ or when the sampling pulse train becomes $$\frac{T_s}{2}$$ periodic :
The sampling pulse train becomes as shown below :
The sampling pulse train can be represented by a sum of two $$T_s$$ periodic pulse trains:
1. One pulse train as if $$T_s$$ periodic and centered at 0.
2. Second pulse train as if $$T_s$$ periodic but shifted by $$\frac{Ts}{2}$$.
Hence, mathematically it will be as follows: $$\sum^{\infty}_{k=-\infty}\delta(t-k\frac{T_s}{2}) = \sum^{\infty}_{k=-\infty}\delta(t-kT_s) + \sum^{\infty}_{k=-\infty}\delta(t-kT_s - \frac{T_s}{2})$$
Hence, the Fourier Transform of the sampling pulse train will also sum of these 2 trains, as convolution is a linear operation. Also, use the time shift property of Fourier transform to get the result as follows:
$$\mathcal F \{ \sum^{\infty}_{k=-\infty}\delta(t-kT_s) + \sum^{\infty}_{k=-\infty}\delta(t-kT_s - \frac{T_s}{2}) \}$$ $$= f_s. \sum^{\infty}_{k=-\infty}\delta(f-kf_s) + f_s.e^{-j\pi \frac{f}{f_s}} \sum^{\infty}_{k=-\infty}\delta(f-kf_s)$$
$$= f_s. \sum^{\infty}_{k=-\infty}\delta(f-kf_s) + f_s.(cos(\pi\frac{f}{f_s})-\mathbb i.sin(\pi \frac{f}{f_s})). \sum^{\infty}_{k=-\infty}\delta(f-kf_s)$$
Notice that the sum is evaluated only at integral multiples of $$f_s$$, because of the $$\delta(f - kf_s)$$. What this means is that $$sin(\pi \frac{f}{f_s})$$ will always be $$0$$, so, no imaginary images of $$X(f)$$ will be seen, and $$cos(\pi \frac{f}{f_s})$$ will be $$1$$ at even multiples of $$f_s$$ and $$-1$$ at odd multiples of $$f_s$$. Pictorially, the fourier transform of pulse train which is used to sample at $$2f_s$$ will look like following :
So, the even multiples of $$f_s$$ will be doubled in magnitude to $$2f_s$$ and odd multiples of $$f_s$$ will cancel out each other to cancel the images of $$X(f)$$. This is the reason you see images of $$X(f)$$ only at multiples of $$2f_s$$ when sampling at double the rate, because images of $$X(f)$$ at odd multiples of $$f_s$$ cancel each other out.
Now, consider the case which you have explained in your question. When you break your pulse train into 4 pulse trains which are $$T_s$$ periodic individually, but shifted as below:
1. $$\sum^{\infty}_{k=-\infty}\delta(t-kT_s)$$
2. $$\sum^{\infty}_{k=-\infty}\delta(t-kT_s -\frac{T_s}{4} - e)$$
3. $$\sum^{\infty}_{k=-\infty}\delta(t-kT_s -\frac{2T_s}{4} - 2e)$$
4. $$\sum^{\infty}_{k=-\infty}\delta(t-kT_s -\frac{3T_s}{4} - 3e)$$
Conclusion:
When you check their Fourier transforms, you will find that images at multiples of $$f_s$$, $$2f_s$$ and $$3f_s$$ will not get cancelled completely because the negative impulses (both real and imaginary) are shifted by $$e$$, $$2e$$ and $$3e$$ respectively. And, image at multiples of $$4f_s$$ will also not be aligned exactly to give a scaling of $$4f_s$$ but they will be fudged around to give a distorted image of $$X(f)$$ except at $$k=0$$, that is the original image of $$X(f)$$ centered around DC.
Depending upon the value of e, the real and imaginary images of $$X(f)$$ will have residuals at $$f_s$$, $$2f_s$$ and $$3f_s$$, and images at non-zero multiples of $$4f_s$$ will be fudged around.
• Fun-fact: you can still reconstruct the original x(t), because X(f) at DC is still untouched. Meaning, you can still throw 3 in 4 samples and interpolate to get x(t) back. – DSP Rookie Apr 7 at 13:40
The spectrum of the oversampled signal as you describe it would look very odd and is likely not what you are intending to do. Let me explain:
First consider each group separately, instead representing what would be the decimated signal if you were sampling at the higher 4x rate as $$f_s$$: Each of the four spectrums would span from frequency = $$0$$ to $$f_s/4$$ and contain all energy within that frequency range as well as all the aliasing from $$f_s/4$$ to $$f_s$$. Each would be translated by a different phase slope in frequency consistent with the time delay each shift represents. First consider the unit delays only and assume the additional time offset $$e = 0$$ to best understand the effect of the unit delays, then we can add in the effect of non-zero $$e$$.
Each delay of one sample would add a linear phase in frequency extending negatively from $$0$$ to $$2\pi$$ as the the normalized radian frequency goes from $$0$$ to $$2\pi$$ (Meaning as the frequency goes from $$0$$ to $$f_s$$).
Consider the z-transform of an m sample delay:
$$\sum_{n=0}^{N-1}x[n-m]z^{-n} = X(z)z^{-m}$$
And the DFT is simply the z transform with z limited to the unit circle; $$z=e^{j2\pi k/N}$$ as k = $$0$$ to $$N-1$$
So here we see that for each delay m, the DFT would be $$X(k)e^{-mj2\pi k/N}$$, with the phase negatively increasing to $$2\pi$$ for m =1, to $$4\pi$$ for m=2, etc. Thus the higher frequencies that are above $$f_s/4$$ are aliased into the $$0$$ to $$f_s/4$$ spectrum with a different phase slope for each of the four groups.
As far as the additional time offsets, which may be fractional samples, consider the Fourier Transform of a time delay given as
$$\mathscr{F}\{x(t-\tau)\} = e^{-\tau}$$
And we see that the time offsets just add an additional slope to the phase slopes given by the unit delays. The time offset as a fraction of a sample given as $$d$$ would introduce a phase slope according to $$z^{-d}$$ using the method above with limiting z to the unit circle.
Importantly, you cannot recreate the interpolated spectrum simply by cascading the samples in frequency as you are describing. As I explained above above you would only be showing the decimated and folded spectrum extending from $$0$$ to $$f_s/4$$, just with four phases of the same bin next to each other. Combining the interpolated spectrum is more complicated as it involves summing all four spectrums with the appropriate phase slope to compensate for the phase introduced as described above.
If your signal is bandlimited to $$[-f_0/2,+f_0/2]$$, the digitized signal sampled at $$f_s$$ can be considered as a summation of 4 different sampled signals with offset of $$0$$, $$e$$, $$2e$$, $$3e$$. Assuming $$3e \le 1/f_s$$.
First imagine $$e=0$$. This means, you have perfectly oversampled 4x. That is $$f_s = 4 \times f_0$$. It is like you have sampled the signal 4 times with sample rate of $$f_0$$, each of them with offset of $$k/(4f_0)$$, where $$k \in \{0,1,2,3\}$$, interleaved them with 3 zeros and added them. Assume $$T_s = 1/f_s = (1/(4f_0))$$. This is equivalent to
$$x_1 = x(0), 0 ,0 ,0, x(4T_s), 0, 0, 0..\\ x_2 = 0 ,x(T_s),0,0,0, x(5T_s),0,0,0.. \\ x_3 = 0 ,0,x(2T_s),0,0,0, x(6T_s),0,0,0.. \\ x_4 = 0 ,0,0,x(3T_s),0,0,0, x(7T_s),0,0,0.. \\ x = x_1 +x_2+x_3+x_4$$ Equivalently in fourier domain $$X(e^{j\omega}) = X_1(e^{j4\omega})+X_2(e^{j4\omega})e^{-j\omega}+X_3(e^{j4\omega})e^{-j2\omega}+X_4(e^{j4\omega})e^{-3\omega}$$ which is same as if you had sampled original signal at $$f_s$$.(This is proved later using MATLAB example for 2x oversampled case). If there were offsets of $$e,2e,3e$$ as you mentioned in each of those copies,
$$x_1 = x(0), 0 ,0 ,0, x(4T_s), 0, 0, 0..\\ x_2 = 0 ,x(T_s+e),0,0,0, x(5T_s+e),0,0,0.. \\ x_3 = 0 ,0,x(2T_s+2e),0,0,0, x(6T_s+2e),0,0,0.. \\ x_4 = 0 ,0,0,x(3T_s+3e),0,0,0, x(7T_s+3e),0,0,0.. \\ \hat{x} = x_1 +x_2+x_3+x_4$$
Equivalently in fourier domain $$\hat{X(e^{j\omega})} = X_1(e^{j4\omega})+X_2(e^{j4\omega})e^{-j\omega(1+e)}+X_2(e^{j4\omega})e^{-j2\omega(1+e)}+X_4(e^{j4\omega})e^{-3\omega(1+e)}$$
APPENDIX: To show that if $$x[n]$$ is an $$N$$ sample signal sampled at rate $$f_s$$, it can be shown that it is sum of 2 signals sampled at rate $$f_s/2$$ but with offset of $$1$$ sample and interleaved with zeros and added. That is $$X(e^{j\omega}) = X_1(e^{j2\omega})+X_2(e^{j2\omega})e^{-j\omega}$$.
That is $$e=0$$ in the below code, the spectrum of $$x$$ and $$X_1(e^{j2\omega})+X_2(e^{j2\omega})e^{-j\omega}$$ would be same. If $$e=0.25$$, spectrum of $$x$$ would not be same as $$X_1(e^{j2\omega})+X_2(e^{j2\omega})e^{-j\omega(1+e)}$$.
clc
clear all
close all
x=randn(1,16)+1i*randn(1,16);
xph1 = x(1:2:end);
xph2 = x(2:2:end);
xph1ups=upsample(xph1,2);
xph2ups=upsample(xph2,2);
e=0.25;
F1= fft(xph1ups);
F2= fft(xph2ups);
F=F1+exp(-1i*2*pi/16*(0+e:1:15+e)).*F2;
plot(1:16,abs(fft(x)),'b',1:16,abs(F),'r')
• @voy82 They are not the same. You cannot take out the exponentials as common factor. You can check that with the MATLAB script I uploaded. It is not same as $X_1(e^{j2\omega})(1+e^{-j\omega})$ – jithin Apr 7 at 5:40
• thanks ! I was also thinking along same lines. Since X1, X2, X3, X4 in my question represent the spectrums of Nyquist sampled x(t), are they not the same ? So, we should be able to take them out, so as to get X(e^(j4ω)) x (1+exp^(-jw(1+e)) + exp^(-j2w(1+e))+exp^(-j3w(1+e))) – voy82 Apr 7 at 5:40
• hmm.. i think, using your notations above, we should get : X(ej4ω)+X(ej4ω) e−j4ω/(1+e) + X(ej4ω) e−j8ω/(1+e) + X(ej4ω) e−j12ω/(1+e); where X(ejω) is the spectrum of the Nyquist sampled original continuous signal x(t). Would you agree ? I am thinking if for e=0, this gives X(ej4ω). – voy82 Apr 7 at 6:00
• sorry I did not mean to have the "/" there in the "e−j4ω/(1+e)" like terms. I was basically repeating my first comment but with the 4ω, 8ω, 12ω instead of ω, 2ω, 3ω. But you clearly do not seem to agree with it (that we can take the X(ej4ω) out) ! Looking at your matlab code now. thanks ! – voy82 Apr 7 at 6:13
• Repeating my comment because of formatting issue earlier : @voy82 No the (1+e) terms are not in denominator. Sorry if my MATLAB code appears like that. The (1+e) terms are all in numerator. Also for the term $X_1(e^{j4\omega)}$ , the $4\omega$ cannot be taken outside. So you would not get 4$\omega$,8$\omega$,12$\omega$ – jithin Apr 7 at 6:14 | {"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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 143, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8819078803062439, "perplexity": 529.9673197465489}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141486017.50/warc/CC-MAIN-20201130192020-20201130222020-00493.warc.gz"} |
https://mathinsight.org/prototypes_more_serious_questions_taylor_polynomials_refresher | # Math Insight
### Prototypes: More serious questions about Taylor polynomials
Beyond just writing out Taylor expansions, we could actually use them to approximate things in a more serious way. There are roughly three different sorts of serious questions that one can ask in this context. They all use similar words, so a careful reading of such questions is necessary to be sure of answering the question asked.
(The word ‘tolerance’ is a synonym for ‘error estimate’, meaning that we know that the error is no worse than such-and-such)
• Given a Taylor polynomial approximation to a function, expanded at some given point, and given a required tolerance, on how large an interval around the given point does the Taylor polynomial achieve that tolerance?
• Given a Taylor polynomial approximation to a function, expanded at some given point, and given an interval around that given point, within what tolerance does the Taylor polynomial approximate the function on that interval?
• Given a function, given a fixed point, given an interval around that fixed point, and given a required tolerance, find how many terms must be used in the Taylor expansion to approximate the function to within the required tolerance on the given interval.
As a special case of the last question, we can consider the question of approximating $f(x)$ to within a given tolerance/error in terms of $f(x_o), f'(x_o), f''(x_o)$ and higher derivatives of $f$ evaluated at a given point $x_o$.
In ‘real life’ this last question is not really so important as the third of the questions listed above, since evaluation at just one point can often be achieved more simply by some other means. Having a polynomial approximation that works all along an interval is a much more substantive thing than evaluation at a single point.
It must be noted that there are also other ways to approach the issue of best approximation by a polynomial on an interval. And beyond worry over approximating the values of the function, we might also want the values of one or more of the derivatives to be close, as well. The theory of splines is one approach to approximation which is very important in practical applications. | {"extraction_info": {"found_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": 0.922443687915802, "perplexity": 277.48325954836196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764500356.92/warc/CC-MAIN-20230206145603-20230206175603-00558.warc.gz"} |
https://www.physicsforums.com/threads/inverse-square-law-help-please.790996/ | Tags:
1. Jan 7, 2015
RichardGib
1. The problem statement, all variables and given/known data
If I measure a sound intensity of 1.0 at distance R from its source, what intensity would I measure at distance 3R in a free, unbounded space? What is the difference in decibels?
&
If I measure a sound pressure of 1.0 at distance R from its source, what pressure would I measure at distance 2R in a free, unbounded space? What is the difference in decibels?
2. Relevant equations
N/A
3. The attempt at a solution
I am stuck on this. i thought the answer to the first question could be a difference of -18db? Really stuck here...
Thankyou!
Last edited: Jan 7, 2015
2. Jan 7, 2015
Fightfish
The inverse square law simply means that the value of the physical observable involved is inversely proportional to the square of the distance from the source. That is to say, for instance if the distance is doubled, then the value falls to a quarter (half squared).
3. Jan 7, 2015
BvU
Hello Richard, welcome to PF :)
Please check the PF guidelines (especially #4); the way you post now actually prevents us from helping you further.
And you can read up on the subject a little here
4. Jan 7, 2015
RichardGib
Thankyou :)
Draft saved Draft deleted
Similar Discussions: Inverse Square Law HELP PLEASE | {"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": 0.8845289945602417, "perplexity": 874.1562313255728}, "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-2017-34/segments/1502886105187.53/warc/CC-MAIN-20170818213959-20170818233959-00161.warc.gz"} |
https://docs.galpy.org/en/v1.6.0/reference/potentialrtide.html | # galpy.potential.Potential.rtide¶
Potential.rtide(R, z, phi=0.0, t=0.0, M=None)[source]
NAME:
rtide
PURPOSE:
Calculate the tidal radius for object of mass M assuming a circular orbit as
$r_t^3 = \frac{GM_s}{\Omega^2-\mathrm{d}^2\Phi/\mathrm{d}r^2}$
where $$M_s$$ is the cluster mass, $$\Omega$$ is the circular frequency, and $$\Phi$$ is the gravitational potential. For non-spherical potentials, we evaluate $$\Omega^2 = (1/r)(\mathrm{d}\Phi/\mathrm{d}r)$$ and evaluate the derivatives at the given position of the cluster.
INPUT:
R - Galactocentric radius (can be Quantity)
z - height (can be Quantity)
phi - azimuth (optional; can be Quantity)
t - time (optional; can be Quantity)
M - (default = None) Mass of object (can be Quantity)
OUTPUT: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8987058997154236, "perplexity": 2606.8140315193837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593655899209.48/warc/CC-MAIN-20200709065456-20200709095456-00566.warc.gz"} |
https://reference.globalspec.com/standard/3842897/astm-e918-83-1999 | # ASTM International - ASTM E918-83(1999)
## Standard Practice for Determining Limits of Flammability of Chemicals at Elevated Temperature and Pressure
historical
Organization: ASTM International Publication Date: 10 April 1999 Status: historical Page Count: 5 ICS Code (Ignitability and burning behaviour of materials and products): 13.220.40 ICS Code (Products of the chemical industry in general): 71.100.01
##### scope:
1.1 This practice covers the determination of the lower and upper concentration limits of flammability of combustible vapor-oxidant mixtures at temperatures up to 200°C and initial pressures up to as much as 1.38 MPa (200 psia). This practice is limited to mixtures which would have explosion pressures less than 13.79 MPa (2000 psia).
1.2 This practice should be used to measure and describe the properties of materials, products, or assemblies in response to heat and flame under controlled laboratory conditions and should not be used to describe or appraise the fire hazard or fire risk of materials, products, or assemblies under actual fire conditions. However, results of this test may be used as elements of a fire risk assessment which takes into account all of the factors which are pertinent to an assessment of the fire hazard of a particular end use.
1.3 This standard may involve hazardous materials, operations, and equipment. This standard does not purport to address all of the safety problems associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
### Document History
August 1, 2011
Standard Practice for Determining Limits of Flammability of Chemicals at Elevated Temperature and Pressure
1.1 This practice covers the determination of the lower and upper concentration limits of flammability of combustible vapor-oxidant mixtures at temperatures up to 200°C and initial pressures up to as...
September 15, 2005
Standard Practice for Determining Limits of Flammability of Chemicals at Elevated Temperature and Pressure
1.1 This practice covers the determination of the lower and upper concentration limits of flammability of combustible vapor-oxidant mixtures at temperatures up to 200°C and initial pressures up to as...
ASTM E918-83(1999)
April 10, 1999
Standard Practice for Determining Limits of Flammability of Chemicals at Elevated Temperature and Pressure
1.1 This practice covers the determination of the lower and upper concentration limits of flammability of combustible vapor-oxidant mixtures at temperatures up to 200°C and initial pressures up to as... | {"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": 0.8131459355354309, "perplexity": 2930.0034784513236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575540530857.12/warc/CC-MAIN-20191211103140-20191211131140-00391.warc.gz"} |
http://libros.duhnnae.com/2017/jul6/150057294359-How-well-can-we-predict-the-total-cross-section-at-the-LHC-High-Energy-Physics-Phenomenology.php | # How well can we predict the total cross section at the LHC - High Energy Physics - Phenomenology
Abstract: Independently of any theory, the possibility that the large value of theTevatron cross section claimed by CDF is correct suggests that the total crosssection at the LHC may be large. Because of the experimental and theoreticaluncertainities, the best prediction is $125\pm 35$ mb.
Author: P V Landshoff
Source: https://arxiv.org/ | {"extraction_info": {"found_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": 0.971699595451355, "perplexity": 1015.4092012598596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195525414.52/warc/CC-MAIN-20190717221901-20190718003901-00393.warc.gz"} |
https://www.physicsforums.com/threads/ring-of-non-uniform-charge.184821/ | # Ring of Non-uniform charge
1. Sep 15, 2007
### Luke1294
1. The problem statement, all variables and given/known data
A ring of radius 'a' has a charge distribution on it that varies as $$\lambda (\theta) = \lambda_0 sin(\theta)$$, where $$\theta$$ is the angle between 'a' and the x axis.
a) What is the direction of the electric field at the center of the ring?
b) What is the magnitude of the field at the center of the ring?
2. Relevant equations
$$d\vec{E}= \frac{k dq}{r^2}\hat{r}$$
$$\lambda (\theta) = \lambda_0 sin(\theta)$$,
3. The attempt at a solution
Okay, here's what I know thus far. For a standard, uniform charge distribution, the electric field is perpendicular to the ring and follows $$\frac{kqz}{(z^2+a^2)^{3/2}}\hat{k}$$, giving us no electric field at the center of the ring, where z=0. But this is not a uniform charge distribution, so I'm not entirely sure this is going to hold true...infact, because sin has a period of 2pi, I would think that the charge on one half of the ring has an opposite charge than the other. Well, here goes nothing...
I read through the derivation for the ring of charge that we went over in class and got to this point-
$$E_z = \frac{k cos(\theta)}{r^2}\int dq$$ Okay, so we re-write dq as
$$E_z = \frac{k cos(\theta)}{r^2}\int \lambda dl$$
Now, at this point in the derivation, we pulled lambda out because it was constant. Not so in this case. So lets leave it in there and see what we can do about that dl. Why not rewrite as $$dL = r d\theta$$? That would make things easier with the integration. We can also pull the 'r' right out of the integral because it is going to be constant.
Okay, so now we have...
$$E_z = \frac{k cos(\theta)}{r^2}\int \lambda d\theta$$
But we know what lambda is, so..
$$E_z = \frac{k cos(\theta)}{r^2}\int \lambda_0 sin(\theta)d \theta$$
Yank the $$\lambda_0$$ out of there...Why not put the limits of the integration on there too? Across the entire circle would be from 0 to 2pi, so...
$$E_z = \frac{k cos(\theta)\lambda_0 }{r^2}\int_0^{2pi} sin(\theta)d \theta$$
Now, when I integrate $$sin \theta$$, I'm going to get zero. Making everything zero, no matter the position on the z axis. Making me think I made a mistake. I have a sneaking suspicion it involves my rewriting of dL as r dtheta, but to be honest, I am VERY fuzzy on how/why it happens. So hopefully someone can shed some light on this...feel free to give a crash course in integrating in non-rectangular coordinates. I have a very weak grasp on it at the moment.
2. Sep 15, 2007
### Staff: Mentor
Several problems here. One, you are asked to find the field smack in the middle of the ring, not somewhere along the z-axis. Two, you should be able to find the direction of the field in the center of the ring just by symmetry--no calculations needed. Three, if you are integrating with respect to theta, you can't just pull cos(theta) out from the integrand and ignore it when integrating. Four, find the proper element of charge and its field:
$$dQ = \lambda dl = \lambda R d\theta = \lambda_0 \sin\theta R d\theta$$
Find the x and y components of the field from that charge element at the center of the ring and integrate around the ring.
3. Sep 15, 2007
### Luke1294
Doc, thank you.
I'll address these things one by one-
1. I was under the impression that if i designated the center of the ring to be the origin, the point x=y=Z=0 WOULD be the dead center of the ring. So that's where that came from.
2. I'm a little confused on this. The direction of field by symmetry alone...The only thing I can come up with is that it should be perpendicular to the ring itself, but to be honest, I can't defend that claim at all.
3) DOH
4) Can you give me a little more guidence as to what you mean by the x and y components from that element? I'm a little confused there.
...am going to go eat and ponder this.
4. Sep 15, 2007
### Staff: Mentor
Yes, the dead center of the ring is at that point. (Not at some off-axis point.)
Why would there be a perpendicular component at all? The charge and the point of interest all lie in the same plane.
See my comment above.
5. Sep 15, 2007
### Luke1294
Okay, so upon further inspection, I had a few elements of this incredibly messed up.
The theta referenced by "cos(theta)" and by the lambda expression are totally different angles. Cosine is referring to the angle between the z axis and to the circle, basically forming the side of a cone. Obviously the sin expression is referring to the angle between the radius of the ring and the z axis. I should have been much more clear there, my apologies. I will refer to the cosine angle as "phi" from here on and correct my original post. Because of that, it can stay out of the integrand.
As far as the symmetry goes, this is all I have come up with- sine has a period of 2 pi. From 0-pi, the value is positive. From pi-2pi, the value is negative. We use the convention that the field points from positive to negative, so the field would be pointing in that direction across the center.
As for the x and y components, the only thing I have come up with is this- I could express the X coordinate as Rsin(theta) and the Y coordinate as Rcos(theta). I don't feel that is what you meant...please bear with me here.
6. Sep 15, 2007
### Staff: Mentor
I see your point about using theta to represent two different angles, but that's not really relevant to this problem. Don't waste time correcting your original post until you realize the following: Assuming that the ring lies in the x-y plane, there is no z-component to consider.
Tell me exactly what direction the field will point at the center of the ring. (Let theta = 0 be the +x direction.)
I'm talking about x & y components of the field at the center. Each element of charge contributes some element of electric field at the center. I'm saying to find the x-components of the field contribution and add them up to get the total x-component of field at the center; then do the same for the y-components. (Take advantage of any symmetry, of course. )
7. Sep 15, 2007
### Luke1294
If my assumptions about the charge distribution are true, I believe the field will be perpendicular to the x axis throughout. Because of this, there are no components of the field flowing in the x- direction.
Will be back with an answer for the Y-components, assuming I'm not way off base on that. Initially, I see that the charge would be symmetrical about the Y-axis.
8. Sep 15, 2007
### Staff: Mentor
So far, so good. Keep at it!
9. Sep 15, 2007
### Luke1294
Haha, well, I just spent a fair amount of time setting up an integral until I realized...wait, I'm not finding the total electric field...just the magnitude at the center of the ring...whhhooops.
Now this is every bit as confusing to me. I can find the force between the two. I can find the dipole moment. But the magnitude of the field through the center...hm.
I know that the charge at the "top" of the ring, where theta= pi/2, is going to be $$\lambda_0$$ . I know at the "bottom" of the ring, where theta = 3pi/2, the charge is $$-\lambda_0$$. I know the distance between the two points is going to be twice the radius of the circle. This seems so simple...gah.
10. Sep 15, 2007
### Luke1294
....Would it just be
$$2\frac{k \lambda_0}{a^2} \hat{j}$$?
I found this by looking at the field from the charge at pi/2 and the field from the charge at 3pi/2 and adding them. Because one is pushing and the other is pulling, the answer is non-zero.
11. Sep 15, 2007
### Staff: Mentor
Sure you're finding the total electric field at the center. You'll need to integrate.
$$\lambda_0$$ is a charge per unit length, not a point charge. It's simple, but not that simple.
Does that expression even have correct units?
Do as I suggested in post #2. Set up the expression for the field components and integrate.
12. Sep 15, 2007
### Luke1294
Well as you pointed out, lambda is charge/length and I just treated it like a point charge...so no, it has the wrong units. I believe the phrase I am looking for is "rats."
Electric field in the x is going to be zero.
$$E_y = \int \frac{k dq}{r^2} sin \theta$$, but $$E_y = E$$, so I can ignore the theta in that expression...espically since it is the angle that the field is pointing at, DIFFERENT than the theta used in my charge distribution expression.
$$E_y = \frac{k \lambda_0}{r^2} \int sin \theta d\theta$$
Symmetrical across the Y axis, so I'm going to integrate from pi/2 to 3pi/2 and double the value...
$$E_y = \frac{2k \lambda_0}{r^2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} sin\theta d\theta$$
Right track?
13. Sep 16, 2007
### Staff: Mentor
Right. By symmetry, the x-component of the total field in the middle will be zero.
OK. (Except for the sign: Realize that the field points away from the positive charge.)
No. The only non-zero component of the total field in the middle will be the y-component, but you must add the y-components: $$E = \Sigma E_y$$. Without the sin(theta), you are not adding y-components. And that theta is the same theta used in the charge distribution expression!
Please review what I gave as the charge element in post #2. (And put back that other sin(theta)!)
Once you get the correct integrand, that will work.
14. Sep 16, 2007
### Luke1294
Ah yes, I left out the R from the charge expression.
Including the extra sin expression, I would get
$$E_y = \frac{k R\lambda_0}{r^2} \int sin^2 \theta d\theta$$
Is the R from the charge expression the same as the radius of the circle? I believe it would be, but they way you wrote it as a capital R is confusing me. If it is the same, obviously that can be simplified, if not, well...then I need to figure out what radius it is referring to.
15. Sep 16, 2007
### Staff: Mentor
Yes.
16. Sep 16, 2007
### genneth
I have a question... Luke1294: have you drawn a diagram?
17. Sep 16, 2007
### Luke1294
Initally ,yes. When I was going through the E_z steps. I'm not sure why I didn't step back and re-draw it after I figured out I was on the wrong track.
Does the integrand look correct now?
18. Sep 16, 2007
### Staff: Mentor
Yes.
19. Sep 16, 2007
### Luke1294
Okay, good. If not I think my head would explode. One last thing I'm not quite understanding... I am looking for the magnitude of the electric field at the center of the circle. Wouldn't this integral give me the field inside the entire ring?
20. Sep 16, 2007
### Staff: Mentor
Why would you think that? We are just adding up the field contributions at the center.
The field anywhere within the ring would be more difficult to find; it would be a function of position. | {"extraction_info": {"found_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": 0.8969619870185852, "perplexity": 472.2917345347275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988719453.9/warc/CC-MAIN-20161020183839-00566-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://rupress.org/jcb/article/75/1/31/52117/Patterns-of-plasminogen-activator-production-in | Cultured normal low-passage embryo fibroblasts, from a number of species, and two untransformed clones of a Balb/3T3 line elaborate increasing amounts of plasminogen activator (PA) as they approach confluence; the low-passage cells then lose this PA activity after reaching confluence, while the 3T3 cells retain it indefinitely. Even at their peaks, however, the PA activities of the low-passage cells remain well below those of the corresponding virally or spontaneously transformed cells. The PA increases in normal cells are probably a result of PA production rather than of adsorption of secreted PA to the cell surface, or of changes in cell-associated protease inhibitors. The elaboration of PA by normal cells is dependent upon their metabolic activity, such that the level of serum supplementation and the growth phase of the culture directly influence the level of cell-associated PA observed. In addition, there may be a component of serum which exerts a negative control on PA production and which is not an acid-labile protease inhibitor.
This content is only available as a PDF. | {"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": 0.8393214344978333, "perplexity": 4636.709916577563}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573760.75/warc/CC-MAIN-20220819191655-20220819221655-00398.warc.gz"} |
http://ipnwww.in2p3.fr/A-new-leading-contribution-to-neutrinoless-double-beta-decay?date=2018-11 | Accueil du site > Actualités et Faits Marquants > A new leading contribution to neutrinoless double-beta decay
A new leading contribution to neutrinoless double-beta decay
V. Cirigliano, W. Dekens, J. de Vries, M.L. Graesser, E. Mereghetti, S. Pastore, and U. van Kolck, Phys. Rev. Lett. 120 (2018) 202001
V. Cirigliano, W. Dekens, J. de Vries, M.L. Graesser, E. Mereghetti, S. Pastore, and U. van Kolck,
Phys. Rev. Lett. 120 (2018) 202001
One of the most important advancements in modern particle physics was the observation of neutrino oscillations and the inference that neutrinos have mass. However, the origin of neutrino masses remains a mystery. They can arise from an interaction with the Higgs field that violates lepton number, makes neutrinos Majorana particles, and potentially explains the observed matter-antimatter asymmetry of the universe. This mechanism is only accessible through neutrinoless double-beta decay experiments, where two neutrons in a nucleus turn into two protons, with the emission of two electrons and no neutrinos. Nuclear physics is required for the interpretation of a non-zero signal (or lack thereof) from the enormous experimental effort which is underway around the world.
Based on renormalization arguments, we have now shown that the leading contribution to neutrinoless double-beta decay, where light Majorana neutrinos are exchanged between nucleons, is not well defined without a short-range interaction. This short-range contribution is missing in all current calculations and should eventually be determined from simulations of Quantum Chromodynamics on a spacetime lattice. It can also be estimated, via chiral symmetry, from isospin-breaking observables in the two-nucleon sector. Using existing data for such an estimate, we have shown explicitly in the decay of 12Be that this new short-range contribution can be comparable to model-dependent estimates of the long-range neutrino exchange. This new leading effect could thus significantly affect the neutrino mass properties extracted from double-beta-decay experiments.
Voir en ligne : Phys. Rev. Lett. 120 (2018) 202001
Institut de Physique Nucléaire Orsay - 15 rue Georges CLEMENCEAU - 91406 ORSAY (FRANCE) UMR 8608 - CNRS/IN2P3 | Connexion SPIP | Fil RSS du site | Crédits et mentions légales | Nous contacter | Ce site est optimisé pour les navigateurs suivants Firefox, Chrome, Internet explore 9 | {"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": 0.9143549203872681, "perplexity": 2889.206163935859}, "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-2018-43/segments/1539583513686.3/warc/CC-MAIN-20181021031444-20181021052944-00376.warc.gz"} |
https://www.physicsforums.com/threads/orthogonal-complements-of-complex-and-continuous-function-subspaces.562266/ | # Homework Help: Orthogonal Complements of complex and continuous function subspaces
1. Dec 22, 2011
### unquantified
1. The problem statement, all variables and given/known data
I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces:
1) span{(1,0,i,1),(0,1,1,-i)},
2) All constant functions in V over the interval [a,b]
2. Relevant equations
I know that for a subspace W of an inner product space, the orthogonal complement is defined as:
W_perp = {vectors in v$\in$V: <v,w> = 0 for all w $\in$ W}
<v,w> is the standard dot product between two vectors;
In the case of constant functions, the dot product is $\int$f(x)g(x)dx;
3. The attempt at a solution
1) I tried putting it in matrix form:
[1 0 i 1]
[0 1 1 -i]
but don't know how to row reduce with complex variables. I actually don' think the matrix needs to be simplified more than it is, but still don't know how to plug in to get two orthogonal vectors (I would like the result to be an orthogonal set)
2) I don't even know where to start ... the book doesn't cover inner products of functions much , let alone how to find the orthogonal complement of them.
Last edited: Dec 22, 2011
2. Dec 22, 2011
### Dick
You doing a pretty good job of ignoring any information in your "Relevant equations" section. If you don't do that the second question should be pretty easy. You can factor a constant function outside of the integral. For the first one you should remember, if you weren't told, that <u,v> for complex vectors involves taking a complex conjugate of one of the vectors. Suppose v=(A,B,C,D). Then what two equations do you have to solve for the four unknowns A, B, C and D?
3. Dec 22, 2011
### unquantified
So, for the first part, the orthogonal complement would just be all f(x) where by ∫f(x) = 0 over the given interval [a,b]?
As for the second, I'm not quite sure what you mean. If I have those equations, and let x =[x_1, x_2, x_3, x_4] be my unknowns, I have:
[x_1,x_2,x_3,x_4] = x_3[-i, -1, 1, 0] + x_4[-1,i,0,1]
Is that correct? I would then plug in x_3=1 and then x_4=1 to each of those vectors to get my orthogonal complement.
4. Dec 23, 2011
### Dick
That's the answer to the first one alright. For the second one, take w=(w1,w2,w3,w4) to be your unknown vector in the orthogonal complement. Now since w need to be orthogonal to the span{(1,0,i,1),(0,1,1,-i)}, it has to be orthogonal to v1=(1,0,i,1) and v2=(0,1,1,-i). So you must have <v1,w>=0 and <v2,w>=0. What do those equations look like when you write them out in terms of w1, w2, w3 and w4? | {"extraction_info": {"found_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": 0.9141865968704224, "perplexity": 573.9570055026123}, "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/1526794865595.47/warc/CC-MAIN-20180523102355-20180523122355-00625.warc.gz"} |
https://undergroundmathematics.org/quadratics | ## How do different representations help us to understand quadratics?
### Key questions
1. 1
How are the graphical and algebraic representations of quadratics connected?
2. 2
What do we mean by the roots of a quadratic equation and how can we find them?
3. 3
What is the discriminant of a quadratic, and what can it tell us?
No resources found.
Title
Lines
Key questions
Related ideas
No resources found.
This resource is in your collection
#### Review questions Click for information about review questions
Title
Lines
Key questions
Related ideas
No questions found.
This resource is in your collection
#### Introducing...
Resource type Title
#### Developing...
Resource type Title
Fluency exercise Inequalities for some occasions
Fluency exercise Pick a card...
Package of problems Discriminating
Package of problems GeoGebra constructions... quadratic edition
Package of problems Irrational roots
Package of problems Name that graph
Package of problems Paired parabolas
Package of problems Which parabola?
Food for thought Parabella
Bigger picture Parabolic mirrors
Resource in action Discriminating - teacher support
#### Review questions
Title Ref
Can we find both roots if one is double the other? R6638
Can we find the midpoint of a chord of this parabola? R8045
Can we find the tangents to a circle from a point? R6804
Can we fit a quadratic to this given data? R8775
Can we prove this inequality involving two square roots? R7533
Can we show $(b^2 - 2ac)x^2 + 4(a + c)x = 8$ always has real roots? R8812
Can we solve $12/(x-3) < x+1$? R7035
Can we solve $\sqrt{3-3x} - \sqrt{2-x} = 1$? R7485
Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$? R6246
Can we solve the simultaneous equations $x + y + \sqrt{xy} = 39$ and $x^2 + y^2 + xy = 741$? R5682
Can we solve these simultaneous equations that involve reciprocals? R8128
Can we solve these simultaneous equations, both of degree two? R7520
Can we solve these simultaneous equations, one linear, one quadratic? R6429
Can we solve this inequality with three square roots? R6387
For what values of $k$ is $x^2 + 6kx +144$ always positive? R6526
Given the minimum point, what's this parabola's equation? R6756
Given two simultaneous equations of degree two, can we solve them? R7588
How do we solve $x + 3\sqrt{x} − 1/2 = 0$? R6215
How do we increase the roots of a quadratic equation by $1$? R6712
How do we solve an equation containing three square roots? R5847
How fast are these particles sliding when they pass? R6864
If $2y = a^x + a^{-x}$, can we find $a^x$? R6816
If $7 - px - x^2 = 16 - (q + x)^2$, what are $p$ and $q$? R9724
If we know two values satisfying a quadratic, can we find the quadratic? R7972
Into how many regions do these parabolas divide the plane? R8781
What does the curve $x^4-y^2=y+1$ look like? R8855
When are the coefficients of a quadratic equal to its roots? R7350
When are the roots for $x^2-bx+c=0$ real and positive? R5138
When are these quadratic inequalities true together? R9989
When can this equation involving algebraic fractions hold? R8407
When do these simultaneous equations have no real solutions? R9658
When does $(p+1)x^2+4px+9=0$ have a repeated root? R7455
When does $x^2 + ax + a = 1$ have distinct real roots? R9659
When does $x^2+(3k-7)x+(2k+6)=0$ have real roots? R6828
When does $x^2-4x-1=2k(x-5)$ have equal roots? R5316
When does $x^4=(x-c)^2$ have four real roots? R9546
When does $y=kx$ intersect the parabola $y=(x-1)^2$? R9614
When does $y=mx-5$ intersect $y=x^2-1$ twice? R5287
When is $12x^2+7x-10$ negative? R5059
When is this line a tangent to this parabola? R7328
Where does $y = px^2 + 8x + p - 6$ cross the $x$-axis? R9742 | {"extraction_info": {"found_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": 0.8421213626861572, "perplexity": 2975.778534573869}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1502886105712.28/warc/CC-MAIN-20170819182059-20170819202059-00125.warc.gz"} |
https://math.stackexchange.com/questions/3509582/circle-envelope-tangent-in-another-circle/3545910 | # circle envelope tangent in another circle
As the picture shows,
One big circle ,$$(0,0)$$ ,radius=R, there is a small circle in it, $$(m,0)$$ ,radius=r .
G is on the big circle. From G ,we can do two tangent lines about the small circle. Get the points of intersection E and F. line EF has a envelope about G , which seem like a circle.
How to prove it? Since calculating it requires much effort.
By picking special points,I get
the radius of envelope circle is $$\frac{R \left(m^4-2 m^2 \left(r^2+R^2\right)-2 r^2 R^2+R^4\right)}{\left(m^2-R^2\right)^2}$$
and the circle center $$\left(\frac{1}{2} \left(\frac{R \left(-m^2+2 m R+2 r^2-R^2\right)}{(R-m)^2}-\frac{R \left(-m^2-2 m R+2 r^2-R^2\right)}{(m+R)^2}\right),0\right)$$
• Your formulas can be simplified : for the radius into $R \left(1-2r^2 \dfrac{m^2+R^2}{(R^2-m^2)^2}\right)$ and even more for the abscissa of the center which can be written : $4m\dfrac{R^2r^2}{(R^2-m^2)^2}$ – Jean Marie Jan 15 at 23:11
• I have been working a lot on your issue without real success. Could we share our advances ? Moreover, could you say something about the origin of this problem ? – Jean Marie Jan 19 at 21:06
• @JeanMarie the origin of this problem is that: prove the envelope is a circle as well as use ruler-and-compass construction to make the tangent point of EF and circle. – wuyudi Jan 20 at 3:25
• From the references in the link from @brainjam: This question is very similar to the one Poncelet started with – Jan-Magnus Økland Feb 14 at 9:40
Here is an analytic geometry solution :
We can assume without loss of generality that $$R=1$$, i.e., we work inside the unit circle. Therefore, we have the following condition : $$0 < m <1$$.
Let us introduce some notations.
Let $$B$$ be the second intersection of line $$GC$$ with the unit circle.
Let $$a$$ be the polar angle of point $$G$$ (i.e., oriented angle between positive $$x$$-axis and $$AG$$).
Let $$b$$ be the polar angle of $$B$$ (i.e., oriented angle between positive $$x$$-axis and $$AB$$).
Let $$\theta$$ be the angle of line $$GE$$ with line $$GB$$ (equal to the angle between $$GB$$ and $$GF$$).
We will use a certain number of trigonometric formulas. An extensive and well structured list of those can be found here.
Fig. 1: $$G(\cos a,\sin a)$$, $$B(\cos b,\sin b)$$, and $$C(m,0)$$, center of the small circle with radius $$r$$. The circle to which lines $$EF$$ are all assumed to be tangent is in red.
Let us look for relationships between angles $$a,b$$ and $$\theta$$ and lengths $$m$$ and $$r$$.
• a) The fact that $$GE$$ and $$GF$$ are tangent to the small circle is expressed by the following relationship :
$$\sin \theta = \dfrac{r}{GC}$$
which is equivalent to :
$$(\sin \theta)^2 = \dfrac{r^2}{GC^2}= \dfrac{r^2}{(\cos a - m)^2+(\sin a - 0)^2}=\dfrac{r^2}{1 - 2m \cos a +m^2}\tag{1}$$
• b) The angle between $$AE$$ and $$AB$$ is $$2 \theta$$ by central angle theorem. The same for the angle between $$AB$$ and $$AF$$. Let $$I$$ be the intersection point of line $$AB$$ and line $$EF$$ ; triangle $$EAF$$ being isosceles, line segment $$AI$$ is orthogonal to $$EF$$, implying that its algebraic measure ("signed distance") is $$AI=\cos(2 \theta)$$ (possibly negative).
Therefore, straight line $$EF$$ whose normal vector is $$\vec{AB} = \binom{\cos b}{\sin b}$$ has equation :
$$x \cos b + y \sin b = \cos 2 \theta \tag{2}$$
Using relationship $$\cos 2 \theta=1-2 \sin^2 \theta$$ with formula (1) :
$$x \cos b + y \sin b = 1-\dfrac{2 r^2}{1 - 2m \cos a +m^2}\tag{3}$$
• c) Let us, finally, express that $$G, C$$ and $$B$$ are aligned. This relationship, written under the form (see http://mathworld.wolfram.com/Collinear.html) :
$$\begin{vmatrix}m &\cos a &\cos b\\0 & \sin a&\sin b\\1&1&1\end{vmatrix}=0\tag{4}$$
i.e.,
$$m=\dfrac{\sin(a-b)}{\sin a - \sin b}$$
$$\iff \ \ m=\dfrac{\color{red}{2 \sin(\tfrac12(b-a))}\cos(\tfrac12(b-a))}{\color{red}{2 \sin(\tfrac12(b-a))}\cos(\tfrac12(b+a))}=\dfrac{1+\tan(a/2) \ \tan(b/2)}{1-\tan(a/2) \ \tan(b/2)}$$
yielding the rather unexpected following condition :
$$\tan(a/2) \ \tan(b/2)=k \ \ \text{where} \ \ k:=\dfrac{m-1}{m+1}\tag{5}$$
from which we deduce (using a "tangent half-angle formula" and setting :
$$t:=\tan(b/2)$$
that :
$$\cos a=\dfrac{1-\tan(a/2)^2}{1+\tan(a/2)^2}=\dfrac{1-\left(\tfrac{k}{t}\right)^2}{1+\left(\tfrac{k}{t}\right)^2}=\dfrac{t^2-k^2}{t^2+k^2}\tag{6}$$
Plugging (6) into formula (3), we get for the RHS of (3) after some algebraic transformations :
$$\cos 2 \theta=1-\dfrac{2r^2(t^2(m+1)^2+(m-1)^2)}{(m^2-1)^2(1+t^2)}$$
Using once again tangent half-angle formulas, we can now express the equation of straight line $$EF$$ under the following parametric form in variable $$t$$ :
$$x \dfrac{1-t^2}{1+t^2} + y \dfrac{2t}{1+t^2} - 1+\dfrac{2r^2(t^2(m+1)^2+(m-1)^2)}{(m^2-1)^2(1+t^2)}=0.\tag{7}$$
It remains to establish that the distance $$d$$ of the point with coordinates
$$C'(x_0,y_0)=\left(4m\dfrac{r^2}{(1-m^2)^2},0\right)\tag{8}$$
(would-be center of envelope circle) to straight line $$EF$$ is constant (i.e., is independent from $$t$$).
This distance, obtained (see http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html) by replacing $$(x,y)$$ in the Left Hand Side of equation (7) by the coordinates $$(x_0,y_0)$$ of $$C'$$ given by (8)
$$d=(2r^2(1+m^2) - (m^2 - 1)^2)/(m^2 - 1)^2\tag{9}$$
is indeed independent from parameter $$t$$ ; moreover it is (up to its sign) the awaited expression (I have used a Computer Algebra System to obtain this expression of $$d$$).
Remarks :
1) It is possible to characterize the tangent point of $$EF$$ on the (red) envelope circle, following the method outlined by @Jan-Magnus Økland in comments following this answer.
2) About a possible more (synthetic) geometry solution. I just discovered in this paper dealing with "exponential pencils of conics" that there is so-called "conjugate conic" of the first circle vs. the second circle (see Theorem 2.5 page 4). But I am not sure it will permit a new solution.
• In maxima CAS with the $r$ you dropped in (7) back in l:x*((1-t^2)/(1+t^2))+y*(2*t/(1+t^2))+(((2*m^2+4*m+2)*r^2-m^4+2*m^2-1)*t^2+(2*m^2-4*m+2)*r^2-m^4+2*m^2-1)/((m^4-2*m^2+1)*t^2+m^4-2*m^2+1); Then the envelope is given by solve([l,diff(l,t)],[x,y]); [[x=(((2*m^2+4*m+2)*r^2-m^4+2*m^2-1)*t^2+((-2*m^2)+4*m-2)*r^2+m^4-2*m^2+1)/((m^4-2*m^2+1)*t^2+m^4-2*m^2+1),y=-(((4*m^2+4)*r^2-2*m^4+4*m^2-2)*t)/((m^4-2*m^2+1)*t^2+m^4-2*m^2+1)]] – Jan-Magnus Økland Jan 27 at 9:41
• Then the implicitization in M2: R=QQ[m,r] S=R[s,t,x,y,z] I=ideal(x-(((2*m^2+4*m+2)*r^2-m^4+2*m^2-1)*t^2+(((-2*m^2)+4*m-2)*r^2+m^4-2*m^2+1)*s^2),y+(((4*m^2+4)*r^2-2*m^4+4*m^2-2)*s*t),z-((m^4-2*m^2+1)*t^2+(m^4-2*m^2+1)*s^2)) gens gb I -- (4*r^4+((-4*m^2)-4)*r^2+m^4-2*m^2+1)*z^2 +8*m*r^2*x*z+((-m^4)+2*m^2-1)*y^2+((-m^4)+2*m^2-1)*x^2 – Jan-Magnus Økland Jan 27 at 9:41
• $z$ is just there to eliminate the fractions. Setting $z=1$ gives the correct answer. – Jan-Magnus Økland Jan 27 at 10:08
• @JeanMarie: I don't (yet?) have a better solution, synthetic or otherwise, but I have an possibly-useful observation: let $e$ and $f$ be the distances from $E$ and $F$ to the points where $\overline{GE}$ and $\overline{GF}$ touch $\bigcirc C$, and let $e'$ and $f'$ be the distances from $E$ and $F$ to the point where $\overline{EF}$ touches the target circle. Then $$\frac{e}{f}=\frac{e'}{f'}$$ – Blue Feb 7 at 19:07
• @JeanMarie I don't think that the "conjugate conic" approach will be fruitful. That's because the red circle is coaxal with the big and small circles, so it is a linear combination of the two (i.e. its equation is a weighted sum of the equations of the other 2). See more context in math.stackexchange.com/a/3545910/1257. If the problem is generalized projectively the red conic will be in the pencil generated by the first two conics. – brainjam Feb 13 at 23:32
The expository paper Poncelet's theorem by András Hraskó very nicely treats this problem and its relation to Poncelet's Closure Theorem.
In the OP diagram call the big and little circles $$e$$ and $$a$$ respectively, and the red circle $$c$$. (this corresponds to the labels in the paper, see Figures 1,4,5). Poncelet's Theorem is concerned with scenarios such as Figures 1 and 4, where a polygon is inscribed in $$e$$ such that the edges touch $$a$$. But the paper speculates that Poncelet studied that situation of an inscribed triangle where one of the sides does not touch $$a$$, and found that the non-touching side generates the envelope of a circle $$c$$ (Figure 5). As @Blue has noted in the comments, $$c$$ is coaxal with the pair $$a$$ and $$e$$.
I leave the details to the paper, but can't resist quoting this:
Poncelet's General Theorem: Let $$e$$ be a circle of a non-intersecting pencil and let $$a_1,a_2,\ldots,a_n$$ be (not necessarily different) oriented circles in the interior of $$e$$ that belong to the same pencil. Starting at an arbitrary point $$A_0$$ of the circle $$e$$, the points $$A_1,A_2,\ldots,A_n$$ are constructed on the same circle, such that the lines $$A_0A_1, A_1A_2, \ldots, A_{n-1}A_n$$ touch the circles $$a_1,a_2,\ldots, a_n$$, respectively, in the appropriate direction. It may happen that at the end of the construction, we get back to the starting point, that is, $$A_n=A_0$$. The theorem states that in that case, we will always get back to the starting point in the $$n$$-th step, whichever point of $$e$$ we start from. We do not even need to take care to draw the tangents to the circles in a fixed order.
The case of $$a_1=a_2=\cdots=a_n=a$$ gives the classic Poncelet's Closure Theorem.
The case $$a_1=a_3=a, a_2=c$$ relates to the OP.
• For more on Poncelet's General Theorem, see Poncelet’s porism: a long story of renewed discoveries, I and II by Andrea Del Centina – Jan-Magnus Økland Feb 18 at 9:51
• @Jan-MagnusØkland, thanks for this reference. It also suggests an answer to a projectively dual problem. See math.stackexchange.com/a/3560237/1257 – brainjam Feb 26 at 17:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 97, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9081491231918335, "perplexity": 308.3536583789576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1590347419639.53/warc/CC-MAIN-20200601211310-20200602001310-00396.warc.gz"} |
https://economics.stackexchange.com/questions/19643/arrows-impossibility-theorem | # Arrow's impossibility theorem
In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency and independence of irrelevant alternatives (IIA)
So, I'm looking for an example where the preferences satisfy unrestricted domain, non dictatorship, IIA and is an ordering, but not meet the Pareto criteria.
Let the set of alternatives be $A = \left\{a_1,a_2,...,a_k\right\}$. Let the number of players be $n$. Let the set of preference orderings over $A$ be $\mathcal{P}$. Then the set of preference profiles is the Cartesian product $$\mathcal{P}^n = \times_{i=1}^{n} \mathcal{P}.$$ Let us denote the preference ordering $$a_1 \succ a_2 \succ ... \succ a_k$$ by $p^*$. Define the Social Choice Function $F$ by $$\forall p \in \mathcal{P}^n: F(p) = p^*.$$ This SCF $F$ clearly has universal domain, is not a dictatorship, and is also independent of irrelevant alternatives. (Or any alternatives for that matter.)
• Hey, thanks. But I am unable to understand the functioning of this social choice rule. How is it not a dictatorship and how does it not meet Pareto criteria? – LUCIFER Dec 10 '17 at 1:07
• @LUCIFER Could you please define what you think a dictatorship SCF is? And can you please also define the Pareto property for SCFs? – Giskard Dec 10 '17 at 7:28
• Dictatorship is when there is an individual such that if he prefers x over y, then so does the society irrespective of what the society feels. – LUCIFER Dec 10 '17 at 21:00
• And Pareto property is that if all individuals prefer x over y, then so should the society. It is about how society preserves total agreement. – LUCIFER Dec 10 '17 at 21:01
• @LUCIFER If you understand these then surely you see how my SCF fulfills neither condition. I will improve notation a bit. – Giskard Dec 10 '17 at 21:16
The Pareto criterion has two effects: It guarantees that every ranking can occur as a social ranking and it connects social rankings to individual rankings. If one drops the Pareto criterion but keeps the assumption that every social ranking is possible, one obtains a generalization in which every SCG corresponds to a dictatorship or an anti-dictatorship in which the social ranking is exactly opposite to the ranking of a specific individual, an anti-dictator. This is known as Wilson's theorem, originally from:
Wilson, Robert. "Social choice theory without the Pareto principle." Journal of Economic Theory 5.3 (1972): 478-486.
• "keeps the assumption that every social ranking is possible" That is not an assumption in this question though. – Giskard Dec 11 '17 at 14:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8690738081932068, "perplexity": 1004.5243806784739}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251678287.60/warc/CC-MAIN-20200125161753-20200125190753-00448.warc.gz"} |
http://math.stackexchange.com/questions/248821/notations-in-group-theory | # Notations in Group theory
I will start by apologizing as many will not like this question.
I am reading the paper COHOMOLOGY THEORY OF GROUPS WITH A SINGLE DEFINING RELATION and having focused on typology throughout my studies i finding myself to be in a lost notation wise.
1. Let $F$ be a free subgroup in $G$ with generators $x_i$ , then elements in $F$ are represented by "words" or a final sequence of $x_i^{\pm1}$. In Page 650 (first one in this article) and 658 they state that every such word $R$ "can be expressed uniquely as a power $R=Q^q$ for $q$ maximal" Nowhere do they explain what is this $Q$, any ideas, links, resources for me to search in?
2. Let $R$ be a normal subgroup in $G$, what does the notation $(R, R)$ symbolize? they used it in pages 650 and 658 without ever explaining.
Am i reading this paper wrong? Or are these notation so common they are never introduced?
-
1. $R=xyz$ can be written as $R=(xyz)^1$, so here $Q=xyz$. $R=xyxy$ can be written as $R=(xy)^2$, so here $Q=xy$. 2. Is probably the commutator. – user641 Dec 1 '12 at 21:38
Oh, that makes sense, thank you very much, What about (R,R)? – user44874 Dec 1 '12 at 21:42
The general references for one-relator groups are two books, both called "combinatorial group theory" (the second was named in honour of the first!). The first was by Magnus, Karrass and Solitar, the second by Lyndon (he who wrote the paper you are reading) and Schupp. I suppose they're pretty out of date now, but they are excellent places to start: the tools laid down by Magnus in his 1931 thesis are still used today! – user1729 Dec 3 '12 at 10:24
1. Maybe you prefer more elaborate would be a formulation like this: "For every word $R$ over the alphabet $A$ there exist $(Q,q)$ such that $Q$ is a word over the alphabet $A$ and $q\in\mathbb N$ and $R=Q^q$ is the $q$-fold concatenation of $Q$ with itself. One example of such a pair for arbitray $R$ is $(R,1)$ as trivially $R=R^1$. If $R$ is not the empty word, then $Q$ in such a pair cannot be the empty word and hence $q$ is bounded from above by the length of the word $R$. Hence there exist such pairs $(Q,q)$ for which $q$ attains te maximal possible value. Since $Q_1\ne Q_2$ implies $Q_1^q\ne Q_2^q$ (if $q\ge 1$), the word $Q$ for which the maximal possible $q$ is attained, is uniquely determined."
2. Since they refer in passing to $R/(R,R)$ as the abelianized group, $(R,R)$ should denote the commutator. Today, writing $[R,R]$ seems to be the preferred notation. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9209490418434143, "perplexity": 361.83587168396656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1432207928102.74/warc/CC-MAIN-20150521113208-00066-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/su-2-pure-ym-on-r-4.585649/ | # SU(2) Pure YM on R^4
1. Mar 10, 2012
### Charles_Henry
1. The problem statement, all variables and given/known data
Derive the pure SU(2) YM theory on $\mathbb{R}^4$ from the action. Let $A_{\mu} (x)$ be a solution to these equations. Show:
$\tilde{A}_{\mu} (cx)$ is also a solution (with the same action).
Background
The Euclidean YM action
$\mathbb{S} = - \int_{\mathbb{R^4}} Tr (F \wedge \ast F)$
yields
$D \ast F = 0$
Let $\ast: \wedge^{p} \rightarrow \wedge^{D+1-p}$ be a linear map, such that
$\ast (dx^{\mu_{1}} \wedge ... \wedge dx^{\mu_{p}} = \frac{\sqrt{|det(n)!|}}{(D+1-p)!} \epsilon^{\mu_{1}...\mu_{p}}_{{\mu}_{p+1}...{\mu}_{D+1}} dx^{\mu}_{p+1} \wedge ... \wedge dx^{{\mu}_{p+1}}$
if $G = SU(2)$ we choose a basis $T_{a}, a = 1, 2, 3$ for an Anti-Hermitian 2 x 2 matrix
$T_{a}, T_{b}$ = $- \epsilon_{abc} T_{c}, T_{a} = \frac{1}{2} i \sigma_{a}$
where $Tr(T_{a}, T_{b}) = - \frac {1}{2} \delta_{ab}$
where $\sigma_{a}$ are pauli matrices, and a general group element $g = exp (\alpha^{a} T_{a} )$ with $\alpha^a$ real. Whence,
$(D_{\mu} \phi)^{a} = \partial_{\mu} \phi^{a} - \epsilon^{abc} A^{b}_{\mu} \phi^{c}$ and $F^{a}_{{\mu} v} = \partial_{\mu} A^{a}_{v} - \partial_{v}A^{a}_{\mu} - \epsilon^{abc}A^{b}_{\mu}A^{c}_{v}$
when $D+1=4$ is a gauge theory in Minkowski space $M$, and $A$ is the gauge potential,
$( \ast F)_{{\mu}v} = \frac{1}{2} \epsilon_{{\mu}v{\alpha}{\beta}}F^{{\alpha}{\beta}}$
A two form $F= \frac{1}{2} F_{{\mu}v}dx^{\mu} \wedge dx^{v}$ is self dual or ASD when $\ast F = F$ and $\ast F = - F$ respectively
$-Tr ( F \wedge \ast F) = - \frac{1}{2} Tr (F_{{\mu}v}F^{{\mu}v}) d^{4}x = \frac{1}{4}F^{a}_{{\mu}v}F^{{\mu}va} d^{a}x$
$d^{4}x = \frac{1}{24} \epsilon_{{\mu}v{\alpha}{\beta}}dx^{\mu} \wedge dx^{v} \wedge dx^{\alpha} \wedge dx^{\beta}$
with identities
$\epsilon_{{\mu}v{\alpha}{\beta}}dx^{4} = - dx^{4} \wedge dx^{v} \wedge dx^{\alpha} \wedge dx^{\beta}$
and $\epsilon_{{\alpha}{\beta}{\rho}{\sigma}} \epsilon^{{\mu}{v}{\rho}{\sigma}} = - 4 (\delta^{\mu}_{\alpha} \delta^{v}_{\beta})$
Instantons are non-singular solutions of classical equations of motion in Euclidean space whose Action is finite.
$F_{{\mu}v} (x)$ ~ $O (\frac{1}{r^3})$
$A_{\mu}$ ~ $\partial_{\mu} gg^{-1} + O \frac{1}{r^2}$ as $r \rightarrow \infty$
note: I understand that the gauge transformations g(x) needs to be defined only asymptotically, so $g: \mathbb{S}^{3}_{\infty} \rightarrow SU(2)$ is extended to $\mathbb{R}^4$ if its degree vanishes:
For example:
If $M_{1}$ and $M_{2}$ are oriented, compact, D-dimensional manifolds without boundary, and $w$ is a volume-form on $M_{2}$. where $deg (f)$ of a smooth map $f: M_{1} \rightarrow M_{2}$ is given by
$\int_{M_{1}} f\ast w = [ deg(f) ] \int_{M_{2}} w$
let $y \in M_{2}$ when $f^{-1}(y) = {x; f(x) = y}$ is finite, and the Jacobian $J(f)$ is not zero (if $x \in U$ with local coordinates $x^{i}$ and $y \in f(u)$ with local co-ordinates $y^{i}$, then we can assume:
$\mathbb{J} = det \frac{\partial y^{i}}{\partial x^{J}}$ if $y^{i} (x^{1}, ... x^{D})$
deg (f) is an integer given by
$deg (f) = \Sigma_{x \in f^{-1} (y)} sign [ \mathbb{J} (x) ]$
(proof withheld)
therefore:
$f: X \rightarrow SU(2) = S^{3}$ where X is closed.
$deg(f) = \frac{1}{24\pi^2} \int_{X} Tr[(f^{-1} df)^3]$
the boundary conditions are understood in terms of one-point compactifications $S^4 = \mathbb{R}^4 \cup {\infty}$ which has a conformally equivalent metric to that of a flat metric in $\mathbb{R}^4$
A solution of YM equations on $S^4$ project stereographically to a connection on $\mathbb{R}^4$ with a curvature which vanishes at infinity.
Scaling Argument:
A Field(s) $(A, \phi)$ given by a potential one-form and a scalar higgs-field:
$E = \int_{\mathbb{R}} d^{D}x [|F|^{2} + |D \phi |^{2} + U(\phi)$
$= E_{F} + E_{D_{\phi}} + E_{U}$
if $A(x)$ and $\phi (x)$ are critical points:
$\phi_{c} (x) = \phi (cx)$
$A_{c} (x) = cA(cx)$
$F_{c} = C^{2} F(cx)$
$D_{c} \phi_{c} = c D\phi (cx)$
$E_{(c)} = \frac{1}{C^{D-4}}E_{F} + \frac{1}{C^{D+2}}E_{D_{\phi}} + \frac{1}{C^{D}}E_{U}$
$(D-4) E_{F} + (D-2) E_{D_{\phi}} + DE_{U} = 0$
note: I believe I am looking for a solution where $E_{D_{\phi}} = E_{U} = 0$ in D=4
3. The attempt at a solution
A YM action S within a given topological sector
$c_{2} = \frac{1}{8 \pi^2} \int_{\mathbb{R}} Tr( F \wedge F) > 0$
bounded from below by $8\pi^2c_{2}$
$F \wedge F = \ast F \wedge \ast F$
$\mathbb{S} = - \frac{1}{2} \int_{\mathbb{R}^4} Tr[(F + \ast F) \wedge (F + \ast F)] + \int_{\mathbb{R}^4} Tr (F \wedge F) = - \frac{1}{2} \int_{\mathbb{R}^4} Tr [(F + \ast F) \wedge \ast (F + \ast F) + 8 \pi^2c_{2} \geq 8 \pi^2c_{2}$
when $F = - \ast F$ hold
some bib: Atiyah, M.F and Ward, R.S (1977) Instantons and Algebraic Geometry, Commun. Math. Phy. 55, 117-124
Sacks, L. and uhlenbeck, K (1981) The existence of minimal immersions of 2-spheres, Ann. Math 113, 1-24
Last edited: Mar 10, 2012
2. Mar 12, 2012
### Charles_Henry
nonsense
Last edited: Mar 12, 2012
3. Mar 12, 2012
### Charles_Henry
more nonsense.
Last edited: Mar 12, 2012
4. Mar 12, 2012
### Charles_Henry
ok, i see it now...derive the euler-lagrange equations from the action that leads to gauge potential. Assume we could derive a solution of pure YM IN R^4 from the Vector Potential by defining invariance along any coordinate of our choosing.
Last edited: Mar 12, 2012
5. Mar 12, 2012
### fzero
This is meant to be much simpler than you're making it out to be.
Literally use the Euler-Lagrange equations to obtain the equation of motion for the gauge potential.
This is meant to follow from the scale-invariance of pure YM. Assume $A_{\mu} (x)$ is a solution. Consider $A_{\mu} (cx)$ and make a change of coordinates, taking into account that $A_\mu$ scales like a tensor of the appropriate 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": 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": 0.9714658856391907, "perplexity": 785.225905329546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647475.66/warc/CC-MAIN-20180320130952-20180320150952-00317.warc.gz"} |
https://www.snapxam.com/problems/40272580/integral-of-x-3-16-1x-2-0-5-dx | Step-by-step Solution
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2
e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
Step-by-step explanation
Problem to solve:
$\int\frac{x^3}{\sqrt{16-x^2}}dx$
Learn how to solve integrals of rational functions problems step by step online.
$x=4\sin\left(\theta \right)$
Learn how to solve integrals of rational functions problems step by step online. Integral of (x^3)/((16-x^2)^0.5) with respect to x. We can solve the integral \int\frac{x^3}{\sqrt{16-x^2}}dx by applying integration method of trigonometric substitution using the substitution. Now, in order to rewrite d\theta in terms of dx, we need to find the derivative of x. We need to calculate dx, we can do that by deriving the equation above. Substituting in the original integral, we get. Factor by the greatest common divisor 16.
$-16\sqrt{16-x^2}+\frac{1}{3}\sqrt{\left(16-x^2\right)^{3}}+C_0$
Problem Analysis
$\int\frac{x^3}{\sqrt{16-x^2}}dx$
Main topic:
Integrals of Rational Functions
~ 0.21 seconds | {"extraction_info": {"found_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": 0.9920560121536255, "perplexity": 635.7226806445492}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897168.4/warc/CC-MAIN-20200714145953-20200714175953-00039.warc.gz"} |
http://math.stackexchange.com/questions/870240/an-inequality-in-numbers | # An inequality in numbers
Which number is larger? $\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits}$ or $\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots67}_\text{68 digits}$? Why? How much is it larger?
-
Let those four numbers be $a,b,c,d$ respectively. Then $a=2c$ and $d=2b+1$. So $cd-ab=c$.
-
note that $$\underbrace{888\cdots8}_\text{19 digits}=2*\underbrace{444\cdots4}_\text{19 digits}$$ and $$2*\underbrace{333\cdots3}_\text{68 digits}=\underbrace{666\cdots6}_\text{68 digits}$$ further $$\underbrace{666\cdots6}_\text{68 digits}+1=\underbrace{666\cdots7}_\text{68 digits}$$ Thus $$\underbrace{888\cdots8}_\text{19 digits}*\underbrace{333\cdots3}_\text{68 digits}=\underbrace{444\cdots4}_\text{19 digits}*2*\underbrace{333\cdots3}_\text{68 digits}= \underbrace{444\cdots4}_\text{19 digits}*\underbrace{666\cdots6}_\text{68 digits}$$ which is by $\underbrace{444\cdots4}_\text{19 digits}$ smaller than $$\underbrace{444\cdots4}_\text{19 digits}*\underbrace{666\cdots7}_\text{68 digits}=\underbrace{444\cdots4}_\text{19 digits}*\underbrace{666\cdots6}_\text{68 digits}+\underbrace{444\cdots4}_\text{19 digits}$$
-
$\underbrace{888\cdots8}_\text{19 digits}\not=2\times\underbrace{444\cdots4}_\text{19 digits}$. Why did you write it???/ – bigli Jul 17 '14 at 20:46
it most truly does. $4*2=8$, $44*2=88$, $444*2=888$ etc. – cirpis Jul 17 '14 at 20:48
Excuse me. You are right. – bigli Jul 17 '14 at 20:52
No problem, looking at my solution i can tell that i shouldve represented those big numbers using letters for a more elegant and less confusing proof. – cirpis Jul 17 '14 at 20:53
In your answer: How did you get the last equality? – bigli Jul 17 '14 at 20:55
You have
$\underbrace{888\cdots8}_\text{19 digits}\times\underbrace{333\cdots3}_\text{68 digits} = 8\cdot 3 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits}) = 24 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits})$.
Similarly we get
$\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots6}_\text{68 digits} = 4\cdot 6 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits}) = 24 \cdot (\underbrace{111\cdots1}_\text{19 digits}\times \underbrace{111\cdots1}_\text{68 digits})$.
So these two numbers are equal.
It is clear that $$\underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots6}_\text{68 digits} \le \underbrace{444\cdots4}_\text{19 digits}\times\underbrace{666\cdots7}_\text{68 digits}.$$ Since the multiplier is increased by one, the difference is exactly $\underbrace{444\cdots4}_\text{19 digits}$.
- | {"extraction_info": {"found_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": 0.9307522177696228, "perplexity": 1311.301352242655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246662032.70/warc/CC-MAIN-20150417045742-00035-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://sbseminar.wordpress.com/2008/08/10/theta-divisors-the-clean-part/ | Theta divisors: the clean part
As often happens, after I do a large computation, a lot more theory makes sense. So, in this post, I’ll talk about the parts that I can explain without getting dirty with formulae. After that, I’ll think there willl be one more post on $\Theta$ functions and divisors.
Suppose we have an abelian variety $A$ of dimension $g$ and we have a cohomology class $\beta$ in $H^2(A, \mathbb{Z})$. Suppose that $\beta$ is the Chern class of a line bundle. We will be interested in the question of to what extent we can canonically choose a line bundle with Chern class $\beta$. (Later, we will ask the same questions regarding whether there is a hypersurface with class Poincare dual to $\beta$.) The application to our original question is when $A$ is the Jacobian of a Riemann surface $X$ and $\beta$ is the class of the $\Theta$ disivor in $Pic^{g-1}(X)$. (The ambiguity in identifying $Pic^0(X)$ with $Pic^{g-1}(X)$ is topologically trivial.)
By the way, when is $\beta$ could be the Chern class of a line bundle? If and only if it is a Hodge class. This is the main case in which the Hodge conjecture has been proved. This is a fascinating subject, but I’ll be ignoring it.
We have a map $Pic(A) \to H^2(A, \mathbb{Z})$ which maps a holomorphic line bundle to its first Chern class. The kernel of this map is called $Pic^0(A)$ and the image is called the Neron-Severi group, $NS(A)$. So we have a short exact sequence:
$0 \to Pic^0(A) \to Pic(A) \to NS(A) \to 0. \ (*)$
The question we are asking is whether we can find a natural way of splitting this sequence. Let ${[-1]}$ be the endomorphism of $A$ which sends every point to its inverse. Then ${[-1]}$ acts on all three terms of $(*)$. The interesting thing is that ${[-1]}$ acts by negation on the left term and acts trivially on the right. If we were dealing with a short exact sequence of real vector spaces, or more generally of $\mathbb{Z}[1/2]$ modules, this would let us split the sequence immediately. We would simply split $Pic(A)$ into its $(-1)$ eigenspace, which would be $Pic^0(A)$, and its $1$ eigenspace, which would be $NS(A)$.
We can still try to mimic this, but life will be trickier because we can’t divide by ${2}$. Choose an arbitrary lift $\gamma$ of $\beta$ to $Pic(A)$. Then $\gamma + [-1]^* \gamma$ is independent of the choice of $\gamma$. There are $2^{2g}$ bundles $L$ such that $2 L = \gamma + [-1]^* \gamma$, each of which has class $\beta$. So, there isn’t one canonical choice of $L$, but we can get down to a discrete $2^{2g}$ choices, as opposed to the entire continuum of choices that we would naively see.
Now, this post has been about line bundles. What happens with divisors? If a line bundle $L$ has Chern class $\beta$, then $\beta$ is the class (Poincare dual to) the zero locus of any section of $L$. In general, $L$ will have many sections. Precisely, assuming that $L$ is ample, $\beta$ induces a skew symmetric form on $H_1(A, \mathbb{Z})$, and the dimension of $H^0(L)$ is the Pfaffian of that form. In particular, as will be the case in our intended application to $\Theta$-functions, if $\beta$ induces a perfect pairing on $H_1(A, \mathbb{Z})$, then $L$ has a unique nonzero section (up to scaling). So, in that case, we can use the methods above to find a canonical divisor $D$, modulo translation by ${2}$-torsion, whose cohomology class is $\beta$.
What is this divisor explicitly? It is determined by the condition that ${[-1]}D=D$. (Note that translating $D$ by a ${2}$-torsion point preserves the truth of this condition.) It’s a good exercise to trace through the previous definitions and see that this works.
Now, all of this was in the context of a general abelian variety with a specified Hodge class that gives an ample line bundle and induces a perfect pairing on $H^1(A, \mathbb{Z})$. (The terminology for this is a principally polarized abelian variety.) In the particular case of the Jacobian, it is a nice exercise (using Riemman-Roch) to check that, if $\kappa$ is any point of $Pic^{g-1}(X)$ with $2 \kappa =K$ then $\Theta - \kappa = - (\Theta - \kappa)$. Thus, the previously described construction does yield the translate of the $\Theta$ divisor by some square root of the canonical bundle. (As several commenters told me.)
That is the end of what can be done nicely, and without reference to explicit $\Theta$ functions (or indeed, any analysis). What $\Theta$ functions let you do, given a choice of basis for $H_1(A,\mathbb{Z})$, is to make a choice of one of the $2^{2g}$ possible line bundles on $A$. More on this in the forthcoming (and final post) “Theta functions: the funky part”!
4 thoughts on “Theta divisors: the clean part”
1. Nice! That clears up some issues I hadn’t been understanding…
I haven’t really worked out the details, but shouldn’t most of this story (all except paragraph -4) go through even if the variety is not principally polarized? The Prym isn’t principally polarized…
Can you say something about the picture in the non-principally-polarized case? (although this has nothing to do with your original question).
2. Yeah, I think that the only importance of principal polarization is that it gives you a unique section of the line bundle. In the general case, everything works the same way on the level of line bundles.
We can then ask, if we are given an isomorphism ${[-1]}^* L \cong L$, how does ${[-1]}$ act on $H^0(L)$? To find a divisor $D$ of class $\beta$, we’d want to find a section in $H^0(L)$ which is an eigenvalue for ${[-1]}$. That divisor will vanish to an even order at the identity (possibly zero) if it comes from a ${1}$-eigenvector and to odd order if it comes from a ${(-1)}$-eignevector. In the principally polarized case, $H^0(L)$ is one dimensional and the zero locus of its section is called an even or odd $\Theta$ divisor according to which of these cases holds.
The above seems to depend on our choice of isomorphism ${[-1]}^* L \cong L$ but we can remove the dependency. Any two such isomorphisms differ by a nonvanishing holomorphic function on $A$, which must be a constant. We can normalize that constant by requiring our isomorphism to act trivially on the fiber over the identity. So don’t worry about that.
As you’ll see in the next post, we can write down a basis for $H^0(L)$ using $\Theta$ functions and that basis is an eigenbasis for $H^0(L)$. So it would be easy to work out how many ${1}$ and how many ${-1}$ eigenvalues occur. But I haven’t found a nice way to formulate the result.
3. Small typo: “When is beta could be…”
I’ve been trying to learn this stuff myself, so I’ll reread this when I’m less jet-lagged and sleepy! Thanks for writing it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 91, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9477786421775818, "perplexity": 139.858440804134}, "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-2016-50/segments/1480698541864.44/warc/CC-MAIN-20161202170901-00358-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://zims-en.kiwix.campusafrica.gos.orange.com/wikipedia_en_all_nopic/A/Euler%E2%80%93Rodrigues_formula | # Euler–Rodrigues formula
In mathematics and mechanics, the EulerRodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization.
The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigues), a method of calculating the position of a rotated point, is used in some software applications, such as flight simulators and computer games.
## Definition
A rotation about the origin is represented by four real numbers, a, b, c, d such that
${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1.}$
When the rotation is applied, a point at position x rotates to its new position
${\displaystyle {\vec {x}}'={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}{\vec {x}}.}$
### Vector formulation
The parameter a may be called the scalar parameter, while ω = (b, c, d) the vector parameter. In standard vector notation, the Rodrigues rotation formula takes the compact form
${\displaystyle {\vec {x}}'={\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)}$
### Symmetry
The parameters (a, b, c, d) and (−a, −b, −c, −d) describe the same rotation. Apart from this symmetry, every set of four parameters describes a unique rotation in three-dimensional space.
### Composition of rotations
The composition of two rotations is itself a rotation. Let (a1, b1, c1, d1) and (a2, b2, c2, d2) be the Euler parameters of two rotations. The parameters for the compound rotation (rotation 2 after rotation 1) are as follows:
{\displaystyle {\begin{aligned}a&=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2};\\b&=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2};\\c&=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2};\\d&=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}.\end{aligned}}}
It is straightforward, though tedious, to check that a2 + b2 + c2 + d2 = 1. (This is essentially Euler's four-square identity, also used by Rodrigues.)
## Rotation angle and rotation axis
Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by a unit vector k = (kx, ky, kz)) and the rotation angle φ. The Euler parameters for this rotation are calculated as follows:
{\displaystyle {\begin{aligned}a&=\cos {\frac {\varphi }{2}};\\b&=k_{x}\sin {\frac {\varphi }{2}};\\c&=k_{y}\sin {\frac {\varphi }{2}};\\d&=k_{z}\sin {\frac {\varphi }{2}}.\end{aligned}}}
Note that if φ is increased by a full rotation of 360 degrees, the arguments of sine and cosine only increase by 180 degrees. The resulting parameters are the opposite of the original values, (−a, −b, −c, −d); they represent the same rotation.
In particular, the identity transformation (null rotation, φ = 0) corresponds to parameter values (a, b, c, d) = (±1, 0, 0, 0). Rotations of 180 degrees about any axis result in a = 0.
## Connection with quaternions
The Euler parameters can be viewed as the coefficients of a quaternion; the scalar parameter a is the real part, the vector parameters b, c, d are the imaginary parts. Thus we have the quaternion
${\displaystyle q=a+bi+cj+dk,}$
which is a quaternion of unit length (or versor) since
${\displaystyle \left\|q\right\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1.}$
Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with composition.
## Connection with SU(2) spin matrices
The Lie group SU(2) can be used to represent three-dimensional rotations in 2 × 2 matrices. The SU(2)-matrix corresponding to a rotation, in terms of its Euler parameters, is
${\displaystyle U={\begin{pmatrix}\ \ \,a+di&b+ci\\-b+ci&a-di\end{pmatrix}}.}$
Alternatively, this can be written as the sum
{\displaystyle {\begin{aligned}U&=a\ {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+b\ {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}+c\ {\begin{pmatrix}0&i\\i&0\end{pmatrix}}+d\ {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}\\&=a\,I+ic\,\sigma _{x}+ib\,\sigma _{y}+id\,\sigma _{z},\end{aligned}}}
where the σi are the Pauli spin matrices. Thus, the Euler parameters are the coefficients for the representation of a three-dimensional rotation in SU(2). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "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": 0.9039814472198486, "perplexity": 782.892373110474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488556482.89/warc/CC-MAIN-20210624171713-20210624201713-00463.warc.gz"} |
https://kk.khanacademy.org/math/algebra/x2f8bb11595b61c86:rational-exponents-radicals/x2f8bb11595b61c86:radicals/v/finding-square-root-of-decimal | If you're seeing this message, it means we're having trouble loading external resources on our website.
Егер веб фильтрлерін қолдансаң, *.kastatic.org мен *.kasandbox.org домендері бұғатталмағанын тексер.
Негізгі бет
# Ондық бөлшектің квадрат түбірі
## Видео транскрипті
- Let's see if we can solve the equation P squared is equal to 0.81. So how could we think about this? Well one thing we could do is we could say, look if P squared is equal to 0.81, another way of expressing this is, that well, that means that P is going to be equal to the positive or negative square root of 0.81. Remember if we just wrote the square root symbol here, that means the principal root, or just the positive square root. But here P could be positive or negative, because if you square it, if you square even a negative number, you're still going to get a positive value. So we could write that P is equal to the plus or minus square root of 0.81, which kind of helps us, it's another way of expressing the same, the same, equation. But still, what could P be? In your brain, you might immediately say, well okay, you know if this was P squared is equal to 81, I kinda know what's going on. Because I know that nine times nine is equal to 81. Or we could write that nine squared is equal to 81, or we could write that nine is equal to the principal root of 81. These are all, I guess, saying the same truth about the universe, but what about 0.81? Well 0.81 has two digits behind, to the right of the decimal and so if I were to multiply something that has one digit to the right of the decimal times itself, I'm gonna have something with two digits to the right of the decimal. And so what happens if I take, instead of nine squared, what happens if I take 0.9 squared? Let me try that out. Zero, I'm gonna use a different color. So let's say I took 0.9 squared. 0.9 squared, well that's going to be 0.9 times 0.9, which is going to be equal to? Well nine times nine is 81, and I have one, two, numbers to the right of the decimal, so I'm gonna have two numbers to the right of the decimal in the product. So one, two. So that indeed is equal to 0.81. In fact we could write 0.81 as 0.9 squared. So we could write this, we could write that P is equal to the plus or minus, the square root of, instead of writing 0.81, I could write that as 0.9 squared. In fact I could also write that as negative 0.9 squared. Cause if you put a negative here and a negative here, it's still not going to change the value. A negative times a negative is going to be a positive. I could, actually I would have put a negative there, which would have implied a negative here and a negative there. So either of those are going to be true. But it's going to work out for us because we are taking the positive and negative square root. So this is going to be, P is going to be equal to plus or minus 0.9. Plus or minus 0.9, or we could write it that P is equal to 0.9, or P could be equal to negative 0.9. And you can verify that, you would square either of these things, you get 0.81. | {"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": 0.8719004392623901, "perplexity": 260.55695709388186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662564830.55/warc/CC-MAIN-20220524045003-20220524075003-00320.warc.gz"} |
https://www.tyrrell4innovation.ca/tag/cosine-distance/ | ## MiWORD of the Day Is…Cosine Distance!
Today we will talk about a way to measure distance, but not about how far away two objects are. Instead, cosine distance, or cosine similarity, is a measure of how similar two non-zero vectors are in terms of orientation, or to put it simply, the direction to which they point. Mathematically, the cosine similarity between two 2-D vectors is equal to the cosine of the angle between them, which can also be calculated using their dot product and magnitudes, as shown on the right. Two vectors pointing in the same direction will have a cosine similarity of 1; two vectors perpendicular to each other will have a similarity of 0; two vectors pointing in opposite direction will have a similarity of -1. Cosine distance is equal to (1 – cosine similarity). In this case, two vectors will have a cosine distance between 0 to 2: 0 when they are pointing in the same direction, and 2 when they are pointing in opposite direction. Cosine similarity and distance essentially measure the same thing, but the distance will convert any negative values to positive.
Cosine distance and similarity also apply to higher dimensions, which makes them useful in analyzing images, texts, and other forms of data. In machine learning, we can use an algorithm to process a dataset of information and store each object as an array of multidimensional vectors, where each vector represents a feature. Then, we can use cosine similarity to compare how similar each pair of vectors are between the two objects and come up with an overall similarity score. In this case, two identical objects will have a similarity score of 1. In higher dimensions, we can rely on the computer to do the calculations for us. For example, we have the distance.cosine function in the SciPy package in Python will compute the cosine distance between two vector arrays in one go.
Here are two examples of how you can use cosine distance in a conversation:
Serious: “I copied an entire essay for my assignment and this online plagiarizing checker says my similarity score is only 1! Time to hand it in.” “It says a COSINE similarity of 1. Please go back and write it yourself…”
Less serious: *during a police car chase* “Check how far are we from the suspect’s car!” “Well, assuming that he doesn’t turn, the distance between us will always be zero. Remember from your math class? Two vectors pointing in the same direction will always have a cosine distance of zero…”
… I’ll see you in the blogosphere.
Jenny Du | {"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": 0.883958101272583, "perplexity": 333.46227094086987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710829.5/warc/CC-MAIN-20221201153700-20221201183700-00030.warc.gz"} |
http://en.wikipedia.org/wiki/Loop_(algebra) | # Quasigroup
(Redirected from Loop (algebra))
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative.
A quasigroup with an identity element is called a loop.
## Definitions
There are at least two equivalent formal definitions of quasigroup. One definition casts quasigroups as a set with one binary operation, and the other is a version from universal algebra which describes a quasigroup by using three primitive operations. We begin with the first definition, which is easier to follow.
A quasigroup (Q, *) is a set Q with a binary operation * (that is, a magma), obeying the Latin square property. This states that, for each a and b in Q, there exist unique elements x and y in Q such that:
• a * x = b ;
• y * a = b .
(In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup is a Latin square.)
The unique solutions to these equations are written x = a \ b and y = b / a. The operations '\' and '/' are called, respectively, left and right division.
The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.[1][2]
### Universal algebra
Given some algebraic structure, an identity is an equation in which all variables are tacitly universally quantified, and in which all operations are among the primitive operations proper to the structure. Algebraic structures axiomatized solely by identities are called varieties. Many standard results in universal algebra hold only for varieties. Quasigroups are varieties if left and right division are taken as primitive.
A quasigroup (Q, *, \, /) is a type (2,2,2) algebra satisfying the identities:
• y = x * (x \ y) ;
• y = x \ (x * y) ;
• y = (y / x) * x ;
• y = (y * x) / x .
Hence if (Q, *) is a quasigroup according to the first definition, then (Q, *, \, /) is the same quasigroup in the sense of universal algebra.
### Loop
A loop is a quasigroup with an identity element, that is, an element e such that:
• x * e = x and e * x = x for all x in Q.
It follows that the identity element e is unique, and that every element of Q has a unique left and right inverse. Since the presence of an identity element is essential, a loop cannot be empty.
A Moufang loop is a loop that satisfies the Moufang identity:
• (x * y) * (z * x) = x * ((y * z) * x) .
### Total antisymmetry
A quasigroup (Q, ∗) is called totally anti-symmetric if for all c, x, yQ, the following implications hold:[3]
1. (cx) ∗ y = (cy) ∗ xx = y
2. xy = yxx = y,
and it is called weakly totally anti-symmetric if only the first implication holds.[3]
This property is required, for example, in the Damm algorithm.
## Examples
• Every group is a loop, because a * x = b if and only if x = a−1 * b, and y * a = b if and only if y = b * a−1.
• The integers Z with subtraction (−) form a quasigroup.
• The nonzero rationals Q× (or the nonzero reals R×) with division (÷) form a quasigroup.
• Any vector space over a field of characteristic not equal to 2 forms an idempotent, commutative quasigroup under the operation x * y = (x + y) / 2.
• Every Steiner triple system defines an idempotent, commutative quasigroup: a * b is the third element of the triple containing a and b. These quasigroups also satisfy (x * y) * y = x for all x and y in the quasigroup. These quasigroups are known as Steiner quasigroups.[4]
• The set {±1, ±i, ±j, ±k} where ii = jj = kk = +1 and with all other products as in the quaternion group forms a nonassociative loop of order 8. See hyperbolic quaternions for its application. (The hyperbolic quaternions themselves do not form a loop or quasigroup).
• The nonzero octonions form a nonassociative loop under multiplication. The octonions are a special type of loop known as a Moufang loop.
• An associative quasigroup is either empty or is a group, since if there is at least one element, the existence of inverses and associativity imply the existence of an identity.
• The following construction is due to Hans Zassenhaus. On the underlying set of the four-dimensional vector space F4 over the 3-element Galois field F = Z/3Z define
(x1, x2, x3, x4) * (y1, y2, y3, y4) = (x1, x2, x3, x4) + (y1, y2, y3, y4) + (0, 0, 0, (x3y3)(x1y2x2y1)).
Then, (F4, *) is a commutative Moufang loop that is not a group.[5]
• More generally, the set of nonzero elements of any division algebra form a quasigroup.
## Properties
In the remainder of the article we shall denote quasigroup multiplication simply by juxtaposition.
Quasigroups have the cancellation property: if ab = ac, then b = c. This follows from the uniqueness of left division of ab or ac by a. Similarly, if ba = ca, then b = c.
### Multiplication operators
The definition of a quasigroup can be treated as conditions on the left and right multiplication operators L(x), R(y): QQ, defined by
\begin{align} L(x)y &= xy \\ R(x)y &= yx \\ \end{align}
The definition says that both mappings are bijections from Q to itself. A magma Q is a quasigroup precisely when all these operators, for every x in Q, are bijective. The inverse mappings are left and right division, that is,
\begin{align} L(x)^{-1}y &= x\backslash y \\ R(x)^{-1}y &= y/x \end{align}
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section on universal algebra) are
\begin{align} L(x)L(x)^{-1} &= 1\qquad&\text{corresponding to}\qquad x(x\backslash y) &= y \\ L(x)^{-1}L(x) &= 1\qquad&\text{corresponding to}\qquad x\backslash(xy) &= y \\ R(x)R(x)^{-1} &= 1\qquad&\text{corresponding to}\qquad (y/x)x &= y \\ R(x)^{-1}R(x) &= 1\qquad&\text{corresponding to}\qquad (yx)/x &= y \end{align}
where 1 denotes the identity mapping on Q.
### Latin squares
The multiplication table of a finite quasigroup is a Latin square: an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. See small Latin squares and quasigroups.
### Inverse properties
Every loop element has a unique left and right inverse given by
$x^{\lambda} = e/x \qquad x^{\lambda}x = e$
$x^{\rho} = x\backslash e \qquad xx^{\rho} = e$
A loop is said to have (two-sided) inverses if $x^{\lambda} = x^{\rho}$ for all x. In this case the inverse element is usually denoted by $x^{-1}$.
There are some stronger notions of inverses in loops which are often useful:
• A loop has the left inverse property if $x^{\lambda}(xy) = y$ for all $x$ and $y$. Equivalently, $L(x)^{-1} = L(x^{\lambda})$ or $x\backslash y = x^{\lambda}y$.
• A loop has the right inverse property if $(yx)x^{\rho} = y$ for all $x$ and $y$. Equivalently, $R(x)^{-1} = R(x^{\rho})$ or $y/x = yx^{\rho}$.
• A loop has the antiautomorphic inverse property if $(xy)^{\lambda} = y^{\lambda}x^{\lambda}$ or, equivalently, if $(xy)^{\rho} = y^{\rho}x^{\rho}$.
• A loop has the weak inverse property when $(xy)z = e$ if and only if $x(yz) = e$. This may be stated in terms of inverses via $(xy)^{\lambda}x = y^{\lambda}$ or equivalently $x(yx)^{\rho} = y^{\rho}$.
A loop has the inverse property if it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop which satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
Any loop which satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
## Morphisms
A quasigroup or loop homomorphism is a map f : QP between two quasigroups such that f(xy) = f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
### Homotopy and isotopy
Main article: Isotopy of loops
Let Q and P be quasigroups. A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that
$\alpha(x)\beta(y) = \gamma(xy)\,$
for all x, y in Q. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
An isotopy is a homotopy for which each of the three maps (α, β, γ) is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ.
An autotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup.
Each quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup which is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by (x + y)/2 is isotopic to the additive group (R, +), but is not itself a group. Every medial quasigroup is isotopic to an abelian group by the Bruck–Toyoda theorem.
### Conjugation (parastrophe)
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation * (i.e., x * y = z) we can form five new operations: x o y := y * x (the opposite operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of *. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
### Paratopy
If the set Q has two quasigroup operations, * and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be paratopic to each other. There are also many other names for this relation of "paratopy", e.g., isostrophe.
## Generalizations
An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: QnQ, such that the equation f(x1,...,xn) = y has a unique solution for any one variable if all the other n variables are specified arbitrarily. Polyadic or multiary means n-ary for some nonnegative integer n.
A 0-ary, or nullary, quasigroup is just a constant element of Q. A 1-ary, or unary, quasigroup is a bijection of Q to itself. A binary, or 2-ary, quasigroup is an ordinary quasigroup.
An example of a multiary quasigroup is an iterated group operation, y = x1 · x2 · ··· · xn; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.
There exist multiary quasigroups that cannot be represented in any of these ways. An n-ary quasigroup is irreducible if its operation cannot be factored into the composition of two operations in the following way:
$f(x_1,\dots,x_n) = g(x_1,\dots,x_{i-1},\,h(x_i,\dots,x_j),\,x_{j+1},\dots,x_n),$
where 1 ≤ i < jn and (i, j) ≠ (1, n). Finite irreducible n-ary quasigroups exist for all n > 2; see Akivis and Goldberg (2001) for details.
An n-ary quasigroup with an n-ary version of associativity is called an n-ary group.
### Right- and left-quasigroups
A right-quasigroup (Q, *, /) is a type (2,2) algebra satisfying the identities:
• y = (y / x) * x;
• y = (y * x) / x.
Similarly, a left-quasigroup (Q, *, \) is a type (2,2) algebra satisfying the identities:
• y = x * (x \ y);
• y = x \ (x * y).
## Number of small quasigroups and loops
The number of isomorphism classes of small quasigroups (sequence A057991 in OEIS) and loops (sequence A057771 in OEIS) is given here:[6]
Order Number of quasigroups Number of loops
0 1 0
1 1 1
2 1 1
3 5 1
4 35 2
5 1,411 6
6 1,130,531 109
7 12,198,455,835 23,746
8 2,697,818,331,680,661 106,228,849
9 15,224,734,061,438,247,321,497 9,365,022,303,540
10 2,750,892,211,809,150,446,995,735,533,513 20,890,436,195,945,769,617
11 19,464,657,391,668,924,966,791,023,043,937,578,299,025 1,478,157,455,158,044,452,849,321,016
## Notes
1. ^ Hala O. Pflugfelder (1990). Quasigroups and loops: introduction. Heldermann Verlag. p. 2.
2. ^ Bruck, Richard Hubert (1971), A survey of binary systems, Springer, p. 1, ISBN 0-387-03497-8
3. ^ a b Damm, H. Michael (2007). "Totally anti-symmetric quasigroups for all orders n≠2,6". Discrete Mathematics 307 (6): 715–729. doi:10.1016/j.disc.2006.05.033. ISSN 0012-365X.
4. ^ Colbourn & Dinitz 2007, pg. 497, definition 28.12
5. ^ Smith, Jonathan D. H.; Romanowska, Anna B. (1999), "Example 4.1.3 (Zassenhaus's Commutative Moufang Loop)", Post-modern algebra, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, p. 93, doi:10.1002/9781118032589, ISBN 0-471-12738-8, MR 1673047.
6. ^ McKay, Brendan D.; Meynert, Alison; Myrvold, Wendy (2007). "Small Latin squares, quasigroups, and loops". J. Comb. Des. 15 (2): 98–119. doi:10.1002/jcd.20105. Zbl 1112.05018.
## References
• Akivis, M. A., and Vladislav V. Goldberg (2001), "Solution of Belousov's problem," Discussiones Mathematicae. General Algebra and Applications 21: 93–103.
• Bruck, R.H. (1958), A Survey of Binary Systems. Springer-Verlag.
• Chein, O., H. O. Pflugfelder, and J.D.H. Smith, eds. (1990), Quasigroups and Loops: Theory and Applications. Berlin: Heldermann. ISBN 3-88538-008-0.
• Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd Edition ed.), Boca Raton: Chapman & Hall/ CRC, ISBN 1-58488-506-8
• Dudek, W.A., and Glazek, K. (2008), "Around the Hosszu-Gluskin Theorem for n-ary groups," Discrete Math. 308: 4861-4876.
• Pflugfelder, H.O. (1990), Quasigroups and Loops: Introduction. Berlin: Heldermann. ISBN 3-88538-007-2.
• Smith, J.D.H. (2007), An Introduction to Quasigroups and their Representations. Chapman & Hall/CRC Press. ISBN 1-58488-537-8.
• Smith, J.D.H. and Anna B. Romanowska (1999), Post-Modern Algebra. Wiley-Interscience. ISBN 0-471-12738-8. | {"extraction_info": {"found_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": 25, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9810258746147156, "perplexity": 1207.2132528868322}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657141651.17/warc/CC-MAIN-20140914011221-00229-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/31651-polar-curve-area-print.html | # Polar Curve Area
• March 21st 2008, 03:42 PM
thegame189
Polar Curve Area
Find the area inside both r=5sin(2theta) and r=5cos(2theta).
I have NO idea how to do it. My book has no examples and I couldn't find anything online.
I ended up trying to do the integral of (5sin(2theta))^2, evaluated from 0 to Pi/8 + the integral of (5cos(2theta))^2 evaluated from 0 to Pi/8 and then multiplying the whole thing by 8 because of symmetry. I don't believe that to be correct.
Any explanations would be great!
• March 21st 2008, 04:34 PM
galactus
One point the curves intersect in at Pi/8. If we find the area inside
$5sin(2{\theta})$ from 0 to Pi/8 and multiply by 16, we can find the area of all
the regions inside the two functions.
$8\int_{0}^{\frac{\pi}{8}}\left[5sin(2{\theta})\right]^{2}d{\theta}$
• March 21st 2008, 04:49 PM
thegame189
First, I don't understand why you disregard the 5cos(2theta) completely. Why do you only use the sin equation?
I also don't understand why you multiply by 16. There are only 8 sections that are in BOTH graphs, so wouldn't you multiply by 8?
Edit: Ah, I see you fixed it. I was wondering why it wasn't squared. Thanks for adding a graph, too. | {"extraction_info": {"found_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": 0.9210138320922852, "perplexity": 645.1436419601646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1461860124045.24/warc/CC-MAIN-20160428161524-00056-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1705308/differentiation-and-integration?noredirect=1 | # Differentiation and integration
Which came first : Differentiation or Integration? If one of them was developed to solve certain types of problems, was the other developed for backward compatibility, or was it an independent development and later discovered that they were inverses?
Also, how were they discovered to be linked?
It was just something that I was thinking, and I would like to clarify it.
(I see that this is related : History of differential and integral calculus, but no answer is present)
• Answering questions like this comes down to what you mean exactly by differentiation and integration. As stated in the question you linked (and elsewhere) the ancient Greeks and Romans used techniques like the method of exhaustion. Is this a proto-integration technique? Is it a proto-limits technique? If you are meaning our modern form linked to functions then you'll get a different date/answer. – Ian Miller Mar 20 '16 at 5:58
• Not an answer, but I do know that they developed more or less independently of each other. For a long time, the connection between the two was not obvious, whereas today they are always taught as being intimately related by the fundamental theorem. Integration developed (at least in part) with the very geometric goal of finding areas under curves. – Elliot G Mar 20 '16 at 5:58
• Check out the website below. It seems like the short answer is that integration came first (in the form of areas under curves) and differentiation later (in the form of tangent lines to curves). math.ucdavis.edu/~temple/MAT16A/ArticlesOnCalculus16A/… – Elliot G Mar 20 '16 at 6:01
In a nutshell, the calculus was "discovered" or "invented" during 17th century independently by Leibniz and Newton who merged brilliantly various techniques developed since ancient Greece to solve geometrical problems.
Following the development of algebra during the Reanaissance and the pubblication of Descartes' Geometry in 1637, those methods were improved and new ones were discovered:
• drawing the normal to a curve: Descartes, Hudde
• finding tangents: Roberval, Fermat
• finding maxima and minima of curves: Fermat
• the method of indivisibles: Cvalieri
• arithmetical methods of integration: Wallis.
See:
The "official" birth of the calculus must be dated with Newton (De analysi of 1669) and Leibniz (various Ms. of 1675) independent developments:
Leibniz's differential and integral calculus and Newton's fluxional calculus, though different in many aspects, each involve a clear recognition of what we now call the inverse relationship between differentiation and integration. Moreover, both men worked out a system of notations, symbols and rules through which their methods could be applied in the form of algorithms performed on formulae, rather than in the form of geometrical arguments presented in prose with reference to figures.
See into: Ivor Grattan-Guinness, cit., Ch.2 Newton, Leibniz and the Leibnizian Tradition, by H.J.M. Bos, page 49-on
This is similar to the chicken-and-the-egg question though relation is not as obvious in the case of integration-and-differentiation. Namely, both types of problems were already studied by Archimedes (areas of figures and volumes of solids), Apollonius (tangents to curves), and others thousands of years ago. However the inverse relationship between the integration and differentiation was not understood until Barrow (some say even earlier by James Gregory), Newton, and Leibniz. | {"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": 0.9168841242790222, "perplexity": 1148.4531694563857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1566027320734.85/warc/CC-MAIN-20190824105853-20190824131853-00514.warc.gz"} |
https://www.physicsforums.com/threads/relativistic-momentum-definition.119162/ | # Relativistic momentum definition
1. Apr 29, 2006
### bernhard.rothenstein
many textbooks start teaching relativistic dynamics by defining the relativistic momentum as
p=dx/dtau
2. Apr 29, 2006
### Meir Achuz
It should be p=m dx/dtau.
U^i=dx^i/dtau are the three spatial components of a 4-vector.
The basic differential space-time 4-vector is dx^mu=(dt;dx,dy,dz)
Its invariant length squared (a 4-scalar) is
dtau^2=dt^2-dx^2-dy^2-dz^2.
To get a 4-vector that behaves like a velocity, the 4-vector is divided by the scalar: U^mu=dx^mu/dtau.
Then a 4-vector momentum is just p^mu=mU^mu. It's x,y,z components are seen to behave for small v like the NR momentum. Its time-like component is the energy.
3. Apr 29, 2006
### bernhard.rothenstein
the authors i mention say "Mermin It's about time]and many others ar far from using four vectors as far as I am. Do ou now an other explanation or it is a simple guess work?
4. Apr 29, 2006
### nrqed
The justification for defining a momentum vector is the same as in classical physics, as far as I know. It is a quantity that is conserved in a collision. This is *the* reason for defining it this way. After all one could define a vector $m {\vec a }$ in Newtonian physics or $m {d^2 x\over d \tau^2 }$ in SR? Of course these could be defined but they would be useless quantities.
EDIT I meant ''One could define a vector $m^2 {\vec a }$ in Newtonian physics or $m^2 {d^2 x\over d \tau^2 }$ in SR but this would be useless...''
Last edited: Apr 29, 2006
5. Apr 29, 2006
### bernhard.rothenstein
you think that we should start with the classical defnition of the momentum p=mdx/dt and to search what should we do in order to bring it in accordance with special relativity? but the author I quote start with
p=mdx?dtau without to offer an explanation for it.
6. Apr 29, 2006
Staff Emeritus
Taylor and Wheeler, in their Spacetime Physics, do specifically derive fourmomentum from collisions under relativistic conditions.
7. Apr 29, 2006
### nrqed
I think that using collisions and looking for a quantity that is conserved is the *logical* way to introduce the expression for relativistic momentum. If the author just quotes the final result it's because he/she did not want to take the time to justify where it comes from.
exactly the same thing happens in Newtonian physics, you know. Why does one introduce this strange vector $m {\vec v}$?? it is because one can show that the total momentum is conserved in a collision. If that was not the case, one would never introduce this quantity in the first place.
Regards
Patrick
8. Apr 29, 2006
### nrqed
it depends what you mean by ''bringing in accordance with SR''. If you just use the argument ''just replaces dt by dtau'' to get to SR, I would say this is unsatisfactory (my personal opinion)
But if the motivation is ''now let's look at a collision in different frames taking into account the Lorentz transformations..then we see that $m {\vec v }$ is not conserved in a collision. Can we figure out a quantity which *is* conserved? (and that quantity will reduce to the usual expression in the nonrelativistic limit). Yes, and here it is.
That's all my personal preference. If I was teaching it, this is the way I would introduce the concept.
9. Apr 29, 2006
### Staff: Mentor
And this is exactly what Mermin takes great pains (and several pages) to do in his nice little book "It's About Time". He certainly doesn't just state the relativistic definition and forget it.
To bernhard.rothenstein: What book were you quoting? The only one you mentioned was Mermin's. (True, Mermin doesn't mention 4-vectors, but it's meant as a popular exposition.)
10. Apr 29, 2006
### bernhard.rothenstein
I quote from Jon Ogborn Introducing special relativity Physics Education 40 (3) 213 2005
In Newtonian mechanics momentum p=mdx/dt. The relativistic idea is that the right way to clock the motion of a particle is to use the proper (or "wristwatch time tau. So Einstein replaced dt by dtau and redefined momentum as p=mdx/dtau. The velocity v is still dx/dt so in the new definition p=m(dx/dt)(dt/dtau). We know that t=gamatau so dt/dtau. Thus the newly defined momentum is given by p=gamamv.
Is that guesswork or intuition?
11. Apr 29, 2006
### Staff: Mentor
I would say brilliant intuition. But I think most of us here agree that just stating that is not sufficient, certainly not if teaching is your goal. I was just responding to a perceived slight against Mermin's book. (Mermin is an pedagogical master!)
12. Apr 29, 2006
### bernhard.rothenstein
I think in the same way. where could I find your response?
13. Apr 29, 2006
### robphy
In introductory physics, momentum is usually introduced by the impulse-momentum theorem... where it appears in an interesting difference of a quantity characterizing the system before and after the impulse. (Kinetic energy is introduced in a similar way in the work-kineticEnergy theorem.) The discussion of conservation laws comes later. Based on this, it would be great if the relativistic motivation retold this story. Otherwise, I'd suggest that the galilean-newtonian story be rewritten to parallel the relativistic one.
From a top-down view, I'd look at the more abstract interpretations of momentum (say, from a Lagrangian or Hamiltonian view), then try to formulate a pedagogical story that could be paralleled in the galilean-newtonian and relativistic cases.
14. May 3, 2006
### bernhard.rothenstein
15. May 4, 2006
### robphy
The "impulse-momentum theorem" essentially says $$\int_{t_i}^{t_f} dt \vec F_{net}=\vec p_f - \vec p_i$$, where the left-hand side is called the impulse. When introducing momentum for the first time [given Newton's Law in the constant-mass form $$\vec F_{net}=m\frac{d\vec v}{dt}$$ for a point particle], I carry out the integration of the left and define momentum as an interesting quantity associated with the system before and after the impulse.
Looking over the relatively new introductory books on my shelf, I see that most books actually state a definition of $$\vec p=m\vec v$$ first, then possibly discuss its conservation, before doing the impulse-momentum theorem. Maybe the approach I used comes from [an old edition of?] Halliday-Resnick? I can't find my [old] copy now.
In any case, my motivation is that it parallels the work-[kinetic-]energy theorem, which [I think] is usually introduced before defining kinetic energy and before discussing any notion of energy conservation.
Until I can find my reference: http://www.google.com/search?q=+impulse+momentum+theorem | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9156693816184998, "perplexity": 1589.5988758449785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280834.29/warc/CC-MAIN-20170116095120-00013-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.conserve-energy-future.com/Advantages_NuclearEnergy.php | # Nuclear Energy
Energy cannot be created nor be destroyed but it can be converted from one form to another. Nearly all the mass of the atom is concentrated in a tiny nucleus in the center. The nucleus is composed principally of two sorts of particles: the proton which carries the positive charge and the neutron which is electrically neutral and has a mass slightly bigger than that of proton. Nuclear energy is the energy released from the nucleus of an atom. When nuclear reaction occurs weather fission or fusion, it produces large amount of energy.
### How Nuclear Energy is Produced
When the heaviest element, uranium was bombarded with neutrons, it was discovered that instead of inducing radioactivity as did other elements, something different happened. This process was named fission. When fission occurred, not only were two lighter elements and a lot of radiation produced, but also more neutrons. It was clear that these neutrons could in turn also cause fission, producing more neutrons and developing a chain reaction which might spread throughout all the uranium present.
In the fission of uranium 235 nucleus, the amount of energy released is about 60,000,000 times as much as when a carbon atom burns. Most of the energy from fission appears as kinetic energy as the fission products shoot apart and quickly share their energy with their surroundings, thus producing heat. The first reactors to produce a usable amount of power were built at Calder hall in England.
With pure fissionable material, atomic bombs can be made. Of the two bombs dropped on Japan to end the World War 2, one contained plutonium and the other very highly enriched uranium 235.
1. Lower Greenhouse Gas Emissions : As per the reports in 1998, it has been calculated the emission of the greenhouse gas has reduced for nearly half due to the popularity in the use of nuclear power. Nuclear energy by far has the lowest impact on the environment since it does not releases any gases like carbon dioxide, methane which are largely responsible for greenhouse effect. There is no adverse effect on water, land or any habitats due to the use of it. Though some greenhouse gases are released while transporting fuel or extracting energy from uranium.
2. Powerful and Efficient : The other main advantage of using nuclear energy is that it is very powerful and efficient than other alternative energy sources. Advancement in technologies has made it more viable option than others. This is one the reason that many countries are putting huge investments in nuclear power. At present, a small portion of world’s electricity comes through it.
3. Reliable : Unlike traditional sources of energy like solar and wind which require sun or wind to produce electricity, nuclear energy can be produced from nuclear power plants even in the cases of rough weather conditions. They can produce power 24/7 and need to be shut down for maintenance purposes only.
4. Cheap Electricity : The cost of uranium which is used as a fuel in generating electricity is quite low. Also, set up costs of nuclear power plants is relatively high while running cost is low. The average life of nuclear reactor range from 4.-60 years depending upon its usage. These factors when combined make the cost of producing electricity very low. Even if the cost of uranium rises, the increase in cost of electricity will be much lower.
5. Low Fuel Cost : The main reason behind the low fuel cost is that it requires little amount of uranium to produce energy. When a nuclear reaction happens, it releases million times more energy as compared to traditional sources of energy.
6. Supply : There are certain economic advantages in setting up nuclear power plants and using nuclear energy in place of conventional energy. It is one of the major sources of electricity throughout the nation. The best part is that this energy has a continuous supply. It is widely available, has huge reserves and expected to last for another 100 years while coal, oil and natural gas are limited and are expected to vanish soon.
7. Easy Transportation : Production of nuclear energy needs very less amount of raw material. This means that only about 28 gram of uranium releases as much energy as produced from 100 metric tons of coal. Since it’s required in small quantities, transportation of fuel is much easier than fossil fuels. Optimal utilization of natural resources in production of energy is a very thoughtful approach for any nation. It not only enhances the socio-economic condition but also sets example for the other countries.
No doubt, nuclear energy has made its way towards the future but like other sources of energy, it also suffers from some serious drawbacks. Let’s take a look at some of its disadvantages.
### Rinkesh
Rinkesh is passionate about clean and green energy. He is running this site since 2009 and writes on various environmental and renewable energy related topics. He lives a green lifestyle and is often looking for ways to improve the environment around him. | {"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": 0.8580474257469177, "perplexity": 766.234767479289}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783402699.36/warc/CC-MAIN-20160624155002-00017-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.hepdata.net/record/82145 | Precision measurement and interpretation of inclusive $W^+$, $W^-$ and $Z/\gamma^*$ production cross sections with the ATLAS detector
The collaboration
Eur.Phys.J.C 77 (2017) 367, 2017.
Abstract (data abstract)
CERN-LHC. High-precision measurements by the ATLAS Collaboration are presented of inclusive $W^+\to\ell^+\nu$, $W^-\to\ell^-\bar{\nu}$ and $Z/\gamma^*\to\ell\ell$ ($\ell=e,\mu$) Drell--Yan production cross sections at the LHC. The data were collected in proton--proton collisions at $\sqrt{s} = 7$ TeV with an integrated luminosity of 4.6 fb$^{-1}$. Differential $W^+$ and $W^-$ cross sections are measured in a lepton pseudorapidity range $|\eta_{\ell}| = 2.5$. Differential $Z/\gamma^*$ cross sections are measured as a function of the absolute dilepton rapidity, for $|y_{\ell\ell}| < 3.6$, for three intervals of dilepton mass, $m_{\ell\ell}$, extending from 46 to 150 GeV. The integrated and differential electron- and muon-channel cross sections are combined and compared to theoretical predictions using recent sets of parton distribution functions. The data, together with the final inclusive $e^{\pm}p$ scattering cross-section data from H1 and ZEUS, are interpreted in a next-to-next-to-leading-order QCD analysis, and a new set of parton distribution functions, ATLAS-epWZ16, is obtained. The ratio of strange-to-light sea-quark densities in the proton is determined more accurately than in previous determinations based on collider data only, and is established to be close to unity in the sensitivity range of the data. A new measurement of the CKM matrix element $|V_{cs}|$ is also provided. The differential cross sections are provided in two forms, a version with summarised uncertainties as in the paper tables and as "attached resource" files in HerAverager/xFitter format detailing all sources of correlated uncertainties separately. In addition to the experimental results, detailed information on the theoretical predictions are provided as required to reproduce the interpretation. | {"extraction_info": {"found_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": 0.9804245233535767, "perplexity": 1031.1520460174302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710218.49/warc/CC-MAIN-20221127073607-20221127103607-00196.warc.gz"} |
https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro/2315350 | # Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it with us?
• @J.M. Thanks. But Euler could very well be a good tag I believe. – AD. Oct 30 '10 at 10:16
• Robin Chapman has a collection of proofs on his homepage: empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf – Hans Lundmark Oct 30 '10 at 10:32
• makes no sense to have an Euler tag... maybe Eulerian but that's pushing it. – anon Oct 30 '10 at 10:46
• Probably Robin should answer with a link to his note. I know I've pointed people to it when they ask precisely this, and they've always been more than satisfied! – Mariano Suárez-Álvarez Oct 30 '10 at 14:09
• What I like the most about this thread is that I know most of the proofs that I've seen posted up to this time, it makes me think that perhaps I was given adequate mathematical education after all :) – Asaf Karagila Nov 1 '10 at 12:57
There is a simple way of proving that $\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ using the following well-known series identity: $$\left(\sin^{-1}(x)\right)^{2} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n^2 \binom{2n}{n}}.$$ From the above equality, we have that $$x^2 = \frac{1}{2}\sum_{n=1}^{\infty}\frac{(2 \sin(x))^{2n}}{n^2 \binom{2n}{n}},$$ and we thus have that: $$\int_{0}^{\pi} x^2 dx = \frac{\pi^3}{12} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{\int_{0}^{\pi} (2 \sin(x))^{2n} dx}{n^2 \binom{2n}{n}}.$$ Since $$\int_{0}^{\pi} \left(\sin(x)\right)^{2n} dx = \frac{\sqrt{\pi} \ \Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n+1)},$$ we thus have that: $$\frac{\pi^3}{12} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{ 4^{n} \frac{\sqrt{\pi} \ \Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n+1)} }{n^2 \binom{2n}{n}}.$$ Simplifying the summand, we have that $$\frac{\pi^3}{12} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{\pi}{n^2},$$ and we thus have that $\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ as desired.
This is, by no measure, the best nor the simplest approach, but I think the approach is pretty peculiar.
We estimate the number $N(x)$ of integer solutions to $a^2+b^2+c^2+d^2\leq x$ as $x\rightarrow\infty$. On one hand, this is the number of lattice points inside the the $4$-ball of radius $\sqrt{x}$, which has volume $\frac{1}{2}\pi^2x^2$, hence $N(x)=\frac{\pi^2}{2}x^2+O(x^{3/2})$.
On the other hand, let $r_4(n)$ be the number of solutions to $a^2+b^2+c^2+d^2=n$. Following the derivation in the book by Iwaniec-Kowalski, by Jacobi's four-square identity we can write $$N(x)=\sum_{n\leq x}r_4(n)=8\sum_{m\leq x}(2+(-1)^m)\sum_{dm\leq x,d\text{ odd}} d \\ =8\sum_{m\leq x}(2+(-1)^m)\left(\frac{x^2}{4m^2}+O\left(\frac{x}{m}\right)\right)\\ =2x^2\sum_{m\leq x}(2+(-1)^m)m^{-2}+O(x\log x)\\ =3x^2\zeta(2)+O(x\log x)$$ (I have copied the steps as they were in the book, it's a neat exercise to justify every transition). In particular, we have $$\zeta(2)=\lim\limits_{x\rightarrow\infty}\frac{N(x)}{3x^2}=\frac{\pi^2}{6}.$$
• (+1) I wonder if one can do the same by only exploiting the fact that the average value of $r_2(n)$ is $\pi$ by Gauss circle problem. – Jack D'Aurizio Nov 9 '17 at 4:50
by using Fourier series of $$f(x)=1, x\in[0,1]$$ $$1=\sum_{n=1}^\infty\frac{4}{(2n-1)\pi}\sin (2n-1)\pi x$$ integrate both sides when integration limits are $$x=0 \rightarrow 1$$ $$\int_{0}^{1}1.dx=\int_{0}^{1} \sum_{n=1}^\infty\frac{4}{(2n-1)\pi}\sin (2n-1)\pi x dx$$ $$1=\sum_{n=1}^\infty\frac{8}{(2n-1)^2\pi^2}$$ $$\sum_{n=1}^\infty\frac{1}{(2n-1)^2}=\frac{\pi^2}{8}$$ then we use the equality series $$\sum_{n=1}^\infty\frac{1}{n^2}=\sum_{n=1}^\infty\frac{1}{(2n-1)^2}+\sum_{n=1}^\infty\frac{1}{(2n)^2}$$ simplify it to get $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{4}{3}\sum_{n=1}^\infty\frac{1}{(2n-1)^2}$$ so, $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{4}{3}\frac{\pi^2}{8}=\frac{\pi^2}{6}$$
Let $X$ be an independent Laplace random variable with $X\sim L(0,1) = \frac12 \exp{(-|x|)}$, then its characteristic function :
$$\varphi_X(t)=\mathbb{E}[e^{itX}]=\frac{1}{1+t^2} \newcommand{\var}[1]{\mathrm{var}\left[#1\right]}$$
By symmetry $\mathbb{E}[X]=0$ we write (generally) :
$$\varphi_X(t)=\mathbb{E}[e^{itX}]=\mathbb{E}[1+itX-t^2X^2+\cdots\,]=1-\var{X}t^2+O(t^3)\tag{A}$$
since $\var{X}=\mathbb{E}[X^2]-\mathbb{E}[X]^2=\mathbb{E}[X^2]$, for our case : $$\frac{1}{1+t^2}=1-t^2+O(t^3) \rightarrow \var{X}=1$$ Now consider a set of such variables $X_n$, independent of each other, for a construction of a new random variable $Y$ as follows :
$$Y=\sum_{n=1}^{\infty}\frac{X_n}{n}$$
Then, taking variance of both sides :
$$\var{Y}=\var{\sum_{n=1}^{\infty}\frac{X_n}{n}}=\sum_{n=1}^{\infty}\var{\frac{X_n}{n}}=\sum_{n=1}^{\infty}\frac{\var{X_n}}{n^2}=\var{X}\zeta(2)=\zeta(2)\tag{B}$$
However, for characteristic function instead, using properities of characteristic function :
$$\varphi_Y(t)=\varphi_{\sum_{n=1}^{\infty}X_n/n}\left(t\right)=\prod_{n=1}^\infty \varphi_{X}\left(\frac{t}{n}\right) = \prod_{n=1}^\infty \frac{1}{1+\frac{t^2}{n^2}}=\frac{\pi t}{\sinh \pi t} = 1-\frac{\pi^2}{6}t^2+O(t^3)\tag{C}$$
Combining it with $(A)$ and $(B)$ we get : $$\zeta(2)=\var{Y}=\frac{\pi^2}{6}$$
NOTE : As long as the set $\{X_n\}_{n\in\mathbb{N}}$ consist of independent variables with idential pdf. the steps are the same upto $\var{X}=1$. So, there might be distributions for which the product in $C$ is easily evaluable.
NOTE2 : by mystake I posted just a fisrst sentence of this answer, so after deleting, this is the second copy
I propose a solution... Consider for $n\in\mathbb N^*$ : $$(1) : \int_0^\pi \left(\alpha t+\beta t^2\right)\cos(nt)\,\mathrm dt = \dfrac{1}{n^2}$$ Integrate by parts : $$\int_0^\pi t\cos(nt)\,\mathrm dt = \underbrace{\left.\dfrac{t\sin(nt)}{n}\right\vert_0^\pi}_{=\,0} -\int_0^\pi \dfrac{\sin(nt)}{n}\,\mathrm dt = -\underbrace{\int_0^{n\pi}\dfrac{\sin x}{n^2}\,\mathrm dx}_{\mathrm{substitution\;by\;}x=nt} = \dfrac{\cos(n\pi)-1}{n^2}$$ and $$\begin{split}\int_0^\pi t^2\cos(nt)\,\mathrm dt &= \underbrace{\left.\dfrac{t^2\sin(nt)}{n}\right\vert_0^\pi}_{=\,0} - \int_0^\pi \dfrac{2t\sin(nt)}{n}\,\mathrm dt = \left.\dfrac{2t\cos(nt)}{n^2}\right\vert_0^\pi - \int_0^\pi \dfrac{2\cos(nt)}{n^2}\,\mathrm dt \\&= \dfrac{2\pi\cos(n\pi)}{n^2} - \underbrace{\int_0^{n\pi}\dfrac{2\cos x}{n^3}\,\mathrm dx}_{\mathrm{substition\;by\;}x=nt} = \dfrac{2\pi\cos(n\pi)}{n^2}- \underbrace{\left.\dfrac{2\sin x}{n^3}\right\vert_0^{n\pi}}_{=\,0} \\&=\dfrac{2\pi\cos(n\pi)}{n^2} \end{split}$$ Thus $$\int_0^\pi \left(\alpha t+\beta t^2\right)\cos(nt)\,\mathrm dt = \alpha \cdot \dfrac{\cos(n\pi)-1}{n^2} + \beta\cdot\dfrac{2\pi\cos(n\pi)}{n^2}$$ We deduce that $\alpha = -1$ and $\beta = 1/2\pi$ satisfies $(1)$.
Since for $x\in\mathbb R\backslash 2\pi\mathbb Z$ : $$\sum_{k=1}^n \cos(kx) =-\dfrac{1}{2} + \dfrac{\sin(nx+x/2)}{2\sin(x/2)}$$ we have $$\begin{split}\sum_{k=1}^n \dfrac{1}{k^2} &= \sum_{k=1}^n \int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\cos(kt)\,\mathrm dt \\&= \int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\sum_{k=1}^n \cos(kt)\,\mathrm dt\\ &= -\dfrac{1}{2}\int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\mathrm dt + \int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\cdot \dfrac{\sin(nt+t/2)}{2\sin(t/2)}\,\mathrm dt \end{split}$$ However $\sin(nt+t/2) = \sin(t/2)\cos(nt)+\sin(nt)\cos(t/2)$. Let $\phi$ and $\psi$ such that $$\phi(t) = \dfrac{t^2}{4\pi}-\dfrac{t}{2} \;\mathrm{and}\; \psi(t) = \left(\dfrac{t^2}{2\pi}-t\right)\cdot\dfrac{\cos(t/2)}{2\sin(t/2)}$$ so that $$\int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\cdot \dfrac{\sin(nt+t/2)}{2\sin(t/2)} = \int_0^\pi \phi(t)\cos(nt)\,\mathrm dt + \int_0^\pi \psi(t)\sin(nt)\,\mathrm dt$$ $\phi$ is continuous on $[0,\pi]$. And $\psi$ can be extended at $t=0$. Indeed as $t\to 0$ $$\psi(t) = \underbrace{\dfrac{t\cos(t/2)}{2\pi}}_{\to\, 0}\cdot\underbrace{\dfrac{t/2}{\sin(t/2)}}_{\to \ 1}- \underbrace{\cos(t/2)}_{\to \, 1}\cdot\underbrace{\dfrac{t/2}{\sin(t/2)}}_{\to\, 1} \xrightarrow[t\to 0]{} -1$$ Therefore, $\psi$ is continuous (by extension) on $[0,\pi]$.
There remains to apply Lebesgue-Riemann Lemma, which tells us that : $$\int_0^\pi \phi(t)\cos(nt)\,\mathrm dt \xrightarrow[n\to \infty]{} 0\;\;\mathrm{and}\;\;\int_0^\pi \psi(t)\sin(nt)\,\mathrm dt\xrightarrow[n\to \infty]{} 0$$ Consequently $$\int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\cdot \dfrac{\sin(nt+t/2)}{2\sin(t/2)} \xrightarrow[n\to \infty]{} 0$$ and $$\sum_{k=1}^n \dfrac{1}{k^2} \xrightarrow[n\to \infty]{} -\dfrac{1}{2}\int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\mathrm dt$$ Now, we can evaluate this integral : $$-\dfrac{1}{2}\int_0^\pi \left(\dfrac{t^2}{2\pi}-t\right)\mathrm dt = -\dfrac{1}{2}\left[\dfrac{t^3}{6\pi}-\dfrac{t^2}{2}\right]_0^\pi = -\dfrac{1}{2}\left[\dfrac{\pi^2}{6}-\dfrac{\pi^2}{2}\right] = \dfrac{\pi^2}{6}$$ Then... the desired result : $$\boxed{\sum_{k=1}^\infty \dfrac{1}{k^2} = \dfrac{\pi^2}{6}}$$
This is a similar proof as posted by Hans Lundmark, but I find it to be a little simpler. I ran across this approach in a Dover copy of The USSR Olympiad Problem Book. It is also based on the observation that $$\cot^2x<\frac{1}{x^2}<\csc^2x\,.$$
We first have the trig identity $$\sin(2n+1)\alpha=\sum_{k=0}^n(-1)^k\binom{2n+1}{2k+1}\cos^{2(n-k)}\alpha\sin^{2k+1}\alpha$$ which is arguably the hardest part of this proof. This directly manipulates into $$\sin(2n+1)\alpha=\sin^{2n+1}\alpha\sum_{k=0}^n(-1)^k\binom{2n+1}{2k+1}\cot^{2(n-k)}\alpha\,.$$ This formula reveals that the $n$ distinct quantities below $$\cot^2\frac{\pi}{2n+1},\quad\cot^2\frac{2\pi}{2n+1},\quad\ldots,\quad\cot^2\frac{n\pi}{2n+1}$$ are the roots of the polynomial $$\sum_{k=0}^n(-1)^k\binom{2n+1}{2k+1}x^{n-k}\,.$$ After scaling by the lead coefficient, Viete's Formulas then imply that $$\sum_{k=1}^n \cot^2\frac{k\pi}{2n+1}=\frac{n(2n-1)}{3}$$ By another elementary trig identity, we also get $$\sum_{k=1}^n \csc^2\frac{k\pi}{2n+1}=\frac{2n(n+1)}{3}$$ The inequality above then gives us $$\frac{n(2n-1)}{3}<\frac{(2n+1)^2}{\pi^2}\left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)<\frac{2n(n+1)}{3}$$ which gives us the desired conclusion after taking limits.
• I later found out that this is one of the three different proofs of the Basel Problem that was presented in "Proofs from the Book". – user123641 May 1 '18 at 15:54
I would like to present you a method I found recently here.
Let $A_n=\int_0^{\pi/2}\cos^{2n}x\;\mathrm{d}x$ and $B_n=\int_0^{\pi/2}x^2\cos^{2n}x\;\mathrm{d}x$.
The first integral is well known, by per partes we get the recurecnce relation :
$$A_{n}=\frac{2n-1}{2n}A_{n-1}\tag{1}$$
By per partes for the second integral:
$$A_n=\int_0^{\pi/2}\cos^{2n}x\;\mathrm{d}x=x\cos^{2n}x\bigg{|}_0^{\pi/2}-\frac{x^2}{2}(\cos^{2n}x)'\bigg{|}_0^{\pi/2}+\frac{1}{2}\int_0^{\pi/2}x^2(\cos^{2n}x)''\;\mathrm{d}x$$
First two terms vanish, so we are left only with the integral and since $(\cos^{2n}x)''=2n(2n-1)\cos^{2n-2}x-4n^2\cos^{2n}x$ we have :
$$A_n=(2n-1)nB_{n-1}-2n^2B_{n}\tag{2}$$
for $n\geq 1$. Rearranging and substituing $(2n-1)=2n\frac{A_n}{A_{n-1}}$ from $(1)$ we get :
$$\frac{1}{n^2}=2\frac{B_{n-1}}{A_{n-1}}-2\frac{B_n}{A_n}\tag{3}$$
Summing from $1$ to some $k$ natural we get by telescoping property
$$\sum_{n=1}^k\frac{1}{n^2}=2\frac{B_0}{A_0}-2\frac{B_k}{A_k}=\frac{\pi^2}{6}-2\frac{B_k}{A_k}\tag{4}$$
Next, using the inequality $\sin x\geq \frac{2x}{\pi}$ on $(0,\frac{\pi}{2})$ and by $(1)$ :
$$\frac{4}{\pi^2}B_{n-1}=\frac{4}{\pi^2}\int_0^{\pi/2}x^2\cos^{2n-2}x\;\mathrm{d}x<\int_0^{\pi/2}\sin^2x\cos^{2n-2}x\;\mathrm{d}x=A_{n-1}-A_n=\frac{A_{n-1}}{2n}$$
so in the limit the last term vanishes by the sqeeze theorem, so we are left with
$$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}\tag{4}$$
That concludes the result.
• I think you mean $\sin x\geq 2x/\pi$. But this is a clever approach. Another reason to appreciate integration by parts. – user123641 Jun 9 '17 at 0:51
• Thanks for sharing, very easy to follow and totally new to me. Fixed the typo mentioned by @Bryan. – AD. Jun 9 '17 at 10:11
Here is an interesting solution that evaluates three sums, one of which is $\zeta(2).$
$$\tag{1}\label{Double Integral} \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \frac{1}{\sqrt{x^2+y^2} \ (1+x^2+y^2)} \ dy \ dx.$$ A quick polar coordinates transformation $x=r \cos(\theta), y= r \sin(\theta)$ transforms \eqref{Double Integral} into $$\int_{0}^{\frac{\pi}{2}}\int_{0}^{1} \frac{1}{1+r^2} \ dr \ d \theta=\frac{\pi^2}{8}.$$
Hence, \eqref{Double Integral} is equal to $\frac{\pi^2}{8}.$
Now integrate \eqref{Double Integral} with respect to $y$ using the fact
$$\int \frac{1}{\sqrt{x^2+y^2} (1+x^2+y^2)} \ dy = \frac{\tanh^{-1} \left( \frac{y}{\sqrt{1+x^2}\sqrt{x^2+y^2}} \right)}{\sqrt{1+x^2}}$$ to see that \eqref{Double Integral} becomes
$$\tag{2} \label{arctanh} \int_{0}^{1} \frac{\tanh^{-1} \left( \frac{\sqrt{1-x^2}}{\sqrt{1+x^2}} \right)}{\sqrt{1+x^2}} \ dx.$$
Next, observe that \eqref{arctanh} is equal to the double integral
$$\tag{3} \label{double integral 2} \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \frac{1}{1+x^2-y^2} \ dy \ dx,$$ which can be confirmed by integrating the inner integral of \eqref{double integral 2} with respect to $y.$ Now here comes the interesting part. Use polar coordinates with $x=r\cos(\theta),y=r\sin(\theta)$ on \eqref{double integral 2} and then with $x=r\sin(\theta),y=r\cos(\theta)$ on \eqref{double integral 2} and average the two to see that \eqref{double integral 2} is the same as
$$\frac{1}{2}\int_{0}^{\frac{\pi}{2}} \int_{0}^{1} \frac{r}{1+r^2\cos(2\theta)} \ dr \ d \theta + \frac{1}{2}\int_{0}^{\frac{\pi}{2}} \int_{0}^{1} \frac{r}{1-r^2\cos(2\theta)} \ dr \ d \theta \,$$ which simplifies down to \begin{align} \int_{0}^{\frac{\pi}{2}} \frac{\ln(1+\cos(2\theta))}{4\cos(2\theta)}-\frac{\ln(1-\cos(2\theta))}{4\cos(2\theta)} \ d \theta & = \int_{0}^{\frac{\pi}{2}} \frac{\ln \left(\frac{1+\cos(2\theta)}{1-\cos(2\theta)} \right)}{4\cos(2\theta)} \ d \theta\\ & = \int_{0}^{\frac{\pi}{2}} -\frac{\ln(\tan^2(\theta))}{4\cos(2\theta)} \ d \theta \tag{4} \label{double angle} \\ & = \int_{0}^{\infty} \frac{\ln(u)}{2(u^2-1)} \ du \tag{5} \label{pi^2/4} \end{align} with \eqref{double angle} following from simplifying the logarithmic term with the double angle formulas $$\sin^2(\theta)=\frac{1-\cos(2\theta)}{2}, \quad \cos^2(\theta)=\frac{1+\cos(2\theta)}{2},$$ and \eqref{pi^2/4} following from the substitution $u=\tan(\theta).$
Splitting \eqref{pi^2/4} into $$\int_{0}^{1} \frac{\ln(u)}{2(u^2-1)} \ du + \int_{1}^{\infty} \frac{\ln(u)}{2(u^2-1)} \ du,$$ a substitution $u=\frac{1}{t}$ on the second term shows \eqref{pi^2/4} is equal to $$2 \int_{0}^{1} \frac{\ln(u)}{2(u^2-1)} \ du = \int_{0}^{1} \frac{\ln(u)}{u^2-1} \ du.$$ Hence, we have \begin{align} \tag{6} \label{pi^2/8} \int_{0}^{1} \frac{\ln(u)}{u^2-1} \ du = \frac{\pi^2}{8} \end{align}
Now following the other users' answers, convert the integrand in the left hand side of \eqref{pi^2/8} into a geometric series, apply the Monotone Convergence Theorem to interchange sum and integral, to see we have $$\sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} = \frac{\pi^2}{8},$$ and observing \begin{align} \zeta(2) & =\sum_{n=1}^{\infty} \frac{1}{(2n)^2}+ \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} \\ & = \frac{1}{4} \zeta(2) + \frac{\pi^2}{8}, \end{align} we see $$\zeta(2)=\frac{\pi^2}{6}.$$
Those are the first two sums. We refer back to \eqref{arctanh}. Make the substitution $u=\sqrt{\frac{{1-x^2}}{{1+x^2}} }$ and simplify to see \eqref{arctanh} becomes to get: $$\tag{7} \label{complicated sub} \int_{0}^{1} \frac{\sqrt{2} u\tanh^{-1}(u)}{\sqrt{1-u^2}(1+u^2)}\ du.$$ Substituting $u=\tanh(\theta)$ transforms \eqref{complicated sub} into \begin{align} \sqrt{2}\int_{0}^{\infty}\frac{\theta(e^{2\theta}-1)}{e^{4\theta}+1}\ d\theta \end{align} and substituting $z=e^{\theta}$ shows that \eqref{arctanh} is the same as $$\sqrt{2}\int_{1}^{\infty}\frac{(z^2-1)\ln(z)}{z^4+1}\ dz,$$ and splitting the region of integration as with \eqref{pi^2/4} to get \eqref{pi^2/8}, we see \eqref{arctanh} is
$$\tag{7} \label {crazy integral} \sqrt{2}\int_{0}^{1}\frac{(t^2-1)\ln(t)}{t^4+1}\ dt.$$
Expanding this integrand into a geometric series and integrating term by term, we see that \begin{align}\frac{\pi^2}{8} & =\sqrt{2}\left(\sum_{n=0}^{\infty}\frac{(-1)^n}{(4n+1)^2}-\frac{(-1)^n}{(4n+3)^2}\right) \\ & =\sqrt{2}\sum_{n=0}^{\infty}\frac{(-1)^n}{(4n+1)^2} +\sqrt{2} \sum_{n=-\infty}^{-1}\frac{(-1)^n}{(4n+1)^2} \\ & = \sqrt{2}\sum_{n=-\infty}^{\infty}\frac{(-1)^n}{(4n+1)^2}. \end{align} Thus, \begin{align} \sum_{n=-\infty}^{\infty}\frac{(-1)^n}{(4n+1)^2}=\frac{\pi^2}{8\sqrt{2}}, \end{align} which is the third sum.
Following proof rely on this integral identity :
$$\int_{a}^{1}\frac{\arccos x}{\sqrt{x^2-a^2}}\mathrm{d}x=-\frac{\pi}{2}\ln a\qquad ;\,a\in(0,1]$$
We will prove it later on. Now, let's make a power series :
$$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\int_0^1\frac{1}{x}\sum_{n=1}^{\infty}\frac{x^n}{n}\,\mathrm{d}x=-\int_0^1\frac{\ln(1-x)}{x}\,\mathrm{d}x=-\int_0^1\frac{\ln x}{1-x}\,\mathrm{d}x$$
Inserting the formula above we get :
$$\zeta(2)=\frac{2}{\pi}\int_0^1\int_{x}^{1}\frac{\arccos y}{(1-x)\sqrt{y^2-x^2}}\,\mathrm{d}y\,\mathrm{d}x$$
Interchanging the order of integration :
$$\zeta(2)=\frac{2}{\pi}\int_0^1\int_{0}^{y}\frac{\arccos y}{(1-x)\sqrt{y^2-x^2}}\,\mathrm{d}x\,\mathrm{d}y\tag{A}$$
But, with help of substitution $x=y \cos{\theta}$ and universal $t=\tan\frac{\theta}{2}$ : $$\int_{0}^{y}\frac{\mathrm{d}x}{(1-x)\sqrt{y^2-x^2}}=\int_{0}^{\frac{\pi}{2}}\frac{\mathrm{d}\theta}{1-y \cos{\theta}}=\int_{0}^{1}\frac{\frac{2\mathrm{d}t}{1+t^2}}{1-y\frac{1-t^2}{1+t^2}}=\frac{\pi-\arccos{y}}{\sqrt{1-y^2}}$$
Plugging this to $(A)$ we get :
\begin{align*}&\zeta(2)=\frac{2}{\pi}\int_{0}^{1}\frac{\pi\arccos{y}-\arccos^2 y}{\sqrt{1-y^2}}\,\mathrm{d}y=\frac{2}{\pi}\left(\frac{\pi}{2}\arccos^2 y- \frac{1}{3}\arccos^3 y\right)\bigg{|}_{1}^{0}= \\ \\ &\frac{2}{\pi}\left(\frac{\pi}{2}\left(\frac{\pi}{2}\right)^2-\frac{1}{3}\left(\frac{\pi}{2}\right)^3\right) = \frac{2}{\pi}\left(\frac{\pi}{2}\right)^3 \left(1-\frac{1}{3}\right) =\frac{\pi^2}{6} \end{align*}
ADDENDUM : Proof of the apriori integral :
\begin{align*}&\int_{a}^{1}\frac{\arccos x}{\sqrt{x^2-a^2}}\mathrm{d}x=\int_{a}^{1}\frac{\arccos\left(\frac{x}{y}\right)}{\sqrt{x^2-a^2}}\bigg{|}_{y=x}^{y=1}\mathrm{d}x=\int_{a}^{1}\int_{x}^{1}\frac{x}{y}\frac{\mathrm{d}y\,\mathrm{d}x}{\sqrt{x^2-a^2}\sqrt{y^2-x^2}} = \\ \\ & \int_{a}^{1}\int_{a}^{y}\frac{x}{y}\frac{\mathrm{d}x\,\mathrm{d}y}{\sqrt{x^2-a^2}\sqrt{y^2-x^2}} = \frac{\pi}{2}\int_{a}^{1}\frac{\mathrm{d}y}{y} = -\frac{\pi}{2}\ln a \end{align*}
Where the inner integral was computed via substitution $x^2=a^2\cos^2\theta+y^2\sin^2\theta$ it is clear, taking differential, that $2x\;\mathrm{d}x=2\left(y^2-a^2\right)\sin\theta\cos\theta\;\mathrm{d}\theta$, then :
$$(x^2-a^2)(y^2-x^2)=(a^2\cos^2\theta+y^2\sin^2\theta-a^2)(y^2-a^2\cos^2\theta-y^2\sin^2\theta)=(y^2\sin^2\theta-a^2\sin^2\theta)(y^2\cos^2\theta-a^2\cos^2\theta)=(y^2-a^2)^2\sin^2\theta\cos^2\theta$$
Or $$\sqrt{x^2-a^2}\sqrt{y^2-x^2}= \left(y^2-a^2\right)\sin\theta\cos\theta\ = x\,\mathrm{d}x$$
Therefore :
$$\int_{a}^{y}\frac{x\,\mathrm{d}x}{\sqrt{x^2-a^2}\sqrt{y^2-x^2}}=\int_{0}^{\frac{\pi}{2}}\frac{x\,\mathrm{d}x}{x\,\mathrm{d}x}=\frac{\pi}{2}$$
\begin{align} \log(2\cos(x)) &=\log\left(e^{ix}+e^{-ix}\right)\tag{1a}\\ &=ix+\log\left(1+e^{-2ix}\right)\tag{1b}\\ &=-ix+\log\left(1+e^{2ix}\right)\tag{1c}\\ &=\cos(2x)-\frac{\cos(4x)}2+\frac{\cos(6x)}3-\cdots\tag{1d} \end{align} Explanation:
$$\text{(1a)}$$: $$2\cos(x)=e^{ix}+e^{-ix}$$
$$\text{(1b)}$$: factor out $$e^{ix}$$
$$\text{(1c)}$$: factor out $$e^{-ix}$$
$$\text{(1d)}$$: average $$\text{(1b)}$$ and $$\text{(1c)}$$ using the power series for $$\log(1+x)$$ \begin{align} \sum_{k=1}^\infty\frac1{k^2} &=\frac1{2\pi}\int_0^{2\pi}\sum_{k=1}^\infty\frac{e^{ikx}}k\sum_{k=1}^\infty\frac{e^{-ikx}}k\,\mathrm{d}x\tag{2a}\\ &=\frac1{2\pi}\int_0^{2\pi}\left|\log(1-e^{ix})\right|^2\,\mathrm{d}x\tag{2b}\\ &=\frac1{2\pi}\int_{-\pi}^\pi\left|\log(1+e^{ix})\right|^2\,\mathrm{d}x\tag{2c}\\ &=\frac1{2\pi}\int_{-\pi}^\pi\left|\,\log\left(2\cos\left(\frac x2\right)\right)+\frac{ix}2\,\right|^{\,2}\,\mathrm{d}x\tag{2d}\\ &=\frac1{2\pi}\int_{-\pi}^\pi\left(\log\left(2\cos\left(\frac x2\right)\right)^2+\frac{x^2}4\right)\,\mathrm{d}x\tag{2e}\\ &=\frac{\pi^2}{12}+\frac1{2\pi}\int_{-\pi}^\pi\left(\cos(x)-\frac{\cos(2x)}2+\frac{\cos(3x)}3-\cdots\right)^2\,\mathrm{d}x\tag{2f}\\ &=\frac{\pi^2}{12}+\frac12\sum_{k=1}^\infty\frac1{k^2}\tag{2g}\\ &=\frac{\pi^2}6\tag{2h} \end{align} Explanation:
$$\text{(2a)}$$: use the orthogonality of $$e^{ijx}$$ and $$e^{ikx}$$ when $$j\ne k$$
$$\text{(2b)}$$: use the power series for $$\log(1+x)$$
$$\text{(2c)}$$: substitute $$x\mapsto x+\pi$$
$$\text{(2d)}$$: $$1+e^{ix}=2\cos(x/2)e^{ix/2}$$
$$\text{(2e)}$$: $$\left|\,x+iy\,\right|^2=x^2+y^2$$
$$\text{(2f)}$$: apply $$(1)$$
$$\text{(2g)}$$: use the orthogonality of $$\cos(jx)$$ and $$\cos(kx)$$ for $$j\ne k$$
$$\text{(2h)}$$: subtract the original from twice $$\text{(2g)}$$
• This is fantastic – Cade Reinberger Jul 9 '19 at 4:28
Using Fourier's expansion of $f(x)=x(1-x)$, we get $$a_{0}=\frac{1}{6}$$ $$a_{n}=\frac{1}{n^2\pi^2}$$ $$b_{n}=0$$ Therefore ,we have $$x(1-x)=\frac{1}{6}-\sum_{n=1}^{\infty}\frac{\cos2xn\pi}{(n\pi)^2}$$ putting $x=0$ we get $$\sum _{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}$$
Define $f$ on $[0;2\pi]$,
$\displaystyle f(a)=\int_0^1 \dfrac{\ln(x^2-2x\cos(a)+1)}{x}dx\tag 0$
Theorem:
For all $a\in [0;2\pi]$,
$f(a)=-\dfrac{1}{2}a^2+\pi a-\dfrac{\pi^2}{3}\tag 1$
For all $a\in [0;2\pi]$,
$\displaystyle f\left(\frac{a}{2}\right)+f\left(\pi-\frac{a}{2}\right)=\frac{f(a)}{2}\tag 2$
Proof:
\begin{align} f\left(\frac{a}{2}\right)+f\left(\pi-\frac{a}{2}\right)&=\int_0^1 \frac{\ln\left(\left(x^2-2x\cos\left(\frac{a}{2}\right)+1\right)\left(x^2+2x\cos\left(\frac{a}{2}\right)+1\right) \right)}{x}dx\\ &=\int_0^1 \frac{\ln\left(x^4-2x^2\cos(a)+1\right)}{x}dx\\ \end{align}
Perform the change of variable $y=x^2$ in the latter integral to obtain (2).
According to theorems about functions défined by integrals $f^{\prime\prime}$ exists and it is continuous.
Derive twice (2),
For all $a\in [0;2\pi]$,
$\displaystyle f^{\prime\prime}\left(\frac{a}{2}\right)+f^{\prime\prime}\left(\pi-\frac{a}{2}\right)=2f^{\prime\prime}(a)\tag 3$
$f^{\prime\prime}$ is continuous on $[0;2\pi]$ therefore this fonction has a maximum $M$ and a minimum $m$ that are obtainable.
Therefore it exists $a_0\in[0;2\pi]$ such that $f^{\prime\prime}(a_0)=M$.
Plug $a_0$ into (3),
$\displaystyle f^{\prime\prime}\left(\frac{a_0}{2}\right)+f^{\prime\prime}\left(\pi-\frac{a_0}{2}\right)=2f^{\prime\prime}(a_0)=2M$
But, $f^{\prime\prime}\left(\frac{a_0}{2}\right)\leq M$ et $f^{\prime\prime}\left(\pi-\frac{a_0}{2}\right)\leq M$ according to the définition of $M$.
therefore $f^{\prime\prime}\left(\frac{a_0}{2}\right)=f^{\prime\prime}\left(\pi-\frac{a_0}{2}\right)=M$
By recurrence reasoning, for all $n\geq 1$, natural integer,
$f^{\prime\prime}\left(\frac{a_0}{2^n}\right)=M\tag 4$
$f^{\prime\prime}$ is continuous in $0$ therefore taking $n$ to infinity in (4) one obtains,
$M=f^{\prime\prime}(0)$.
Considering $m$ the minimum of $f^{\prime\prime}$ using the same way it can be proved that,
$m=f^{\prime\prime}(0)$
Since $m=M$, therefore $f^{\prime\prime}$ is a constant function.
therefore, there exist $\alpha,\beta,\gamma$ real such that,
For all $a\in[0;2\pi]$,
$f(a)=\alpha a^2+\beta a+\gamma\tag 5$
Plug (5) into (3), one obtains:
$\alpha\pi+\dfrac{\beta}{2}=0$ and $\alpha \pi^2 +\beta \pi+\dfrac{3}{2}\gamma=0$
On the other hand, for all $a\in [0;2\pi]$,
$\displaystyle f^\prime(a)= 2\sin a\int_0^1\dfrac{1}{x^2-2x\cos a+1}dx$
If $a=\dfrac{\pi}{2}$ one obtains,
\begin{align} f^\prime\left(\dfrac{\pi}{2}\right)&=2\int_0^1 \dfrac{1}{x^2+1}dx\\ &=2\times \dfrac{\pi}{4}\\ &=\dfrac{\pi}{2} \end{align}
Taking derivative of (5), one obtains for all $a\in [0;2\pi]$,
$f^\prime(a)=2\alpha a+\beta$
Therefore,
$\alpha \pi +\beta=\dfrac{\pi}{2}$
One have obtained a linear system of three equations in $\alpha,\beta,\gamma$.
To achieve the proof of the theorem solve it.
To get the value of $\zeta(2)$, apply the theorem with $a=0$, one obtains,
$\displaystyle \int_0^1 \dfrac{\ln(1-x)}{x}dx=-\dfrac{\pi^2}{6}$
And then, continue in usual way, expand the integrand...
From, Euler's integrals, H. Haruki and S. Haruki, The mathematical gazette, 1983.
• This one is very simple and straightforward to follow. However, I would like to see the argument for continuity for the second derivative $f''(x)$ to be more explained (or am I so blind to see it?). – Machinato Aug 25 '17 at 12:20
• It uses general theorems about derivation under the integral sign as Lebesgue's dominated convergence theorem (but weaker theorems do exist for the Riemann integral). – FDP Aug 25 '17 at 17:21
I really like this one. Consider $f(x)=x^2-\pi^2$. Compute it's Fourier expansion to obtain $$f(x)=\frac{2}{3}\pi^2-4\sum_{n=1}^\infty\frac{(-1)^n}{n^2}\cos nx.$$ Now let $x=\pi$, then it quickly follows that $$4\zeta(2)=\frac{2}{3}\pi^2\implies \zeta(2)=\frac{\pi^2}{6}.$$
• This is the same as the first part of this one. – AD. Oct 24 '16 at 12:31
I found this proof on YouTube but I did little changes:
\begin{align} I&=\int_0^{\pi/2}\ln(2\cos x)\ dx=\int_0^{\pi/2}\ln\left(e^{ix}(1+e^{-2ix})\right)\ dx\\ &=\int_0^{\pi/2}ix\ dx-\sum_{n=1}^\infty \frac{(-1)^n}{n}\int_0^{\pi/2}e^{-2ix}\ dx\\ &=\frac{\pi^2}{8}i-\sum_{n=1}^\infty\frac{(-1)^n}{n}\left(-\frac{(-1)^n-1}{2in}\right)\\ &=\frac{\pi^2}{8}i-\frac12i\left(\zeta(2)-\operatorname{Li}_2(-1)\right)\\ &=\frac{\pi^2}{8}i-\frac12i\left(\zeta(2)+\frac12\zeta(2)\right)\\ &=i\left(\frac{\pi^2}{8}-\frac34\zeta(2)\right) \end{align}
By comparing the imaginary parts, we have
$$0=\frac{\pi^2}{8}-\frac34\zeta(2)\Longrightarrow\zeta(2)=\frac{\pi^2}{6}$$
Let $$f(x)=\frac 12-x$$ on the interval $$[0, 1)$$, and extend $$f$$ to be periodic on $$\mathbb{R}$$.
By definition, \begin{align*} \hat f(0)=\int_0^1 f(x)dx=\int_0^1 \left(\frac 12-x\right)dx=0. \end{align*} And for $$\kappa\ne 0$$: \begin{align*} \hat f(\kappa)&=\int_{0}^{1}f(x)e^{-2\pi i\kappa x }dx=\int_0^1\left( \frac 12 -x \right)e^{-2\pi i\kappa x}dx=-\int_0^1xe^{-2\pi i \kappa x}dx\\ &=\frac{1}{2\pi i\kappa }\int_{0}^{1}xd(e^{-2\pi i\kappa x})=\left.\frac{1}{2\pi i\kappa}xe^{-2\pi i\kappa x}\right|_0^1+\frac{1}{2\pi i\kappa}\int_0^1 e^{-2\pi i\kappa x}dx\\ &=\frac{1}{2\pi i\kappa}. \end{align*}
By the Parseval identity \begin{align*} \int_{0}^{1}|f(x)|^2dx=\sum_{k=-\infty}^{\infty}|\hat{f}(k)|^2=|\hat{f}(0)|^2+2\sum_{k=1}^{\infty}|\hat{f}(k)|^2=2\sum_{k=1}^{\infty}\frac{1}{4\pi^2 k^2}. \end{align*} On the other hand, \begin{align*} \int_{0}^{1}|f(x)|^2dx&=\int_{0}^{1}\left( \frac{1}{2}-x \right)^2 dx=\frac 14-\frac 12+\frac 13=\frac 1{12}. \end{align*} Hence, we have $$\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}.$$
Remark: This is an exercise(Chapter 8.13 on page 254 ) in Folland's book.
Here's mine. I'm answering late, I know that, but I am still answering it.
We'll use the expansion of $$\tanh^{-1}$$: $$\frac{1}{2}\log\frac{1+y}{1-y}=\sum_{n\geq0}\frac{y^{2n+1}}{2n+1},\quad|y|<1$$ We start with this inequality:
$$\int_{-1}^{1}\int_{-1}^{1}\frac{1}{1+2xy+y^2}dy\,dx=\int_{-1}^{1}\frac{1}{1+2xy+y^2}dx\,dy$$ The LHS of this equality gives: $$\int_{-1}^{1}\int_{-1}^{1}\frac{1}{1+2xy+y^2}dy\,dx=\int_{-1}^{1}\frac{\arctan \frac{x+y}{\sqrt{1-x^2}}}{\sqrt{1-x^2}}dx\Biggr|_{y=-1}^{y=1}\\ \quad\,\,\quad\quad\quad\quad\quad\quad\quad\quad\quad=\int_{-1}^{1}\frac{\pi}{2\sqrt{1-x^2}}dx=\frac{\pi^2}{2}$$ The RHS of the former equality yields: \begin{align} \int_{-1}^{1}\int_{-1}^{1}\frac{1}{1+2xy+y^2}dy\,dx&=\int_{-1}^{1}\frac{\log(1+2xy+y^2)}{2y}dy\Biggr|_{x=-1}^{x=1}\\ &=\int_{-1}^{1}\frac{\log\frac{1+y}{1-y}}{y}dy\\ &=2\int_{-1}^{1}\sum_{n\geq0}\frac{y^{2n}}{2n+1}dy\\ &=4\sum_{n\geq0}\frac{1}{(2n+1)^2} \end{align} Hence, $$\sum_{r\geq0}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}$$ Now $$\frac{3}{4}\zeta(2)=\zeta(2)-\frac{1}{4}\zeta(2)=\sum_{n\geq 1}\frac{1}{n^2}=\sum_{m\geq1}\frac{1}{(2m)^2}=\sum_{r\geq0}\frac{1}{(2r+1)^2}=\frac{\pi^2}{8}$$ Solving this we get $$\zeta(2)=\frac{\pi^2}{6}$$ as desired. Source:https://www.emis.de/journals/GM/vol16nr4/ivan/ivan.pdf
Here are more proofs. | {"extraction_info": {"found_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": 61, "wp-katex-eq": 0, "align": 12, "equation": 5, "x-ck12": 0, "texerror": 0, "math_score": 0.9971717596054077, "perplexity": 676.5329122045805}, "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-2020-50/segments/1606141188899.42/warc/CC-MAIN-20201126171830-20201126201830-00716.warc.gz"} |
http://mathhelpforum.com/calculus/23764-basic-complex-analysis-question.html | Math Help - Basic Complex Analysis question
1. Basic Complex Analysis question
This question appears really easy, I am just not sure I am undestanding it correctly.
I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?
2. Originally Posted by thecow
This question appears really easy, I am just not sure I am undestanding it correctly.
I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?
You need the version of the residue theorem with winding number.
RonL
3. We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)
4. Originally Posted by CaptainBlank
You need the version of the residue theorem with winding number.
RonL
Originally Posted by thecow
We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)
You can do what CaptainBlank said and use the residue theorem. But here is a special case, you can rely on the Cauchy integral formula but you still need winding numbers because $\Gamma$ is not a contour (i.e. a simple closed curve).
5. Yea I mean, I get that its a winding number. I am just curious if I am reading it correctly, so for instance the pole at 0 would be a winding number of 3?
6. Originally Posted by thecow
Yea I mean, I get that its a winding number. I am just curious if I am reading it correctly, so for instance the pole at 0 would be a winding number of 3?
Yes. Just count the number of times it "winds" around the number.
7. Alright thanks. I thought it might be 2 because if you follow the path with your finger you really only go around twice it seems. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9466578364372253, "perplexity": 207.39555784944767}, "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-2014-41/segments/1410657125488.38/warc/CC-MAIN-20140914011205-00222-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://mathoverflow.net/questions/92865/is-reflexivity-an-open-condition/92880 | # Is reflexivity an open condition?
Is the condition that a module is reflexive an open condition?
That is, if $X$ is a smooth projective complex variety, $T$ a quasi-projective variety, and $F$ a finitely presented module on $X \times T$ that is $T$-flat, then we can form the locus $T' \subset T$ of points $t$ such that the restriction of $F$ to $X \times t$ is reflexive.
Is $T' \subset T$ open?
If not, is it locally closed?
Recall that a coherent sheaf $F$ is said to be reflexive if the natural map $F \to (F^{\vee})^{\vee}$ to the double dual is an isomorphism.
-
This locus is indeed open. I will explain why using Kollar's "Hulls and Husks" (arXiv:0805.0576). More generally, this article studies in great detail when taking the double dual commutes with base change.
First, we may restrict to the open locus of $T$ where $F_t$ is torsion-free (because reflexive sheaves are torsion-free). Then, we choose an ample line bundle $H$ on $X$, and we compute the Hilbert polynomials relatively to $H$. The Hilbert polynomial $P(F_t)$ of $F_t$ is constant by flatness. From the exact sequence $0\to F_t\to F^{\vee\vee}_t\to F^{\vee\vee}_t/F_t\to 0$, we see that $F_t$ is reflexive exactly when $P(F^{\vee\vee}_t)=P(F_t)$, i.e. exactly when $P(F^{\vee\vee}_t)$ takes its minimal value. But the polynomial $P(F^{\vee\vee}_t)$ is constructible and upper semicontinuous by Proposition 28 (3) of Hulls ans Husks. This proves that this locus is open.
-
@Olivier Benoist: Thanks! – jlk Apr 2 '12 at 21:10
Olivier, how do you know that the polynomial $P(F_t^{\vee\vee})$ is constructible? The problem is that a priori $F_t^{\vee\vee}$ is just a collection of sheaves on the fibers, it is NOT given by restrictions of one coherent sheaf to the fibers... – Sasha Apr 3 '12 at 2:46
@Sasha : It is part of the statement of Kollar's article (arXiv:0805.0576 Proposition 28 (3)), and it is proven there. By noetherian induction, you only need to show that it is constant on an open subset of $T$, hence, you will only have to show that it is "given by restrictions of one coherent sheaf to the fibers" on an open subset of $T$. – Olivier Benoist Apr 3 '12 at 6:54
[EDIT: as Sasha points out, this does not answer the question. Please see it as a comment explaining why $X$ should be proper!]
The answer is no in general if $X$ is not proper: take $X=\mathrm{Spec}\,\mathbb{C}[x]$, $T=\mathrm{Spec}\,\mathbb{C}[t]$, and $F=$ the structure sheaf of $Z=\mathrm{Spec}\,(\mathbb{C}[t,x]/(1-tx))$. Then $T'$ is just the origin.
Variant: if instead you take $T=\mathrm{Spec}\,\mathbb{C}[t,u]$ and $Z=\mathrm{Spec}\,(\mathbb{C}[t,u,x]/(u,1-tx))$ (i.e. the same $Z$ as before, but embedded in 3-space), then $T'$ is the union of the origin and the complement of the $t$-axis, hence not locally closed (but still constructible).
Of course the point here is that $Z$ "goes to infinity" at the origin. I don't have a counterexample where $X$ is proper, but the main problem then is "taking the dual in the fibers", as in Sasha's comment above.
-
In original question $X$ was proper (even projective). – Sasha Apr 2 '12 at 15:23
@Sasha: oops, sorry! I ha read "quasiprojective" but that was about $T$. – Laurent Moret-Bailly Apr 2 '12 at 17:02
I'll assume you meant to say that $F$ was locally finitely presented or coherent in your second sentence. The locus where $F$ is reflexive is the complement of the union of the supports of the kernel and cokernel of $F\to (F^\vee)^\vee$. This will be open.
Postscript As Sasha points out the argument is incomplete because it is not clear that duals commute with restriction to the fibres.
PPS Now Laurent has a counter example, so I guess that finishes it.
-
Donu, the problem is that operation of taking the dual sheaf does not commute with restriction to the fiber over $T$. So, it is not clear why $((F^\vee)^\vee)_{|X\times\{t\}} \cong ((F_{|X\times\{t\}})^\vee)^\vee$. – Sasha Apr 2 '12 at 12:00
what happens if you take the derived dual? – Yosemite Sam Apr 2 '12 at 15:28
Well, in the bounded derived category of a smooth variety, everything is reflexive in the sense that $$F\cong RHom(RHom(F, O_X), O_X)$$ – Donu Arapura Apr 2 '12 at 16:08
ah, of course, silly me. it's an open condition then! :) – Yosemite Sam Apr 3 '12 at 23:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9201835989952087, "perplexity": 315.4205510983626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1422119446463.10/warc/CC-MAIN-20150124171046-00076-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.arxiv-vanity.com/papers/1305.7509/ | arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.
# On a local mass dimension one Fermi field of spin one-half and the theoretical crevice that allows it
Dharam Vir Ahluwalia Institute of Mathematics Statistics and Scientific Computation, Unicamp, 13083-859 Campinas, São Paulo, Brazil, and Department of Physics and Astronomy, Rutherford Building, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand Electronic address:
28 May 2013
###### Abstract
Since the 1928 seminal work of Dirac, and its subsequent development by Weinberg, a view is held that there is a unique Fermi field of spin one-half. It is endowed with mass dimension three-half. Combined, these characteristics profoundly affect the phenomenology of the high energy physics, astrophysics, and cosmology. We here present a counter example by providing a local, mass dimension one, Fermi field of spin one-half. The theory, inter alia, thus allows dimensionless quartic self interaction for the new fermions, and its only other dimensionless coupling is quadratic in the new fermions and in the standard-model scalar field. For these reasons, the immediate application of the new theory resides in the dark-matter sector of physical reality. The lowest-mass associated new particle may leave its unique signature at the Large Hadron Collider. We discuss in detail the theoretical crevice that allows the existence of the new quantum field.
## I Introduction and background
To report the existence of a local spin one-half fermion field with mass dimension one is tantamount to claiming an element of incompleteness in our knowledge of quantum fields at a basic level. If true, it would, for example, allow dimensionless couplings of the type ; where and are spin one-half fermionic and spin-zero bosonic fields respectively, and is an appropriate adjoint. That the existence of such a field would have escaped even so careful an analysis as that of Weinberg’s indicates that there is either something non-trivial, or something non-trivially wrong, with such a construct. Tentatively, we assume the former and narrate the circumstances under which the claimed new quantum field came to exist and then proceed to systematically present the construct. Because of the nature of the claim we make an effort to be explicit about every known element that may affect our results. This approach serves the dual purpose of making the presentation pedagogic and to make it more accessible to scrutiny.
The rest of this section is presented in the first person singular. This departure from the convention seems necessitated by the subject at hand.
During the years 1992-98 I was surrounded by experimental physicists at Los Alamos Meson Physics Facility (LAMPF)111Which since then has undergone several changes in its mission and its name. on the one hand and theoretical colleagues at the Theory Division of the Los Alamos National Laboratory on the other. Initially my interests were what I later called ‘mathematical science fiction’ Ahluwalia (1999) but because of the new results that were emerging from the neutrino experiments my colleagues informally encouraged me to explore Majorana neutrinos. The subject of Majorana field and Majorana spinors was confusing, and somewhat of a mess. And what had one to do with the other, I asked. This realization arose in a conversation with Peter Herczeg.
I asked the library to get Majorana’s 1937 paper Majorana (1937) translated into English, and two weeks later a professional translation was in my office. I found that in the standard language Majorana started with the Dirac field and then identified with . An intrinsically neutral field was thus introduced for the first time. There was no mention of any new spinors.
My first exposure to Majorana spinors came through two papers McLennan (1957); Case (1957) that appeared some twenty years after Majorana’s original paper on the subject. At the same time I found a very nice group theoretical introduction to these spinors, but in their Grassmannian incarnation, in Ramond’s primer Ramond (1989). My personal exploration of c-number Majorana spinors began with an observation of Ramond on the ‘magic of Pauli matrices’ and how it resulted in the existence of Majorana spinors (Ramond, 1989, Section 1.4). That ‘magic’ confined the Majorana spinors to spin one-half and concealed some of their real content. The realization that Ramond’s argument could be readily generalized to higher spins if the said magic was, instead, associated with the Wigner time reversal operator, led to the writing of two exploratory papers Ahluwalia (1996); Ahluwalia et al. (1994b) and other presentations; see, for example, Ahluwalia et al. (1994a); Ahluwalia (2003). The notion of a complete set of eigenspinors of the spin one-half charge conjugation operator, which was later dubbed Elko in Ahluwalia-Khalilova and Grumiller (2005a, b), originated in those early papers.
In the fall of 1998 I left Los Alamos National Laboratory to join the Universidad Autónoma de Zacatecas, México. There, on his way from MIT to Leipzig, Daniel Grumiller came for a short visit and it resulted in an unexpected collaboration. Preprint Ahluwalia (2003) was our starting point, and we now asked: What are the properties of a quantum field constructed with Elko as its expansion coefficients. Had any one of us fully appreciated Weinberg’s work of the sixties Weinberg (1964, 1969), or his later monograph Weinberg (1995), we would have never dared to ask such a question. So with certain element of innocence, and ignorance, we two wrote a paper which opened with the line Ahluwalia-Khalilova and Grumiller (2005a), “we report an unexpected theoretical discovery of a spin one-half matter field with mass dimension one,” soon to be followed by another paper that opened similarly Ahluwalia-Khalilova and Grumiller (2005b), “we provide the first details on the unexpected theoretical discovery of a spin-one-half matter field with mass dimension one.” Our excitement was quite apparent! It was also very clear that the new quantum field provided a very natural dark matter candidate with a quartic self interaction, and it coupled to EB-GHK-H scalar field Englert and Brout (1964); Higgs (1964); Guralnik et al. (1964) but its interactions with other standard-model fields was suppressed.
However, when the locality-determining anticommutators were calculated this fermion field of spin one-half exhibited non-locality.
In the middle of 2006, I moved to the University of Canterbury in New Zealand. There, with my ever cheerful and hard working students,222Cheng-Yang Lee, Dimitri Schritt, Tom Watson, and later Sebastian Horvath we learned, to our collective surprise, that the Dirac quantum field as presented in many, though not all, textbooks did not transform properly under Poincaré space-time transformations! Not only that, when we identified (in the usual notation), with à la Majorana’s 1937 paper we discovered that the resulting field exhibited nonlocal anticommutators! The fields presented in the monographs of Weinberg and Srednicki, on the other hand, were completely free from such inconsistencies Weinberg (1995); Srednicki (2007).
With the insights gained from these monographs we discovered the culprits: there is a freedom of certain global phases associated with each of the expansion coefficients, and this freedom affects various properties of the quantum field; and designation also matter.333That is, what one calls or matters. With a similar remark for Elko. Gradual understanding of these elements led to a much improved locality structure for the Elko-based quantum field Ahluwalia et al. (2010, 2011), but non-locality still remained stubborn and showed up in one additive integral. Then in the 2012-2013 period I reached Instituto de Matemática, Estatística e Computacão Científica (IMECC), Brasil, for a sabbatical year, and the last hurdle evaporated. Under the IMECC expertise, the mischievous integral magically evaluated to zero de Oliveira and Rodrigues (2012).
It was also towards the end of my stay at University of Canterbury444The successes at IMECC, and post-earthquake academic turmoil of the University of Canterbury, made my sabbatical end in a transition to Brasil. that a new connection with the Very Special Relativity (VSR) began to emerge Cohen and Glashow (2006b); Ahluwalia and Horvath (2010). Again, the mathematical expertise at IMECC, and other Brazilian institutes, helped me to better understand the symmetry properties of the new field.
What follows is a detailed crystallization of these insights. The ‘Abstract’ above, and the opening paragraph of this section, places the mass dimension one Fermi field in the historical context and briefly describes the outcome of this research. The ‘Contents’ serve as a brief summary of the general flow of the paper: in II below resides the necessary background for constructing the local mass dimension one Fermi field of spin one-half in III.
## Ii Coefficient functions for a local mass dimension one Fermi field of spin one-half
### ii.1 Parity, Charge conjugation, and Elko: A review
We begin our exposition by setting up the notation. The right- and left-handed Weyl spinors transform under Lorentz boost as
ϕR(pμ)=exp(+σ2⋅φ)ϕR(kμ) (1a) ϕL(pμ)=exp(−σ2⋅φ)ϕL(kμ). (1b)
The boost parameter is defined so that acting on the standard four momentum555This definition allows us to introduce the helicity basis below. We have verified that the results reported here hold even if we do not work in the helicity basis. These alternate approaches require giving freedom of certain phases to . A freedom that is ultimately constrained by working out the relevant equal-time anticommutators.
kμdef=(m,limp→0pp),p=|p|. (2)
equals the general four momentum
pμ=(E,psinθcosϕ,psinθsinϕ,pcosθ). (3)
This yields , and . are the matrices for the generators of boosts (in the vector representation).666We shall use the conventions of Ryder, L H (1985) with and set to unity. Equations (1a) and (1b) follow from the fact that are the generators of the boosts for the right-handed Weyl representation space, while are for the left-handed Weyl representation space. For the direct sum of the right- and left-Weyl representation spaces, to be motivated below, the boost generator thus reads
κ=(−iσ/200+iσ/2). (4)
#### ii.1.1 Parity
It is immediately clear from the transformations (1a) and (1b) that parity, which in the vector space corresponds to , interchanges the right- and left-handed Weyl representation spaces. Thus the operation of parity, up to a global phase, for the 4-component spinors
ψ(pμ)=(ϕR(pμ)ϕL(pμ))=exp(iκ⋅φ)ψ(kμ) (5)
must contain purely off-diagonal identity matrices , and in addition an operation that implements the action of on . Up to a global phase, it is thus defined as
(6)
Here, is the transformed ; and is a null matrix. Now may be related to as follows
ψ(pμ′) =exp(−iκ⋅φ)ψ(kμ) (7) =exp(−iκ⋅φ)exp(−iκ⋅φ)ψ(pμ) =exp(−2iκ⋅φ)ψ(pμ).
Substituting (7) in (6), and on using the anti-commutativity of with each of the generators of the boost, , with , (6) becomes
Pψ(pμ)=exp(2iκ⋅φ)γ0ψ(pμ). (8)
A direct evaluation of the exponential in (8) gives
exp(2iκ⋅φ)=m−1γμpμγ0 (9)
where are the Dirac matrices in the Weyl representation. Finally, a substitution of the expansion (9) in (8), on combining with the identity (where is an identity matrix in the space of four-component spinors), results in
Pψ(pμ)=m−1γμpμψ(pμ). (10)
Thus, we have the desired expression for the parity operator777In obtaining this result we have followed a recent work of Speranca Speranca (2013).
P=m−1γμpμ. (11)
Its eigenvalues are . Each of these has a two fold degeneracy
PξS(pμ)=+ξS(pμ),PξA(pμ)=−ξA(pμ). (12)
The superscripts refer to self and anti-self conjugacy of under . The Dirac’s and spinors are thus seen as the eigenspinors of the parity operator, , with eigenvalues and , respectively:
ξS(pμ)→uσ(pμ),ξA(pμ)→vσ(pμ), (13)
where the designation represents the degeneracy index. Seen from this perspective, with the help of (11), (12) translates to
(γμpμ−m\openone4)u(pμ)=0,(γμpμ+m\openone4)v(pμ)=0, (14)
which are the 1928 Dirac equations in the momentum space.
This stage still leaves the global phases associated with each of the Dirac spinors to be still free. They contain important elements of physics affecting locality and transformation properties of the single-particle states associated with the Dirac and the 1937 Majorana quantum fields Majorana (1937).888Both of these fields carry Dirac’s and as their expansion coefficients. These can be read off from Weinberg’s construction of these fields Weinberg (1995).
Since the boost operator is not unitary, the Dirac spinors cannot in any sense be considered to represent quantum states. Instead, these enter as expansion coefficients in a quantum field which is specifically built to circumvent this problem.
In the configuration space the operator annihilates the Dirac and Majorana quantum fields, generically denoted by the same symbol . But, so does the spinorial Klein-Gordon operator. The question which determines that it is the former, and not the latter, that enters the Dirac/Majorana Lagrangian density is related to the structure of the , where is the usual time ordering operator. So the Lagrangian density is to be seen as a derived object after one has constructed a quantum field.999A reader who may not be familiar with this aspect of quantum field theory may wish to consult III below for an outline of this argument. This view is similar and consistent with the formalism presented in Weinberg (1995). We shall work in this spirit.
Incorporating the four-component spinor introduced in (5) is a necessary, but not sufficient, condition for preserving parity symmetry. We end this discussion of parity operator by noting that .
There exists an operator that transmutes the self-conjugate eigenspinors of the parity operator to the anti-self conjugates eigenspinors, and vice versa. This operator we discuss next, and arrive at the counterpart of (12) for this operator in (23b) below. We shall discover that eigenspinors of this operator play a central role in constructing the new mass dimension one quantum field of spin one-half.
#### ii.1.2 Charge conjugation operator and its eigenspinors (Elko)
Introduction of the four-component spinors (5), not necessarily as eigenspinors of the parity operator, doubles the degrees of freedom. In a quantum field theoretic formalism this doubling introduces the notion of antiparticles. The relative charges of the particles and antiparticles are then determined by the type of local gauge symmetries that the underlying kinematic framework supports. The particle-antiparticle symmetry enters via charge conjugation operator. For the four-component spinors, it may be similarly constructed as without first invoking a wave equation or a Lagrangian density. To see this we begin with the observation that the Wigner time reversal operator for spin one-half, , acts on the Pauli matrices as follows101010For any spin, ; with and . For convenience, we abbreviate to .
ΘσΘ−1=−σ∗, (15)
with
Θ=(0−110). (16)
It allows the following ‘magic’ to happen (cf. ‘magic of Pauli matrices’ in (Ramond, 1989, Section 1.4)). First complex conjugate (1a) and (1b), then multiply from the left by , and use the above defining feature of the Wigner time reversal operator. This sequence of manipulations gives
Θϕ∗R(pμ)=exp(−σ2⋅φ)Θϕ∗R(kμ)) (17a) Θϕ∗L(pμ)=exp(+σ2⋅φ)Θϕ∗L(kμ)). (17b)
That is, if transforms as a left-handed Weyl spinor then transforms as a right-handed Weyl spinor, where is an undetermined phase.111111Note: multiplication by a phase does not affect the Lorentz transformation properties of the spinors. Similarly, if transforms as a right-handed Weyl spinor then transforms as a left-handed Weyl spinor, where is an undetermined phase.
This crucial observation motivates the introduction of two sets of four-component spinors Ahluwalia (1996)
λ(pμ)=(ζλΘϕ∗L(pμ)ϕL(pμ)) (18)
and
ρ(pμ)=(ϕR(pμ)ζρΘϕ∗R(pμ)). (19)
The do not provide an additional independent set of spinors from that in (18) and for that reason we do not consider them further.
Generally, this result is introduced as a ‘magic of Pauli matrices’ (Ramond, 1989, Section 1.4) where gets concealed in Pauli’s , which equals . Our argument in terms of the Wigner time reversal operator has the advantage that it immediately generalizes to higher spins. Furthermore, the recognition that there is an element of freedom in the indicated phases, , makes escape their interpretation as Weyl spinors in a four-component disguise. It will become apparent below that we may now have four, rather than two, four-component spinors of the general form carried by .
With these observations at hand we are led to entertain the possibility that in addition to the symmetry operator , there may exist a second symmetry operator. Up to a global phase, it has the form
Cdef=(0αΘβΘ0)K (20)
where complex conjugates to its right. The arguments that leads to (20) are similar to the ones that give (6). Requiring to be an identity operator determines ; giving
C=(0iΘ−iΘ0)K=γ2K. (21)
There also exists a second solution with . But this does not result is a physically different operator and in any case the additional minus sign can be absorbed in the indicated global phase. This is the same operator that appears in the particle-antiparticle symmetry associated with the 1928 Dirac equation Dirac (1928).
We have thus arrived at the charge conjugation operator from the analysis of the symmetries of the -component representation space of spinors. This perspective has the advantage of immediate generalization to any spin: if is taken as a spin Wigner time reversal operator in footnote 10, then the resulting becomes the charge conjugation operator in the dimensional representation space.
To construct the eigenspinors of for spin one-half we act it on and re-write the result as
Cλ(pμ)=(iζ∗λ)−1(−ζ∗λΘϕ∗L(pμ)ϕL(pμ)). (22)
A comparison of the right-hand side of the above expression with the definition (18) shows that become eigenspinors of with eigenvalues if we demand . This requirement translates to, , resulting in .
In order that both the right- and left-handed components of remain on the same footing we shall here onwards study the case where we set .121212Departures from may induce violations of discrete symmetries; a subject we do not pursue here. Each of the signs provides a doubly degenerate set of ; and, additionally, these ensure that do not become Weyl spinors in disguise. This discussion adds new insights to the self and anti-self conjugate spinors first introduced in Ahluwalia et al. (1994b); Ahluwalia (1996); Ahluwalia-Khalilova and Grumiller (2005a, b)
λ(pμ)={λS(pμ) for ζλ=+iλA(pμ) for ζλ=−i (23a) with CλS(pμ)=+λS(pμ),CλA(pμ)=−λA(pμ). (23b)
The physics literature identifies in (23a) with ‘Majorana spinors’ (often, as Grassmann numbers). The seem to have been entirely overlooked.
A failure to construct a Lagrangian density using Majorana spinors is noted in (Aitchison and Hey, 2004, Appendix P). We shall see below that this only reflects that the considered Majorana spinors do not satisfy Dirac equation.
As Weinberg has emphasized in his monograph on the subject and in his other writings Weinberg (2012), the modern version of Dirac equation in quantum field theory has a much richer physics and is to be separated from its original motivations and interpretations confined to the finite dimensional representation space. It is in this latter sense, as expansion coefficients of a quantum field, that we treat the four-component spinors in this communication. In a quantum field theoretic setting the inevitability of antiparticles is elegantly argued by Feynman in Feynman and Weinberg (1987).
In order to avoid confusion with the folklore on Majorana spinors we refer to as Elko (German acronym for Eigenspinoren des Ladungskonjugationsoperators first introduced in Ahluwalia-Khalilova and Grumiller (2005a, b)).
### ii.2 Global phase transformations for Elko
In general, a global unitary transformation of the type
λ(pμ)→λ′(pμ)=exp(iaϑ)λ(pμ) (24)
with, and , does not preserve the self/anti-self conjugacy of under given in (23b) unless the matrix satisfies the condition
γ2a∗+aγ2=0 (25)
This is due to the presence of the operator in (21). The general form of satisfying these requirements is found to be
a =⎛⎜ ⎜ ⎜⎝αβλ0βδ0λλ0−δβ0λβ−α⎞⎟ ⎟ ⎟⎠ (27) =λγ0+i4(α−δ)[γ1,γ2]+i2β[γ2,γ3] −12(α+δ)γ5
with (with no association with the same symbols used elsewhere in this work) and .
In II.8 below we will discover that certain symmetries of the spin sums require , and . With the scale factor absorbed in , reduces to .
The counterpart of for the Dirac case that preserves the self/anti-self conjugacy under , (12) as opposed to (23b), is simply a identity matrix.
### ii.3 Explicit construction of Elko and on the choice of certain phases
To obtain an explicit form of Elko calls for a choice of the ‘rest’ spinors with defined in (2). That done, one then has for an arbitrary
λ(pμ)=exp(iκ⋅φ)λ(kμ). (28)
In principle, the boosted spinors reside in the boosted frames. But since no frame is a preferred frame they must also exist in all frames (an argument originally due to E. P. Wigner). It is this interpretation that we attach to . With the generator of the boost, , defined in (4), the boost operator in (28) can be readily evaluated using , to the effect that
exp(iκ⋅φ)=(e(σ/2)⋅φ00e−(σ/2)⋅φ) =√E+m2m⎛⎝\openone+σ⋅pE+m00\openone−σ⋅pE+m⎞⎠. (29)
To provide a concrete example of a mass dimension one quantum field, we confine our attention to the -defining as eigenspinors of
σ⋅^pϕ±L(kμ)=±ϕ±L(kμ) (30)
with . Furthermore, we adopt the phases given below
ϕ+L(kμ) =√m(cos(θ/2)exp(−iϕ/2)sin(θ/2)exp(+iϕ/2)) =ϕ+L(0)∣∣Eq. (A.2) of AG ϕ−L(kμ) =√m(−sin(θ/2)exp(−iϕ/2)cos(θ/2)exp(+iϕ/2)) =−ϕ−L(0)∣∣Eq. (A.3) of AG.
The abbreviation AG stands for Ahluwalia-Khalilova and Grumiller (2005a).131313The dictionary of comparison for Ahluwalia (1996) is the same as for AG modulo a minor change of notation: AG’s and . In writing the above ansatz, we have explicitly noted the differences from the most-often used earlier work.
There is a second choice of phases, and designations (that is, the indices are assigned), which is invoked when Elko are used as expansion coefficients of a quantum field. This choice we make explicit below in defining the
λS+(kμ) =+(iΘ[ϕ+L(kμ)]∗ϕ+L(kμ)) (32a) =λS{−,+}(0)∣∣Of (3.9) of AG λS−(kμ) =+(iΘ[ϕ−L(kμ)]∗ϕ−L(kμ)) (32b) =−[λS{+,−}(0)∣∣Of (3.9) of % AG] and λA+(kμ)=+(−iΘ[ϕ−L(kμ)]∗ϕ−L(kμ)) =−[λA{+,−}(0)andnotλA{−,+}(0)∣∣Of (3.10) of AG] (32c) λA−(kμ)=−(−iΘ[ϕ+L(kμ)]∗ϕ+L(kμ)) =−[λA{−,+}(0)andnotλA{+,−}(0)∣∣Of (3.10) of AG] (32d)
If one wishes one can keep the here-chosen phases free and fix them later by demanding locality for the resulting quantum field. It is worth noting that the freedom of global phases associated with each of the four is restricted to as (23b) does not allow a general replacement, , without affecting the self/ant-self conjugacy under .141414This observation was first made by in Schritt, D (circa 2007).. This freedom, and its restriction to , affects the locality properties of the quantum field we shall construct, and its judicious incorporation, in part, lies behind the removal of the non-locality encountered in Ahluwalia-Khalilova and Grumiller (2005a, b).
To obtain the explicit form of we need one last piece of information. It is deciphered by complex conjugating (30), then replacing in accord with (15), using , and finally multiplying from the left by . This exercise yields
σ⋅^p(Θ[ϕ±L(kμ)]∗)=∓(Θ[ϕ±L(kμ)]∗) (33)
and shows that the helicity of is opposite to that of .
The interplay of the result (33) with the boost (29) and the chosen form of in (32a) to (32d) gives the following analytically compact forms for
λS+(pμ)=√E+m2m(1−pE+m)λS+(kμ) (34a) λS−(pμ)=√E+m2m(1+pE+m)λS−(kμ) (34b) and λA+(pμ)=√E+m2m(1+pE+m)λA+(kμ) (34c) λA−(pμ)=√E+m2m(1−pE+m)λA−(kμ) (34d)
These are the expansion coefficients of a quantum field to be introduced below.
To construct an adjoint of the field we shall need the spinorial duals of the enumerated in (34a) to (34d). This task is undertaken below in Section II.5 after making a few remarks in II.4 on the Weinberg formalism for the construction of quantum fields and a departure necessitated by a circumstance encountered in constructing a quantum field with Elko as its expansion coefficients.
### ii.4 Misconceptions in Literature and On a departure from the Weinberg formalism
The choice for a set of -like objects appears as a trivial task in most textbooks: pick up a wave equation, like that of Dirac, set to , and solve. This straightforward exercise immediately gives the required objects. For the Dirac case, these are and .
For the construction of quantum fields, this exercise does not tell which of these coefficients, when appropriately boosted, shall accompany which one of the creation and destruction operators, nor does it tell us about the set of phases to be picked when making this pairing. This is where one of the first errors occurs in many of the modern textbooks on quantum fields. The other important error resides, as already alluded to above, in the lack of appreciation for the phases associated with each of the expansion coefficients. Our 2005 publications Ahluwalia-Khalilova and Grumiller (2005a, b) were not immune to these errors.151515A reader interested in specific examples of these errors in the modern textbooks may correspond with the author in private; or write to any of the former students of mine: Cheng-Yang Lee, Sebastian Horvath, and Dimitri Schritt.
Another question that escapes attention in these presentations is as to what tells us, for example, that the spin one-half particles are described by the Dirac fermions, and as to what is the deeper origin of the Dirac operator . The argument that it is the square root of the Klein-Gordon operator is historically correct, but in the modern context it carries with it an element of triviality.
The answer to these questions emerges elegantly in the Weinberg formalism Weinberg (1995). It, for example, establishes the uniqueness of the Dirac field assuming Lorentz symmetry along with symmetry under four space-time translations, validity of the cluster decomposition principle, and certain additional assumptions on discrete symmetries. The quantum field is obtained first, and then by calculating the Feynman-Dyson propagator through the evaluation of one arrives at the Dirac operator, and the Lagrangian density. The Dirac spinors appear naturally as expansion coefficients in without any recourse, direct or indirect (except certain symmetries), to Dirac or any other wave equation.
The Weinberg formalism however is not designed to furnish a local quantum field if the Lagrangian density is Poincaré covariant and the Feynman-Dyson propagator carries covariance under certain subgroup of Poincaré. This hitherto unknown type of symmetry breaking can arise without violating the null result of Michelson-Morley type experiments Cohen and Glashow (2006b). It allows the charge conjugation symmetry, but incorporating any one of the discrete symmetries of P, T, CP, or CT enlarges the subgroups to the full Poincaré group of Special Relativity (SR).
The noted null result, and the existence of massive particles, only requires the breaking of the conformal symmetry to a group containing four space-time translations adjoined to certain - or -parameter subgroups of Lorentz. A theory of relativity based on these subgroups was termed Very Special Relativity (VSR) by its authors. This remarkable result became widely known only in 2006 through the above cited Cohen-Glashow paper. It will become apparent below that it plays a significant role in the interpretational aspects of our work.
### ii.5 Spinorial dual for Elko
Under the Dirac dual
¯λ(pμ)def=λ(pμ)†γ0 (35)
Elko for massive particles have a null norm Ahluwalia (1996):
¯λS±(pμ)λS±(pμ)=0,¯λS±(pμ)λA±(pμ)=0 ¯λS±(pμ)λA∓(pμ)=0 (36a) ¯λA±(pμ)λA±(pμ)=0,¯λA±(pμ)λS±(pμ)=0 ¯λA±(pμ)λS∓(pμ)=0 (36b) and ¯λS±(pμ)λS∓(pμ)=∓2im ¯λA±(pμ)λA∓(pμ)=±2im. (36c)
A lack of full appreciation of this fact in physics literature, in part, leads to the classic problem of constructing a Lagrangian density for c-number Majorana spinors (Aitchison and Hey, 2004, Appendix P).
To take an ab initio look at the problem of defining a spinorial dual, let us take a general -component spinor defined for massive particles. Call it . It does not have to be an eigenspinor of , or an eigenspinor of . And ask: What is the dual spinor, denoted by , such that it yields a non-null Lorentz invariant norm, , under boosts as well as rotations. To answer this question we examine a general form of the dual defined as
\lx@overaccentset∼ϱα(pμ)def=[Ξϱα(pμ)]†η (37)
where is to be so defined that its action on any one of the yields one of the spinors from the same set. It is not necessary that the indices and be the same. We require to define an invertible map, with (possibly, up to a phase). The requirement of a Lorentz invariant norm then translates to the statement that in (37) must anti-commute with the generators of boosts, and commute with the generators of the rotations Ahluwalia et al. (2010, 2011)
{κi,η}=0,[ζi,η]=0,i=x,y,z. (38)
The three generators of boosts are given by (4). The three generators of rotation are
ζ=(σ/200σ/2). (39)
A slightly lengthy but a straight forward calculation satisfying the constraints (38) shows to have the form
η=(0a\openoneb\openone0),a,b∈R. (40)
In order that the right and the left transforming components of a are treated symmetrically, we set .161616The last equality is unimportant. It simply sets a scale of the norms. This is an additional assumption that we explicitly note. The standard Dirac dual corresponds to . Notationally, if represents a Dirac spinor, .
For Elko, the results (36a), (36b), and (36c) suggest that we define171717In obtaining this result we have followed a recent e-print of Speranca Speranca (2013).
Ξdef=12m( λS+(pμ)¯λS+(pμ)+λS−(pμ)¯λS−(pμ) (41) −λA+(pμ)¯λA+(pμ)−λA−(pμ)¯λA−(pμ)).
It is readily seen that and indeed exists and equals itself. We may thus introduce a spinorial dual for Elko in accordance with the general definition (37) and an Elko-specific notation that distinguishes it from the Dirac dual
\lx@overaccentset¬λα(pμ)=[Ξλα(pμ)]†η,a=b=1 (42)
with given by (41). This definition allows us to rewrite results (36a), (36b), and (36c) into the following orthonormality relations
\lx@overaccentset¬λSα(pμ)λSα′(pμ)=2mδαα′ (43a) \lx@overaccentset¬λAα(pμ)λAα′(pμ)=−2mδαα′ (43b) \lx@overaccentset¬λSα(pμ)λAα′(pμ)=0,\lx@overaccentset¬λAα(pμ)λSα′(pμ)=0. (43c)
The Elko dual, by construction, does precisely what it is intended to do. It provides a non-null Lorentz invariant norm. Mathematically, as well as physically, it encodes exactly the same information as that contained in (36a), (36b), and (36c). As shall be seen below, in its present incarnation, as opposed to all earlier works since the 2005 publications, it is a much more powerful tool in investigating the symmetry structure that will appear for the mass dimension one quantum fields introduced below. We will discover that this structure is intrinsic to Elko, and to its Majorana cousin. The Elko dual simply makes it manifest through the here-introduced operator . All results that we now obtain can be obtained in a much more cumbersome manner without invoking the Elko dual or the operator .
As a consistency check, we find that the Elko dual defined using the operator yields exactly the same dual as in Ahluwalia et al. (2010, 2011). To see this we act (41) from the right by . Use of (36a) and (36b) then gives
ΞλS+(pμ)=12mλS−(pμ)¯λS−(pμ)λS+(pμ)=2im=iλS−(pμ) (44)
where the last substitution is due to the relevant part of equations (36c). Definition (42) thus yields
\lx@overaccentset¬λS+(pμ)=−i[λS−(pμ)]†η. (45a) Repeating similar evaluations with λS−(pμ), λA+(pμ), and λA−(pμ) in succession gives \lx@overaccentset¬λS−(pμ)=i[λS+(pμ)]†η (45b) \lx@overaccentset¬λA+(pμ)=−i[λA−(pμ)]†η (45c) \lx@overaccentset¬λA−(pμ)=i[λA+(pμ)]†η. (45d)
A comparison of these results with those given in (Ahluwalia et al., 2010, Eq. 15) and (Ahluwalia et al., 2011, Eq. 22) establishes the equivalence of the Elko dual introduced here and the one introduced in the previous works. This, however, happens with the benefit of providing new insights (see below).
The knowledge of the Elko dual will help us define an appropriate adjoint for the quantum fields constructed with Elko as expansion coefficients. The calculation of the Feynman-Dyson propagator associated with these fields would require spin sums for Elko. We, therefore, evaluate these next and study their symmetry properties. The latter lie behind the departure from the Weinberg formalism for the construction of quantum fields noted in Section II.4.
### ii.6 Spin sums and projectors for Elko
The spin sums
∑αλSα(pμ)\lx@overaccentset¬λSα(pμ)and∑αλAα(pμ)\lx@overaccentset¬λAα(pμ) (46)
can now be readily evaluated using (34a) to (34d) for the and , and (45a) to (45d) for their duals. The first of the two spin sums evaluates to
i=1[E+m2m(1−p2(E+m)2)] (47) ×(−λS+(kμ)[λS−(kμ)]†+λS−(kμ)[λS+(kμ)]†)η=−im⎛⎜ ⎜ ⎜ ⎜⎝100−ie−iϕ01ieiϕ00−ie−iϕ10ieiϕ001⎞⎟ ⎟ ⎟ ⎟⎠
which suggests introducing181818Sometimes it may be convenient to use the functional dependence of on , and write it as .
G(ϕ)def=⎛⎜ ⎜ ⎜ ⎜⎝cccc000−ie−iϕ00ieiϕ00−ie−iϕ00ieiϕ000⎞⎟ ⎟ ⎟ ⎟⎠. (48)
In the spherical polar coordinate system parity is implemented by . The definition (48) thus shows that is an odd function under parity
G(ϕ)=−G(ϕ+π). (49)
The second of the spin sums can be evaluated in exactly the same manner. The combined result is
∑αλSα(pμ)\lx@overaccentset¬λSα(pμ)=m[G(ϕ)+\openone4] (50a) ∑αλAα(pμ)\lx@overaccentset¬λAα(pμ)=m[G(ϕ)−\openone4]. (50b)
These spin sums have the eigenvalues , and , respectively. Since eigenvalues of projectors must be either zero or one (Weinberg, 2012, Section 3.3), we define
and confirm that indeed they are projectors and furnish the completeness relation
S2=S,A2=A,S+A=\openone4 (52)
We have thus arrived at one of the most intriguing and subtle aspects of Elko: Manifestly, through the projectors break Lorentz symmetry. A detailed analysis found in Ahluwalia and Horvath (2010), and reviewed afresh below, shows that respects symmetries of the theory of very special relativity (VSR). For reasons given in the seminal paper on VSR Cohen and Glashow (2006b), the theory thus immediately evades the usual sensitive searches devoted to look for departures from Lorentz invariance (that is, from the symmetries underlying the theory of special relativity (SR)).
### ii.7 The SR→VSR breaking of the Lorentz symmetry
It is now necessary to first provide a brief summary of the Cohen-Glashow VSR which asserts, ‘‘invariance under HOM(2),191919HOM(2) is a 3-parameter subgroup of Lorentz to be defined below in II.7.1. rather than (as is often taught) the Lorentz group, is both necessary and sufficient to ensure that the speed of light is the same for all observers, and inter alia, to explain the null result to the Michelson-Morley experiment and its more sensitive successors.” Besides the just mentioned constancy of speed light, VSR also shares with SR the same time dilation, the same law of velocity addition, the same existence of a center-of-mass frame, and the same universal and isotropic maximal attainable velocity. These observations have been explicitly made in the original VSR paper Cohen and Glashow (2006b).
#### ii.7.1 Cohen-Glashow VSR: a brief summary
The theory of VSR has its origin in the above-quoted observation on necessity and sufficiency of certain subgroups of Lorentz to accommodate the null result of Michelson-Morley experiment and its more sensitive successors. There are four avatars of VSR Cohen and Glashow (2006b); Ahluwalia and Horvath (2010). They are defined through the associated Lie algebras as follows.
: Generated by
T1def=Kx+Jy,T2def=Ky−Jx (53)
where and are generators of rotations and boots, respectively. These provide an Abelian Lie algebra which is isomorphic to the algebra associated with translations in a plane. The two generate the T(2) group transformations
exp(iT1ϵ)={\openone4+iT1ϵ−12T21ϵ2for vectors\openone4+iτ1ϵfor spinors (54)
where the parameter of transformation is given by
ϵ=pxE−pz (55)
and and are the four-vector and the spinor representations of , respectively; and similarly
exp(iT2ε)={\openone4+iT2ε−12T22ε2for vectors\openone4+iτ2εfor spinors (56)
where the parameter of transformation is given by
ε=pyE−pz. (57)
and and are the four-vector and the spinor representations of , respectively.
To obtain the above group transformations the following identities were found helpful
T31=T32=04,τ21=τ22=04. (58)
1. yields , an algebra which is isomorphic to the 3-parameter algebra associated with the group of Euclidean motions E(2).
2. yields (2), an algebra which is isomorphic to the 3-parameter algebra associated with the group of orientation preserving similarity transformations, or homotheties, of HOM(2).
3. and simultaneously yields (2), an algebra which is isomorphic to the algebra associated with the four-parameter similitude group, SIM(2).
The counterparts of (54) and (56) for rotation about, and boost along, the VSR preferred direction (taken here as the ) are found to be
exp(iJzΦ)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩\openone4+iJzsinΦ+J2z(cosΦ−1) for vectors\openone+i2ζzsin(Φ/2)+4ζ2z(cos(Φ/2)−1) for spinors (59) | {"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": 0.9272290468215942, "perplexity": 1037.7157397262906}, "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-2020-40/segments/1600401600771.78/warc/CC-MAIN-20200928104328-20200928134328-00184.warc.gz"} |
https://www.physicsforums.com/threads/heisenberg-uncertainty-need-some-clarification-time-sesitive-help.395910/ | # Homework Help: Heisenberg Uncertainty, Need Some Clarification. TIME SESITIVE, HELP
1. Apr 16, 2010
### asl3589
Heisenberg Uncertainty, Need Some Clarification. TIME SESITIVE, HELP!!
1. The problem statement, all variables and given/known data
Use the Heisenberg uncertainty principle to calculate Deltax for a ball (mass = 100 g, diameter = 6.65 cm) with Deltav = 0.645 m/s.
2. Relevant equations
PX = h/(4*3.14)
3. The attempt at a solution
So, I took the equation and converted the values with my
numbers:((h/4π)/(.1kg * .645 m/s))/(.0665m) and yielded an
10^-32 m. This answer is wrong and I am not sure why. I only have two hours left to answer this question. I would really appreciate it if someone could guide me on what my mistake is?
2. Apr 16, 2010
### collinsmark
Re: Heisenberg Uncertainty, Need Some Clarification. TIME SESITIVE, HELP!!
Hello asl3589,
Why did you divide by the diameter of the ball?
[Edit: Also, you seem to be using the more formal $$\sigma _x \sigma _p \geq \frac{\hbar}{2}$$ where $$\hbar = \frac{h}{2 \pi}$$. But keep in mind that relationship is not an equality. If you want an approximate value with a $$\approx$$ sign and using $$\Delta x$$ and $$\Delta p$$, there is a slightly different version of the relation.]
Last edited: Apr 16, 2010
3. Apr 16, 2010
### asl3589
Re: Heisenberg Uncertainty, Need Some Clarification. TIME SESITIVE, HELP!!
Thanks for looking at it. I just assumed that the diameter factors into somehow. Is that unneccesary. If I take that step out the answer is just 8.174 * 10^-34 m. Is that right?
4. Apr 16, 2010
### collinsmark
Re: Heisenberg Uncertainty, Need Some Clarification. TIME SESITIVE, HELP!!
That would give you the minimum possible uncertainty in position. But that's not necessarily the approximate uncertainty. The minimum possible uncertainty in position might be the answer your instructor is looking for, but I'm uncertain about that. | {"extraction_info": {"found_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": 0.9115045070648193, "perplexity": 978.5407777752707}, "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-2018-51/segments/1544376832330.93/warc/CC-MAIN-20181219130756-20181219152756-00537.warc.gz"} |
https://www.physicsforums.com/threads/jumping-physics-problem.362338/ | # Jumping physics problem
1. Dec 10, 2009
### PearlyD
1. The problem statement, all variables and given/known data
An exceptional standing jump would raise a person 0.80m off the ground. To do this, what force must a 66 kg person exert against the ground?
Assume the person crouches a distance of 0.20 m prior to jumping,and thus the upward force has this distance to act over before he leaves the ground.
This question i dont even know how to start
can some one explain how to even start it?
2. Dec 10, 2009
### nickdk
Re: Jumping
Work = Force * Distance
Force = (9.81)(66)
Work = (9.81)(66)(.80)
Work = 517.968 J
Doesn't account for the crouching distance, but it may be irrelevant, someone else will probably cover that. I hope I went in the right direction with this.
Similar Discussions: Jumping physics problem | {"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": 0.8522143959999084, "perplexity": 2284.7425798374734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1502886105712.28/warc/CC-MAIN-20170819182059-20170819202059-00536.warc.gz"} |
http://www.onlineprediction.net/?n=Main.KernelMethods | # Kernel Methods
Kernel methods are a powerful tool of modern learning. This article provides an introduction to kernel methods through a motivating example of kernel ridge regression, defines reproducing kernel Hilbert spaces (RKHS), and then sketches a proof of the fundamental existence theorem.
Some results that appear to be important in the context of learning are also discussed.
# 1. A Motivating Example: Kernel Ridge Regression
In this section we will introduce kernels in the context of ridge regression. The reader may skip this section and proceed straight to the next session if he is only interested in the formal theory of RKHSs.
A more detailed discussion of Ridge Regression and kernels can be found in Section 3 of Steve Busuttil's dissertation.
## 1.1 The Problem
Suppose we are given a set of examples , where are signals and are outcomes or labels. We want to find a dependency between signals and outcomes and to be able to predict given a new . This problem is often referred to as the regression problem 1.
Let us start by restricting ourselves to linear dependencies of the form , where , is the standard scalar product in , and the prime stands for transposition (by default all vectors are assumed to be column vectors). The class of linear functions is not too reach, and we will need to progress to more sophisticated classes later.
## 1.2 Least Squares and Ridge Regression
The least squares is a natural, popular, and time-honoured (apparently going back to Legendre and Gauss) approach to finding . Let us take an minimising the sum of squared discrepancies
Minimising this expression is equivalent to projecting of the vector on the subspace of generated by vectors , where consists of -th coordinates of s.
The projection on the subspace is always unique and well-defined. However if the vectors are linearly dependent (which always happens if the sample is small and ) this unique projection has multiple representations as their linear combination. The method of least squares thus does not necessarily define a unique regressor.
Consider a more general problem. Let us take and consider the expression
A vector minimising this is called a solution of the ridge regression problem (for reasons that will become apparent later). The least squares approach is a special case of the ridge regression approach, namely, that of .
Why would anyone want to use ? There are three main reasons. First, ridge regression with always specifies a unique regressor as will be shown below. Secondly, a positive value of makes the problem easier computationally. These properties will be shown later in this section. Thirdly, the term performs the regularisation function. It penalises the growth of coefficients of and urges us to look for "simpler" solutions.
## 1.3 Solution: Primary Form
Let us find the solution of the ridge regression problem with . It is convenient to introduce a matrix ; the rows of are vectors (and the columns are vectors mentioned above). We get
By differentiating w.r.t. and equating the result to 0 we get
and
where is the identity matrix. This must be a solution; indeed, as coefficients of approach infinity, and therefore must go to infinity.
Let us analyse this expression. The matrices , , and have the size . The matrix is positive semi-definite, i.e., for all (this follows from ). By adding we make the matrix positive definite, i.e., we have for all (indeed, ). Because every positive definite matrix is non-singular 2, must have the inverse. If , a solution to the ridge regression problem always exists and it is unique.
If and , the matrix becomes singular 3. The geometrical interpretation of this situation was discussed above.
As approach 0, the matrix may become close to singular. The numerical routines for finding will then become less and less stable: they will have to deal with very big or very small values and make large round-up errors. Taking a larger thus stabilises the computation as mentioned above.
Let us attempt a rough hypothetical analysis of the predictive performance of ridge regression for different values of . If is very big, the term completely overshadows and the predictive performance deteriorates. If is very small, we may encounter numerical problems. An optimal value should thus be neither too big no too small. In some sense it must be comparable in size to elements of . The exact choice of depends on the particular dataset.
Finally, let us go back to the term "ridge regression". One of the versions of its etymology is that the diagonal of forms a "ridge" added on top of the least squares matrix .
## 1.4 Solution: Dual Form
Using the matrix identity we can rewrite the ridge regression solution as follows. For an arbitrary the outcome suggested by ridge regression is and this can be rewritten as
This formula is called the dual form of the ridge regression solution.
Similar arguments concerning non-singularity apply to . The matrix has the size . This might seem a disadvantage compared to the primary form: it is natural to expect that in practice would be fixed and not too big, while the size of the sample may be quite large. However this formula allows us to develop important generalisations.
We can say that in the dual form. However there is a more interesting way to interpret the dual form formula. We have
where is the matrix of mutual scalar products
and is the vector of scalar products of by :
Note that all s and appear in this formula only in mutual scalar product. This observation has important consequences.
## 1.5 Non-linear Regression
Now let us try to extend the class of functions we use and consider a wider class. Suppose that , i.e., all s are numbers and we are interested in approximations by polynomials of degree , i.e., functions of the form . Of course we can write down for this case, perform the differentiation and find the solution as we did before. However there is a simpler argument based on the dual form.
Let us map into as follows: . Once we have done this, we can do linear regression on new "long" signals. If we use the dual form, we do not even have to perform the transformations explicitly. Because we only need scalar products, we can compute all the necessary products and substitute them into the dual form formula.
Let us write down a formal generalisation. The signals do not have to come from any longer. Let them be drawn from some arbitrary set 4 . Suppose that we have a mapping , where is some vector space equipped with a scalar product (dot-product space); the space is sometimes referred to as the feature space. We can use ridge regression in the feature space. The prediction of ridge regression on a signal can be written as
where
and
the function is given by .
The space does not have to be finite-dimensional. However since every vector space with a scalar product can be embedded into a Hilbert space (see below for a definition) we can assume that it is Hilbert.
The transformation is of no particular importance to us. Once we know the kernel , we can perform regression with it. A justification for the ridge regression in the kernel case will be obtained below in Subsection 5.3.
## 1.6 Mappings and Kernels
It would be nice to have a characterisation of all without a reference to . A characterisation of this kind can be given.
It is easy to see that has the following properties:
• it is symmetric: for all ; this follows from the symmetry of the scalar product ;
• it is positive semi-definite: for every positive integer and every the matrix
is positive semi-definite 5; indeed, is the Gram matrix of the images 6.
Surprisingly these two simple properties are sufficient. Let us call a function satisfying these two properties a kernel. Then the following theorem can be formulated.
Theorem 1. For any set a function is a kernel, i.e., it is symmetric and positive semi-definite, if and only if there is a mapping from into a Hilbert space with a scalar product such that for all .
We proved the "if" part when we defined kernels. The "only if" part follows from the results of the next sections, where we will show that the class of kernels coincides with the class of so called reproducing kernels.
# 2. Reproducing Kernel Hilbert Spaces
In this section we introduce reproducing kernel Hilbert spaces (RKHS) and show some of their basic properties. The presentation is based mainly on [Aronszajn, 1943] and [Aronszajn, 1950] and the reader may consult these papers for more details; note that the former paper is in French.
## 2.1 Hilbert Space
A set of some elements is a Hilbert space if
1. is a vector space over (Hilbert spaces over the complex plain can also be considered, but we shall restrict ourselves to in this article);
2. is equipped with a scalar product (i.e., with a symmetric positive definite bilinear form);
3. is complete w.r.t. the metric generated by the scalar product, i.e., every fundamental sequence of elements of converges.
Some authors require a Hilbert space to be separable, t.e., to have a countable dense subset. For example, [Aronszajn, 1943] reserves the name "Hilbert". We shall not impose this requirement by default.
As a matter of fact all separable Hilbert spaces are isomorphic (the situation is similar to that with finite-dimensional spaces; the separable Hilbert space is "countable-dimensional").
A typical (though not particularly relevant to this article) example of a Hilbert space is provided by , which is the space of all real-valued functions
on such that is Lebesgue-integrable w.r.t. the measure on with the scalar product . Another example is given by , which is the set of infinite sequences , , such that the sum converges. Both and on with the standard Lebesgue measure are separable; therefore they are isomorphic.
## 2.2 Reproducing Kernel Hilbert Spaces: a Definition
Let be a Hilbert space consisting of functions on a set . A function is a reproducing kernel (r.k.) for if
• for every the function (i.e., as the function of the second argument with fixed) belongs to
• the reproducing property holds: for every and every we have . A space admitting a reproducing kernel is called a reproducing kernel Hilbert space (RKHS).
## 2.3 Reproducing Kernel Hilbert Spaces: Some Properties
Let us formulate and prove some basic properties of reproducing kernels.
Theorem 2.
1. If a r.k. for exists, it is unique.
2. If is a reproducing kernel for , then for all and we have .
3. If is a RKHS, then convergence in implies pointwise convergence of corresponding functions.
Proof: In order to prove (1) suppose that there are two r.k. and for the same space . For every the function belongs to and, applying linearity and the reproducing property, we get
The definition of a Hilbert space implies that coincides with and therefore they are equal everywhere as functions.
Property (2) follows immediately from the reproducing property and the Cauchy(-Schwarz-Bunyakovsky) inequality.
Property (3) follows from (2). Indeed, for all and we have
We shall now give an important "internal" characterisation of reproducing kernel Hilbert spaces.
Let consisting of real-valued functions on be a Hilbert space. Take and consider the functional mapping into . It is linear (in ) and is called the evaluation functional.
Note that the evaluation functional is not defined on : the elements of are in fact equivalence classes of functions that coincide everywhere up to a set of measure 0, and thus they are not really defined at every point 7.
Theorem 3. A Hilbert space consisting of real-valued functions on is a RKHS if and only if for every the corresponding evaluation functional is continuous.
Proof: The "only if" part follows from (2) from the previous theorem.
In order to prove the "if" part we need the Riess-Fischer Representation Theorem, which states that every continuous linear functional on a Hilbert space can be represented as the scalar product by some element of the space.
Take . Because the evaluation functional is continuous, there is a unique such that . We can define a mapping by . Let .
We have
and thus . On the other hand for every and we have
Therefore is a r.k. for .
This criterion is quite important. The continuity of the evaluation functional means that it is consistent with the norm: functions and that are close with respect to the norm evaluate to values and that are close. If we consider functions from some space as hypotheses in machine learning and the norm on the space as a measure of complexity, it is natural to require the continuity of the evaluation functional. The theorem shows that all "natural" Hilbert spaces of functions are in fact reproducing kernel Hilbert spaces.
## 2.4 Existence Theorem
We have shown that a r.k. can be represented as . This implies that is
• symmetric due to the symmetry of the scalar product;
• positive semi-definite, i.e., for all the matrix
is positive semi-definite; this holds since is the Gram matrix. Thus is a kernel according to the definition from the previous section. The following theorem shows that the classes of kernels and reproducing kernels coincide.
Theorem 4. Let is a real-valued function of two arguments on . Then is a reproducing kernel for some Hilbert space of functions on if and only if
• is symmetric
• is positive semi-definite. If there is a space admitting as its reproducing kernel, it is unique.
# 3. Proof of the Existence Theorem
In this section we will prove the existence theorem. Let be a kernel.
## 3.1 Linear Combinations: A Dot Product Space
We start the proof by constructing a linear space of functions consisting of linear combinations , where is a positive integer, and . The linearity follows by construction.
The scalar product is defined after the following fashion. Let
(by adding terms with zero coefficients we can ensure that the linear combinations have equal numbers of terms and that all in the combinations are the same). We need to prove that the scalar product is well-defined, i.e., to show that it is independent of particular representations of factors (recall that we are constructing a space of functions rather than formal linear combinations).
Let and . We have
We see that the scalar product can be expressed in terms of values of and thus is independent of a particular representation of as a linear combination. A similar argument can be applied to . The independence follows.
The function is symmetric because is symmetric. For from above we have
because is positive semi-definite. Therefore is positive semi-definite. We have shown that it is a positive semi-definite symmetric bilinear form. One final step is necessary to prove that it is positive definite and therefore a scalar product.
Let us evaluate , where and is some element from . We get
The form and thus satisfy the reproducing property.
Because the form is positive semi-definite, the Cauchy(-Schwarz-Bunyakovsky) inequality holds for it and
where is defined as . Combining this with the reproducing property yields
Therefore implies that for an arbitrary . We have thus shown that is actually positive definite and therefore a scalar product.
The construction is not finished yet because is not necessarily complete. It remains to construct a completion of . It is well known that every linear space with a scalar product has a completion, which is a Hilbert space. However this argument cannot be applied here 8: we need a completion of a specific form, namely, consisting of functions . Note that, however, we have already proved Theorem 1 from Section 1: we can map into some Hilbert space so that the value of the kernel is given by the scalar product of images. The mapping is given by the obvious .
## 3.2 Completion
In this subsection we will construct a completion of .
Let be a fundamental sequence. For every the inequalities
which follow from the previous subsection, imply that the sequence of values is fundamental and therefore has a limit. We can define a function by .
Let consist of all functions thus obtained. Clearly, since each from is the pointwise limit of the sequence .
The scalar product on can be introduced as follows. If is the pointwise limit of and is the pointwise limit of , then .
Let us show that this limit exists. For all positive integers , , and we have
Because the norms of elements of a fundamental sequence are uniformly bounded, the difference can be made as close to 0 as necessary for sufficiently large , , and . Thus there is even a double limit in the sense that for all sufficiently big and the difference becomes arbitrarily small.
Let us show that the scalar product is independent of a choice of fundamental sequences converging to and . Consider two pairs of fundamental sequences, and converging to and and converging to .
Consider the expression . The sequence consisting of functions is clearly fundamental, therefore, as shown above, there must exist a limit . Let us evaluate this limit. There are coefficients and elements such that ( may change as varies). We have
Since and converge pointwise to 0, this expression converges to zero as . Thus
Similarly
Therefore the difference
converges to 0 as . Our definition is thus independent of a particular choice of fundamental sequences.
The bilinearity of on is easy to check. The number is non-negative as a limit of non-negative numbers. More precisely, let be a fundamental sequence converging to pointwise. Because
the equality implies that for all .
We have shown that is indeed a linear space with a scalar product. Clearly, and the scalar product on extends that on .
Let us show that is complete. First, let be a fundamental sequence of elements of converging pointwise to . We have
This converges to 0 as and thus is the limit of in . Secondly, consider a fundamental sequence of elements of . For each there is such that . The sequence is fundamental in and therefore has a limit in . It must be the limit of too.
It remains to show that the reproducing property holds of . It follows by continuity. Let be a fundamental sequence of elements of converging pointwise to . We have
We have constructed a RKHS for . Note that constructed in the previous subsection is dense in it.
## 3.3 Uniqueness
Let us show that the RKHS for a particular kernel is unique. Let be the RKHS constructed above and be some other RKHS for the same kernel .
The definition of an RKHS implies that all functions must belong to . The same must be true of their linear combinations . Thus as a set.
Since the reproducing property holds on , on elements of the scalar product must coincide with scalar product we constructed above. Thus is a subspace of .
Because is complete, all fundamental sequences from should have limits in . In RKHSs convergence implies pointwise convergence and thus all pointwise limits of fundamental sequences from belong to . Thus as a set. Because the scalar product is continuous w.r.t. itself, we have
for all sequences and such that and in as . Thus the scalar product on coincides with that on , or, in other terms, is a closed subspace of .
Let . We can represent it as , where and is orthogonal to and therefore to all functions , which belong to . Because the reproducing property holds on , we get
Thus coincides with everywhere on and .
# 4. What RKHSs are and What They are not
In this section we provide a selection of minor results and counterexamples concerning RKHSs. They will help the reader to get a better intuition of an RKHS and perhaps dissolve some misconceptions.
## 4.1 Continuity and Separability
Suppose that is a topological space and is continuous (in both the arguments, i.e., on ). One can claim that any feature mapping associated with is continuous. Indeed,
The continuity of implies that as approach , the expression approaches 0. We have shown that is continuous 9.
If moreover is separable, then the RKHS is separable. Indeed we have shown in the proof of the existence theorem that finite linear combinations are dense in . The coefficients can be taken to be rational. Take a countable dense subset . Because the mapping is a feature mapping, it is continuous, and we can approximate any with some .
In most natural cases signals are strings of reals, i.e., and the kernel is continuous and therefore the RKHS is separable. Note however that inseparable RKHSs exist. Indeed, consider the space . It consists of functions such that everywhere except, perhaps, for a countable set and
The scalar product is defined by
It is easy to see that this space is Hilbert. The only non-trivial bit is completeness. Let be a fundamental sequence. By taking the union of all supports we can obtain a countable set such that all vanish outside . The sequences belong to the "usual" and we can reduce the question of completeness of to completeness of .
The space is an RKHS. Indeed, consider
(here is the indicator function of a set ). This is clearly a kernel.
On the other hand is not separable. Indeed, the functions , , form an uncountable orthonormal system.
## 4.2 Pointwise Convergence in RKHSs
As we know from Theorem 2, convergence in an RKHS implies pointwise convergence of functions. Let us show that the opposite is not true.
A simple counterexample is provided by considered as a set of functions on the set of positive integers . It is an RKHS with
as above. Consider the sequence of functions
We have as , which implies pointwise and even uniform convergence to equal to 0 everywhere. On the other hand .
Requiring to be continuous and the domain to be connected will not change the situation; the counterexample can be easily extended to cover this possibility. Indeed, take . For each positive integer consider the "tooth" function equal to 0 outside , 1 at and linear on and . Let the space consist of functions , where . A function is continuous on , its values at integer points form a sequence from and the value at is given by , . Let the scalar product be induced by : . This space is isomorphic to .
The space is an RKHS with the following kernel. For a positive integer we have and for all . For , where is a positive integer, . This kernel is continuous on . On the other hand pointwise (and uniform) convergence does not imply convergence in the norm.
It is easy to do a further step and show that can be connected and compact. Note that for any sequence we have as and therefore for any we have as . We can define by continuity. The set can be mapped onto by .
The following intuition may be helpful. As explained above, an RKHS may be thought of as a Hilbert space of hypotheses and the norm as the hypothesis complexity. The functions uniformly close to zero but having non-zero norms correspond to overcomplicated hypotheses with no real explanatory power.
## 4.3 RKHSs with the Same Sets of Functions
We know from part 1 of Theorem 2 that for each RKHS there is a unique kernel, i.e., if two kernels, and , specify the same RKHS, then they coincide. Here the words "the same RKHS" mean the same set of functions with the same scalar product on them.
What if the scalar product is different? Can two different kernels specify RKHSs with the same sets of functions?
It is easy to construct a trivial example. The kernel specifies the same set of functions as the kernel . Indeed, recall the construction from the existence theorem. The kernels and specify the same set of finite linear combinations with equivalent norms. A sequence is fundamental in one space if and only if it is fundamental in another and the completions, consisting of pointwise limits, will coincide.
A slightly less trivial example may be constructed as follows. Let and is given by . If is equipped with the standard Euclidean scalar product , we get a kernel . The RKHS corresponding to this kernel consists of functions of the form , where .
Let is given by . It specifies the kernel , which is not a multiple of . The RKHS corresponding to consists of functions of the form , where . It is easy to see that the set of functions is the set of polynomials of degree 1, just as in the first case. The kernels specifying the RKHSs with the same sets of functions thus do not have to be multiples of each other.
This argument can be generalised further as follows. Let specify kernel and let be a bounded linear operator in the Hilbert space . Consider the mapping and the kernel . The RKHS corresponding to this kernel consists of functions of the form . This can be rewritten as , where is the operator adjoint to . If (and therefore ) is invertible, ranges over the whole space and the RKHS for consists of the same functions as that for .
The following observation can also be made.
Theorem 5. The set of kernels on specifying the RKHS with the same set of functions is closed under multiplication by a positive constant and addition, i.e.,
1. for every kernel and , the kernel specifies the RKHS with the same set of functions as the RKHS for ;
2. for every two kernels and on specifying the RKHSs with the same set of functions, the kernel specifies the RKHS with the same set of functions.
Proof: The multiplicative part has been already proved above. Let us prove the additive part (the multiplicative part can be proved in the same fashion).
Let and be feature mappings corresponding to and . Consider the Hilbert space with the "componentwise" scalar product (here and are the scalar products on and , respectively). Clearly the mapping given by specifies the kernel . The corresponding RKHS consists of functions of the form , where and . Since or can be equal to 0, the RKHS contains the RKHSs for and as sets. Since the two sets of functions are equal, adding them up does not produce anything new.
# 5. RKHSs and Prediction in Feature Spaces
We have shown that the three definitions of a kernel are equivalent:
• a positive semi-definite symmetric function;
• a reproducing kernel;
• the scalar product in a feature space, i.e., , where maps into a Hilbert space . The RKHS for a particular kernel is unique. Note that uniqueness holds in a very strong sense: it is a unique set of functions with a uniquely defined scalar product; there are no isomorphisms or equivalences involved.
The mapping in the third definition is by no means unique. Indeed, let . Consider a right shift defined by . The composition will produce the same kernel as .
However there is some degree of uniqueness. Let be the closure of the linear span of all images , . It is isomorphic to the RKHS.
## 5.1 RKHS Inside a Feature Space
Theorem 6. For every mapping , where is a Hilbert space, the closure of the linear span of the image of , i.e., , is isomorphic to the RKHS of the kernel . There is a canonical isomorphism mapping onto from the RKHS.
Proof: Let us denote the closure of the span by and the RKHS by . Let be the set of finite sums of the form , where and are some coefficients, as in the construction above.
We start by constructing the isomorphism of and . Put and, by linearity, . We need to show that is well-defined. Let for some coefficients and and elements . Then for every we have
i.e.,
by the definition of . This means that the functions and coincide everywhere and thus is well-defined.
The mapping preserves the scalar product:
The mapping is surjective. Indeed, each sum has an inverse image. The mapping is also injective. Assume the converse. Then there is a point such that but in the RKHS. This is a contradiction because preserves the scalar product and therefore the norm. Thus is a bijection.
Let us extend to the isomorphism of and . Let converge to . The sequence is fundamental in . Since preserves the scalar product on , the images form a fundamental sequence in . It should converge. Put .
Suppose that there are two sequences and converging to . Let us mix the sequences into (e.g., by letting and , ). The sequence converges to and is therefore fundamental. The images of must form a fundamental sequence in and must have a limit. All its subsequences should converge to the same limit. Thus and is well-defined.
The scalar product is preserved by continuity. The surjectivity can be shown as follows. Let and let converge to . The inverse images of must form a fundamental sequence in and must have a limit . It follows from the definition that . The injectivity on follows from the same argument as on .
The theorem follows.
The mapping can be extended to the mapping of the whole by letting , where is the projection operator. The mapping is no longer injective (unless coincides with ) and no longer preserves the scalar product. However we have , where the minimum is attained on the projection .
## 5.2 Another Definition of RKHS
The above construction allows us to construct an interpretation of RKHS important for machine learning.
In competitive prediction we prove consistency results of the following type. We do not assume the existence of a "correct" hypothesis but rather show that our method competes well with a class of some other predictors, such as all linear regressors. Therefore identifying and describing such natural classes is an important task.
In Hilbert spaces we have a natural equivalent of linear regressors. Those are linear functionals, or, as the Riess-Fischer Representation Theorem shows, scalar products by an element . After we have mapped into a Hilbert space , we can consider predictors of the form .
Theorem 6 immediately implies that the class of such functions coincides with the RKHS. Indeed, there is a unique decomposition , where is the projection of on and is orthogonal to . We have
where belongs to the RKHS.
We may want to assign the norm to the predictor ; clearly . The space of predictors thus obtained does not exceed the RKHS and the norms of predictors are equal to or greater than those of the elements of the RKHS. Thus regressors in the feature space have no more power than functions from the RKHS.
We get the following theorem as a bonus.
Theorem 7. Let be a mapping into a Hilbert space . The space of functions defined by , where , equipped with the norm coincides with the reproducing kernel Hilbert space for the kernel defined by .
## 5.3 Ridge Regression in Feature Spaces; the Representer Theorem
In this section we revisit the ridge regression problem from Section 1 and present one argument of great importance for competitive prediction.
Suppose that we have a sequence of signals and outcomes as in Section 1. On top of that suppose that we have a mapping from the set of signals into a Hilbert feature space . Take ; as we said before, it can be considered as a regressor yielding the dependency . We may ask if there is minimising the expression
with .
Consider the predictor
where
and
It qualifies as a regressor of the above type. Indeed, it is a linear combination of . Let us show that it minimises .
Let be a subspace of consisting of all linear combinations of (it is closed because it is finite-dimensional). Take . It can be decomposed into a sum , where is orthogonal to . For all we have ; we also have . Therefore . When minimising we can restrict ourselves to predictors from , i.e., linear combinations of !
Because is finite-dimensional, the arguments from Section 1 apply and the ridge regression turns out to provide the minimum of over (or the minimum of over the corresponding RKHS).
This argument can be generalised to the representer theorem. We shall formulate two variants of it.
Theorem 8. Let be a mapping into a Hilbert space and be given by
where is a function from to nondecreasing in the last argument and are some fixed elements. Then for every there is a linear combination , where are constants, such that . If is strictly increasing in the last argument and does not itself have the form , there is a linear combination such that .
Proof: The proof is by observing that the projection of on the subspace has the same scalar products with elements , and a smaller norm .
Corollary 1. Let be a kernel and the corresponding RHKS; let be given by
where is a function from to nondecreasing in the last argument and . Then for every there is a linear combination , where are some constants, such that . If is strictly increasing in the last argument and does not itself have the form , there is a linear combination such that .
# 6. Hierarchies of Kernels
## 6.1 Subspaces of RKHSs and Sums of Kernels
Consider an RKHS corresponding to a kernel . Let be a subspace of . Clearly, is a RKHS. This can be shown as follows. The evaluation functional is continuous on . Its restriction on should remain continuous and therefore is a RKHS. This does not contradict the uniqueness theorem. If is a proper subspace of , it is an RKHS for a different kernel .
Suppose that itself is a subspace of some Hilbert space of functions . As we discussed above, in applications such as machine learning it does not make much sense to consider spaces where the evaluation functional is not continuous, and therefore should be an RKHS with its own kernel too.
One can say that RKHSs form hierarchies: larger spaces have more power than smaller spaces. However each of them has its own kernel. In competitive prediction the natural competitors of a kernel method are the functions from the corresponding RKHS. Other RKHSs require the use of a different kernel.
The rest of this section contains some results clarifying the structure of this hierarchy.
Theorem 9. Let a space of real-valued functions on be the RKHS corresponding to a kernel . If is a subspace of , then is an RKHS and has a kernel . If is the orthogonal complement of , then it is also an RKHS and it has the kernel such that for all .
Proof: Let and be the projection operators from to and , respectively.
Take a point . The evaluation functional on equals the scalar product by . It is easy to see that plays the same role in . Indeed, and for every function we have
Put . Let us prove that it is the kernel for
We do this by showing that as a function of coincides with . Fix and denote the function by . We have . The above argument implies that for every and thus . The reproducing property follows.
Similarly is the kernel for .
Let . We have and ; therefore
By taking and we get .
Theorem 10. Let be three kernels on such that for all . Then the RKHSs and corresponding to the kernels and are subsets (but not necessarily subspaces) of the RKHS corresponding to the kernel . For each there are functions and such that and for its norm we have the equality
If and have only the identical zero function in common (i.e., ), then they are subspaces of and each one is the orthogonal complement of the other.
It is easy to see that does not have to be a subspace of . Indeed, let and . Clearly if we take the set of functions constituting and equip it with the scalar product , we get the RKHS for . It is a subset but not a subspace of .
Proof: Let and mapping into Hilbert spaces and , respectively, be feature maps giving rise to the kernels and , i.e.,
Let be the Hilbert space consisting of pairs such that and with the scalar product given by
Take defined by
It is easy to see that
The results of Subsect. 5.2 imply that the RKHS for coincides with the set of functions , where . Similarly, the RKHSs for and consist of all functions and , respectively, where and .
For every the element belongs to ( is the zero element of here). We have
and therefore as a set; the same argument applies to . The decomposition
implies that each can be decomposed into the sum of and .
We have
The minimum is taken over pairs of ; clearly, we can take the minimum over all pairs of and such that ; indeed, .
Now let . Under this assumption every has a unique decomposition , where and . Indeed, if , then . The function of the left-hand side belongs to and the function on the right-hand side belongs to and therefore they are both equal to zero. Thus for every pair and we have .
Take . Then this equality implies that . Taking leads to . The norms on and coincide with the norm on ; the same should apply to the scalar product and thus and are subspaces rather than just subsets of .
Picking arbitrary and and applying Pythagoras theorem yields . Comparing this with the above equality implies that , i.e., and are orthogonal subspaces.
Let us introduce a relation on kernels on a set . We will write if the difference is a kernel.
If this relation holds, then for all . Indeed, since is a kernel, the matrix is positive semi-definite, i.e., . This observation justifies the notation to some extent and implies that is antisymmetric. Indeed, if and then for we get for all . Theorem 2 implies that for every from the RKHS of and every we have and thus is identically zero. This implies that .
Clearly, is transitive: if , then .
The theorems above imply that for kernels is closely linked with the relation on their RKHSs. However no direct correspondence has been shown. The following results close the gap.
Theorem 11. Let and be two kernels on the same set and let and be their RKHSs. If , then as a set and for every we have
This theorem follows from our previous results. Indeed, the square is given by the minimum of taken over all decompositions , where is the RKHS corresponding to the difference . Every can be represented as , which implies the inequality in the theorem.
The opposite result holds.
Theorem 12. Let and be its RKHSs. If as a set and forms a Hilbert space w.r.t. a norm such that
for every , then is an RKHS and its reproducing kernel satisfies .
It is easy to see that the inequality on the norms cannot be omitted. Consider some kernel on and let . Let be the RKHS for . For every we have
and therefore that coincides with as a set and has the scalar product is the RKHS for . We have . However let us consider as a subset of . It satisfies the conditions of the theorem apart from the norm clause but .
The proof of the theorem is beyond the scope of this article and can be found in [Aronszajn, 1950], pp.355-356.
## 6.2 Subsets of the Domain
The representer theorem provides a motivation to the following analysis important for learning.
Suppose that we have a kernel and let be its RKHS. Consider a subset and let be the restriction of to . It is easy to see that is still a kernel. What can one say about the RKHS corresponding to ?
This question is important for learning for the following reason. Suppose that we are given a training sample (or a history of past signals and outcomes, if we are using an on-line protocol) and a new signal . One can consider different sets containing the observed signals . What effect will the choice of a particular have on the consistency analysis of a learning method? The representer theorem above suggests that the choice of is essentially irrelevant. Padding does not really change much. (Of course the situation becomes different if we need to make a statement valid for all s that can possibly occur.) We shall see that this is generally true.
Let us try and clarify the situation. First, does contain more functions on ? Are restrictions of functions from form a set richer than ?
This is easy to answer if we use the definition of a RKHS provided by Theorem 7. Let be a mapping such that for all (here is a Hilbert space as usual). Then coincides with the set of functions of the form , where ranges over and the norm of is the minimum of the norms of all elements corresponding to . Clearly represents as well as : we have for all . Thus consists of functions of the form restricted to .
This implies that the restrictions of all functions from to belong to . On the other hand for every there is such that and we can extend to by considering scalar products with the same . Therefore each function from is a restriction of some function from . The norm of does not exceed that of each extension; on the other hand it equals to the norm of some and therefore to the norm of some extension.
Theorem 13. Let a space of real-valued functions on be the RKHS corresponding to a kernel and let . Then the RKHS corresponding to the kernel coincides with the set of restrictions to of functions from and the norm on it is given by .
The mapping from to provided by the restriction operator does not have to be injective. Indeed two elements can have the same scalar products with all elements from but different scalar products with some elements of . One can easily identify a subspace of isomorphic to though.
Theorem 14. Let a space of real-valued functions on be the RKHS corresponding to a kernel and let . Then the closure of the set of finite linear combinations of the form , where and , i.e., the subspace is isomorphic to the RKHS corresponding to the kernel and an isomorphism is given by the restriction of functions from to .
Proof: Let us denote the restriction operator by . It maps each into its restriction .
We already know that maps into and it does not increase the norm. It is easy to see that it is linear. We need to show that as a mapping from into is injective, surjective, and actually preserves the norm.
Consider the set of finite linear combinations of the form , where . We can consider them as elements of and elements of . It is easy to see that "" these linear combinations, i.e., (note that on the left the combination is considered as an element of and on the right the same combination is an element of ).
Let us show that restricted to functions that are finite linear combinations of the above form is an isomorphism. Its surjectivity follows from the above. On the other hand we have
The mapping thus preserves the norm of a linear combination. This implies that is injective because cannot map a non-zero element into zero.
Now consider on the subspace . Let ; it can be approximated by a linear combination to any degree of precision w.r.t. the norm . However does not increase the norm and we therefore have
Because , the norm of is also preserved, which immediately implies that is an injection on .
To prove the surjectivity we need to remember that the finite linear combinations are dense in . If , it is the limit of a sequence of finite linear combinations. Their inverse images (i.e., the same combinations) in will form a fundamental sequence because the norm is preserved and therefore will converge. The image of the limit has to be the original .
Corollary 2. The orthogonal complement of w.r.t. consists of all functions such that for all .
Proof: Consider and . We have . If is orthogonal to , it is orthogonal to each and therefore for all . If on , then it is orthogonal to all , where , all their linear combinations and therefore to all elements of .
# 7. Tensor Products of Kernels
Suppose that we have two kernels and . Let be the Cartesian product . We can consider the function defined by
We will call it the tensor product 10 of and and use the notation . We will show that the tensor product is also a kernel and construct the corresponding RKHS.
Theorem 15. If and are kernels then their tensor product , where , defined by
is also a kernel.
We shall prove this theorem later. Let us first derive some corollaries.
Corollary 3. Let and be two kernels defined on the same domain . Then their componentwise product defined by
is also a kernel.
Proof: Inside the direct product consider the "diagonal" set defined by . The restriction of the tensor product coincides with the componentwise product on . Clearly, it is therefore a kernel.
Theorem 13 implies that the RKHS corresponding to the componentwise product consists of restrictions of functions from the RKHS corresponding to the tensor product to .
We obtain the following linear algebra result as a bonus.
Corollary 4. The componentwise product of two symmetric positive-definite matrices is symmetric and positive-definite.
The corollary can be proved by considering a symmetric positive-definite matrix as a kernel on a finite set.
Let us prove the theorem about the tensor product. We shall prove it by constructing the RKHS for the product . The space is the tensor product of RKHSs with the scalar product corresponding to and with the scalar product corresponding to . We cannot employ the standard construction (see e.g., [Weidmann, 1980], Section 3.4) directly because we need to ensure that the resulting space is still a space of functions rather than a generic Hilbert space.
Consider the set of functions on of the form , where and . As often happens when we try to construct a tensor product of some spaces, this set is not necessarily closed under addition. Therefore let us consider the space of finite sums , where is a positive integer, and . It is clearly a vector space.
Let us define the scalar product of and by
We need to show that the scalar product is well-defined, i.e., it depends on functions rather than their representations in the form of sums of products. We have
The inner sum is a function on . We have
We can thus get rid of a particular representation of as a sum of products; only the values of matter. Clearly, the same argument applies to . The scalar product is thus well-defined.
It still remains to show that satisfies the properties of a scalar product. It is easy to check that it is a bilinear form. We need to show that and vanishes only for . Let us take and apply an orthonormalisation procedure (e.g., the Gram-Schmidt method) to in and to in . After renaming the functions, we get a representation , where, for both and , the equality holds for all and holds for all . For the scalar product we have
If , all are 0 and therefore .
Let us now find out the role of the product of kernels defined by
Fix and . The function
is a product of functions from and , i.e., has the form , where and .
For we get
The resulting space with a scalar product can be incomplete, so we need to construct its completion; the situation is somehow similar to that with the existence theorem. We cannot use the standard result about completions of pre-Hilbert spaces, because we need to obtain a space of functions.
In order to construct the completion, we need to recall some facts about orthonormal bases in Hilbert spaces. Let be a (pre-)Hilbert space with a scalar product . An orthonormal system is a system of vectors , where is some index set, not necessarily countable, such that equals 0 for and 1 for . It is easy to check that all finite subsets of an orthonormal system are linearly independent.
If is an orthonormal system, then for each only countably many scalar products are non-zero and the Bessel inequality holds.
An orthonormal system is an orthonormal base, if its span (the set of all finite linear combinations of its elements) is dense in , i.e., . An orthonormal system is an orthonormal base if and only if for every the Parseval equality holds. The Parseval equality implies that . (The convergence of this series can be understood as follows. If , is a sequence of (pairwise different) indexes including all such that , then as .) The scalar product of is given by . This generalises the finite-dimensional facts about the coordinate decomposition of a vector and its length.
Each Hilbert space has an orthonormal base; this follows from Zorn's lemma. This system is countable if and only if is separable.
Let vectors , form an orthonormal system in a Hilbert space. The series converges if and only if the sequence belongs to . Each separable Hilbert space is thus isomorphic to . As a matter of fact, any Hilbert space is isomorphic to , where is the index set of its orthonormal base. The space consists of systems of reals , where only countably many elements are non-zero and . The addition and multiplication by a constant are defined componentwise. The scalar product of and is .
Let us now build the completion. Take an orthonormal base in and an orthonormal base in . The index sets and do not have to be countable 11 Consider the class of functions of the form , where only countably many coefficients are non-zero and . Let us show that this series is absolutely convergent for each . Fix ; the Cauchy(-Schwarz-Bunyakovsky) inequality implies that
where the first sum on the left-hand side must converge and second is still to be analysed. Recall that is an RKHS and the evaluation functional at a point corresponds to the scalar product by . Thus . The Bessel inequality implies that
and therefore
Repeating the same argument, we get
Take and fix the first coordinate. The function belongs to as a converging sum of . (Indeed, the sum converges and .) We can calculate the scalar product
This expression considered as a function of belongs to . Clearly its scalar product with equals . The coordinates are thus uniquely defined by the function . We can define a scalar product as the sum of the products of corresponding coordinates. The resulting space is isomorphic to . Let us denote if by . It is easy to see that it is complete and therefore Hilbert.
Let us show that contains all products of the form , where and . Indeed, if and then and the sum
converges. Finite linear combinations of functions of this form are dense in ; indeed, the expression for can be truncated to a finite number of terms. The scalar product we introduced on functions coincides with the product in . Indeed, let , , , and . We have
By linearly this property can be extended to finite linear combinations.
One can conclude, by continuity, that
is a reproducing kernel for the Hilbert space of functions .
# 8. Some Popular Kernels
In this section we summarise some sufficient conditions for a function to be a kernel and obtain some popular kernels.
Let us recall the facts we know and formulate simple corollaries.
Theorem 16. A function is a kernel if and only if it is symmetric and positive-semidefinite.
This is Theorem 4 or the existence theorem.
Corollary 5. If is a kernel and , then is a kernel.
Corollary 6. If and are kernels on the same set, then their sum is a kernel on the set.
Corollary 7. The pointwise limit of kernels is a kernel.
Proof: Let be the pointwise limit of and each is a kernel.
Because is symmetric, is symmetric too. For any finite array and we have . As , we get and thus is positive semi-definite.
Theorem 17. If and are kernels on the same set, then their product is a kernel on the set.
This was proven as Corollary 3.
Corollary 8. Let be a kernel and is a polynomial with positive coefficients. Then is a kernel.
Note the requirement that the coefficients should be positive. Indeed, is not a kernel unless is identically equal to zero.
Corollary 9. If is a kernel, then is a kernel.
Proof: The exponent can be decomposed into Taylor's series on . The function is a pointwise limit of polynomials on and all these polynomials have positive coefficients.
Theorem 18. A function is a kernel if and only if there is a mapping , where is a vector space with a scalar product , such that .
Corollary 10. If is a real vector-function on , then
is a kernel on .
Corollary 11. The function is a kernel on , .
This fact is obvious and can be derived by many different ways.
We can now obtain Vapnik's inhomogeneous polynomial kernel.
Corollary 12. For any positive integer the function is a kernel on .
Proof: Indeed, is a kernel and is a polynomial with positive coefficients.
The following corollary introduces radial-basis function (rbf) kernel.
Corollary 13. For any , the function , where is the Euclidean norm, is a kernel on .
Proof: We have
The product of the first two terms is a kernel generated by the mapping . The third term is a kernel as the exponent of a kernel. The product of two kernels is also a kernel.
1 As different, for example, to the classification problem, where s belong to a finite set.
2 Indeed, let be positive definite. If is singular, for some , but this implies .
3 The condition is not just sufficient, but necessary for to be singular. Indeed, if is singular, it is not positive definite and for some . This implies , i.e., columns of are linearly dependent.
4 It is important that no particular structure is postulated on ; throughout the most of this article it is just a set.
5 A definition and a discussion were given above.
6 We have .
7 Thus is not an RKHS according to our definition: it is disqualified without a substantive consideration.
8 The book [Schölkopf and Smola 2002] uses this argument in Section 2.2.3, p. 35-36, in a rather misleading way.
9 Proposition 2.14 from [Schölkopf and Smola 2002] states that continuous maps exist with a reference to [Parthasary and Smith, 1972], where the stronger statement is proven, but the weaker statement is claimed.
10 Aronszajn uses the term "direct product". The reason why we use the term "tensor product" will become apparent later
11 Paper [Aronszajn, 1950] considers only the countable case and its proof is formally valid only for separable RKHSs. However minor amendments can make it valid everywhere. | {"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": 0.9875653386116028, "perplexity": 276.39847266850103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1500549423785.29/warc/CC-MAIN-20170721142410-20170721162410-00376.warc.gz"} |
http://ufldl.stanford.edu/wiki/index.php?title=Softmax_Regression&diff=next&oldid=610 | # Softmax Regression
(Difference between revisions)
Revision as of 23:24, 7 May 2011 (view source)Zellyn (Talk | contribs)m (→Parameterization)← Older edit Revision as of 23:26, 7 May 2011 (view source)Zellyn (Talk | contribs) m (→Binary Logistic Regression)Newer edit → Line 190: Line 190: &= &= - \frac{e^{ \Theta_1^T x^{(1)} } }{ 1 + e^{ \Theta_1^T x^{(i)} } } + \frac{e^{ \Theta_1^T x^{(i)} } }{ 1 + e^{ \Theta_1^T x^{(i)} } } \cdot \cdot - \frac{1}{e^{ \Theta_1^T x^{(1)} } } + \frac{1}{e^{ \Theta_1^T x^{(i)} } } \begin{bmatrix} \begin{bmatrix} e^{ \Theta_1^T x^{(i)} } \\ e^{ \Theta_1^T x^{(i)} } \\
## Introduction
Softmax regression, also known as multinomial logistic regression, is a generalisation of logistic regression to problems where there are more than 2 class labels.
Recall that in logistic regression, our hypothesis was of the form:
\begin{align} h_\theta(x) = \frac{1}{1+\exp(-\theta^Tx)}, \end{align}
where we trained the logistic regression weights to optimize the (conditional) log-likelihood of the dataset using p(y | x) = hθ(x). In softmax regression, we are interested in multi-class problems where each example (input image) is assigned to one of k labels. One example of a multi-class classification problem would be classifying digits on the MNIST dataset where each example has label 1 of 10 possible labels (i.e., where it is the digit 0, 1, ... or 9).
To extend the logistic regression framework which only outputs a single probability value, we consider a hypothesis that outputs K values (summing to 1) that represent the predicted probability distribution. Formally, let us consider the classification problem where we have m k-dimensional inputs $x^{(1)}, x^{(2)}, \ldots, x^{(m)}$ with corresponding class labels $y^{(1)}, y^{(2)}, \ldots, y^{(m)}$, where $y^{(i)} \in \{1, 2, \ldots, n\}$, with n being the number of classes.
Our hypothesis hθ(x), returns a vector of probabilities, such that
\begin{align} h(x^{(i)}) = \begin{bmatrix} P(y^{(i)} = 1 | x^{(i)}) \\ P(y^{(i)} = 2 | x^{(i)}) \\ \vdots \\ P(y^{(i)} = n | x^{(i)}) \end{bmatrix} = \frac{1}{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } \begin{bmatrix} e^{ \theta_1^T x^{(i)} } \\ e^{ \theta_2^T x^{(i)} } \\ \vdots \\ e^{ \theta_n^T x^{(i)} } \\ \end{bmatrix} \end{align}
where $\theta_1, \theta_2, \ldots, \theta_n$ are each k-dimensional column vectors that constitute the parameters of our hypothesis. Notice that $\frac{1}{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} }$ normalizes the distribution so that it sums to one.
Strictly speaking, we only need n − 1 parameters for n classes, but for convenience, we use n parameters in our derivation.
Now, this hypothesis defines a predicted probability distribution given some x, P(y | x(i)) = h(x(i)). Thus to train the model, a natural choice is to maximize the (conditional) log-likelihood of the data, $l(\theta; x, y) = \sum_{i=1}^{m} \ln { P(y^{(i)} | x^{(i)}) }$.
Motivation: One reason for selecting this form of hypotheses comes from connections to linear discriminant analysis. In particular, if one assumes a generative model for the data in the form $p(x,y) = p(y) \times p(x | y)$ and selects for p(x | y) a member of the exponential family (which includes Gaussians, Poissons, etc.) it is possible to show that the conditional probability p(y | x) has the same form as our chosen hypotheses h(x). For more details, see the CS 229 Lecture 2 Notes.
## Optimizing Softmax Regression
Expanding the log-likelihood expression, we find that:
\begin{align} \ell(\theta) &= \ln L(\theta; x, y) \\ &= \ln \prod_{i=1}^{m}{ P(y^{(i)} | x^{(i)}) } \\ &= \sum_{i=1}^{m}{ \ln \frac{ e^{ \theta^T_{y^{(i)}} x^{(i)} } }{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } } \\ &= \theta^T_{y^{(i)}} x^{(i)} - \ln \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} \end{align}
Unfortunately, there is no closed form solution to this optimization problem (although it is concave), and we usually use an off-the-shelf optimization method (e.g., L-BFGS, stochastic gradient descent) to find the optimal parameters. Using these optimization methods require computing the gradient ($\ell(\theta)$ w.r.t. θk), which can can be derived as follows:
\begin{align} \frac{\partial \ell(\theta)}{\partial \theta_k} &= \frac{\partial}{\partial \theta_k} \theta^T_{y^{(i)}} x^{(i)} - \ln \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} \\ &= I_{ \{ y^{(i)} = k\} } x^{(i)} - \frac{1}{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } \cdot e^{ \theta_k^T x^{(i)} } \cdot x^{(i)} \qquad \text{(where } I_{ \{ y^{(i)} = k\} } \text{is 1 when } y^{(i)} = k \text{ and 0 otherwise) } \\ &= x^{(i)} ( I_{ \{ y^{(i)} = k\} } - P(y^{(i)} = k | x^{(i)}) ) \end{align}
With this, we can now find a set of parameters that maximizes $\ell(\theta)$, for instance by using L-BFGS with minFunc.
### Weight Regularization
When using softmax regression in practice, it is important to use weight regularization. In particular, if there exists a linear separator that perfectly classifies all the data points, then the softmax-objective is unbounded (given any θ that separates the data perfectly, one can always scale θ to be larger and obtain a better objective value). With weight regularization, one penalizes the weights for being large and thus avoids these degenerate situations.
Weight regularization is also important as it often results in models that generalize better. In particular, one can view weight regularization as placing a (Gaussian) prior on θ so as to prefer θ with smaller values.
In practice, we often use a L2 weight regularization on the weights where we penalize the squared value of each element of θ. Formally, we use:
\begin{align} w(\theta) = \frac{\lambda}{2} \sum_{i}{ \sum_{j}{ \theta_{ij}^2 } } \end{align}
This regularization term is added together with the log-likelihood function to give a cost function, J(θ), which we want to minimize (note that we want to minimize the negative log-likelihood, which corresponds to maximizing the log-likelihood):
\begin{align} J(\theta) = -\ell(\theta) + \frac{\lambda}{2} \sum_{i}{ \sum_{j}{ \theta_{ij}^2 } } \end{align}
The gradients with respect to the cost function must then be adjusted to account for the weight decay term:
\begin{align} \frac{\partial J(\theta)}{\partial \theta_k} &= x^{(i)} ( I_{ \{ y^{(i)} = k\} } - P(y^{(i)} = k | x^{(i)}) ) + \lambda \theta_k \end{align}
Minimizing J(θ) now performs regularized softmax regression.
## Parameterization
We noted earlier that we actually only need n − 1 parameters to model n classes. To see why this is so, consider our hypothesis again:
\begin{align} h(x^{(i)}) &= \frac{1}{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } \begin{bmatrix} e^{ \theta_1^T x^{(i)} } \\ e^{ \theta_2^T x^{(i)} } \\ \vdots \\ e^{ \theta_n^T x^{(i)} } \\ \end{bmatrix} \\ &= \frac{e^{ \theta_n^T x^{(i)} } }{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } \cdot \frac{1}{e^{ \theta_n^T x^{(i)} } } \begin{bmatrix} e^{ \theta_1^T x^{(i)} } \\ e^{ \theta_2^T x^{(i)} } \\ \vdots \\ e^{ \theta_n^T x^{(i)} } \\ \end{bmatrix} \\ &= \frac{1}{ \sum_{j=1}^{n}{e^{ (\theta_j^T - \theta_n^T) x^{(i)} }} } \begin{bmatrix} e^{ (\theta_1^T - \theta_n^T) x^{(i)} } \\ e^{ (\theta_2^T - \theta_n^T) x^{(i)} } \\ \vdots \\ e^{ (\theta_n^T - \theta_n^T) x^{(i)} } \\ \end{bmatrix} \\ \end{align}
Letting Θj = θj − θn for $j = 1, 2 \ldots n - 1$ gives
\begin{align} h(x^{(i)}) &= \frac{1}{ 1 + \sum_{j=1}^{n-1}{e^{ \Theta_j^T x^{(i)} }} } \begin{bmatrix} e^{ \Theta_1^T x^{(i)} } \\ e^{ \Theta_2^T x^{(i)} } \\ \vdots \\ 1 \\ \end{bmatrix} \\ \end{align}
Showing that only n − 1 parameters are required.
In practice, however, it is often easier to implement the version which is over-parametrized although both methods will lead to the same classifier.
### Binary Logistic Regression
In the special case where n = 2, one can also show that softmax regression reduces to logistic regression:
\begin{align} h(x^{(i)}) &= \frac{1}{ 1 + e^{ \Theta_1^T x^{(i)} } } \begin{bmatrix} e^{ \Theta_1^T x^{(i)} } \\ 1 \\ \end{bmatrix} \\ &= \frac{e^{ \Theta_1^T x^{(i)} } }{ 1 + e^{ \Theta_1^T x^{(i)} } } \cdot \frac{1}{e^{ \Theta_1^T x^{(i)} } } \begin{bmatrix} e^{ \Theta_1^T x^{(i)} } \\ 1 \\ \end{bmatrix} \\ &= \frac{1}{ e^{ -\Theta_1^T x^{(i)} } + 1 } \begin{bmatrix} 1 \\ e^{ -\Theta_1^T x^{(i)} } \\ \end{bmatrix} \\ \end{align} | {"extraction_info": {"found_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": 20, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000097751617432, "perplexity": 3002.1959747435826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221214713.45/warc/CC-MAIN-20180819070943-20180819090943-00228.warc.gz"} |
http://www.dolfin-adjoint.org/en/latest/documentation/maths/1-foreword.html | # Foreword¶
Written by Patrick E. Farrell
Far too often, maths books launch into their subject without explaining to the novice reader why he or she should care about it in the first place. So, before diving into the details, let’s take a few minutes to motivate why adjoint techniques were invented.
Suppose an aeronautical engineer wishes to design a wing. The wing is parametrised by a vector $$m$$; for example, suppose each entry of $$m$$ is the coefficient of a Bézier curve. For any potential wing design $$m$$, the Euler equations can be solved, and the lift-to-drag ratio $$J$$ of the design computed. With an adjoint, the engineer can do far more: the adjoint computes the derivative of the drag with respect to the design parameters. This can be used to guide a human designer, or can be passed to an automated optimisation algorithm to automatically compute an optimal shape. [jameson1988] [giles2000]. In the literature, this concept is referred to as adjoint design optimisation.
Suppose a meteorologist wishes to improve a forecast by constraining the weather model to match atmospheric observations. The state of the atmosphere at the initial time is partially known (from weather stations), but in order to initialise the model an initial condition for the whole world is required. For any guess of the (unknown) initial state of the atmosphere $$m$$, the Navier-Stokes and related equations can be solved, and the weighted misfit $$J$$ between the observed values and the simulation results can be computed. With an adjoint, the meteorologist can systematically update their guess for the initial state of the atmosphere to match the observations [ledimet1986] [talagrand1987]. In the literature, this concept is referred to as variational data assimilation, 3D-Var and 4D-Var.
Suppose an oceanographer wishes to understand the impact of bottom topography on transport through the Drake passage. Bottom topography (the shape of the sea floor) is quite poorly known; many areas of the world are sparsely observed, and observations from over a century ago are still used in some places. The bottom topography is represented as a scalar field $$m$$, the Navier-Stokes and related equations are solved, and the average net transport through the Drake passage $$J$$ computed. With an adjoint, the oceanographer can see where the transport is most sensitive to the topography, and so quantify where the uncertainty matters most [losch2007]. In the literature, this concept is referred to as sensitivity analysis.
Suppose a nuclear engineer working for a government regulator wishes to examine a proposed new nuclear reactor design. To do this, a forward model of the Boltzmann transport equations will be used to simulate the proposed design and verify its safety. However, all simulations inherently come with discretisation errors, and unless those errors are quantified, the simulations cannot be relied upon to make decisions upon which millions of lives and billions of pounds depend. With an adjoint, the engineer can quantify the impact of discretisation errors on the criticality rate, and decide to what extent the simulations may be trusted [becker2001]. In the literature, this concept is referred to as goal-based error estimation, or goal-based adaptivity.
Suppose a mathematician wishes to understand the stability of some physical system. The traditional approach to this problem is to linearise the operator and investigate its eigenvalues, which determine the long-term behaviour of the system (as $$t \rightarrow \infty$$). However, systems that are eigenvalue-stable can exhibit unexpected transient growth of small perturbations, which in turn can cause the system to become unstable (through nonlinear effects) [trefethen1993]. By computing the singular value decomposition of the tangent linear model, the transient growth of the system to such perturbations can be quantified, and the optimally growing perturbations identified [farrell1996]. The computation of the singular value decomposition in turn requires the adjoint. In the literature, this approach is referred to as generalised stability theory.
As you can see, adjoints show up in many applications, and in many computational techniques. One of the reasons why adjoints have a reputation for being difficult is because their discussion is performed in many different areas of science, usually with their own specialised terminology. Reading the literature, there are almost as many ways to approach the topic as there are practitioners! With this introduction, I hope to strike to the heart of the matter, and clear some of the confusion with the minimum of application– or technique–specific lingo.
## A note on the exposition¶
I have chosen to motivate adjoints via a discussion of PDE-constrained optimisation for two reasons. The first is that this approach encapsulates many important applications of adjoints in a general way, and so the reader will be well-equipped to understand much adjoint-related mathematics in the literature. The second is the elegance of the result: most people are amazed when they first learn that it is possible to compute the gradient of a functional $$\widehat{J}(m)$$ in a cost independent of the number of parameters $$\textrm{dim}(m)$$! The topic of adjoints is intriguing, counterintuitive and beautiful; any exposition should try to live up to that.
The focus of the exposition will be on getting the core ideas across, and for this reason the discussion will sometimes neglect technicalities. For example, I will implicitly assume that all problems are well-posed, that all necessary derivatives exist and are sufficiently smooth, etc. Occasionally, to build intuition, I will refer to objects as matrices and vectors, although the exposition holds in exactly the same way for their analogues in functional analysis. For an advanced in-depth technical treatment of PDE-constrained optimisation, see the excellent book of Hinze et al. [hinze2009].
## Notation¶
The notation is mostly inspired by Gunzburger [gunzburger2003].
Symbol Meaning
$$m$$ the vector of parameters
$$u$$ the solution of the PDE
$$F(u, m)$$ the PDE relating $$u$$ and $$m$$: $$F \equiv 0$$
$$J(u, m)$$ a functional of interest
$$\widehat{J}(m)$$ the functional considered as a pure function of $$m$$: $$\widehat{J}(m) = J(u(m), m)$$
In the next section, we introduce the PDE-constrained optimisation problem and give a broad overview of how it may be tackled.
References
[1M-BR01] R. Becker and R. Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10:1–102, 2001. doi:10.1017/S0962492901000010.
[1M-FI96] B. F. Farrell and P. J. Ioannou. Generalized stability theory. Part I: Autonomous operators. Journal of the Atmospheric Sciences, 53(14):2025–2040, 1996. doi:10.1175/1520-0469(1996)053<2025:GSTPIA>2.0.CO;2.
[1M-GP00] M. B. Giles and N. A. Pierce. An introduction to the adjoint approach to design. Flow, Turbulence and Combustion, 65(3-4):393–415, 2000. doi:10.1023/A:1011430410075.
[1M-Gun03] M. D. Gunzburger. Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM, 2003. ISBN 089871527X.
[1M-HPUU09] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints. Volume 23 of Mathematical Modelling: Theory and Applications. Springer, 2009. ISBN 978-1-4020-8838-4.
[1M-Jam88] A. Jameson. Aerodynamic design via control theory. Journal of Scientific Computing, 3(3):233–260, 1988. doi:10.1007/BF01061285.
[1M-LDT86] F.-X. Le Dimet and O. Talagrand. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A, 38A(2):97–110, 1986. doi:10.1111/j.1600-0870.1986.tb00459.x.
[1M-LH07] M. Losch and P. Heimbach. Adjoint sensitivity of an ocean general circulation model to bottom topography. Journal of Physical Oceanography, 37(2):377–393, 2007. doi:10.1175/JPO3017.1.
[1M-TC87] O. Talagrand and P. Courtier. Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. Quarterly Journal of the Royal Meteorological Society, 113(478):1311–1328, 1987. doi:10.1002/qj.49711347812.
[1M-TTRD93] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll. Hydrodynamic stability without eigenvalues. Science, 261(5121):578–584, 1993. doi:10.1126/science.261.5121.578. | {"extraction_info": {"found_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": 0.8665491342544556, "perplexity": 945.0198298603748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1590347396089.30/warc/CC-MAIN-20200528104652-20200528134652-00159.warc.gz"} |
https://math.stackexchange.com/questions/58890/similarity-between-special-matrices-and-special-complex-numbers | # Similarity between special matrices and special complex numbers
From Wikipedia:
It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers:
• Invertible matrices are analogous to non-zero complex numbers
• Unitary matrices are analogous to complex numbers whose absolute value is 1
• Hermitian matrices are analogous to real numbers
• Hermitian positive definite matrices are analogous to positive real numbers
• Skew Hermitian matrices are analogous to purely imaginary numbers
I also happen to see that the number analogy of each kind of special matrices also agree with their eigenvalues:
• eigenvalues of invertible matrices are non-zero complex numbers
• eigenvalues of Unitary matrices are complex numbers whose absolute value is 1
• eigenvalues of Hermitian matrices are real numbers
• eigenvalues of Hermitian positive definite matrices are positive real numbers
• eigenvalues of Skew Hermitian matrices are purely imaginary numbers
I wonder
1. if the relation between the eigenvalues of special matrices and their number analogy is just coincidence, or there are something inherent, fundamental and more than analogy?
2. if there are other ways than eigenvalues by which the special matrices and special numbers are similar to each other? (I actually don't quite understand what Wikipedia means by analogy. The example in terms of eigenvalues is just my guess. The author may have other things in mind and there may be other possibilities.)
3. if there are other kinds of special matrices not mentioned in the list have similar analogy to special numbers? (Hermitian negative definite is too trivial to mention.)
4. if there are some relevant references? (It seems that this kind of analogy is mentioned in Halmos "Linear Algebra", but cannot find where it is.)
Thanks and regards!
• Your entire list is based on the identification of $\mathbb{C}$ with the complex $1 \times 1$ matrices. The analogy is obtained by replacing $1$ by $n$ throughout. I'll remove the (elementary-number-theory) tag and before you ask: elementary number theory is generally used for properties of natural numbers or integers, divisibility, etc. – t.b. Aug 21 '11 at 22:56
• Something related... – J. M. is a poor mathematician Aug 21 '11 at 22:56
• Re: reference in Halmos: section 71, page 139 "[...] Hermitian matrices play the same role as real numbers." It seems that much of the part on orthogonality is built around the analogy you ask about. – t.b. Aug 21 '11 at 23:01
• @Theo: Thanks! The first quote mentioned "different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers" followed by "invertible matrices are analogous to non-zero complex numbers". I was wondering if invertible matrices are not all normal matrices, and the quote meant "invertible normal matrices" when saying "invertible matrices"? – Tim Aug 23 '11 at 22:26
• Normal matrices are diagonalizable, invertible matrices aren't in general: e.g. $\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$ is invertible but not normal (check this!). No, I don't think Halmos meant to write invertible normal matrices, as the invertible normal $n \times n$-matrices don't form a group under matrix multiplication if $n \geq 2$ while the invertible matrices do. – t.b. Aug 23 '11 at 22:31
The complex numbers and the algebra of all $n \times n$ complex matrices are particular examples of (complex) Banach algebras with involution. The involution is somewhat (though not always perfectly) analogous to complex conjugation. The algebra of bounded linear operators on a Hilbert space is one example of such an algebra (mentioned in the reference of Halmos in Theo's comment), which includes both these cases as subcase. In that case, the involution sends an operator to its adjoint, and the analogy with complex conjugation is quite strong. The behaviour of an operator with respect to this involution is strongly reflected in the behaviour of its spectrum (in the case of finite dimensional vector spaces, spectrum of a matrix equates to the set of eigenvalues of that matrix). However, the complex numbers is unique among (complex) Banach algebras, in that the only Banach division algebra (that is, Banach algebra in which every non-zero element has a multiplicative inverse) up to isomorphism is the complex numbers itself (this is the Banach-Mazur theorem).
I think what you're noticing about eigenvalues comes down to this: if $\lambda$ is an eigenvalue of $A$, with corresponding eigenvector $\bf v$, then multiplying $\bf v$ by $A$ is the same as multiplying $\bf v$ by $\lambda$, that is, $A{\bf v}=\lambda{\bf v}$. Also, if $A$ has a dominant eigenvalue $\lambda$ (an eigenvalue exceeding all the others in modulus), then the entries of $A^n$ grow as $\lambda^n$ as $n$ increases, and the length of $A^n{\bf v}$ is (in the limit) proportional to $\lambda^n$ for all $\bf v$ outside of a proper subspace. So in these senses, matrices behave like their eigenvalues. | {"extraction_info": {"found_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": 0.9314733743667603, "perplexity": 381.0924428389173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526489.6/warc/CC-MAIN-20190720070937-20190720092937-00071.warc.gz"} |
https://www.science.gov/topicpages/n/nuclear+hamiltonian+ab.html | #### Sample records for nuclear hamiltonian ab
1. The nuclear monopole Hamiltonian
Duflo, J.; Zuker, A. P.
1999-05-01
The monopole Hamiltonian Hm is defined as the part of the interaction that reproduces the average energies of configurations. After separating the bulk contributions, we propose a minimal form for Hm containing six parameters adjusted to reproduce the spectra of particle and hole states on doubly magic cores. The mechanism of shell formation is then explained. The reliability of the parametrization is checked by showing that the predicted particle-hole gaps are consistent with experimental data, and that the monopole matrix elements obtained provide the phenomenological cure made necessary by the bad saturation and shell properties of the realistic NN interaction. Predictions are made for the yet unobserved levels around 132Sn, 22O, 34,42Si, 68,78Ni, and 100Sn and for the particle-hole gaps in these nuclei.
2. Construction of diabatic Hamiltonian matrix from ab initio calculated molecular symmetry adapted nonadiabatic coupling terms and nuclear dynamics for the excited states of Na3 cluster.
PubMed
2013-04-25
We present the molecular symmetry (MS) adapted treatment of nonadiabatic coupling terms (NACTs) for the excited electronic states (2(2)E' and 1(2)A1') of Na3 cluster, where the adiabatic potential energy surfaces (PESs) and the NACTs are calculated at the MRCI level by using an ab initio quantum chemistry package (MOLPRO). The signs of the NACTs at each point of the configuration space (CS) are determined by employing appropriate irreducible representations (IREPs) arising due to MS group, and such terms are incorporated into the adiabatic to diabatic transformation (ADT) equations to obtain the ADT angles. Since those sign corrected NACTs and the corresponding ADT angles demonstrate the validity of curl condition for the existence of three-state (2(2)E' and 1(2)A1') sub-Hilbert space, it becomes possible to construct the continuous, single-valued, symmetric, and smooth 3 × 3 diabatic Hamiltonian matrix. Finally, nuclear dynamics has been carried out on such diabatic surfaces to explore whether our MS-based treatment of diabatization can reproduce the pattern of the experimental spectrum for system B of Na3 cluster.
3. Ab initio no core calculations of light nuclei and preludes to Hamiltonian quantum field theory
SciTech Connect
Vary, J. P.; Maris, P.; Honkanen, H.; Li, J.; Shirokov, A. M.; Brodsky, S. J.; Harindranath, A.
2009-12-17
Recent advances in ab initio quantum many-body methods and growth in computer power now enable highly precise calculations of nuclear structure. The precision has attained a level sufficient to make clear statements on the nature of 3-body forces in nuclear physics. Total binding energies, spin-dependent structure effects, and electroweak properties of light nuclei play major roles in pinpointing properties of the underlying strong interaction. Eventually, we anticipate a theory bridge with immense predictive power from QCD through nuclear forces to nuclear structure and nuclear reactions. Light front Hamiltonian quantum field theory offers an attractive pathway and we outline key elements.
4. Ab initio no core calculations of light nuclei and preludes to Hamiltonian quantum field theory
SciTech Connect
Vary, J.P.; Maris, P.; Shirokov, A.M.; Honkanen, H.; li, J.; Brodsky, S.J.; Harindranath, A.; Teramond, G.F.de; /Costa Rica U.
2009-08-03
Recent advances in ab initio quantum many-body methods and growth in computer power now enable highly precise calculations of nuclear structure. The precision has attained a level sufficient to make clear statements on the nature of 3-body forces in nuclear physics. Total binding energies, spin-dependent structure effects, and electroweak properties of light nuclei play major roles in pinpointing properties of the underlying strong interaction. Eventually,we anticipate a theory bridge with immense predictive power from QCD through nuclear forces to nuclear structure and nuclear reactions. Light front Hamiltonian quantum field theory offers an attractive pathway and we outline key elements.
5. Density-matrix based determination of low-energy model Hamiltonians from ab initio wavefunctions.
PubMed
Changlani, Hitesh J; Zheng, Huihuo; Wagner, Lucas K
2015-09-14
We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding. For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard U(∗)/t to be 1.3 ± 0.2, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large-scale calculations using techniques designed for lattice models.
6. Ab initio calculation of excitonic Hamiltonian of light-harvesting complex LH1 of Thermochromatium tepidum
Kozlov, Maxim I.; Poddubnyy, Vladimir V.; Glebov, Ilya O.; Belov, Aleksandr S.; Khokhlov, Daniil V.
2016-02-01
The electronic properties of light-harvesting complexes determine the efficiency of energy transfer in photosynthetic antennae. Ab initio calculations of the electronic properties of bacteriochlorophylls (composing the LH1 complex of the purple bacteria Thermochromatium tepidum) were performed. Based on these calculations, the excitonic Hamiltonian of a native cyclic complex and the Hamiltonians of open complexes with several removed bacteriochlorophylls were constructed. Absorption spectra calculated based on these Hamiltonians agree well with the experimental data. We found that the parameters of interaction between the neighboring bacteriochlorophylls are significantly larger than the empirical parameters suggested previously.
7. Relativistic k .p Hamiltonians for centrosymmetric topological insulators from ab initio wave functions
Nechaev, I. A.; Krasovskii, E. E.
2016-11-01
We present a method to microscopically derive a small-size k .p Hamiltonian in a Hilbert space spanned by physically chosen ab initio spinor wave functions. Without imposing any complementary symmetry constraints, our formalism equally treats three- and two-dimensional systems and simultaneously yields the Hamiltonian parameters and the true Z2 topological invariant. We consider bulk crystals and thin films of Bi2Se3 , Bi2Te3 , and Sb2Te3 . It turns out that the effective continuous k .p models with open boundary conditions often incorrectly predict the topological character of thin films.
8. Density-matrix based determination of low-energy model Hamiltonians from ab initio wavefunctions
SciTech Connect
Changlani, Hitesh J.; Zheng, Huihuo; Wagner, Lucas K.
2015-09-14
We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding. For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard U{sup ∗}/t to be 1.3 ± 0.2, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large-scale calculations using techniques designed for lattice models.
9. Unified ab initio approaches to nuclear structure and reactions
DOE PAGES
Navratil, Petr; Quaglioni, Sofia; Hupin, Guillaume; ...
2016-04-13
The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In addition to the complex nature of the nuclear forces, with two-, three- and possibly higher many-nucleon components, one faces the quantum-mechanical many-nucleon problem governed by an interplay between bound and continuum states. In recent years, significant progress has been made in ab initio nuclear structure and reaction calculations based on input from QCD-employing Hamiltonians constructed within chiral effective field theory. After a brief overview of the field, we focus on ab initio many-body approaches—built upon the no-core shell model—that are capable of simultaneously describing both bound and scattering nuclear states, and present results for resonances in light nuclei, reactions important for astrophysics and fusion research. In particular, we review recent calculations of resonances in the 6He halo nucleus, of five- and six-nucleon scattering, and an investigation of the role of chiral three-nucleon interactions in the structure of 9Be. Further, we discuss applications to the 7Bemore » $${({\\rm{p}},\\gamma )}^{8}{\\rm{B}}$$ radiative capture. Lastly, we highlight our efforts to describe transfer reactions including the 3H$${({\\rm{d}},{\\rm{n}})}^{4}$$He fusion.« less
10. Ab Initio Effective Rovibrational Hamiltonians for Non-Rigid Molecules via Curvilinear VMP2
Changala, Bryan; Baraban, Joshua H.
2017-06-01
Accurate predictions of spectroscopic constants for non-rigid molecules are particularly challenging for ab initio theory. For all but the smallest systems, brute force'' diagonalization of the full rovibrational Hamiltonian is computationally prohibitive, leaving us at the mercy of perturbative approaches. However, standard perturbative techniques, such as second order vibrational perturbation theory (VPT2), are based on the approximation that a molecule makes small amplitude vibrations about a well defined equilibrium structure. Such assumptions are physically inappropriate for non-rigid systems. In this talk, we will describe extensions to curvilinear vibrational Møller-Plesset perturbation theory (VMP2) that account for rotational and rovibrational effects in the molecular Hamiltonian. Through several examples, we will show that this approach provides predictions to nearly microwave accuracy of molecular constants including rotational and centrifugal distortion parameters, Coriolis coupling constants, and anharmonic vibrational and tunneling frequencies.
11. Ab initio calculation of anisotropic magnetic properties of complexes. I. Unique definition of pseudospin Hamiltonians and their derivation
Chibotaru, L. F.; Ungur, L.
2012-08-01
A methodology for the rigorous nonperturbative derivation of magnetic pseudospin Hamiltonians of mononuclear complexes and fragments based on ab initio calculations of their electronic structure is described. It is supposed that the spin-orbit coupling and other relativistic effects are already taken fully into account at the stage of quantum chemistry calculations of complexes. The methodology is based on the establishment of the correspondence between the ab initio wave functions of the chosen manifold of multielectronic states and the pseudospin eigenfunctions, which allows to define the pseudospin Hamiltonians in the unique way. Working expressions are derived for the pseudospin Zeeman and zero-field splitting Hamiltonian corresponding to arbitrary pseudospins. The proposed calculation methodology, already implemented in the SINGLE_ANISO module of the MOLCAS-7.6 quantum chemistry package, is applied for a first-principles evaluation of pseudospin Hamiltonians of several complexes exhibiting weak, moderate, and very strong spin-orbit coupling effects.
12. Comparison of nuclear Hamiltonians using spectral function sum rules
Rios, A.; Carbone, A.; Polls, A.
2017-07-01
Background: The energy weighted sum rules of the single-particle spectral functions provide a quantitative understanding of the fragmentation of nuclear states due to short-range and tensor correlations. Purpose: The aim of this paper is to compare on a quantitative basis the single-particle spectral function generated by different nuclear Hamiltonians in symmetric nuclear matter using the first three energy-weighted moments. Method: The spectral functions are calculated in the framework of the self-consistent Green's function approach at finite temperature within a ladder resummation scheme. We analyze the first three moments of the spectral function and connect these to the correlations induced by the interactions between the nucleons in symmetric nuclear matter. In particular, the variance of the spectral function is directly linked to the dispersive contribution of the self-energy. The discussion is centered around two- and three-body chiral nuclear interactions, with and without renormalization, but we also provide results obtained with the traditional phase-shift-equivalent CD-Bonn and Av18 potentials. Results: The variance of the spectral function is particularly sensitive to the short-range structure of the force, with hard-core interactions providing large variances. Chiral forces yield variances which are an order of magnitude smaller and, when tamed using the similarity renormalization group, the variance reduces significantly and in proportion to the renormalization scale. The presence of three-body forces does not substantially affect the results. Conclusions: The first three moments of the spectral function are useful tools in analyzing the importance of correlations in nuclear ground states. In particular, the second-order moment provides a direct insight into dispersive contributions to the self-energy and its value is indicative of the fragmentation of single-particle states.
13. Ab-Initio Hamiltonian Approach to Light Nuclei And to Quantum Field Theory
SciTech Connect
Vary, J.P.; Honkanen, H.; Li, Jun; Maris, P.; Shirokov, A.M.; Brodsky, S.J.; Harindranath, A.; de Teramond, G.F.; Ng, E.G.; Yang, C.; Sosonkina, M.; /Ames Lab
2012-06-22
Nuclear structure physics is on the threshold of confronting several long-standing problems such as the origin of shell structure from basic nucleon-nucleon and three-nucleon interactions. At the same time those interactions are being developed with increasing contact to QCD, the underlying theory of the strong interactions, using effective field theory. The motivation is clear - QCD offers the promise of great predictive power spanning phenomena on multiple scales from quarks and gluons to nuclear structure. However, new tools that involve non-perturbative methods are required to build bridges from one scale to the next. We present an overview of recent theoretical and computational progress with a Hamiltonian approach to build these bridges and provide illustrative results for the nuclear structure of light nuclei and quantum field theory.
14. Ab initio calculations of nuclear reactions important for astrophysics
Navratil, Petr; Dohet-Eraly, Jeremy; Calci, Angelo; Horiuchi, Wataru; Hupin, Guillaume; Quaglioni, Sofia
2016-09-01
In recent years, significant progress has been made in ab initio nuclear structure and reaction calculations based on input from QCD employing Hamiltonians constructed within chiral effective field theory. One of the newly developed approaches is the No-Core Shell Model with Continuum (NCSMC), capable of describing both bound and scattering states in light nuclei simultaneously. We will present NCSMC results for reactions important for astrophysics that are difficult to measure at relevant low energies, such as 3He(α,γ)7Be and 3H(α,γ)7Li and 11C(p,γ)12N radiative capture, as well as the 3H(d,n)4He fusion. We will also address prospects of calculating the 2H(α,γ)6Li capture reaction within the NCSMC formalism. Prepared in part by LLNL under Contract DE-AC52-07NA27344. Supported by the U.S. DOE, OS, NP, under Work Proposal No. SCW1158, and by the NSERC Grant No. SAPIN-2016-00033. TRIUMF receives funding from the NRC Canada.
15. Toward eliminating the electronic structure bottleneck in nonadiabatic dynamics on the fly: An algorithm to fit nonlocal, quasidiabatic, coupled electronic state Hamiltonians based on ab initio electronic structure data
Zhu, Xiaolei; Yarkony, David R.
2010-03-01
An algorithm for constructing a quasidiabatic, coupled electronic state Hamiltonian, in a localized region of nuclear coordinate space, suitable for determining bound state spectra, is generalized to determine a nonlocal Hamiltonian capable of describing, for example, multichannel nonadiabatic photodissociation. For Nstate coupled electronic states, the Hamiltonian, Hd, is a symmetric Nstate×Nstate matrix whose elements are polynomials involving: decaying exponentials exp(-ari,jn) n =1,2, where ri,j=Ri-Rj, ri,j=|ri,j|, Rj locates the jth nucleus; and scaled dot-cross product coordinates, proportional to ri,j×ri,k•ri,l. The constructed Hamiltonian is constrained to reproduce, exactly, the ab initio data, energies, gradients, and derivative coupling at selected points, or nodes, in nuclear coordinate space. The remainder of the ab initio data is approximated in a least-squares sense using a normal equations approach. The fitting procedure includes a damping term that precludes oscillations due to the nodal constraints or local excesses of parameters. To illustrate the potential of the fitting procedure an Hd is constructed, with the full nuclear permutation-inversion symmetry, which describes portions of the 1,2 A1 potential energy surfaces of NH3, including the minimum energy point on the 1,2 A1 seam of conical intersection and the NH2+H asymptote. Ab initio data at 239 nuclear configurations was used in the construction which was tested at 48 additional nuclear configurations. While the energy range on the ground and excited potential energy surface is each individually ˜45 000 cm-1, the root mean square error for the energies at all points is only 93.6 cm-1. The location and local conical topography of the minimum energy conical intersection is exactly reproduced. The derivative couplings are shown to be well reproduced, justifying the attribute quasidiabatic.
16. From Model Hamiltonians to ab Initio Hamiltonians and Back Again: Using Single Excitation Quantum Chemistry Methods To Find Multiexciton States in Singlet Fission Materials.
PubMed
Mayhall, Nicholas J
2016-09-13
Due to the promise of significantly enhanced photovoltaic efficiencies, significant effort has been directed toward understanding and controlling the singlet fission mechanism. Although accurate quantum chemical calculations would provide a detail-rich view of the singlet fission mechanism, this is complicated by the multiexcitonic nature of one of the key intermediates, the (1)(TT) state. Being described as two simultaneous and singlet-coupled triplet excitations on a pair of nearest neighbor monomers, the (1)(TT) state is inherently a multielectronic excitation. This fact renders most single-reference ab initio quantum chemical methods incapable of providing accurate results. This paper serves two purposes: (1) to demonstrate that the multiexciton states in singlet fission materials can be described using a spin-only Hamiltonian and with each monomer treated as a biradical and (2) to propose a very simple procedure for extracting the values for this Hamiltonian from single-reference calculations. Numerical examples are included for a number of different systems, including dimers, trimers, tetramers, and a cluster comprised of seven chromophores.
17. Dipole and transition moments of SiH, PH and SH by ab initio effective valence shell Hamiltonian method
Park, Jong Keun; Sun, Hosung
1992-07-01
The ab initio effective valence shell Hamiltonian method, based on quasi-degenerate many-body perturbation theory, is generalized to calculate molecular properties as well as the valence state energies. The procedure requires the evaluation of effective operators for each molecular property. Effective operators are perturbatively expanded in powers of correlation and contain contributions from excitations outside of the multireference valence space. To demonstrate the validity of this method, calculations for dipole moments of and transition moments between several low lying valence states of SiH, SiH +, PH, PH +, SH, and SH + to the lowest nontrivial order in the correlations have been performed and compared with other theoretical calculations.
18. Understanding nuclear motions in molecules: Derivation of Eckart frame ro-vibrational Hamiltonian operators via a gateway Hamiltonian operator
SciTech Connect
Szalay, Viktor
2015-05-07
A new ro-vibrational Hamiltonian operator, named gateway Hamiltonian operator, with exact kinetic energy term, T-hat, is presented. It is in the Eckart frame and it is of the same form as Watson’s normal coordinate Hamiltonian. However, the vibrational coordinates employed are not normal coordinates. The new Hamiltonian is shown to provide easy access to Eckart frame ro-vibrational Hamiltonians with exact T-hat given in terms of any desired set of vibrational coordinates. A general expression of the Eckart frame ro-vibrational Hamiltonian operator is given and some of its properties are discussed.
19. Unified ab initio approaches to nuclear structure and reactions
SciTech Connect
Navratil, Petr; Quaglioni, Sofia; Hupin, Guillaume; Romero-Redondo, Carolina; Calci, Angelo
2016-04-13
The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In addition to the complex nature of the nuclear forces, with two-, three- and possibly higher many-nucleon components, one faces the quantum-mechanical many-nucleon problem governed by an interplay between bound and continuum states. In recent years, significant progress has been made in ab initio nuclear structure and reaction calculations based on input from QCD-employing Hamiltonians constructed within chiral effective field theory. After a brief overview of the field, we focus on ab initio many-body approaches—built upon the no-core shell model—that are capable of simultaneously describing both bound and scattering nuclear states, and present results for resonances in light nuclei, reactions important for astrophysics and fusion research. In particular, we review recent calculations of resonances in the 6He halo nucleus, of five- and six-nucleon scattering, and an investigation of the role of chiral three-nucleon interactions in the structure of 9Be. Further, we discuss applications to the 7Be ${({\\rm{p}},\\gamma )}^{8}{\\rm{B}}$ radiative capture. Lastly, we highlight our efforts to describe transfer reactions including the 3H${({\\rm{d}},{\\rm{n}})}^{4}$He fusion.
20. Unified ab initio approaches to nuclear structure and reactions
SciTech Connect
Navratil, Petr; Quaglioni, Sofia; Hupin, Guillaume; Romero-Redondo, Carolina; Calci, Angelo
2016-04-13
The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In addition to the complex nature of the nuclear forces, with two-, three- and possibly higher many-nucleon components, one faces the quantum-mechanical many-nucleon problem governed by an interplay between bound and continuum states. In recent years, significant progress has been made in ab initio nuclear structure and reaction calculations based on input from QCD-employing Hamiltonians constructed within chiral effective field theory. After a brief overview of the field, we focus on ab initio many-body approaches—built upon the no-core shell model—that are capable of simultaneously describing both bound and scattering nuclear states, and present results for resonances in light nuclei, reactions important for astrophysics and fusion research. In particular, we review recent calculations of resonances in the 6He halo nucleus, of five- and six-nucleon scattering, and an investigation of the role of chiral three-nucleon interactions in the structure of 9Be. Further, we discuss applications to the 7Be ${({\\rm{p}},\\gamma )}^{8}{\\rm{B}}$ radiative capture. Lastly, we highlight our efforts to describe transfer reactions including the 3H${({\\rm{d}},{\\rm{n}})}^{4}$He fusion.
1. Some implications of the Hartree product treatment of the quantum nuclei in the ab initio nuclear-electronic orbital methodology
Gharabaghi, Masumeh; Shahbazian, Shant
2016-12-01
In this letter the conceptual and computational implications of the Hartree product type nuclear wavefunction introduced recently within the context of the ab initio non-Born-Oppenheimer Nuclear-electronic orbital (NEO) methodology are considered. It is demonstrated that this wavefunction may imply a pseudo-adiabatic separation of the nuclei and electrons and each nucleus is conceived as a quantum oscillator while a non-Coulombic effective Hamiltonian is deduced for electrons. Using the variational principle this Hamiltonian is employed to derive a modified set of single-component Hartree-Fock equations which are equivalent to the multi-component version derived previously within the context of the NEO and, easy to be implemented computationally.
2. Ab initio relaxation times and time-dependent Hamiltonians within the steepest-entropy-ascent quantum thermodynamic framework
Kim, Ilki; von Spakovsky, Michael R.
2017-08-01
Quantum systems driven by time-dependent Hamiltonians are considered here within the framework of steepest-entropy-ascent quantum thermodynamics (SEAQT) and used to study the thermodynamic characteristics of such systems. In doing so, a generalization of the SEAQT framework valid for all such systems is provided, leading to the development of an ab initio physically relevant expression for the intrarelaxation time, an important element of this framework and one that had as of yet not been uniquely determined as an integral part of the theory. The resulting expression for the relaxation time is valid as well for time-independent Hamiltonians as a special case and makes the description provided by the SEAQT framework more robust at the fundamental level. In addition, the SEAQT framework is used to help resolve a fundamental issue of thermodynamics in the quantum domain, namely, that concerning the unique definition of process-dependent work and heat functions. The developments presented lead to the conclusion that this framework is not just an alternative approach to thermodynamics in the quantum domain but instead one that uniquely sheds new light on various fundamental but as of yet not completely resolved questions of thermodynamics.
3. Advances in ab initio theories for nuclear reactions
Quaglioni, Sofia
2016-09-01
Driven by high-performance computing and new ideas, in recent years ab initio theory has made great strides in achieving a unified description of nuclear structure, clustering and reactions from the constituent nucleons and their strong and electroweak interactions. This is giving access to forefront tools and new fertile grounds to further our understanding of the nuclear force and electroweak currents in nuclei in terms of effective degrees of freedom. A fundamental understanding of nuclear reaction mechanisms and a new capability to accurately compute their properties is also relevant for nuclear astrophysics, terrestrial applications of nuclear fusion, and for using nuclei as probes of fundamental physics through, for example, neutrino-nucleus scattering. In this talk, I will present recent highlights and reflect on future perspectives for this area of nuclear theory. Prepared by LLNL under Contract No. DE-AC52-07NA27344.
4. Ab initio calculations of nuclear structure and reactions with chiral two- and three-nucleon interactions
Navratil, Petr; Langhammer, Joachim; Hupin, Guillaume; Quaglioni, Sofia; Calci, Angelo; Roth, Robert; Soma, Vittorio; Cipollone, Andrea; Barbieri, Carlo; Duguet, T.
2014-09-01
The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In recent years, a significant progress has been made in developing ab initio many-body approaches capable of describing both bound and scattering states in light and medium mass nuclei based on input from QCD employing Hamiltonians constructed within chiral effective field theory. We will present calculations of proton-10C scattering and resonances of the exotic nuclei 11N and 9He within the no-core shell model with continuum. Also, we will discuss calculations of binding and separation energies of neutron rich isotopes of Ar, K, Ca, Sc and Ti within the self-consistent Gorkov-Green's function approach. The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In recent years, a significant progress has been made in developing ab initio many-body approaches capable of describing both bound and scattering states in light and medium mass nuclei based on input from QCD employing Hamiltonians constructed within chiral effective field theory. We will present calculations of proton-10C scattering and resonances of the exotic nuclei 11N and 9He within the no-core shell model with continuum. Also, we will discuss calculations of binding and separation energies of neutron rich isotopes of Ar, K, Ca, Sc and Ti within the self-consistent Gorkov-Green's function approach. Support from the NSERC Grant No. 401945-2011 is acknowledged. This work was prepared in part by the LLNL under Contract No. DE-AC52-07NA27344.
5. Study of Nuclear Clustering from an Ab Initio Perspective
Kravvaris, Konstantinos; Volya, Alexander
2017-08-01
We put forward a new ab initio approach that seamlessly bridges the structure, clustering, and reactions aspects of the nuclear quantum many-body problem. The configuration interaction technique combined with the resonating group method based on a harmonic oscillator basis allows us to treat the reaction and multiclustering dynamics in a translationally invariant way and preserve the Pauli principle. Our presentation includes studies of Be,108 and an exploration of 3 α clustering in 12C.
6. Ab initio study of the Zener polaron spectrum of half-doped manganites: Comparison of several model Hamiltonians
Bastardis, Roland; Guihéry, Nathalie; de Graaf, Coen
2006-07-01
The low-energy spectrum of the Zener polaron in half-doped manganite is studied by means of correlated ab initio calculations. It is shown that the electronic structure of the low-energy states results from a subtle interplay between double-exchange configurations and O 2pσ to Mn 3d charge-transfer configurations that obey a Heisenberg logic. The comparison of the calculated spectrum to those predicted by the Zener Hamiltonian reveals that this simple description does not correctly reproduces the Zener polaron physics. A better reproduction of the calculated spectrum is obtained with either a Heisenberg model that considers a purely magnetic oxygen or the Girerd-Papaefthymiou double-exchange model. An additional significant improvement is obtained when different antiferromagnetic contributions are combined with the double-exchange model, showing that the Zener polaron spectrum is actually ruled by a refined double-exchange mechanism where non-Hund atomic states play a non-negligible role. Finally, eight states of a different nature have been found to be intercalated in the double-exchange spectrum. These states exhibit an O to Mn charge transfer, implying a second O 2p orbital of approximate π character instead of the usual σ symmetry. A small mixing of the two families of states occurs, accounting for the final ordering of the states.
7. Ab Initio Nuclear Structure and Reaction Calculations for Rare Isotopes
SciTech Connect
Draayer, Jerry P.
2014-09-28
We have developed a novel ab initio symmetry-adapted no-core shell model (SA-NCSM), which has opened the intermediate-mass region for ab initio investigations, thereby providing an opportunity for first-principle symmetry-guided applications to nuclear structure and reactions for nuclear isotopes from the lightest p-shell systems to intermediate-mass nuclei. This includes short-lived proton-rich nuclei on the path of X-ray burst nucleosynthesis and rare neutron-rich isotopes to be produced by the Facility for Rare Isotope Beams (FRIB). We have provided ab initio descriptions of high accuracy for low-lying (including collectivity-driven) states of isotopes of Li, He, Be, C, O, Ne, Mg, Al, and Si, and studied related strong- and weak-interaction driven reactions that are important, in astrophysics, for further understanding stellar evolution, X-ray bursts and triggering of s, p, and rp processes, and in applied physics, for electron and neutrino-nucleus scattering experiments as well as for fusion ignition at the National Ignition Facility (NIF).
8. Nuclear motion effects on the density matrix of crystals: An ab initio Monte Carlo harmonic approach
Pisani, Cesare; Erba, Alessandro; Ferrabone, Matteo; Dovesi, Roberto
2012-07-01
In the frame of the Born-Oppenheimer approximation, nuclear motions in crystals can be simulated rather accurately using a harmonic model. In turn, the electronic first-order density matrix (DM) can be expressed as the statistically weighted average over all its determinations each resulting from an instantaneous nuclear configuration. This model has been implemented in a computational scheme which adopts an ab initio one-electron (Hartree-Fock or Kohn-Sham) Hamiltonian in the CRYSTAL program. After selecting a supercell of reasonable size and solving the corresponding vibrational problem in the harmonic approximation, a Metropolis algorithm is adopted for generating a sample of nuclear configurations which reflects their probability distribution at a given temperature. For each configuration in the sample the "instantaneous" DM is calculated, and its contribution to the observables of interest is extracted. Translational and point symmetry of the crystal as reflected in its average DM are fully exploited. The influence of zero-point and thermal motion of nuclei on such important first-order observables as x-ray structure factors and Compton profiles can thus be estimated.
9. Nuclear motion effects on the density matrix of crystals: an ab initio Monte Carlo harmonic approach.
PubMed
Pisani, Cesare; Erba, Alessandro; Ferrabone, Matteo; Dovesi, Roberto
2012-07-28
In the frame of the Born-Oppenheimer approximation, nuclear motions in crystals can be simulated rather accurately using a harmonic model. In turn, the electronic first-order density matrix (DM) can be expressed as the statistically weighted average over all its determinations each resulting from an instantaneous nuclear configuration. This model has been implemented in a computational scheme which adopts an ab initio one-electron (Hartree-Fock or Kohn-Sham) Hamiltonian in the CRYSTAL program. After selecting a supercell of reasonable size and solving the corresponding vibrational problem in the harmonic approximation, a Metropolis algorithm is adopted for generating a sample of nuclear configurations which reflects their probability distribution at a given temperature. For each configuration in the sample the "instantaneous" DM is calculated, and its contribution to the observables of interest is extracted. Translational and point symmetry of the crystal as reflected in its average DM are fully exploited. The influence of zero-point and thermal motion of nuclei on such important first-order observables as x-ray structure factors and Compton profiles can thus be estimated.
10. Ab initio calculations of nuclear widths via an integral relation
Nollett, Kenneth M.
2012-10-01
I describe the computation of energy widths of nuclear states using an integral over the interaction region of ab initio variational Monte Carlo wave functions, and I present calculated widths for many states. I begin by presenting relations that connect certain short-range integrals to widths. I then present predicted widths for 5⩽A⩽9 nuclei, and I compare them against measured widths. They match the data more closely and with less ambiguity than estimates based on spectroscopic factors. I consider the consequences of my results for identification of observed states in 8B, 9He, and 9Li. I also examine failures of the method and conclude that they generally involve broad states and variational wave functions that are not strongly peaked in the interaction region. After examining bound-state overlap functions computed from a similar integral relation, I conclude that overlap calculations can diagnose cases in which computed widths should not be trusted.
11. Isoscalar monopole resonance of the alpha particle: a prism to nuclear Hamiltonians.
PubMed
Bacca, Sonia; Barnea, Nir; Leidemann, Winfried; Orlandini, Giuseppina
2013-01-25
We present an ab initio study of the isoscalar monopole excitations of (4)He using different realistic nuclear interactions, including modern effective field theory potentials. In particular we concentrate on the transition form factor F(M) to the narrow 0(+) resonance close to threshold. F(M) exhibits a strong potential model dependence, and can serve as a kind of prism to distinguish among different nuclear force models. Compared to the measurements obtained from inelastic electron scattering off ^{4}He, one finds that the state-of-the-art theoretical transition form factors are at variance with experimental data, especially in the case of effective field theory potentials. We discuss some possible reasons for such a discrepancy, which still remains a puzzle.
12. Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules
Rey, M.; Nikitin, A. V.; Tyuterev, Vl. G.
2010-08-01
A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY3 molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH3 where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].
13. Ab initio effective rotational and rovibrational Hamiltonians for non-rigid systems via curvilinear second order vibrational Møller-Plesset perturbation theory
Changala, P. Bryan; Baraban, Joshua H.
2016-11-01
We present a perturbative method for ab initio calculations of rotational and rovibrational effective Hamiltonians of both rigid and non-rigid molecules. Our approach is based on a curvilinear implementation of second order vibrational Møller-Plesset perturbation theory extended to include rotational effects via a second order contact transformation. Though more expensive, this approach is significantly more accurate than standard second order vibrational perturbation theory for systems that are poorly described to zeroth order by rectilinear normal mode harmonic oscillators. We apply this method to and demonstrate its accuracy on two molecules: Si2C, a quasilinear triatomic with significant bending anharmonicity, and CH3NO2, which contains a completely unhindered methyl rotor. In addition to these two examples, we discuss several key technical aspects of the method, including an efficient implementation of Eckart and quasi-Eckart frame embedding that does not rely on numerical finite differences.
14. Ab initio effective rotational and rovibrational Hamiltonians for non-rigid systems via curvilinear second order vibrational Møller-Plesset perturbation theory.
PubMed
Changala, P Bryan; Baraban, Joshua H
2016-11-07
We present a perturbative method for ab initio calculations of rotational and rovibrational effective Hamiltonians of both rigid and non-rigid molecules. Our approach is based on a curvilinear implementation of second order vibrational Møller-Plesset perturbation theory extended to include rotational effects via a second order contact transformation. Though more expensive, this approach is significantly more accurate than standard second order vibrational perturbation theory for systems that are poorly described to zeroth order by rectilinear normal mode harmonic oscillators. We apply this method to and demonstrate its accuracy on two molecules: Si2C, a quasilinear triatomic with significant bending anharmonicity, and CH3NO2, which contains a completely unhindered methyl rotor. In addition to these two examples, we discuss several key technical aspects of the method, including an efficient implementation of Eckart and quasi-Eckart frame embedding that does not rely on numerical finite differences.
15. Vibrational dynamics of zero-field-splitting hamiltonian in gadolinium-based MRI contrast agents from ab initio molecular dynamics
SciTech Connect
Lasoroski, Aurélie; Vuilleumier, Rodolphe; Pollet, Rodolphe
2014-07-07
The electronic relaxation of gadolinium complexes used as MRI contrast agents was studied theoretically by following the short time evolution of zero-field-splitting parameters. The statistical analysis of ab initio molecular dynamics trajectories provided a clear separation between static and transient contributions to the zero-field-splitting. For the latter, the correlation time was estimated at approximately 0.1 ps. The influence of the ligand was also probed by replacing one pendant arm of our reference macrocyclic complex by a bulkier phosphonate arm. In contrast to the transient contribution, the static zero-field-splitting was significantly influenced by this substitution.
16. Accurate ab initio tight-binding Hamiltonians: Effective tools for electronic transport and optical spectroscopy from first principles
D'Amico, Pino; Agapito, Luis; Catellani, Alessandra; Ruini, Alice; Curtarolo, Stefano; Fornari, Marco; Nardelli, Marco Buongiorno; Calzolari, Arrigo
2016-10-01
The calculations of electronic transport coefficients and optical properties require a very dense interpolation of the electronic band structure in reciprocal space that is computationally expensive and may have issues with band crossing and degeneracies. Capitalizing on a recently developed pseudoatomic orbital projection technique, we exploit the exact tight-binding representation of the first-principles electronic structure for the purposes of (i) providing an efficient strategy to explore the full band structure En(k ) , (ii) computing the momentum operator differentiating directly the Hamiltonian, and (iii) calculating the imaginary part of the dielectric function. This enables us to determine the Boltzmann transport coefficients and the optical properties within the independent particle approximation. In addition, the local nature of the tight-binding representation facilitates the calculation of the ballistic transport within the Landauer theory for systems with hundreds of atoms. In order to validate our approach we study the multivalley band structure of CoSb3 and a large core-shell nanowire using the ACBN0 functional. In CoSb3 we point the many band minima contributing to the electronic transport that enhance the thermoelectric properties; for the core-shell nanowire we identify possible mechanisms for photo-current generation and justify the presence of protected transport channels in the wire.
17. Spin-orbit decomposition of ab initio nuclear wave functions
Johnson, Calvin W.
2015-03-01
Although the modern shell-model picture of atomic nuclei is built from single-particle orbits with good total angular momentum j , leading to j -j coupling, decades ago phenomenological models suggested that a simpler picture for 0 p -shell nuclides can be realized via coupling of the total spin S and total orbital angular momentum L . I revisit this idea with large-basis, no-core shell-model calculations using modern ab initio two-body interactions and dissect the resulting wave functions into their component L - and S -components. Remarkably, there is broad agreement with calculations using the phenomenological Cohen-Kurath forces, despite a gap of nearly 50 years and six orders of magnitude in basis dimensions. I suggest that L -S decomposition may be a useful tool for analyzing ab initio wave functions of light nuclei, for example, in the case of rotational bands.
18. Ground-state properties of Na2IrO3 determined from an ab initio Hamiltonian and its extensions containing Kitaev and extended Heisenberg interactions
Okubo, Tsuyoshi; Shinjo, Kazuya; Yamaji, Youhei; Kawashima, Naoki; Sota, Shigetoshi; Tohyama, Takami; Imada, Masatoshi
2017-08-01
We investigate the ground state properties of Na2IrO3 based on numerical calculations of the recently proposed ab initio Hamiltonian represented by Kitaev and extended Heisenberg interactions. To overcome the limitation posed by small tractable system sizes in the exact diagonalization study employed in a previous study [Y. Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014), 10.1103/PhysRevLett.113.107201], we apply a two-dimensional density matrix renormalization group and an infinite-size tensor-network method. By calculating at much larger system sizes, we critically test the validity of the exact diagonalization results. The results consistently indicate that the ground state of Na2IrO3 is a magnetically ordered state with zigzag configuration in agreement with experimental observations and the previous diagonalization study. Applications of the two independent methods in addition to the exact diagonalization study further uncover a consistent and rich phase diagram near the zigzag phase beyond the accessibility of the exact diagonalization. For example, in the parameter space away from the ab initio value of Na2IrO3 controlled by the trigonal distortion, we find three phases: (i) an ordered phase with the magnetic moment aligned mutually in 120 degrees orientation on every third hexagon, (ii) a magnetically ordered phase with a 16-site unit cell, and (iii) an ordered phase with presumably incommensurate periodicity of the moment. It suggests that potentially rich magnetic structures may appear in A2IrO3 compounds for A other than Na. The present results also serve to establish the accuracy of the first-principles approach in reproducing the available experimental results thereby further contributing to finding a route to realize the Kitaev spin liquid.
19. Ab Initio Calculations Of Nuclear Reactions And Exotic Nuclei
SciTech Connect
Quaglioni, S.
2014-05-05
Our ultimate goal is to develop a fundamental theory and efficient computational tools to describe dynamic processes between nuclei and to use such tools toward supporting several DOE milestones by: 1) performing predictive calculations of difficult-to-measure landmark reactions for nuclear astrophysics, such as those driving the neutrino signature of our sun; 2) improving our understanding of the structure of nuclei near the neutron drip line, which will be the focus of the DOE’s Facility for Rare Isotope Beams (FRIB) being constructed at Michigan State University; but also 3) helping to reveal the true nature of the nuclear force. Furthermore, these theoretical developments will support plasma diagnostic efforts at facilities dedicated to the development of terrestrial fusion energy.
20. Ab initio nuclear structure from lattice effective field theory
SciTech Connect
Lee, Dean
2014-11-11
This proceedings article reviews recent results by the Nuclear Lattice EFT Collaboration on an excited state of the {sup 12}C nucleus known as the Hoyle state. The Hoyle state plays a key role in the production of carbon via the triple-alpha reaction in red giant stars. We discuss the structure of low-lying states of {sup 12}C as well as the dependence of the triple-alpha reaction on the masses of the light quarks.
1. Extension of the MIRS computer package for the modeling of molecular spectra: From effective to full ab initio ro-vibrational Hamiltonians in irreducible tensor form
Nikitin, A. V.; Rey, M.; Champion, J. P.; Tyuterev, Vl. G.
2012-07-01
The MIRS software for the modeling of ro-vibrational spectra of polyatomic molecules was considerably extended and improved. The original version [Nikitin AV, Champion JP, Tyuterev VlG. The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules. J Quant Spectrosc Radiat Transf 2003;82:239-49.] was especially designed for separate or simultaneous treatments of complex band systems of polyatomic molecules. It was set up in the frame of effective polyad models by using algorithms based on advanced group theory algebra to take full account of symmetry properties. It has been successfully used for predictions and data fitting (positions and intensities) of numerous spectra of symmetric and spherical top molecules within the vibration extrapolation scheme. The new version offers more advanced possibilities for spectra calculations and modeling by getting rid of several previous limitations particularly for the size of polyads and the number of tensors involved. It allows dealing with overlapping polyads and includes more efficient and faster algorithms for the calculation of coefficients related to molecular symmetry properties (6C, 9C and 12C symbols for C3v, Td, and Oh point groups) and for better convergence of least-square-fit iterations as well. The new version is not limited to polyad effective models. It also allows direct predictions using full ab initio ro-vibrational normal mode Hamiltonians converted into the irreducible tensor form. Illustrative examples on CH3D, CH4, CH3Cl, CH3F and PH3 are reported reflecting the present status of data available. It is written in C++ for standard PC computer operating under Windows. The full package including on-line documentation and recent data are freely available at http://www.iao.ru/mirs/mirs.htm or http://xeon.univ-reims.fr/Mirs/ or http://icb.u-bourgogne.fr/OMR/SMA/SHTDS/MIRS.html and as supplementary data from the online version of the article.
2. Quantifying statistical uncertainties in ab initio nuclear physics using Lagrange multipliers
Carlsson, B. D.
2017-03-01
Theoretical predictions need quantified uncertainties for a meaningful comparison to experimental results. This is an idea which presently permeates the field of theoretical nuclear physics. In light of the recent progress in estimating theoretical uncertainties in ab initio nuclear physics, I here present and compare methods for evaluating the statistical part of the uncertainties. A special focus is put on the (for the field) novel method of Lagrange multipliers (LM). Uncertainties from the fit of the nuclear interaction to experimental data are propagated to a few observables in light-mass nuclei to highlight any differences between the presented methods. The main conclusion is that the LM method is more robust, while covariance-based methods are less demanding in their evaluation.
3. Realistic collective nuclear Hamiltonian
Dufour, Marianne; Zuker, Andrés P.
1996-10-01
The residual part of the realistic forces-obtained after extracting the monopole terms responsible for bulk properties-is strongly dominated by pairing and quadrupole interactions, with important στ.στ, octupole, and hexadecapole contributions. Their forms retain the simplicity of the traditional pairing plus multipole models, while eliminating their flaws through a normalization mechanism dictated by a universal A-1/3 scaling. Coupling strengths and effective charges are calculated and shown to agree with empirical values. Comparisons between different realistic interactions confirm the claim that they are very similar.
4. Nuclear quantum effect on intramolecular hydrogen bond of hydrogen maleate anion: An ab initio path integral molecular dynamics study
Kawashima, Yukio; Tachikawa, Masanori
2013-05-01
Ab initio path integral molecular dynamics simulation was performed to understand the nuclear quantum effect on the hydrogen bond of hydrogen malonate anion. Static calculation predicted the proton transfer barrier as 0.12 kcal/mol. Conventional ab initio molecular dynamics simulation at 300 K found proton distribution with a double peak on the proton transfer coordinate. Inclusion of thermal effect alone elongates the hydrogen bond length, which increases the barrier height. Inclusion of nuclear quantum effect washes out this barrier, and distributes a single broad peak in the center. H/D isotope effect on the proton transfer is also discussed.
5. Symmetry-Adapted Ab Initio Shell Model for Nuclear Structure Calculations
Draayer, J. P.; Dytrych, T.; Launey, K. D.; Langr, D.
2012-05-01
An innovative concept, the symmetry-adapted ab initio shell model, that capitalizes on partial as well as exact symmetries that underpin the structure of nuclei, is discussed. This framework is expected to inform the leading features of nuclear structure and reaction data for light and medium mass nuclei, which are currently inaccessible by theory and experiment and for which predictions of modern phenomenological models often diverge. We use powerful computational and group-theoretical algorithms to perform ab initio CI (configuration-interaction) calculations in a model space spanned by SU(3) symmetry-adapted many-body configurations with the JISP16 nucleon-nucleon interaction. We demonstrate that the results for the ground states of light nuclei up through A = 16 exhibit a strong dominance of low-spin and high-deformation configurations together with an evident symplectic structure. This, in turn, points to the importance of using a symmetry-adapted framework, one based on an LS coupling scheme with the associated spatial configurations organized according to deformation.
6. Hamiltonian purification
SciTech Connect
Orsucci, Davide; Burgarth, Daniel; Facchi, Paolo; Pascazio, Saverio; Nakazato, Hiromichi; Yuasa, Kazuya; Giovannetti, Vittorio
2015-12-15
The problem of Hamiltonian purification introduced by Burgarth et al. [Nat. Commun. 5, 5173 (2014)] is formalized and discussed. Specifically, given a set of non-commuting Hamiltonians (h{sub 1}, …, h{sub m}) operating on a d-dimensional quantum system ℋ{sub d}, the problem consists in identifying a set of commuting Hamiltonians (H{sub 1}, …, H{sub m}) operating on a larger d{sub E}-dimensional system ℋ{sub d{sub E}} which embeds ℋ{sub d} as a proper subspace, such that h{sub j} = PH{sub j}P with P being the projection which allows one to recover ℋ{sub d} from ℋ{sub d{sub E}}. The notions of spanning-set purification and generator purification of an algebra are also introduced and optimal solutions for u(d) are provided.
7. Quantum wavepacket ab initio molecular dynamics: an approach for computing dynamically averaged vibrational spectra including critical nuclear quantum effects.
PubMed
Sumner, Isaiah; Iyengar, Srinivasan S
2007-10-18
We have introduced a computational methodology to study vibrational spectroscopy in clusters inclusive of critical nuclear quantum effects. This approach is based on the recently developed quantum wavepacket ab initio molecular dynamics method that combines quantum wavepacket dynamics with ab initio molecular dynamics. The computational efficiency of the dynamical procedure is drastically improved (by several orders of magnitude) through the utilization of wavelet-based techniques combined with the previously introduced time-dependent deterministic sampling procedure measure to achieve stable, picosecond length, quantum-classical dynamics of electrons and nuclei in clusters. The dynamical information is employed to construct a novel cumulative flux/velocity correlation function, where the wavepacket flux from the quantized particle is combined with classical nuclear velocities to obtain the vibrational density of states. The approach is demonstrated by computing the vibrational density of states of [Cl-H-Cl]-, inclusive of critical quantum nuclear effects, and our results are in good agreement with experiment. A general hierarchical procedure is also provided, based on electronic structure harmonic frequencies, classical ab initio molecular dynamics, computation of nuclear quantum-mechanical eigenstates, and employing quantum wavepacket ab initio dynamics to understand vibrational spectroscopy in hydrogen-bonded clusters that display large degrees of anharmonicities.
8. Integration of ab-initio nuclear calculation with derivative free optimization technique
SciTech Connect
Sharda, Anurag
2008-01-01
Optimization techniques are finding their inroads into the field of nuclear physics calculations where the objective functions are very complex and computationally intensive. A vast space of parameters needs searching to obtain a good match between theoretical (computed) and experimental observables, such as energy levels and spectra. Manual calculation defies the scope of such complex calculation and are prone to error at the same time. This body of work attempts to formulate a design and implement it which would integrate the ab initio nuclear physics code MFDn and the VTDIRECT95 code. VTDIRECT95 is a Fortran95 suite of parallel code implementing the derivative-free optimization algorithm DIRECT. Proposed design is implemented for a serial and parallel version of the optimization technique. Experiment with the initial implementation of the design showing good matches for several single-nucleus cases are conducted. Determination and assignment of appropriate number of processors for parallel integration code is implemented to increase the efficiency and resource utilization in the case of multiple nuclei parameter search.
9. Semi-empirical and ab initio DFT modeling of the spin-Hamiltonian parameters for Fe6+: K2MO4 (M = S, Cr, Se)
Avram, N. M.; Brik, M. G.; Andreici, E.-L.
2014-09-01
In this paper we calculated the spin-Hamiltonian parameters (g factors {{g}||}, {{g}\\bot } and zero field splitting parameter D) for Fe6+ ions doped in K2MO4 (M = S, Cr, Se) crystals, taking into account the actual site symmetry of the Fe6+ impurity ion. The suggested method is based on the successful application of two different approaches: the crystal field theory (CFT) and density functional based (DFT). Within the CFT model we used the cluster approach and the perturbation theory method, based on the crystal field parameters, which were calculated in the superposition model. Within the DFT approach the calculations were done at the self-consistent field (SCF) by solving the coupled perturbed SCF equations. Comparison with experimental data shows that the obtained results are quite satisfactory, which proves applicability of the suggested calculating technique.
10. Input/Output of ab-initio nuclear structure calculations for improved performance and portability
SciTech Connect
Laghave, Nikhil
2010-01-01
Many modern scientific applications rely on highly computation intensive calculations. However, most applications do not concentrate as much on the role that input/output operations can play for improved performance and portability. Parallelizing input/output operations of large files can significantly improve the performance of parallel applications where sequential I/O is a bottleneck. A proper choice of I/O library also offers a scope for making input/output operations portable across different architectures. Thus, use of parallel I/O libraries for organizing I/O of large data files offers great scope in improving performance and portability of applications. In particular, sequential I/O has been identified as a bottleneck for the highly scalable MFDn (Many Fermion Dynamics for nuclear structure) code performing ab-initio nuclear structure calculations. We develop interfaces and parallel I/O procedures to use a well-known parallel I/O library in MFDn. As a result, we gain efficient I/O of large datasets along with their portability and ease of use in the down-stream processing. Even situations where the amount of data to be written is not huge, proper use of input/output operations can boost the performance of scientific applications. Application checkpointing offers enormous performance improvement and flexibility by doing a negligible amount of I/O to disk. Checkpointing saves and resumes application state in such a manner that in most cases the application is unaware that there has been an interruption to its execution. This helps in saving large amount of work that has been previously done and continue application execution. This small amount of I/O provides substantial time saving by offering restart/resume capability to applications. The need for checkpointing in optimization code NEWUOA has been identified and checkpoint/restart capability has been implemented in NEWUOA by using simple file I/O.
11. Ab Initio Enhanced calphad Modeling of Actinide-Rich Nuclear Fuels
SciTech Connect
Morgan, Dane; Yang, Yong Austin
2013-10-28
The process of fuel recycling is central to the Advanced Fuel Cycle Initiative (AFCI), where plutonium and the minor actinides (MA) Am, Np, and Cm are extracted from spent fuel and fabricated into new fuel for a fast reactor. Metallic alloys of U-Pu-Zr-MA are leading candidates for fast reactor fuels and are the current basis for fast spectrum metal fuels in a fully recycled closed fuel cycle. Safe and optimal use of these fuels will require knowledge of their multicomponent phase stability and thermodynamics (Gibbs free energies). In additional to their use as nuclear fuels, U-Pu-Zr-MA contain elements and alloy phases that pose fundamental questions about electronic structure and energetics at the forefront of modern many-body electron theory. This project will validate state-of-the-art electronic structure approaches for these alloys and use the resulting energetics to model U-Pu-Zr-MA phase stability. In order to keep the work scope practical, researchers will focus on only U-Pu-Zr-{Np,Am}, leaving Cm for later study. The overall objectives of this project are to: Provide a thermodynamic model for U-Pu-Zr-MA for improving and controlling reactor fuels; and, Develop and validate an ab initio approach for predicting actinide alloy energetics for thermodynamic modeling.
12. Spectroscopic fingerprints of toroidal nuclear quantum delocalization via ab initio path integral simulations.
PubMed
Schütt, Ole; Sebastiani, Daniel
2013-04-05
We investigate the quantum-mechanical delocalization of hydrogen in rotational symmetric molecular systems. To this purpose, we perform ab initio path integral molecular dynamics simulations of a methanol molecule to characterize the quantum properties of hydrogen atoms in a representative system by means of their real-space and momentum-space densities. In particular, we compute the spherically averaged momentum distribution n(k) and the pseudoangular momentum distribution n(kθ). We interpret our results by comparing them to path integral samplings of a bare proton in an ideal torus potential. We find that the hydroxyl hydrogen exhibits a toroidal delocalization, which leads to characteristic fingerprints in the line shapes of the momentum distributions. We can describe these specific spectroscopic patterns quantitatively and compute their onset as a function of temperature and potential energy landscape. The delocalization patterns in the projected momentum distribution provide a promising computational tool to address the intriguing phenomenon of quantum delocalization in condensed matter and its spectroscopic characterization. As the momentum distribution n(k) is also accessible through Nuclear Compton Scattering experiments, our results will help to interpret and understand future measurements more thoroughly.
13. Ab initio statistical mechanics of surface adsorption and desorption. II. Nuclear quantum effects.
PubMed
Alfè, D; Gillan, M J
2010-07-28
We show how the path-integral formulation of quantum statistical mechanics can be used to construct practical ab initio techniques for computing the chemical potential of molecules adsorbed on surfaces, with full inclusion of quantum nuclear effects. The techniques we describe are based on the computation of the potential of mean force on a chosen molecule and generalize the techniques developed recently for classical nuclei. We present practical calculations based on density functional theory with a generalized-gradient exchange-correlation functional for the case of H(2)O on the MgO (001) surface at low coverage. We note that the very high vibrational frequencies of the H(2)O molecule would normally require very large numbers of time slices (beads) in path-integral calculations, but we show that this requirement can be dramatically reduced by employing the idea of thermodynamic integration with respect to the number of beads. The validity and correctness of our path-integral calculations on the H(2)O/MgO(001) system are demonstrated by supporting calculations on a set of simple model systems for which quantum contributions to the free energy are known exactly from analytic arguments.
14. Ab initio molecular dynamics with nuclear quantum effects at classical cost: Ring polymer contraction for density functional theory.
PubMed
Marsalek, Ondrej; Markland, Thomas E
2016-02-07
Path integral molecular dynamics simulations, combined with an ab initio evaluation of interactions using electronic structure theory, incorporate the quantum mechanical nature of both the electrons and nuclei, which are essential to accurately describe systems containing light nuclei. However, path integral simulations have traditionally required a computational cost around two orders of magnitude greater than treating the nuclei classically, making them prohibitively costly for most applications. Here we show that the cost of path integral simulations can be dramatically reduced by extending our ring polymer contraction approach to ab initio molecular dynamics simulations. By using density functional tight binding as a reference system, we show that our ring polymer contraction scheme gives rapid and systematic convergence to the full path integral density functional theory result. We demonstrate the efficiency of this approach in ab initio simulations of liquid water and the reactive protonated and deprotonated water dimer systems. We find that the vast majority of the nuclear quantum effects are accurately captured using contraction to just the ring polymer centroid, which requires the same number of density functional theory calculations as a classical simulation. Combined with a multiple time step scheme using the same reference system, which allows the time step to be increased, this approach is as fast as a typical classical ab initio molecular dynamics simulation and 35× faster than a full path integral calculation, while still exactly including the quantum sampling of nuclei. This development thus offers a route to routinely include nuclear quantum effects in ab initio molecular dynamics simulations at negligible computational cost.
15. Ab initio molecular dynamics with nuclear quantum effects at classical cost: Ring polymer contraction for density functional theory
SciTech Connect
Marsalek, Ondrej; Markland, Thomas E.
2016-02-07
Path integral molecular dynamics simulations, combined with an ab initio evaluation of interactions using electronic structure theory, incorporate the quantum mechanical nature of both the electrons and nuclei, which are essential to accurately describe systems containing light nuclei. However, path integral simulations have traditionally required a computational cost around two orders of magnitude greater than treating the nuclei classically, making them prohibitively costly for most applications. Here we show that the cost of path integral simulations can be dramatically reduced by extending our ring polymer contraction approach to ab initio molecular dynamics simulations. By using density functional tight binding as a reference system, we show that our ring polymer contraction scheme gives rapid and systematic convergence to the full path integral density functional theory result. We demonstrate the efficiency of this approach in ab initio simulations of liquid water and the reactive protonated and deprotonated water dimer systems. We find that the vast majority of the nuclear quantum effects are accurately captured using contraction to just the ring polymer centroid, which requires the same number of density functional theory calculations as a classical simulation. Combined with a multiple time step scheme using the same reference system, which allows the time step to be increased, this approach is as fast as a typical classical ab initio molecular dynamics simulation and 35× faster than a full path integral calculation, while still exactly including the quantum sampling of nuclei. This development thus offers a route to routinely include nuclear quantum effects in ab initio molecular dynamics simulations at negligible computational cost.
16. Why Dynamic Simulations are Needed to Calculate Thermally Averaged Spin Hamiltonians
Weitekamp, Daniel P.; Mueller, Leonard J.
1998-03-01
The spin Hamiltonian needed to describe nearly all magnetic resonance experiments is an average over rapidly relaxing spatial degrees of freedom. This has previously been taken to be a Boltzmann average of quantities calculable from the time-independent Hamiltonian describing the system. We show why this approach is conceptually flawed and describe the physics of previously unsuspected, intrinsically dynamic, contributions to the spin Hamiltonian for this ubiquitous situation. Numerical estimates indicate that these new terms are required in order to simulate nuclear magnetic resonance spectra at the resolution with which they are routinely measured. An approach is outlined in which ab initio electronic structures may be combined with a tractable semi-classical description of rovibrational relaxation to give the necessary dynamic corrections, which are described by an average Liouvillian born as the result of spatial susceptibility (ALBATROSS).
17. Ab initio calculations of the intermolecular chemical shift in nuclear magnetic resonance in the gas phase and for adsorbed species
Jameson, Cynthia J.; de Dios, Angel C.
1992-07-01
The chemical shifts observed in nuclear magnetic resonance experiments are the differences in shielding of the nuclear spin in different electronic environments. These are known to depend on intermolecular interactions as evidenced by density-dependent chemical shifts in the gas phase, gas-to-liquid shifts, and adsorption shifts on surfaces. We present the results of the first ab initio intermolecular chemical shielding function calculated for a pair of interacting atoms for a wide range of internuclear separations. We used the localized orbital local origin (LORG) approach of Hansen and Bouman and also investigated the second-order electron correlation contributions using second-order LORG (SOLO). The 39Ar shielding in Ar2 passes through zero at some very short distance, going through a minimum, and asymptotically approaches zero at larger separations. The 21Ne shielding function in Ne2 has a similar shape. The Drude model suggests a method of scaling that portion of the shielding function that is weighted most heavily by exp[-V(R)/kT]. The scaling factors, which have been verified in the comparison of 21Ne in Ne2 against 39Ar in Ar2 ab initio results, allows us to project out from the same 39Ar in Ar2 ab initio values the appropriate 129Xe shielding functions in the Xe-Ar, Xe-Kr, and Xe-Xe interacting pairs. These functions lead to temperature-dependent second virial coefficients of chemical shielding which agree with experiments in the gas phase. Ab initio calculations of 39Ar shielding in clusters of argon are used to model the observed 129Xe chemical shifts of Xe, Xe2,...,Xe8 trapped in the cages of zeolite NaA.
18. Determining quasidiabatic coupled electronic state Hamiltonians using derivative couplings: A normal equations based method.
PubMed
Papas, Brian N; Schuurman, Michael S; Yarkony, David R
2008-09-28
A self-consistent procedure for constructing a quasidiabatic Hamiltonian representing N(state) coupled electronic states in the vicinity of an arbitrary point in nuclear coordinate space is described. The matrix elements of the Hamiltonian are polynomials of arbitrary order. Employing a crude adiabatic basis, the coefficients of the linear terms are determined exactly using analytic gradient techniques. The remaining polynomial coefficients are determined from the normal form of a system of pseudolinear equations, which uses energy gradient and derivative coupling information obtained from reliable multireference configuration interaction wave functions. In a previous implementation energy gradient and derivative coupling information were employed to limit the number of nuclear configurations at which ab initio data were required to determine the unknown coefficients. Conversely, the key aspect of the current approach is the use of ab initio data over an extended range of nuclear configurations. The normal form of the system of pseudolinear equations is introduced here to obtain a least-squares fit to what would have been an (intractable) overcomplete set of data in the previous approach. This method provides a quasidiabatic representation that minimizes the residual derivative coupling in a least-squares sense, a means to extend the domain of accuracy of the diabatic Hamiltonian or refine its accuracy within a given domain, and a way to impose point group symmetry and hermiticity. These attributes are illustrated using the 1 (2)A(1) and 1 (2)E states of the 1-propynyl radical, CH(3)CC.
19. Communication: XFAIMS—eXternal Field Ab Initio Multiple Spawning for electron-nuclear dynamics triggered by short laser pulses
SciTech Connect
Mignolet, Benoit; Curchod, Basile F. E.; Martinez, Todd J.
2016-11-17
Attoscience is an emerging field where attosecond pulses or few cycle IR pulses are used to pump and probe the correlated electron-nuclear motion of molecules. We present the trajectory-guided eXternal Field Ab Initio Multiple Spawning (XFAIMS) method that models such experiments “on-the-fly,” from laser pulse excitation to fragmentation or nonadiabatic relaxation to the ground electronic state. For the photoexcitation of the LiH molecule, we show that XFAIMS gives results in close agreement with numerically exact quantum dynamics simulations, both for atto- and femtosecond laser pulses. As a result, we then show the ability of XFAIMS to model the dynamics in polyatomic molecules by studying the effect of nuclear motion on the photoexcitation of a sulfine (H2CSO).
20. Communication: XFAIMS—eXternal Field Ab Initio Multiple Spawning for electron-nuclear dynamics triggered by short laser pulses
DOE PAGES
Mignolet, Benoit; Curchod, Basile F. E.; Martinez, Todd J.
2016-11-17
Attoscience is an emerging field where attosecond pulses or few cycle IR pulses are used to pump and probe the correlated electron-nuclear motion of molecules. We present the trajectory-guided eXternal Field Ab Initio Multiple Spawning (XFAIMS) method that models such experiments “on-the-fly,” from laser pulse excitation to fragmentation or nonadiabatic relaxation to the ground electronic state. For the photoexcitation of the LiH molecule, we show that XFAIMS gives results in close agreement with numerically exact quantum dynamics simulations, both for atto- and femtosecond laser pulses. As a result, we then show the ability of XFAIMS to model the dynamics inmore » polyatomic molecules by studying the effect of nuclear motion on the photoexcitation of a sulfine (H2CSO).« less
1. Weakly Hamiltonian actions
Martínez Torres, David; Miranda, Eva
2017-05-01
In this paper we generalize constructions of non-commutative integrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamiltonian actions on symplectic manifolds split into Hamiltonian and non-Hamiltonian factors, and explore generalizations in the Poisson setting.
2. Ab Initio Calculations Of Light-Ion Reactions
SciTech Connect
Navratil, P; Quaglioni, S; Roth, R; Horiuchi, W
2012-03-12
The exact treatment of nuclei starting from the constituent nucleons and the fundamental interactions among them has been a long-standing goal in nuclear physics. In addition to the complex nature of nuclear forces, one faces the quantum-mechanical many-nucleon problem governed by an interplay between bound and continuum states. In recent years, significant progress has been made in ab initio nuclear structure and reaction calculations based on input from QCD employing Hamiltonians constructed within chiral effective field theory. In this contribution, we present one of such promising techniques capable of describing simultaneously both bound and scattering states in light nuclei. By combining the resonating-group method (RGM) with the ab initio no-core shell model (NCSM), we complement a microscopic cluster approach with the use of realistic interactions and a microscopic and consistent description of the clusters. We discuss applications to light nuclei scattering, radiative capture and fusion reactions.
3. The isotropic Hamiltonian formalism
SciTech Connect
Vaisman, Izu
2011-02-10
A Hamiltonian formalism is a procedure that allows to associate a dynamical system to a function and that includes classical Hamiltonian mechanics as a particular case. The present, expository paper gives a survey of the Hamiltonian formalism defined by an isotropic subbundle of TM+T*M, in particular, by a Dirac structure. We discuss reduction and geometric quantization of the Hamiltonian dynamical systems provided by this formalism.
4. Precise Lifetime Measurements in Light Nuclei for Benchmarking Modern Ab-initio Nuclear Structure Models
SciTech Connect
Lister, C.J.; McCutchan, E.A.
2014-06-15
A new generation of ab-initio calculations, based on realistic two- and three-body forces, is having a profound impact on our view of how nuclei work. To improve the numerical methods, and the parameterization of 3-body forces, new precise data are needed. Electromagnetic transitions are very sensitive to the dynamics which drive mixing between configurations. We have made a series of precise (< 3%) measurements of electromagnetic transitions in the A=10 nuclei {sup 10}C and {sup 10}Be by using the Doppler Shift Attenuation method carefully. Many interesting features can be reproduced including the strong α clustering. New measurements on {sup 8}Be and {sup 12}Be highlight the interplay between the alpha clusters and their valence neutrons.
5. Ab initio determination of the nuclear relaxation contribution to the second hyperpolarizability of carbon disulfide
Champagne, Benoı̂t
1998-04-01
Although basis set saturation, electron correlation and frequency dispersion have been addressed thoroughly, the electronic second hyperpolarizability of carbon disulfide computed by K. Ohta, T. Sakaguchi, K. Kamada and T. Fukumi (Chem. Phys. Lett. 274 (1997) 306) is not in agreement with experiment. In this Letter the potentially substantial nuclear relaxation contribution is evaluated within the Møller-Plesset scheme limited to second order by using the 6-31G * basis set augmented by three diffuse functions (1p and 2d). Within the enhanced approximation, the nuclear relaxation contribution to the static, dc-Kerr and ESHG second hyperpolarizability turns out to amount to 26.5%, 6.8% and -0.8% of the pure static electronic counterpart, respectively. The remaining gap between theory and experiment suggests new experiments should be carried out.
6. Ab initio simulation of radiation damage in nuclear reactor pressure vessel materials
2012-02-01
Using Kinetic Monte Carlo we developed a code to study point defect hopping in BCC metallic alloys using energetics and attempt frequencies calculated using VASP, an electronic structure software package. Our code provides a way of simulating the effects of neutron radiation on potential reactor materials. Specifically we will compare the Molybdenum-Chromium alloy system to steel alloys for use in nuclear reactor pressure vessels.
7. Modelling the local atomic structure of molybdenum in nuclear waste glasses with ab initio molecular dynamics simulations.
PubMed
Konstantinou, Konstantinos; Sushko, Peter V; Duffy, Dorothy M
2016-09-21
The nature of chemical bonding of molybdenum in high level nuclear waste glasses has been elucidated by ab initio molecular dynamics simulations. Two compositions, (SiO2)57.5-(B2O3)10-(Na2O)15-(CaO)15-(MoO3)2.5 and (SiO2)57.3-(B2O3)20-(Na2O)6.8-(Li2O)13.4-(MoO3)2.5, were considered in order to investigate the effect of ionic and covalent components on the glass structure and the formation of the crystallisation precursors (Na2MoO4 and CaMoO4). The coordination environments of Mo cations and the corresponding bond lengths calculated from our model are in excellent agreement with experimental observations. The analysis of the first coordination shell reveals two different types of molybdenum host matrix bonds in the lithium sodium borosilicate glass. Based on the structural data and the bond valence model, we demonstrate that the Mo cation can be found in a redox state and the molybdate tetrahedron can be connected with the borosilicate network in a way that inhibits the formation of crystalline molybdates. These results significantly extend our understanding of bonding in Mo-containing nuclear waste glasses and demonstrate that tailoring the glass composition to specific heavy metal constituents can facilitate incorporation of heavy metals at high concentrations.
8. Photoexcited Nuclear Dynamics with Ab Initio Electronic Structure Theory: Is TD-DFT Ready For the Challenge?
Subotnik, Joseph
In this talk, I will give a broad overview of our work in nonadiabatic dynamics, i.e. the dynamics of strongly coupled nuclear-electronic motion whereby the relaxation of a photo-excited electron leads to the heating up of phonons. I will briefly discuss how to model such nuclear motion beyond mean field theory. Armed with the proper framework, I will then focus on how to calculate one flavor of electron-phonon couplings, known as derivative couplings in the chemical literature. Derivative couplings are the matrix elements that couple adiabatic electronic states within the Born-Oppenheimer treatment, and I will show that these matrix elements show spurious poles using formal (frequency-independent) time-dependent density functional theory. To correct this TD-DFT failure, a simple approximation will be proposed and evaluated. Finally, time permitting, I will show some ab initio calculations whereby one can use TD-DFT derivative couplings to study electronic relaxation through a conical intersection.
9. Direct assessment of quantum nuclear effects on hydrogen bond strength by constrained-centroid ab initio path integral molecular dynamics
Walker, Brent; Michaelides, Angelos
2010-11-01
The impact of quantum nuclear effects on hydrogen (H-) bond strength has been inferred in earlier work from bond lengths obtained from path integral molecular dynamics (PIMD) simulations. To obtain a direct quantitative assessment of such effects, we use constrained-centroid PIMD simulations to calculate the free energy changes upon breaking the H-bonds in dimers of HF and water. Comparing ab initio simulations performed using PIMD and classical nucleus molecular dynamics (MD), we find smaller dissociation free energies with the PIMD method. Specifically, at 50 K, the H-bond in (HF)2 is about 30% weaker when quantum nuclear effects are included, while that in (H2O)2 is about 15% weaker. In a complementary set of simulations, we compare unconstrained PIMD and classical nucleus MD simulations to assess the influence of quantum nuclei on the structures of these systems. We find increased heavy atom distances, indicating weakening of the H-bond consistent with that observed by direct calculation of the free energies of dissociation.
10. Significance of symmetry in the nuclear spin Hamiltonian for efficient heteronuclear dipolar decoupling in solid-state NMR: A Floquet description of supercycled rCW schemes.
PubMed
Equbal, Asif; Shankar, Ravi; Leskes, Michal; Vega, Shimon; Nielsen, Niels Chr; Madhu, P K
2017-03-14
Symmetry plays an important role in the retention or annihilation of a desired interaction Hamiltonian in NMR experiments. Here, we explore the role of symmetry in the radio-frequency interaction frame Hamiltonian of the refocused-continuous-wave (rCW) pulse scheme that leads to efficient (1)H heteronuclear decoupling in solid-state NMR. It is demonstrated that anti-periodic symmetry of single-spin operators (Ix, Iy, Iz) in the interaction frame can lead to complete annihilation of the (1)H-(1)H homonuclear dipolar coupling effects that induce line broadening in solid-state NMR experiments. This symmetry also plays a critical role in cancelling or minimizing the effect of (1)H chemical-shift anisotropy in the effective Hamiltonian. An analytical description based on Floquet theory is presented here along with experimental evidences to understand the decoupling efficiency of supercycled (concatenated) rCW scheme.
11. Significance of symmetry in the nuclear spin Hamiltonian for efficient heteronuclear dipolar decoupling in solid-state NMR: A Floquet description of supercycled rCW schemes
Equbal, Asif; Shankar, Ravi; Leskes, Michal; Vega, Shimon; Nielsen, Niels Chr.; Madhu, P. K.
2017-03-01
Symmetry plays an important role in the retention or annihilation of a desired interaction Hamiltonian in NMR experiments. Here, we explore the role of symmetry in the radio-frequency interaction frame Hamiltonian of the refocused-continuous-wave (rCW) pulse scheme that leads to efficient 1H heteronuclear decoupling in solid-state NMR. It is demonstrated that anti-periodic symmetry of single-spin operators (Ix, Iy, Iz) in the interaction frame can lead to complete annihilation of the 1H-1H homonuclear dipolar coupling effects that induce line broadening in solid-state NMR experiments. This symmetry also plays a critical role in cancelling or minimizing the effect of 1H chemical-shift anisotropy in the effective Hamiltonian. An analytical description based on Floquet theory is presented here along with experimental evidences to understand the decoupling efficiency of supercycled (concatenated) rCW scheme.
12. Hamiltonian Light-Front Ffield Theory in a Basis Function Approach
SciTech Connect
Vary, J.P.; Honkanen, H.; Li, Jun; Maris, P.; Brodsky, S.J.; Harindranath, A.; de Teramond, G.F.; Sternberg, P.; Ng, E.G.; Yang, C.
2009-05-15
Hamiltonian light-front quantum field theory constitutes a framework for the non-perturbative solution of invariant masses and correlated parton amplitudes of self-bound systems. By choosing the light-front gauge and adopting a basis function representation, we obtain a large, sparse, Hamiltonian matrix for mass eigenstates of gauge theories that is solvable by adapting the ab initio no-core methods of nuclear many-body theory. Full covariance is recovered in the continuum limit, the infinite matrix limit. There is considerable freedom in the choice of the orthonormal and complete set of basis functions with convenience and convergence rates providing key considerations. Here, we use a two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall AdS/QCD model obtained from light-front holography. We outline our approach, present illustrative features of some non-interacting systems in a cavity and discuss the computational challenges.
13. Open-shell nuclei and excited states from multireference normal-ordered Hamiltonians
Gebrerufael, Eskendr; Calci, Angelo; Roth, Robert
2016-03-01
We discuss the approximate inclusion of three-nucleon (3 N ) interactions into ab initio nuclear structure calculations using a multireference formulation of normal ordering and Wick's theorem. Following the successful application of single-reference normal ordering for the study of ground states of closed-shell nuclei, e.g., in coupled-cluster theory, multireference normal ordering opens a path to open-shell nuclei and excited states. Based on different multideterminantal reference states we benchmark the truncation of the normal-ordered Hamiltonian at the two-body level in no-core shell-model calculations for p -shell nuclei, including 6Li,12C, and 10B. We find that this multireference normal-ordered two-body approximation is able to capture the effects of the 3 N interaction with sufficient accuracy, both for ground-state and excitation energies, at the computational cost of a two-body Hamiltonian. It is robust with respect to the choice of reference states and has a multitude of applications in ab initio nuclear structure calculations of open-shell nuclei and their excitations as well as in nuclear reaction studies.
14. Fitting and using model Hamiltonian in non-adiabatic molecular dynamics simulations
Smale, Jonathan Ross
In order to study computationally increasingly complex systems using theoretical methods model, Hamiltonians are required to accurately describe the potential energy surface they represent. Also ab-initio methods improve the calculation of the excited states of these complex systems becomes increasingly feasible. One such model Hamiltonian described herein, the Vibronic Coupling Hamiltonian, has previously shown its versatility and ability to describe a variety of non-adiabatic problems. This thesis describes a new method, a genetic algorithm, for the parameterisation of the Vibronic Coupling Hamiltonian to describe both previously calculated potential energy surfaces (allene and pentatetraene) and newly calculated (cyclo-butadiene and toluene) potential energy surfaces. In order to test this genetic algorithm, quantum nuclear dynamics calculations were performed using the multi-configurational time dependent Hartree method and the results compared to experiment..
15. Ab initio Study of Nuclear Quadrupole Interactions in Selenium and Tellurium
Maharjan, N. B.; Paudyal, D. D.; Mishra, D. R.; Byahut, S. P.; Cho, Hwa-Suck; Scheicher, R. H.; Jeong, Junho; Das, T. P.
2004-03-01
We are systematically studying the influence of impurities in calcogenide glasses on the glass transition temperature using the first-principles Hartree-Fock cluster method. Results of our calculations on the electronic structures of pure selenium and tellurium chain systems, and with Te and Se impurities respectively, will be reported. By comparing the theoretically obtained nuclear quadrupole interaction (NQI) tensors for ^77Se and ^125Te with available experimental NQI tensors, we were able to test the accuracy of the calculated electronic structures. Good agreement for both the pure and the impurity systems has been found. We have also studied ^125Te NQI tensors in Te-Thiourea and compared our result with experimental data to check on the choice of the ^125Te quadrupole moment used.
16. MO-AB-207-03: ACR Update in Nuclear Medicine
SciTech Connect
Harkness, B.
2015-06-15
A goal of an imaging accreditation program is to ensure adequate image quality, verify appropriate staff qualifications, and to assure patient and personnel safety. Currently, more than 35,000 facilities in 10 modalities have been accredited by the American College of Radiology (ACR), making the ACR program one of the most prolific accreditation options in the U.S. In addition, ACR is one of the accepted accreditations required by some state laws, CMS/MIPPA insurance and others. Familiarity with the ACR accreditation process is therefore essential to clinical diagnostic medical physicists. Maintaining sufficient knowledge of the ACR program must include keeping up-to-date as the various modality requirements are refined to better serve the goals of the program and to accommodate newer technologies and practices. This session consists of presentations from authorities in four ACR accreditation modality programs, including magnetic resonance imaging, computed tomography, nuclear medicine, and mammography. Each speaker will discuss the general components of the modality program and address any recent changes to the requirements. Learning Objectives: To understand the requirements of the ACR MR Accreditation program. The discussion will include accreditation of whole-body general purpose magnets, dedicated extremity systems well as breast MRI accreditation. Anticipated updates to the ACR MRI Quality Control Manual will also be reviewed. To understand the requirements of the ACR CT accreditation program, including updates to the QC manual as well as updates through the FAQ process. To understand the requirements of the ACR nuclear medicine accreditation program, and the role of the physicist in annual equipment surveys and the set up and supervision of the routine QC program. To understand the current ACR MAP Accreditation requirement and present the concepts and structure of the forthcoming ACR Digital Mammography QC Manual and Program.
17. MO-AB-207-00: ACR Update in MR, CT, Nuclear Medicine, and Mammography
SciTech Connect
2015-06-15
A goal of an imaging accreditation program is to ensure adequate image quality, verify appropriate staff qualifications, and to assure patient and personnel safety. Currently, more than 35,000 facilities in 10 modalities have been accredited by the American College of Radiology (ACR), making the ACR program one of the most prolific accreditation options in the U.S. In addition, ACR is one of the accepted accreditations required by some state laws, CMS/MIPPA insurance and others. Familiarity with the ACR accreditation process is therefore essential to clinical diagnostic medical physicists. Maintaining sufficient knowledge of the ACR program must include keeping up-to-date as the various modality requirements are refined to better serve the goals of the program and to accommodate newer technologies and practices. This session consists of presentations from authorities in four ACR accreditation modality programs, including magnetic resonance imaging, computed tomography, nuclear medicine, and mammography. Each speaker will discuss the general components of the modality program and address any recent changes to the requirements. Learning Objectives: To understand the requirements of the ACR MR Accreditation program. The discussion will include accreditation of whole-body general purpose magnets, dedicated extremity systems well as breast MRI accreditation. Anticipated updates to the ACR MRI Quality Control Manual will also be reviewed. To understand the requirements of the ACR CT accreditation program, including updates to the QC manual as well as updates through the FAQ process. To understand the requirements of the ACR nuclear medicine accreditation program, and the role of the physicist in annual equipment surveys and the set up and supervision of the routine QC program. To understand the current ACR MAP Accreditation requirement and present the concepts and structure of the forthcoming ACR Digital Mammography QC Manual and Program.
18. Quantum ring-polymer contraction method: Including nuclear quantum effects at no additional computational cost in comparison to ab initio molecular dynamics
John, Christopher; Spura, Thomas; Habershon, Scott; Kühne, Thomas D.
2016-04-01
We present a simple and accurate computational method which facilitates ab initio path-integral molecular dynamics simulations, where the quantum-mechanical nature of the nuclei is explicitly taken into account, at essentially no additional computational cost in comparison to the corresponding calculation using classical nuclei. The predictive power of the proposed quantum ring-polymer contraction method is demonstrated by computing various static and dynamic properties of liquid water at ambient conditions using density functional theory. This development will enable routine inclusion of nuclear quantum effects in ab initio molecular dynamics simulations of condensed-phase systems.
19. Ab initio nuclear many-body perturbation calculations in the Hartree-Fock basis
Hu, B. S.; Xu, F. R.; Sun, Z. H.; Vary, J. P.; Li, T.
2016-07-01
Starting from realistic nuclear forces, the chiral N3LO and JISP16, we have applied many-body perturbation theory (MBPT) to the structure of closed-shell nuclei, 4He and 16O. The two-body N3LO interaction is softened by a similarity renormalization group transformation while JISP16 is adopted without renormalization. The MBPT calculations are performed within the Hartree-Fock (HF) bases. The angular momentum coupled scheme is used, which can reduce the computational task. Corrections up to the third order in energy and up to the second order in radius are evaluated. Higher-order corrections in the HF basis are small relative to the leading-order perturbative result. Using the antisymmetrized Goldstone diagram expansions of the wave function, we directly correct the one-body density for the calculation of the radius, rather than calculate corrections to the occupation probabilities of single-particle orbits as found in other treatments. We compare our results with other methods where available and find good agreement. This supports the conclusion that our methods produce reasonably converged results with these interactions. We also compare our results with experimental data.
20. The Role of Anharmonicity and Nuclear Quantum Effects in the Pyridine Molecular Crystal: An ab initio Molecular Dynamics Study
Ko, Hsin-Yu; Distasio, Robert A., Jr.; Santra, Biswajit; Car, Roberto
Molecular crystal structure prediction has posed a substantial challenge to first-principles methods and requires sophisticated electronic structure methods to determine the stabilities of nearly degenerate polymorphs. In this work, we demonstrate that the anharmonicity from van der Waals interactions is relevant to the finite-temperature structures of pyridine and pyridine-like molecular crystals. Using such an approach, we find that the equilibrium structures are well captured with classical ab initio molecular dynamics (AIMD), despite the presence of light atoms such as hydrogen. Employing path integral AIMD simulations, we demonstrate that the success of classical AIMD results from a separation of nuclear quantum effects between the intermolecular and intramolecular degrees of freedom. In this separation, the quasiclassical and anharmonic intermolecular degrees of freedom determine the equilibrium structure, while the quantum and harmonic intramolecular degrees of freedom are averaging to the correct intramolecular structure. This work has been supported by the Department of Energy under Grants No. DE-FG02-05ER46201 and DE-SC0008626.
1. Vibrational circular dichroism from ab initio molecular dynamics and nuclear velocity perturbation theory in the liquid phase.
PubMed
Scherrer, Arne; Vuilleumier, Rodolphe; Sebastiani, Daniel
2016-08-28
We report the first fully ab initio calculation of dynamical vibrational circular dichroism spectra in the liquid phase using nuclear velocity perturbation theory (NVPT) derived electronic currents. Our approach is rigorous and general and thus capable of treating weak interactions of chiral molecules as, e.g., chirality transfer from a chiral molecule to an achiral solvent. We use an implementation of the NVPT that is projected along the dynamics to obtain the current and magnetic dipole moments required for accurate intensities. The gauge problem in the liquid phase is resolved in a twofold approach. The electronic expectation values are evaluated in a distributed origin gauge, employing maximally localized Wannier orbitals. In a second step, the gauge invariant spectrum is obtained in terms of a scaled molecular moments, which allows to systematically include solvent effects while keeping a significant signal-to-noise ratio. We give a thorough analysis and discussion of this choice of gauge for the liquid phase. At low temperatures, we recover the established double harmonic approximation. The methodology is applied to chiral molecules ((S)-d2-oxirane and (R)-propylene-oxide) in the gas phase and in solution. We find an excellent agreement with the theoretical and experimental references, including the emergence of signals due to chirality transfer from the solute to the (achiral) solvent.
2. Nuclear Quantum Effects in Liquid Water: A Highly Accurate ab initio Path-Integral Molecular Dynamics Study
Distasio, Robert A., Jr.; Santra, Biswajit; Ko, Hsin-Yu; Car, Roberto
2014-03-01
In this work, we report highly accurate ab initio path-integral molecular dynamics (AI-PIMD) simulations on liquid water at ambient conditions utilizing the recently developed PBE0+vdW(SC) exchange-correlation functional, which accounts for exact exchange and a self-consistent pairwise treatment of van der Waals (vdW) or dispersion interactions, combined with nuclear quantum effects (via the colored-noise generalized Langevin equation). The importance of each of these effects in the theoretical prediction of the structure of liquid water will be demonstrated by a detailed comparative analysis of the predicted and experimental oxygen-oxygen (O-O), oxygen-hydrogen (O-H), and hydrogen-hydrogen (H-H) radial distribution functions as well as other structural properties. In addition, we will discuss the theoretically obtained proton momentum distribution, computed using the recently developed Feynman path formulation, in light of the experimental deep inelastic neutron scattering (DINS) measurements. DOE: DE-SC0008626, DOE: DE-SC0005180.
3. Modelling the local atomic structure of molybdenum in nuclear waste glasses with ab initio molecular dynamics simulations
SciTech Connect
None, None
2016-01-01
The nature of chemical bonding of molybdenum in high level nuclear waste glasses has been elucidated by ab initio molecular dynamics simulations. Two compositions, (SiO2)57.5 – (B2O3)10 – (Na2O)15 – (CaO)15 – (MoO3)2.5 and (SiO2)57.3 – (B2O3)20 – (Na2O)6.8 – (Li2O)13.4 – (MoO3)2.5 , were considered in order to investigate the effect of ionic and covalent components on the glass structure and the formation of the crystallisation precursors (Na2MoO4 and CaMoO4). The coordination environments of Mo cations and the corresponding bond lengths calculated from our model are in excellent agreement with experimental observations. The analysis of the first coordination shell reveals two different types of molybdenum host matrix bonds in the lithium sodium borosilicate glass. Based on the structural data and the bond valence model, we demonstrate that the Mo cation can be found in a redox state and the molybdate tetrahedron can be connected with the borosilicate network in a way that inhibits the formation of crystalline molybdates. These results significantly extend our understanding of bonding in Mo-containing nuclear waste glasses and demonstrate that tailoring the glass composition to specific heavy metal constituents can facilitate incorporation of heavy metals at high concentrations. K.K. was supported through the Impact Studentship scheme at UCL co-funded by the IHI Corporation and UCL. P.V.S. thanks the Royal Society, which supported preliminary work on this project, and the Laboratory Directed Research and Development program at PNNL, a multiprogram national laboratory operated by Battelle for the U.S. Department of Energy. Via our membership of the UK's HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202), this work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk).
4. Effective Floquet Hamiltonians for dipolar and quadrupolar coupled N-spin systems in solid state nuclear magnetic resonance under magic angle spinning.
PubMed
Pandey, Manoj Kumar; Krishnan, Mangala Sunder
2010-11-07
Spin dynamics under magic angle spinning has been studied using different theoretical approaches and also by extensive numerical simulation programs. In this article we present a general theoretical approach that leads to analytic forms for effective Hamiltonians for an N-spin dipolar and quadrupolar coupled system under magic angle spinning (MAS) conditions, using a combination of Floquet theory and van Vleck (contact) transformation. The analytic forms presented are shown to be useful for the study of MAS spin dynamics in solids with the help of a number of simulations in two, three, and four coupled, spin-1/2 systems as well as spins in which quadrupolar interactions are also present.
5. Path Integrals and Hamiltonians
Baaquie, Belal E.
2014-03-01
1. Synopsis; Part I. Fundamental Principles: 2. The mathematical structure of quantum mechanics; 3. Operators; 4. The Feynman path integral; 5. Hamiltonian mechanics; 6. Path integral quantization; Part II. Stochastic Processes: 7. Stochastic systems; Part III. Discrete Degrees of Freedom: 8. Ising model; 9. Ising model: magnetic field; 10. Fermions; Part IV. Quadratic Path Integrals: 11. Simple harmonic oscillators; 12. Gaussian path integrals; Part V. Action with Acceleration: 13. Acceleration Lagrangian; 14. Pseudo-Hermitian Euclidean Hamiltonian; 15. Non-Hermitian Hamiltonian: Jordan blocks; 16. The quartic potential: instantons; 17. Compact degrees of freedom; Index.
6. Branched Hamiltonians and supersymmetry
DOE PAGES
Curtright, Thomas L.; Zachos, Cosmas K.
2014-03-21
Some examples of branched Hamiltonians are explored both classically and in the context of quantum mechanics, as recently advocated by Shapere and Wilczek. These are in fact cases of switchback potentials, albeit in momentum space, as previously analyzed for quasi-Hamiltonian chaotic dynamical systems in a classical setting, and as encountered in analogous renormalization group flows for quantum theories which exhibit RG cycles. In conclusion, a basic two-worlds model, with a pair of Hamiltonian branches related by supersymmetry, is considered in detail.
7. Supersymmetry of tridiagonal Hamiltonians
Yamani, Hashim A.; Mouayn, Zouhair
2014-07-01
A positive semi-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set, allows a definition of forward- and backward-shift operators that can be used to define the matrix representation of its supersymmetric partner Hamiltonian H( + ) with respect to the same basis. We find explicit relationships connecting the matrix elements of both Hamiltonians. We present a method to obtain the orthogonal polynomials in the eigenstate expansion problem attached to H( + ) starting from those polynomials arising in the same problem for H. This connection is established by using the notion of kernel polynomials. We apply the obtained results to two known solvable models with different kinds of spectrum.
8. Dynamical supersymmetric Dirac Hamiltonians
SciTech Connect
Ginocchio, J.N.
1986-01-01
Using the language of quantum electrodynamics, the Dirac Hamiltonian of a neutral fermion interacting with a tensor field is examined. A supersymmetry found for a general Dirac Hamiltonian of this type is discussed, followed by consideration of the special case of a harmonic electric potential. The square of the Dirac Hamiltonian of a neutral fermion interacting via an anomalous magnetic moment in an electric potential is shown to be equivalent to a three-dimensional supersymmetric Schroedinger equation. It is found that for a potential that grows as a power of r, the lowest energy of the Hamiltonian equals the rest mass of the fermion, and the Dirac eigenfunction has only an upper component which is normalizable. It is also found that the higher energy states have upper and lower components which form a supersymmetric doublet. 15 refs. (LEW)
9. Stimulated Raman signals at conical intersections: Ab initio surface hopping simulation protocol with direct propagation of the nuclear wave function.
PubMed
Kowalewski, Markus; Mukamel, Shaul
2015-07-28
Femtosecond Stimulated Raman Spectroscopy (FSRS) signals that monitor the excited state conical intersections dynamics of acrolein are simulated. An effective time dependent Hamiltonian for two C-H vibrational marker bands is constructed on the fly using a local mode expansion combined with a semi-classical surface hopping simulation protocol. The signals are obtained by a direct forward and backward propagation of the vibrational wave function on a numerical grid. Earlier work is extended to fully incorporate the anharmonicities and intermode couplings.
10. Stimulated Raman signals at conical intersections: Ab initio surface hopping simulation protocol with direct propagation of the nuclear wave function
SciTech Connect
Kowalewski, Markus Mukamel, Shaul
2015-07-28
Femtosecond Stimulated Raman Spectroscopy (FSRS) signals that monitor the excited state conical intersections dynamics of acrolein are simulated. An effective time dependent Hamiltonian for two C—H vibrational marker bands is constructed on the fly using a local mode expansion combined with a semi-classical surface hopping simulation protocol. The signals are obtained by a direct forward and backward propagation of the vibrational wave function on a numerical grid. Earlier work is extended to fully incorporate the anharmonicities and intermode couplings.
11. Ab initio path integral simulations for the fluoride ion-water clusters: competitive nuclear quantum effect between F(-)-water and water-water hydrogen bonds.
PubMed
Kawashima, Yukio; Suzuki, Kimichi; Tachikawa, Masanori
2013-06-20
Small hydrated fluoride ion complexes, F(-)(H2O)n (n = 1-3), have been studied by ab initio hybrid Monte Carlo (HMC) and ab initio path integral hybrid Monte Carlo (PIHMC) simulations. Because of the quantum effect, our simulation shows that the average hydrogen-bonded F(-)···HO distance in the quantum F(-)(H2O) is shorter than that in the classical one, while the relation inverts at the three water molecular F(-)(H2O)3 cluster. In the case of F(-)(H2O)3, we have found that the nuclear quantum effect enhances the formation of hydrogen bonds between two water molecules. In F(-)(H2O)2 and F(-)(H2O)3, the nuclear quantum effect on two different kinds of hydrogen bonds, F(-)-water and water-water hydrogen bonds, competes against each other. In F(-)(H2O)3, thus, the nuclear quantum effect on the water-water hydrogen bond leads to the elongation of hydrogen-bonded F(-)···HO distance, which we suggest this as the possible origin of the structural inversion from F(-)(H2O) to F(-)(H2O)3.
12. Nuclear polarization corrections to the μ4He+ Lamb shift.
PubMed
Ji, C; Nevo Dinur, N; Bacca, S; Barnea, N
2013-10-04
Stimulated by the proton radius conundrum, measurements of the Lamb shift in various light muonic atoms are planned at PSI. The aim is to extract the rms charge radius with high precision, limited by the uncertainty in the nuclear polarization corrections. We present an ab initio calculation of the nuclear polarization for μ(4)He(+) leading to an energy correction in the 2S-2P transitions of δ(pol)(A)=-2.47 meV ±6%. We use two different state-of-the-art nuclear Hamiltonians and utilize the Lorentz integral transform with hyperspherical harmonics expansion as few-body methods. We take into account the leading multipole contributions, plus Coulomb, relativistic, and finite-nucleon-size corrections. Our main source of uncertainty is the nuclear Hamiltonian, which currently limits the attainable accuracy. Our predictions considerably reduce the uncertainty with respect to previous estimates and should be instrumental to the μ(4)He(+) experiment planned for 2013.
13. Allene and pentatetraene cations as models for intramolecular charge transfer: vibronic coupling Hamiltonian and conical intersections.
PubMed
Markmann, Andreas; Worth, Graham A; Cederbaum, Lorenz S
2005-04-08
We consider the vibronic coupling effects involving cationic states with degenerate components that can be represented as charge localized at either end of the short cumulene molecules allene and pentatetraene. Our aim is to simulate dynamically the charge transfer process when one component is artificially depopulated. We model the Jahn-Teller vibronic interaction within these states as well as their pseudo-Jahn-Teller coupling with some neighboring states. For the manifold of these states, we have calculated cross sections of the ab initio adiabatic potential energy surfaces along all nuclear degrees of freedom, including points at large distances from the equilibrium to increase the physical significance of our model. Ab initio calculations for the cationic states of allene and pentatetraene were based on the fourth-order Møller-Plesset method and the outer valence Green's function method. In some cases we had to go beyond this method and use the more involved third-order algebraic diagrammatic construction method to include intersections with satellite states. The parameters for a five-state, all-mode diabatic vibronic coupling model Hamiltonian were least-square fitted to these potentials. The coupling parameters for the diabatic model Hamiltonian are such that, in comparison to allene, an enhanced preference for indirect charge transfer is predicted for pentatetraene.
14. Determination of nuclear quadrupole moments – An example of the synergy of ab initio calculations and microwave spectroscopy
SciTech Connect
2015-01-22
Highly correlated scalar relativistic calculations of electric field gradients at nuclei in diatomic molecules in combination with accurate nuclear quadrupole coupling constants obtained from microwave spectroscopy are used for determination of nuclear quadrupole moments.
15. Stochastic surrogate Hamiltonian
Katz, Gil; Gelman, David; Ratner, Mark A.; Kosloff, Ronnie
2008-07-01
The surrogate Hamiltonian is a general scheme to simulate the many body quantum dynamics composed of a primary system coupled to a bath. The method has been based on a representative bath Hamiltonian composed of two-level systems that is able to mimic the true system-bath dynamics up to a prespecified time. The original surrogate Hamiltonian method is limited to short time dynamics since the size of the Hilbert space required to obtain convergence grows exponentially with time. By randomly swapping bath modes with a secondary thermal reservoir, the method can simulate quantum dynamics of the primary system from short times to thermal equilibrium. By averaging a small number of realizations converged values of the system observables are obtained avoiding the exponential increase in resources. The method is demonstrated for the equilibration of a molecular oscillator with a thermal bath.
16. Stochastic surrogate Hamiltonian
SciTech Connect
Katz, Gil; Kosloff, Ronnie; Gelman, David; Ratner, Mark A.
2008-07-21
The surrogate Hamiltonian is a general scheme to simulate the many body quantum dynamics composed of a primary system coupled to a bath. The method has been based on a representative bath Hamiltonian composed of two-level systems that is able to mimic the true system-bath dynamics up to a prespecified time. The original surrogate Hamiltonian method is limited to short time dynamics since the size of the Hilbert space required to obtain convergence grows exponentially with time. By randomly swapping bath modes with a secondary thermal reservoir, the method can simulate quantum dynamics of the primary system from short times to thermal equilibrium. By averaging a small number of realizations converged values of the system observables are obtained avoiding the exponential increase in resources. The method is demonstrated for the equilibration of a molecular oscillator with a thermal bath.
17. Experimental quantum Hamiltonian learning
Wang, Jianwei; Paesani, Stefano; Santagati, Raffaele; Knauer, Sebastian; Gentile, Antonio A.; Wiebe, Nathan; Petruzzella, Maurangelo; O'Brien, Jeremy L.; Rarity, John G.; Laing, Anthony; Thompson, Mark G.
2017-06-01
The efficient characterization of quantum systems, the verification of the operations of quantum devices and the validation of underpinning physical models, are central challenges for quantum technologies and fundamental physics. The computational cost of such studies could be improved by machine learning enhanced by quantum simulators. Here we interface two different quantum systems through a classical channel--a silicon-photonics quantum simulator and an electron spin in a diamond nitrogen-vacancy centre--and use the former to learn the Hamiltonian of the latter via Bayesian inference. We learn the salient Hamiltonian parameter with an uncertainty of approximately 10-5. Furthermore, an observed saturation in the learning algorithm suggests deficiencies in the underlying Hamiltonian model, which we exploit to further improve the model. We implement an interactive version of the protocol and experimentally show its ability to characterize the operation of the quantum photonic device.
18. Machine-learned approximations to Density Functional Theory Hamiltonians
PubMed Central
Hegde, Ganesh; Bowen, R. Chris
2017-01-01
Large scale Density Functional Theory (DFT) based electronic structure calculations are highly time consuming and scale poorly with system size. While semi-empirical approximations to DFT result in a reduction in computational time versus ab initio DFT, creating such approximations involves significant manual intervention and is highly inefficient for high-throughput electronic structure screening calculations. In this letter, we propose the use of machine-learning for prediction of DFT Hamiltonians. Using suitable representations of atomic neighborhoods and Kernel Ridge Regression, we show that an accurate and transferable prediction of DFT Hamiltonians for a variety of material environments can be achieved. Electronic structure properties such as ballistic transmission and band structure computed using predicted Hamiltonians compare accurately with their DFT counterparts. The method is independent of the specifics of the DFT basis or material system used and can easily be automated and scaled for predicting Hamiltonians of any material system of interest. PMID:28198471
19. Machine-learned approximations to Density Functional Theory Hamiltonians
Hegde, Ganesh; Bowen, R. Chris
2017-02-01
Large scale Density Functional Theory (DFT) based electronic structure calculations are highly time consuming and scale poorly with system size. While semi-empirical approximations to DFT result in a reduction in computational time versus ab initio DFT, creating such approximations involves significant manual intervention and is highly inefficient for high-throughput electronic structure screening calculations. In this letter, we propose the use of machine-learning for prediction of DFT Hamiltonians. Using suitable representations of atomic neighborhoods and Kernel Ridge Regression, we show that an accurate and transferable prediction of DFT Hamiltonians for a variety of material environments can be achieved. Electronic structure properties such as ballistic transmission and band structure computed using predicted Hamiltonians compare accurately with their DFT counterparts. The method is independent of the specifics of the DFT basis or material system used and can easily be automated and scaled for predicting Hamiltonians of any material system of interest.
20. Machine-learned approximations to Density Functional Theory Hamiltonians.
PubMed
Hegde, Ganesh; Bowen, R Chris
2017-02-15
Large scale Density Functional Theory (DFT) based electronic structure calculations are highly time consuming and scale poorly with system size. While semi-empirical approximations to DFT result in a reduction in computational time versus ab initio DFT, creating such approximations involves significant manual intervention and is highly inefficient for high-throughput electronic structure screening calculations. In this letter, we propose the use of machine-learning for prediction of DFT Hamiltonians. Using suitable representations of atomic neighborhoods and Kernel Ridge Regression, we show that an accurate and transferable prediction of DFT Hamiltonians for a variety of material environments can be achieved. Electronic structure properties such as ballistic transmission and band structure computed using predicted Hamiltonians compare accurately with their DFT counterparts. The method is independent of the specifics of the DFT basis or material system used and can easily be automated and scaled for predicting Hamiltonians of any material system of interest.
1. Hamiltonian Engineering for High Fidelity Quantum Operations
Ribeiro, Hugo; Baksic, Alexandre; Clerk, Aashish
High-fidelity gates and operations are crucial to almost every aspect of quantum information processing. In recent experiments, fidelity is mostly limited by unwanted couplings with states living out of the logical subspace. This results in both leakage and phase errors. Here, we present a general method to deal simultaneously with both these issues and improve the fidelity of quantum gates and operations. Our method is applicable to a wide variety of systems. As an example, we can correct gates for superconducting qubits, improve coherent state transfer between a single NV centre electronic spin and a single nitrogen nuclear spin, improve control over a nuclear spin ensemble, etc. Our method is intimately linked to the Magnus expansion. By modifying the Magnus expansion of an initially given Hamiltonian Hi, we find analytically additional control Hamiltonians Hctrl such that Hi +Hctrl leads to the desired gate while minimizing both leakage and phase errors.
2. Which grids are Hamiltonian
SciTech Connect
Hedetniemi, S. M.; Hedetniemi, S. T.; Slater, P. J.
1980-01-01
A complete grid G/sub m,n/ is a graph having m x n pertices that are connected to form a rectangular lattice in the plane, i.e., all edges of G/sub m,n/ connect vertices along horizontal or vertical lines. A grid is a subgraph of a complete grid. As an illustration, complete grids describe the basic pattern of streets in most cities. This paper examines the existence of Hamiltonian cycles in complete grids and complete grids with one or two vertices removed. It is determined for most values of m,n greater than or equal to 1, which grids G/sub m,n/ - (u) and G/sub m,n/ - (u,v) are Hamiltonian. 12 figures. (RWR)
3. Hamiltonian spinfoam gravity
Wieland, Wolfgang M.
2014-01-01
This paper presents a Hamiltonian formulation of spinfoam gravity, which leads to a straightforward canonical quantization. To begin with, we derive a continuum action adapted to a simplicial decomposition of space-time. The equations of motion admit a Hamiltonian formulation, allowing us to perform the constraint analysis. We do not find any secondary constraints, but only get restrictions on the Lagrange multipliers enforcing the reality conditions. This comes as a surprise—in the continuum theory, the reality conditions are preserved in time, only if the torsionless condition (a secondary constraint) holds true. Studying an additional conservation law for each spinfoam vertex, we discuss the issue of torsion and argue that spinfoam gravity may still miss an additional constraint. Finally, we canonically quantize and recover the EPRL (Engle-Pereira-Rovelli-Livine) face amplitudes. Communicated by P R L V Moniz
4. An electromechanical Ising Hamiltonian
PubMed Central
Mahboob, Imran; Okamoto, Hajime; Yamaguchi, Hiroshi
2016-01-01
Solving intractable mathematical problems in simulators composed of atoms, ions, photons, or electrons has recently emerged as a subject of intense interest. We extend this concept to phonons that are localized in spectrally pure resonances in an electromechanical system that enables their interactions to be exquisitely fashioned via electrical means. We harness this platform to emulate the Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from the parametric resonances of multiple modes. The coupling between the mechanical spins is created by generating two-mode squeezed states, which impart correlations between modes that can imitate a random, ferromagnetic state or an antiferromagnetic state on demand. These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling. PMID:28861469
5. Approximate symmetries of Hamiltonians
Chubb, Christopher T.; Flammia, Steven T.
2017-08-01
We explore the relationship between approximate symmetries of a gapped Hamiltonian and the structure of its ground space. We start by considering approximate symmetry operators, defined as unitary operators whose commutators with the Hamiltonian have norms that are sufficiently small. We show that when approximate symmetry operators can be restricted to the ground space while approximately preserving certain mutual commutation relations. We generalize the Stone-von Neumann theorem to matrices that approximately satisfy the canonical (Heisenberg-Weyl-type) commutation relations and use this to show that approximate symmetry operators can certify the degeneracy of the ground space even though they only approximately form a group. Importantly, the notions of "approximate" and "small" are all independent of the dimension of the ambient Hilbert space and depend only on the degeneracy in the ground space. Our analysis additionally holds for any gapped band of sufficiently small width in the excited spectrum of the Hamiltonian, and we discuss applications of these ideas to topological quantum phases of matter and topological quantum error correcting codes. Finally, in our analysis, we also provide an exponential improvement upon bounds concerning the existence of shared approximate eigenvectors of approximately commuting operators under an added normality constraint, which may be of independent interest.
6. Chaotic Hamiltonian Dynamics.
Bialek, James Mark
Chaotic behavior may be observed in deterministic Hamiltonian Systems with as few as three dimensions, i.e., X, P, and t. The amount of chaotic behavior depends on the relative influence of the integrable and non-integrable parts of the Hamiltonian. The Standard Map is such a system and the amount of chaotic behavior may be varied by adjusting a single parameter. The global phase space portrait is a complicated mixture of quiescent and chaotic regions. First a new calculational method, characterized by a Fractal Diagram, is presented. This allows the quantitative prediction of the boundaries between regular and chaotic regions in phase space. Where these barriers are located gives qualitative insight into diffusion in phase space. The method is illustrated with the Standard Map but may be applied to any Hamiltonian System. The second phenomenon is the Universal Behavior predicted to occur for all area preserving maps. As a parameter is varied causing the mapping to become more chaotic a pattern is observed in the location and stability of the fixed points of the maps. The fixed points undergo an infinite sequence of period doubling bifurcations in a finite range of the parameter. The relative locations of the fixed point bifurcation and the parameter intervals between bifurcations both asymptotically approach constants which are Universal in that the same constants keep appearing in different problems. Predictions of Universal Behavior have been based on the study of algebraic mappings. The problem we examine has a Hamiltonian given by H = p^2 over {2} - lambda over{2pi}sin(2pi x)sin(2pit). This Hamiltonian describes the motion of a compass needle in a sinusoidally varying magnetic field or, equally well, the one dimensional motion of a particle in a standing wave potential. By treating the magnitude(lambda ) of the time dependent potential as a parameter and by examining the trajectories of the system in a Poincare surface of section, the resulting differential
7. Hamiltonian cosmology of bigravity
Soloviev, V. O.
2017-03-01
This article is written as a review of the Hamiltonian formalism for the bigravity with de Rham-Gabadadze-Tolley (dRGT) potential, and also of applications of this formalism to the derivation of the background cosmological equations. It is demonstrated that the cosmological scenarios are close to the standard ΛCDM model, but they also uncover the dynamical behavior of the cosmological term. This term arises in bigravity regardless on the choice of the dRGT potential parameters, and its scale is given by the graviton mass. Various matter couplings are considered.
8. Ab-initio calculations of electric field gradient in Ru compounds and their implication on the nuclear quadrupole moments of ^{99}Ru and ^{101}Ru
Mishra, S. N.
2017-08-01
The nuclear quadrupole moments, Q, for the ground and first excited states in ^{99}Ru and ground state of ^{101}Ru have been determined by comparing the experimentally observed quadrupole interaction frequencies ν _Q with calculated electric field gradient (EFG) for a large number of Ru-based compounds. The ab-initio calculations of EFG were performed using the all-electron augmented plane wave + local orbital (APW + lo) method of the density functional theory (DFT). From the slope of the linear correlation between theoretically calculated EFGs and experimentally observed ν _Q, we obtain the quadrupole moment for the (5/2^+) ground state in ^{99}Ru and ^{101}Ru as 0.0734(17) b and 0.431(14) b respectively, showing excellent agreement with the values reported in literature. For 3/2^+, the quadrupole moment of the first excited state in ^{99}Ru is obtained as +0.203(3) b, which is considerably lower than the commonly accepted literature value of +0.231(12) b. The results presented in this paper would be useful for the precise determination of quadrupole moment of high spin states in other Ru isotopes and is likely to stimulate further shell model calculations for an improved understanding of nuclear shape in these nuclei.
9. Nuclear Zero Point Effects as a Function of Density in Ice-like Structures and Liquid Water from vdW-DF Ab Initio Calculations
Pamuk, Betül; Allen, Philip B.; Soler, Jose M.; Fernández-Serra, Marivi
2014-03-01
The contributions of nuclear zero point vibrations to the structures of liquid water and ice are not negligible. Recently, we have explained the source of an anomalous isotope shift in hexagonal ice, representing itself as an increase in the lattice volume when H is replaced by D, by calculating free energy within the quasiharmonic approximation, with ab initio density functional theory. In this work, we extend our studies to analyze the zero point effect in other ice-like structures under different densities: clathrate hydrates, LDL and HDL-like amorphous ices with different densities, and a highly dense ice phase, ice VIII. We show that there is a transition from anomalous isotope effect to normal isotope effect as the density increases. We also analyze nuclear zero point effects in liquid water using different vdW-DFs and make connections to this anomalous-normal isotope effect transition in ice. This work is supported by DOE Early Career Award No. DE-SC0003871.
10. Hamiltonian light-front field theory in a basis function approach
SciTech Connect
Vary, J. P.; Honkanen, H.; Li Jun; Maris, P.; Brodsky, S. J.; Harindranath, A.; Sternberg, P.; Ng, E. G.; Yang, C.
2010-03-15
Hamiltonian light-front quantum field theory constitutes a framework for the nonperturbative solution of invariant masses and correlated parton amplitudes of self-bound systems. By choosing the light-front gauge and adopting a basis function representation, a large, sparse, Hamiltonian matrix for mass eigenstates of gauge theories is obtained that is solvable by adapting the ab initio no-core methods of nuclear many-body theory. Full covariance is recovered in the continuum limit, the infinite matrix limit. There is considerable freedom in the choice of the orthonormal and complete set of basis functions with convenience and convergence rates providing key considerations. Here we use a two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall anti-de Sitter/quantum chromodynamics (AdS/QCD) model obtained from light-front holography. We outline our approach and present illustrative features of some noninteracting systems in a cavity. We illustrate the first steps toward solving quantum electrodynamics (QED) by obtaining the mass eigenstates of an electron in a cavity in small basis spaces and discuss the computational challenges.
11. Ab initio studies of the nuclear magnetic resonance chemical shifts of a rare gas atom in a zeolite
Jameson, Cynthia J.; Lim, Hyung-Mi
1995-09-01
The intermolecular chemical shift of a rare gas atom inside a zeolite cavity is calculated by ab initio analytical derivative theory using gauge-including atomic orbitals (GIAO) at the Ar atom and the atoms of selected neutral clusters each of which is a 4-, 6-, or 8-ring fragment of the zeolite cage. The Si, Al, O atoms and the charge-balancing counterions (Na+, K+, Ca2+) of the clusters (from 24 to 52 atoms) are at coordinates taken from the refined single crystal x-ray structure of the NaA, KA, and CaA zeolites. Terminating OH groups place the H atom at an appropriate O-H distance along the bond to the next Si or Al atom in the crystal. The chemical shift of the Ar atom located at various positions relative to the cluster is calculated using Boys-Bernardi counterpoise correction at each position. The dependence of the rare gas atom chemical shift on the Al/Si ratio of the clusters is investigated. The resulting shielding values are fitted to a pairwise additive form to elicit effective individual Ar-O, Ar-Na, Ar-K, Ar-Ca intermolecular shielding functions of the form σ(39Ar, Ar...Ozeol)= a6r-6+a8r-8+a10r-10+a12r -12, where r is the distance between the Ar and the O atom. A similar form is used for the counterions. The dependence of the Ar shielding on the Al/Si ratio is established (the greater the Al content, the higher the Ar chemical shift), which is in agreement with the few experimental cases where the dependence of the 129Xe chemical shift on the Al/Si ratio of the zeolite has been observed.
12. Quantum Dynamics and Spectroscopy of Ab Initio Liquid Water: The Interplay of Nuclear and Electronic Quantum Effects.
PubMed
Marsalek, Ondrej; Markland, Thomas E
2017-04-06
Understanding the reactivity and spectroscopy of aqueous solutions at the atomistic level is crucial for the elucidation and design of chemical processes. However, the simulation of these systems requires addressing the formidable challenges of treating the quantum nature of both the electrons and nuclei. Exploiting our recently developed methods that provide acceleration by up to 2 orders of magnitude, we combine path integral simulations with on-the-fly evaluation of the electronic structure at the hybrid density functional theory level to capture the interplay between nuclear quantum effects and the electronic surface. Here we show that this combination provides accurate structure and dynamics, including the full infrared and Raman spectra of liquid water. This allows us to demonstrate and explain the failings of lower-level density functionals for dynamics and vibrational spectroscopy when the nuclei are treated quantum mechanically. These insights thus provide a foundation for the reliable investigation of spectroscopy and reactivity in aqueous environments.
13. Drift Hamiltonian in magnetic coordinates
SciTech Connect
White, R.B.; Boozer, A.H.; Hay, R.
1982-02-01
A Hamiltonian formulation of the guiding-center drift in arbitrary, steady state, magnetic and electric fields is given. The canonical variables of this formulation are simply related to the magnetic coordinates. The modifications required to treat ergodic magnetic fields using magnetic coordinates are explicitly given in the Hamiltonian formulation.
14. Robust online Hamiltonian learning
Granade, Christopher E.; Ferrie, Christopher; Wiebe, Nathan; Cory, D. G.
2012-10-01
In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer-Rao lower bound, certifying its own performance.
15. Heterogeneous nuclear ribonucleoprotein A/B and G inhibits the transcription of gonadotropin-releasing-hormone 1
PubMed Central
Zhao, Sheng; Korzan, Wayne J.; Chen, Chun-Chun; Fernald, Russell D.
2008-01-01
Gonadotropin releasing hormone 1 (GnRH1) causes the release of gonadotropins from the pituitary to control reproduction. Here we report that two heterogeneous nuclear ribonucleoproteins (hnRNP-A/B and hnRNP-G) bind to the GnRH-I upstream promoter region in a cichlid fish, Astatotilapia burtoni. We identified these binding proteins using a newly developed homology based method of mass spectrometric peptide mapping. We show that both hnRNP-A/B and hnRNP-G co-localize with GnRH1 in the pre-optic area of the hypothalamus in the brain. We also demonstrated that these ribonucleoproteins exhibit similar binding capacity in vivo, using immortalized mouse GT1-7 cells where overexpression of either hnRNP-A/B or hnRNP-G significantly down-regulate GnRH1 mRNA levels in GT1-7 cells, suggesting that both act as repressors in GnRH1 transcriptional regulation. PMID:17920292
16. Ab initio no core shell model
SciTech Connect
Barrett, Bruce R.; Navrátil, Petr; Vary, James P.
2012-11-17
A long-standing goal of nuclear theory is to determine the properties of atomic nuclei based on the fundamental interactions among the protons and neutrons (i.e., nucleons). By adopting nucleon-nucleon (NN), three-nucleon (NNN) and higher-nucleon interactions determined from either meson-exchange theory or QCD, with couplings fixed by few-body systems, we preserve the predictive power of nuclear theory. This foundation enables tests of nature's fundamental symmetries and offers new vistas for the full range of complex nuclear phenomena. Basic questions that drive our quest for a microscopic predictive theory of nuclear phenomena include: (1) What controls nuclear saturation; (2) How the nuclear shell model emerges from the underlying theory; (3) What are the properties of nuclei with extreme neutron/proton ratios; (4) Can we predict useful cross sections that cannot be measured; (5) Can nuclei provide precision tests of the fundamental laws of nature; and (6) Under what conditions do we need QCD to describe nuclear structure, among others. Along with other ab initio nuclear theory groups, we have pursued these questions with meson-theoretical NN interactions, such as CD-Bonn and Argonne V18, that were tuned to provide high-quality descriptions of the NN scattering phase shifts and deuteron properties. We then add meson-theoretic NNN interactions such as the Tucson-Melbourne or Urbana IX interactions. More recently, we have adopted realistic NN and NNN interactions with ties to QCD. Chiral perturbation theory within effective field theory ({chi}EFT) provides us with a promising bridge between QCD and hadronic systems. In this approach one works consistently with systems of increasing nucleon number and makes use of the explicit and spontaneous breaking of chiral symmetry to expand the strong interaction in terms of a dimensionless constant, the ratio of a generic small momentum divided by the chiral symmetry breaking scale taken to be about 1 GeV/c. The resulting NN
17. Multistage ab initio quantum wavepacket dynamics for electronic structure and dynamics in open systems: momentum representation, coupled electron-nuclear dynamics, and external fields.
PubMed
Pacheco, Alexander B; Iyengar, Srinivasan S
2011-02-21
We recently proposed a multistage ab initio wavepacket dynamics (MS-AIWD) treatment for the study of delocalized electronic systems as well as electron transport through donor-bridge-acceptor systems such as those found in molecular-wire/electrode networks. In this method, the full donor-bridge-acceptor open system is treated through a rigorous partitioning scheme that utilizes judiciously placed offsetting absorbing and emitting boundary conditions. In this manner, the electronic coupling between the bridge molecule and surrounding electrodes is accounted. Here, we extend MS-AIWD to include the dynamics of open-electronic systems in conjunction with (a) simultaneous treatment of nuclear dynamics and (b) external electromagnetic fields. This generalization is benchmarked through an analysis of wavepackets propagated on a potential modeled on an Al(27) - C(7) - Al(27) nanowire. The wavepacket results are inspected in the momentum representation and the dependence of momentum of the wavepacket as well as its transmission probabilities on the magnitude of external bias are analyzed.
18. Multistage ab initio quantum wavepacket dynamics for electronic structure and dynamics in open systems: Momentum representation, coupled electron-nuclear dynamics, and external fields
Pacheco, Alexander B.; Iyengar, Srinivasan S.
2011-02-01
We recently proposed a multistage ab initio wavepacket dynamics (MS-AIWD) treatment for the study of delocalized electronic systems as well as electron transport through donor-bridge-acceptor systems such as those found in molecular-wire/electrode networks. In this method, the full donor-bridge-acceptor open system is treated through a rigorous partitioning scheme that utilizes judiciously placed offsetting absorbing and emitting boundary conditions. In this manner, the electronic coupling between the bridge molecule and surrounding electrodes is accounted. Here, we extend MS-AIWD to include the dynamics of open-electronic systems in conjunction with (a) simultaneous treatment of nuclear dynamics and (b) external electromagnetic fields. This generalization is benchmarked through an analysis of wavepackets propagated on a potential modeled on an Al27 - C7 - Al27 nanowire. The wavepacket results are inspected in the momentum representation and the dependence of momentum of the wavepacket as well as its transmission probabilities on the magnitude of external bias are analyzed.
19. Nuclear, Virescent Mutants of Zea mays L. with High Levels of Chlorophyll (a/b) Light-Harvesting Complex during Thylakoid Assembly 1
PubMed Central
Polacco, Mary L.; Chang, M. T.; Neuffer, M. Gerald
1985-01-01
We have found nuclear, recessive mutants in Zea mays L. where assembly of the major chlorophyll (a/b) light-harvesting complex (LHC) was not delayed relative to most other thylakoid protein complexes during thylakoid biogenesis. This contrasts with the normal development of maize chloroplasts (NR Baker, R Leech 1977 Plant Physiol 60: 640-644). All four mutants examined were allelic and virescent, and displayed visibly higher yields of leaf Chl fluorescence during greening. Fully greened mutants had normal leaf Chl fluorescence yield and normal levels of LHC, and grew to maturity under field conditions. Therefore, delayed LHC assembly is not an obligate feature of thylakoid differentiation. Assigning the molecular basis for the mutation should provide information concerning reguation of LHC assembly. Several possibilities are discussed. The pleiotropic mutant phenotype is not attributable to defects in thylakoid glycerolipid synthesis. Thylakoids isolated from greening mutant leaf sections had elevated glycerolipid/Chl ratios. In addition, both the molar distribution and acyl composition of four major glycerolipids were normal for developing mutant thylakoids. Images Fig. 2 PMID:16664140
20. Nuclear, Virescent Mutants of Zea mays L. with High Levels of Chlorophyll (a/b) Light-Harvesting Complex during Thylakoid Assembly.
PubMed
Polacco, M L; Chang, M T; Neuffer, M G
1985-04-01
We have found nuclear, recessive mutants in Zea mays L. where assembly of the major chlorophyll (a/b) light-harvesting complex (LHC) was not delayed relative to most other thylakoid protein complexes during thylakoid biogenesis. This contrasts with the normal development of maize chloroplasts (NR Baker, R Leech 1977 Plant Physiol 60: 640-644). All four mutants examined were allelic and virescent, and displayed visibly higher yields of leaf Chl fluorescence during greening. Fully greened mutants had normal leaf Chl fluorescence yield and normal levels of LHC, and grew to maturity under field conditions. Therefore, delayed LHC assembly is not an obligate feature of thylakoid differentiation.Assigning the molecular basis for the mutation should provide information concerning reguation of LHC assembly. Several possibilities are discussed. The pleiotropic mutant phenotype is not attributable to defects in thylakoid glycerolipid synthesis. Thylakoids isolated from greening mutant leaf sections had elevated glycerolipid/Chl ratios. In addition, both the molar distribution and acyl composition of four major glycerolipids were normal for developing mutant thylakoids.
1. Effective Hamiltonians for phosphorene and silicene
DOE PAGES
Lew Yan Voon, L. C.; Lopez-Bezanilla, A.; Wang, J.; ...
2015-02-01
We derived the effective Hamiltonians for silicene and phosphorene with strain, electric field and magnetic field using the method of invariants. Our paper extends the work of Geissler et al 2013 (New J. Phys. 15 085030) on silicene, and Li and Appelbaum 2014 (Phys. Rev. B 90, 115439) on phosphorene. Our Hamiltonians are compared to an equivalent one for graphene. For silicene, the expression for band warping is obtained analytically and found to be of different order than for graphene.We prove that a uniaxial strain does not open a gap, resolving contradictory numerical results in the literature. For phosphorene, itmore » is shown that the bands near the Brillouin zone center only have terms in even powers of the wave vector.We predict that the energies change quadratically in the presence of a perpendicular external electric field but linearly in a perpendicular magnetic field, as opposed to those for silicene which vary linearly in both cases. Preliminary ab initio calculations for the intrinsic band structures have been carried out in order to evaluate some of the k · p parameters.« less
2. Effective Hamiltonians for phosphorene and silicene
SciTech Connect
Lew Yan Voon, L. C.; Lopez-Bezanilla, A.; Wang, J.; Zhang, Y.; Willatzen, M.
2015-02-01
We derived the effective Hamiltonians for silicene and phosphorene with strain, electric field and magnetic field using the method of invariants. Our paper extends the work of Geissler et al 2013 (New J. Phys. 15 085030) on silicene, and Li and Appelbaum 2014 (Phys. Rev. B 90, 115439) on phosphorene. Our Hamiltonians are compared to an equivalent one for graphene. For silicene, the expression for band warping is obtained analytically and found to be of different order than for graphene.We prove that a uniaxial strain does not open a gap, resolving contradictory numerical results in the literature. For phosphorene, it is shown that the bands near the Brillouin zone center only have terms in even powers of the wave vector.We predict that the energies change quadratically in the presence of a perpendicular external electric field but linearly in a perpendicular magnetic field, as opposed to those for silicene which vary linearly in both cases. Preliminary ab initio calculations for the intrinsic band structures have been carried out in order to evaluate some of the k · p parameters.
3. Effective Hamiltonians of polymethineimine, polyazine and polyazoethene: A density matrix variation approach
Chen, GuanHua; Su, ZhongMin; Shen, ZhenWen; Yan, YiJing
1998-08-01
A new variation method is proposed to determine the effective Hamiltonians for conjugated π-electron systems. This method is based on the minimization of the difference between the ground state reduced single electron density matrix calculated from the effective Hamiltonian and its ab initio counterpart under a set of well-defined constraints. Applications are made to various oligomers of polymethineimine (PMI), polyazine (PAZ) and polyazoethene (PAE) at the Hartree-Fock level. Calculated are also the optical gaps of these oligomers. The effective Hamiltonians contain electron-electron Coulomb interactions and are suitable for the study of excited state dynamic processes such as nonlinear optical properties in π-conjugated systems.
4. A partial Hamiltonian approach for current value Hamiltonian systems
Naz, R.; Mahomed, F. M.; Chaudhry, Azam
2014-10-01
We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.
5. Robust Online Hamiltonian Learning
Granade, Christopher; Ferrie, Christopher; Wiebe, Nathan; Cory, David
2013-05-01
In this talk, we introduce a machine-learning algorithm for the problem of inferring the dynamical parameters of a quantum system, and discuss this algorithm in the example of estimating the precession frequency of a single qubit in a static field. Our algorithm is designed with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online, during experimental data collection, or can be used as a tool for post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. Finally, we discuss the performance of the our algorithm by appeal to the Cramer-Rao bound. This work was financially supported by the Canadian government through NSERC and CERC and by the United States government through DARPA. NW would like to acknowledge funding from USARO-DTO.
6. Simulating highly nonlocal Hamiltonians with less nonlocal Hamiltonians
Subasi, Yigit; Jarzynski, Christopher
The need for Hamiltonians with many-body interactions arises in various applications of quantum computing. However, interactions beyond two-body are difficult to realize experimentally. Perturbative gadgets were introduced to obtain arbitrary many-body effective interactions using Hamiltonians with two-body interactions only. Although valid for arbitrary k-body interactions, their use is limited to small k because the strength of interaction is k'th order in perturbation theory. Here we develop a nonperturbative technique for obtaining effective k-body interactions using Hamiltonians consisting of at most l-body interactions with l < k . This technique works best for Hamiltonians with a few interactions with very large k and can be used together with perturbative gadgets to embed Hamiltonians of considerable complexity in proper subspaces of two-local Hamiltonians. We describe how our technique can be implemented in a hybrid (gate-based and adiabatic) as well as solely adiabatic quantum computing scheme. We gratefully acknowledge financial support from the Lockheed Martin Corporation under Contract U12001C.
7. Solutions of the Bohr Hamiltonian, a compendium
Fortunato, L.
2005-10-01
The Bohr Hamiltonian, also called collective Hamiltonian, is one of the cornerstones of nuclear physics and a wealth of solutions (analytic or approximated) of the associated eigenvalue equation have been proposed over more than half a century (confining ourselves to the quadrupole degree of freedom). Each particular solution is associated with a peculiar form for the V(β,γ) potential. The large number and the different details of the mathematical derivation of these solutions, as well as their increased and renewed importance for nuclear structure and spectroscopy, demand a thorough discussion. It is the aim of the present monograph to present in detail all the known solutions in γ-unstable and γ-stable cases, in a taxonomic and didactical way. In pursuing this task we especially stressed the mathematical side leaving the discussion of the physics to already published comprehensive material. The paper contains also a new approximate solution for the linear potential, and a new solution for prolate and oblate soft axial rotors, as well as some new formulae and comments. The quasi-dynamical SO(2) symmetry is proposed in connection with the labeling of bands in triaxial nuclei.
8. Collective Hamiltonian for wobbling modes
Chen, Q. B.; Zhang, S. Q.; Zhao, P. W.; Meng, J.
2014-10-01
The simple, longitudinal, and transverse wobblers are systematically studied within the framework of a collective Hamiltonian, where the collective potential and mass parameter included are obtained based on the tilted axis cranking approach. Solving the collective Hamiltonian by diagonalization, the energies and the wave functions of the wobbling states are obtained. The obtained results are compared with those by the harmonic approximation formula and particle rotor model. The wobbling energies calculated by the collective Hamiltonian are closer to the exact solutions by the particle rotor model than the harmonic approximation formula. It is confirmed that the wobbling frequency increases with the rotational frequency in simple and longitudinal wobbling motions while decreases in transverse wobbling motion. These variation trends are related to the stiffness of the collective potential in the collective Hamiltonian.
9. Time-dependent drift Hamiltonian
SciTech Connect
Boozer, A.H.
1983-03-01
The lowest-order drift equations are given in a canonical magnetic coordinate form for time-dependent magnetic and electric fields. The advantages of the canonical Hamiltonian form are also discussed.
10. Hamiltonian description of the ideal fluid
SciTech Connect
Morrison, P.J.
1994-01-01
Fluid mechanics is examined from a Hamiltonian perspective. The Hamiltonian point of view provides a unifying framework; by understanding the Hamiltonian perspective, one knows in advance (within bounds) what answers to expect and what kinds of procedures can be performed. The material is organized into five lectures, on the following topics: rudiments of few-degree-of-freedom Hamiltonian systems illustrated by passive advection in two-dimensional fluids; functional differentiation, two action principles of mechanics, and the action principle and canonical Hamiltonian description of the ideal fluid; noncanonical Hamiltonian dynamics with examples; tutorial on Lie groups and algebras, reduction-realization, and Clebsch variables; and stability and Hamiltonian systems.
11. Hamiltonian formulation of general relativity.
Teitelboim, Claudio
The following sections are included: * INTRODUCTION * HAMILTONIAN FORMULATION OF GAUGE THEORIES (PRE-BRST) * BRST HAMILTONIAN FORMULATION OF GAUGE THEORIES * DYNAMICS OF GRAVITATIONAL FIELD * DOES THE HAMILTONIAN VANISH? GENERAL COVARIANCE AS AN "ORDINARY" GAUGE INVARIANCE * GENERALLY COVARIANT SYSTEMS * TIME AS A CANONICAL VARIABLE. ZERO HAMILTONIAN * Parametrized Systems * Zero Hamiltonian * Parametrization and Explicit Time Dependence * TIME REPARAMETRIZATION INVARIANCE * Form of Gauge Transformations * Must the Hamiltonian be Zero for a Generally Covariant System? * Simple Example of a Generally Covariant System with a Nonzero Hamiltonian * "TRUE DYNAMICS" VERSUS GAUGE TRANSFORMATIONS * Interpretation of the Formalism * Reduced Phase Space * MUST TIME FLOW? * GAUGE INDEPENDENCE OF PATH INTEGRAL FOR A PARAMETRIZED SYSTEM ILLUSTRATED. EQUIVALENCE OF THE GAUGES t = τ AND t = 0 * Reduced Phase Space Transition Amplitude as a Reduced Phase Space Path Integral * Canonical Gauge Conditions * Gauge t = 0 * Gauge t α τ * BRST CHARGE OF GRAVITATIONAL FIELD * ELEMENTS OF BRST THEORY * THE GHOST, YOU'VE COME A LONG WAY BABY * Introduction * Quantum mechanics, the art of finding and combining simple elementary processes * Ghosts necessary to keep elementary processes simple * BRST symmetry: ghosts and matter become different components of single geometrical object * BRST SYMMETRY IN CLASSICAL MECHANICS * Ghosts have role in classical mechanics * Gauge invariance and constraints * Classical mechanics over Grassmann algebra necessary * Higher order structure functions * Rank defined. Open algebras * Ghosts. Ghost number. BRST generator as generating function for structure functions * A belianization of constraints. Existence of Ω * Uniqueness of Ω * Classical BRST cohomology * QUANTUM BRST THEORY * States and operators * Ghost number * BRST invariant states * Quantum BRST cohomology * Equivalence of the BRST physical subspace with the conventional gauge
12. Ab-Initio Shell Model with a Core
SciTech Connect
Lisetskiy, A F; Barrett, B R; Kruse, M; Navratil, P; Stetcu, I; Vary, J P
2008-06-04
We construct effective 2- and 3-body Hamiltonians for the p-shell by performing 12{h_bar}{Omega} ab initio no-core shell model (NCSM) calculations for A=6 and 7 nuclei and explicitly projecting the many-body Hamiltonians onto the 0{h_bar}{Omega} space. We then separate these effective Hamiltonians into 0-, 1- and 2-body contributions (also 3-body for A=7) and analyze the systematic behavior of these different parts as a function of the mass number A and size of the NCSM basis space. The role of effective 3- and higher-body interactions for A > 6 is investigated and discussed.
13. Hamiltonian approach to frame dragging
Epstein, Kenneth J.
2008-07-01
A Hamiltonian approach makes the phenomenon of frame dragging apparent “up front” from the appearance of the drag velocity in the Hamiltonian of a test particle in an arbitrary metric. Hamiltonian (1) uses the inhomogeneous force equation (4), which applies to non-geodesic motion as well as to geodesics. The Hamiltonian is not in manifestly covariant form, but is covariant because it is derived from Hamilton’s manifestly covariant scalar action principle. A distinction is made between manifest frame dragging such as that in the Kerr metric, and hidden frame dragging that can be made manifest by a coordinate transformation such as that applied to the Robertson-Walker metric in Sect. 2. In Sect. 3 a zone of repulsive gravity is found in the extreme Kerr metric. Section 4 treats frame dragging in special relativity as a manifestation of the equivalence principle in accelerated frames. It answers a question posed by Bell about how the Lorentz contraction can break a thread connecting two uniformly accelerated rocket ships. In Sect. 5 the form of the Hamiltonian facilitates the definition of gravitomagnetic and gravitoelectric potentials.
14. Three-cluster dynamics within an ab initio framework
DOE PAGES
Quaglioni, Sofia; Romero-Redondo, Carolina; Navratil, Petr
2013-09-26
In this study, we introduce a fully antisymmetrized treatment of three-cluster dynamics within the ab initio framework of the no-core shell model/resonating-group method. Energy-independent nonlocal interactions among the three nuclear fragments are obtained from realistic nucleon-nucleon interactions and consistent ab initio many-body wave functions of the clusters. The three-cluster Schrödinger equation is solved with bound-state boundary conditions by means of the hyperspherical-harmonic method on a Lagrange mesh. We discuss the formalism in detail and give algebraic expressions for systems of two single nucleons plus a nucleus. Using a soft similarity-renormalization-group evolved chiral nucleon-nucleon potential, we apply the method to amore » 4He+n+n description of 6He and compare the results to experiment and to a six-body diagonalization of the Hamiltonian performed within the harmonic-oscillator expansions of the no-core shell model. Differences between the two calculations provide a measure of core (4He) polarization effects.« less
15. Can we perturbatively expand the \\Qcirc -box in the Bloch-Horowitz Hamiltonian?
Shimizu, Genki; Takayanagi, Kazuo; Otsuka, Takaharu
2014-09-01
In nuclear many-body problems, it is impossible to diagonalize the Hamiltonian directly because of the huge Hilbert space. We introduce, therefore, the concept of the effective interaction. We first partition the whole Hilbert space into the model space of tractable size and its complement, and then look for the effective Hamiltonian defined in the model space that reproduces exact eigenenergies and model space projections of the corresponding eigenstates. Effective Hamiltonians are categorized into energy-independent and energy-dependent groups. The energy-independent effective Hamiltonian has been calculated by iterative methods, and has been used widely for a long time. The energy-dependent effective Hamiltonian is known as the Bloch-Horowitz (BH) Hamiltonian. Though it requires a self-consistent solution, it can, in principle, give all the eigenenergies of the Hamiltonian, if provided with the exact BH Hamiltonian. In actual calculations, however, we can calculate the \\Qcirc -box only up to a finite order of perturbation expansion. In this work, we clarify its convergence condition and examine what we can obtain with the approximate BH Hamiltonian, and what we cannot.
16. Characterization of DNA sequences that mediate nuclear protein binding to the regulatory region of the Pisum sativum (pea) chlorophyl a/b binding protein gene AB80: identification of a repeated heptamer motif.
PubMed
Argüello, G; García-Hernández, E; Sánchez, M; Gariglio, P; Herrera-Estrella, L; Simpson, J
1992-05-01
Two protein factors binding to the regulatory region of the pea chlorophyl a/b binding protein gene AB80 have been identified. One of these factors is found only in green tissue but not in etiolated or root tissue. The second factor (denominated ABF-2) binds to a DNA sequence element that contains a direct heptamer repeat TCTCAAA. It was found that presence of both of the repeats is essential for binding. ABF-2 is present in both green and etiolated tissue and in roots and factors analogous to ABF-2 are present in several plant species. Computer analysis showed that the TCTCAAA motif is present in the regulatory region of several plant genes.
17. First principles of Hamiltonian medicine.
PubMed
Crespi, Bernard; Foster, Kevin; Úbeda, Francisco
2014-05-19
We introduce the field of Hamiltonian medicine, which centres on the roles of genetic relatedness in human health and disease. Hamiltonian medicine represents the application of basic social-evolution theory, for interactions involving kinship, to core issues in medicine such as pathogens, cancer, optimal growth and mental illness. It encompasses three domains, which involve conflict and cooperation between: (i) microbes or cancer cells, within humans, (ii) genes expressed in humans, (iii) human individuals. A set of six core principles, based on these domains and their interfaces, serves to conceptually organize the field, and contextualize illustrative examples. The primary usefulness of Hamiltonian medicine is that, like Darwinian medicine more generally, it provides novel insights into what data will be productive to collect, to address important clinical and public health problems. Our synthesis of this nascent field is intended predominantly for evolutionary and behavioural biologists who aspire to address questions directly relevant to human health and disease.
18. First principles of Hamiltonian medicine
PubMed Central
Crespi, Bernard; Foster, Kevin; Úbeda, Francisco
2014-01-01
We introduce the field of Hamiltonian medicine, which centres on the roles of genetic relatedness in human health and disease. Hamiltonian medicine represents the application of basic social-evolution theory, for interactions involving kinship, to core issues in medicine such as pathogens, cancer, optimal growth and mental illness. It encompasses three domains, which involve conflict and cooperation between: (i) microbes or cancer cells, within humans, (ii) genes expressed in humans, (iii) human individuals. A set of six core principles, based on these domains and their interfaces, serves to conceptually organize the field, and contextualize illustrative examples. The primary usefulness of Hamiltonian medicine is that, like Darwinian medicine more generally, it provides novel insights into what data will be productive to collect, to address important clinical and public health problems. Our synthesis of this nascent field is intended predominantly for evolutionary and behavioural biologists who aspire to address questions directly relevant to human health and disease. PMID:24686937
19. Hamiltonians defined by biorthogonal sets
Bagarello, Fabio; Bellomonte, Giorgia
2017-04-01
In some recent papers, studies on biorthogonal Riesz bases have found renewed motivation because of their connection with pseudo-Hermitian quantum mechanics, which deals with physical systems described by Hamiltonians that are not self-adjoint but may still have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed in some previous papers. However, in many physical models, one has to deal not with orthonormal bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of G -quasi basis, and we show a series of conditions under which a definition of non-self-adjoint Hamiltonian with purely point real spectra is still possible.
20. Variational identities and Hamiltonian structures
SciTech Connect
Ma Wenxiu
2010-03-08
This report is concerned with Hamiltonian structures of classical and super soliton hierarchies. In the classical case, basic tools are variational identities associated with continuous and discrete matrix spectral problems, targeted to soliton equations derived from zero curvature equations over general Lie algebras, both semisimple and non-semisimple. In the super case, a supertrace identity is presented for constructing Hamiltonian structures of super soliton equations associated with Lie superalgebras. We illustrate the general theories by the KdV hierarchy, the Volterra lattice hierarchy, the super AKNS hierarchy, and two hierarchies of dark KdV equations and dark Volterra lattices. The resulting Hamiltonian structures show the commutativity of each hierarchy discussed and thus the existence of infinitely many commuting symmetries and conservation laws.
1. Generalized James' effective Hamiltonian method
Shao, Wenjun; Wu, Chunfeng; Feng, Xun-Li
2017-03-01
James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method only corresponds to the second-order perturbation theory and cannot be exploited to treat problems which should be solved by using the third- or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two recently published examples [L. Garziano et al., Phys. Rev. Lett. 117, 043601 (2016), 10.1103/PhysRevLett.117.043601; Ken K. W. Ma and C. K. Law, Phys. Rev. A 92, 023842 (2015), 10.1103/PhysRevA.92.023842]; our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method, respectively. For some specific problems, this method can simplify the calculating procedure and the resultant effective Hamiltonian is more general.
2. Ab initio calculation of the potential bubble nucleus 34Si
Duguet, T.; Somà, V.; Lecluse, S.; Barbieri, C.; Navrátil, P.
2017-03-01
the many-body correlations included in the calculation, is studied in detail. We eventually compare our predictions to state-of-the-art multireference energy density functional and shell model calculations. Results: The prediction regarding the (non)existence of the bubble structure in 34Si varies significantly with the nuclear Hamiltonian used. However, demanding that the experimental charge density distribution and the root-mean-square radius of 36S be well reproduced, along with 34Si and 36S binding energies, only leaves the NNLOsat Hamiltonian as a serious candidate to perform this prediction. In this context, a bubble structure, whose fingerprint should be visible in an electron scattering experiment of 34Si, is predicted. Furthermore, a clear correlation is established between the occurrence of the bubble structure and the weakening of the 1 /2--3 /2- splitting in the spectrum of 35Si as compared to 37S. Conclusions: The occurrence of a bubble structure in the charge distribution of 34Si is convincingly established on the basis of state-of-the-art ab initio calculations. This prediction will have to be reexamined in the future when improved chiral nuclear Hamiltonians are constructed. On the experimental side, present results act as a strong motivation to measure the charge density distribution of 34Si in future electron scattering experiments on unstable nuclei. In the meantime, it is of interest to perform one-neutron removal on 34Si and 36S in order to further test our theoretical spectral strength distributions over a wide energy range.
3. Collective Hamiltonian for chiral modes
Chen, Q. B.; Zhang, S. Q.; Zhao, P. W.; Jolos, R. V.; Meng, J.
2013-02-01
A collective model is proposed to describe the chiral rotation and vibration and applied to a system with one h11/2 proton particle and one h11/2 neutron hole coupled to a triaxial rigid rotor. The collective Hamiltonian is constructed from the potential energy and mass parameter obtained in the tilted axis cranking approach. By diagonalizing the collective Hamiltonian with a box boundary condition, it is found that for the chiral rotation, the partner states become more degenerate with the increase of the cranking frequency, and for the chiral vibrations, their important roles for the collective excitation are revealed at the beginning of the chiral rotation region.
4. Quasilocal Hamiltonians in general relativity
SciTech Connect
Anderson, Michael T.
2010-10-15
We analyze the definition of quasilocal energy in general relativity based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular, the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consistent definition of quasilocal energy in general relativity requires, at a minimum, a framework based on the (currently unknown) geometric well-posedness of the initial boundary value problem for the Einstein equations.
5. Lowest eigenvalues of random Hamiltonians
SciTech Connect
Shen, J. J.; Zhao, Y. M.; Arima, A.; Yoshinaga, N.
2008-05-15
In this article we study the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues is applicable to many different systems. We improve the accuracy of the formula by considering the third central moment. We show that these formulas are applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.
6. Remembering AB
Belyayev, S. T.
2013-06-01
In 1947 I became a second-year student at Moscow State University's Physics and Engineering Department, where a part of the week's classes were taught at base organizations. Our group's base was the future Kurchatov Institute, at that time known as the mysterious "Laboratory N^circ 2," and later as LIPAN. . Besides group lectures and practical work at the experimental laboratories, we also had access to the general seminars which Igor Vasilyevich Kurchatov tried to hold, with Leonid Vasilyevich Groshev filling in when he was absent. At the seminar, theorists spoke as welcome co-presenters and commentators. In 1949 I felt ready to approach A. B. Migdal to ask if I could transfer to his theoretical sector. In response, he suggested a number of simple qualitative problems, which I then successfully solved. (Incidentally, AB used the very same "introductory problems" for screening many generations of students.) So I wound up among AB's students. From 1952 on (for 10 years) I also served as an employee of the Migdal Sector. My memoirs here are mainly inspired by these years of constant communication with AB. After my departure for Novosibirsk in 1962, although our meetings still took place, they became occasional....
7. Operator evolution for ab initio electric dipole transitions of 4He
DOE PAGES
Schuster, Micah D.; Quaglioni, Sofia; Johnson, Calvin W.; ...
2015-07-24
A goal of nuclear theory is to make quantitative predictions of low-energy nuclear observables starting from accurate microscopic internucleon forces. A major element of such an effort is applying unitary transformations to soften the nuclear Hamiltonian and hence accelerate the convergence of ab initio calculations as a function of the model space size. The consistent simultaneous transformation of external operators, however, has been overlooked in applications of the theory, particularly for nonscalar transitions. We study the evolution of the electric dipole operator in the framework of the similarity renormalization group method and apply the renormalized matrix elements to the calculationmore » of the 4He total photoabsorption cross section and electric dipole polarizability. All observables are calculated within the ab initio no-core shell model. Furthermore, we find that, although seemingly small, the effects of evolved operators on the photoabsorption cross section are comparable in magnitude to the correction produced by including the chiral three-nucleon force and cannot be neglected.« less
8. A Note on Hamiltonian Graphs
ERIC Educational Resources Information Center
Skurnick, Ronald; Davi, Charles; Skurnick, Mia
2005-01-01
Since 1952, several well-known graph theorists have proven numerous results regarding Hamiltonian graphs. In fact, many elementary graph theory textbooks contain the theorems of Ore, Bondy and Chvatal, Chvatal and Erdos, Posa, and Dirac, to name a few. In this note, the authors state and prove some propositions of their own concerning Hamiltonian…
9. Derivation of Hamiltonians for accelerators
SciTech Connect
Symon, K.R.
1997-09-12
In this report various forms of the Hamiltonian for particle motion in an accelerator will be derived. Except where noted, the treatment will apply generally to linear and circular accelerators, storage rings, and beamlines. The generic term accelerator will be used to refer to any of these devices. The author will use the usual accelerator coordinate system, which will be introduced first, along with a list of handy formulas. He then starts from the general Hamiltonian for a particle in an electromagnetic field, using the accelerator coordinate system, with time t as independent variable. He switches to a form more convenient for most purposes using the distance s along the reference orbit as independent variable. In section 2, formulas will be derived for the vector potentials that describe the various lattice components. In sections 3, 4, and 5, special forms of the Hamiltonian will be derived for transverse horizontal and vertical motion, for longitudinal motion, and for synchrobetatron coupling of horizontal and longitudinal motions. Hamiltonians will be expanded to fourth order in the variables.
10. A Note on Hamiltonian Graphs
ERIC Educational Resources Information Center
Skurnick, Ronald; Davi, Charles; Skurnick, Mia
2005-01-01
Since 1952, several well-known graph theorists have proven numerous results regarding Hamiltonian graphs. In fact, many elementary graph theory textbooks contain the theorems of Ore, Bondy and Chvatal, Chvatal and Erdos, Posa, and Dirac, to name a few. In this note, the authors state and prove some propositions of their own concerning Hamiltonian…
11. On third order integrable vector Hamiltonian equations
Meshkov, A. G.; Sokolov, V. V.
2017-03-01
A complete list of third order vector Hamiltonian equations with the Hamiltonian operator Dx having an infinite series of higher conservation laws is presented. A new vector integrable equation on the sphere is found.
12. Collective Hamiltonian for Chiral and Wobbling Modes
Chen, Q. B.; Zhang, S. Q.; Zhao, P. W.; Jolos, R. V.; Meng, J.
The recent progresses of the collective Hamiltonian for chiral and wobbling modes are briefly introduced. The collective Hamiltonian is constructed from the collective potential and mass parameter obtained in the tilted axis cranking approach. The collective Hamiltonian can reproduce the exact solutions by the particle rotor model very well for both chiral and wobbling modes.
13. Systematic method for deriving effective Hamiltonians
Swain, S.
1994-04-01
A systematic procedure for deriving effective Hamiltonians to any order is presented, which is applicable to any time-independent Hamiltonian. The method is based on a continued-fraction approach and avoids the singularities which may occur with perturbation theory. It is illustrated by deriving the effective Hamiltonian for the one-photon, dressed-state laser to second order.
14. Constructing Dense Graphs with Unique Hamiltonian Cycles
ERIC Educational Resources Information Center
Lynch, Mark A. M.
2012-01-01
It is not difficult to construct dense graphs containing Hamiltonian cycles, but it is difficult to generate dense graphs that are guaranteed to contain a unique Hamiltonian cycle. This article presents an algorithm for generating arbitrarily large simple graphs containing "unique" Hamiltonian cycles. These graphs can be turned into dense graphs…
15. Geometric Hamiltonian structures and perturbation theory
SciTech Connect
Omohundro, S.
1984-08-01
We have been engaged in a program of investigating the Hamiltonian structure of the various perturbation theories used in practice. We describe the geometry of a Hamiltonian structure for non-singular perturbation theory applied to Hamiltonian systems on symplectic manifolds and the connection with singular perturbation techniques based on the method of averaging.
16. Lagrangian and Hamiltonian constraints for guiding-center Hamiltonian theories
SciTech Connect
Tronko, Natalia; Brizard, Alain J.
2015-11-15
A consistent guiding-center Hamiltonian theory is derived by Lie-transform perturbation method, with terms up to second order in magnetic-field nonuniformity. Consistency is demonstrated by showing that the guiding-center transformation presented here satisfies separate Jacobian and Lagrangian constraints that have not been explored before. A new first-order term appearing in the guiding-center phase-space Lagrangian is identified through a calculation of the guiding-center polarization. It is shown that this new polarization term also yields a simpler expression of the guiding-center toroidal canonical momentum, which satisfies an exact conservation law in axisymmetric magnetic geometries. Finally, an application of the guiding-center Lagrangian constraint on the guiding-center Hamiltonian yields a natural interpretation for its higher-order corrections.
17. Hamiltonian formulation of teleparallel gravity
Ferraro, Rafael; Guzmán, María José
2016-11-01
The Hamiltonian formulation of the teleparallel equivalent of general relativity is developed from an ordinary second-order Lagrangian, which is written as a quadratic form of the coefficients of anholonomy of the orthonormal frames (vielbeins). We analyze the structure of eigenvalues of the multi-index matrix entering the (linear) relation between canonical velocities and momenta to obtain the set of primary constraints. The canonical Hamiltonian is then built with the Moore-Penrose pseudoinverse of that matrix. The set of constraints, including the subsequent secondary constraints, completes a first-class algebra. This means that all of them generate gauge transformations. The gauge freedoms are basically the diffeomorphisms and the (local) Lorentz transformations of the vielbein. In particular, the Arnowitt, Deser, and Misner algebra of general relativity is recovered as a subalgebra.
18. A Hamiltonian approach to Thermodynamics
SciTech Connect
Baldiotti, M.C.; Fresneda, R.; Molina, C.
2016-10-15
In the present work we develop a strictly Hamiltonian approach to Thermodynamics. A thermodynamic description based on symplectic geometry is introduced, where all thermodynamic processes can be described within the framework of Analytic Mechanics. Our proposal is constructed on top of a usual symplectic manifold, where phase space is even dimensional and one has well-defined Poisson brackets. The main idea is the introduction of an extended phase space where thermodynamic equations of state are realized as constraints. We are then able to apply the canonical transformation toolkit to thermodynamic problems. Throughout this development, Dirac’s theory of constrained systems is extensively used. To illustrate the formalism, we consider paradigmatic examples, namely, the ideal, van der Waals and Clausius gases. - Highlights: • A strictly Hamiltonian approach to Thermodynamics is proposed. • Dirac’s theory of constrained systems is extensively used. • Thermodynamic equations of state are realized as constraints. • Thermodynamic potentials are related by canonical transformations.
19. Computational power of symmetric Hamiltonians
Kay, Alastair
2008-07-01
The presence of symmetries, be they discrete or continuous, in a physical system typically leads to a reduction in the problem to be solved. Here we report that neither translational invariance nor rotational invariance reduce the computational complexity of simulating Hamiltonian dynamics; the problem is still bounded error, quantum polynomial time complete, and is believed to be hard on a classical computer. This is achieved by designing a system to implement a universal quantum interface, a device which enables control of an arbitrary computation through the control of a fixed number of spins, and using it as a building block to entirely remove the need for control, except in the system initialization. Finally, it is shown that cooling such Hamiltonians to their ground states in the presence of random magnetic fields solves a Quantum-Merlin-Arthur-complete problem.
20. Core polarization and modern realistic shell-model Hamiltonians
Coraggio, L.; Covello, A.; Gargano, A.; Itaco, N.
The understanding of the convergence properties of the shell-model effective Hamiltonian, within the framework of the many-body perturbation theory, is a long-standing problem. The infinite summation of a certain class of diagrams, the so-called "bubble diagrams," may be provided calculating the Kirson-Babu-Brown induced interaction, and provides a valid instrument to study whether or not the finite summation of the perturbative series is well-grounded. Here, we perform an application of the calculation of the Kirson-Babu-Brown induced interaction to derive the shell-model effective Hamiltonian for p-shell nuclei starting from a modern nucleon-nucleon potential, obtained by way of the chiral perturbation theory. The outcome of our calculation is compared with a standard calculation of the shell-model Hamiltonian, where the core-polarization effects are calculated only up to third-order in perturbation theory. The results of the two calculations are very close to each other, evidencing that the perturbative approach to the derivation of the shell-model Hamiltonian is still a valid tool for nuclear structure studies.
1. Core polarization and modern realistic shell-model Hamiltonians
Coraggio, L.; Covello, A.; Gargano, A.; Itaco, N.
The understanding of the convergence properties of the shell-model effective Hamiltonian, within the framework of the many-body perturbation theory, is a long-standing problem. The infinite summation of a certain class of diagrams, the so-called “bubble diagrams,” may be provided calculating the Kirson-Babu-Brown induced interaction, and provides a valid instrument to study whether or not the finite summation of the perturbative series is well-grounded. Here, we perform an application of the calculation of the Kirson-Babu-Brown induced interaction to derive the shell-model effective Hamiltonian for p-shell nuclei starting from a modern nucleon-nucleon potential, obtained by way of the chiral perturbation theory. The outcome of our calculation is compared with a standard calculation of the shell-model Hamiltonian, where the core-polarization effects are calculated only up to third-order in perturbation theory. The results of the two calculations are very close to each other, evidencing that the perturbative approach to the derivation of the shell-model Hamiltonian is still a valid tool for nuclear structure studies.
2. Higher-dimensional Wannier functions of multiparameter Hamiltonians
Hanke, Jan-Philipp; Freimuth, Frank; Blügel, Stefan; Mokrousov, Yuriy
2015-05-01
When using Wannier functions to study the electronic structure of multiparameter Hamiltonians H(k ,λ ) carrying a dependence on crystal momentum k and an additional periodic parameter λ , one usually constructs several sets of Wannier functions for a set of values of λ . We present the concept of higher-dimensional Wannier functions (HDWFs), which provide a minimal and accurate description of the electronic structure of multiparameter Hamiltonians based on a single set of HDWFs. The obstacle of nonorthogonality of Bloch functions at different λ is overcome by introducing an auxiliary real space, which is reciprocal to the parameter λ . We derive a generalized interpolation scheme and emphasize the essential conceptual and computational simplifications in using the formalism, for instance, in the evaluation of linear response coefficients. We further implement the necessary machinery to construct HDWFs from ab initio within the full potential linearized augmented plane-wave method (FLAPW). We apply our implementation to accurately interpolate the Hamiltonian of a one-dimensional magnetic chain of Mn atoms in two important cases of λ : (i) the spin-spiral vector q and (ii) the direction of the ferromagnetic magnetization m ̂. Using the generalized interpolation of the energy, we extract the corresponding values of magnetocrystalline anisotropy energy, Heisenberg exchange constants, and spin stiffness, which compare very well with the values obtained from direct first principles calculations. For toy models we demonstrate that the method of HDWFs can also be used in applications such as the virtual crystal approximation, ferroelectric polarization, and spin torques.
3. Contact symmetries and Hamiltonian thermodynamics
SciTech Connect
Bravetti, A.; Lopez-Monsalvo, C.S.; Nettel, F.
2015-10-15
It has been shown that contact geometry is the proper framework underlying classical thermodynamics and that thermodynamic fluctuations are captured by an additional metric structure related to Fisher’s Information Matrix. In this work we analyse several unaddressed aspects about the application of contact and metric geometry to thermodynamics. We consider here the Thermodynamic Phase Space and start by investigating the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics. Then we present a novel mathematical characterization of first order phase transitions as equilibrium processes on the Thermodynamic Phase Space for which the Legendre symmetry is broken. Moreover, we use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics and we show that the relevant Hamiltonian coincides with the irreversible entropy production along thermodynamic processes. Therefore, we use such property to give a geometric definition of thermodynamically admissible fluctuations according to the Second Law of thermodynamics. Finally, we show that the length of a curve describing a thermodynamic process measures its entropy production.
4. Ab Initio and Ab Exitu No-Core Shell Model
SciTech Connect
Vary, J P; Navratil, P; Gueorguiev, V G; Ormand, W E; Nogga, A; Maris, P; Shirokov, A
2007-10-02
We outline two complementary approaches based on the no core shell model (NCSM) and present recent results. In the ab initio approach, nuclear properties are evaluated with two-nucleon (NN) and three-nucleon interactions (TNI) derived within effective field theory (EFT) based on chiral perturbation theory (ChPT). Fitting two available parameters of the TNI generates good descriptions of light nuclei. In a second effort, an ab exitu approach, results are obtained with a realistic NN interaction derived by inverse scattering theory with off-shell properties tuned to fit light nuclei. Both approaches produce good results for observables sensitive to spin-orbit properties.
5. Ab initio calculations of ^12C and neutron drops
Pieper, Steven C.
2009-10-01
Ab initio calculations of nuclei, which treat a nucleus as a system of A nucleons interacting by realistic two- and three-nucleon forces, have made tremendous progress in the last 15 years. This is a result of better Hamiltonians, rapidly increasing computer power, and new or improved many-body methods. Three methods are principally being used: Green's function Monte Carlo (GFMC), no-core shell model, and coupled cluster. In the limit of large computer resources, all three methods produce exact eigenvalues of a given nuclear Hamiltonian. With DOE SciDAC and INCITE support, all three methods are using the largest computers available today. Under the UNEDF SciDAC grant, the Argonne GFMC program was modified to efficiently use more than 2000 processors. E. Lusk (Argonne), R.M. Butler (Middle Tennessee State U.) and I have developed an Asynchronous Dynamic Load-Balancing (ADLB) library. In addition all the cores in a node are used via OpenMP as one ADLB/MPI client. In this way we obtain very good scalability up to 30,000 processors on Argonne's IBM Blue Gene/P. Two systems of particular interest that require this computer power are ^12C and neutron drops. V.R. Pandharipande (UIUC, deceased), J. Carlson (LANL), R.B. Wiringa (Argonne), and I have developed new trial wave functions that explicitly contain the three-alpha particle structure of ^12C. These are being used with the Argonne V18 and Illinois-7 potentials which reproduce the energies of 51 states in 3<=A<=12 nuclei with an rms error of 600,eV. Neutron drops are collections of neutrons confined in an artificial external well and interacting with realistic NN and NNN potentials. Their properties can be used as experimental data'' for developing energy-density functionals.
6. Emergent properties of nuclei from ab initio coupled-cluster calculations
SciTech Connect
Hagen, G.; Hjorth-Jensen, M.; Jansen, G. R.; Papenbrock, T.
2016-05-17
Emergent properties such as nuclear saturation and deformation, and the effects on shell structure due to the proximity of the scattering continuum and particle decay channels are fascinating phenomena in atomic nuclei. In recent years, ab initio approaches to nuclei have taken the first steps towards tackling the computational challenge of describing these phenomena from Hamiltonians with microscopic degrees of freedom. Our endeavor is now possible due to ideas from effective field theories, novel optimization strategies for nuclear interactions, ab initio methods exhibiting a soft scaling with mass number, and ever-increasing computational power. We review some of the recent accomplishments. We also present new results. The recently optimized chiral interaction NNLO${}_{{\\rm{sat}}}$ is shown to provide an accurate description of both charge radii and binding energies in selected light- and medium-mass nuclei up to 56Ni. We derive an efficient scheme for including continuum effects in coupled-cluster computations of nuclei based on chiral nucleon–nucleon and three-nucleon forces, and present new results for unbound states in the neutron-rich isotopes of oxygen and calcium. Finally, the coupling to the continuum impacts the energies of the ${J}^{\\pi }=1/{2}^{-},3/{2}^{-},7/{2}^{-},3/{2}^{+}$ states in ${}^{\\mathrm{17,23,25}}$O, and—contrary to naive shell-model expectations—the level ordering of the ${J}^{\\pi }=3/{2}^{+},5/{2}^{+},9/{2}^{+}$ states in ${}^{\\mathrm{53,55,61}}$Ca.
7. Emergent properties of nuclei from ab initio coupled-cluster calculations
SciTech Connect
Hagen, G.; Hjorth-Jensen, M.; Jansen, G. R.; Papenbrock, T.
2016-05-17
Emergent properties such as nuclear saturation and deformation, and the effects on shell structure due to the proximity of the scattering continuum and particle decay channels are fascinating phenomena in atomic nuclei. In recent years, ab initio approaches to nuclei have taken the first steps towards tackling the computational challenge of describing these phenomena from Hamiltonians with microscopic degrees of freedom. Our endeavor is now possible due to ideas from effective field theories, novel optimization strategies for nuclear interactions, ab initio methods exhibiting a soft scaling with mass number, and ever-increasing computational power. We review some of the recent accomplishments. We also present new results. The recently optimized chiral interaction NNLO${}_{{\\rm{sat}}}$ is shown to provide an accurate description of both charge radii and binding energies in selected light- and medium-mass nuclei up to 56Ni. We derive an efficient scheme for including continuum effects in coupled-cluster computations of nuclei based on chiral nucleon–nucleon and three-nucleon forces, and present new results for unbound states in the neutron-rich isotopes of oxygen and calcium. Finally, the coupling to the continuum impacts the energies of the ${J}^{\\pi }=1/{2}^{-},3/{2}^{-},7/{2}^{-},3/{2}^{+}$ states in ${}^{\\mathrm{17,23,25}}$O, and—contrary to naive shell-model expectations—the level ordering of the ${J}^{\\pi }=3/{2}^{+},5/{2}^{+},9/{2}^{+}$ states in ${}^{\\mathrm{53,55,61}}$Ca.
8. Emergent properties of nuclei from ab initio coupled-cluster calculations
Hagen, G.; Hjorth-Jensen, M.; Jansen, G. R.; Papenbrock, T.
2016-06-01
Emergent properties such as nuclear saturation and deformation, and the effects on shell structure due to the proximity of the scattering continuum and particle decay channels are fascinating phenomena in atomic nuclei. In recent years, ab initio approaches to nuclei have taken the first steps towards tackling the computational challenge of describing these phenomena from Hamiltonians with microscopic degrees of freedom. This endeavor is now possible due to ideas from effective field theories, novel optimization strategies for nuclear interactions, ab initio methods exhibiting a soft scaling with mass number, and ever-increasing computational power. This paper reviews some of the recent accomplishments. We also present new results. The recently optimized chiral interaction NNLO{}{{sat}} is shown to provide an accurate description of both charge radii and binding energies in selected light- and medium-mass nuclei up to 56Ni. We derive an efficient scheme for including continuum effects in coupled-cluster computations of nuclei based on chiral nucleon-nucleon and three-nucleon forces, and present new results for unbound states in the neutron-rich isotopes of oxygen and calcium. The coupling to the continuum impacts the energies of the {J}π =1/{2}-,3/{2}-,7/{2}-,3/{2}+ states in {}{17,23,25}O, and—contrary to naive shell-model expectations—the level ordering of the {J}π =3/{2}+,5/{2}+,9/{2}+ states in {}{53,55,61}Ca. ).
9. Killing symmetries as Hamiltonian constraints
Lusanna, Luca
2016-02-01
The existence of a Killing symmetry in a gauge theory is equivalent to the addition of extra Hamiltonian constraints in its phase space formulation, which imply restrictions both on the Dirac observables (the gauge invariant physical degrees of freedom) and on the gauge freedom. When there is a time-like Killing vector field only pure gauge electromagnetic fields survive in Maxwell theory in Minkowski space-time, while in ADM canonical gravity in asymptotically Minkowskian space-times only inertial effects without gravitational waves survive.
10. Staggered quantum walks with Hamiltonians
Portugal, R.; de Oliveira, M. C.; Moqadam, J. K.
2017-01-01
Quantum walks are recognizably useful for the development of new quantum algorithms, as well as for the investigation of several physical phenomena in quantum systems. Actual implementations of quantum walks face technological difficulties similar to the ones for quantum computers, though. Therefore, there is a strong motivation to develop new quantum-walk models which might be easier to implement. In this work we present an extension of the staggered quantum walk model that is fitted for physical implementations in terms of time-independent Hamiltonians. We demonstrate that this class of quantum walk includes the entire class of staggered quantum walk model, Szegedy's model, and an important subset of the coined model.
11. Canonical form of Hamiltonian matrices
Zuker, A. P.; Waha Ndeuna, L.; Nowacki, F.; Caurier, E.
2001-08-01
On the basis of shell model simulations, it is conjectured that the Lanczos construction at fixed quantum numbers defines-within fluctuations and behavior very near the origin-smooth canonical matrices whose forms depend on the rank of the Hamiltonian, dimensionality of the vector space, and second and third moments. A framework emerges that amounts to a general Anderson model capable of dealing with ground state properties and strength functions. The smooth forms imply binomial level densities. A simplified approach to canonical thermodynamics is proposed.
12. Chasing Hamiltonian structure in gyrokinetic theory
SciTech Connect
Burby, J. W.
2015-09-01
Hamiltonian structure is pursued and uncovered in collisional and collisionless gyrokinetic theory. A new Hamiltonian formulation of collisionless electromagnetic theory is presented that is ideally suited to implementation on modern supercomputers. The method used to uncover this structure is described in detail and applied to a number of examples, where several well-known plasma models are endowed with a Hamiltonian structure for the first time. The first energy- and momentum-conserving formulation of full-F collisional gyrokinetics is presented. In an effort to understand the theoretical underpinnings of this result at a deeper level, a stochastic Hamiltonian modeling approach is presented and applied to pitch angle scattering. Interestingly, the collision operator produced by the Hamiltonian approach is equal to the Lorentz operator plus higher-order terms, but does not exactly conserve energy. Conversely, the classical Lorentz collision operator is provably not Hamiltonian in the stochastic sense.
13. Hamiltonian thermostats fail to promote heat flow
Hoover, Wm. G.; Hoover, Carol G.
2013-12-01
Hamiltonian mechanics can be used to constrain temperature simultaneously with energy. We illustrate the interesting situations that develop when two different temperatures are imposed within a composite Hamiltonian system. The model systems we treat are ϕ4 chains, with quartic tethers and quadratic nearest-neighbor Hooke's-law interactions. This model is known to satisfy Fourier's law. Our prototypical problem sandwiches a Newtonian subsystem between hot and cold Hamiltonian reservoir regions. We have characterized four different Hamiltonian reservoir types. There is no tendency for any of these two-temperature Hamiltonian simulations to transfer heat from the hot to the cold degrees of freedom. Evidently steady heat flow simulations require energy sources and sinks, and are therefore incompatible with Hamiltonian mechanics.
14. Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions
Bonnaillie-Noël, V.; Helffer, B.; Hoffmann-Ostenhof, T.
2009-05-01
The spectral analysis of Aharonov-Bohm Hamiltonians with flux \\frac12 leads surprisingly to a new insight on some questions of isospectrality appearing for example in Jakobson et al (2006 J. Comput. Appl. Math. 194 141-55) and Levitin et al (J. Phys. A: Math. Gen. 39 2073-82) and of minimal partitions (Helffer et al 2009 Ann. Inst. H. Poincaré Anal. Non Linéaire 26 101-38). We will illustrate this point of view by discussing the question of spectral minimal 3-partitions for the rectangle \\big]{-}\\frac a2,\\frac a2\\big[\\times \\big]{-}\\frac b2,\\frac b2\\big[ , with 0 < a <= b. It has been observed in Helffer et al (2009 Ann. Inst. H. Poincaré Anal. Non Linéaire 26 101-38) that when 0<\\frac ab < \\sqrt{\\vphantom{A^A}\\smash{\\\\frac 38}} the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles \\big]{-}\\frac a2,\\frac a2\\big[\\times \\big] {-}\\frac b2,-\\frac b6\\big[, \\big]{-}\\frac a2,\\frac a2\\big[\\times \\big]{-}\\frac b6,\\frac b6\\big[ and \\big]{-}\\frac a2,\\frac a2\\big[\\times \\big] \\frac b6, \\frac b2\\big[ . We will describe a possible mechanism of transition for increasing \\frac ab between these nodal minimal 3-partitions and non-nodal minimal 3-partitions at the value \\sqrt{\\vphantom{A^A}\\smash{\\\\frac 38}} and discuss the existence of symmetric candidates for giving minimal 3-partitions when \\sqrt{\\vphantom{A^A}\\smash{\\\\frac 38}} <\\frac ab \\leq 1 . Numerical analysis leads very naturally to nice questions of isospectrality which are solved by the introduction of Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle.
15. Hamiltonian approach to slip-stacking dynamics
DOE PAGES
Lee, S. Y.; Ng, K. Y.
2017-06-29
Hamiltonian dynamics has been applied to study the slip-stacking dynamics. The canonical-perturbation method is employed to obtain the second-harmonic correction term in the slip-stacking Hamiltonian. The Hamiltonian approach provides a clear optimal method for choosing the slip-stacking parameter and improving stacking efficiency. The dynamics are applied specifically to the Fermilab Booster-Recycler complex. As a result, the dynamics can also be applied to other accelerator complexes.
16. Local bulk physics from intersecting modular Hamiltonians
2017-06-01
We show that bulk quantities localized on a minimal surface homologous to a boundary region correspond in the CFT to operators that commute with the modular Hamiltonian associated with the boundary region. If two such minimal surfaces intersect at a point in the bulk then CFT operators which commute with both extended modular Hamiltonians must be localized at the intersection point. We use this to construct local bulk operators purely from CFT considerations, without knowing the bulk metric, using intersecting modular Hamiltonians. For conformal field theories at zero and finite temperature the appropriate modular Hamiltonians are known explicitly and we recover known expressions for local bulk observables.
17. Hamiltonian decomposition for bulk and surface states.
PubMed
Sasaki, Ken-Ichi; Shimomura, Yuji; Takane, Yositake; Wakabayashi, Katsunori
2009-04-10
We demonstrate that a tight-binding Hamiltonian with nearest- and next-nearest-neighbor hopping integrals can be decomposed into bulk and boundary parts for honeycomb lattice systems. The Hamiltonian decomposition reveals that next-nearest-neighbor hopping causes sizable changes in the energy spectrum of surface states even if the correction to the energy spectrum of bulk states is negligible. By applying the Hamiltonian decomposition to edge states in graphene systems, we show that the next-nearest-neighbor hopping stabilizes the edge states. The application of Hamiltonian decomposition to a general lattice system is discussed.
18. Moment methods and nuclear level densities
SciTech Connect
Johnson, Calvin W.
2008-04-17
Working in a shell-model framework, I use moments of the nuclear many-body Hamiltonian to illustrate the importance of the residual interaction to microscopic calculations of the nuclear level density.
19. Ab initio excited states from the in-medium similarity renormalization group
Parzuchowski, N. M.; Morris, T. D.; Bogner, S. K.
2017-04-01
We present two new methods for performing ab initio calculations of excited states for closed-shell systems within the in-medium similarity renormalization group (IMSRG) framework. Both are based on combining the IMSRG with simple many-body methods commonly used to target excited states, such as the Tamm-Dancoff approximation (TDA) and equations-of-motion (EOM) techniques. In the first approach, a two-step sequential IMSRG transformation is used to drive the Hamiltonian to a form where a simple TDA calculation (i.e., diagonalization in the space of 1 p 1 h excitations) becomes exact for a subset of eigenvalues. In the second approach, EOM techniques are applied to the IMSRG ground-state-decoupled Hamiltonian to access excited states. We perform proof-of-principle calculations for parabolic quantum dots in two dimensions and the closed-shell nuclei 16O and 22O. We find that the TDA-IMSRG approach gives better accuracy than the EOM-IMSRG when calculations converge, but it is otherwise lacking the versatility and numerical stability of the latter. Our calculated spectra are in reasonable agreement with analogous EOM-coupled-cluster calculations. This work paves the way for more interesting applications of the EOM-IMSRG approach to calculations of consistently evolved observables such as electromagnetic strength functions and nuclear matrix elements, and extensions to nuclei within one or two nucleons of a closed shell by generalizing the EOM ladder operator to include particle-number nonconserving terms.
20. Operator evolution for ab initio electric dipole transitions of 4He
SciTech Connect
Schuster, Micah D.; Quaglioni, Sofia; Johnson, Calvin W.; Jurgenson, Eric D.; Navartil, Petr
2015-07-24
A goal of nuclear theory is to make quantitative predictions of low-energy nuclear observables starting from accurate microscopic internucleon forces. A major element of such an effort is applying unitary transformations to soften the nuclear Hamiltonian and hence accelerate the convergence of ab initio calculations as a function of the model space size. The consistent simultaneous transformation of external operators, however, has been overlooked in applications of the theory, particularly for nonscalar transitions. We study the evolution of the electric dipole operator in the framework of the similarity renormalization group method and apply the renormalized matrix elements to the calculation of the 4He total photoabsorption cross section and electric dipole polarizability. All observables are calculated within the ab initio no-core shell model. Furthermore, we find that, although seemingly small, the effects of evolved operators on the photoabsorption cross section are comparable in magnitude to the correction produced by including the chiral three-nucleon force and cannot be neglected.
1. Hamiltonian tomography of photonic lattices
Ma, Ruichao; Owens, Clai; LaChapelle, Aman; Schuster, David I.; Simon, Jonathan
2017-06-01
In this paper we introduce an approach to Hamiltonian tomography of noninteracting tight-binding photonic lattices. To begin with, we prove that the matrix element of the low-energy effective Hamiltonian between sites α and β may be obtained directly from Sα β(ω ) , the (suitably normalized) two-port measurement between sites α and β at frequency ω . This general result enables complete characterization of both on-site energies and tunneling matrix elements in arbitrary lattice networks by spectroscopy, and suggests that coupling between lattice sites is a topological property of the two-port spectrum. We further provide extensions of this technique for measurement of band projectors in finite, disordered systems with good band flatness ratios, and apply the tool to direct real-space measurement of the Chern number. Our approach demonstrates the extraordinary potential of microwave quantum circuits for exploration of exotic synthetic materials, providing a clear path to characterization and control of single-particle properties of Jaynes-Cummings-Hubbard lattices. More broadly, we provide a robust, unified method of spectroscopic characterization of linear networks from photonic crystals to microwave lattices and everything in between.
2. Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian
Ripoche, J.; Lacroix, D.; Gambacurta, D.; Ebran, J.-P.; Duguet, T.
2017-01-01
internucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage of automatic access to the low-lying spectroscopy, which it does with high accuracy. Conclusions: The numerical cost of the newly designed variational method is polynomial (N6) in system size. This method achieves unprecedented accuracy for the ground-state correlation energy, effective pairing gap, and one-body entropy as well as for the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a sufficiently strong motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate, and versatile ab initio description of mid-mass open-shell nuclei in the future.
3. Theoretical gas to liquid shift of (15)N isotropic nuclear magnetic shielding in nitromethane using ab initio molecular dynamics and GIAO/GIPAW calculations.
PubMed
Gerber, Iann C; Jolibois, Franck
2015-05-14
Chemical shift requires the knowledge of both the sample and a reference magnetic shielding. In few cases as nitrogen (15N), the standard experimental reference corresponds to its liquid phase. Theoretical estimate of NMR magnetic shielding parameters of compounds in their liquid phase is then mandatory but usually replaced by an easily-get gas phase value, forbidding direct comparisons with experiments. We propose here to combine ab initio molecular dynamic simulations with the calculations of magnetic shielding using GIAO approach on extracted cluster's structures from MD. Using several computational strategies, we manage to accurately calculate 15N magnetic shielding of nitromethane in its liquid phase. Theoretical comparison between liquid and gas phase allows us to extrapolate an experimental value for the 15N magnetic shielding of nitromethane in gas phase between -121.8 and -120.8 ppm.
4. Implicit variational principle for contact Hamiltonian systems
Wang, Kaizhi; Wang, Lin; Yan, Jun
2017-02-01
We establish an implicit variational principle for the contact Hamiltonian systems generated by the Hamiltonian H(x, u, p) with respect to the contact 1-form α =\\text{d}u-p\\text{d}x under Tonelli and Lipschitz continuity conditions.
5. Bohr Hamiltonian with time-dependent potential
2016-04-01
In this paper, Bohr Hamiltonian has been studied with the time-dependent potential. Using the Lewis-Riesenfeld dynamical invariant method appropriate dynamical invariant for this Hamiltonian has been constructed and the exact time-dependent wave functions of such a system have been derived due to this dynamical invariant.
6. Generalized seniority from random Hamiltonians
SciTech Connect
Johnson, C. W.; Bertsch, G. F.; Dean, D. J.; Talmi, I.
2000-01-01
We investigate the generic pairing properties of shell-model many-body Hamiltonians drawn from ensembles of random two-body matrix elements. Many features of pairing that are commonly attributed to the interaction are in fact seen in a large part of the ensemble space. Not only do the spectra show evidence of pairing with favored J=0 ground states and an energy gap, but the relationship between ground-state wave functions of neighboring nuclei shows signatures of pairing as well. Matrix elements of pair creation-annihilation operators between ground states tend to be strongly enhanced. Furthermore, the same or similar pair operators connect several ground states along an isotopic chain. This algebraic structure is reminiscent of the generalized seniority model. Thus pairing may be encoded to a certain extent in the Fock space connectivity of the interacting shell model even without specific features of the interaction required. (c) 1999 The American Physical Society.
7. Dynamical manifestations of Hamiltonian monodromy
SciTech Connect
Delos, J.B. Dhont, G. Sadovskii, D.A. Zhilinskii, B.I.
2009-09-15
Monodromy is the simplest obstruction to the existence of global action-angle variables in integrable Hamiltonian dynamical systems. We consider one of the simplest possible systems with monodromy: a particle in a circular box containing a cylindrically symmetric potential-energy barrier. Systems with monodromy have nontrivial smooth connections between their regular Liouville tori. We consider a dynamical connection produced by an appropriate time-dependent perturbation of our system. This turns studying monodromy into studying a physical process. We explain what aspects of this process are to be looked upon in order to uncover the interesting and somewhat unexpected dynamical behavior resulting from the nontrivial properties of the connection. We compute and analyze this behavior.
8. Universal two-body-Hamiltonian quantum computing
Nagaj, Daniel
2012-03-01
We present a Hamiltonian quantum-computation scheme universal for quantum computation. Our Hamiltonian is a sum of a polynomial number (in the number of gates L in the quantum circuit) of constant-norm, time-independent, two-body interaction terms. Furthermore, each qubit in the system interacts only with a constant number of other qubits in a three-layer, geometrically local layout. The computer runs in three steps—it starts in a simple initial product state, evolves according to a time-independent Hamiltonian for time of order L2 (up to logarithmic factors), and finishes with a two-qubit measurement. Our model improves previous universal two-local-Hamiltonian constructions, as it avoids using perturbation gadgets and large energy-penalty terms in the Hamiltonian, which would result in a large required run time.
9. Building A Universal Nuclear Energy Density Functional (UNEDF)
SciTech Connect
Joe Carlson; Dick Furnstahl; Mihai Horoi; Rusty Lusk; Witek Nazarewicz; Esmond Ng; Ian Thompson; James Vary
2012-09-30
During the period of Dec. 1 2006 - Jun. 30, 2012, the UNEDF collaboration carried out a comprehensive study of all nuclei, based on the most accurate knowledge of the strong nuclear interaction, the most reliable theoretical approaches, the most advanced algorithms, and extensive computational resources, with a view towards scaling to the petaflop platforms and beyond. The long-term vision initiated with UNEDF is to arrive at a comprehensive, quantitative, and unified description of nuclei and their reactions, grounded in the fundamental interactions between the constituent nucleons. We seek to replace current phenomenological models of nuclear structure and reactions with a well-founded microscopic theory that delivers maximum predictive power with well-quantified uncertainties. Specifically, the mission of this project has been three-fold: first, to find an optimal energy density functional (EDF) using all our knowledge of the nucleonic Hamiltonian and basic nuclear properties; second, to apply the EDF theory and its extensions to validate the functional using all the available relevant nuclear structure and reaction data; third, to apply the validated theory to properties of interest that cannot be measured, in particular the properties needed for reaction theory. The main physics areas of UNEDF, defined at the beginning of the project, were: ab initio structure; ab initio functionals; DFT applications; DFT extensions; reactions.
10. Applications of Floquet-Magnus expansion, average Hamiltonian theory and Fer expansion to study interactions in solid state NMR when irradiated with the magic-echo sequence.
PubMed
Mananga, Eugene Stephane
2013-01-01
This work presents the possibility of applying the Floquet-Magnus expansion and the Fer expansion approaches to the most useful interactions known in solid-state nuclear magnetic resonance using the magic-echo scheme. The results of the effective Hamiltonians of these theories and average Hamiltonian theory are presented. © 2013 Elsevier Inc. All rights reserved.
11. Applications of Floquet-Magnus expansion, average Hamiltonian theory and Fer expansion to study interactions in solid state NMR when irradiated with the magic-echo sequence
PubMed Central
Mananga, Eugene Stephane
2015-01-01
This work presents the possibility of applying the Floquet-Magnus expansion and the Fer expansion approaches to the most useful interactions known in solid-state nuclear magnetic resonance using the magic-echo scheme. The results of the effective Hamiltonians of these theories and average Hamiltonian theory are presented. PMID:24034855
12. Analytic derivative couplings and first-principles exciton/phonon coupling constants for an ab initio Frenkel-Davydov exciton model: Theory, implementation, and application to compute triplet exciton mobility parameters for crystalline tetracene
Morrison, Adrian F.; Herbert, John M.
2017-06-01
Recently, we introduced an ab initio version of the Frenkel-Davydov exciton model for computing excited-state properties of molecular crystals and aggregates. Within this model, supersystem excited states are approximated as linear combinations of excitations localized on molecular sites, and the electronic Hamiltonian is constructed and diagonalized in a direct-product basis of non-orthogonal configuration state functions computed for isolated fragments. Here, we derive and implement analytic derivative couplings for this model, including nuclear derivatives of the natural transition orbital and symmetric orthogonalization transformations that are part of the approximation. Nuclear derivatives of the exciton Hamiltonian's matrix elements, required in order to compute the nonadiabatic couplings, are equivalent to the "Holstein" and "Peierls" exciton/phonon couplings that are widely discussed in the context of model Hamiltonians for energy and charge transport in organic photovoltaics. As an example, we compute the couplings that modulate triplet exciton transport in crystalline tetracene, which is relevant in the context of carrier diffusion following singlet exciton fission.
13. Ab Initio Infrared and Raman Spectra.
DTIC Science & Technology
1982-08-01
tions. For parameters not depending on momenta, a parallel ab fhti Monte Carlo approach would use electronic energies and other parameters of... Monte Carlo approach. Specifically, as one of us has suggested,t I classical molecular dynamics may be integrated with ab iniHo quan- tum force...alternative approach, for phenomena which are not explicitly time dependent, is a Monte Carlo procedure in which at each trial nuclear configuration
14. Periodic ab initio calculation of nuclear quadrupole parameters as an assignment tool in solid-state NMR spectroscopy: applications to 23Na NMR spectra of crystalline materials.
PubMed
Johnson, Clive; Moore, Elaine A; Mortimer, Michael
2005-05-01
Periodic ab initio HF calculations using the CRYSTAL code have been used to calculate (23)Na NMR quadrupole parameters for a wide range of crystalline sodium compounds including Na(3)OCl. An approach is developed that can be used routinely as an alternative to point-charge modelling schemes for the assignment of distinct lines in (23)Na NMR spectra to specific crystallographic sodium sites. The calculations are based on standard 3-21 G and 6-21 G molecular basis sets and in each case the same modified basis set for sodium is used for all compounds. The general approach is extendable to other quadrupolar nuclei. For the 3-21 G calculations a 1:1 linear correlation between experimental and calculated values of C(Q)((23)Na) is obtained. The 6-21 G calculations, including the addition of d-polarisation functions, give better accuracy in the calculation of eta((23)Na). The sensitivity of eta((23)Na) to hydrogen atom location is shown to be useful in testing the reported hydrogen-bonded structure of Na(2)HPO(4).
15. Gauge-invariant hydrogen-atom Hamiltonian
SciTech Connect
Sun Weimin; Wang Fan; Chen Xiangsong; Lue Xiaofu
2010-07-15
For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this puzzle was recently provided by us [X.-S. Chen et al., Phys. Rev. Lett. 100, 232002 (2008)]. Based on the separation of the electromagnetic potential into pure-gauge and gauge-invariant parts, we have proposed a new set of momentum and Hamiltonian operators which satisfy both the requirement of gauge invariance and the relevant commutation relations. In this paper we report a check for the case of the hydrogen-atom problem: Starting from the Hamiltonian of the coupled electron, proton, and electromagnetic field, under the infinite proton mass approximation, we derive the gauge-invariant hydrogen-atom Hamiltonian and verify explicitly that this Hamiltonian is different from the Dirac Hamiltonian, which is the time translation generator of the system. The gauge-invariant Hamiltonian is the energy operator, whose eigenvalue is the energy of the hydrogen atom. It is generally time dependent. In this case, one can solve the energy eigenvalue equation at any specific instant of time. It is shown that the energy eigenvalues are gauge independent, and by suitably choosing the phase factor of the time-dependent eigenfunction, one can ensure that the time-dependent eigenfunction satisfies the Dirac equation.
16. WE-AB-204-11: Development of a Nuclear Medicine Dosimetry Module for the GPU-Based Monte Carlo Code ARCHER
SciTech Connect
Liu, T; Lin, H; Xu, X; Stabin, M
2015-06-15
Purpose: To develop a nuclear medicine dosimetry module for the GPU-based Monte Carlo code ARCHER. Methods: We have developed a nuclear medicine dosimetry module for the fast Monte Carlo code ARCHER. The coupled electron-photon Monte Carlo transport kernel included in ARCHER is built upon the Dose Planning Method code (DPM). The developed module manages the radioactive decay simulation by consecutively tracking several types of radiation on a per disintegration basis using the statistical sampling method. Optimization techniques such as persistent threads and prefetching are studied and implemented. The developed module is verified against the VIDA code, which is based on Geant4 toolkit and has previously been verified against OLINDA/EXM. A voxelized geometry is used in the preliminary test: a sphere made of ICRP soft tissue is surrounded by a box filled with water. Uniform activity distribution of I-131 is assumed in the sphere. Results: The self-absorption dose factors (mGy/MBqs) of the sphere with varying diameters are calculated by ARCHER and VIDA respectively. ARCHER’s result is in agreement with VIDA’s that are obtained from a previous publication. VIDA takes hours of CPU time to finish the computation, while it takes ARCHER 4.31 seconds for the 12.4-cm uniform activity sphere case. For a fairer CPU-GPU comparison, more effort will be made to eliminate the algorithmic differences. Conclusion: The coupled electron-photon Monte Carlo code ARCHER has been extended to radioactive decay simulation for nuclear medicine dosimetry. The developed code exhibits good performance in our preliminary test. The GPU-based Monte Carlo code is developed with grant support from the National Institute of Biomedical Imaging and Bioengineering through an R01 grant (R01EB015478)
17. Three-cluster dynamics within an ab initio framework
SciTech Connect
Quaglioni, Sofia; Romero-Redondo, Carolina; Navratil, Petr
2013-09-26
In this study, we introduce a fully antisymmetrized treatment of three-cluster dynamics within the ab initio framework of the no-core shell model/resonating-group method. Energy-independent nonlocal interactions among the three nuclear fragments are obtained from realistic nucleon-nucleon interactions and consistent ab initio many-body wave functions of the clusters. The three-cluster Schrödinger equation is solved with bound-state boundary conditions by means of the hyperspherical-harmonic method on a Lagrange mesh. We discuss the formalism in detail and give algebraic expressions for systems of two single nucleons plus a nucleus. Using a soft similarity-renormalization-group evolved chiral nucleon-nucleon potential, we apply the method to a 4He+n+n description of 6He and compare the results to experiment and to a six-body diagonalization of the Hamiltonian performed within the harmonic-oscillator expansions of the no-core shell model. Differences between the two calculations provide a measure of core (4He) polarization effects.
18. Incomplete Dirac reduction of constrained Hamiltonian systems
SciTech Connect
Chandre, C.
2015-10-15
First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.
19. Hamiltonian and Lagrangian theory of viscoelasticity
Hanyga, A.; Seredyńska, M.
2008-03-01
The viscoelastic relaxation modulus is a positive-definite function of time. This property alone allows the definition of a conserved energy which is a positive-definite quadratic functional of the stress and strain fields. Using the conserved energy concept a Hamiltonian and a Lagrangian functional are constructed for dynamic viscoelasticity. The Hamiltonian represents an elastic medium interacting with a continuum of oscillators. By allowing for multiphase displacement and introducing memory effects in the kinetic terms of the equations of motion a Hamiltonian is constructed for the visco-poroelasticity.
20. The gravity duals of modular Hamiltonians
Jafferis, Daniel L.; Suh, S. Josephine
2016-09-01
In this work, we investigate modular Hamiltonians defined with respect to arbitrary spatial regions in quantum field theory states which have semi-classical gravity duals. We find prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its defining state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum field theory acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-like to the causal completion of the region.
1. The gravity duals of modular Hamiltonians
SciTech Connect
Jafferis, Daniel L.; Suh, S. Josephine
2016-09-12
In this study, we investigate modular Hamiltonians defined with respect to arbitrary spatial regions in quantum field theory states which have semi-classical gravity duals. We find prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its defining state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum field theory acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-like to the causal completion of the region.
2. Symmetries and regular behavior of Hamiltonian systems.
PubMed
Kozlov, Valeriy V.
1996-03-01
The behavior of the phase trajectories of the Hamilton equations is commonly classified as regular and chaotic. Regularity is usually related to the condition for complete integrability, i.e., a Hamiltonian system with n degrees of freedom has n independent integrals in involution. If at the same time the simultaneous integral manifolds are compact, the solutions of the Hamilton equations are quasiperiodic. In particular, the entropy of the Hamiltonian phase flow of a completely integrable system is zero. It is found that there is a broader class of Hamiltonian systems that do not show signs of chaotic behavior. These are systems that allow n commuting "Lagrangian" vector fields, i.e., the symplectic 2-form on each pair of such fields is zero. They include, in particular, Hamiltonian systems with multivalued integrals. (c) 1996 American Institute of Physics.
3. Compressed quantum metrology for the Ising Hamiltonian
Boyajian, W. L.; Skotiniotis, M.; Dür, W.; Kraus, B.
2016-12-01
We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase-transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on N spins, such a metrology protocol can be efficiently simulated on an exponentially smaller quantum computer while maintaining the same Heisenberg scaling for the squared error, i.e., O (N-2) precision, and derive the explicit circuit that accomplishes the simulation.
4. Quasi-Hamiltonian structure and Hojman construction
Carinena, Jose F.; Guha, Partha; Ranada, Manuel F.
2007-08-01
Given a smooth vector field [Gamma] and assuming the knowledge of an infinitesimal symmetry X, Hojman [S. Hojman, The construction of a Poisson structure out of a symmetry and a conservation law of a dynamical system, J. Phys. A Math. Gen. 29 (1996) 667-674] proposed a method for finding both a Poisson tensor and a function H such that [Gamma] is the corresponding Hamiltonian system. In this paper, we approach the problem from geometrical point of view. The geometrization leads to the clarification of several concepts and methods used in Hojman's paper. In particular, the relationship between the nonstandard Hamiltonian structure proposed by Hojman and the degenerate quasi-Hamiltonian structures introduced by Crampin and Sarlet [M. Crampin, W. Sarlet, Bi-quasi-Hamiltonian systems, J. Math. Phys. 43 (2002) 2505-2517] is unveiled in this paper. We also provide some applications of our construction.
5. Momentum and Hamiltonian in Complex Action Theory
Nagao, Keiichi; Nielsen, Holger Bech
In the complex action theory (CAT) we explicitly examine how the momentum and Hamiltonian are defined from the Feynman path integral (FPI) point of view based on the complex coordinate formalism of our foregoing paper. After reviewing the formalism briefly, we describe in FPI with a Lagrangian the time development of a ξ-parametrized wave function, which is a solution to an eigenvalue problem of a momentum operator. Solving this eigenvalue problem, we derive the momentum and Hamiltonian. Oppositely, starting from the Hamiltonian we derive the Lagrangian in FPI, and we are led to the momentum relation again via the saddle point for p. This study confirms that the momentum and Hamiltonian in the CAT have the same forms as those in the real action theory. We also show the third derivation of the momentum relation via the saddle point for q.
6. Quantum mechanical hamiltonian models of turing machines
Benioff, Paul
1982-11-01
Quantum mechanical Hamiltonian models, which represent an aribtrary but finite number of steps of any Turing machine computation, are constructed here on a finite lattice of spin-1/2 systems. Different regions of the lattice correspond to different components of the Turing machine (plus recording system). Successive states of any machine computation are represented in the model by spin configuration states. Both time-independent and time-dependent Hamiltonian models are constructed here. The time-independent models do not dissipate energy or degrade the system state as they evolve. They operate close to the quantum limit in that the total system energy uncertainty/computation speed is close to the limit given by the time-energy uncertainty relation. However, the model evolution is time global and the Hamiltonian is more complex. The time-dependent models do not degrade the system state. Also they are time local and the Hamiltonian is less complex.
7. A Student's Guide to Lagrangians and Hamiltonians
Hamill, Patrick
2013-11-01
Part I. Lagrangian Mechanics: 1. Fundamental concepts; 2. The calculus of variations; 3. Lagrangian dynamics; Part II. Hamiltonian Mechanics: 4. Hamilton's equations; 5. Canonical transformations: Poisson brackets; 6. Hamilton-Jacobi theory; 7. Continuous systems; Further reading; Index.
8. Hamiltonian surface charges using external sources
SciTech Connect
Troessaert, Cédric
2016-05-15
In this work, we interpret part of the boundary conditions as external sources in order to partially solve the integrability problem present in the computation of surface charges associated to gauge symmetries in the hamiltonian formalism. We start by describing the hamiltonian structure of external symmetries preserving the action up to a transformation of the external sources of the theory. We then extend these results to the computation of surface charges for field theories with non-trivial boundary conditions.
9. Nonperturbative embedding for highly nonlocal Hamiltonians
Subaşı, Yiǧit; Jarzynski, Christopher
2016-07-01
The need for Hamiltonians with many-body interactions arises in various applications of quantum computing. However, interactions beyond two-body are difficult to realize experimentally. Perturbative gadgets were introduced to obtain arbitrary many-body effective interactions using Hamiltonians with at most two-body interactions. Although valid for arbitrary k -body interactions, their use is limited to small k because the strength of interaction is k th order in perturbation theory. In this paper we develop a nonperturbative technique for obtaining effective k -body interactions using Hamiltonians consisting of at most l -body interactions with l Hamiltonian which is more local than the original one (using an analog device), and finally reverse the unitary transformation. The net effect of this procedure is shown to be equivalent to evolving the system with the original nonlocal Hamiltonian. This technique does not suffer from the aforementioned shortcoming of perturbative methods and requires only one ancilla qubit for each k -body interaction irrespective of the value of k . It works best for Hamiltonians with a few many-body interactions involving a large number of qubits and can be used together with perturbative gadgets to embed Hamiltonians of considerable complexity in proper subspaces of two-local Hamiltonians. We describe how our technique can be implemented in a hybrid (gate-based and adiabatic) as well as solely adiabatic quantum computing scheme.
10. Hamiltonian formulation of string field theory
Siopsis, George
1987-09-01
Witten's string field theory is quantized in the hamiltonian formalism. The constraints are solved and the hamiltonian is expressed in terms of only physical degrees of freedom. Thus, no Faddeev-Popov ghosts are introduced. Instead, the action contains terms of arbitrarily high order in the string functionals. Agreement with the standard results is demonstrated by an explicit calculation of the residues of the first few poles of the four-tachyon tree amplitude.
11. Hamiltonian vector fields on almost symplectic manifolds
Vaisman, Izu
2013-09-01
Let (M, ω) be an almost symplectic manifold (ω is a nondegenerate, not closed, 2-form). We say that a vector field X of M is locally Hamiltonian if LXω = 0, d(i(X)ω) = 0, and it is Hamiltonian if, furthermore, the 1-form i(X)ω is exact. Such vector fields were considered in Fassò and Sansonetto ["Integrable almost-symplectic Hamiltonian systems," J. Math. Phys. 48, 092902 (2007)], 10.1063/1.2783937, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, nonzero, Hamiltonian vector fields, or even none. Therefore, it is important to have examples and it is our aim to provide such examples here. We also obtain some new general results. In particular, we show that the locally Hamiltonian vector fields generate a Dirac structure on M and we state a reduction theorem of the Marsden-Weinstein type. A final section is dedicated to almost symplectic structures on tangent bundles.
12. Reaction Hamiltonian and state-to-state description of chemical reactions
SciTech Connect
Ruf, B.A.; Kresin, V.Z.; Lester, W.A. Jr.
1985-08-01
A chemical reaction is treated as a quantum transition from reactants to products. A specific reaction Hamiltonian (in second quantization formalism) is introduced. The approach leads to Franck-Condon-like factor, and adiabatic method in the framework of the nuclear motion problems. The influence of reagent vibrational state on the product energy distribution has been studied following the reaction Hamiltonian method. Two different cases (fixed available energy and fixed translational energy) are distinguished. Results for several biomolecular reactions are presented. 40 refs., 5 figs.
13. Geometric Construction of Quantum Hall Clustering Hamiltonians
Lee, Ching Hua; Papić, Zlatko; Thomale, Ronny
2015-10-01
Many fractional quantum Hall wave functions are known to be unique highest-density zero modes of certain "pseudopotential" Hamiltonians. While a systematic method to construct such parent Hamiltonians has been available for the infinite plane and sphere geometries, the generalization to manifolds where relative angular momentum is not an exact quantum number, i.e., the cylinder or torus, remains an open problem. This is particularly true for non-Abelian states, such as the Read-Rezayi series (in particular, the Moore-Read and Read-Rezayi Z3 states) and more exotic nonunitary (Haldane-Rezayi and Gaffnian) or irrational (Haffnian) states, whose parent Hamiltonians involve complicated many-body interactions. Here, we develop a universal geometric approach for constructing pseudopotential Hamiltonians that is applicable to all geometries. Our method straightforwardly generalizes to the multicomponent SU (n ) cases with a combination of spin or pseudospin (layer, subband, or valley) degrees of freedom. We demonstrate the utility of our approach through several examples, some of which involve non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and we verify the results by numerically computing their entanglement properties.
14. Hamiltonian Description of Multi-fluid Streaming
Valls, C.; de La Llave, R.; Morrison, P. J.
2001-10-01
The general noncanonical Hamiltonian description of interpenetrating fluids coupled by electrostatic, gravitational, or other forces is presented. This formalism is used to describe equilibrium and nonlinear stability using techniques of Hamiltonian dynamics theory. For example, we study the stability of two warm counter-streaming electron beams in a neutralizing ion background. The normal modes are obtained from an energy functional by computing the lowest-order expression for the perturbed energy about an equilibrium, and transforming the corresponding system into action-angle variables. Higher-order terms in the Hamiltonian provide coupling between normal modes and can lead to instability because of the presence of negative energy modes (NEM's). (The signature of the NEM's is determined by the signature of the Hamiltonian, Moser's bracket definition, or the conventional plasma definition in terms of the dielectric function, all of which are shown to be equivalent.) The possible nonlinear behavior is discovered by constructing the Birkhoff normal form. Accounting for resonances, we transform away terms in the Hamiltonian to address the question of long-time stability for such systems.
15. Geometric construction of quantum hall clustering Hamiltonians
DOE PAGES
Lee, Ching Hua; Papić, Zlatko; Thomale, Ronny
2015-10-08
In this study, many fractional quantum Hall wave functions are known to be unique highest-density zero modes of certain “pseudopotential” Hamiltonians. While a systematic method to construct such parent Hamiltonians has been available for the infinite plane and sphere geometries, the generalization to manifolds where relative angular momentum is not an exact quantum number, i.e., the cylinder or torus, remains an open problem. This is particularly true for non-Abelian states, such as the Read-Rezayi series (in particular, the Moore-Read and Read-Rezayi Z3 states) and more exotic nonunitary (Haldane-Rezayi and Gaffnian) or irrational (Haffnian) states, whose parent Hamiltonians involve complicated many-bodymore » interactions. Here, we develop a universal geometric approach for constructing pseudopotential Hamiltonians that is applicable to all geometries. Our method straightforwardly generalizes to the multicomponent SU(n) cases with a combination of spin or pseudospin (layer, subband, or valley) degrees of freedom. We demonstrate the utility of our approach through several examples, some of which involve non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and we verify the results by numerically computing their entanglement properties.« less
16. Generalized index for Hamiltonian systems with applications
Zevin, Alexandr A.
2005-09-01
Some classes of linear Hamiltonian equations with periodic coefficients (nondegenerate, strongly stable, completely unstable) are determined by the disposition of the Floquet multipliers. Periodic solutions of nonlinear equations (e.g. elliptic or hyperbolic solutions) are also defined by the multipliers of the corresponding variational equation. In this paper, we consider a general set M of linear Hamiltonian equations with multipliers satisfying some arbitrary conditions and a specific condition on the multiplier lying at some point of the unit circle (all known sets admit such a definition). We show that the set M consists of a finite number of subsets Mi which comprise a countable number of domains M_q^i within which any two Hamiltonians can be continuously deformed into each other. The corresponding integer index q is expressed through the eigenvalues of some self-adjoint problem. It is shown that this index (and, therefore, the known indices relating to specific sets) increases on increasing the Hamiltonian. Using the obtained results, some known and new sets are studied from the unified point of view. It is shown that for the sets of nondegenerate and completely unstable equations, the domains M_q^i are directionally convex; for strongly stable equations, necessary and sufficient conditions for directional convexity are found. The results are applied to problems of existence and stability of periodic solutions of nonlinear Hamiltonian equations.
17. Rovibrational molecular hamiltonian in mixed bond-angle and umbrella-like coordinates.
PubMed
Makarewicz, Jan; Skalozub, Alexander
2007-08-16
A new exact quantum mechanical rovibrational Hamiltonian operator for molecules exhibiting large amplitude inversion and torsion motions is derived. The derivation is based on a division of a molecule into two parts: a frame and a top. The nuclei of the frame only are used to construct a molecular system of axes. The inversion motion of the frame is described in the umbrella-like coordinates, whereas the torsion motion of the top is described by the nonstandard torsion angle defined in terms of the nuclear vectors and one of the molecular axes. The internal coordinates chosen take into account the properties of the inversion and torsion motions. Vibrational s and rotational Omega vectors obtained for the introduced internal coordinates determine the rovibrational tensor G defined by simple scalar products of these vectors. The Jacobian of the transformation from the Cartesian to the internal coordinates considered and the G tensor specify the rovibrational Hamiltonian. As a result, the Hamiltonian for penta-atomic molecules like NH2OH with one inverter is presented and a complete set of the formulas necessary to write down the Hamiltonian of more complex molecules, like NH2NH2 with two inverters, is reported. The approach considered is essentially general and sufficiently simple, as demonstrated by derivation of a polyatomic molecule Hamiltonian in polyspherical coordinates, obtained by other methods with much greater efforts.
18. Emergent properties of nuclei from ab initio coupled-cluster calculations
DOE PAGES
Hagen, G.; Hjorth-Jensen, M.; Jansen, G. R.; ...
2016-05-17
Emergent properties such as nuclear saturation and deformation, and the effects on shell structure due to the proximity of the scattering continuum and particle decay channels are fascinating phenomena in atomic nuclei. In recent years, ab initio approaches to nuclei have taken the first steps towards tackling the computational challenge of describing these phenomena from Hamiltonians with microscopic degrees of freedom. Our endeavor is now possible due to ideas from effective field theories, novel optimization strategies for nuclear interactions, ab initio methods exhibiting a soft scaling with mass number, and ever-increasing computational power. We review some of the recent accomplishments. We also present new results. The recently optimized chiral interaction NNLOmore » $${}_{{\\rm{sat}}}$$ is shown to provide an accurate description of both charge radii and binding energies in selected light- and medium-mass nuclei up to 56Ni. We derive an efficient scheme for including continuum effects in coupled-cluster computations of nuclei based on chiral nucleon–nucleon and three-nucleon forces, and present new results for unbound states in the neutron-rich isotopes of oxygen and calcium. Finally, the coupling to the continuum impacts the energies of the $${J}^{\\pi }=1/{2}^{-},3/{2}^{-},7/{2}^{-},3/{2}^{+}$$ states in $${}^{\\mathrm{17,23,25}}$$O, and—contrary to naive shell-model expectations—the level ordering of the $${J}^{\\pi }=3/{2}^{+},5/{2}^{+},9/{2}^{+}$$ states in $${}^{\\mathrm{53,55,61}}$$Ca.« less
19. Geometric Hamiltonian quantum mechanics and applications
Pastorello, Davide
2016-08-01
Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.
20. Hamiltonian boundary term and quasilocal energy flux
SciTech Connect
Chen, C.-M.; Nester, James M.; Tung, R.-S.
2005-11-15
The Hamiltonian for a gravitating region includes a boundary term which determines not only the quasilocal values but also, via the boundary variation principle, the boundary conditions. Using our covariant Hamiltonian formalism, we found four particular quasilocal energy-momentum boundary term expressions; each corresponds to a physically distinct and geometrically clear boundary condition. Here, from a consideration of the asymptotics, we show how a fundamental Hamiltonian identity naturally leads to the associated quasilocal energy flux expressions. For electromagnetism one of the four is distinguished: the only one which is gauge invariant; it gives the familiar energy density and Poynting flux. For Einstein's general relativity two different boundary condition choices correspond to quasilocal expressions which asymptotically give the ADM energy, the Trautman-Bondi energy and, moreover, an associated energy flux (both outgoing and incoming). Again there is a distinguished expression: the one which is covariant.
1. Equivalent Hamiltonians with additional discrete states
Chinn, C. R.; Thaler, R. M.
1991-01-01
Given a particular Hamiltonian H, we present a method to generate a new Hamiltonian H~, which has the same discrete energy eigenvalues and the same continuum phase shifts as H, but which also has additional given discrete eigenstates. This method is used to generate a Hamiltonian h1, which gives rise to a complete orthonormal set of basis states, which contain a given set of biorthonormal discrete states, the continuum states of which are asymptotic to plane waves (have zero phase shifts). Such a set of states may be helpful in representing the medium modification of the Green's function due to the Pauli principle, as well as including Pauli exclusion effects into scattering calculations.
2. Gravitational surface Hamiltonian and entropy quantization
Bakshi, Ashish; Majhi, Bibhas Ranjan; Samanta, Saurav
2017-02-01
The surface Hamiltonian corresponding to the surface part of a gravitational action has xp structure where p is conjugate momentum of x. Moreover, it leads to TS on the horizon of a black hole. Here T and S are temperature and entropy of the horizon. Imposing the hermiticity condition we quantize this Hamiltonian. This leads to an equidistant spectrum of its eigenvalues. Using this we show that the entropy of the horizon is quantized. This analysis holds for any order of Lanczos-Lovelock gravity. For general relativity, the area spectrum is consistent with Bekenstein's observation. This provides a more robust confirmation of this earlier result as the calculation is based on the direct quantization of the Hamiltonian in the sense of usual quantum mechanics.
3. The gravity duals of modular Hamiltonians
DOE PAGES
Jafferis, Daniel L.; Suh, S. Josephine
2016-09-12
In this study, we investigate modular Hamiltonians defined with respect to arbitrary spatial regions in quantum field theory states which have semi-classical gravity duals. We find prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its defining state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum field theory acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-likemore » to the causal completion of the region.« less
4. Charge transfer in strongly correlated systems: An exact diagonalization approach to model Hamiltonians
SciTech Connect
Schöppach, Andreas; Gnandt, David; Koslowski, Thorsten
2014-04-07
We study charge transfer in bridged di- and triruthenium complexes from a theoretical and computational point of view. Ab initio computations are interpreted from the perspective of a simple empirical Hamiltonian, a chemically specific Mott-Hubbard model of the complexes' π electron systems. This Hamiltonian is coupled to classical harmonic oscillators mimicking a polarizable dielectric environment. The model can be solved without further approximations in a valence bond picture using the method of exact diagonalization and permits the computation of charge transfer reaction rates in the framework of Marcus' theory. In comparison to the exact solution, the Hartree-Fock mean field theory overestimates both the activation barrier and the magnitude of charge-transfer excitations significantly. For triruthenium complexes, we are able to directly access the interruthenium antiferromagnetic coupling strengths.
5. Charge transfer in strongly correlated systems: an exact diagonalization approach to model Hamiltonians.
PubMed
Schöppach, Andreas; Gnandt, David; Koslowski, Thorsten
2014-04-07
We study charge transfer in bridged di- and triruthenium complexes from a theoretical and computational point of view. Ab initio computations are interpreted from the perspective of a simple empirical Hamiltonian, a chemically specific Mott-Hubbard model of the complexes' π electron systems. This Hamiltonian is coupled to classical harmonic oscillators mimicking a polarizable dielectric environment. The model can be solved without further approximations in a valence bond picture using the method of exact diagonalization and permits the computation of charge transfer reaction rates in the framework of Marcus' theory. In comparison to the exact solution, the Hartree-Fock mean field theory overestimates both the activation barrier and the magnitude of charge-transfer excitations significantly. For triruthenium complexes, we are able to directly access the interruthenium antiferromagnetic coupling strengths.
6. The Legendre transformations in Hamiltonian optics
Gitin, A. V.
2010-04-01
The Legendre transformations are an important tool in theoretical physics. They play a critical role in mechanics, optics, and thermodynamics. In Hamiltonian optics the Legendre transformations appear twice: as the connection between the Lagrangian and the Hamiltonian and as relations among eikonals. In this article interconnections between these two types of Legendre transformations have been investigated. Using the method of "transition to the centre and difference coordinates'' it is shown that four Legendre transformations which connect point, point-angle, angle-point, and angle eikonals of an optical system correspond to four Legendre transformations which connect four systems of equations: Euler's equations, Hamilton's equations, and two unknown before pairs of equations.
7. Exploring Hamiltonian dielectric solvent molecular dynamics
Bauer, Sebastian; Tavan, Paul; Mathias, Gerald
2014-09-01
Hamiltonian dielectric solvent (HADES) is a recent method [7,25], which enables Hamiltonian molecular dynamics (MD) simulations of peptides and proteins in dielectric continua. Sample simulations of an α-helical decapeptide with and without explicit solvent demonstrate the high efficiency of HADES-MD. Addressing the folding of this peptide by replica exchange MD we study the properties of HADES by comparing melting curves, secondary structure motifs and salt bridges with explicit solvent results. Despite the unoptimized ad hoc parametrization of HADES, calculated reaction field energies correlate well with numerical grid solutions of the dielectric Poisson equation.
8. Hamiltonian dynamics for complex food webs.
PubMed
Kozlov, Vladimir; Vakulenko, Sergey; Wennergren, Uno
2016-03-01
We investigate stability and dynamics of large ecological networks by introducing classical methods of dynamical system theory from physics, including Hamiltonian and averaging methods. Our analysis exploits the topological structure of the network, namely the existence of strongly connected nodes (hubs) in the networks. We reveal new relations between topology, interaction structure, and network dynamics. We describe mechanisms of catastrophic phenomena leading to sharp changes of dynamics and hence completely altering the ecosystem. We also show how these phenomena depend on the structure of interaction between species. We can conclude that a Hamiltonian structure of biological interactions leads to stability and large biodiversity.
9. Canonical transformations and Hamiltonian evolutionary systems
SciTech Connect
Al-Ashhab, Samer
2012-06-15
In many Lagrangian field theories, one has a Poisson bracket defined on the space of local functionals. We find necessary and sufficient conditions for a transformation on the space of local functionals to be canonical in three different cases. These three cases depend on the specific dimensions of the vector bundle of the theory and the associated Hamiltonian differential operator. We also show how a canonical transformation transforms a Hamiltonian evolutionary system and its conservation laws. Finally, we illustrate these ideas with three examples.
10. Reduction of pre-Hamiltonian actions
De Nicola, Antonio; Esposito, Chiara
2017-05-01
We prove a reduction theorem for the tangent bundle of a Poisson manifold (M , π) endowed with a pre-Hamiltonian action of a Poisson-Lie group (G ,πG) . In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of M. If the manifold M is symplectic and simply connected, the reduced tangent bundle is integrable and its integral symplectic groupoid is the Marsden-Weinstein reduction of the pair groupoid M × M ¯ .
11. Hamiltonians generating optimal-speed evolutions
2009-01-01
We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric (inner product) dependence of the lower bound on the travel time and the universality (metric independence) of the upper bound on the speed of unitary evolutions.
12. Hamiltonian mechanics and planar fishlike locomotion
Kelly, Scott; Xiong, Hailong; Burgoyne, Will
2007-11-01
A free deformable body interacting with a system of point vortices in the plane constitutes a Hamiltonian system. A free Joukowski foil with variable camber shedding point vortices in an ideal fluid according to a periodically applied Kutta condition provides a model for fishlike locomotion which bridges the gap between inviscid analytical models that sacrifice realism for tractability and viscous computational models inaccessible to tools from nonlinear control theory. We frame such a model in the context of Hamiltonian mechanics and describe its relevance both to the study of hydrodynamic interactions within schools of fish and to the realization of model-based control laws for biomimetic autonomous robotic vehicles.
13. Surface terms and the Gauss Bonnet Hamiltonian
2003-07-01
We derive the gravitational Hamiltonian starting from the Gauss Bonnet action, keeping track of all surface terms. This is done using the language of orthonormal frames and forms to keep things as tidy as possible. The surface terms in the Hamiltonian give a remarkably simple expression for the total energy of a spacetime. This expression is consistent with energy expressions found in Preprint hep-th/0212292. However, we can apply our results whatever the choice of background and whatever the symmetries of the spacetime.
14. Hamiltonian dynamics for complex food webs
Kozlov, Vladimir; Vakulenko, Sergey; Wennergren, Uno
2016-03-01
We investigate stability and dynamics of large ecological networks by introducing classical methods of dynamical system theory from physics, including Hamiltonian and averaging methods. Our analysis exploits the topological structure of the network, namely the existence of strongly connected nodes (hubs) in the networks. We reveal new relations between topology, interaction structure, and network dynamics. We describe mechanisms of catastrophic phenomena leading to sharp changes of dynamics and hence completely altering the ecosystem. We also show how these phenomena depend on the structure of interaction between species. We can conclude that a Hamiltonian structure of biological interactions leads to stability and large biodiversity.
15. Ostrogradski Hamiltonian approach for geodetic brane gravity
SciTech Connect
2010-12-07
We present an alternative Hamiltonian description of a branelike universe immersed in a flat background spacetime. This model is named geodetic brane gravity. We set up the Regge-Teitelboim model to describe our Universe where such field theory is originally thought as a second order derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the system to its quantization. This approach comprize the manage of both first- and second-class constraints and the counting of degrees of freedom follows accordingly.
16. Convergence to equilibrium under a random Hamiltonian.
PubMed
Brandão, Fernando G S L; Ćwikliński, Piotr; Horodecki, Michał; Horodecki, Paweł; Korbicz, Jarosław K; Mozrzymas, Marek
2012-09-01
We analyze equilibration times of subsystems of a larger system under a random total Hamiltonian, in which the basis of the Hamiltonian is drawn from the Haar measure. We obtain that the time of equilibration is of the order of the inverse of the arithmetic average of the Bohr frequencies. To compute the average over a random basis, we compute the inverse of a matrix of overlaps of operators which permute four systems. We first obtain results on such a matrix for a representation of an arbitrary finite group and then apply it to the particular representation of the permutation group under consideration.
17. Ab initio valence-space theory for exotic nuclei
Holt, Jason
2015-10-01
Recent advances in ab initio nuclear structure theory have led to groundbreaking predictions in the exotic medium-mass region, from the location of the neutron dripline to the emergence of new magic numbers far from stability. Playing a key role in this progress has been the development of sophisticated many-body techniques and chiral effective field theory, which provides a systematic basis for consistent many-nucleon forces and electroweak currents. Within the context of valence-space Hamiltonians derived from the nonperturbative in-medium similarity renormalization group (IM-SRG) approach, I will discuss the importance of 3N forces in understanding and making new discoveries in the exotic sd -shell region. Beginning in oxygen, we find that the effects of 3N forces are decisive in explaining why 24O is the last bound oxygen isotope, validating first predictions of this phenomenon from several years ago. Furthermore, 3N forces play a key role in reproducing spectroscopy, including signatures of doubly magic 22,24O, and physics beyond the dripline. Similar improvements are obtained in new spectroscopic predictions for exotic fluorine and neon isotopes, where agreement with recent experimental data is competitive with state-of-the-art phenomenology. Finally, I will discuss first applications of the IM-SRG to effective valence-space operators, such as radii and E 0 transitions, as well as extensions to general operators crucial for our future understanding of electroweak processes, such as neutrinoless double-beta decay. This work was supported by NSERC and the NRC Canada.
18. Suppressing qubit dephasing using real-time Hamiltonian estimation
PubMed Central
Harvey, S. P.; Nichol, J. M.; Bartlett, S. D.; Doherty, A. C.; Umansky, V.; Yacoby, A.
2014-01-01
Unwanted interaction between a quantum system and its fluctuating environment leads to decoherence and is the primary obstacle to establishing a scalable quantum information processing architecture. Strategies such as environmental and materials engineering, quantum error correction and dynamical decoupling can mitigate decoherence, but generally increase experimental complexity. Here we improve coherence in a qubit using real-time Hamiltonian parameter estimation. Using a rapidly converging Bayesian approach, we precisely measure the splitting in a singlet-triplet spin qubit faster than the surrounding nuclear bath fluctuates. We continuously adjust qubit control parameters based on this information, thereby improving the inhomogenously broadened coherence time from tens of nanoseconds to >2 μs. Because the technique demonstrated here is compatible with arbitrary qubit operations, it is a natural complement to quantum error correction and can be used to improve the performance of a wide variety of qubits in both meteorological and quantum information processing applications. PMID:25295674
19. Hamiltonian Approach to Nonlinear Travelling Whistler Waves
SciTech Connect
Webb, G.M.; McKenzie, J.F.; Dubinin, E.; Sauer, K.
2005-08-01
A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is discussed. The model is based on the equations of two-fluid electron-proton plasmas. In the cold gas limit, the complete system of equations reduces to two coupled differential equations for the transverse electron speed u and a phase variable {phi} = {phi}p - {phi}e representing the difference in the phases of the transverse complex velocities of the protons and the electrons. Two integrals of the equations are obtained. The Hamiltonian integral H, is used to classify the trajectories in the ({phi}, w) phase plane, where {phi} and w = u2 are the canonical coordinates. Periodic, oscillation solitary wave and compacton solutions are obtained, depending on the value of the Hamiltonian integral H and the Alfven Mach number M of the travelling wave. The individual electron and proton phase variables {phi}e and {phi}p are determined in terms of {phi} and w. An alternative Hamiltonian formulation in which {phi}-tilde = {phi}p + {phi}e is the new independent variable replacing x is used to write the travelling wave solutions parametrically in terms of {phi}-tilde.
20. Mapping between dissipative and Hamiltonian systems
Xing, Jianhua
2010-09-01
Theoretical studies of nonequilibrium systems are complicated by the lack of a general framework. In this work we first show that a transformation recently introduced by Ao (2004 J. Phys. A: Math. Gen. 37 L25) is related to previous works of Graham (1977 Z. Phys. B 26 397) and Eyink et al (1996 J. Stat. Phys. 83 385), which can also be viewed as the generalized application of the Helmholtz theorem in vector calculus. We then show that systems described by ordinary stochastic differential equations with white noise can be mapped to thermostated Hamiltonian systems. A steady-state of a dissipative system corresponds to the equilibrium state of the corresponding Hamiltonian system. These results provide a solid theoretical ground for corresponding studies on nonequilibrium dynamics, especially on nonequilibrium steady state. Mapping permits the application of established techniques and results for Hamiltonian systems to dissipative non-Hamiltonian systems, those for thermodynamic equilibrium states to nonequilibrium steady states. We discuss several implications of this work.
1. Periodic Solutions of Hamiltonian Systems: A Survey.
DTIC Science & Technology
1980-12-01
auto - nomous Hamiltonian system has the form: (0.) aH 8Hp -S-(p,q) q ( where d denotes This system can be represented more concisely as (HS) z = ZHz(Z...oscillazioni periodiche d’une sistema dinamico, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 19, (1934), 234-237. [15] Arnold, V. I
2. An underlying geometrical manifold for Hamiltonian mechanics
Horwitz, L. P.; Yahalom, A.; Levitan, J.; Lewkowicz, M.
2017-02-01
We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture), that can be put into correspondence with the usual Hamilton-Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamiltonian-Lagrange picture, defined on the Hamilton-Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton-Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton-Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the the original Hamiltonian motion.
3. Lagrangian tetragons and instabilities in Hamiltonian dynamics
Entov, Michael; Polterovich, Leonid
2017-01-01
We present a new existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. The method involves function theory on symplectic manifolds combined with rigidity of Lagrangian submanifolds. Applications include superconductivity channels in nearly integrable systems and dynamics near a perturbed unstable equilibrium.
4. From Nonlinear to Hamiltonian via Feedback1
DTIC Science & Technology
2002-01-01
distribution unlimited. 13. Abstract Mechanical control systems are a very important class of nonlinear control systems . They posses a rich mathematical...methodologies developed for mechanical control systel logically rendering nonlinear control systems , mechanical by a proper choice of feedback. In particular, w...OF PA Nonlinear mechanical control systems , Hamiltonian Control Systems x 16. PRICE CODE 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19
5. Eigenfunction expansions for time dependent hamiltonians
Jauslin, H. R.; Guerin, S.; Deroussiaux, A.
We describe a generalization of Floquet theory for non periodic time dependent Hamiltonians. It allows to express the time evolution in terms of an expansion in eigenfunctions of a generalized quasienergy operator. We discuss a conjecture on the extension of the adiabatic theorem to this type of systems, which gives a procedure for the physical preparation of Floquet states. *** DIRECT SUPPORT *** A3418380 00004
6. Hamiltonian constraint in polymer parametrized field theory
2011-01-01
Recently, a generally covariant reformulation of two-dimensional flat spacetime free scalar field theory known as parametrized field theory was quantized using loop quantum gravity (LQG) type “polymer” representations. Physical states were constructed, without intermediate regularization structures, by averaging over the group of gauge transformations generated by the constraints, the constraint algebra being a Lie algebra. We consider classically equivalent combinations of these constraints corresponding to a diffeomorphism and a Hamiltonian constraint, which, as in gravity, define a Dirac algebra. Our treatment of the quantum constraints parallels that of LQG and obtains the following results, expected to be of use in the construction of the quantum dynamics of LQG: (i) the (triangulated) Hamiltonian constraint acts only on vertices, its construction involves some of the same ambiguities as in LQG and its action on diffeomorphism invariant states admits a continuum limit, (ii) if the regulating holonomies are in representations tailored to the edge labels of the state, all previously obtained physical states lie in the kernel of the Hamiltonian constraint, (iii) the commutator of two (density weight 1) Hamiltonian constraints as well as the operator correspondent of their classical Poisson bracket converge to zero in the continuum limit defined by diffeomorphism invariant states, and vanish on the Lewandowski-Marolf habitat, (iv) the rescaled density 2 Hamiltonian constraints and their commutator are ill-defined on the Lewandowski-Marolf habitat despite the well-definedness of the operator correspondent of their classical Poisson bracket there, (v) there is a new habitat which supports a nontrivial representation of the Poisson-Lie algebra of density 2 constraints.
7. Hamiltonian constraint in polymer parametrized field theory
SciTech Connect
2011-01-15
Recently, a generally covariant reformulation of two-dimensional flat spacetime free scalar field theory known as parametrized field theory was quantized using loop quantum gravity (LQG) type ''polymer'' representations. Physical states were constructed, without intermediate regularization structures, by averaging over the group of gauge transformations generated by the constraints, the constraint algebra being a Lie algebra. We consider classically equivalent combinations of these constraints corresponding to a diffeomorphism and a Hamiltonian constraint, which, as in gravity, define a Dirac algebra. Our treatment of the quantum constraints parallels that of LQG and obtains the following results, expected to be of use in the construction of the quantum dynamics of LQG: (i) the (triangulated) Hamiltonian constraint acts only on vertices, its construction involves some of the same ambiguities as in LQG and its action on diffeomorphism invariant states admits a continuum limit, (ii) if the regulating holonomies are in representations tailored to the edge labels of the state, all previously obtained physical states lie in the kernel of the Hamiltonian constraint, (iii) the commutator of two (density weight 1) Hamiltonian constraints as well as the operator correspondent of their classical Poisson bracket converge to zero in the continuum limit defined by diffeomorphism invariant states, and vanish on the Lewandowski-Marolf habitat, (iv) the rescaled density 2 Hamiltonian constraints and their commutator are ill-defined on the Lewandowski-Marolf habitat despite the well-definedness of the operator correspondent of their classical Poisson bracket there, (v) there is a new habitat which supports a nontrivial representation of the Poisson-Lie algebra of density 2 constraints.
8. Hamiltonian gadgets with reduced resource requirements
Cao, Yudong; Babbush, Ryan; Biamonte, Jacob; Kais, Sabre
2015-01-01
9. Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree
SciTech Connect
Lepskii, Timur A
2010-12-07
Complex Hamiltonian systems with one degree of freedom on C{sup 2} with the standard symplectic structure {omega}C=dz and dw and a polynomial Hamiltonian function f=z{sup 2}+P{sub n}(w), n=1,2,3,4, are studied. Two Hamiltonian systems (M{sub i}, Re{omega}{sub C,i}, H{sub i}=Ref{sub i}), i=1,2, are said to be Hamiltonian equivalent if there exists a complex symplectomorphism M{sub 1}{yields}M{sub 2} taking the vector field sgradH{sub 1} to sgradH{sub 2}. Hamiltonian equivalence classes of systems are described in the case n=1,2,3,4, a completed system is defined for n=3,4, and it is proved that it is Liouville integrable as a real Hamiltonian system. By restricting the real action-angle coordinates defined for the completed system in a neighbourhood of any nonsingular leaf, real canonical coordinates are obtained for the original system. Bibliography: 9 titles.
10. The 2-alkyl-2H-indazole regioisomers of synthetic cannabinoids AB-CHMINACA, AB-FUBINACA, AB-PINACA, and 5F-AB-PINACA are possible manufacturing impurities with cannabimimetic activities.
PubMed
Longworth, Mitchell; Banister, Samuel D; Mack, James B C; Glass, Michelle; Connor, Mark; Kassiou, Michael
2016-01-01
Indazole-derived synthetic cannabinoids (SCs) featuring an alkyl substituent at the 1-position and l-valinamide at the 3-carboxamide position (e.g., AB-CHMINACA) have been identified by forensic chemists around the world, and are associated with serious adverse health effects. Regioisomerism is possible for indazole SCs, with the 2-alkyl-2H-indazole regioisomer of AB-CHMINACA recently identified in SC products in Japan. It is unknown whether this regiosiomer represents a manufacturing impurity arising as a synthetic byproduct, or was intentionally synthesized as a cannabimimetic agent. This study reports the synthesis, analytical characterization, and pharmacological evaluation of commonly encountered indazole SCs AB-CHMINACA, AB-FUBINACA, AB-PINACA, 5F-AB-PINACA and their corresponding 2-alkyl-2H-indazole regioisomers. Both regioisomers of each SC were prepared from a common precursor, and the physical properties, (1)H and (13)C nuclear magnetic resonance spectroscopy, gas chromatography-mass spectrometry, and ultraviolet-visible spectroscopy of all SC compounds are described. Additionally, AB-CHMINACA, AB-FUBINACA, AB-PINACA, and 5F-AB-PINACA were found to act as high potency agonists at CB1 (EC50 = 2.1-11.6 nM) and CB2 (EC50 = 5.6-21.1 nM) receptors in fluorometric assays, while the corresponding 2-alkyl-2H-indazole regioisomers demonstrated low potency (micromolar) agonist activities at both receptors. Taken together, these data suggest that 2-alkyl-2H-indazole regioisomers of AB-CHMINACA, AB-FUBINACA, AB-PINACA, and 5F-AB-PINACA are likely to be encountered by forensic chemists and toxicologists as the result of improper purification during the clandestine synthesis of 1-alkyl-1H-indazole regioisomers, and can be distinguished by differences in gas chromatography-mass spectrometry fragmentation pattern.
11. Hamiltonian time integrators for Vlasov-Maxwell equations
He, Yang; Qin, Hong; Sun, Yajuan; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian
2015-12-01
Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.
12. Hamiltonian time integrators for Vlasov-Maxwell equations
SciTech Connect
He, Yang; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian; Qin, Hong; Sun, Yajuan
2015-12-15
Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.
13. Conformal killing tensors and covariant Hamiltonian dynamics
SciTech Connect
Cariglia, M.; Gibbons, G. W.; Holten, J.-W. van; Horvathy, P. A.; Zhang, P.-M.
2014-12-15
A covariant algorithm for deriving the conserved quantities for natural Hamiltonian systems is combined with the non-relativistic framework of Eisenhart, and of Duval, in which the classical trajectories arise as geodesics in a higher dimensional space-time, realized by Brinkmann manifolds. Conserved quantities which are polynomial in the momenta can be built using time-dependent conformal Killing tensors with flux. The latter are associated with terms proportional to the Hamiltonian in the lower dimensional theory and with spectrum generating algebras for higher dimensional quantities of order 1 and 2 in the momenta. Illustrations of the general theory include the Runge-Lenz vector for planetary motion with a time-dependent gravitational constant G(t), motion in a time-dependent electromagnetic field of a certain form, quantum dots, the Hénon-Heiles and Holt systems, respectively, providing us with Killing tensors of rank that ranges from one to six.
14. Tangent lifts of bi-Hamiltonian structures
Dobrogowska, Alina; Jakimowicz, Grzegorz
2017-08-01
We construct two Poisson structures πT M and π˜ T M on the tangent bundle TM to a Poisson manifold (M ,π ) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold (T M ,πT M ,λ ) from a bi-Hamiltonian manifold (M ,π1,π2 ) . This structure can be considered as a deformation of the Poisson structure related to an algebroid structure. We show that the bi-Hamiltonian structure from M can be transferred to the manifold TM. Moreover we present how to construct the Casimir functions for structures πT M, πT M ,λ, π˜ T g*, and π˜ T g*,λ from the Casimir functions for π1 and π2 and discuss some particular examples.
15. Hamiltonian deformations of Gabor frames: First steps.
PubMed
de Gosson, Maurice A
2015-03-01
Gabor frames can advantageously be redefined using the Heisenberg-Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies (i.e. arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. We will thereafter discuss possible applications and extensions of our method, which can be viewed - as the title suggests - as the very first steps towards a general deformation theory for Gabor frames.
16. Hamiltonian deformations of Gabor frames: First steps
PubMed Central
de Gosson, Maurice A.
2015-01-01
Gabor frames can advantageously be redefined using the Heisenberg–Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies (i.e. arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. We will thereafter discuss possible applications and extensions of our method, which can be viewed – as the title suggests – as the very first steps towards a general deformation theory for Gabor frames. PMID:25892903
17. General formalism for singly thermostated Hamiltonian dynamics.
PubMed
Ramshaw, John D
2015-11-01
A general formalism is developed for constructing modified Hamiltonian dynamical systems which preserve a canonical equilibrium distribution by adding a time evolution equation for a single additional thermostat variable. When such systems are ergodic, canonical ensemble averages can be computed as dynamical time averages over a single trajectory. Systems of this type were unknown until their recent discovery by Hoover and colleagues. The present formalism should facilitate the discovery, construction, and classification of other such systems by encompassing a wide class of them within a single unified framework. This formalism includes both canonical and generalized Hamiltonian systems in a state space of arbitrary dimensionality (either even or odd) and therefore encompasses both few- and many-particle systems. Particular attention is devoted to the physical motivation and interpretation of the formalism, which largely determine its structure. An analogy to stochastic thermostats and fluctuation-dissipation theorems is briefly discussed.
18. Diffusion in very chaotic hamiltonian systems
SciTech Connect
Abarbanel, Henry D. I.; Crawford, John David
1981-04-20
In this paper, we study nonintegrable hamiltonian dynamics: H(I,θ) = H0(I) + kH1(I,θ), for large k, that is, far from integrability. An integral representation is given for the conditional probability P(I,θ, t¦I0, θ0, t0) that the system is at I, θ at t, given it was at I0, θ0 at t0. By discretizing time into steps of size ϵ, we show how to evaluate physical observables for large k, fixed ϵ. An explicit calculation of a diffusion coefficient in a two degrees of freedom problem is reported. Finally, passage to ϵ = 0, the original hamiltonian flow, is discussed.
19. Hamiltonian learning and certification using quantum resources.
PubMed
Wiebe, Nathan; Granade, Christopher; Ferrie, Christopher; Cory, D G
2014-05-16
In recent years quantum simulation has made great strides, culminating in experiments that existing supercomputers cannot easily simulate. Although this raises the possibility that special purpose analog quantum simulators may be able to perform computational tasks that existing computers cannot, it also introduces a major challenge: certifying that the quantum simulator is in fact simulating the correct quantum dynamics. We provide an algorithm that, under relatively weak assumptions, can be used to efficiently infer the Hamiltonian of a large but untrusted quantum simulator using a trusted quantum simulator. We illustrate the power of this approach by showing numerically that it can inexpensively learn the Hamiltonians for large frustrated Ising models, demonstrating that quantum resources can make certifying analog quantum simulators tractable.
20. Hamiltonian Approach To Dp-Brane Noncommutativity
Nikolic, B.; Sazdovic, B.
2010-07-01
In this article we investigate Dp-brane noncommutativity using Hamiltonian approach. We consider separately open bosonic string and type IIB superstring which endpoints are attached to the Dp-brane. From requirement that Hamiltonian, as the time translation generator, has well defined derivatives in the coordinates and momenta, we obtain boundary conditions directly in the canonical form. Boundary conditions are treated as canonical constraints. Solving them we obtain initial coordinates in terms of the effective ones as well as effective momenta. Presence of momenta implies noncommutativity of the initial coordinates. Effective theory, defined as initial one on the solution of boundary conditions, is its Ω even projection, where Ω is world-sheet parity transformation Ω:σ→-σ. The effective background fields are expressed in terms of Ω even and squares of the Ω odd initial background fields.
1. NUCLEAR REACTORS
DOEpatents
Long, E.; Ashby, J.W.
1958-09-16
ABS>A graphite moderator structure is presented for a nuclear reactor compriscd of an assembly of similarly orientated prismatic graphite blocks arranged on spaced longitudinal axes lying in common planes wherein the planes of the walls of the blocks are positioned so as to be twisted reintive to the planes of said axes so thatthe unlmpeded dtrect paths in direction wholly across the walls of the blocks are limited to the width of the blocks plus spacing between the blocks.
2. Controlling Hamiltonian chaos via Gaussian curvature.
PubMed
Oloumi, A; Teychenné, D
1999-12-01
We present a method allowing one to partly stabilize some chaotic Hamiltonians which have two degrees of freedom. The purpose of the method is to avoid the regions of V(q(1),q(2)) where its Gaussian curvature becomes negative. We show the stabilization of the Hénon-Heiles system, over a wide area, for the critical energy E=1/6. Total energy of the system varies only by a few percent.
3. Hamiltonian anomalies of bound states in QED
SciTech Connect
Shilin, V. I.; Pervushin, V. N.
2013-10-15
The Bound State in QED is described in systematic way by means of nonlocal irreducible representations of the nonhomogeneous Poincare group and Dirac's method of quantization. As an example of application of this method we calculate triangle diagram Para-Positronium {yields} {gamma}{gamma}. We show that the Hamiltonian approach to Bound State in QED leads to anomaly-type contribution to creation of pair of parapositronium by two photon.
4. Hamiltonian theory of guiding-center motion
SciTech Connect
Littlejohn, R.G.
1980-05-01
A Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation of the unperturbed system from the perturbation is achieved by using a coordinate transformation suggested by a theorem of Darboux. As a model to illustrate the method, motion in the magnetic field B=B(x,y)z is studied. Lie transforms are used to carry out the perturbation expansion.
5. The Hamiltonian Mechanics of Stochastic Acceleration
SciTech Connect
Burby, J. W.
2013-07-17
We show how to nd the physical Langevin equation describing the trajectories of particles un- dergoing collisionless stochastic acceleration. These stochastic di erential equations retain not only one-, but two-particle statistics, and inherit the Hamiltonian nature of the underlying microscopic equations. This opens the door to using stochastic variational integrators to perform simulations of stochastic interactions such as Fermi acceleration. We illustrate the theory by applying it to two example problems.
6. mAbs
PubMed Central
2009-01-01
The twenty two monoclonal antibodies (mAbs) currently marketed in the U.S. have captured almost half of the top-20 U.S. therapeutic biotechnology sales for 2007. Eight of these products have annual sales each of more than \$1 B, were developed in the relatively short average period of six years, qualified for FDA programs designed to accelerate drug approval, and their cost has been reimbursed liberally by payers. With growth of the product class driven primarily by advancements in protein engineering and the low probability of generic threats, mAbs are now the largest class of biological therapies under development. The high cost of these drugs and the lack of generic competition conflict with a financially stressed health system, setting reimbursement by payers as the major limiting factor to growth. Advances in mAb engineering are likely to result in more effective mAb drugs and an expansion of the therapeutic indications covered by the class. The parallel development of biomarkers for identifying the patient subpopulations most likely to respond to treatment may lead to a more cost-effective use of these drugs. To achieve the success of the current top-tier mAbs, companies developing new mAb products must adapt to a significantly more challenging commercial environment. PMID:20061824
7. Room temperature line lists for CO2 symmetric isotopologues with ab initio computed intensities
Zak, Emil J.; Tennyson, Jonathan; Polyansky, Oleg L.; Lodi, Lorenzo; Zobov, Nikolay F.; Tashkun, Sergei A.; Perevalov, Valery I.
2017-03-01
Remote sensing experiments require high-accuracy, preferably sub-percent, line intensities and in response to this need we present computed room temperature line lists for six symmetric isotopologues of carbon dioxide: 13C16O2, 14C16O2, 12C17O2, 12C18O2, 13C17O2 and 13C18O2, covering the range 0-8000 cm-1. Our calculation scheme is based on variational nuclear motion calculations and on a reliability analysis of the generated line intensities. Rotation-vibration wavefunctions and energy levels are computed using the DVR3D software suite and a high quality semi-empirical potential energy surface (PES), followed by computation of intensities using an ab initio dipole moment surface (DMS). Four line lists are computed for each isotopologue to quantify sensitivity to minor distortions of the PES/DMS. Reliable lines are benchmarked against recent state-of-the-art measurements and against the HITRAN2012 database, supporting the claim that the majority of line intensities for strong bands are predicted with sub-percent accuracy. Accurate line positions are generated using an effective Hamiltonian. We recommend the use of these line lists for future remote sensing studies and their inclusion in databases.
8. Ab initio Approach to Effective Single-Particle Energies in Doubly Closed Shell Nuclei
SciTech Connect
Duguet, T.
2012-01-01
The present work discusses, from an ab initio standpoint, the definition, the meaning, and the usefulness of effective single-particle energies (ESPEs) in doubly closed shell nuclei. We perform coupled-cluster calculations to quantify to what extent selected closed-shell nuclei in the oxygen and calcium isotopic chains can effectively be mapped onto an effective independent-particle picture. To do so, we revisit in detail the notion of ESPEs in the context of strongly correlated many-nucleon systems and illustrate the necessity of extracting ESPEs through the diagonalization of the centroid matrix, as originally argued by Baranger. For the purpose of illustration, we analyze the impact of correlations on observable one-nucleon separation energies and nonobservable ESPEs in selected closed-shell oxygen and calcium isotopes. We then state and illustrate the nonobservability of ESPEs. Similarly to spectroscopic factors, ESPEs can indeed be modified by a redefinition of inaccessible quantities while leaving actual observables unchanged. This leads to the absolute necessity of employing consistent structure and reaction models based on the same nuclear Hamiltonian to extract the shell structure in a meaningful fashion from experimental data.
9. Optimal Hamiltonian Simulation by Quantum Signal Processing
Low, Guang Hao; Chuang, Isaac L.
2017-01-01
The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d -sparse Hamiltonian H ^ for time-interval t with error ɛ is O [t d ∥H ^ ∥max+log (1 /ɛ ) /log log (1 /ɛ ) ] , which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H ^ into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.
10. Invariants for time-dependent Hamiltonian systems.
PubMed
Struckmeier, J; Riedel, C
2001-08-01
An exact invariant is derived for n-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special Ansatz for the invariant and determine its time-dependent coefficients. In the second approach, we perform a two-step canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. The invariant is found to contain a function of time f(2)(t), defined as a solution of a linear third-order differential equation whose coefficients depend in general on the explicitly known configuration space trajectory that follows from the system's time evolution. It is shown that the invariant can be interpreted as the time integral of an energy balance equation. Our result is applied to a one-dimensional, time-dependent, damped non-linear oscillator, and to a three-dimensional system of Coulomb-interacting particles that are confined in a time-dependent quadratic external potential. We finally show that our results can be used to assess the accuracy of numerical simulations of time-dependent Hamiltonian systems.
11. Effective Hamiltonian for edge states in graphene.
DOE PAGES
Deshpande, H.; Winkler, R.
2017-06-03
Edge states in topological insulators (TIs) disperse symmetrically about one of the time-reversal invariant momenta Lambda in the Brillouin zone (BZ) with protected degeneracies at Lambda. Commonly TIs are distinguished from trivial insulators by the values of one or multiple topological invariants that require an analysis of the bulk band structure across the BZ. We propose an effective two-band Hamiltonian for the electronic states in graphene based on a Taylor expansion of the tight-binding Hamiltonian about the time-reversal invariant M point at the edge of the BZ. This Hamiltonian provides a faithful description of the protected edge states for bothmore » zigzag and armchair ribbons, though the concept of a BZ is not part of such an effective model. We show that the edge states are determined by a band inversion in both reciprocal and real space, which allows one to select Lambda for the edge states without affecting the bulk spectrum.« less
12. Numerical Continuation of Hamiltonian Relative Periodic Orbits
Wulff, Claudia; Schebesch, Andreas
2008-08-01
The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it.
13. Redesign of the DFT/MRCI Hamiltonian
SciTech Connect
Lyskov, Igor; Kleinschmidt, Martin; Marian, Christel M.
2016-01-21
The combined density functional theory and multireference configuration interaction (DFT/MRCI) method of Grimme and Waletzke [J. Chem. Phys. 111, 5645 (1999)] is a well-established semi-empirical quantum chemical method for efficiently computing excited-state properties of organic molecules. As it turns out, the method fails to treat bi-chromophores owing to the strong dependence of the parameters on the excitation class. In this work, we present an alternative form of correcting the matrix elements of a MRCI Hamiltonian which is built from a Kohn-Sham set of orbitals. It is based on the idea of constructing individual energy shifts for each of the state functions of a configuration. The new parameterization is spin-invariant and incorporates less empirism compared to the original formulation. By utilizing damping techniques together with an algorithm of selecting important configurations for treating static electron correlation, the high computational efficiency has been preserved. The robustness of the original and redesigned Hamiltonians has been tested on experimentally known vertical excitation energies of organic molecules yielding similar statistics for the two parameterizations. Besides that, our new formulation is free from artificially low-lying doubly excited states, producing qualitatively correct and consistent results for excimers. The way of modifying matrix elements of the MRCI Hamiltonian presented here shall be considered as default choice when investigating photophysical processes of bi-chromophoric systems such as singlet fission or triplet-triplet upconversion.
14. Cylindrical coordinate representation for multiband Hamiltonians
Takhtamirov, Eduard
2012-10-01
Rotationally invariant combinations of the Brillouin zone-center Bloch functions are used as basis function to express in cylindrical coordinates the valence-band and Kane envelope-function Hamiltonians for wurtzite and zinc-blende semiconductor heterostructures. For cylindrically symmetric systems, this basis allows to treat the envelope functions as eigenstates of the operator of projection of total angular momentum on the symmetry axis, with the operator's eigenvalue conventionally entering the Hamiltonians as a parameter. Complementing the Hamiltonians with boundary conditions for the envelope functions on the symmetry axis, we present for the first time a complete formalism for efficient modeling and description of multiband electron states in low-dimensional semiconductor structures with cylindrical symmetry. To demonstrate the potency of the cylindrical symmetry approximation and establish a criterion of its applicability for actual structures, we map the ground and several excited valence-band states in an isolated wurtzite GaN quantum wire of a hexagonal cross-section to the states in an equivalent quantum wire of a circular cross-section.
15. Dynamics of Hamiltonian Systems and Memristor Circuits
Itoh, Makoto; Chua, Leon
In this paper, we show that any n-dimensional autonomous systems can be regarded as subsystems of 2n-dimensional Hamiltonian systems. One of the two subsystems is identical to the n-dimensional autonomous system, which is called the driving system. Another subsystem, called the response system, can exhibit interesting behaviors in the neighborhood of infinity. That is, the trajectories approach infinity with complicated nonperiodic (chaotic-like) behaviors, or periodic-like behavior. In order to show the above results, we project the trajectories of the Hamiltonian systems onto n-dimensional spheres, or n-dimensional balls by using the well-known central projection transformation. Another interesting behavior is that the transient regime of the subsystems can exhibit Chua corsage knots. We next show that generic memristors can be used to realize the above Hamiltonian systems. Finally, we show that the internal state of two-element memristor circuits can have the same dynamics as n-dimensional autonomous systems.
16. Ab initio infrared and Raman spectra
NASA Technical Reports Server (NTRS)
Fredkin, D. R.; White, S. R.; Wilson, K. R.; Komornicki, A.
1983-01-01
It is pointed out that with increased computer power and improved computational techniques, such as the gradients developed in recent years, it is becoming practical to compute spectra ab initio, from the fundamental constants of nature, for systems of increasing complexity. The present investigation has the objective to explore several possible ab initio approaches to spectra, giving particular attention to infrared and nonresonance Raman. Two approaches are discussed. The sequential approach, in which first the electronic part and then later the nuclear part of the Born-Oppenheimer approximation is solved, is appropriate for small systems. The simultaneous approach, in which the electronic and nuclear parts are solved at the same time, is more appropriate for many-atom systems. A review of the newer quantum gradient techniques is provided, and the infrared and Raman spectral band contours for the water molecule are computed.
17. Formalisms for Electron Exchange Kinetics in Aqueous Solution, and the Role of Ab Initio Techniques in Their Implementation
SciTech Connect
Newton, M. D.
1980-01-01
Formalisms suitable for calculating the rate of electron exchange between transition metal complexes in aqueous solution are reviewed and implemented in conjunction with ab initio quantum chemical calculations which provide crucial off-diagonal Hamiltonian matrix elements as well as other relevant electronic structural data. Rate constants and activation parameters are calculated for the hex-aquo Fe2 +-Fe3+ system, using a simple activated complex theory, a non-adiabatic semi-classical model which includes nuclear tunnelling, and a more detailed quantum mechanical method based on the Golden Rule. Comparisons are made between calculated results and those obtained by extrapolating experimental data to zero ionic strength. All methods yield similar values for the overall rate constant (∾ 0.1 L/(mol-sec)). The experimental activation parameters (δH and δS) are in somewhat better agreement with the semi classical and quantum mechanical results than with those from the simple activated complex theory, thereby providing some indication that non-adiabaticity and nuclear tunnelling may be important in the Fe2+/3+ exchange reaction. It is concluded that a model based on direct metal-metal overlap can account for the observed reaction kinetics provided the reactants are allowed to approach well within the traditional contact distance of 6.9 Å. 6 figures, 7 tables.
18. A self-consistent and environment-dependent Hamiltonian for large-scale simulations of complex nanostructures
Yu, Ming; Wu, S. Y.; Jayanthi, C. S.
2009-11-01
This review is devoted to the development of a robust semi-empirical Hamiltonian for quantum-mechanics-based simulations. The Hamiltonian referred as self-consistent (SC) and environment-dependent (ED) Hamiltonian is developed in the framework of linear combination of atomic orbitals (LCAO) and includes multi-center electron-ion and electron-electron interactions. Furthermore, the framework allows for a self-consistent treatment of charge-redistributions. The parameterized Hamiltonian matrix elements and overlap functions are obtained by fitting them to accurate first-principles database of properties corresponding to clusters and bulk phases of materials. The total energy includes the band structure energy, the correction term from the double-counting of electrons, and ion-ion repulsions, where the band structure energy is obtained by solving a generalized eigenvalue equation. Linear scaling algorithms for large-scale simulations of materials have also been incorporated. The present approach goes beyond the traditional two-center tight-binding Hamiltonians in terms of its accuracy and transferability and allows the study of system sizes that are beyond the scope of ab-initio simulations. We have studied a wide-variety of complex materials and complex phenomena using the SCED-LCAO MD that include: (i) the structure and stability of bucky-diamond carbon clusters and their phase transformations upon annealing, (ii) the initial stage of growth of single-wall carbon nanotubes (SWCNTs), and (iii) structural and electronic properties of bucky-diamond SiC clusters and SiC nanowires (NWs). The successful outcome of these case studies is a testament to the transferability of the Hamiltonian to different types of atomic environments ( i.e., co-ordinations and bonding configurations).
19. The Use of Ab Initio Wavefunctions in Line-Shape Calculations for Water Vapor
Gamache, Robert R.; Lamouroux, Julien; Schwenke, David W.
2014-06-01
In semi-classical line-shape calculations, the internal motions of the colliding pair are treated via quantum mechanics and the collision trajectory is determined by classical dynamics. The quantum mechanical component, i.e. the determination of reduced matrix elements (RME) for the colliding pair, requires the wavefunctions of the radiating and the perturbing molecules be known. Here the reduced matrix elements for collisions in the ground vibrational state of water vapor are calculated by two methods and compared. First, wavefunctions determined by diagonalizing an effective (Watson) Hamiltonian are used to calculate the RMEs and, second, the ab initio wavefunctions of Partridge and Schwenke are used. While the ground vibrational state will yield the best approximation of the wavefunctions from the effective Hamiltonian approach, this study clearly identifies problems for states not included in the fit of the Hamiltonian and for extrapolated states. RMEs determined using ab initio wavefunctions use ˜100000 times more computational time; however, all ro-vibrational interactions are included. Hence, the ab initio approach will yield better RMEs as the number of vibrational quanta exchanged in the optical transition increases, resulting in improvements in calculated half-widths and line shifts. It is important to note that even for pure rotational transitions the use of ab initio wavefunctions will yield improved results.
20. Ab initio infrared and Raman spectra
Fredkin, Donald R.; Komornicki, Andrew; White, Steven R.; Wilson, Kent R.
1983-06-01
We discuss several ways in which molecular absorption and scattering spectra can be computed ab initio, from the fundamental constants of nature. These methods can be divided into two general categories. In the first, or sequential, type of approach, one first solves the electronic part of the Schrödinger equation in the Born-Oppenheimer approximation, mapping out the potential energy, dipole moment vector (for infrared absorption) and polarizability tensor (for Raman scattering) as functions of nuclear coordinates. Having completed the electronic part of the calculation, one then solves the nuclear part of the problem either classically or quantum mechanically. As an example of the sequential ab initio approach, the infrared and Raman rotational and vibrational-rotational spectral band contours for the water molecule are computed in the simplest rigid rotor, normal mode approximation. Quantum techniques are used to calculate the necessary potential energy, dipole moment, and polarizability information at the equilibrium geometry. A new quick, accurate, and easy to program classical technique involving no reference to Euler angles or special functions is developed to compute the infrared and Raman band contours for any rigid rotor, including asymmetric tops. A second, or simultaneous, type of ab initio approach is suggested for large systems, particularly those for which normal mode analysis is inappropriate, such as liquids, clusters, or floppy molecules. Then the curse of dimensionality prevents mapping out in advance the complete potential, dipole moment, and polarizability functions over the whole space of nuclear positions of all atoms, and a solution in which the electronic and nuclear parts of the Born-Oppenheimer approximation are simultaneously solved is needed. A quantum force classical trajectory (QFCT) molecular dynamic method, based on linear response theory, is described, in which the forces, dipole moment, and polarizability are computed quantum
1. Hierarchical structure of noncanonical Hamiltonian systems
Yoshida, Z.; Morrison, P. J.
2016-02-01
Topological constraints play a key role in the self-organizing processes that create structures in macro systems. In fact, if all possible degrees of freedom are actualized on equal footing without constraint, the state of ‘equipartition’ may bear no specific structure. Fluid turbulence is a typical example—while turbulent mixing seems to increase entropy, a variety of sustained vortical structures can emerge. In Hamiltonian formalism, some topological constraints are represented by Casimir invariants (for example, helicities of a fluid or a plasma), and then, the effective phase space is reduced to the Casimir leaves. However, a general constraint is not necessarily integrable, which precludes the existence of an appropriate Casimir invariant; the circulation is an example of such an invariant. In this work, we formulate a systematic method to embed a Hamiltonian system in an extended phase space; we introduce phantom fields and extend the Poisson algebra. A phantom field defines a new Casimir invariant, a cross helicity, which represents a topological constraint that is not integrable in the original phase space. Changing the perspective, a singularity of the extended system may be viewed as a subsystem on which the phantom fields (though they are actual fields, when viewed from the extended system) vanish, i.e., the original system. This hierarchical relation of degenerate Poisson manifolds enables us to see the ‘interior’ of a singularity as a sub Poisson manifold. The theory can be applied to describe bifurcations and instabilities in a wide class of general Hamiltonian systems (Yoshida and Morrison 2014 Fluid Dyn. Res. 46 031412).
2. Erythema ab igne.
PubMed
Miller, Kristen; Hunt, Raegan; Chu, Julie; Meehan, Shane; Stein, Jennifer
2011-10-15
Erythema ab igne is a reticulated, erythematous or hyperpigmented dermatosis that results from chronic and repeated exposure to low levels of infrared radiation. Multiple heat sources have been reported to cause this condition, which include heated reclining chairs, heating pads, hot water bottles, car heaters, electric space heaters, and, more recently, laptop computers. Treatment consists of withdrawing the inciting heat source. Although erythema ab igne carries a good prognosis, it is not necessarily a self-limited diagnosis as patients are at long-term risk of developing subsequent cutaneous malignant conditions, which include squamous cell and merkel-cell carcinomas.
3. The quantization of the Rabi Hamiltonian
Vandaele, Eva R. J.; Arvanitidis, Athanasios; Ceulemans, Arnout
2017-03-01
The Rabi Hamiltonian addresses the proverbial paradigmatic case of a two-level fermionic system coupled to a single bosonic mode. It is expressed by a system of two coupled first-order differential equations in the complex field, which may be rewritten in a canonical form under the Birkhoff transformation. The transformation gives rise to leapfrog recurrence relations, from which the eigenvalues and eigenvectors could be obtained. The interesting feature of this approach is that it generates integer quantum numbers, which rationalize the spectrum by relating the solutions to the Juddian baselines. The relationship with Braak’s integrability claim (Braak 2011 Phys. Rev. Lett. 107 100401) is discussed.
4. Quantum Hamiltonian Identification from Measurement Time Traces
Zhang, Jun; Sarovar, Mohan
2014-08-01
Precise identification of parameters governing quantum processes is a critical task for quantum information and communication technologies. In this Letter, we consider a setting where system evolution is determined by a parametrized Hamiltonian, and the task is to estimate these parameters from temporal records of a restricted set of system observables (time traces). Based on the notion of system realization from linear systems theory, we develop a constructive algorithm that provides estimates of the unknown parameters directly from these time traces. We illustrate the algorithm and its robustness to measurement noise by applying it to a one-dimensional spin chain model with variable couplings.
5. Hamiltonian analysis of BHT massive gravity
Blagojević, M.; Cvetković, B.
2011-01-01
We study the Hamiltonian structure of the Bergshoeff-Hohm-Townsend (BHT) massive gravity with a cosmological constant. In the space of coupling constants ( Λ 0, m 2), our canonical analysis reveals the special role of the condition Λ 0/ m 2 ≠ -1. In this sector, the dimension of the physical phase space is found to be N ∗ = 4, which corresponds to two Lagrangian degree of freedom. When applied to the AdS asymptotic region, the canonical approach yields the conserved charges of the BTZ black hole, and central charges of the asymptotic symmetry algebra.
6. Forced Oscillations of Nonlinear Hamiltonian Systems, II.
DTIC Science & Technology
1979-12-01
Rabinowitz (J8], 9]). The author obtained similar results ([6]), by using a variational method devised b.: F. Clarke and himself for convex subquadratic...and satisfying, for some constants bl > a’ > 0 and 5 > 2: (39) a’ ixi-21Y 2 _< (V"(x)yy) < b’ lxiy-21yj 2, all x ev , y c ip Then for any T > 0, there is...34Linear operators", Wiley. [6] I. Ekeland, "Periodic Hamiltonian trajectories and a theorem of P. Rabinowitz ", 1978, to appear in Journal of Differential
7. Exact two-component Hamiltonians revisited.
PubMed
Liu, Wenjian; Peng, Daoling
2009-07-21
Two routes for deriving the exact two-component Hamiltonians are compared. In the first case, as already known, we start directly from the matrix representation of the Dirac operator in a restricted kinetically balanced basis and make a single block diagonalization. In the second case, not considered before, we start instead from the Foldy-Wouthuysen operator and make proper use of resolutions of the identity. The expressions are surprisingly different. It turns out that a mistake was made in the former formulation when going from the Dirac to the Schrodinger picture. The two formulations become equivalent after the mistake is corrected.
8. Exact two-component Hamiltonians revisited
SciTech Connect
Liu Wenjian; Peng Daoling
2009-07-21
Two routes for deriving the exact two-component Hamiltonians are compared. In the first case, as already known, we start directly from the matrix representation of the Dirac operator in a restricted kinetically balanced basis and make a single block diagonalization. In the second case, not considered before, we start instead from the Foldy-Wouthuysen operator and make proper use of resolutions of the identity. The expressions are surprisingly different. It turns out that a mistake was made in the former formulation when going from the Dirac to the Schroedinger picture. The two formulations become equivalent after the mistake is corrected.
9. Hamiltonian Description of Convective-cell Generation
SciTech Connect
J.A. Krommes and R.A. Kolesnikov
2004-03-11
The nonlinear statistical growth rate eq for convective cells driven by drift-wave (DW) interactions is studied with the aid of a covariant Hamiltonian formalism for the gyrofluid nonlinearities. A statistical energy theorem is proven that relates eq to a second functional tensor derivative of the DW energy. This generalizes to a wide class of systems of coupled partial differential equations a previous result for scalar dynamics. Applications to (i) electrostatic ion-temperature-gradient-driven modes at small ion temperature, and (ii) weakly electromagnetic collisional DW's are noted.
10. Perturbation Theory for Parent Hamiltonians of Matrix Product States
Szehr, Oleg; Wolf, Michael M.
2015-05-01
This article investigates the stability of the ground state subspace of a canonical parent Hamiltonian of a Matrix product state against local perturbations. We prove that the spectral gap of such a Hamiltonian remains stable under weak local perturbations even in the thermodynamic limit, where the entire perturbation might not be bounded. Our discussion is based on preceding work by Yarotsky that develops a perturbation theory for relatively bounded quantum perturbations of classical Hamiltonians. We exploit a renormalization procedure, which on large scale transforms the parent Hamiltonian of a Matrix product state into a classical Hamiltonian plus some perturbation. We can thus extend Yarotsky's results to provide a perturbation theory for parent Hamiltonians of Matrix product states and recover some of the findings of the independent contributions (Cirac et al in Phys Rev B 8(11):115108, 2013) and (Michalakis and Pytel in Comm Math Phys 322(2):277-302, 2013).
11. Wigner quantization of some one-dimensional Hamiltonians
SciTech Connect
Regniers, G.; Van der Jeugt, J.
2010-12-15
Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H=xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H{sub f}=p{sup 2}/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2).
12. Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians
SciTech Connect
Buljubasich, Lisandro; Dente, Axel D.; Levstein, Patricia R.; Chattah, Ana K.; Pastawski, Horacio M.; Sánchez, Claudia M.
2015-10-28
We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.
13. Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians
Buljubasich, Lisandro; Sánchez, Claudia M.; Dente, Axel D.; Levstein, Patricia R.; Chattah, Ana K.; Pastawski, Horacio M.
2015-10-01
We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.
14. Experimental quantification of decoherence via the Loschmidt echo in a many spin system with scaled dipolar Hamiltonians.
PubMed
Buljubasich, Lisandro; Sánchez, Claudia M; Dente, Axel D; Levstein, Patricia R; Chattah, Ana K; Pastawski, Horacio M
2015-10-28
We performed Loschmidt echo nuclear magnetic resonance experiments to study decoherence under a scaled dipolar Hamiltonian by means of a symmetrical time-reversal pulse sequence denominated Proportionally Refocused Loschmidt (PRL) echo. The many-spin system represented by the protons in polycrystalline adamantane evolves through two steps of evolution characterized by the secular part of the dipolar Hamiltonian, scaled down with a factor |k| and opposite signs. The scaling factor can be varied continuously from 0 to 1/2, giving access to a range of complexity in the dynamics. The experimental results for the Loschmidt echoes showed a spreading of the decay rates that correlate directly to the scaling factors |k|, giving evidence that the decoherence is partially governed by the coherent dynamics. The average Hamiltonian theory was applied to give an insight into the spin dynamics during the pulse sequence. The calculations were performed for every single radio frequency block in contrast to the most widely used form. The first order of the average Hamiltonian numerically computed for an 8-spin system showed decay rates that progressively decrease as the secular dipolar Hamiltonian becomes weaker. Notably, the first order Hamiltonian term neglected by conventional calculations yielded an explanation for the ordering of the experimental decoherence rates. However, there is a strong overall decoherence observed in the experiments which is not reflected by the theoretical results. The fact that the non-inverted terms do not account for this effect is a challenging topic. A number of experiments to further explore the relation of the complete Hamiltonian with this dominant decoherence rate are proposed.
15. Nucleus-Dependent Valence-Space Approach to Nuclear Structure
Stroberg, S. R.; Calci, A.; Hergert, H.; Holt, J. D.; Bogner, S. K.; Roth, R.; Schwenk, A.
2017-01-01
We present a nucleus-dependent valence-space approach for calculating ground and excited states of nuclei, which generalizes the shell-model in-medium similarity renormalization group to an ensemble reference with fractionally filled orbitals. Because the ensemble is used only as a reference, and not to represent physical states, no symmetry restoration is required. This allows us to capture three-nucleon (3 N ) forces among valence nucleons with a valence-space Hamiltonian specifically targeted to each nucleus of interest. Predicted ground-state energies from carbon through nickel agree with results of other large-space ab initio methods, generally to the 1% level. In addition, we show that this new approach is required in order to obtain convergence for nuclei in the upper p and s d shells. Finally, we address the 1+/3+ inversion problem in 22Na and 46V. This approach extends the reach of ab initio nuclear structure calculations to essentially all light- and medium-mass nuclei.
16. Norm-square localization for Hamiltonian LG-spaces
Loizides, Yiannis
2017-04-01
We prove a formula for twisted Duistermaat-Heckman distributions associated to a Hamiltonian LG-space. The terms of the formula are localized at the critical points of the norm-square of the moment map, and can be computed in cross-sections. Our main tools are the theory of quasi-Hamiltonian G-spaces, as well as the Hamiltonian cobordism approach to norm-square localization introduced recently by Harada and Karshon.
17. Hamiltonian and non-Hamiltonian perturbation theory for nearly periodic motion
1986-02-01
Kruskal's asymptotic theory of nearly period motion [M. Kruskal, J. Math. Phys. 4, 806 (1962)] (with applications to nonlinear oscillators, guiding center motion, etc.) is generalized and modified. A new more natural recursive formula, with considerable advantages in applications, determining the averaging transformations and the drift equations is derived. Also almost quasiperiodic motion is considered. For a Hamiltonian system, a manifestly Hamiltonian extension of Kruskal's theory is given by means of the phase-space Lagrangian formulation of Hamiltonian mechanics. By performing an averaging transformation on the phase-space Lagrangian for the system (L → L¯) and adding a total derivative dS/dτ, a nonoscillatory Lagrangian Λ=L¯+dS/dτ is obtained. The drift equations and the adiabatic invariant are now obtained from Λ. By truncating Λ to some finite order in the small parameter ɛ, manifestly Hamiltonian approximating systems are obtained. The utility of the method for treating the guiding-center motion is demonstrated in a separate paper.
18. Ab initio derivation of model energy density functionals
Dobaczewski, Jacek
2016-08-01
I propose a simple and manageable method that allows for deriving coupling constants of model energy density functionals (EDFs) directly from ab initio calculations performed for finite fermion systems. A proof-of-principle application allows for linking properties of finite nuclei, determined by using the nuclear nonlocal Gogny functional, to the coupling constants of the quasilocal Skyrme functional. The method does not rely on properties of infinite fermion systems but on the ab initio calculations in finite systems. It also allows for quantifying merits of different model EDFs in describing the ab initio results.
19. Hamiltonian description of closed configurations of the vacuum magnetic field
SciTech Connect
Skovoroda, A. A.
2015-05-15
Methods of obtaining and using the Hamiltonians of closed vacuum magnetic configurations of fusion research systems are reviewed. Various approaches to calculate the flux functions determining the Hamiltonian are discussed. It is shown that the Hamiltonian description allows one not only to reproduce all traditional results, but also to study the behavior of magnetic field lines by using the theory of dynamic systems. The potentialities of the Hamiltonian formalism and its close relation to traditional methods are demonstrated using a large number of classical examples adopted from the fundamental works by A.I. Morozov, L.S. Solov’ev, and V.D. Shafranov.
20. Entanglement Hamiltonians for Chiral Fermions with Zero Modes
Klich, Israel; Vaman, Diana; Wong, Gabriel
2017-09-01
In this Letter, we study the effect of topological zero modes on entanglement Hamiltonians and the entropy of free chiral fermions in (1 +1 )D . We show how Riemann-Hilbert solutions combined with finite rank perturbation theory allow us to obtain exact expressions for entanglement Hamiltonians. In the absence of the zero mode, the resulting entanglement Hamiltonians consist of local and bilocal terms. In the periodic sector, the presence of a zero mode leads to an additional nonlocal contribution to the entanglement Hamiltonian. We derive an exact expression for this term and for the resulting change in the entanglement entropy.
1. Position-dependent mass quantum Hamiltonians: general approach and duality
Rego-Monteiro, M. A.; Rodrigues, Ligia M. C. S.; Curado, E. M. F.
2016-03-01
We analyze a general family of position-dependent mass (PDM) quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a general family of self-adjoint Hamiltonians which are quantum mechanically equivalent to the non-self-adjoint proposed ones. Inspired by the probability density of the problem, we construct an ansatz for the solutions of the family of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the time independent Schrödinger equations generated by the non-self-adjoint Hamiltonians into the Hilbert space of the solutions of the respective dual self-adjoint Hamiltonians. This mapping depends on both the PDM and on a function of position satisfying a condition that assures the existence of a consistent continuity equation. We identify the non-self-adjoint Hamiltonians here studied with a very general family of Hamiltonians proposed in a seminal article of Harrison (1961 Phys. Rev. 123 85) to describe varying band structures in different types of metals. Therefore, we have self-adjoint Hamiltonians that correspond to the non-self-adjoint ones found in Harrison’s article.
2. Fractional Hamiltonian monodromy from a Gauss-Manin monodromy
SciTech Connect
Sugny, D.; Jauslin, H. R.; Mardesic, P.; Pelletier, M.; Jebrane, A.
2008-04-15
Fractional Hamiltonian monodromy is a generalization of the notion of Hamiltonian monodromy, recently introduced by [Nekhoroshev, Sadovskii, and Zhilinskii, C. R. Acad. Sci. Paris, Ser. 1 335, 985 (2002); and Ann. Henri Poincare 7, 1099 (2006)] for energy-momentum maps whose image has a particular type of nonisolated singularities. In this paper, we analyze the notion of fractional Hamiltonian monodromy in terms of the Gauss-Manin monodromy of a Riemann surface constructed from the energy-momentum map and associated with a loop in complex space which bypasses the line of singularities. We also prove some propositions on fractional Hamiltonian monodromy for 1:-n and m:-n resonant systems.
3. How is Lorentz invariance encoded in the Hamiltonian?
Kajuri, Nirmalya
2016-07-01
One of the disadvantages of the Hamiltonian formulation is that Lorentz invariance is not manifest in the former. Given a Hamiltonian, there is no simple way to check whether it is relativistic or not. One would either have to solve for the equations of motion or calculate the Poisson brackets of the Noether charges to perform such a check. In this paper we show that, for a class of Hamiltonians, it is possible to check Lorentz invariance directly from the Hamiltonian. Our work is particularly useful for theories where the other methods may not be readily available.
4. Hamiltonian theory of nonlinear waves in planetary rings
NASA Technical Reports Server (NTRS)
Stewart, G. R.
1987-01-01
The derivation of a Hamiltonian field theory for nonlinear density waves in Saturn's rings is discussed. Starting with a Hamiltonian for a discrete system of gravitating streamlines, an averaged Hamiltonian is obtained by successive applications of Lie transforms. The transformation may be carried out to any desired order in q, where q is the nonlinearity parameter defined in the work of Shu, et al (1985) and Borderies et al (1985). Subsequent application of the Wentzel-Kramer-Brillouin Method approximation yields an asymptotic field Hamiltonian. Both the nonlinear dispersion relation and the wave action transport equation are easily derived from the corresponding Lagrangian by the standard variational principle.
5. Action with Acceleration i: Euclidean Hamiltonian and Path Integral
Baaquie, Belal E.
2013-10-01
An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.
6. Dressed qubits in nuclear spin baths
SciTech Connect
Wu Lianao
2010-04-15
We present a method to encode a dressed qubit into the product state of an electron spin localized in a quantum dot and its surrounding nuclear spins via a dressing transformation. In this scheme, the hyperfine coupling and a portion of a nuclear dipole-dipole interaction become logic gates, while they are the sources of decoherence in electron-spin qubit proposals. We discuss errors and corrections for the dressed qubits. Interestingly, the effective Hamiltonian of nuclear spins is equivalent to a pairing Hamiltonian, which provides the microscopic mechanism to protect dressed qubits against decoherence.
7. Classical Hamiltonian structures in wave packet dynamics
Gray, Stephen K.; Verosky, John M.
1994-04-01
The general, N state matrix representation of the time-dependent Schrödinger equation is equivalent to an N degree of freedom classical Hamiltonian system. We describe how classical mechanical methods and ideas can be applied towards understanding and modeling exact quantum dynamics. Two applications are presented. First, we illustrate how qualitative insights may be gained by treating the two state problem with a time-dependent coupling. In the case of periodic coupling, Poincaré surfaces of section are used to view the quantum dynamics, and features such as the Floquet modes take on interesting interpretations. The second application illustrates computational implications by showing how Liouville's theorem, or more generally the symplectic nature of classical Hamiltonian dynamics, provides a new perspective for carrying out numerical wave packet propagation. We show how certain simple and explicit symplectic integrators can be used to numerically propagate wave packets. The approach is illustrated with an application to the problem of a diatomic molecule interacting with a laser, although it and related approaches may be useful for describing a variety of problems.
8. Strong coupling expansions for nonintegrable hamiltonian systems
SciTech Connect
Abarbanel, Henry D. I.; Crawford, John David
1982-09-01
In this paper, we present a method for studying nonintegrable Hamiltonian systems H(I,θ) = H0(I) + kH1(I,θ) (I, θ are action-angle variables) in the regime of large k. Our central tool is the conditional probability P(I,θ,t | I00,t0) that the system is at I. θ at time t given that it resided at I0, θ0 at t0. An integral representation is given for this conditional probability. By discretizing the Hamiltonian equations of motion in small time steps, ϵ, we arrive at a phase volume-preserving mapping which replaces the actual flow. When the motion on the energy surface E = H(I,θ) is bounded we are able to evaluate physical quantities of interest for large k and fixed ϵ. We also discuss the representation of P (I,θ,t | I00t0) when an external random forcing is added in order to smooth the singular functions associated with the deterministic flow. Explicit calculations of a “diffusion” coefficient are given for a non-integrable system with two degrees of freedom. Finally, the limit ϵ → 0, which returns us to the actual flow, is subtle and is discussed.
9. Thermalization Time Bounds for Pauli Stabilizer Hamiltonians
Temme, Kristan
2017-03-01
We prove a general lower bound to the spectral gap of the Davies generator for Hamiltonians that can be written as the sum of commuting Pauli operators. These Hamiltonians, defined on the Hilbert space of N-qubits, serve as one of the most frequently considered candidates for a self-correcting quantum memory. A spectral gap bound on the Davies generator establishes an upper limit on the life time of such a quantum memory and can be used to estimate the time until the system relaxes to thermal equilibrium when brought into contact with a thermal heat bath. The bound can be shown to behave as {λ ≥ O(N^{-1} exp(-2β overline{ɛ}))}, where {overline{ɛ}} is a generalization of the well known energy barrier for logical operators. Particularly in the low temperature regime we expect this bound to provide the correct asymptotic scaling of the gap with the system size up to a factor of N -1. Furthermore, we discuss conditions and provide scenarios where this factor can be removed and a constant lower bound can be proven.
10. A Hamiltonian Five-Field Gyrofluid Model
Keramidas Charidakos, Ioannis; Waelbroeck, Francois; Morrison, Philip
2015-11-01
Reduced fluid models constitute versatile tools for the study of multi-scale phenomena. Examples include magnetic islands, edge localized modes, resonant magnetic perturbations, and fishbone and Alfven modes. Gyrofluid models improve over Braginskii-type models by accounting for the nonlocal response due to particle orbits. A desirable property for all models is that they not only have a conserved energy, but also that they be Hamiltonian in the ideal limit. Here, a Lie-Poisson bracket is presented for a five-field gyrofluid model, thereby showing the model to be Hamiltonian. The model includes the effects of magnetic field curvature and describes the evolution of electron and ion densities, the parallel component of ion and electron velocities and ion temperature. Quasineutrality and Ampere's law determine respectively the electrostatic potential and magnetic flux. The Casimir invariants are presented, and shown to be associated to five Lagrangian invariants advected by distinct velocity fields. A linear, local study of the model is conducted both with and without Landau and diamagnetic resonant damping terms. Stability criteria and dispersion relations for the electrostatic and the electromagnetic cases are derived and compared with their analogs for fluid and kinetic models. This work was funded by U.S. DOE Contract No. DE-FG02-04ER-54742.
11. AB initio infrared and Raman spectra
Fredkin, D. R.; Komornicki, A.; White, S. R.; Wilson, K. R.
1982-08-01
We discuss several ways in which molecular absorption and scattering spectra can be computed ab initio, from the fundamental constants of nature. These methods can be divided into two general categories. In the first, or sequential, type of approach, one first solves the electronic part of the Schroedinger equation in the Born-Oppenheimer approximation, mapping out the potential energy, dipole moment vector (for infrared absorption) and polarizability tensor (for Raman scattering) as functions of nuclear coordinates. Having completed the electronic part of the calculation, one then solves the nuclear part of the problem either classically or quantum mechanically. As an example of the sequential ab initio approach, the infrared and Raman rotational and vibrational-rotational spectral band contours for the water molecule are computed in the simplest rigid rotor, normal mode approximation. Quantum techniques, are used to calculate the necessary potential energy, dipole moment, and polarizability information at the equilibrium geometry. A new quick, accurate, and easy to program classical technique involving no reference to Euler angles or special functions is developed to compute the infrared and Raman angles or special functions is developed to compute the infrared and Raman band contours for any rigid rotor, including asymmetric tops. A second, or simultaneous, type of ab initio approach is suggested for large systems, particularly those for which normal mode analysis is inappropriate, such as liquids, clusters, or floppy molecules.
12. Multiple time step integrators in ab initio molecular dynamics
SciTech Connect
Luehr, Nathan; Martínez, Todd J.; Markland, Thomas E.
2014-02-28
Multiple time-scale algorithms exploit the natural separation of time-scales in chemical systems to greatly accelerate the efficiency of molecular dynamics simulations. Although the utility of these methods in systems where the interactions are described by empirical potentials is now well established, their application to ab initio molecular dynamics calculations has been limited by difficulties associated with splitting the ab initio potential into fast and slowly varying components. Here we present two schemes that enable efficient time-scale separation in ab initio calculations: one based on fragment decomposition and the other on range separation of the Coulomb operator in the electronic Hamiltonian. We demonstrate for both water clusters and a solvated hydroxide ion that multiple time-scale molecular dynamics allows for outer time steps of 2.5 fs, which are as large as those obtained when such schemes are applied to empirical potentials, while still allowing for bonds to be broken and reformed throughout the dynamics. This permits computational speedups of up to 4.4x, compared to standard Born-Oppenheimer ab initio molecular dynamics with a 0.5 fs time step, while maintaining the same energy conservation and accuracy.
13. Vibrational analysis of HOCl up to 98{percent} of the dissociation energy with a Fermi resonance Hamiltonian
SciTech Connect
Jost, R.; Joyeux, M.; Skokov, S.; Bowman, J.
1999-10-01
We have analyzed the vibrational energies and wave functions of HOCl obtained from previous {ital ab initio} calculations [J. Chem. Phys. {bold 109}, 2662 (1998); {bold 109}, 10273 (1998)]. Up to approximately 13&hthinsp;000 cm{sup {minus}1}, the normal modes are nearly decoupled, so that the analysis is straightforward with a Dunham model. In contrast, above 13&hthinsp;000 cm{sup {minus}1} the Dunham model is no longer valid for the levels with no quanta in the OH stretch (v{sub 1}=0). In addition to v{sub 1}, these levels can only be assigned a so-called polyad quantum number P=2v{sub 2}+v{sub 3}, where 2 and 3 denote, respectively, the bending and OCl stretching normal modes. In contrast, the levels with v{sub 1}{ge}2 remain assignable with three v{sub i} quantum numbers up to the dissociation (D{sub 0}=19&hthinsp;290&hthinsp;cm{sup {minus}1}). The interaction between the bending and the OCl stretch ({omega}{sub 2}{congruent}2{omega}{sub 3}) is well described with a simple, fitted Fermi resonance Hamiltonian. The energies and wave functions of this model Hamiltonian are compared with those obtained from {ital ab initio} calculations, which in turn enables the assignment of many additional {ital ab initio} vibrational levels. Globally, among the 809 bound levels calculated below dissociation, 790 have been assigned, the lowest unassigned level, No. 736, being located at 18&hthinsp;885 cm{sup {minus}1} above the (0,0,0) ground level, that is, at about 98{percent} of D{sub 0}. In addition, 84 {open_quotes}resonances{close_quotes} located above D{sub 0} have also been assigned. Our best Fermi resonance Hamiltonian has 29 parameters fitted with 725 {ital ab initio} levels, the rms deviation being of 5.3 cm{sup {minus}1}. This set of 725 fitted levels includes the full set of levels up to No. 702 at 18&hthinsp;650 cm{sup {minus}1}. The {ital ab initio} levels, which are assigned but not included in the fit, are reasonably predicted by the model Hamiltonian, but with a
14. Spectral Properties of Fractional Quantum Hall Hamiltonians
Weerasinghe, Amila
The fractional quantum Hall (FQH) effect plays a prominent role in the study of topological phases of matter and of strongly correlated electron systems in general. FQH systems have been demonstrated to show many interesting novel properties such as fractional charges, and are believed to harbor even more intriguing phenomena such as fractional statistics. However, there remain many interesting questions to be addressed in this regime. The work reported in this thesis aims to push the envelope of our understanding of the low-energy properties of FQH states using microscopic principles. In the first part of the thesis, we present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the "one-dimensional" thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case. In the second part of the thesis, we study the low energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of FQH states. Starting from the observation that positive many-body Hamiltonians must have ground state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with "center-of-mass conservation" symmetry, both in the presence and absence of particle
15. New relativistic Hamiltonian: the angular magnetoelectric coupling
Paillard, Charles; Mondal, Ritwik; Berritta, Marco; Dkhil, Brahim; Singh, Surendra; Oppeneer, Peter M.; Bellaiche, Laurent
2016-10-01
Spin-Orbit Coupling (SOC) is a ubiquitous phenomenon in the spintronics area, as it plays a major role in allowing for enhancing many well-known phenomena, such as the Dzyaloshinskii-Moriya interaction, magnetocrystalline anisotropy, the Rashba effect, etc. However, the usual expression of the SOC interaction ħ/4m2c2 [E×p] • σ (1) where p is the momentum operator, E the electric field, σ the vector of Pauli matrices, breaks the gauge invariance required by the electronic Hamiltonian. On the other hand, very recently, a new phenomenological interaction, coupling the angular momentum of light and magnetic moments, has been proposed based on symmetry arguments: ξ/2 [r × (E × B)] M, (2) with M the magnetization, r the position, and ξ the interaction strength constant. This interaction has been demonstrated to contribute and/or give rise, in a straightforward way, to various magnetoelectric phenomena,such as the anomalous Hall effect (AHE), the anisotropic magnetoresistance (AMR), the planar Hall effect and Rashba-like effects, or the spin-current model in multiferroics. This last model is known to be the origin of the cycloidal spin arrangement in bismuth ferrite for instance. However, the coupling of the angular momentum of light with magnetic moments lacked a fundamental theoretical basis. Starting from the Dirac equation, we derive a relativistic interaction Hamiltonian which linearly couples the angular momentum density of the electromagnetic (EM) field and the electrons spin. We name this coupling the Angular MagnetoElectric (AME) coupling. We show that in the limit of uniform magnetic field, the AME coupling yields an interaction exactly of the form of Eq. (2), thereby giving a firm theoretical basis to earlier works. The AME coupling can be expressed as: ξ [E × A] • σ (3) with A being the vector potential. Interestingly, the AME coupling was shown to be complementary to the traditional SOC, and together they restore the gauge invariance of the
16. On the physical applications of hyper-Hamiltonian dynamics
Gaeta, Giuseppe; Rodríguez, Miguel A.
2008-05-01
An extension of Hamiltonian dynamics, defined on hyper-Kahler manifolds ('hyper-Hamiltonian dynamics') and sharing many of the attractive features of standard Hamiltonian dynamics, was introduced in previous work. In this paper, we discuss applications of the theory to physically interesting cases, dealing with the dynamics of particles with spin 1/2 in a magnetic field, i.e. the Pauli and the Dirac equations. While the free Pauli equation corresponds to a hyper-Hamiltonian flow, it turns out that the hyper-Hamiltonian description of the Dirac equation, and of the full Pauli one, is in terms of two commuting hyper-Hamiltonian flows. In this framework one can use a factorization principle discussed here (which is a special case of a general phenomenon studied by Walcher) and provide an explicit description of the resulting flow. On the other hand, by applying the familiar Foldy-Wouthuysen and Cini-Tousheck transformations (and the one recently introduced by Mulligan) which separate—in suitable limits—the Dirac equation into two equations, each of these turn out to be described by a single hyper-Hamiltonian flow. Thus the hyper-Hamiltonian construction is able to describe the fundamental dynamics for particles with spin.
17. Higher-order Hamiltonian fluid reduction of Vlasov equation
SciTech Connect
Perin, M.; Chandre, C.; Morrison, P.J.; Tassi, E.
2014-09-15
From the Hamiltonian structure of the Vlasov equation, we build a Hamiltonian model for the first three moments of the Vlasov distribution function, namely, the density, the momentum density and the specific internal energy. We derive the Poisson bracket of this model from the Poisson bracket of the Vlasov equation, and we discuss the associated Casimir invariants.
18. Boson Hamiltonians and stochasticity for the vorticity equation
NASA Technical Reports Server (NTRS)
Shen, Hubert H.
1990-01-01
The evolution of the vorticity in time for two-dimensional inviscid flow and in Lagrangian time for three-dimensional viscous flow is written in Hamiltonian form by introducing Bose operators. The addition of the viscous and convective terms, respectively, leads to an interpretation of the Hamiltonian contribution to the evolution as Langevin noise.
19. A HAMILTONIAN FORMULATION FOR SPIRAL-SECTOR ACCELERATORS.
SciTech Connect
BERG,J.S.
2007-11-05
I develop a formulation for Hamiltonian dynamics in an accelerator with magnets whose edges follow a spiral. I demonstrate using this Hamiltonian that a spiral FFAG can be made perfectly 'scaling'. I examine the effect of tilting an RF cavity with respect a radial line from the center of the machine, potentially with a different angle than the spiral of the magnets.
20. Non-self-adjoint hamiltonians defined by Riesz bases
SciTech Connect
Bagarello, F.; Inoue, A.; Trapani, C.
2014-03-15
We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, we give conditions under which these Hamiltonians can be factorized in terms of generalized lowering and raising operators.
1. Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings
de León, Manuel; Jiménez, Fernando; Martín de Diego, David
2012-05-01
The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to the geometrical integration of Hamiltonian systems are obtained.
2. Hamiltonian structures for the Ostrovsky-Vakhnenko equation
Brunelli, J. C.; Sakovich, S.
2013-01-01
We obtain a bi-Hamiltonian formulation for the Ostrovsky-Vakhnenko (OV) equation using its higher order symmetry and a new transformation to the Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Central to this derivation is the relation between Hamiltonian structures when dependent and independent variables are transformed.
3. Friction in a Model of Hamiltonian Dynamics
Fröhlich, Jürg; Gang, Zhou; Soffer, Avy
2012-10-01
We study the motion of a heavy tracer particle weakly coupled to a dense ideal Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations describing a process of emission of Cerenkov radiation of sound waves into the Bose-Einstein condensate along the particle's trajectory. The emission of Cerenkov radiation results in a friction force with memory acting on the tracer particle and causing it to decelerate until it comes to rest. "A moving body will come to rest as soon as the force pushing it no longer acts on it in the manner necessary for its propulsion."—— Aristotle
4. Nonperturbative light-front Hamiltonian methods
Hiller, J. R.
2016-09-01
We examine the current state-of-the-art in nonperturbative calculations done with Hamiltonians constructed in light-front quantization of various field theories. The language of light-front quantization is introduced, and important (numerical) techniques, such as Pauli-Villars regularization, discrete light-cone quantization, basis light-front quantization, the light-front coupled-cluster method, the renormalization group procedure for effective particles, sector-dependent renormalization, and the Lanczos diagonalization method, are surveyed. Specific applications are discussed for quenched scalar Yukawa theory, ϕ4 theory, ordinary Yukawa theory, supersymmetric Yang-Mills theory, quantum electrodynamics, and quantum chromodynamics. The content should serve as an introduction to these methods for anyone interested in doing such calculations and as a rallying point for those who wish to solve quantum chromodynamics in terms of wave functions rather than random samplings of Euclidean field configurations.
5. A Hamiltonian five-field gyrofluid model
SciTech Connect
Keramidas Charidakos, I.; Waelbroeck, F. L.; Morrison, P. J.
2015-11-15
A Lie-Poisson bracket is presented for a five-field gyrofluid model, thereby showing the model to be Hamiltonian. The model includes the effects of magnetic field curvature and describes the evolution of the electron and ion gyro-center densities, the parallel component of the ion and electron velocities, and the ion temperature. The quasineutrality property and Ampère's law determine, respectively, the electrostatic potential and magnetic flux. The Casimir invariants are presented, and shown to be associated with five Lagrangian invariants advected by distinct velocity fields. A linear, local study of the model is conducted both with and without Landau and diamagnetic resonant damping terms. Stability criteria and dispersion relations for the electrostatic and the electromagnetic cases are derived and compared with their analogs for fluid and kinetic models.
6. Hamiltonian formalism and path entropy maximization
Davis, Sergio; González, Diego
2015-10-01
Maximization of the path information entropy is a clear prescription for constructing models in non-equilibrium statistical mechanics. Here it is shown that, following this prescription under the assumption of arbitrary instantaneous constraints on position and velocity, a Lagrangian emerges which determines the most probable trajectory. Deviations from the probability maximum can be consistently described as slices in time by a Hamiltonian, according to a nonlinear Langevin equation and its associated Fokker-Planck equation. The connections unveiled between the maximization of path entropy and the Langevin/Fokker-Planck equations imply that missing information about the phase space coordinate never decreases in time, a purely information-theoretical version of the second law of thermodynamics. All of these results are independent of any physical assumptions, and thus valid for any generalized coordinate as a function of time, or any other parameter. This reinforces the view that the second law is a fundamental property of plausible inference.
7. Hamiltonian formulations and symmetries in rod mechanics
SciTech Connect
Dichmann, D.J.; Li, Yiwei; Maddocks, J.H.
1996-12-31
This article provides a survey of contemporary rod mechanics, including both dynamic and static theories. Much of what we discuss is regarded as classic material within the mechanics community, but the objective here is to provide a self-contained account accessible to workers interested in modelling DNA. We also describe a number of recent results and computations involving rod mechanics that have been obtained by our group at the University of Maryland. This work was largely motivated by applications to modelling DNA, but our approach reflects a background of research in continuum mechanics. In particular, we emphasize the role that Hamiltonian formulations and symmetries play in the effective computation of special solutions, conservation laws of dynamics and integrals of statics. 41 refs., 10 figs.
8. Hamiltonian inclusive fitness: a fitter fitness concept
PubMed Central
Costa, James T.
2013-01-01
In 1963–1964 W. D. Hamilton introduced the concept of inclusive fitness, the only significant elaboration of Darwinian fitness since the nineteenth century. I discuss the origin of the modern fitness concept, providing context for Hamilton's discovery of inclusive fitness in relation to the puzzle of altruism. While fitness conceptually originates with Darwin, the term itself stems from Spencer and crystallized quantitatively in the early twentieth century. Hamiltonian inclusive fitness, with Price's reformulation, provided the solution to Darwin's ‘special difficulty’—the evolution of caste polymorphism and sterility in social insects. Hamilton further explored the roles of inclusive fitness and reciprocation to tackle Darwin's other difficulty, the evolution of human altruism. The heuristically powerful inclusive fitness concept ramified over the past 50 years: the number and diversity of ‘offspring ideas’ that it has engendered render it a fitter fitness concept, one that Darwin would have appreciated. PMID:24132089
9. Using Hamiltonian control to desynchronize Kuramoto oscillators
Gjata, Oltiana; Asllani, Malbor; Barletti, Luigi; Carletti, Timoteo
2017-02-01
Many coordination phenomena are based on a synchronization process, whose global behavior emerges from the interactions among the individual parts. Often in nature, such self-organized mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are, however, cases where synchronization acts against the stability of the system; for instance in some neurodegenerative diseases or epilepsy or the famous case of Millennium Bridge where the crowd synchronization of the pedestrians seriously endangered the stability of the structure. In this paper we propose an innovative control method to tackle the synchronization process based on the application of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronization. We present our results on a generalized class of the paradigmatic Kuramoto model.
10. Discrete variable representation for singular Hamiltonians
Schneider, Barry I.; Nygaard, Nicolai
2004-11-01
We discuss the application of the discrete variable representation (DVR) to Schrödinger problems which involve singular Hamiltonians. Unlike recent authors who invoke transformations to rid the eigenvalue equation of singularities at the cost of added complexity, we show that an approach based solely on an orthogonal polynomial basis is adequate, provided the Gauss-Lobatto or Gauss-Radau quadrature rule is used. This ensures that the mesh contains the singular points and by simply discarding the DVR functions corresponding to those points, all matrix elements become well behaved, the boundary conditions are satisfied, and the calculation is rapidly convergent. The accuracy of the method is demonstrated by applying it to the hydrogen atom. We emphasize that the method is equally capable of describing bound states and continuum solutions.
11. Fourier series expansion for nonlinear Hamiltonian oscillators.
PubMed
Méndez, Vicenç; Sans, Cristina; Campos, Daniel; Llopis, Isaac
2010-06-01
The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relationship. We compare our results with other recent approaches such as variational methods or heuristic approximations, in particular the Ren-He's method. Based on its application to the Duffing oscillator, the nonlinear pendulum and the eardrum equation, it is shown that the Fourier series expansion method is the most accurate.
12. Finite Hamiltonian Systems: Linear Transformations and Aberrations
Wolf, Kurt Bernardo
2008-11-01
In finite Hamiltonian systems, the operators of position, momentum, and energy have a finite number N of eigenvalues. These operators can be naturally realized as generators of the Lie algebra su(2), in a representation of spin j, of dimension N = 2j+1. Time evolution is rotation of a phase space sphere, whose projections perform the harmonic motion of an oscillator. The (centrally extended) group of rigid—linear—motions of this phase space is then U(2). On the other hand, N-point wavefunctions—signals—can be subject to a U(N) group of unitary matrices, containing the linear U(2); aberrations are transformations outside that subgroup. As in geometric optics, we classify the aberration multiplets by order and weight. Their action on phase space is displayed by means of a Wigner function on the sphere, to be compared with the corresponding geometric canonical transformations.
13. Hamiltonian formalism of minimal massive gravity
Mahdavian Yekta, Davood
2015-09-01
In this paper, we study the three-dimensional minimal massive gravity (MMG) in the Hamiltonian formalism. At first, we define the canonical gauge generators as building blocks in this formalism and then derive the canonical expressions for the asymptotic conserved charges. The construction of a consistent asymptotic structure of MMG requires introducing suitable boundary conditions. In the second step, we show that the Poisson bracket algebra of the improved canonical gauge generators produces an asymptotic gauge group, which includes two separable versions of the Virasoro algebras. For instance, we study the Banados-Teitelboim-Zanelli (BTZ) black hole as a solution of the MMG field equations, and the conserved charges give the energy and angular momentum of the BTZ black hole. Finally, we compute the black hole entropy from the Cardy formula in the dual conformal field theory and show our result is consistent with the value obtained by using the Smarr formula from the holographic principle.
14. Discrete variable representation for singular Hamiltonians.
PubMed
Schneider, Barry I; Nygaard, Nicolai
2004-11-01
We discuss the application of the discrete variable representation (DVR) to Schrödinger problems which involve singular Hamiltonians. Unlike recent authors who invoke transformations to rid the eigenvalue equation of singularities at the cost of added complexity, we show that an approach based solely on an orthogonal polynomial basis is adequate, provided the Gauss-Lobatto or Gauss-Radau quadrature rule is used. This ensures that the mesh contains the singular points and by simply discarding the DVR functions corresponding to those points, all matrix elements become well behaved, the boundary conditions are satisfied, and the calculation is rapidly convergent. The accuracy of the method is demonstrated by applying it to the hydrogen atom. We emphasize that the method is equally capable of describing bound states and continuum solutions.
15. Hamiltonian description of composite fermions: Magnetoexciton dispersions
Murthy, Ganpathy
1999-11-01
A microscopic Hamiltonian theory of the FQHE, developed by Shankar and myself based on the fermionic Chern-Simons approach, has recently been quite successful in calculating gaps in fractional quantum hall states, and in predicting approximate scaling relations between the gaps of different fractions. I now apply this formalism towards computing magnetoexciton dispersions (including spin-flip dispersions) in the ν=13, 25, and 37 gapped fractions, and find approximate agreement with numerical results. I also analyze the evolution of these dispersions with increasing sample thickness, modelled by a potential soft at high momenta. New results are obtained for instabilities as a function of thickness for 25 and 37, and it is shown that the spin-polarized 25 state, in contrast to the spin-polarized 13 state, cannot be described as a simple quantum ferromagnet.
16. Hamiltonian Approach to the Dynamical Casimir Effect
SciTech Connect
Haro, Jaume; Elizalde, Emilio
2006-09-29
A Hamiltonian approach is introduced in order to address some severe problems associated with the physical description of the dynamical Casimir effect at all times. For simplicity, the case of a neutral scalar field in a one-dimensional cavity with partially transmitting mirrors (an essential proviso) is considered, but the method can be extended to fields of any kind and higher dimensions. The motional force calculated in our approach contains a reactive term--proportional to the mirrors' acceleration - which is fundamental in order to obtain (quasi)particles with a positive energy all the time during the movement of the mirrors - while always satisfying the energy conservation law. Comparisons with other approaches and a careful analysis of the interrelations among the different results previously obtained in the literature are carried out.
17. Geometric solitons of Hamiltonian flows on manifolds
SciTech Connect
Song, Chong; Sun, Xiaowei; Wang, Youde
2013-12-15
It is well-known that the LIE (Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of isometries of the domain and the target space, respectively. With this insight, we propose the new concept of geometric solitons of Hamiltonian flows on manifolds, such as geometric Schrödinger flows and KdV flows for maps. Moreover, we give several examples of geometric solitons of the Schrödinger flow and geometric KdV flow, including magnetic curves as geometric Schrödinger solitons and explicit geometric KdV solitons on surfaces of revolution.
18. Modal decomposition of Hamiltonian variational equations
NASA Technical Reports Server (NTRS)
Wiesel, William E.
1994-01-01
Over any finite arc of trajectory, the variational equations of a Hamiltonian system can be separated into 'normal' modes. This transformation is canonical, and the Lyapunov exponents over the trajectory arc occur as positive/negative pairs for conjugate modes, while the modal vectors remain unit vectors. This decomposition effectively solves the variational equations for any canonical, linear-dependent system. As an example, we study the Voyager I trajectory. In an interplanetary flyby, some of the modal variables increase by very large multiplicative factors, but this means that their conjugate modal variables decrease by those same very large multiplicative vectors. Maneuver strategies for this case are explored, and the minimum delta upsilon maneuver is found.
SciTech Connect
Malitsky, N.; Bourianoff, G.; Severgin, Yu.
1993-11-01
For the many applied tasks of accelerator physics, the 4D single particle pseudo-Hamiltonian may be presented as the Hamiltonian of the near-integrable system consisting of integrable and perturbed terms. The KAM theorem states that for sufficiently small perturbation the invariant surfaces continue to exist and, for the systems with two degrees of freedom, completely isolate the thin stochastic layers. As the perturbation strength increases, a transition can occur in which these surfaces disappear and the stochastic layers connect, resulting in globally stochastic motion. One of the important problems is to determine this {open_quotes}boundary{close_quotes} invariant surface. There are several approaches that may be used to describe the regular trajectories in the small limited region. The most powerful method is the perturbation theory which allows us to study the combined influence of the different multipoles. The inclusion of Lie operators improved this method and developed it up to high order perturbation. But the perturbation theory failed to describe the change in topology and since the regular trajectories depend discontinuously on choice of initial coordinates, it cannot be used in the whole region of the stable motion. The authors suggest to limit the attention to the study of the {open_quotes}boundary{close_quotes} invariant and implement the additional {open_quotes}local{close_quotes} transformation. The authors briefly review the well known theories, their advantages and imperfections, and the necessity of the {open_quotes}local{close_quotes} transformation. They present the comparison of the map tracking with the invariants determined by the perturbation methods.
20. Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation
SciTech Connect
Fernández, Francisco M.
2016-06-15
We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. -- Highlights: •Symmetric quadratic operators are useful models for many physical applications. •Any such operator exhibits a pseudo-Hermitian matrix representation. •Its eigenvalues are the natural frequencies of the Hamiltonian operator. •The eigenvalues may be real or complex and describe a phase transition.
1. Simulating typical entanglement with many-body Hamiltonian dynamics
SciTech Connect
Nakata, Yoshifumi; Murao, Mio
2011-11-15
We study the time evolution of the amount of entanglement generated by one-dimensional spin-1/2 Ising-type Hamiltonians composed of many-body interactions. We investigate sets of states randomly selected during the time evolution generated by several types of time-independent Hamiltonians by analyzing the distributions of the amount of entanglement of the sets. We compare such entanglement distributions with that of typical entanglement, entanglement of a set of states randomly selected from a Hilbert space with respect to the unitarily invariant measure. We show that the entanglement distribution obtained by a time-independent Hamiltonian can simulate the average and standard deviation of the typical entanglement, if the Hamiltonian contains suitable many-body interactions. We also show that the time required to achieve such a distribution is polynomial in the system size for certain types of Hamiltonians.
2. Remarks on the Lagrangian representation of bi-Hamiltonian equations
Pavlov, M. V.; Vitolo, R. F.
2017-03-01
The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair A1, A2, where A1 is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field τ such that A2 =LτA1. We use this result in order to find the Lagrangian representation when A2 is a homogeneous third-order Hamiltonian operator, although the method that we use can be applied to any other homogeneous Hamiltonian operator. As an example we provide the Lagrangian representation of a WDVV hydrodynamic-type system in 3 components.
3. Hamiltonian analysis of higher derivative scalar-tensor theories
Langlois, David; Noui, Karim
2016-07-01
We perform a Hamiltonian analysis of a large class of scalar-tensor Lagrangians which depend quadratically on the second derivatives of a scalar field. By resorting to a convenient choice of dynamical variables, we show that the Hamiltonian can be written in a very simple form, where the Hamiltonian and the momentum constraints are easily identified. In the case of degenerate Lagrangians, which include the Horndeski and beyond Horndeski quartic Lagrangians, our analysis confirms that the dimension of the physical phase space is reduced by the primary and secondary constraints due to the degeneracy, thus leading to the elimination of the dangerous Ostrogradsky ghost. We also present the Hamiltonian formulation for nondegenerate theories and find that they contain four degrees of freedom, including a ghost, as expected. We finally discuss the status of the unitary gauge from the Hamiltonian perspective.
4. Investigation of non-Hermitian Hamiltonians in the Heisenberg picture
Miao, Yan-Gang; Xu, Zhen-Ming
2016-05-01
The Heisenberg picture for non-Hermitian but η-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but η-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-Hermitian Hamiltonian systems. As an application, two new non-Hermitian Hamiltonians are constructed and investigated, where one is non-Hermitian and non-PT-symmetric and the other is non-Hermitian but PT-symmetric. Moreover, the complementarity and compatibility between our treatment and the PT symmetry are discussed.
5. Action with Acceleration II: Euclidean Hamiltonian and Jordan Blocks
Baaquie, Belal E.
2013-10-01
The Euclidean action with acceleration has been analyzed in Ref. 1, and referred to henceforth as Paper I, for its Hamiltonian and path integral. In this paper, the state space of the Hamiltonian is analyzed for the case when it is pseudo-Hermitian (equivalent to a Hermitian Hamiltonian), as well as the case when it is inequivalent. The propagator is computed using both creation and destruction operators as well as the path integral. A state space calculation of the propagator shows the crucial role played by the dual state vectors that yields a result impossible to obtain from a Hermitian Hamiltonian. When it is not pseudo-Hermitian, the Hamiltonian is shown to be a direct sum of Jordan blocks.
6. Quantum control by means of hamiltonian structure manipulation.
PubMed
Donovan, A; Beltrani, V; Rabitz, H
2011-04-28
A traditional quantum optimal control experiment begins with a specific physical system and seeks an optimal time-dependent field to steer the evolution towards a target observable value. In a more general framework, the Hamiltonian structure may also be manipulated when the material or molecular 'stockroom' is accessible as a part of the controls. The current work takes a step in this direction by considering the converse of the normal perspective to now start with a specific fixed field and employ the system's time-independent Hamiltonian structure as the control to identify an optimal form. The Hamiltonian structure control variables are taken as the system energies and transition dipole matrix elements. An analysis is presented of the Hamiltonian structure control landscape, defined by the observable as a function of the Hamiltonian structure. A proof of system controllability is provided, showing the existence of a Hamiltonian structure that yields an arbitrary unitary transformation when working with virtually any field. The landscape analysis shows that there are no suboptimal traps (i.e., local extrema) for controllable quantum systems when unconstrained structural controls are utilized to optimize a state-to-state transition probability. This analysis is corroborated by numerical simulations on model multilevel systems. The search effort to reach the top of the Hamiltonian structure landscape is found to be nearly invariant to system dimension. A control mechanism analysis is performed, showing a wide variety of behavior for different systems at the top of the Hamiltonian structure landscape. It is also shown that reducing the number of available Hamiltonian structure controls, thus constraining the system, does not always prevent reaching the landscape top. The results from this work lay a foundation for considering the laboratory implementation of optimal Hamiltonian structure manipulation for seeking the best control performance, especially with limited
7. Optimization of quantum Hamiltonian evolution: From two projection operators to local Hamiltonians
Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the Lie-Trotter formula. The strategy is then extended first to simulation of any Hamiltonian that is a linear combination of two projection operators, and then to any local efficiently computable Hamiltonian. The key feature is to construct an evolution in terms of the largest possible steps instead of taking small time steps. Reflection operations and Chebyshev expansions are used to efficiently control the total error on the overall evolution, without worrying about discretization errors for individual steps. We also use a digital implementation of quantum states that makes linear algebra operations rather simple to perform.
8. Trinucleon Electromagnetic Form Factors and the Light-Front Hamiltonian Dynamics
SciTech Connect
Baroncini, F.; Kievsky, A.; Pace, E.; Salme, G.
2008-10-13
This contribution briefly illustrates preliminary calculations of the electromagnetic form factors of {sup 3}He and {sup 3}H, obtained within the Light-front Relativistic Hamiltonian Dynamics, adopting i) a Poincare covariant current operator, without dynamical two-body currents, and ii) realistic nuclear bound states with S, P and D waves. The kinematical region of few (GeV/c){sup 2}, relevant for forthcoming TJLAB experiments, has been investigated, obtaining possible signatures of relativistic effects for Q{sup 2}>2.5(GeV/c){sup 2}.
9. Five-dimensional collective Hamiltonian with the Gogny force: An ongoing saga
Libert, J.; Delaroche, J.-P.; Girod, M.
2016-07-01
We provide a sample of analyses for nuclear spectroscopic properties based on the five-dimensional collective Hamiltonian (5DCH) implemented with the Gogny force. The very first illustration is dating back to the late 70's. It is next followed by others, focusing on shape coexistence, shape isomerism, superdeformation, and systematics over the periodic table. Finally, the inclusion of Thouless-Valatin dynamical contributions to vibrational mass parameters is briefly discussed as a mean of strengthening the basis of the 5DCH theory.
10. Ab initio alpha-alpha scattering
Elhatisari, Serdar; Lee, Dean; Rupak, Gautam; Epelbaum, Evgeny; Krebs, Hermann; Lähde, Timo A.; Luu, Thomas; Meißner, Ulf-G.
2015-12-01
Processes such as the scattering of alpha particles (4He), the triple-alpha reaction, and alpha capture play a major role in stellar nucleosynthesis. In particular, alpha capture on carbon determines the ratio of carbon to oxygen during helium burning, and affects subsequent carbon, neon, oxygen, and silicon burning stages. It also substantially affects models of thermonuclear type Ia supernovae, owing to carbon detonation in accreting carbon-oxygen white-dwarf stars. In these reactions, the accurate calculation of the elastic scattering of alpha particles and alpha-like nuclei—nuclei with even and equal numbers of protons and neutrons—is important for understanding background and resonant scattering contributions. First-principles calculations of processes involving alpha particles and alpha-like nuclei have so far been impractical, owing to the exponential growth of the number of computational operations with the number of particles. Here we describe an ab initio calculation of alpha-alpha scattering that uses lattice Monte Carlo simulations. We use lattice effective field theory to describe the low-energy interactions of protons and neutrons, and apply a technique called the ‘adiabatic projection method’ to reduce the eight-body system to a two-cluster system. We take advantage of the computational efficiency and the more favourable scaling with system size of auxiliary-field Monte Carlo simulations to compute an ab initio effective Hamiltonian for the two clusters. We find promising agreement between lattice results and experimental phase shifts for s-wave and d-wave scattering. The approximately quadratic scaling of computational operations with particle number suggests that it should be possible to compute alpha scattering and capture on carbon and oxygen in the near future. The methods described here can be applied to ultracold atomic few-body systems as well as to hadronic systems using lattice quantum chromodynamics to describe the interactions of
11. Ab initio alpha-alpha scattering.
PubMed
Elhatisari, Serdar; Lee, Dean; Rupak, Gautam; Epelbaum, Evgeny; Krebs, Hermann; Lähde, Timo A; Luu, Thomas; Meißner, Ulf-G
2015-12-03
Processes such as the scattering of alpha particles ((4)He), the triple-alpha reaction, and alpha capture play a major role in stellar nucleosynthesis. In particular, alpha capture on carbon determines the ratio of carbon to oxygen during helium burning, and affects subsequent carbon, neon, oxygen, and silicon burning stages. It also substantially affects models of thermonuclear type Ia supernovae, owing to carbon detonation in accreting carbon-oxygen white-dwarf stars. In these reactions, the accurate calculation of the elastic scattering of alpha particles and alpha-like nuclei--nuclei with even and equal numbers of protons and neutrons--is important for understanding background and resonant scattering contributions. First-principles calculations of processes involving alpha particles and alpha-like nuclei have so far been impractical, owing to the exponential growth of the number of computational operations with the number of particles. Here we describe an ab initio calculation of alpha-alpha scattering that uses lattice Monte Carlo simulations. We use lattice effective field theory to describe the low-energy interactions of protons and neutrons, and apply a technique called the 'adiabatic projection method' to reduce the eight-body system to a two-cluster system. We take advantage of the computational efficiency and the more favourable scaling with system size of auxiliary-field Monte Carlo simulations to compute an ab initio effective Hamiltonian for the two clusters. We find promising agreement between lattice results and experimental phase shifts for s-wave and d-wave scattering. The approximately quadratic scaling of computational operations with particle number suggests that it should be possible to compute alpha scattering and capture on carbon and oxygen in the near future. The methods described here can be applied to ultracold atomic few-body systems as well as to hadronic systems using lattice quantum chromodynamics to describe the interactions of
12. Computational studies of competing phases in model Hamiltonians
Jiang, Mi
Model Hamiltonians play an important role in our understanding of both quantum and classical systems, such as strongly correlated unconventional superconductivity, quantum magnetism, non-fermi liquid heavy fermion materials and classical magnetic phase transitions. The central problem is how models with many degrees of freedom choose between competing ground states, e.g. magnetic, superconducting, metallic, insulating as the degree of thermal and quantum fluctuations is varied. This dissertation focuses on the numerical investigation of several important model Hamiltonians. Specifically, we used the determinant Quantum Monte Carlo (DQMC) to study three Hubbard-like models: the Fermi-Hubbard model with two regions of different interaction strength, the Fermi-Hubbard model with a spin-dependent band structure, and the related periodic Anderson model (PAM). The first model used was to explore inter-penetration of metallic and Mott insulator physics across a Metal-Mott Insulator interface by computing the magnetic properties and spectral functions. As a minimal model of a half metallic magnet, the second model was used to explore the impact of on-site Hubbard interaction U, finite temperature, and an external (Zeeman) magnetic field on a bilayer tight-binding model with spin-dependent hybridization. We use PAM to study the Knight shift anomaly in heavy fermion materials found in Nuclear magnetic resonance (NMR) experiments and confirm several predictions of the two-fluid theory accounting for the anomaly. Another application of the Hubbard model described in this dissertation is the investigation on the effects of spin-dependent disorder on s-wave superconductors based on the attractive Hubbard model. Here we used the Bogoliubov-de Gennes (BdG) self-consistent approach instead of quantum simulations. The spin-dependent random potential was shown to induce distinct transitions at which the energy gap and average order parameter vanish, generating an intermediate gapless
13. Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula with applications to graphene-type systems
Vogl, M.; Pankratov, O.; Shallcross, S.
2017-07-01
We present a tractable and physically transparent semiclassical theory of matrix-valued Hamiltonians, i.e., those that describe quantum systems with internal degrees of freedoms, based on a generalization of the Gutzwiller trace formula for a n ×n dimensional Hamiltonian H (p ̂,q ̂) . The classical dynamics is governed by n Hamilton-Jacobi (HJ) equations that act in a phase space endowed with a classical Berry curvature encoding anholonomy in the parallel transport of the eigenvectors of H (p ,q ) ; these vectors describe the internal structure of the semiclassical particles. At the O (ℏ1) level and for nondegenerate HJ systems, this curvature results in an additional semiclassical phase composed of (i) a Berry phase and (ii) a dynamical phase resulting from the classical particles "moving through the Berry curvature". We show that the dynamical part of this semiclassical phase will, generally, be zero only for the case in which the Berry phase is topological (i.e., depends only on the winding number). We illustrate the method by calculating the Landau spectrum for monolayer graphene, the four-band model of AB bilayer graphene, and for a more complicated matrix Hamiltonian describing the silicene band structure. Finally, we apply our method to an inhomogeneous system consisting of a strain engineered one-dimensional moiré in bilayer graphene, finding localized states near the Dirac point that arise from electron trapping in a semiclassical moiré potential. The semiclassical density of states of these localized states we show to be in perfect agreement with an exact quantum mechanical calculation of the density of states.
14. Low-energy effective Hamiltonians for correlated electron systems beyond density functional theory
Hirayama, Motoaki; Miyake, Takashi; Imada, Masatoshi; Biermann, Silke
2017-08-01
We propose a refined scheme of deriving an effective low-energy Hamiltonian for materials with strong electronic Coulomb correlations beyond density functional theory (DFT). By tracing out the electronic states away from the target degrees of freedom in a controlled way by a perturbative scheme, we construct an effective Hamiltonian for a restricted low-energy target space incorporating the effects of high-energy degrees of freedom in an effective manner. The resulting effective Hamiltonian can afterwards be solved by accurate many-body solvers. We improve this "multiscale ab initio scheme for correlated electrons" (MACE) primarily in two directions by elaborating and combining two frameworks developed by Hirayama et al. [M. Hirayama, T. Miyake, and M. Imada, Phys. Rev. B 87, 195144 (2013), 10.1103/PhysRevB.87.195144] and Casula et al. [M. Casula, P. Werner, L. Vaugier, F. Aryasetiawan, T. Miyake, A. J. Millis, and S. Biermann, Phys. Rev. Lett. 109, 126408 (2012), 10.1103/PhysRevLett.109.126408]: (1) Double counting of electronic correlations between the DFT and the low-energy solver is avoided by using the constrained G W scheme; and (2) the frequency dependent interactions emerging from the partial trace summation are successfully separated into a nonlocal part that is treated following ideas by Hirayama et al. and a local part treated nonperturbatively in the spirit of Casula et al. and are incorporated into the renormalization of the low-energy dispersion. The scheme is favorably tested on the example of SrVO3.
15. Novel Exciton States in Monolayer MoS2: Unconventional Effective Hamiltonian
da Jornada, Felipe; Qiu, Diana; Louie, Steven
2014-03-01
Recent well-converged ab inito GW-BSE calculations show that monolayer MoS2 has a large number of strongly bound excitons with varying characters. We show that these excitonic states cannot be even qualitatively described by an effective mass hydrogenic model without a detailed understanding of the 2D screening. Additionally, we identify and analyze one exciton series having an unusually high binding energy, which originates around the Γ point of the Brillouin zone. We show that this excitonic series arises from a fundamentally different effective Hamiltonian with a kinetic energy term resembling a Mexican hat in momentum space, which is responsible for the unusual ordering of the energy levels and distribution of oscillator strength. This work was supported by NSF grant No. DMR10-1006184 and the U.S. DOE under Contract No. DE-AC02-05CH11231.
16. Electronic properties, low-energy Hamiltonian, and superconducting instabilities in CaKFe4As4
Lochner, Felix; Ahn, Felix; Hickel, Tilmann; Eremin, Ilya
2017-09-01
We analyze the electronic properties of the recently discovered stoichiometric superconductor CaKFe4As4 by combining an ab initio approach and a projection of the band structure to a low-energy tight-binding Hamiltonian, based on the maximally localized Wannier orbitals of the 3 d Fe states. We identify the key symmetries as well as differences and similarities in the electronic structure between CaKFe4As4 and the parent systems CaFe2As2 and KFe2As2 . In particular, we find CaKFe4As4 to have a significantly more quasi-two-dimensional electronic structure than the latter systems. Finally, we study the superconducting instabilities in CaKFe4As4 by employing the leading angular harmonics approximation and find two potential A1 g-symmetry representations of the superconducting gap to be the dominant instabilities in this system.
17. Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian
Wahlen-Strothman, Jacob M.; Henderson, Thomas M.; Hermes, Matthew R.; Degroote, Matthias; Qiu, Yiheng; Zhao, Jinmo; Dukelsky, Jorge; Scuseria, Gustavo E.
2017-02-01
Coupled cluster and symmetry projected Hartree-Fock are two central paradigms in electronic structure theory. However, they are very different. Single reference coupled cluster is highly successful for treating weakly correlated systems but fails under strong correlation unless one sacrifices good quantum numbers and works with broken-symmetry wave functions, which is unphysical for finite systems. Symmetry projection is effective for the treatment of strong correlation at the mean-field level through multireference non-orthogonal configuration interaction wavefunctions, but unlike coupled cluster, it is neither size extensive nor ideal for treating dynamic correlation. We here examine different scenarios for merging these two dissimilar theories. We carry out this exercise over the integrable Lipkin model Hamiltonian, which despite its simplicity, encompasses non-trivial physics for degenerate systems and can be solved via diagonalization for a very large number of particles. We show how symmetry projection and coupled cluster doubles individually fail in different correlation limits, whereas models that merge these two theories are highly successful over the entire phase diagram. Despite the simplicity of the Lipkin Hamiltonian, the lessons learned in this work will be useful for building an ab initio symmetry projected coupled cluster theory that we expect to be accurate in the weakly and strongly correlated limits, as well as the recoupling regime.
18. An effective Hamiltonian approach to quantum random walk
Sarkar, Debajyoti; Paul, Niladri; Bhattacharya, Kaushik; Ghosh, Tarun Kanti
2017-03-01
In this article we present an effective Hamiltonian approach for discrete time quantum random walk. A form of the Hamiltonian for one-dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative (see Chandrashekar, Sci. Rep. 3, 2829 (18)). We showed that in the case of two-step walk, the time evolution operator effectively can have multiplicative form. In the case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the graphene Hamiltonian, the walk has been studied on a graphene lattice and we conclude the preference of additive approach over the multiplicative one.
19. Local Hamiltonians for quantitative Green's function embedding methods
Rusakov, Alexander A.; Phillips, Jordan J.; Zgid, Dominika
2014-11-01
Embedding calculations that find approximate solutions to the Schrödinger equation for large molecules and realistic solids are performed commonly in a three step procedure involving (i) construction of a model system with effective interactions approximating the low energy physics of the initial realistic system, (ii) mapping the model system onto an impurity Hamiltonian, and (iii) solving the impurity problem. We have developed a novel procedure for parametrizing the impurity Hamiltonian that avoids the mathematically uncontrolled step of constructing the low energy model system. Instead, the impurity Hamiltonian is immediately parametrized to recover the self-energy of the realistic system in the limit of high frequencies or short time. The effective interactions parametrizing the fictitious impurity Hamiltonian are local to the embedded regions, and include all the non-local interactions present in the original realistic Hamiltonian in an implicit way. We show that this impurity Hamiltonian can lead to excellent total energies and self-energies that approximate the quantities of the initial realistic system very well. Moreover, we show that as long as the effective impurity Hamiltonian parametrization is designed to recover the self-energy of the initial realistic system for high frequencies, we can expect a good total energy and self-energy. Finally, we propose two practical ways of evaluating effective integrals for parametrizing impurity models.
20. Nonunitary quantum computation in the ground space of local Hamiltonians
Usher, Naïri; Hoban, Matty J.; Browne, Dan E.
2017-09-01
A central result in the study of quantum Hamiltonian complexity is that the k -local Hamiltonian problem is quantum-Merlin-Arthur-complete. In that problem, we must decide if the lowest eigenvalue of a Hamiltonian is bounded below some value, or above another, promised one of these is true. Given the ground state of the Hamiltonian, a quantum computer can determine this question, even if the ground state itself may not be efficiently quantum preparable. Kitaev's proof of QMA-completeness encodes a unitary quantum circuit in QMA into the ground space of a Hamiltonian. However, we now have quantum computing models based on measurement instead of unitary evolution; furthermore, we can use postselected measurement as an additional computational tool. In this work, we generalize Kitaev's construction to allow for nonunitary evolution including postselection. Furthermore, we consider a type of postselection under which the construction is consistent, which we call tame postselection. We consider the computational complexity consequences of this construction and then consider how the probability of an event upon which we are postselecting affects the gap between the ground-state energy and the energy of the first excited state of its corresponding Hamiltonian. We provide numerical evidence that the two are not immediately related by giving a family of circuits where the probability of an event upon which we postselect is exponentially small, but the gap in the energy levels of the Hamiltonian decreases as a polynomial.
1. Hamiltonian of a spinning test particle in curved spacetime
SciTech Connect
Barausse, Enrico; Racine, Etienne; Buonanno, Alessandra
2009-11-15
Using a Legendre transformation, we compute the unconstrained Hamiltonian of a spinning test particle in a curved spacetime at linear order in the particle spin. The equations of motion of this unconstrained Hamiltonian coincide with the Mathisson-Papapetrou-Pirani equations. We then use the formalism of Dirac brackets to derive the constrained Hamiltonian and the corresponding phase space algebra in the Newton-Wigner spin supplementary condition, suitably generalized to curved spacetime, and find that the phase space algebra (q,p,S) is canonical at linear order in the particle spin. We provide explicit expressions for this Hamiltonian in a spherically symmetric spacetime, both in isotropic and spherical coordinates, and in the Kerr spacetime in Boyer-Lindquist coordinates. Furthermore, we find that our Hamiltonian, when expanded in post-Newtonian (PN) orders, agrees with the Arnowitt-Deser-Misner canonical Hamiltonian computed in PN theory in the test particle limit. Notably, we recover the known spin-orbit couplings through 2.5PN order and the spin-spin couplings of type S{sub Kerr}S (and S{sub Kerr}{sup 2}) through 3PN order, S{sub Kerr} being the spin of the Kerr spacetime. Our method allows one to compute the PN Hamiltonian at any order, in the test particle limit and at linear order in the particle spin. As an application we compute it at 3.5PN order.
2. Hamiltonian of a spinning test-particle in curved spacetime
Barausse, Enrico; Racine, Etienne; Buonanno, Alessandra
2010-02-01
Using a Legendre transformation, we compute the unconstrained Hamiltonian of a spinning test-particle in a curved spacetime at linear order in the particle spin. The equations of motion of this unconstrained Hamiltonian coincide with the Mathisson-Papapetrou-Pirani equations. We then use the formalism of Dirac brackets to derive the constrained Hamiltonian and the corresponding phase-space algebra in the Newton-Wigner spin supplementary condition (SSC), suitably generalized to curved spacetime, and find that the phase-space algebra (q,p,S) is canonical at linear order in the particle spin. We provide explicit expressions for this Hamiltonian in a spherically symmetric spacetime, both in isotropic and spherical coordinates, and in the Kerr spacetime in Boyer-Lindquist coordinates. Furthermore, we find that our Hamiltonian, when expanded in Post-Newtonian (PN) orders, agrees with the Arnowitt-Deser-Misner (ADM) canonical Hamiltonian computed in PN theory in the test-particle limit. Notably, we recover the known spin-orbit couplings through 2.5PN order and the spin-spin couplings of type SKerr, (and SKerr^2) through 3PN order, SKerr being the spin of the Kerr spacetime. Our method allows one to compute the PN Hamiltonian at any order, in the test-particle limit and at linear order in the particle spin. As an application we compute it at 3.5PN order. )
3. Hamiltonian of a spinning test particle in curved spacetime
Barausse, Enrico; Racine, Etienne; Buonanno, Alessandra
2009-11-01
Using a Legendre transformation, we compute the unconstrained Hamiltonian of a spinning test particle in a curved spacetime at linear order in the particle spin. The equations of motion of this unconstrained Hamiltonian coincide with the Mathisson-Papapetrou-Pirani equations. We then use the formalism of Dirac brackets to derive the constrained Hamiltonian and the corresponding phase space algebra in the Newton-Wigner spin supplementary condition, suitably generalized to curved spacetime, and find that the phase space algebra (q,p,S) is canonical at linear order in the particle spin. We provide explicit expressions for this Hamiltonian in a spherically symmetric spacetime, both in isotropic and spherical coordinates, and in the Kerr spacetime in Boyer-Lindquist coordinates. Furthermore, we find that our Hamiltonian, when expanded in post-Newtonian (PN) orders, agrees with the Arnowitt-Deser-Misner canonical Hamiltonian computed in PN theory in the test particle limit. Notably, we recover the known spin-orbit couplings through 2.5PN order and the spin-spin couplings of type SKerrS (and SKerr2) through 3PN order, SKerr being the spin of the Kerr spacetime. Our method allows one to compute the PN Hamiltonian at any order, in the test particle limit and at linear order in the particle spin. As an application we compute it at 3.5PN order.
4. Enhancing sensitivity in quantum metrology by Hamiltonian extensions
Fraïsse, Julien Mathieu Elias; Braun, Daniel
2017-06-01
A well-studied scenario in quantum parameter estimation theory arises when the parameter to be estimated is imprinted on the initial state by a Hamiltonian of the form θ G . For such "phase-shift Hamiltonians" it has been shown that one cannot improve the channel quantum Fisher information by adding ancillas and letting the system interact with them. Here we investigate the general case, where the Hamiltonian is not necessarily a phase shift, and show that in this case in general it is possible to increase the quantum channel information and to reach an upper bound. This can be done by adding a term proportional to the derivative of the Hamiltonian, or by subtracting a term from the original Hamiltonian. Neither method makes use of any ancillas, which shows that, for quantum channel estimation with an arbitrary parameter-dependent Hamiltonian, entanglement with an ancillary system is not necessary to reach the best possible sensitivity. By adding an operator to the Hamiltonian we can also modify the time scaling of the channel quantum Fisher information. We illustrate our techniques with nitrogen vacancy center magnetometry and the estimation of the direction of a magnetic field in a given plane using a single spin-1 as probe.
5. Uncertainty relation for non-Hamiltonian quantum systems
SciTech Connect
Tarasov, Vasily E.
2013-01-15
General forms of uncertainty relations for quantum observables of non-Hamiltonian quantum systems are considered. Special cases of uncertainty relations are discussed. The uncertainty relations for non-Hamiltonian quantum systems are considered in the Schroedinger-Robertson form since it allows us to take into account Lie-Jordan algebra of quantum observables. In uncertainty relations, the time dependence of quantum observables and the properties of this dependence are discussed. We take into account that a time evolution of observables of a non-Hamiltonian quantum system is not an endomorphism with respect to Lie, Jordan, and associative multiplications.
6. Stability of Gabor Frames Under Small Time Hamiltonian Evolutions
de Gosson, Maurice A.; Gröchenig, Karlheinz; Romero, José Luis
2016-06-01
We consider Hamiltonian deformations of Gabor systems, where the window evolves according to the action of a Schrödinger propagator and the phase-space nodes evolve according to the corresponding Hamiltonian flow. We prove the stability of the frame property for small times and Hamiltonians consisting of a quadratic polynomial plus a potential in the Sjöstrand class with bounded second-order derivatives. This answers a question raised in de Gosson (Appl Comput Harmonic Anal 38(2):196-221, 2015)
7. Transformation of Hamiltonians to near action-angle form
SciTech Connect
Boozer, A.H.
1985-04-01
A classical Hamiltonian would be solved by a transformation to action-angle variables I,theta in which the Hamiltonian is H-bar(I). Generally, such a transformation does not exist, and, at best, the Hamiltonian can be transformed to H-bar(I) + H(I,theta,t) with H being a sum of Fourier terms that resonate with H-bar. We give a set of ordinary differential equations in a parameter epsilon that carry out this transformation as the set is integrated from epsilon equal to zero to one. Although the differential equations can be integrated numerically, approximations give classical perturbation theory.
8. Covariant Hamiltonian for the electromagnetic two-body problem
De Luca, Jayme
2005-09-01
We give a Hamiltonian formalism for the delay equations of motion of the electromagnetic two-body problem with arbitrary masses and with either repulsive or attractive interaction. This dynamical system based on action-at-a-distance electrodynamics appeared 100 years ago and it was popularized in the 1940s by the Wheeler and Feynman program to quantize it as a means to overcome the divergencies of perturbative QED. Our finite-dimensional implicit Hamiltonian is closed and involves no series expansions. As an application, the Hamiltonian formalism is used to construct a semiclassical canonical quantization based on the numerical trajectories of the attractive problem.
9. Covariant Hamiltonian for the electromagnetic two-body problem.
PubMed
De Luca, Jayme
2005-09-01
We give a Hamiltonian formalism for the delay equations of motion of the electromagnetic two-body problem with arbitrary masses and with either repulsive or attractive interaction. This dynamical system based on action-at-a-distance electrodynamics appeared 100 years ago and it was popularized in the 1940s by the Wheeler and Feynman program to quantize it as a means to overcome the divergencies of perturbative QED. Our finite-dimensional implicit Hamiltonian is closed and involves no series expansions. As an application, the Hamiltonian formalism is used to construct a semiclassical canonical quantization based on the numerical trajectories of the attractive problem.
10. Generic perturbations of linear integrable Hamiltonian systems
Bounemoura, Abed
2016-11-01
In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ɛ -1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).
11. Surface Lifshits tails for random quantum Hamiltonians
Kirsch, Werner; Raikov, Georgi
2017-03-01
We consider Schrödinger operators on L2(ℝd) ⊗L2 (ℝℓ) of the form Hω=H⊥⊗I∥ +I⊥⊗H∥ +Vω , where H⊥ and H∥ are Schrödinger operators on L2(ℝd) and L2(ℝℓ) , respectively, and Vω(x ,y ) :=∑ξ∈ℤdλξ(ω ) v (x -ξ ,y ) ,x ∈ℝd ,y ∈ℝℓ is a random "surface potential." We investigate the behavior of the integrated density of surface states of Hω near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of Hω can be read off from the integrated density of states of a reduced Hamiltonian H⊥+Wω where Wω is a quantum mechanical average of Vω with respect to y ∈ℝℓ . We are particularly interested in cases when H⊥ is a magnetic Schrödinger operator, but we also recover some of the results from Kirsch and Warzel [J. Funct. Anal. 230, 222-250 (2006)] for non-magnetic H⊥.
12. Hamiltonian chaos in nonlinear optical polarization dynamics
David, D.; Holm, D. D.; Tratnik, M. V.
1990-03-01
This paper applies Hamiltonian methods to the Stokes representation of the one-beam and two-beam problems of polarized optical pulses propagating as travelling waves in nonlinear media. We treat these two dynamical systems as follows. First, we use the reduction method of Marsden and Weinstein to map each of the systems to the two-dimensional sphere, S 2. The resulting reduced systems are then analyzed from the viewpoints of their stability properties and of bifurcations with symmetry; in particular, several degenerate bifurcations are found and described. We also establish the presence of chaotic dynamics in these systems by demonstrating the existence of Smale horseshoe maps in the three- and four-dimensional cases, as well as Arnold diffusion in the higher-dimensional cases. The method we use to establish such complex dynamics is the Mel'nikov technique, as extended by Holmes and Marsden, and Wiggins for the higher-dimensional cases. These results apply to perturbations of homoclinic and heteroclinic orbits of the reduced integrable problems for static, as well as travelling-wave, solutions describing either a single opt ical beam, or two such beams counterpropagating. Thus, we show that these optics problems exhibit complex dynamics and predict the experimental consequences of this dynamics.
13. Hamiltonian formalism for Perturbed Black Hole Spacetimes
Mihaylov, Deyan; Gair, Jonathan
2017-01-01
Present and future gravitational wave observations provide a new mechanism to probe the predictions of general relativity. Observations of extreme mass ratio inspirals with millihertz gravitational wave detectors such as LISA will provide exquisite constraints on the spacetime structure outside astrophysical black holes, enabling tests of the no-hair property that all general relativistic black holes are described by the Kerr metric. Previous work to understand what constraints LISA observations will be able to place has focussed on specific alternative theories of gravity, or generic deviations that preserve geodesic separability. We describe an alternative approach to this problem--a technique that employs canonical perturbations of the Hamiltonian function describing motion in the Kerr metric. We derive this new approach and demonstrate its application to the cases of a slowly rotating Kerr black hole which is viewed as a perturbation of a Schwarzschild black hole, of coupled perturbations of black holes in the second-order Chern-Simons modified gravity theory, and several more indicative scenarios. Deyan Mihaylov is funded by STFC.
14. Laptop Induced Erythema Ab Igne
PubMed Central
Nayak, Sudhir U K; Shenoi, Shrutakirthi D; Prabhu, Smitha
2012-01-01
Erythema ab igne is a reticular, pigmented dermatosis caused by prolonged and repeated exposure to infrared radiation that is insufficient to produce a burn. The use of laptop computers has increased manifold in the recent past. Prolonged contact of the laptop with the skin can lead to the development of erythema ab igne. We present a case of erythema ab igne secondary to laptop use in an Indian student. PMID:22615512
15. Laptop induced erythema ab igne.
PubMed
Nayak, Sudhir U K; Shenoi, Shrutakirthi D; Prabhu, Smitha
2012-03-01
Erythema ab igne is a reticular, pigmented dermatosis caused by prolonged and repeated exposure to infrared radiation that is insufficient to produce a burn. The use of laptop computers has increased manifold in the recent past. Prolonged contact of the laptop with the skin can lead to the development of erythema ab igne. We present a case of erythema ab igne secondary to laptop use in an Indian student.
16. The Hamiltonian of Einstein affine-metric formulation of General Relativity
Kiriushcheva, N.; Kuzmin, S. V.
2010-11-01
It is shown that the Hamiltonian of the Einstein affine-metric (first-order) formulation of General Relativity (GR) leads to a constraint structure that allows the restoration of its unique gauge invariance, four-diffeomorphism, without the need of any field dependent redefinition of gauge parameters as in the case of the second-order formulation. In the second-order formulation of ADM gravity the need for such a redefinition is the result of the non-canonical change of variables ( arXiv:0809.0097 abs/arXiv:0809.0097" TargetType="URL"/> ). For the first-order formulation, the necessity of such a redefinition "to correspond to diffeomorphism invariance" (reported by Ghalati, arXiv:0901.3344 abs/arXiv:0901.3344" TargetType="URL"/> ) is just an artifact of using the Henneaux-Teitelboim-Zanelli ansatz (Nucl. Phys. B 332:169, 1990), which is sensitive to the choice of linear combination of tertiary constraints. This ansatz cannot be used as an algorithm for finding a gauge invariance, which is a unique property of a physical system, and it should not be affected by different choices of linear combinations of non-primary first class constraints. The algorithm of Castellani (Ann. Phys. 143:357, 1982) is free from such a deficiency and it leads directly to four-diffeomorphism invariance for first, as well as for second-order Hamiltonian formulations of GR. The distinct role of primary first class constraints, the effect of considering different linear combinations of constraints, the canonical transformations of phase-space variables, and their interplay are discussed in some detail for Hamiltonians of the second- and first-order formulations of metric GR. The first-order formulation of Einstein-Cartan theory, which is the classical background of Loop Quantum Gravity, is also discussed.
17. PREFACE: 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics
Fring, Andreas; Jones, Hugh; Znojil, Miloslav
2008-06-01
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS5 x S5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the
18. Computing pKa Values with a Mixing Hamiltonian Quantum Mechanical/Molecular Mechanical Approach.
PubMed
Liu, Yang; Fan, Xiaoli; Jin, Yingdi; Hu, Xiangqian; Hu, Hao
2013-09-10
Accurate computation of the pKa value of a compound in solution is important but challenging. Here, a new mixing quantum mechanical/molecular mechanical (QM/MM) Hamiltonian method is developed to simulate the free-energy change associated with the protonation/deprotonation processes in solution. The mixing Hamiltonian method is designed for efficient quantum mechanical free-energy simulations by alchemically varying the nuclear potential, i.e., the nuclear charge of the transforming nucleus. In pKa calculation, the charge on the proton is varied in fraction between 0 and 1, corresponding to the fully deprotonated and protonated states, respectively. Inspired by the mixing potential QM/MM free energy simulation method developed previously [H. Hu and W. T. Yang, J. Chem. Phys. 2005, 123, 041102], this method succeeds many advantages of a large class of λ-coupled free-energy simulation methods and the linear combination of atomic potential approach. Theory and technique details of this method, along with the calculation results of the pKa of methanol and methanethiol molecules in aqueous solution, are reported. The results show satisfactory agreement with the experimental data.
19. Estimation of many-body quantum Hamiltonians via compressive sensing
SciTech Connect
Shabani, A.; Rabitz, H.; Mohseni, M.; Lloyd, S.; Kosut, R. L.
2011-07-15
We develop an efficient and robust approach for quantum measurement of nearly sparse many-body quantum Hamiltonians based on the method of compressive sensing. This work demonstrates that with only O(sln(d)) experimental configurations, consisting of random local preparations and measurements, one can estimate the Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly s sparse in a known basis. The classical postprocessing is a convex optimization problem on the total Hilbert space which is generally not scalable. We numerically simulate the performance of this algorithm for three- and four-body interactions in spin-coupled quantum dots and atoms in optical lattices. Furthermore, we apply the algorithm to characterize Hamiltonian fine structure and unknown system-bath interactions.
20. Construction of Lagrangians and Hamiltonians from the Equation of Motion
ERIC Educational Resources Information Center
Yan, C. C.
1978-01-01
Demonstrates that infinitely many Lagrangians and Hamiltonians can be constructed from a given equation of motion. Points out the lack of an established criterion for making a proper selection. (Author/GA)
1. Periodic equatorial water flows from a Hamiltonian perspective
Ionescu-Kruse, Delia; Martin, Calin Iulian
2017-04-01
The main result of this paper is a Hamiltonian formulation of the nonlinear governing equations for geophysical periodic stratified water flows in the equatorial f-plane approximation allowing for piecewise constant vorticity.
2. Effective Hamiltonians of 2D Spin Glass Clusters
Clement, Colin; Liarte, Danilo; Middleton, Alan; Sethna, James
2015-03-01
We have a method for directly identifying the clusters which are thought to dominate the dynamics of spin glasses. We also have a method for generating an effective Hamiltonian treating each cluster as an individual spin. We used these methods on a 2D Ising spin glass with Gaussian bonds. We study these systems by generating samples and correlation functions using a combination of Monte Carlo and high-performance numerically exact Pfaffian methods. With effective cluster Hamiltonians we can calculate the free energy asymmetry of the original clusters and perform a scaling analysis. The scaling exponents found are consistent with Domain-Wall Renormalization Group methods, and probe all length scales. We can also study the flow of these effective Hamiltonians by clustering the clustered spins, and we find that our hard spin Hamiltonians at high temperature retain accurate low-temperature fluctuations when compared to their parent models.
3. Fractional Hamiltonian monodromy from a Gauss-Manin monodromy
Sugny, D.; Mardešić, P.; Pelletier, M.; Jebrane, A.; Jauslin, H. R.
2008-04-01
Fractional Hamiltonian monodromy is a generalization of the notion of Hamiltonian monodromy, recently introduced by [Nekhoroshev, Sadovskií, and Zhilinskií, C. R. Acad. Sci. Paris, Ser. 1 335, 985 (2002); Nekhoroshev, Sadovskií, and Zhilinskií, Ann. Henri Poincare 7, 1099 (2006)] for energy-momentum maps whose image has a particular type of nonisolated singularities. In this paper, we analyze the notion of fractional Hamiltonian monodromy in terms of the Gauss-Manin monodromy of a Riemann surface constructed from the energy-momentum map and associated with a loop in complex space which bypasses the line of singularities. We also prove some propositions on fractional Hamiltonian monodromy for 1:-n and m :-n resonant systems.
4. Two time physics and Hamiltonian Noether theorem for gauge systems
SciTech Connect
Nieto, J. A.; Ruiz, L.; Silvas, J.; Villanueva, V. M.
2006-09-25
Motivated by two time physics theory we revisited the Noether theorem for Hamiltonian constrained systems. Our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints.
5. Room Temperature Line Lists for CO_2 Isotopologues with AB Initio Computed Intensities
Zak, Emil; Tennyson, Jonathan; Polyansky, Oleg; Lodi, Lorenzo; Zobov, Nikolay Fedorovich; Tashkun, Sergey; Perevalov, Valery
2016-06-01
We report 13 room temperature line lists for all major CO_2 isotopologues, covering 0-8000 wn. These line lists are a response to the need for line intensities of high, preferably sub-percent, accuracy by remote sensing experiments. Our scheme encompasses nuclear motion calculations supported by critical reliability analysis of the generated line intensities. Rotation-vibration wavefunctions and energy levels are computed using DVR3D and a high quality semi-empirical potential energy surface (PES) [1], followed by computation of intensities using a fully ab initio dipole moment surface (DMS). Cross comparison of line lists calculated using pairs of high-quality PES's and DMS's is used to assess imperfections in the PES, which lead to unreliable transition intensities between levels involved in resonance interactions. Four line lists are computed for each isotopologue to quantify sensitivity to minor distortions of the PES/DMS. This provides an estimate of the contribution to the overall line intensity error introduced by the underlying PES. Reliable lines are benchmarked against recent state-of-the-art measurements [2] and HITRAN-2012 supporting the claim that the majority of line intensities for strong bands are predicted with sub-percent accuracy [3]. Accurate line positions are generated using an effective Hamiltonian [4]. We recommend use of these line lists for future remote sensing studies and inclusions in databases. X. Huang, D. W. Schwenke, S. A. Tashkun, T. J. Lee, J. Chem. Phys. 136, 124311, 2012. O. L. Polyansky, K. Bielska, M. Ghysels, L. Lodi, N. F. Zobov, J. T. Hodges, J. Tennyson, PRL, 114, 243001, 2015. E. Zak, J. Tennyson, O. L. Polyansky, L. Lodi, S. A. Tashkun, V. I. Perevalov, JQSRT, in press and to be submitted. S. A. Tashkun, V. I. Perevalov, R. R. Gamache, J. Lamouroux, JQSRT, 152, 45-73, 2015.
6. Hamiltonian Light-front Field Theory Within an AdS/QCD Basis
SciTech Connect
Vary, J.P.; Honkanen, H.; Li, Jun; Maris, P.; Brodsky, S.J.; Harindranath, A.; de Teramond, G.F.; Sternberg, P.; Ng, E.G.; Yang, C.; /LBL, Berkeley
2009-12-16
Non-perturbative Hamiltonian light-front quantum field theory presents opportunities and challenges that bridge particle physics and nuclear physics. Fundamental theories, such as Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) offer the promise of great predictive power spanning phenomena on all scales from the microscopic to cosmic scales, but new tools that do not rely exclusively on perturbation theory are required to make connection from one scale to the next. We outline recent theoretical and computational progress to build these bridges and provide illustrative results for nuclear structure and quantum field theory. As our framework we choose light-front gauge and a basis function representation with two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall AdS/QCD model obtained from light-front holography.
7. Global Calculations Using Potential Functions and Effective Hamiltonian Models for Vibration-Rotation Spectroscopy and Dynamics of Small Polyatomic Molecules
2010-06-01
It has become increasingly common to use accurate potential energy surfaces and dipole moment surfaces for predictions and assignment of high-resolution vibration-rotation molecular spectra. These surfaces are obtained either from high-level ab initio electronic structure calculations or from a direct fit to experimental spectroscopic data. The talk will continue a discussion of some recent advances in the domain of the "potentiology". The role of basis extrapolations, of the Born-Oppenheimer breakdown corrections , in particular for very highly excited vibration states will be considered. As effective polyad Hamiltonians and band transition moment operators are still widely used for data reductions in high-resolutions molecular spectroscopy, experimental spectra analyses invoke a need for accurate methods of building physically meaningful effective models from ab initio surfaces. This involves predictions for various spectroscopic constants, including vibration dependence of rotational and centrifugal distortion and resonance coupling parameters. Topics planned for discussion include: high-order Contact Transformations of rovibrational Hamiltonians and of the dipole moment for small polyatomic molecules; convergence issues; the role of the anharmonicity in a potential energy function and of resonance couplings on the normal mode mixing and on vib-rot assignments with application to high energy vibration levels of SO_2 and to ozone near the dissociation limit; intensity anomalies in H_2S / HDS / D_2S spectra, relation of the shape of ab initio dipole moment surfaces with a "mystery" of nearly vanishing symmetry allowed bands. A full account for symmetry properties requires efficient theoretical tools for transformations of molecular Hamiltonians such as irreducible tensor formalism, applications using phosphine and methane potentials will be discussed. Both potential functions and effective polyad Hamiltonians allow studying changes in quasi-classical vibration
8. Time and a physical Hamiltonian for quantum gravity.
PubMed
Husain, Viqar; Pawłowski, Tomasz
2012-04-06
We present a nonperturbative quantization of general relativity coupled to dust and other matter fields. The dust provides a natural time variable, leading to a physical Hamiltonian with spatial diffeomorphism symmetry. The surprising feature is that the Hamiltonian is not a square root. This property, together with the kinematical structure of loop quantum gravity, provides a complete theory of quantum gravity, and puts applications to cosmology, quantum gravitational collapse, and Hawking radiation within technical reach.
9. Hyperbolic tori in Hamiltonian systems with slowly varying parameter
SciTech Connect
Medvedev, Anton G
2013-05-31
This paper looks at a Hamiltonian system which depends periodically on a parameter. For each value of the parameter the system is assumed to have a hyperbolic periodic solution. Using the methods in KAM-theory it is proved that if the Hamiltonian is perturbed so that the value of the parameter varies with constant small frequency, then the nonautonomous system will have hyperbolic 2-tori in the extended phase space. Bibliography: 12 titles.
10. Noncanonical Hamiltonian density formulation of hydrodynamics and ideal MHD
SciTech Connect
Morrison, P.J.; Greene, J.M.
1980-04-01
A new Hamiltonian density formulation of a perfect fluid with or without a magnetic field is presented. Contrary to previous work the dynamical variables are the physical variables, rho, v, B, and s, which form a noncanonical set. A Poisson bracket which satisfies the Jacobi identity is defined. This formulation is transformed to a Hamiltonian system where the dynamical variables are the spatial Fourier coefficients of the fluid variables.
11. Applications of Noether conservation theorem to Hamiltonian systems
Mouchet, Amaury
2016-09-01
The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the boundary conditions under a canonical transformation and this paper proposes to address this issue. Then, the unified treatment of Hamiltonian systems offered by Noether's approach is illustrated on several examples, including classical field theory and quantum dynamics.
12. Applications of Noether conservation theorem to Hamiltonian systems
SciTech Connect
Mouchet, Amaury
2016-09-15
The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the boundary conditions under a canonical transformation and this paper proposes to address this issue. Then, the unified treatment of Hamiltonian systems offered by Noether’s approach is illustrated on several examples, including classical field theory and quantum dynamics.
13. The Lagrangian-Hamiltonian formalism for higher order field theories
Vitagliano, Luca
2010-06-01
We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to higher order field theories on fiber bundles. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for higher order Lagrangian field theories. Namely, our formalism does only depend on the action functional and, therefore, unlike previously proposed ones, is free from any relevant ambiguity.
14. MARKOV CHAIN MONTE CARLO POSTERIOR SAMPLING WITH THE HAMILTONIAN METHOD
SciTech Connect
K. HANSON
2001-02-01
The Markov Chain Monte Carlo technique provides a means for drawing random samples from a target probability density function (pdf). MCMC allows one to assess the uncertainties in a Bayesian analysis described by a numerically calculated posterior distribution. This paper describes the Hamiltonian MCMC technique in which a momentum variable is introduced for each parameter of the target pdf. In analogy to a physical system, a Hamiltonian H is defined as a kinetic energy involving the momenta plus a potential energy {var_phi}, where {var_phi} is minus the logarithm of the target pdf. Hamiltonian dynamics allows one to move along trajectories of constant H, taking large jumps in the parameter space with relatively few evaluations of {var_phi} and its gradient. The Hamiltonian algorithm alternates between picking a new momentum vector and following such trajectories. The efficiency of the Hamiltonian method for multidimensional isotropic Gaussian pdfs is shown to remain constant at around 7% for up to several hundred dimensions. The Hamiltonian method handles correlations among the variables much better than the standard Metropolis algorithm. A new test, based on the gradient of {var_phi}, is proposed to measure the convergence of the MCMC sequence.
15. The rovibrational Hamiltonian for ammonia-like molecules.
PubMed
Makarewicz, Jan; Skalozub, Alexander
2002-03-01
A new exact quantum mechanical rovibrational Hamiltonian operator for ammonia-like molecules is derived. The Hamiltonian is constructed in a molecular system of axes, such that its z' axis makes a trisection of the pyramidal angle formed by three bond vectors with the vertex on the central atom. The introduced set of the internal rovibrational coordinates is adapted to facilitate a convenient description of the inversion motion. These internal coordinates and the molecular axis system have a remarkable property, namely, the internal vibrational angular momentum of the molecule equals zero. This property significantly reduces the Coriolis coupling and simplifies the form of the Hamiltonian. The correctness of this Hamiltonian is proved by a numerical procedure. The orthogonal Radau vectors allowing us to define a similar molecular axis system and the internal coordinates are considered. The Hamiltonian for the Radau parameterization takes a form simple enough to carry out effectively variational calculations of the molecular rovibrational states. Under the appropriate choice of the variational basis functions, the Hamiltonian matrix elements are fully factorizable and do not have any singularities. A convenient method of symmetrization of the basis functions is proposed.
16. Ab initio calculation of (hyper)polarizabilities using a sum-over-states formalism.
Taylor, Caroline M.; Chaudhuri, Rajat K.; Potts, Davin M.; Freed, Karl F.
2001-03-01
Hyperpolarizabilities are relevant to a wide range of non-linear optical properties. Ab initio computations often require a high level of correlation for accurate determination of β and γ , and especially of thier frequency dependence. While sum-over-states methods are widely used within semi-empirical frameworks, they have not been employed with high level ab initio methods because of the computational costs associated with calculating a sufficient number of states. The effective valence shell Hamiltonian method (H^v) is a highly correlated, size-extensive, ab initio, multireference, perturbative (perturb-then-diagonalize'') method. A single H^v calculation yields a large number of states, making it ideal for use with the sum-over-states fomalism for determination of molecular properties. The method has been used to calculate the (hyper)polarizabilities of small polyene systems.
17. Leading-order relativistic effects on nuclear magnetic resonance shielding tensors.
PubMed
Manninen, Pekka; Ruud, Kenneth; Lantto, Perttu; Vaara, Juha
2005-03-15
We present perturbational ab initio calculations of the nuclear-spin-dependent relativistic corrections to the nuclear magnetic resonance shielding tensors that constitute, together with the other relativistic terms reported by us earlier, the full leading-order perturbational set of results for the one-electron relativistic contributions to this observable, based on the (Breit-)Pauli Hamiltonian. These contributions are considered for the H(2)X (X = O,S,Se,Te,Po) and HX (X = F,Cl,Br,I,At) molecules, as well as the noble gas (Ne, Ar, Kr, Xe, Rn) atoms. The corrections are evaluated using the relativistic and magnetic operators as perturbations on an equal footing, calculated using analytical linear and quadratic response theory applied on top of a nonrelativistic reference state provided by self-consistent field calculations. The (1)H and heavy-atom nuclear magnetic shielding tensors are compared with four component, nearly basis-set-limit Dirac-Hartree-Fock calculations that include positronic excitations, as well as available literature data. Besides the easy interpretability of the different contributions in terms of familiar nonrelativistic concepts, the accuracy of the present perturbational scheme is striking for the isotropic part of the shielding tensor, for systems including elements up to Xe.
18. Global exploration and inversion of quantum Hamiltonian- Observable relationships
Geremia, John Michael
2001-09-01
High-precision, quantitative knowledge of quantum Hamiltonians is a crucial prerequisite for accurately predicting and controlling molecular behavior. Understanding the physical connection between the quantum equations of motion and the physical observables measured in the laboratory is fundamental to bridging theory and experiment. However, Hamiltonian-Observable relationships are generally complex and nonlinear, making them difficult to represent, explore, and invert. A functional mapping concept that allows many nonlinear Hamiltonian-Observable relationships to be learned using a relatively small set of representative Hamiltonians is developed. Maps provide rationally organized response information that details how non-perturbative variations in the Hamiltonian affect physical observables. Nonlinear mapping techniques permit the application of global search algorithms to quantum inverse problems, such as Hamiltonian identification and control. The resulting map-facilitated inversion framework provides a new capability for identifying the full family of Hamiltonians consistent with a finite, noise-contaminated molecular objective. As a demonstration, potential energy surfaces for He-Ne, Na2, and Ar-HCl are extracted from spectral and scattering data. A facilitated laboratory control algorithm is introduced and it is demonstrated that maps provide a reliable dynamic analysis of strong-field control mechanisms. The framework of map-facilitated closed-loop inversion unifies the previously distinct fields of quantum Hamiltonian identification and coherent control. It also leads to a hybrid laboratory/computational algorithm that systematically determines the best experiments and data for global Hamiltonian identification. Conventional inversions have been limited to treating available data. Here, it is shown that optimal data, i.e., measurements with maximum information content, provides a superior means for studying molecular Hamiltonians. The concept of optimal
19. Nuclear Safety Information Center, Its Products and Services
ERIC Educational Resources Information Center
Buchanan, J. R.
1970-01-01
The Nuclear Safety Information Center (NSIC) serves as a focal point for the collection, analysis and dissemination of information related to safety problems encountered in the design, analysis, and operation of nuclear facilities. (Author/AB)
20. Nuclear Safety Information Center, Its Products and Services
ERIC Educational Resources Information Center
Buchanan, J. R.
1970-01-01
The Nuclear Safety Information Center (NSIC) serves as a focal point for the collection, analysis and dissemination of information related to safety problems encountered in the design, analysis, and operation of nuclear facilities. (Author/AB)
1. Toward spectroscopically accurate global ab initio potential energy surface for the acetylene-vinylidene isomerization
SciTech Connect
Han, Huixian; Li, Anyang; Guo, Hua
2014-12-28
A new full-dimensional global potential energy surface (PES) for the acetylene-vinylidene isomerization on the ground (S{sub 0}) electronic state has been constructed by fitting ∼37 000 high-level ab initio points using the permutation invariant polynomial-neural network method with a root mean square error of 9.54 cm{sup −1}. The geometries and harmonic vibrational frequencies of acetylene, vinylidene, and all other stationary points (two distinct transition states and one secondary minimum in between) have been determined on this PES. Furthermore, acetylene vibrational energy levels have been calculated using the Lanczos algorithm with an exact (J = 0) Hamiltonian. The vibrational energies up to 12 700 cm{sup −1} above the zero-point energy are in excellent agreement with the experimentally derived effective Hamiltonians, suggesting that the PES is approaching spectroscopic accuracy. In addition, analyses of the wavefunctions confirm the experimentally observed emergence of the local bending and counter-rotational modes in the highly excited bending vibrational states. The reproduction of the experimentally derived effective Hamiltonians for highly excited bending states signals the coming of age for the ab initio based PES, which can now be trusted for studying the isomerization reaction.
2. PREFACE: International Symposium on Exotic Nuclear Structure From Nucleons (ENSFN 2012)
Honma, Michio; Utsuno, Yutaka; Shimizu, Noritaka
2013-07-01
The International Symposium on 'Exotic Nuclear Structure From Nucleons (ENSFN2012)' was held at the Koshiba Hall, the University of Tokyo, Japan, from October 10th to 12th, 2012. This symposium was supported by RIKEN Nishina Center (RNC) and the Center for Nuclear Study (CNS), University of Tokyo. This symposium was devoted to discussing recent achievement and perspectives in the structure of exotic nuclei from the viewpoint of the nuclear force. The following subjects were covered in this symposium from both theoretical and experimental sides: Evolution of shell structure and collectivity in exotic nuclei Ab-initio theory and its application to exotic nuclei Advancement in large-scale nuclear-structure calculations Effective Hamiltonian and energy density functional Spin-isospin responses New aspects of two- and three-body forces Impact on nuclear astrophysics Emphasis was placed on the development of large-scale nuclear-structure calculations and the new experimental information on exotic nuclei. Around 80 participants attended this symposium and we enjoyed 37 excellent invited talks and 9 selected oral presentations. A special talk was presented to celebrate the 60th birthday of professor Takaharu Otsuka, who has made invaluable contribution to the progress in the fields covered in this symposium. The organizing committee consisted of T Abe (Tokyo), M Honma (Aizu; chair), N Itagaki (YITP, Kyoto), T Mizusaki (Senshu), T Nakatsukasa (RIKEN), H Sakurai (Tokyo/RIKEN), N Shimizu (CNS, Tokyo; scientific secretary), S Shimoura (CNS, Tokyo), Y Utsuno (JAEA/CNS, Tokyo; scientific secretary). Finally, we would like to thank all the speakers and the participants as well as the other organizers for their contributions which made the symposium very successful. Michio Honma, Yutaka Utsuno and Noritaka Shimizu Editors Tokyo, April 2013 Sponsors logo1 Sponsors logo2 The PDF also contains the conference program.
3. Ab initio study of electron-phonon coupling in rubrene
Ordejón, P.; Boskovic, D.; Panhans, M.; Ortmann, F.
2017-07-01
The use of ab initio methods for accurate simulations of electronic, phononic, and electron-phonon properties of molecular materials such as organic crystals is a challenge that is often tackled stepwise based on molecular properties calculated in gas phase and perturbatively treated parameters relevant for solid phases. In contrast, in this work we report a full first-principles description of such properties for the prototypical rubrene crystals. More specifically, we determine a Holstein-Peierls-type Hamiltonian for rubrene, including local and nonlocal electron-phonon couplings. Thereby, a recipe for circumventing the issue of numerical inaccuracies with low-frequency phonons is presented. In addition, we study the phenyl group motion with a molecular dynamics approach.
4. Exotic Nuclear Shapes:
Dudek, J.; Schunck, N.; Dubray, N.; Góźdź, A.
After recalling some in principle known but seldom mentioned facts about variety of concepts/notions of the nuclear shapes, we briefly summarize the results of the recent microscopic calculations predicting the existence of the large-elongation (hyper-deformed) nuclear configurations — as well as another series of calculations predicting that some nuclei should exhibit high-rank symmetries: the tetrahedral and the octahedral ones. The latter are associated with 48- and 96- symmetry elements, respectively, of the nuclear mean-field Hamiltonian. Obviously the physics motivations behind the hyper-deformation and the high-rank symmetry studies are not the observations of the new geometrical forms as such; in our opinion these motivations are much deeper and are given in the text.
5. Probing the symmetries of the Dirac Hamiltonian with axially deformed scalar and vector potentials by similarity renormalization group.
PubMed
Guo, Jian-You; Chen, Shou-Wan; Niu, Zhong-Ming; Li, Dong-Peng; Liu, Quan
2014-02-14
Symmetry is an important and basic topic in physics. The similarity renormalization group theory provides a novel view to study the symmetries hidden in the Dirac Hamiltonian, especially for the deformed system. Based on the similarity renormalization group theory, the contributions from the nonrelativistic term, the spin-orbit term, the dynamical term, the relativistic modification of kinetic energy, and the Darwin term are self-consistently extracted from a general Dirac Hamiltonian and, hence, we get an accurate description for their dependence on the deformation. Taking an axially deformed nucleus as an example, we find that the self-consistent description of the nonrelativistic term, spin-orbit term, and dynamical term is crucial for understanding the relativistic symmetries and their breaking in a deformed nuclear system.
6. Magnetic Coupling Constants in Three Electrons Three Centers Problems from Effective Hamiltonian Theory and Validation of Broken Symmetry-Based Approaches.
PubMed
Reta, Daniel; Moreira, Ibério de P R; Illas, Francesc
2016-07-12
In the most general case of three electrons in three symmetry unrelated centers with Ŝ1 = Ŝ2 = Ŝ3 = 1/2 localized magnetic moments, the low energy spectrum consists of one quartet (Q) and two doublet (D1, D2) pure spin states. The energy splitting between these spin states can be described with the well-known Heisenberg-Dirac-Van Vleck (HDVV) model spin Hamiltonian, and their corresponding energy expressions are expressed in terms of the three different two-body magnetic coupling constants J12, J23, and J13. However, the values of all three magnetic coupling constants cannot be extracted using the calculated energy of the three spin-adapted states since only two linearly independent energy differences between pure spin states exist. This problem has been recently investigated by Reta et al. (J. Chem. Theory Comput. 2015, 11, 3650), resulting in an alternative proposal to the original Noodleman's broken symmetry mapping approach. In the present work, this proposal is validated by means of ab initio effective Hamiltonian theory, which allows a direct extraction of all three J values from the one-to-one correspondence between the matrix elements of both effective and HDVV Hamiltonian. The effective Hamiltonian matrix representation has been constructed from configuration interaction wave functions for the three spin states obtained for two model systems showing a different degree of delocalization of the unpaired electrons. These encompass a trinuclear Cu(II) complex and a π-conjugated purely organic triradical.
7. Hamiltonian thermodynamics of three-dimensional dilatonic black holes
Dias, Gonçalo A. S.; Lemos, José P. S.
2008-08-01
The action for a class of three-dimensional dilaton-gravity theories with a negative cosmological constant can be recast in a Brans-Dicke type action, with its free ω parameter. These theories have static spherically symmetric black holes. Those with well formulated asymptotics are studied through a Hamiltonian formalism, and their thermodynamical properties are found out. The theories studied are general relativity (ω→∞), a dimensionally reduced cylindrical four-dimensional general relativity theory (ω=0), and a theory representing a class of theories (ω=-3). The Hamiltonian formalism is set up in three dimensions through foliations on the right region of the Carter-Penrose diagram, with the bifurcation 1-sphere as the left boundary, and anti de Sitter infinity as the right boundary. The metric functions on the foliated hypersurfaces are the canonical coordinates. The Hamiltonian action is written, the Hamiltonian being a sum of constraints. One finds a new action which yields an unconstrained theory with one pair of canonical coordinates {M,PM}, M being the mass parameter and PM its conjugate momenta The resulting Hamiltonian is a sum of boundary terms only. A quantization of the theory is performed. The Schrödinger evolution operator is constructed, the trace is taken, and the partition function of the canonical ensemble is obtained. The black hole entropies differ, in general, from the usual quarter of the horizon area due to the dilaton.
8. Periodic pseudo-Hermitian Hamiltonian: Nonadiabatic geometric phase
Maamache, M.
2015-09-01
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians H are indeed real if they are invariant under the combined action of self-adjoint parity P and time reversal T . The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the P T -symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t ) =ZH(t ) exp(i MHt ) associated with the periodic pseudo-hermitian Hamiltonian H (t )=H (t +T ) . One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.
9. Ab Interno Trabeculectomy
PubMed Central
Pantcheva, Mina B.; Kahook, Malik Y.
2010-01-01
Anterior chamber drainage angle surgery, namely trabeculotomy and goniotomy, has been commonly utilized in children for many years. Its’ reported success has ranged between 68% and 100% in infants and young children with congenital glaucoma. However, the long-term success of these procedures has been limited in adults presumably due to the formation of anterior synechiae (AS) in the postoperative phase. Recently, ab interno trabeculectomy with the Trabectome™ has emerged as a novel surgical approach to effectively and selectively remove and ablate the trabecular meshwork and the inner wall of the Schlemm’s canal in an attempt to avoid AS formation or other forms of wound healing with resultant closure of the cleft. This procedure seems to have an appealing safety profile with respect to early hypotony or infection if compared to trabeculectomy or glaucoma drainage device implantation. This might be advantageous in some of the impoverish regions of the Middle East and Africa where patients experience difficulties keeping up with their postoperative visits. It is important to note that no randomized trial comparing the Trabectome to other glaucoma procedures appears to have been published to date. Trabectome surgery is not a panacea, however, and it is associated with early postoperative intraocular pressure spikes that may require additional glaucoma surgery as well as a high incidence of hyphema. Reported results show that postoperative intraocular pressure (IOP) remains, at best, in the mid-teen range making it undesirable in patients with low-target IOP goals. A major advantage of Trabectome surgery is that it does not preclude further glaucoma surgery involving the conjunctiva, such as a trabeculectomy or drainage device implantation. As prospective randomized long-term clinical data become available, we will be better positioned to elucidate the exact role of this technique in the glaucoma surgical armamentarium. PMID:21180426
10. Ab initio downfolding for electron-phonon-coupled systems: Constrained density-functional perturbation theory
Nomura, Yusuke; Arita, Ryotaro
2015-12-01
We formulate an ab initio downfolding scheme for electron-phonon-coupled systems. In this scheme, we calculate partially renormalized phonon frequencies and electron-phonon coupling, which include the screening effects of high-energy electrons, to construct a realistic Hamiltonian consisting of low-energy electron and phonon degrees of freedom. We show that our scheme can be implemented by slightly modifying the density functional-perturbation theory (DFPT), which is one of the standard methods for calculating phonon properties from first principles. Our scheme, which we call the constrained DFPT, can be applied to various phonon-related problems, such as superconductivity, electron and thermal transport, thermoelectricity, piezoelectricity, dielectricity, and multiferroicity. We believe that the constrained DFPT provides a firm basis for the understanding of the role of phonons in strongly correlated materials. Here, we apply the scheme to fullerene superconductors and discuss how the realistic low-energy Hamiltonian is constructed.
11. Many-body ab initio study of antiferromagnetic {Cr7M } molecular rings
Chiesa, A.; Carretta, S.; Santini, P.; Amoretti, G.; Pavarini, E.
2016-12-01
Antiferromagnetic molecular rings are widely studied both for fundamental quantum-mechanical issues and for technological applications, particularly in the field of quantum information processing. Here we present a detailed first-principles study of two families—purple and green—of {Cr7M } antiferromagnetic rings, where M is a divalent transition metal ion (M =Ni2 + , Mn2 +, and Zn2 +). We employ a recently developed flexible and efficient scheme to build ab initio system-specific Hubbard models. From such many-body models we systematically derive the low-energy effective spin Hamiltonian for the rings. Our approach allows us to calculate isotropic as well as anisotropic terms of the spin Hamiltonian, without any a priori assumption on its form. For each compound we calculate magnetic exchange couplings, zero-field splitting tensors, and gyromagnetic tensors, finding good agreement with experimental results.
12. Quantum Monte Carlo Calculations in Solids with Downfolded Hamiltonians
Ma, Fengjie; Purwanto, Wirawan; Zhang, Shiwei; Krakauer, Henry
2015-06-01
We present a combination of a downfolding many-body approach with auxiliary-field quantum Monte Carlo (AFQMC) calculations for extended systems. Many-body calculations operate on a simpler Hamiltonian which retains material-specific properties. The Hamiltonian is systematically improvable and allows one to dial, in principle, between the simplest model and the original Hamiltonian. As a by-product, pseudopotential errors are essentially eliminated using frozen orbitals constructed adaptively from the solid environment. The computational cost of the many-body calculation is dramatically reduced without sacrificing accuracy. Excellent accuracy is achieved for a range of solids, including semiconductors, ionic insulators, and metals. We apply the method to calculate the equation of state of cubic BN under ultrahigh pressure, and determine the spin gap in NiO, a challenging prototypical material with strong electron correlation effects.
13. Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms
Xin-Lei, Kong; Hui-Bin, Wu; Feng-Xiang, Mei
2016-01-01
In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoffian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities. Project supported by the National Natural Science Foundation of China (Grant No. 11272050), the Excellent Young Teachers Program of North China University of Technology (Grant No. XN132), and the Construction Plan for Innovative Research Team of North China University of Technology (Grant No. XN129).
14. Limit of small exits in open Hamiltonian systems.
PubMed
Aguirre, Jacobo; Sanjuán, Miguel A F
2003-05-01
The nature of open Hamiltonian systems is analyzed, when the size of the exits decreases and tends to zero. Fractal basins appear typically in open Hamiltonian systems, but we claim that in the limit of small exits, the invariant sets tend to fill up the whole phase space with the strong consequence that a new kind of basin appears, where the unpredictability grows indefinitely. This means that for finite, arbitrarily small accuracy, we can find uncertain basins, where any information about the future of the system is lost. This total indeterminism had only been reported in dissipative systems, in particular in the so-called intermingled riddled basins, as well as in the riddledlike basins. We show that this peculiar, behavior is a general feature of open Hamiltonian systems.
15. Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory
Cang, Shijian; Wu, Aiguo; Wang, Zenghui; Chen, Zengqiang
Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.
16. Reverse engineering of a Hamiltonian by designing the evolution operators
PubMed Central
Kang, Yi-Hao; Chen, Ye-Hong; Wu, Qi-Cheng; Huang, Bi-Hua; Xia, Yan; Song, Jie
2016-01-01
We propose an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators to eliminate the terms of Hamiltonian which are hard to be realized in practice. Different from transitionless quantum driving (TQD), the present scheme is focus on only one or parts of moving states in a D-dimension (D ≥ 3) system. The numerical simulation shows that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. An example is given by using this scheme to realize the population transfer for a Rydberg atom. The influences of various decoherence processes are discussed by numerical simulation and the result shows that the scheme is fast and robust against the decoherence and operational imperfection. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments. PMID:27444137
17. On the Hamiltonian formalism of the tetrad-connection gravity
Lagraa, M. H.; Lagraa, M.; Touhami, N.
2017-06-01
We present a detailed analysis of the Hamiltonian constraints of the d-dimensional tetrad-connection gravity where the non-dynamic part of the spatial connection is fixed to zero by an adequate gauge transformation. This new action leads to a coherent Hamiltonian formalism where the Lorentz, scalar and vectorial first-class constraints obey a closed algebra in terms of Poisson brackets. This algebra closes with structure constants instead of structure functions resulting from the Hamiltonian formalisms based on the A.D.M. decomposition. The same algebra of the reduced first-class constraints, where the second-class constraints are eliminated as strong equalities, is obtained in terms of Dirac brackets. These first-class constraints lead to the same physical degrees of freedom of the general relativity.
18. Riemannian geometry of Hamiltonian chaos: Hints for a general theory
Cerruti-Sola, Monica; Ciraolo, Guido; Franzosi, Roberto; Pettini, Marco
2008-10-01
We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam β model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down. The breakdown is expected whenever the topology of the mechanical manifolds is nontrivial. This is an important step forward in view of developing a geometric theory of Hamiltonian chaos of general validity.
19. Effective Hamiltonian for protected edge states in graphene
Winkler, R.; Deshpande, H.
2017-06-01
Edge states in topological insulators (TIs) disperse symmetrically about one of the time-reversal invariant momenta Λ in the Brillouin zone (BZ) with protected degeneracies at Λ . Commonly TIs are distinguished from trivial insulators by the values of one or multiple topological invariants that require an analysis of the bulk band structure across the BZ. We propose an effective two-band Hamiltonian for the electronic states in graphene based on a Taylor expansion of the tight-binding Hamiltonian about the time-reversal invariant M point at the edge of the BZ. This Hamiltonian provides a faithful description of the protected edge states for both zigzag and armchair ribbons, though the concept of a BZ is not part of such an effective model. We show that the edge states are determined by a band inversion in both reciprocal and real space, which allows one to select Λ for the edge states without affecting the bulk spectrum.
20. Hamiltonian formalism of two-dimensional Vlasov kinetic equation
PubMed Central
Pavlov, Maxim V.
2014-01-01
In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo–Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo–Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented. PMID:25484603
1. Hamiltonian formalism of two-dimensional Vlasov kinetic equation.
PubMed
Pavlov, Maxim V
2014-12-08
In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo-Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.
2. Reverse engineering of a Hamiltonian by designing the evolution operators
Kang, Yi-Hao; Chen, Ye-Hong; Wu, Qi-Cheng; Huang, Bi-Hua; Xia, Yan; Song, Jie
2016-07-01
We propose an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators to eliminate the terms of Hamiltonian which are hard to be realized in practice. Different from transitionless quantum driving (TQD), the present scheme is focus on only one or parts of moving states in a D-dimension (D ≥ 3) system. The numerical simulation shows that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. An example is given by using this scheme to realize the population transfer for a Rydberg atom. The influences of various decoherence processes are discussed by numerical simulation and the result shows that the scheme is fast and robust against the decoherence and operational imperfection. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments.
3. Reverse engineering of a Hamiltonian by designing the evolution operators.
PubMed
Kang, Yi-Hao; Chen, Ye-Hong; Wu, Qi-Cheng; Huang, Bi-Hua; Xia, Yan; Song, Jie
2016-07-22
We propose an effective and flexible scheme for reverse engineering of a Hamiltonian by designing the evolution operators to eliminate the terms of Hamiltonian which are hard to be realized in practice. Different from transitionless quantum driving (TQD), the present scheme is focus on only one or parts of moving states in a D-dimension (D ≥ 3) system. The numerical simulation shows that the present scheme not only contains the results of TQD, but also has more free parameters, which make this scheme more flexible. An example is given by using this scheme to realize the population transfer for a Rydberg atom. The influences of various decoherence processes are discussed by numerical simulation and the result shows that the scheme is fast and robust against the decoherence and operational imperfection. Therefore, this scheme may be used to construct a Hamiltonian which can be realized in experiments.
4. Riemannian geometry of Hamiltonian chaos: hints for a general theory.
PubMed
Cerruti-Sola, Monica; Ciraolo, Guido; Franzosi, Roberto; Pettini, Marco
2008-10-01
We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam beta model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down. The breakdown is expected whenever the topology of the mechanical manifolds is nontrivial. This is an important step forward in view of developing a geometric theory of Hamiltonian chaos of general validity.
5. Non-Hermitian Hamiltonians with unitary and antiunitary symmetries
Fernández, Francisco M.; Garcia, Javier
2014-03-01
We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary for the diagonalization of the Hamiltonian in a given basis set. We can also classify the solutions according to the irreducible representations of the point group and thus analyse their properties separately. One of the main results of this paper is that some PT-symmetric Hamiltonians with point-group symmetry C2v exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit which suggests that the phenomenon is not robust. Point-group symmetry enables us to explain such anomalous behaviour and to choose a suitable antiunitary operator for the PT symmetry.
6. Effective Hamiltonians for fastly driven tight-binding chains
Itin, A. P.; Neishtadt, A. I.
2014-02-01
We consider a single particle tunnelling in a tight-binding model with nearest-neighbour couplings, in the presence of a periodic high-frequency force. An effective Hamiltonian for the particle is derived using an averaging method resembling classical canonical perturbation theory. Three cases are considered: uniform lattice with periodic and open boundary conditions, and lattice with a parabolic potential. We find that in the latter case, interplay of the potential and driving leads to appearance of the effective next-nearest neighbour couplings. In the uniform case with periodic boundary conditions the second- and third-order corrections to the averaged Hamiltonian are completely absent, while in the case with open boundary conditions they have a very simple form, found before in some particular cases by S. Longhi (2008) [10]. These general results may found applications in designing effective Hamiltonian models in experiments with ultracold atoms in optical lattices, e.g. for simulating solid-state phenomena.
7. Autonomous Biological System (ABS) experiments.
PubMed
MacCallum, T K; Anderson, G A; Poynter, J E; Stodieck, L S; Klaus, D M
1998-12-01
Three space flight experiments have been conducted to test and demonstrate the use of a passively controlled, materially closed, bioregenerative life support system in space. The Autonomous Biological System (ABS) provides an experimental environment for long term growth and breeding of aquatic plants and animals. The ABS is completely materially closed, isolated from human life support systems and cabin atmosphere contaminants, and requires little need for astronaut intervention. Testing of the ABS marked several firsts: the first aquatic angiosperms to be grown in space; the first higher organisms (aquatic invertebrate animals) to complete their life cycles in space; the first completely bioregenerative life support system in space; and, among the first gravitational ecology experiments. As an introduction this paper describes the ABS, its flight performance, advantages and disadvantages.
8. Ab initio two-component Ehrenfest dynamics
SciTech Connect
Ding, Feizhi; Goings, Joshua J.; Liu, Hongbin; Lingerfelt, David B.; Li, Xiaosong
2015-09-21
We present an ab initio two-component Ehrenfest-based mixed quantum/classical molecular dynamics method to describe the effect of nuclear motion on the electron spin dynamics (and vice versa) in molecular systems. The two-component time-dependent non-collinear density functional theory is used for the propagation of spin-polarized electrons while the nuclei are treated classically. We use a three-time-step algorithm for the numerical integration of the coupled equations of motion, namely, the velocity Verlet for nuclear motion, the nuclear-position-dependent midpoint Fock update, and the modified midpoint and unitary transformation method for electronic propagation. As a test case, the method is applied to the dissociation of H{sub 2} and O{sub 2}. In contrast to conventional Ehrenfest dynamics, this two-component approach provides a first principles description of the dynamics of non-collinear (e.g., spin-frustrated) magnetic materials, as well as the proper description of spin-state crossover, spin-rotation, and spin-flip dynamics by relaxing the constraint on spin configuration. This method also holds potential for applications to spin transport in molecular or even nanoscale magnetic devices.
9. Algorithmic approach to simulate Hamiltonian dynamics and an NMR simulation of quantum state transfer
Ajoy, Ashok; Rao, Rama Koteswara; Kumar, Anil; Rungta, Pranaw
2012-03-01
We propose an iterative algorithm to simulate the dynamics generated by any n-qubit Hamiltonian. The simulation entails decomposing the unitary time evolution operator U (unitary) into a product of different time-step unitaries. The algorithm product-decomposes U in a chosen operator basis by identifying a certain symmetry of U that is intimately related to the number of gates in the decomposition. We illustrate the algorithm by first obtaining a polynomial decomposition in the Pauli basis of the n-qubit quantum state transfer unitary by Di Franco [Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.101.230502 101, 230502 (2008)] that transports quantum information from one end of a spin chain to the other, and then implement it in nuclear magnetic resonance to demonstrate that the decomposition is experimentally viable. We further experimentally test the resilience of the state transfer to static errors in the coupling parameters of the simulated Hamiltonian. This is done by decomposing and simulating the corresponding imperfect unitaries.
10. Ab Initio Multiple Spawning Method for Intersystem Crossing Dynamics: Spin-Forbidden Transitions between (3)B1 and (1)A1 States of GeH2.
PubMed
Fedorov, Dmitry A; Pruitt, Spencer R; Keipert, Kristopher; Gordon, Mark S; Varganov, Sergey A
2016-05-12
Dynamics at intersystem crossings are fundamental to many processes in chemistry, physics, and biology. The ab initio multiple spawning (AIMS) method was originally developed to describe internal conversion dynamics at conical intersections where derivative coupling is responsible for nonadiabatic transitions between electronic states with the same spin multiplicity. Here, the applicability of the AIMS method is extended to intersystem crossing dynamics in which transitions between electronic states with different spin multiplicities are mediated by relativistic spin-orbit coupling. In the direct AIMS dynamics, the nuclear wave function is expanded in the basis of frozen multidimensional Gaussians propagating on the coupled electronic potential energy surfaces calculated on the fly. The AIMS method for intersystem crossing is used to describe the nonadiabatic transitions between the (3)B1 and (1)A1 states of GeH2. The potential energies and gradients were obtained at the CASSCF(6,6)/6-31G(d) level of theory. The spin-orbit coupling matrix elements were calculated with the configuration interaction method using the two-electron Breit-Pauli Hamiltonian. The excited (3)B1 state lifetime and intersystem crossing rate constants were estimated by fitting the AIMS state population with the first-order kinetics equation for a reversible unimolecular reaction. The obtained rate constants are compared with the values predicted by the statistical nonadiabatic transition state theory with transition probabilities calculated using the Landau-Zener and weak coupling formulas.
11. Bounded stabilisation of stochastic port-Hamiltonian systems
Satoh, Satoshi; Saeki, Masami
2014-08-01
This paper proposes a stochastic bounded stabilisation method for a class of stochastic port-Hamiltonian systems. Both full-actuated and underactuated mechanical systems in the presence of noise are considered in this class. The proposed method gives conditions for the controller gain and design parameters under which the state remains bounded in probability. The bounded region and achieving probability are both assignable, and a stochastic Lyapunov function is explicitly provided based on a Hamiltonian structure. Although many conventional stabilisation methods assume that the noise vanishes at the origin, the proposed method is applicable to systems under persistent disturbances.
12. Evolution-Free Hamiltonian Parameter Estimation through Zeeman Markers
Burgarth, Daniel; Ajoy, Ashok
2017-07-01
We provide a protocol for Hamiltonian parameter estimation which relies only on the Zeeman effect. No time-dependent quantities need to be measured; it fully suffices to observe spectral shifts induced by fields applied to local "markers." We demonstrate the idea with a simple tight-binding Hamiltonian and numerically show stability with respect to Gaussian noise on the spectral measurements. Then we generalize the result to show applicability to a wide range of systems, including quantum spin chains, networks of qubits, and coupled harmonic oscillators, and suggest potential experimental implementations.
13. Evolution-Free Hamiltonian Parameter Estimation through Zeeman Markers.
PubMed
Burgarth, Daniel; Ajoy, Ashok
2017-07-21
We provide a protocol for Hamiltonian parameter estimation which relies only on the Zeeman effect. No time-dependent quantities need to be measured; it fully suffices to observe spectral shifts induced by fields applied to local "markers." We demonstrate the idea with a simple tight-binding Hamiltonian and numerically show stability with respect to Gaussian noise on the spectral measurements. Then we generalize the result to show applicability to a wide range of systems, including quantum spin chains, networks of qubits, and coupled harmonic oscillators, and suggest potential experimental implementations.
14. Hamiltonian and Godunov structures of the Grad hierarchy.
PubMed
Grmela, Miroslav; Hong, Liu; Jou, David; Lebon, Georgy; Pavelka, Michal
2017-03-01
The time evolution governed by the Boltzmann kinetic equation is compatible with mechanics and thermodynamics. The former compatibility is mathematically expressed in the Hamiltonian and Godunov structures, the latter in the structure of gradient dynamics guaranteeing the growth of entropy and consequently the approach to equilibrium. We carry all three structures to the Grad reformulation of the Boltzmann equation (to the Grad hierarchy). First, we recognize the structures in the infinite Grad hierarchy and then in several examples of finite hierarchies representing extended hydrodynamic equations. In the context of Grad's hierarchies, we also investigate relations between Hamiltonian and Godunov structures.
15. Lagrangian-Hamiltonian unified formalism for field theory
Echeverría-Enríquez, Arturo; López, Carlos; Marín-Solano, Jesús; Muñoz-Lecanda, Miguel C.; Román-Roy, Narciso
2004-01-01
The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems (including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.). In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations.
16. Continuation of periodic orbits in symmetric Hamiltonian and conservative systems
Galan-Vioque, J.; Almaraz, F. J. M.; Macías, E. F.
2014-12-01
We present and review results on the continuation and bifurcation of periodic solutions in conservative, reversible and Hamiltonian systems in the presence of symmetries. In particular we show how two-point boundary value problem continuation software can be used to compute families of periodic solutions of symmetric Hamiltonian systems. The technique is introduced with a very simple model example (the mathematical pendulum), justified with a theoretical continuation result and then applied to two non trivial examples: the non integrable spring pendulum and the continuation of the figure eight solution of the three body problem.
17. Comments on HKT supersymmetric sigma models and their Hamiltonian reduction
Fedoruk, Sergey; Smilga, Andrei
2015-05-01
Using complex notation, we present new simple expressions for two pairs of complex supercharges in HKT (‘hyper-Kähler with torsion’) supersymmetric sigma models. The second pair of supercharges depends on the holomorphic antisymmetric ‘hypercomplex structure’ tensor {{I}jk} which plays the same role for the HKT models as the complex structure tensor for the Kähler models. When the Hamiltonian and supercharges commute with the momenta conjugate to the imaginary parts of the complex coordinates, one can perform a Hamiltonian reduction. The models thus obtained represent a special class of quasicomplex sigma models introduced recently by Ivanov and Smilga (2013 SIGMA 9 069)
18. A Geometrical Version of the Maxwell-Vlasov Hamiltonian Structure
Vittot, Michel; Morrison, Philip
2014-10-01
We present a geometrization of the Hamiltonian approach of classical electrodynamics, via (non-canonical) Poisson structures. This relativistic Hamiltonian framework (introduced by Morrison, Marsden, Weinstein) is a field theory written in terms of differential forms, independently of the gauge potentials. This algebraic and geometric description of the Vlasov kinetics is well suited for a perturbation theory, in a strong inhomogeneous magnetic field (expansion in 1/B, with all the curvature terms...), like in magnetically confined plasmas, and in any coordinates, for instance adapted to a Tokamak (toroidal coordinates, or else...).
19. Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics.
PubMed
Ilinskii, Yurii A; Hamilton, Mark F; Zabolotskaya, Evgenia A
2007-02-01
Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.
20. Phase equilibria in polymer blend thin films: a Hamiltonian approach.
PubMed
Souche, M; Clarke, N
2009-12-28
We propose a Hamiltonian formulation of the Flory-Huggins-de Gennes theory describing a polymer blend thin film. We then focus on the case of 50:50 polymer blends confined between antisymmetric walls. The different phases of the system and the transitions between them, including finite-size effects, are systematically studied through their relation with the geometry of the Hamiltonian flow in phase space. This method provides an easy and efficient way, with strong graphical insight, to infer the qualitative physical behavior of polymer blend thin films.
1. Characterizing Ground and Thermal States of Few-Body Hamiltonians
Huber, Felix; Gühne, Otfried
2016-07-01
The question whether a given quantum state is a ground or thermal state of a few-body Hamiltonian can be used to characterize the complexity of the state and is important for possible experimental implementations. We provide methods to characterize the states generated by two- and, more generally, k -body Hamiltonians as well as the convex hull of these sets. This leads to new insights into the question of which states are uniquely determined by their marginals and to a generalization of the concept of entanglement. Finally, certification methods for quantum simulation can be derived.
2. The Hamilton-Jacobi method and Hamiltonian maps
Abdullaev, S. S.
2002-03-01
A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton-Jacobi method and Jacobi's theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method.
3. Effective Hamiltonian for non-minimally coupled scalar fields
Meşe, Emine; Pirinççiog˜Lu, Nurettin; Açıkgöz, Irfan; Binbay, Figen
2009-01-01
In the post Newtonian limit, a non-relativistic Hamiltonian is derived for scalar fields with quartic self-interaction and non-minimal coupling to the curvature scalar of the background spacetime. These effects are found to contribute to the non-relativistic Hamiltonian by adding nonlinearities and by modifying the gravitational Darwin term. As we discuss briefly in the text, the impact of these novel structures can be sizable in dense media like neutron star core, and can have observable signatures in phase transitions, for example.
4. Nonclassical degrees of freedom in the Riemann Hamiltonian.
PubMed
Srednicki, Mark
2011-09-02
The Hilbert-Pólya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum Hamiltonian. If so, conjectures by Katz and Sarnak put this Hamiltonian in the Altland-Zirnbauer universality class C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of -1. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.
5. A parity breaking Ising chain Hamiltonian as a Brownian motor
Cornu, F.; Hilhorst, H. J.
2014-10-01
We consider the translationally invariant but parity (left-right symmetry) breaking Ising chain Hamiltonian {\\cal H} =-{U_2}\\sumk sksk+1 - {U_3}\\sumk sksk+1sk+3 and let this system evolve by Kawasaki spin exchange dynamics. Monte Carlo simulations show that perturbations forcing this system off equilibrium make it act as a Brownian molecular motor which, in the lattice gas interpretation, transports particles along the chain. We determine the particle current under various different circumstances, in particular as a function of the ratio {U_3}/{U_2} and of the conserved magnetization M=\\sum_ksk . The symmetry of the U3 term in the Hamiltonian is discussed.
6. Effective Hamiltonian for a microwave billiard with attached waveguide.
PubMed
Stöckmann, H-J; Persson, E; Kim, Y-H; Barth, M; Kuhl, U; Rotter, I
2002-06-01
In a recent work the resonance widths in a microwave billiard with attached waveguide were studied in dependence on the coupling strength [E. Persson et al., Phys. Rev. Lett. 85, 2478 (2000)], and resonance trapping was experimentally found. In the present paper an effective Hamiltonian is derived that depends exclusively on billiard and waveguide geometry. Its eigenvalues give the poles of the scattering matrix provided that the system and environment are defined adequately. Further, we present the results of resonance trapping measurements where, in addition to our previous work, the position of the slit aperture within the waveguide was varied. Numerical simulations with the derived Hamiltonian qualitatively reproduce the experimental data.
7. Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories
Román-Roy, Narciso
2009-11-01
This review paper is devoted to presenting the standard multisymplectic formulation for describing geometrically classical field theories, both the regular and singular cases. First, the main features of the Lagrangian formalism are revisited and, second, the Hamiltonian formalism is constructed using Hamiltonian sections. In both cases, the variational principles leading to the Euler-Lagrange and the Hamilton-De Donder-Weyl equations, respectively, are stated, and these field equations are given in different but equivalent geometrical ways in each formalism. Finally, both are unified in a new formulation (which has been developed in the last years), following the original ideas of Rusk and Skinner for mechanical systems.
8. Quantifying the Effects of Higher Order Jahn-Teller Coupling Terms on a Quadratic Jahn-Teller Hamiltonian in the Case of NO_3 and Li_3.
Tran, Henry; Stanton, John F.; Miller, Terry A.
2016-06-01
The Jahn-Teller (JT) effect represents an enormous complication in the understanding of many molecules. We have been able to assign ˜20 vibronic bands in the tilde{A}^2E'' ← tilde{X}^2A_2' transition of NO_3 and determine the linear and quadratic JT coupling terms for ν_3 and ν_4, indicating strong and weak JT coupling along these modes respectively. It was found that the experimental results quantitatively disagree with ones determined from a vibronic Hamiltonian based on high-level ab-initio theory. Typical analyses of experimental data use the quadratic JT Hamiltonian because limited measured levels tend to allow fitting only to coupling terms up to quadratic JT coupling. Hence, these analyses may neglect key contributions from cubic and quartic terms. To quantify this limitation, we have fit artificial spectra calculated with up to fourth order terms in the potential using a quadratic JT Hamiltonian and analyzed the results. The parameters chosen for this analysis are determined from ab-initio potentials for the tilde{A} state of NO_3 and tilde{X} state of Li_3 to gain further insight on these molecules. Our initial results concerning the limitations of the quadratic JT Hamiltonian will be presented. T. Codd, M.-W. Chen, M. Roudjane, J. F. Stanton, and T. A. Miller. Jet cooled cavity ringdown spectroscopy of the tilde{A}^2E'' ← tilde{X}^2A'_2 Transition of the NO_3 Radical. J. Chem. Phys., 142:184305, 2015
9. Ab initio calculations of light-ion fusion reactions
SciTech Connect
Hupin, G.; Quaglioni, S.; Navratil, P.
2012-10-20
The exact treatment of nuclei starting from the constituent nucleons and the fundamental interactions among them has been a long-standing goal in nuclear physics. Above all nuclear scattering and reactions, which require the solution of the many-body quantum-mechanical problem in the continuum, represent an extraordinary theoretical as well as computational challenge for ab initio approaches. The ab initio No-Core Shell Model/Resonating-Group Method (NCSM/RGM) complements a microscopic cluster technique with the use of realistic interactions, and a microscopic and consistent description of the nucleon clusters. This approach is capable of describing simultaneously both bound and scattering states in light nuclei. Recent applications to light nuclei scattering and fusion reactions relevant to energy production in stars and Earth based fusion facilities, such as the deuterium-{sup 3}He fusion, are presented. Progress toward the inclusion of the three nucleon force into the formalism is outlined.
10. Ab Initio Calculation of the Hoyle State
SciTech Connect
Epelbaum, Evgeny; Krebs, Hermann; Lee, Dean; Meissner, Ulf-G.
2011-05-13
The Hoyle state plays a crucial role in the helium burning of stars heavier than our Sun and in the production of carbon and other elements necessary for life. This excited state of the carbon-12 nucleus was postulated by Hoyle as a necessary ingredient for the fusion of three alpha particles to produce carbon at stellar temperatures. Although the Hoyle state was seen experimentally more than a half century ago nuclear theorists have not yet uncovered the nature of this state from first principles. In this Letter we report the first ab initio calculation of the low-lying states of carbon-12 using supercomputer lattice simulations and a theoretical framework known as effective field theory. In addition to the ground state and excited spin-2 state, we find a resonance at -85(3) MeV with all of the properties of the Hoyle state and in agreement with the experimentally observed energy.
11. Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
Fring, Andreas; Frith, Thomas
2017-01-01
We propose a procedure to obtain exact analytical solutions to the time-dependent Schrödinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.
12. Bohr Hamiltonian with a deformation-dependent mass term for the Davidson potential
SciTech Connect
Bonatsos, Dennis; Georgoudis, P. E.; Lenis, D.; Minkov, N.; Quesne, C.
2011-04-15
Analytical expressions for spectra and wave functions are derived for a Bohr Hamiltonian, describing the collective motion of deformed nuclei, in which the mass is allowed to depend on the nuclear deformation. Solutions are obtained for separable potentials consisting of a Davidson potential in the {beta} variable, in the cases of {gamma}-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. The solution, called the deformation-dependent mass (DDM) Davidson model, is achieved by using techniques of supersymmetric quantum mechanics (SUSYQM), involving a deformed shape invariance condition. Spectra and B(E2) transition rates are compared to experimental data. The dependence of the mass on the deformation, dictated by SUSYQM for the potential used, reduces the rate of increase of the moment of inertia with deformation, removing a main drawback of the model.
13. Shapes and stability of algebraic nuclear models
NASA Technical Reports Server (NTRS)
Lopez-Moreno, Enrique; Castanos, Octavio
1995-01-01
A generalization of the procedure to study shapes and stability of algebraic nuclear models introduced by Gilmore is presented. One calculates the expectation value of the Hamiltonian with respect to the coherent states of the algebraic structure of the system. Then equilibrium configurations of the resulting energy surface, which depends in general on state variables and a set of parameters, are classified through the Catastrophe theory. For one- and two-body interactions in the Hamiltonian of the interacting Boson model-1, the critical points are organized through the Cusp catastrophe. As an example, we apply this Separatrix to describe the energy surfaces associated to the Rutenium and Samarium isotopes.
14. Path-integral description of combined Hamiltonian and non-Hamiltonian dynamics in quantum dissipative systems
Barth, A. M.; Vagov, A.; Axt, V. M.
2016-09-01
We present a numerical path-integral iteration scheme for the low-dimensional reduced density matrix of a time-dependent quantum dissipative system. Our approach simultaneously accounts for the combined action of a microscopically modeled pure-dephasing-type coupling to a continuum of harmonic oscillators representing, e.g., phonons, and further environmental interactions inducing non-Hamiltonian dynamics in the inner system represented, e.g., by Lindblad-type dissipation or relaxation. Our formulation of the path-integral method allows for a numerically exact treatment of the coupling to the oscillator modes and moreover is general enough to provide a natural way to include Markovian processes that are sufficiently described by rate equations. We apply this new formalism to a model of a single semiconductor quantum dot which includes the coupling to longitudinal acoustic phonons for two cases: (a) external laser excitation taking into account a phenomenological radiative decay of the excited dot state and (b) a coupling of the quantum dot to a single mode of an optical cavity taking into account cavity photon losses.
15. TOPICAL REVIEW: Quadrupole collective states within the Bohr collective Hamiltonian
Próchniak, L.; Rohoziński, S. G.
2009-12-01
The article reviews the general version of the Bohr collective model for the description of quadrupole collective states, including a detailed discussion of the model's kinematics. The quadrupole coordinates, momenta and angular momenta are defined and the structure of the isotropic tensor fields as functions of the tensor variables is investigated. After a comprehensive discussion of the quadrupole kinematics, the general form of the classical and quantum Bohr Hamiltonian is presented. The electric and magnetic multipole moment operators acting in the collective space are constructed and the collective sum rules are given. A discussion of the tensor structure of the collective wavefunctions and a review of various methods of solving the Bohr Hamiltonian eigenvalue equation are also presented. Next, the methods of derivation of the classical and quantum Bohr Hamiltonian from the microscopic many-body theory are recalled. Finally, the microscopic approach to the Bohr Hamiltonian is applied to interpret collective properties of 12 heavy even-even nuclei in the Hf-Hg region. Calculated energy levels and E2 transition probabilities are compared with experimental data.
16. A separable shadow Hamiltonian hybrid Monte Carlo method
Sweet, Christopher R.; Hampton, Scott S.; Skeel, Robert D.; Izaguirre, Jesús A.
2009-11-01
Hybrid Monte Carlo (HMC) is a rigorous sampling method that uses molecular dynamics (MD) as a global Monte Carlo move. The acceptance rate of HMC decays exponentially with system size. The shadow hybrid Monte Carlo (SHMC) was previously introduced to reduce this performance degradation by sampling instead from the shadow Hamiltonian defined for MD when using a symplectic integrator. SHMC's performance is limited by the need to generate momenta for the MD step from a nonseparable shadow Hamiltonian. We introduce the separable shadow Hamiltonian hybrid Monte Carlo (S2HMC) method based on a formulation of the leapfrog/Verlet integrator that corresponds to a separable shadow Hamiltonian, which allows efficient generation of momenta. S2HMC gives the acceptance rate of a fourth order integrator at the cost of a second-order integrator. Through numerical experiments we show that S2HMC consistently gives a speedup greater than two over HMC for systems with more than 4000 atoms for the same variance. By comparison, SHMC gave a maximum speedup of only 1.6 over HMC. S2HMC has the additional advantage of not requiring any user parameters beyond those of HMC. S2HMC is available in the program PROTOMOL 2.1. A Python version, adequate for didactic purposes, is also in MDL (http://mdlab.sourceforge.net/s2hmc).
17. A separable shadow Hamiltonian hybrid Monte Carlo method.
PubMed
Sweet, Christopher R; Hampton, Scott S; Skeel, Robert D; Izaguirre, Jesús A
2009-11-07
Hybrid Monte Carlo (HMC) is a rigorous sampling method that uses molecular dynamics (MD) as a global Monte Carlo move. The acceptance rate of HMC decays exponentially with system size. The shadow hybrid Monte Carlo (SHMC) was previously introduced to reduce this performance degradation by sampling instead from the shadow Hamiltonian defined for MD when using a symplectic integrator. SHMC's performance is limited by the need to generate momenta for the MD step from a nonseparable shadow Hamiltonian. We introduce the separable shadow Hamiltonian hybrid Monte Carlo (S2HMC) method based on a formulation of the leapfrog/Verlet integrator that corresponds to a separable shadow Hamiltonian, which allows efficient generation of momenta. S2HMC gives the acceptance rate of a fourth order integrator at the cost of a second-order integrator. Through numerical experiments we show that S2HMC consistently gives a speedup greater than two over HMC for systems with more than 4000 atoms for the same variance. By comparison, SHMC gave a maximum speedup of only 1.6 over HMC. S2HMC has the additional advantage of not requiring any user parameters beyond those of HMC. S2HMC is available in the program PROTOMOL 2.1. A Python version, adequate for didactic purposes, is also in MDL (http://mdlab.sourceforge.net/s2hmc).
18. Non-Hermitian Hamiltonians with unitary and antiunitary symmetries
SciTech Connect
Fernández, Francisco M. Garcia, Javier
2014-03-15
We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary for the diagonalization of the Hamiltonian in a given basis set. We can also classify the solutions according to the irreducible representations of the point group and thus analyse their properties separately. One of the main results of this paper is that some PT-symmetric Hamiltonians with point-group symmetry C{sub 2v} exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit which suggests that the phenomenon is not robust. Point-group symmetry enables us to explain such anomalous behaviour and to choose a suitable antiunitary operator for the PT symmetry. -- Highlights: •PT-symmetric Hamiltonians exhibit real eigenvalues when PT symmetry is unbroken. •PT-symmetric multidimensional oscillators appear to show PT phase transitions. •This transition was conjectured to be a high-energy phenomenon. •We show that point group symmetry is useful for predicting broken PT symmetry in multidimensional oscillators. •PT-symmetric oscillators with C{sub 2v} symmetry exhibit phase transitions at the trivial Hermitian limit.
19. Hamiltonian Noether theorem for gauge systems and two time physics
Villanueva, V. M.; Nieto, J. A.; Ruiz, L.; Silvas, J.
2005-08-01
The Noether theorem for Hamiltonian constrained systems is revisited. In particular, our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints. We apply our results to the relativistic point particle, to the Friedberg et al model and, with special emphasis, to two time physics.
20. Simulating Hamiltonian Dynamics with a Truncated Taylor Series
Somma, Rolando
2015-03-01
One of the main motivations for quantum computers is their ability to efficiently simulate the dynamics of quantum systems. Since the mid-1990s, many algorithms have been developed to simulate Hamiltonian dynamics on a quantum computer, with applications to problems such as simulating spin models and quantum chemistry. While it is now well known that quantum computers can efficiently simulate Hamiltonian dynamics, ongoing work has improved the performance and expanded the scope of such simulations. In this talk, I will describe a very simple and efficient algorithm for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. This algorithm can simulate the time evolution of a wide variety of physical systems. The cost of this algorithm depends only logarithmically on the inverse of the desired precision, and can be shown to be optimal. Such a cost also represents an exponential improvement over known methods for Hamiltonian simulation based on, e.g., Trotter-Suzuki approximations. Roughly speaking, doubling the number of digits of accuracy of the simulation only doubles the complexity. The new algorithm and its analysis are highly simplified due to a technique for implementing linear combinations of unitary operations to directly apply the truncated Taylor series. This is joint work with Dominic Berry, Andrew Childs, Richard Cleve, and Robin Kothari.
1. The Electromagnetic Dipole Radiation Field through the Hamiltonian Approach
ERIC Educational Resources Information Center
Likar, A.; Razpet, N.
2009-01-01
The dipole radiation from an oscillating charge is treated using the Hamiltonian approach to electrodynamics where the concept of cavity modes plays a central role. We show that the calculation of the radiation field can be obtained in a closed form within this approach by emphasizing the role of coherence between the cavity modes, which is…
2. The Hamiltonian property of the flow of singular trajectories
SciTech Connect
Lokutsievskiy, L V
2014-03-31
Pontryagin's maximum principle reduces optimal control problems to the investigation of Hamiltonian systems of ordinary differential equations with discontinuous right-hand side. An optimal synthesis is the totality of solutions to this system with a fixed terminal (or initial) condition, which fill a region in the phase space one-to-one. In the construction of optimal synthesis, singular trajectories that go along the discontinuity surface N of the right-hand side of the Hamiltonian system of ordinary differential equations, are crucial. The aim of the paper is to prove that the system of singular trajectories makes up a Hamiltonian flow on a submanifold of N. In particular, it is proved that the flow of singular trajectories in the problem of control of the magnetized Lagrange top in a variable magnetic field is completely Liouville integrable and can be embedded in the flow of a smooth superintegrable Hamiltonian system in the ambient space. Bibliography: 17 titles.
3. Warped product Finsler manifolds from Hamiltonian point of view
In this paper, the Finslerian warped product structures are introduced as Hamiltonian formalism without restricting Finsler functions to be absolutely homogeneous. Afterwards, the constituents of the related variational problem and Finslerian connections of this warped product are obtained according to those of its constructing Finsler manifolds.
4. The Electromagnetic Dipole Radiation Field through the Hamiltonian Approach
ERIC Educational Resources Information Center
Likar, A.; Razpet, N.
2009-01-01
The dipole radiation from an oscillating charge is treated using the Hamiltonian approach to electrodynamics where the concept of cavity modes plays a central role. We show that the calculation of the radiation field can be obtained in a closed form within this approach by emphasizing the role of coherence between the cavity modes, which is…
5. Regularization of the Hamiltonian constraint compatible with the spinfoam dynamics
Alesci, Emanuele; Rovelli, Carlo
2010-08-01
We introduce a new regularization for Thiemann’s Hamiltonian constraint. The resulting constraint can generate the 1-4 Pachner moves and is therefore more compatible with the dynamics defined by the spinfoam formalism. We calculate its matrix elements and observe the appearance of the 15j Wigner symbol in these.
6. Fractional Hamiltonian analysis of higher order derivatives systems
SciTech Connect
Baleanu, Dumitru; Muslih, Sami I.; Tas, Kenan
2006-10-15
The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski's formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part and a damped oscillator are analyzed. The classical results are obtained when fractional derivatives are replaced with the integer order derivatives.
7. New bi-Hamiltonian systems on the plane
Tsiganov, A. V.
2017-06-01
We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and recursion operators are also presented in the framework of the Jacobi method.
8. Estimating equilibrium properties from non-Hamiltonian dynamics
VandeVondele, Joost; Rothlisberger, Ursula
2001-11-01
We derive an expression that enables the accurate estimation of equilibrium properties using non-Hamiltonian dynamics. The major advantage of our scheme is that a time average over a single non-Hamiltonian trajectory can be employed instead of an ensemble average. Hence, it can directly be used in standard molecular dynamics simulations. The connection between non-Hamiltonian dynamics and equilibrium properties is established by assigning to the individual frames of the trajectory a weight that is based on the fluctuations of the phase space compression factor. Additionally, a simple scheme that takes into account only fluctuation of a given maximum duration is introduced to reduce the statistical error. By systematically extending the duration of the allowed fluctuations, increasingly accurate results can be obtained. Non-Hamiltonian dynamics schemes that are capable to enhance sampling efficiency are applied to two model systems in order to demonstrate the practical performance of our approach for the calculation of equilibrium free energy differences and probability density profiles.
9. Model spin-orbit coupling Hamiltonians for graphene systems
Kochan, Denis; Irmer, Susanne; Fabian, Jaroslav
2017-04-01
We present a detailed theoretical study of effective spin-orbit coupling (SOC) Hamiltonians for graphene-based systems, covering global effects such as proximity to substrates and local SOC effects resulting, for example, from dilute adsorbate functionalization. Our approach combines group theory and tight-binding descriptions. We consider structures with global point group symmetries D6 h, D3 d, D3 h, C6 v, and C3 v that represent, for example, pristine graphene, graphene miniripple, planar boron nitride, graphene on a substrate, and free standing graphone, respectively. The presence of certain spin-orbit coupling parameters is correlated with the absence of the specific point group symmetries. Especially in the case of C6 v—graphene on a substrate, or transverse electric field—we point out the presence of a third SOC parameter, besides the conventional intrinsic and Rashba contributions, thus far neglected in literature. For all global structures we provide effective SOC Hamiltonians both in the local atomic and Bloch forms. Dilute adsorbate coverage results in the local point group symmetries C6 v, C3 v, and C2 v, which represent the stable adsorption at hollow, top and bridge positions, respectively. For each configuration we provide effective SOC Hamiltonians in the atomic orbital basis that respect local symmetries. In addition to giving specific analytic expressions for model SOC Hamiltonians, we also present general (no-go) arguments about the absence of certain SOC terms.
10. Translation-Invariant Parent Hamiltonians of Valence Bond Crystals
Huerga, Daniel; Greco, Andrés; Gazza, Claudio; Muramatsu, Alejandro
2017-04-01
We present a general method to construct translation-invariant and SU(2) symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBCs). The method is based on a canonical mapping transforming S =1 /2 spin operators into a bilinear form of a new set of dimer fermion operators. We construct parent Hamiltonians of the columnar and the staggered VBCs on the square lattice, for which the VBC is an eigenstate in all regimes and the exact ground state in some region of the phase diagram. We study the departure from the exact VBC regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters. In both Hamiltonians, the VBC phase extends over the exact regime and transits to a columnar antiferromagnet (CAFM) through a window of intermediate phases, revealing an intriguing competition of correlation lengths at the VBC-CAFM transition. The method can be readily applied to construct other VBC parent Hamiltonians in different lattices and dimensions.
11. Davidson potential and SUSYQM in the Bohr Hamiltonian
SciTech Connect
Georgoudis, P. E.
2013-06-10
The Bohr Hamiltonian is modified through the Shape Invariance principle of SUper-SYmmetric Quantum Mechanics for the Davidson potential. The modification is equivalent to a conformal transformation of Bohr's metric, generating a different {beta}-dependence of the moments of inertia.
12. Stochastic diffusion and Kolmogorov entropy in regular and random Hamiltonians
SciTech Connect
Isichenko, M.B. . Inst. for Fusion Studies Kurchatov Inst. of Atomic Energy, Moscow ); Horton, W. . Inst. for Fusion Studies); Kim, D.E.; Heo, E.G.; Choi, D.I. )
1992-05-01
The scalings of the E x B turbulent diffusion coefficient D and the Kolmogorov entropy K with the potential amplitude {phi} {sup {approximately}} of the fluctuation are studied using the geometrical analysis of closed and extended particle orbits for several types of drift Hamiltonians. The high-amplitude scalings , D {proportional to} {phi} {sup {approximately} 2} or {phi} {sup {approximately} 0} and K {proportional to} log {phi} {sup {approximately}}, are shown to arise from different forms of a periodic (four-wave) Hamiltonian {phi}{sup {approximately}} (x,y,t), thereby explaining the controversy in earlier numerical results. For a quasi-random (six-wave) Hamiltonian numerical data for the diffusion D {proportional to} {phi} {sup {approximately} 0.92 {plus minus} 0.04} and the Kolmogorov entropy K {proportional to} {phi} {sup {approximately} 0.56 {plus minus} 0.17} are presented and compared with the percolation theory predictions D {sub p} {proportional to} {phi} {sup {approximately} 0.7}, K {sub p} {proportional to} {phi} {sup {approximately} 0.5}. To study the turbulent diffusion in a general form of Hamiltonian, a new approach of the series expansion of the Lagrangian velocity correlation function is proposed and discussed.
13. Stochastic diffusion and Kolmogorov entropy in regular and random Hamiltonians
SciTech Connect
Isichenko, M.B. |; Horton, W.; Kim, D.E.; Heo, E.G.; Choi, D.I.
1992-05-01
The scalings of the E x B turbulent diffusion coefficient D and the Kolmogorov entropy K with the potential amplitude {phi} {sup {approximately}} of the fluctuation are studied using the geometrical analysis of closed and extended particle orbits for several types of drift Hamiltonians. The high-amplitude scalings , D {proportional_to} {phi} {sup {approximately} 2} or {phi} {sup {approximately} 0} and K {proportional_to} log {phi} {sup {approximately}}, are shown to arise from different forms of a periodic (four-wave) Hamiltonian {phi}{sup {approximately}} (x,y,t), thereby explaining the controversy in earlier numerical results. For a quasi-random (six-wave) Hamiltonian numerical data for the diffusion D {proportional_to} {phi} {sup {approximately} 0.92 {plus_minus} 0.04} and the Kolmogorov entropy K {proportional_to} {phi} {sup {approximately} 0.56 {plus_minus} 0.17} are presented and compared with the percolation theory predictions D {sub p} {proportional_to} {phi} {sup {approximately} 0.7}, K {sub p} {proportional_to} {phi} {sup {approximately} 0.5}. To study the turbulent diffusion in a general form of Hamiltonian, a new approach of the series expansion of the Lagrangian velocity correlation function is proposed and discussed.
14. Adaptive molecular resolution approach in Hamiltonian form: An asymptotic analysis.
PubMed
Zhu, Jinglong; Klein, Rupert; Delle Site, Luigi
2016-10-01
Adaptive molecular resolution approaches in molecular dynamics are becoming relevant tools for the analysis of molecular liquids characterized by the interplay of different physical scales. The essential difference among these methods is in the way the change of molecular resolution is made in a buffer (transition) region. In particular a central question concerns the possibility of the existence of a global Hamiltonian which, by describing the change of resolution, is at the same time physically consistent, mathematically well defined, and numerically accurate. In this paper we present an asymptotic analysis of the adaptive process complemented by numerical results and show that under certain mathematical conditions a Hamiltonian, which is physically consistent and numerically accurate, may exist. Such conditions show that molecular simulations in the current computational implementation require systems of large size, and thus a Hamiltonian approach such as the one proposed, at this stage, would not be practical from the numerical point of view. However, the Hamiltonian proposed provides the basis for a simplification and generalization of the numerical implementation of adaptive resolution algorithms to other molecular dynamics codes.
15. Translation-Invariant Parent Hamiltonians of Valence Bond Crystals.
PubMed
Huerga, Daniel; Greco, Andrés; Gazza, Claudio; Muramatsu, Alejandro
2017-04-21
We present a general method to construct translation-invariant and SU(2) symmetric antiferromagnetic parent Hamiltonians of valence bond crystals (VBCs). The method is based on a canonical mapping transforming S=1/2 spin operators into a bilinear form of a new set of dimer fermion operators. We construct parent Hamiltonians of the columnar and the staggered VBCs on the square lattice, for which the VBC is an eigenstate in all regimes and the exact ground state in some region of the phase diagram. We study the departure from the exact VBC regime upon tuning the anisotropy by means of the hierarchical mean field theory and exact diagonalization on finite clusters. In both Hamiltonians, the VBC phase extends over the exact regime and transits to a columnar antiferromagnet (CAFM) through a window of intermediate phases, revealing an intriguing competition of correlation lengths at the VBC-CAFM transition. The method can be readily applied to construct other VBC parent Hamiltonians in different lattices and dimensions.
16. Hamiltonian evolutions of twisted polygons in {RP}^n
Marì Beffa, Gloria; Wang, Jing Ping
2013-09-01
In this paper we find a discrete moving frame and their associated invariants along projective polygons in {RP}^n , and we use them to describe invariant evolutions of projective N-gons. We then apply a reduction process to obtain a natural Hamiltonian structure on the space of projective invariants for polygons, establishing a close relationship between the projective N-gon invariant evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that any Hamiltonian evolution is induced on invariants by an invariant evolution of N-gons—what we call a projective realization—and both evolutions are connected explicitly in a very simple way. Finally, we provide a completely integrable evolution (the Boussinesq lattice related to the lattice W3-algebra), its projective realization in {RP}^2 and its Hamiltonian pencil. We generalize both structures to n-dimensions and we prove that they are Poisson, defining explicitly the n-dimensional generalization of the planar evolution (a discretization of the Wn-algebra). We prove that the generalization is completely integrable, and we also give its projective realization, which turns out to be very simple.
17. Integrable Hamiltonian systems on low-dimensional Lie algebras
SciTech Connect
Korotkevich, Aleksandr A
2009-12-31
For any real Lie algebra of dimension 3, 4 or 5 and any nilpotent algebra of dimension 6 an integrable Hamiltonian system with polynomial coefficients is found on its coalgebra. These systems are constructed using Sadetov's method for constructing complete commutative families of polynomials on a Lie coalgebra. Bibliography: 17 titles.
18. Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups
DTIC Science & Technology
1981-07-28
478. V. I. Arnold [1964]. Inst"ility of dynamical systenms with several degrees of freedom, Dokl . Akad . Riuk. SSSR 156,9-12. V. I. Ar’nold [1966...a rigid body, Trans, oscow Math. Soc. 41, 287. S.L. Ziglin [1981]. Branching of solutions and nonexistence of integrals in Hamiltonian systems. Doklady Akad . Nauk . SSSR 257, 26-29. - J. I
19. LETTER TO THE EDITOR: On optimum Hamiltonians for state transformations
Brody, Dorje C.; Hook, Daniel W.
2006-03-01
For a prescribed pair of quantum states |ψIrang and |ψFrang we establish an elementary derivation of the optimum Hamiltonian, under constraints on its eigenvalues, that generates the unitary transformation |ψIrang → |ψFrang in the shortest duration. The derivation is geometric in character and does not rely on variational calculus.
20. Evolutionary approach for determining first-principles hamiltonians.
PubMed
Hart, Gus L W; Blum, Volker; Walorski, Michael J; Zunger, Alex
2005-05-01
Modern condensed-matter theory from first principles is highly successful when applied to materials of given structure-type or restricted unit-cell size. But this approach is limited where large cells or searches over millions of structure types become necessary. To treat these with first-principles accuracy, one 'coarse-grains' the many-particle Schrodinger equation into 'model hamiltonians' whose variables are configurational order parameters (atomic positions, spin and so on), connected by a few 'interaction parameters' obtained from a microscopic theory. But to construct a truly quantitative model hamiltonian, one must know just which types of interaction parameters to use, from possibly 10(6)-10(8) alternative selections. Here we show how genetic algorithms, mimicking biological evolution ('survival of the fittest'), can be used to distil reliable model hamiltonian parameters from a database of first-principles calculations. We demonstrate this for a classic dilemma in solid-state physics, structural inorganic chemistry and metallurgy: how to predict the stable crystal structure of a compound given only its composition. The selection of leading parameters based on a genetic algorithm is general and easily applied to construct any other type of complex model hamiltonian from direct quantum-mechanical results.
1. Hamiltonian flow in Coulomb gauge Yang-Mills theory
SciTech Connect
Leder, Markus; Reinhardt, Hugo; Pawlowski, Jan M.; Weber, Axel
2011-01-15
We derive a new functional renormalization group equation for Hamiltonian Yang-Mills theory in Coulomb gauge. The flow equations for the static gluon and ghost propagators are solved under the assumption of ghost dominance within different diagrammatic approximations. The results are compared to those obtained in the variational approach and the reliability of the approximations is discussed.
2. Evolution of a Spin System Under a Periodic Hamiltonian
Goldman, M.
The expression of the density matrix for a spin system subjected to a periodic Hamiltonian is derived in the form of an expansion in powers of the inverse modulation frequency, an extension of a method devised by Ruishvili and Menabde and by Mehring. Its application to MAS experiments, as regards the contribution of the dipolar interactions to the sideband intensities, is discussed.
3. Spectral and resonance properties of the Smilansky Hamiltonian
Exner, Pavel; Lotoreichik, Vladimir; Tater, Miloš
2017-02-01
We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically in the subcritical case. Furthermore, we show that the model then has a rich resonance structure.
4. Effective Hamiltonians by optimal control: solid-state NMR double-quantum planar and isotropic dipolar recoupling.
PubMed
Tosner, Zdenek; Glaser, Steffen J; Khaneja, Navin; Nielsen, Niels Chr
2006-11-14
We report the use of optimal control algorithms for tailoring the effective Hamiltonians in nuclear magnetic resonance (NMR) spectroscopy through sophisticated radio-frequency (rf) pulse irradiation. Specifically, we address dipolar recoupling in solid-state NMR of powder samples for which case pulse sequences offering evolution under planar double-quantum and isotropic mixing dipolar coupling Hamiltonians are designed. The pulse sequences are constructed numerically to cope with a range of experimental conditions such as inhomogeneous rf fields, spread of chemical shifts, the intrinsic orientation dependencies of powder samples, and sample spinning. While the vast majority of previous dipolar recoupling sequences are operating through planar double-or zero-quantum effective Hamiltonians, we present here not only improved variants of such experiments but also for the first time homonuclear isotropic mixing sequences which transfers all I(x), I(y), and I(z) polarizations from one spin to the same operators on another spin simultaneously and with equal efficiency. This property may be exploited to increase the signal-to-noise ratio of two-dimensional experiments by a factor of square root 2 compared to conventional solid-state methods otherwise showing the same efficiency. The sequences are tested numerically and experimentally for a powder of (13)C(alpha),(13)C(beta)-L-alanine and demonstrate substantial sensitivity gains over previous dipolar recoupling experiments.
5. Fock-space diagonalization of the state-dependent pairing Hamiltonian with the Woods-Saxon mean field
Molique, H.; Dudek, J.
1997-10-01
A particle-number conserving approach is presented to solve the nuclear mean-field plus pairing Hamiltonian problem with a realistic deformed Woods-Saxon single-particle potential. The method is designed for the state-dependent monopole pairing Hamiltonian H⁁pair=∑αβGαβc†αc†α ¯cβ ¯cβ with an arbitrary set of matrix elements Gαβ. Symmetries of the Hamiltonians on the many-body level are discussed using the language of P symmetry introduced earlier in the literature and are employed to diagonalize the problem; the only essential approximation used is a many-body (Fock-space) basis cutoff. An optimal basis construction is discussed and the stability of the final result with respect to the basis cutoff is illustrated in details. Extensions of the concept of P symmetry are introduced and their consequences for an optimal many-body basis cutoff construction are exploited. An algorithm is constructed allowing to solve the pairing problems in the many-body spaces corresponding to p~40 particles on n~80 levels and for several dozens of lowest lying states with precision ~(1-2) % within seconds of the CPU time on a CRAY computer. Among applications, the presence of the low-lying seniority s=0 solutions, that are usually poorly described in terms of the standard approximations (BCS, HFB), is discussed and demonstrated to play a role in the interpretation of the spectra of rotating nuclei.
6. The (100), (111) and (110) surfaces of diamond: an ab initio B3LYP study
De La Pierre, Marco; Bruno, Marco; Manfredotti, Chiara; Nestola, Fabrizio; Prencipe, Mauro; Manfredotti, Claudio
2014-04-01
We present an accurate ab initio study of the structure and surface energy of the low-index (100), (111) and (110) diamond faces, by using the hybrid Hartree-Fock/density functional B3LYP Hamiltonian and a localised all-electron Gaussian-type basis set. A two-dimensional periodic slab model has been adopted, for which convergence on both structural and energetic parameters has been thoroughly investigated. For all the three surfaces, possible relaxations and reconstructions have been considered; a detailed geometrical characterisation is provided for the most stable structure of each orientation. Surface energy is discussed for all the investigated faces.
7. Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems
Gong, Yuezheng; Wang, Qi; Wang, Zhu
2017-03-01
The proper orthogonal decomposition reduced-order models (POD-ROMs) have been widely used as a computationally efficient surrogate models in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy its Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduce-order modeling approach for the Hamiltonian system, which uses the traditional framework of Galerkin projection-based model reduction but modifies the ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use the POD basis from shifted snapshots to improve the Hamiltonian function approximation. We further derive a rigorous a priori error estimate of the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems etc.
8. Two-dimensional surrogate Hamiltonian investigation of laser-induced desorption of NO/NiO(100)
SciTech Connect
Dittrich, Soeren; Freund, Hans-Joachim; Koch, Christiane P.; Kosloff, Ronnie; Kluener, Thorsten
2006-01-14
The photodesorption of NO from NiO(100) is studied from first principles, with electronic relaxation treated by the use of the surrogate Hamiltonian approach. Two nuclear degrees of freedom of the adsorbate-substrate system are taken into account. To perform the quantum dynamical wave-packet calculations, a massively parallel implementation with a one-dimensional data decomposition had to be introduced. The calculated desorption probabilities and velocity distributions are in qualitative agreement with experimental data. The results are compared to those of stochastic wave-packet calculations where a sufficiently large number of quantum trajectories is propagated within a jumping wave-packet scenario.
9. Hamiltonian thermodynamics of charged three-dimensional dilatonic black holes
Dias, Gonçalo A. S.; Lemos, José P. S.
2008-10-01
The action for a class of three-dimensional dilaton-gravity theories, with an electromagnetic Maxwell field and a cosmological constant, can be recast in a Brans-Dicke-Maxwell type action, with its free ω parameter. For a negative cosmological constant, these theories have static, electrically charged, and spherically symmetric black hole solutions. Those theories with well formulated asymptotics are studied through a Hamiltonian formalism, and their thermodynamical properties are found out. The theories studied are general relativity (ω→±∞), a dimensionally reduced cylindrical four-dimensional general relativity theory (ω=0), and a theory representing a class of theories (ω=-3), all with a Maxwell term. The Hamiltonian formalism is set up in three dimensions through foliations on the right region of the Carter-Penrose diagram, with the bifurcation 1-sphere as the left boundary, and anti-de Sitter infinity as the right boundary. The metric functions on the foliated hypersurfaces and the radial component of the vector potential one-form are the canonical coordinates. The Hamiltonian action is written, the Hamiltonian being a sum of constraints. One finds a new action which yields an unconstrained theory with two pairs of canonical coordinates {M,PM;Q,PQ}, where M is the mass parameter, which for ω<-(3)/(2) and for ω=±∞ needs a careful renormalization, PM is the conjugate momenta of M, Q is the charge parameter, and PQ is its conjugate momentum. The resulting Hamiltonian is a sum of boundary terms only. A quantization of the theory is performed. The Schrödinger evolution operator is constructed, the trace is taken, and the partition function of the grand canonical ensemble is obtained, where the chemical potential is the scalar electric field ϕ¯. Like the uncharged cases studied previously, the charged black hole entropies differ, in general, from the usual quarter of the horizon area due to the dilaton.
10. First integrals of generalized Ermakov systems via the Hamiltonian formulation
Mahomed, K. S.; Moitsheki, R. J.
2016-07-01
We obtain first integrals of the generalized two-dimensional Ermakov systems, in plane polar form, via the Hamiltonian approaches. There are two methods used for the construction of the first integrals, viz. the standard Hamiltonian and the partial Hamiltonian approaches. In the first approach, F(𝜃) and G(𝜃) in the Ermakov system are related as G(𝜃) + F‧(𝜃)/2 = 0. In this case, we deduce four first integrals (three of which are functionally independent) which correspond to the Lie algebra sl(2,R) ⊕ A1 in a direct constructive manner. We recover the results of earlier work that uses the relationship between symmetries and integrals. This results in the complete integrability of the Ermakov system. By use of the partial Hamiltonian method, we discover four new cases: F(𝜃) = G(𝜃)(c1sin 𝜃 + c3cos 𝜃)/(c1cos 𝜃 - c3sin 𝜃) with c2c3 = c1c4, c1≠0, c3≠0; F(𝜃) = G(𝜃)(c2sin 𝜃 + c4cos 𝜃)/(c2cos 𝜃 - c4sin 𝜃) with c1 = c3 = 0, c2≠0, c4≠0; F(𝜃) = -G(𝜃)cot 𝜃 with c1 = c2 = 0, c3, c4 arbitrary and F(𝜃) = G(𝜃)tan 𝜃 with c3 = c4 = 0, c1, c2 arbitrary, where the cis are constants in all cases. In the last two cases, we find that there are three operators each which give rise to three first integrals each. In both these cases, we have complete integrability of the Ermakov system. The first two cases each result in two first integrals each. For every case, both for the standard and partial Hamiltonian, the angular momentum type first integral arises and this is a consequence of the operator which depends on a momentum coordinate which is a generalized symmetry in the Lagrangian context.
11. Contact Transformations and Determinable Parameters in Spectroscopic Fitting Hamiltonians
Mekhtiev, Mirza A.; Hougen, Jon T.
2000-02-01
In recent least-squares fits of torsion-rotation spectra of acetaldehyde and methanol it was found possible to adjust more fourth-order parameters than would be expected from traditional contact-transformation considerations. To investigate this discrepancy between theory and practice we have carried out numerical fitting experiments on the simpler three-dimensional (three-Eulerian-angle) asymmetric rotor problem, using J ≤ 20 unitless energy levels generated artificially from a full orthorhombic Hamiltonian with quadratic through octic operators in the angular momentum components. Results are analyzed using the condition number κ of the least-squares matrix, which is a measure of its invertibility in the presence of round-off and other errors. When κ is very large, parameters must be removed from the fit until κ becomes acceptably small, corresponding to procedures which lead to reduced Hamiltonians in molecular spectroscopy. We find that under certain circumstances κ can be decreased to an acceptable level for Hamiltonians which are only partially reduced when compared to Watson A and S reductions. Some insight into this behavior is obtained from classical mechanics and from the concept of delayed contact transformations. Transferring this numerical and algebraic understanding to the more complicated four-dimensional methyl-top internal rotor problem supports the empirical observation that presently existing data sets for methanol and acetaldehyde are most efficiently fit using partially reduced Hamiltonians and further suggests that expanding the methanol data set to transitions involving levels of higher J, K, and vt would favor even more strongly the use of partially reduced fourth-order Hamiltonians.
12. When a local Hamiltonian must be frustration-free.
PubMed
Sattath, Or; Morampudi, Siddhardh C; Laumann, Chris R; Moessner, Roderich
2016-06-07
A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion-a sufficient condition-under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
13. Moreau-Yosida approximation and convergence of Hamiltonian systems on Wasserstein space
Kim, Hwa Kil
In this paper, we study the stability property of Hamiltonian systems on the Wasserstein space. Let H be a given Hamiltonian satisfying certain properties. We regularize H using the Moreau-Yosida approximation and denote it by Hτ. We show that solutions of the Hamiltonian system for Hτ converge to a solution of the Hamiltonian system for H as τ converges to zero. We provide sufficient conditions on H to carry out this process.
14. Differential evolution algorithm for global optimizations in nuclear physics
Qi, Chong
2017-04-01
We explore the applicability of the differential evolution algorithm in finding the global minima of three typical nuclear structure physics problems: the global deformation minimum in the nuclear potential energy surface, the optimization of mass model parameters and the lowest eigenvalue of a nuclear Hamiltonian. The algorithm works very effectively and efficiently in identifying the minima in all problems we have tested. We also show that the algorithm can be parallelized in a straightforward way.
15. Two integrable Hamiltonian hierarchies in sl(2 ,R ) and so(3 ,R ) with three potentials
Gu, Xiang; Ma, Wen-Xiu; Zhang, Wen-Ying
2017-05-01
By introducing two specific matrix spectral problems associated with sl(2 ,R ) and so(3 ,R ) matrix Lie algebras, we generate two integrable Hamiltonian hierarchies with three potentials. The computation and analysis on their Hamiltonian structures by means of the trace identity show that the resulting hierarchies are Liouville integrable, namely, that each hierarchy consists of commuting Hamiltonian soliton equations.
16. On bi-Hamiltonian structure of two-component Novikov equation
Li, Nianhua; Liu, Q. P.
2013-01-01
In this Letter, we present a bi-Hamiltonian structure for the two-component Novikov equation. We also show that proper reduction of this bi-Hamiltonian structure leads to the Hamiltonian operators found by Hone and Wang for the Novikov equation.
17. Targeting transcription factor Stat5a/b as a therapeutic strategy for prostate cancer
PubMed Central
Liao, Zhiyong; Nevalainen, Marja T
2011-01-01
Signal transducer and activator of transcription 5 (Stat5) is critical for the viability and growth of human prostate cancer cells in culture and for prostate xenograft tumors in nude mice. The expression of nuclear active Stat5a/b is associated with high histological grades of clinical prostate cancers, and the presence of active Stat5a/b in prostate cancer predicts early disease recurrence. Stat5a/b and androgen receptor signaling pathways functionally synergize in prostate cancer cells, and recent work suggests that Stat5a/b may be involved in the progression of prostate cancer to metastatic disease. Here, we review the biological functions of Stat5a/b in prostate cancer and potential strategies to target the prolactin receptor (PrlR)/Jak2/Stat5 signaling pathway for therapy development for prostate cancer. PMID:21416055
18. An investigation of ab initio shell-model interactions derived by no-core shell model
Wang, XiaoBao; Dong, GuoXiang; Li, QingFeng; Shen, CaiWan; Yu, ShaoYing
2016-09-01
The microscopic shell-model effective interactions are mainly based on the many-body perturbation theory (MBPT), the first work of which can be traced to Brown and Kuo's first attempt in 1966, derived from the Hamada-Johnston nucleon-nucleon potential. However, the convergence of the MBPT is still unclear. On the other hand, ab initio theories, such as Green's function Monte Carlo (GFMC), no-core shell model (NCSM), and coupled-cluster theory with single and double excitations (CCSD), have made many progress in recent years. However, due to the increasing demanding of computing resources, these ab initio applications are usually limited to nuclei with mass up to A = 16. Recently, people have realized the ab initio construction of valence-space effective interactions, which is obtained through a second-time renormalization, or to be more exactly, projecting the full-manybody Hamiltonian into core, one-body, and two-body cluster parts. In this paper, we present the investigation of such ab initio shell-model interactions, by the recent derived sd-shell effective interactions based on effective J-matrix Inverse Scattering Potential (JISP) and chiral effective-field theory (EFT) through NCSM. In this work, we have seen the similarity between the ab initio shellmodel interactions and the interactions obtained by MBPT or by empirical fitting. Without the inclusion of three-body (3-bd) force, the ab initio shell-model interactions still share similar defects with the microscopic interactions by MBPT, i.e., T = 1 channel is more attractive while T = 0 channel is more repulsive than empirical interactions. The progress to include more many-body correlations and 3-bd force is still badly needed, to see whether such efforts of ab initio shell-model interactions can reach similar precision as the interactions fitted to experimental data.
19. Deuteron distribution in nuclear matter
Benhar, O.; Fabrocini, A.; Fantoni, S.; Illarionov, A. Yu.; Lykasov, G. I.
2002-05-01
We analyze the properties of deuteron-like structures in infinite, correlated nuclear matter, described by a realistic hamiltonian containing the Urbana v14 two-nucleon and the Urbana TNI many-body potentials. The distribution of neutron-proton pairs, carrying the deuteron quantum numbers, is obtained as a function of the total momentum by computing the overlap between the nuclear matter in its ground state and the deuteron wave functions in correlated basis functions theory. We study the differences between the S- and D-wave components of the deuteron and those of the deuteron-like pair in the nuclear medium. The total number of deuteron type pairs is computed and compared with the predictions of Levinger's quasideuteron model. The resulting Levinger's factor in nuclear matter at equilibrium density is 11.63. We use the local density approximation to estimate the Levinger's factor for heavy nuclei, obtaining results which are consistent with the available experimental data from photoreactions.
20. Astrophysical jet dynamos based on spheromak, dusty plasma, and Hamiltonian concepts | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8473686575889587, "perplexity": 1470.9712621487988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934806225.78/warc/CC-MAIN-20171120203833-20171120223833-00665.warc.gz"} |
http://digre.pmf.unizg.hr/3956/ | # Stable homotopy theory of dendroidal sets
Bašić, Matija (2015) Stable homotopy theory of dendroidal sets. Doctoral thesis, Faculty of Science > Department of Mathematics.
PDF Restricted to Repository staff only Language: English Download (829kB) | Request a copy
## Abstract
The main topic of this thesis is the stable homotopy theory of dendroidal sets. This topic belongs to the area of mathematics called algebraic topology. Algebraic topology studies the interaction between the algebraic and topological structures. Examples of topological spaces with a very rich algebraic structure are (iterated) loop spaces. Loop spaces carry an algebraic structure which is called an $A_{\infty}$-structure, while infinite loop spaces carry an $E_{\infty}$-structure. These structures consist of an infinite sequence of operations that satisfy various coherence laws. As it is difficult to grasp all these data, one usually uses topological operads to efficiently describe this information. One can think of operads as carrying “blueprints” for the algebraic structure which is realized in every space with that structure. The characterization results for (iterated) loop spaces using topological operads have been established in the early 1970’s by the work of P. May, M. Boardman and R. Vogt. In the 1990’s it became evident that it is important to understand the homotopy theory operads. The theory of dendroidal sets provides a new context for studying operads up to homotopy. Dendroidal sets were introduced in 2007 by I. Moerdijk and I. Weiss. Subsequent work of I. Moerdijk and D.-C. Cisinski shows that dendroidal sets indeed model topological/simplicial operads. An important advantage of dendroidal sets is that the theory is built in a natural way as a generalization of the theory of simplicial sets. The study of dendroidal sets is very combinatorial in its nature since it is based on the notion of trees (graphs with no loops). Also, as a category of presheaves, the category of dendroidal sets has nice categorical properties. Simplicial sets provide combinatorial models for spaces (think of it in terms of triangulations of spaces given by simplicial approximations) and dendroidal sets provide combinatorial models for infinite loops spaces as spaces together with complicated algebraic structure. In fact, the precise formulation of this idea is one of the main topics of this thesis. A precise formulation of our results is given in the language of Quillen’s model categories. Model categories provide a formalism to study and compare homotopy theories in various contexts (topological spaces, chain complexes, simplicial sets, operads etc.) One of the main results of this thesis is that the category of dendroidal sets admits a model structure such that the underlying homotopy theory is equivalent to the homotopy theory of infinite loop spaces (equivalently, of grouplike $E_{\infty}$-algebras or connective spectra). We call this model structure the stable model structure on dendroidal sets. Constructing a model structure is a tedious job. In our case it requires a great deal of technical combinatorial results about dendroidal sets (i.e. ab out trees). In order to simplify our arguments, in Chapter 4 we develop a combinatorial technique for proving results about dendroidal anodyne extensions. This technique can be viewed as a result in its own right as one might apply it also in different ways than it is used in the later chapters of the thesis. We give two constructions of the stable model model structure. The first construction is more elementary and has an advantage of providing a characterization of fibrations between fibrant objects. This construction is based on standard mo del-theoretical arguments and it is given in Chapter 5. The second construction, given in Chapter 6, is based on the work of G. Heuts. This approach makes it possible to show that the stable model structure on dendroidal sets is Quillen equivalent to a model structure on $E_{\infty}$-spaces with grouplike $E_{\infty}$-spaces as fibrant objects. The equivalence to grouplike $E_{\infty}$-objects (i.e. connective spectra) might be considered as a solution to the problem of geometric realization of dendroidal sets. Also, these results open new possibilities to investigate the connective part of classical stable homotopy theory. The results of the thesis presented in Chapter 7 go in that direction. In that final chapter we discuss homology groups of dendroidal sets. This homology theory generalizes the well-known homology theory of simplicial sets (i.e. the singular homology of spaces). The generalization is not straightforward because we work with non-planar trees, but we want to use a certain sign-convention for planar trees. After giving the definition, we establish that these homology groups are homotopy invariant and that they compute the standard homology of the corresponding connective spectrum. The results of Chapters 6 and 7 are joint work with T. Nikolaus.
Item Type: Thesis (Doctoral thesis) Moerdijk, Ieke 2015 170 NATURAL SCIENCES > Mathematics > AlgebraNATURAL SCIENCES > Mathematics > Geometry and Topology ISBN 978-94-6259-634-4 Faculty of Science > Department of Mathematics Iva Prah 02 Jun 2015 09:16 02 Jun 2015 09:16 http://digre.pmf.unizg.hr/id/eprint/3956 | {"extraction_info": {"found_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": 0.9169481992721558, "perplexity": 359.18281474604584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866733.77/warc/CC-MAIN-20180524170605-20180524190605-00195.warc.gz"} |
https://www.arxiv-vanity.com/papers/1302.5209/ | # The Herschel††thanks: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA PEP/HerMES Luminosity Function – I: Probing the Evolution of PACS selected Galaxies to z≃4
C. Gruppioni, F. Pozzi, G. Rodighiero, I. Delvecchio, S. Berta, L. Pozzetti, G. Zamorani, P. Andreani, A. Cimatti, O. Ilbert, E. Le Floc’h, D. Lutz, B. Magnelli, L. Marchetti, P. Monaco, R. Nordon, S. Oliver, P. Popesso, L. Riguccini, I. Roseboom, D.J. Rosario, M. Sargent, M. Vaccari, B. Altieri, H. Aussel, A. Bongiovanni, J. Cepa, E. Daddi, H. Domínguez-Sánchez, D. Elbaz, N. Förster Schreiber, R. Genzel, A. Iribarrem, M. Magliocchetti, R. Maiolino, A. Poglitsch, A. Pérez García, M. Sanchez-Portal, E. Sturm, L. Tacconi, I. Valtchanov, A. Amblard, V. Arumugam, M. Bethermin, J. Bock, A. Boselli, V. Buat, D. Burgarella, N. Castro-Rodríguez, A. Cava, P. Chanial, D.L. Clements, A. Conley, A. Cooray, C.D. Dowell, E. Dwek, S. Eales, A. Franceschini, J. Glenn, M. Griffin, E. Hatziminaoglou, E. Ibar, K. Isaak, R.J. Ivison, G. Lagache, L. Levenson, N. Lu, S. Madden, B. Maffei, G. Mainetti, H.T. Nguyen, B. O’Halloran, M.J. Page, P. Panuzzo, A. Papageorgiou C.P. Pearson, I. Pérez-Fournon, M. Pohlen, D. Rigopoulou, M. Rowan-Robinson, B. Schulz, D. Scott, N. Seymour, D.L. Shupe, A.J. Smith, J.A. Stevens, M. Symeonidis, M. Trichas, K.E. Tugwell, L. Vigroux, L. Wang, G. Wright, C.K. Xu, M. Zemcov, S. Bardelli, M. Carollo, T. Contini, O. Le Févre, S. Lilly, V. Mainieri, A. Renzini, M. Scodeggio, E. Zucca
Accepted 2013 February 18. Received 2013 February 13; in original form 2012 October 23
###### Abstract
We exploit the deep and extended far-infrared data-sets (at 70, 100 and 160 m) of the Herschel GTO PACS Evolutionary Probe (PEP) Survey, in combination with the HERschel Multi-tiered Extragalactic Survey (HerMES) data at 250, 350 and 500 m, to derive the evolution of the rest-frame 35-m, 60-m, 90-m, and total infrared (IR) luminosity functions (LFs) up to 4. We detect very strong luminosity evolution for the total IR LF (L(1) up to 2, and (1) at 24) combined with a density evolution ((1) up to 1 and (1) at 14). In agreement with previous findings, the IR luminosity density () increases steeply to 1, then flattens between 1 and 3 to decrease at 3. Galaxies with different SEDs, masses and sSFRs evolve in very different ways and this large and deep statistical sample is the first one allowing us to separately study the different evolutionary behaviours of the individual IR populations contributing to . Galaxies occupying the well established SFR–stellar mass main sequence (MS) are found to dominate both the total IR LF and at all redshifts, with the contribution from off-MS sources (0.6 dex above MS) being nearly constant (20% of the total ) and showing no significant signs of increase with increasing over the whole 0.82.2 range. Sources with mass in the range 10log(/M)11 are found to dominate the total IR LF, with more massive galaxies prevailing at the bright end of the high- (2) LF. A two-fold evolutionary scheme for IR galaxies is envisaged: on the one hand, a starburst-dominated phase in which the SMBH grows and is obscured by dust , is followed by an AGN-dominated phase, then evolving toward a local elliptical. On the other hand, moderately star-forming galaxies containing a low-luminosity AGN have various properties suggesting they are good candidates for systems in a transition phase preceding the formation of steady spiral galaxies.
###### keywords:
cosmology: observations – galaxies: active – galaxies: evolution – galaxies: luminosity function – galaxies: starburst – infrared: galaxies.
pagerange: The Herschelthanks: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA PEP/HerMES Luminosity Function – I: Probing the Evolution of PACS selected Galaxies to z4The Herschelthanks: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA PEP/HerMES Luminosity Function – I: Probing the Evolution of PACS selected Galaxies to z4pubyear: 2012
## 1 Introduction
Understanding the origin and growth of the galaxies we observe today is one of the main problems of current cosmology. The luminosity function (LF) provides one of the fundamental tools to probe the distribution of galaxies over cosmological time, since it allows us to assess the statistical nature of galaxy formation and evolution. When computed at different redshifts, the LF constitutes the most direct method for exploring the evolution of a galaxy population, describing the relative number of sources of different luminosities counted in representative volumes of the Universe. The LF computed for different samples of galaxies can provide a crucial comparison between the distribution of different galaxy types, i.e. galaxies at different redshifts, in different enviroments or selected at different wavelengths.
It has now become clear that we cannot understand galaxy evolution without accounting for the energy absorbed by dust and re-emitted at longer wavelengths (e.g, Genzel & Cesarsky 2000), in the infrared (IR) or sub-millimetre (sub-mm). Dust is responsible for obscuring the ultraviolet (UV) and optical light from galaxies: since star-formation occurs within dusty molecular clouds, far-IR and sub-mm data, where the absorbed radiation is re-emitted, are essential for providing a complete picture of the history of star-formation through cosmic time, which is one of the fundamental instruments we have to reconstruct how galaxies have evolved since their formation epoch. For these reasons, extragalactic surveys in the rest-frame IR represent a key tool for understanding galaxy formation and evolution.
Surveys of dust emission performed with the former satellites exploring the Universe in the mid- and far-IR domain, i.e. the Infrared Astronomical Satellite (IRAS; Neugebauer 1984) and the Infrared Space Observatory (ISO; Kessler et al. 1996), allowed the first studies of the IR-galaxy LF at 0.3 (Saunders et al. 1990) and 1 (Pozzi et al. 2004), respectively. With Spitzer 24-m data, it was possible to study the evolution of the mid-IR LF up to 2 (e.g. Le Floc’h et al. 2005, Caputi et al. 2007, Rodighiero et al. 2010a), while, even with the deepest Spitzer Space Telescope (Werner et al. 2004) 70-m data, only 1–1.2 could be reached in the far-IR (Magnelli et al. 2009; Patel et al. 2012) ) – though Magnelli et al. (2011) reached 2 through stacking. Since the rest-frame IR spectral energy distributions (SEDs) of star-forming galaxies and AGN peak at 60–200 m, to measure their bolometric luminosity and evolution with we need to observe in the far-IR/sub-mm regime. However, the detection of large numbers of high- sources at the peak of their IR SED was not achievable before the Herschel Space Observatory (Pilbratt et al. 2010), due to source confusion and/or low detector sensitivity, and our knowledge of the far-IR luminosity function in the distant Universe is still affected by substantial uncertainties. Ground-based and balloon-borne observations in the mm/sub-mm range, probing the evolution of the most distant (2) and luminous dusty galaxies, have so far been limited to the identification of sources at the very bright end of the luminosity function (e.g., Chapman et al. 2005). All of these works detected strong evolution in both luminosity and/or density, indicating that IR galaxies were more luminous and/or more numerous in the past. Strong observational evidence of high rates of evolution for IR galaxies has been obtained also through the detection of a large amount of energy contained in the Cosmic Infrared Background (CIRB; Hauser & Dwek 2001), and the source counts from several deep cosmological surveys (from 15 m to 850 m) largely exceeding the no-evolution expectations (e.g. Smail et al. 1997; Elbaz et al. 1999; Papovich et al. 2004; Bethermin et al. 2010; Marsden et al. 2011). Both the CIRB and the source counts require a strong increase in the IR energy density between the present time and 1–2. At higher redshifts the total emissivity of IR galaxies is poorly constrained, due to the scarcity of Spitzer galaxies at 2, the large spectral extrapolations to derive the total IR luminosity from the mid-IR (see e.g. Elbaz et al. 2010, Nordon et al. 2010 and Nordon et al. 2012 for descriptions of the failure, at least at 1.5, of previous total IR luminosity extrapolations from the mid-IR, although we must note that this failure mainly affects luminosity-dependent methods like, e.g., that of Chary & Elbaz 2001) and the incomplete information on the -distribution of sub-mm sources (Chapman et al. 2005).
Herschel, with its 3.5-m mirror, is the first telescope which allows us to detect the far-IR population to high redshifts (4–5) and to derive its rate of evolution through a detailed LF analysis. The new extragalactic surveys provided by Herschel in the far-IR/sub-mm domain, like the wide and shallow Herschel-ATLAS (Eales et al. 2010; Dunne et al. 2011), the complementary Herschel Multi-tiered Extragalactic Survey (HerMES; Oliver et al. 2012) and PACS Evolutionary Probe (PEP; Lutz et al. 2011) covering the most popular cosmological fields, and the deep, pencil beam, Herschel-GOODS project (Elbaz et al. 2011), will be crucial to assess galaxy and AGN evolution in the IR at 2. They will give us the opportunity to study in detail the population of IR galaxies and their evolution with cosmic time since the Universe was about a billion years old. In particular, the Photodetector Array Camera & Spectrometer (PACS; Poglitsch et al. 2010), with its high sensitivity and resolution at 70-m, 100-m and 160-m, is the best suited instrument to detect faint IR sources by overcoming the source confusion and blending problems that affected the previous far-IR missions.
This is the first of two papers aiming at deriving the far- and total IR LFs from the Herschel PACSSpectral and Photometric Imaging Receiver (SPIRE; Griffin et al. 2010) data obtained within the PEP and HerMES extragalactic survey projects. In the present paper, we derive the rest-frame 35-m, 60-m, 90-m and total IR (8–1000 m) LFs from a sample selected at PACS 70, 100 and 160 m wavelengths in the GOODS (GOODS-S and GOODS-N), Extended Chandra Deep Field South (ECDFS) and COSMOS areas. We use the full 70–500 m PACSSPIRE data to determine and SED properties of the PACS selected sources. In a related paper, Vaccari et al. (in prep.) derive rest-frame 100-, 160- and 250-m and total IR LFs for a SPIRE selected sample. In addition, a third work aimed at studying the total IR LF based on the 24-m selected sample, using all the PEPHerMES data in the COSMOS field, is ongoing (Le Floc’h et al., in preparation).
PEP is one of the major Herschel Guaranteed Time extragalactic key-projects, designed specifically to determine the cosmic evolution of dusty star-formation and of the IR luminosity function. It is structured as a “wedding cake”, based on four different layers covering different areas to different depths at 100 and 160 m (in the GOODS-S field also at 70 m), from the large and shallow COSMOS field to the deep, pencil beam GOODS-S field. PEP includes the most popular and widely studied extragalactic fields with extensive multi-wavelength coverage available, in particular deep optical, near-IR and Spitzer imaging and spectroscopic and photometric redshifts: COSMOS; Lockman Hole; Extended Groth Streep (EGS); ECDFS; GOODS-N; and GOODS-S (see Berta et al. 2010, Berta et al. 2011 and Lutz et al. 2011 for a detailed description of the fields and observations). Coordinated observations of the PEP fields at 250, 350 and 500 m with SPIRE have been obtained by the HerMES Survey (Oliver et al. 2012). HerMES, analogously to PEP but extending to a much wider area, is a legacy programme designed to map a set of nested fields (380 deg in total) of different sizes and depths, using SPIRE (at 250, 350 and 500 m), and PACS (at 100 and 160 m, shallower than PEP), with the widest component of 270 deg with SPIRE alone. In the fields covered by PEP, the two surveys are closely coordinated to provide an optimized sampling over wavelength.
In Gruppioni et al. (2010) we started to determine the evolution with redshift of the galaxy and AGN LF in the far-IR domain by exploiting the PEP data obtained in GOODS-N by the PEP Science Demonstration Programme (SDP). Here we extend the analysis to the wider and shallower fields – COSMOS and ECDFS – and to the deepest field – GOODS-S – observed by PEP, and we also take advantage of the HerMES sub-mm data in the same fields to derive improved SED classifications and accurate total IR luminosities for our sources. This allows us to have statistically significant samples of IR galaxies at different redshifts and over a broad range of luminosities, to make a detailed study of the LF at several intervals, all the way from 0 to 4. The measure of the total IR luminosity obtained by integrating the SEDs, well constrained over the entire mid- and far-IR domain (and also in the sub-mm thanks to the available SPIRE data), allows us to derive the total IR LF and its evolution directly from far-IR data for unbiased samples selected at wavelengths close to the peak of dust emission. Moreover, the availability of deep multi-wavelength catalogues in the PEP fields is crucial for analysing the SEDs, obtaining k-corrections and total IR luminosities, and classifying the PEP sources into different IR populations, in order to separately study their LFs and evolutionary behaviour. This is the first study ever based on such a statistically wide and deep far-IR sample, to be able to provide LFs for different IR populations of galaxies and AGN. Here the evolution of the far- and total IR LFs (and luminosity density, hereafter ) are derived up to unprecedented high redshifts (4) both globally (e.g. for all the populations together) and separately, for each SED class.
Despite the abundance of information available in the literature about the stellar mass function (MF; Fontana et al. 2004; Pozzetti et al. 2010; Ilbert et al. 2010; Dominguez-Sanchez et al. 2011), very little is known about the corresponding total IR LFs and star-formation rate (SFR) densities at different masses (an attempt based on Spitzer data was made by Pérez-González et al. 2005). From stellar MF studies one finds a clear increase with of the relative fraction of massive (log(/M11) star-forming objects, starting to contribute significantly to the massive-end of the MFs at 1 (Fontana et al. 2004; Ilbert et al. 2010). Their evolution and contribution to the total SFR history is however still uncertain, since only few studies have tried to reconstruct the evolution of the SF history of massive objects from optical/near-IR or mid-IR surveys (Juneau et al. 2005; Pérez-González et al. 2005; Santini et al. 2009; Fontanot et al. 2012) but none from far-IR selected surveys (providing a more direct indicator of the galaxy SF activity). In this work, we have derived the IR luminosity function and density in three different mass ranges (from log(/M)8.5 to log(/M)12), extending previous studies (limited to 1.8–2 for the most massive galaxies) to 4.
Finally, our PEP data-sets have allowed us to quantify the relative contribution of the two main modes of star formation (a relatively steady one in disk-like galaxies, defining a tight SFR-stellar mass sequence, and a starburst mode in outliers) to the total IR LF and in three redshift intervals (0.81.25, 1.251.8 and 1.82.2) and to test the SED-classes belonging to each mode.
The paper is structured as follows. The PEP Survey with the far-IR and multi-wavelength data, together with the SED characterisation and redshift distribution of the PEP sources, are described in Sect. 2. The LFs (rest-frame 35-m, 60-m, 90-m and total IR), their evolution (derived for different SED-classes, mass and specific star-formation rate intervals) are discussed in Sect. 3. In Sect. 4 we present the number and IR luminosity densities of the different galaxy types, while in Sect. 5 we discuss our results. In Sect. 6 we present our conclusions.
Throughout this paper, we use a Chabrier initial mass function (IMF) and we assume a CDM cosmology with = 71 km s Mpc, = 0.27, and .
## 2 The Data
The PEP fields where we computed the LFs are: COSMOS, 2 deg observed down to 3 depths of 5 mJy and 10.2 mJy at 100 m and 160 m, respectively; ECDFS, 700 arcmin down to 3 depths of 4.5 mJy and 8.5 mJy at 100 m and 160 m, respectively; GOODS-N, 300 arcmin to 3 and 5.7 mJy at 100 m and 160 m, respectively; and GOODS-S, 300 arcmin to 1.2 mJy, 1.2 mJy and 2.4 mJy at 70 m, 100 m and 160 m, respectively. Our reference samples are the blind catalogues at 70 (in GOODS-S only), 100 and 160 m to the level, which contain 373 (all in GOODS-S), 7176 (GOODS-S: 717, GOODS-N: 291, ECDFS: 813, COSMOS: 5355) and 7376 (GOODS-S: 867, GOODS-N: 316, ECDFS: 688, COSMOS: 5105) sources at 70, 100 and 160 m, respectively. We refer to Berta et al. (2010) and Berta et al. (2011) for a detailed description of the data catalogues and source counts.
### 2.1 Multi-wavelength Identification
The PEP fields benefit from an extensive multi-wavelength coverage. We have therefore associated our sources to the ancillary catalogues by means of a multi-band likelihood ratio technique (Sutherland & Saunders 1992; Ciliegi et al. 2001), starting from the longest available wavelength (160 m, PACS) and progressively matching 100 m (PACS), 70 m (PACS, GOODS-S only) and 24 m (Spitzer/MIPS). In the GOODS-S field, we have associated to our PEP sources the 24-m catalogue by Magnelli et al. (2009), that we have matched with the opticalnear-IRIRAC MUSIC catalogue of Grazian et al. (2006), revised by Santini et al. (2009), which includes spectroscopic and photometric redshifts. To maximise the fraction of identifications, we limited our study to the area covered by the MUSIC catalogue (196 arcmin), obtaining 233, 468 and 492 sources at 70, 100 and 160 m, respectively, with flux density greater than the flux limits reported above (all with either spectroscopic or photometric redshifts). In the GOODS-N field, as described in Berta et al. (2010), Berta et al. (2011) and Gruppioni et al. (2010), a PSF-matched multi-wavelength catalogue111publicly available at
http://www.mpe.mpg.de/ir/Research/PEP/publicdatareleases.php
was created, including photometry from the far-UV (GALEX) to the mid-IR (Spitzer). As in GOODS-S, to maximise the identifications, we limited our study in GOODS-N to the area covered by the ACS (150 arcmin), obtaining 176 and 186 sources with flux density greater than the flux limit at 100 and 160 m, respectively (all with redshifts). We have matched our sources in the ECDFS with the multi-wavelength Survey by Yale-Chile (MUSYC) by Cardamone et al. (2010), obtaining 687 sources at 100 m and 625 sources at 160 m (578 and 547 with redshifts, 45% spectroscopic). Finally, in COSMOS, we have matched our catalogue with the deep 24-m sample of Le Floc’h et al. (2009) and with the IRAC-based catalogue of Ilbert et al. (2010), including optical and near-IR photometry and photometric redshifts. After the removal of PEP sources within flagged areas of the optical and/or IRAC COSMOS catalogues, we ended up with two catalogues consisting of 4110 and 4118 sources, with flux densities 5.0 and 10.2 mJy at 100 and 160 m respectively (3817 and 3849 with either spectroscopic or photometric redshifts). Throughout this paper and specifically for the SED fits described in Section 2.2, we adopt these spectroscopic or rest-frame UV to near-IR photometric redshifts for the various fields.
The HerMES extragalactic survey (Oliver et al. 2012) performed coordinated observations with SPIRE at 250, 350 and 500 m in the same fields covered by PEP. In particular, in HerMES a prior source extraction was performed using the method presented in Roseboom et al. (2011), based on MIPS-24 m positions. The 24-m sources used as priors for SPIRE source extraction are the same as those associated with our PEP sources through the likelihood ratio technique. We have therefore associated the HerMES sources with the PEP sources by means of the 24-m sources matched to both samples. For most of our PEP sources (87 per cent) we found a 3 SPIRE counterpart in the HerMES catalogues.
### 2.2 Galaxy Classification
We made use of all the available multi-wavelength data to derive the SEDs of our PEP sources, which we interpreted and classified by performing a fit (using the Le Phare code222available at
http://www.cfht.hawaii.edu/arnouts/LEPHARE/lephare.html
; Arnouts et al. 2002 and Ilbert et al. 2006) with the semi-empirical template library of Polletta et al. (2007), representative of different classes of IR galaxies and AGN. To this library we added some templates modified in their far-IR part to better reproduce the observed Herschel data (see Gruppioni et al. 2010), and three starburst templates from Rieke et al. (2009). If required to improve the fit, different extinction values ( from 0.0 to 0.5) have been applied to the templates, by letting the code free to choose the most suitable extinction curve. The considered set of templates included SEDs of elliptical galaxies of different ages, lenticular, spirals (from Sa to Sdm), starburst galaxies (SB), type 1 QSOs, type 2 QSOs, Seyferts, LINERs and composite ULIRGs (containing both starburst and obscured AGN component), in the wavelength range between 0.1 and 1000 m. The latter templates, are empirical ones created to reproduce the SEDs of the heavily obscured AGN. Two of these SEDs (the broad absorption-line QSO Markarian 231 (Berta, 2005) and the Seyfert 2 galaxy IRAS 192547245 South (Berta et al., 2003)) are similar in shape, containing a powerful starburst component, mainly responsible for their far-IR emission, and an AGN component that contributes to – and dominates – the mid-IR (Farrah et al. 2003), reproducing the SEDs of “obscured” AGN regardless of their optical spectra (i.e. broad or narrow lines in the optical; Gruppioni et al. 2008). Hereafter, we will refer to this class of templates and to the sources reproduced by them as to type 2 AGN (AGN2). Three other empirical templates, reproducing the observed SEDs of nearby ULIRGs containing an obscured AGN (i.e. IRAS 20551-4250; IRAS 22491-1808; NGC 6240) have been associated to the Seyfert 1.8/2, LINER ones, since they all contain an AGN, but this AGN does not dominate the observed energetic output at any wavelength (from UV to far-IR/sub-mm), showing up just in the range where the host galaxy SED has a minimum (i.e. the mid-IR). The AGN in these objects is either obscured or of low luminosity. We refer to this class as to star-forming galaxies containing an AGN (SF-AGN), since their IR luminosity is largely dominated by star-formation.
In our analysis, we make the basic assumption that the SED shapes seen at low redshifts are also able to represent the higher redshift objects. In any case, to further increase the range of SEDs in the fit, we have applied additional extinction with different extinction curves to our templates. All SED fits adopt fixed spectroscopic or photometric redshifts described in Section 2.1.
The template library used to fit our data contains a finite number of SEDs (38), representative of given classes of local infrared objects, which do not vary with continuity from one class to another (there are large gaps in the parameter space). Therefore, the quality of the fit depends not only on the photometric errors, but also on the template SED uncertainties. For this reason, in our fitting procedure, in addition to the photometric errors on data, we need to take into account also the uncertainties due to the template SEDs discretisation and additional extinction. To do this, we have proceeded as described in detail by Gruppioni et al. (2008) and summarised as follows. First, we have run Le Phare on our PEP SEDs considering the nominal errors from catalogues, computing the distributions of the values in each of the considered photometric band (where and are the flux density and the relative error of the source, and the flux density of the template in the considered band), iteratively increasing the photometric errors until we have obtained a Gaussian distribution with 1. This corresponds to reduced distributions peaked around 1 (as expected in the case of good fit). With the new photometric uncertainties (on average, significantly increased mainly in the optical/near-IR and SPIRE bands), we have run Le Phare on our sources for the second time, obtaining what we have taken as the final SED-fitting results.
The majority of our PEP sources are reproduced by templates of normal spiral galaxies (spiral), SB galaxies (starburst), and Seyfert2/1.8/LINERS/ULIRGsAGN (SF-AGN), although different classes prevail at different redshifts and luminosities. The spiral SEDs show no clear signs of enhanced SF or nuclear activity (see Fig. 1), the far-IR bump being characterised by relatively cold dust (20 K). On the other hand, SB templates are characterised by warmer (40–45 K), more pronounced far-IR bumps and significant UV extinction, indicative of intense star-formation activity. Templates of star-forming galaxies containing either a low-luminosity or obscured AGN (SF-AGN) are characterised by a “flattening” in the 3–10 m spectrum (suggesting detection of an AGN in the wavelength range where the host galaxy SED has a minimum) and a far-IR bump dominated by star-formation, which is intermediate (in terms of both energy and ) between spirals and SBs. Although they can be considered as star-forming galaxies at the wavelengths relevant to this work, we prefer to refer to them as SF-AGN throughout the paper, to keep in mind that they probably contain an AGN, whose presence, though not dominant in the far-IR, might be very important for analysis in other bands (e.g., in the X-rays or the mid-IR).
We note that the well-studied high-z (2.3) SED of the strongly lensed sub-mm galaxy SMM J2135-0102, known as “the Cosmic Eyelash” (Ivison et al. 2010; Swinbank et al. 2010), best-fitted with our procedure by an extincted (E(B-V)0.2) IRAS 22491-1808 template (though rather poorly in the near-IR), does not represent the bulk of our population at high- (1.5), whose SEDs are indeed well reproduced by our library of templates.
In fact, the considered template set provides very good fits to the SEDs of our PEP sources. In Fig. 1 we show the rest-frame SEDs (black dots) of the PEP sources belonging to the different “broad” SED classes (spiral, starburst, SF-AGN, AGN1 and AGN2), compared to the template SEDs of those classes normalised to the -band flux density. In Table 1 we report the fraction of sources belonging to each SED class: we find that in all the fields 41(38) per cent of the 100(160)-m sources are reproduced by a spiral template SED, 7(7) per cent with a starburst template SED, 45(48) per cent with a SF-AGN template SED, 2(3) per cent with an AGN2 SED and 5(4) per cent with an AGN1 SED. We note that the fraction of SF-AGN derived in this work is in agreement with results from mid-IR spectroscopy (with Spitzer-IRS) of local star-forming galaxies from the SINGS sample by Smith et al. (2007), who found that 50 per cent of local galaxies (though of lower luminosities than ours) do harbour low-luminosity AGN (of LINER or Seyfert types). Recently, Sajina et al. (2012) found an even higher fraction (70 per cent) of objects hosting an AGN in the mid-IR (Spitzer 24-m) selected samples (23 per cent AGN-dominated and 47 per cent showing both AGN and starburst activity). However, since the far-IR SED of the SF-AGN is dominated by star-formation and at these wavelengths resembles either starburst or spiral galaxy templates, we have also divided the SF-AGN class into SF-AGN(SB) and SF-AGN(Spiral) sub-classes, based of their far-IR/near-IR colours (e.g. /) and SED resemblance (apart from the rest-frame mid-IR flattening, which is detected in all of the SF-AGN SEDs). Specifically, galaxies best-fitted by the Seyfert2/1.8 templates (either the original ones from Polletta et al. 2007 or those modified by Gruppioni et al. 2010) have been classified as SF-AGN(Spiral), while galaxies best-fitted by the NGC 6240, IRAS 20551-4250 or IRAS 22491-1808 templates have been classified as SF-AGN(SB). The number of sources belonging to the former and the latter sub-classes are also reported in Table 1 as additional information.
### 2.3 Redshift Distribution
A large number of spectroscopic redshifts have been measured in the GOODS, ECDFS and COSMOS regions. In the GOODS-S and ECDFS area a collection of more than 5000 spectroscopic redshifts are available (Cristiani et al. 2000; Croom et al. 2001; Bunker et al. 2003; Dickinson et al. 2004; Stanway et al. 2004; Strolger et al. 2004; Szokoly et al. 2004; van der Wel et al. 2004; Doherty et al. 2005; Le Fèvre et al. 2005; Mignoli et al. 2005; Vanzella et al. 2008; Popesso et al. 2009; Santini et al. 2009; Balestra et al. 2010; Cooper et al. 2012). In the GOODS-N area more than 2000 spectroscopic redshifts come from various observations (Cohen et al. 2000; Wirth et al. 2004; Cowie et al. 2004; Barger et al. 2008). Finally, in COSMOS we could use a collection of 3000 spectroscopic redshifts from either the public zCOSMOS bright database or the non-public zCOSMOS deep database (Lilly, S.J. et al. 2007; Lilly et al. 2009). For the PEP sources without spectroscopic redshift available, we have adopted the photometric redshifts derived from multi-wavelength (UV to near-IR) photometry by different authors in the different fields, as mentioned in Section 2.1. In the GOODS-S field the MUSIC photometric redshift catalogue (Grazian et al. 2006; Santini et al. 2009) provided photo-s for most of our PEP sources without spectroscopic data, while in the GOODS-N field, photo-s were obtained by Berta et al. (2010) for almost all the PEP sources within the ACS area. The Cardamone et al. (2010) and Ilbert et al. (2009) catalogues provided photometric redshifts for a large fraction of the PEP sources in the ECDFS and COSMOS areas, respectively. When considering both the spectroscopic and photometric redshifts, in our PEP fields the redshift incompleteness is very low. In particular, in the GOODS-S field we have either a spec- or a photo- for 100 per cent of the PEP sample within the MUSIC areas (80 per cent spectroscopic, though most of them lie at 2.5; see Berta et al. 2011). In the GOODS-N field we have a redshift completeness of 100 per cent of sources (70 per cent spectroscopic) within the ACS area. In the ECDFS and COSMOS fields we have a redshift completeness of 88 per cent and 93 per cent respectively (45 per cent and 40 per cent spectroscopic).
The uncertainty in the photometric redshifts has been evaluated by means of a comparison with the available spec-s by the different authors providing photo- catalogues in the PEP fields. In particular, Berta et al. (2011) have compared the photometric and the available spectroscopic redshifts in GOODS-S, GOODS-N and COSMOS, finding a fraction of outliers, defined as objects having /(1)0.2, of 2 per cent for sources with a PACS detection. Most of these outliers are sources with few photometric points available, or SEDs not well reproduced by the available templates. The median absolute deviation of the /(1) distribution in the three fields analised by Berta et al. (2011) is 0.04 for the whole catalogue, and 0.038 for PACS-detected objects. In GOODS-S, Grazian et al. (2006) found an excellent agreement between photometric and spectroscopic redshifts over the fully accessible redshift range 06 (0.045), with a very limited number of catastrophic errors. In COSMOS, Ilbert et al. (2010) estimated the photometric redshift uncertainties of their 3.6-m catalogue matched with the COSMOS photo- multi-wavelength catalogue of Ilbert et al. (2009), finding 0.008 (and 1 per cent of catastrophic failures) at 22.5, 0.011 at 22.524 and 0.053 at 2425. In the ECDFS, by comparing non-X-ray sources with high-quality spectroscopic redshifts, Cardamone et al. 2010 found 0.008 to 1.2 0.027 at 1.23.7 and 0.016 at 3.7. Note that we have checked all the 2.5 photometric redshifts through Le Phare, assigning the Le Phare derived value in case of significant disagreement with that from the catalogue (though most resulted in very good agreement). The fractions of spectroscopic redshifts in the 2.53.0 interval amount to just 6% and 25% in COSMOS and GOODSS respectively, dropping to 4% and 6% at 3.04.2. We note that from our comparison between photo- and spec- (when available) we find a general good agreement in all fields.
Photometric redshift errors may, in principle, affect the shape of the luminosity function at the bright end: by scattering objects to higher redshifts they make the steep fall-off at high luminosities appear shallower (e.g. Drory et al. 2003). To study the impact of redshift uncertainties on the inferred infrared LF, we have performed Monte Carlo simulations, as discussed in detail in Section 3.3. It is indeed very difficult to estimate the effect of catastrophic failures at 2.5, where mainly photometric redshifts are available and very little reliable spectroscopic data can be used to validate them. Moreover, for the limited high- samples with spectroscopic information, different results are found in the different fields: i.e., in GOODS-S Berta et al. (2011) found about 25 per cent of catastrophic failures for PACS detected sources above 2, with the tendency to have a higher than real photometric redshift, while in COSMOS the catastrophic failures (20 per cent) found by Ilbert et al. (2009) for MIPS selected sources at 23 were mostly for photo-’s smaller than spectroscopic ones. In addition to that, sometimes high- spectroscopic redshifts can be even more uncertain than photometric ones and great care must be taken when selecting spec-s for comparison (i.e. we need to choose those with high quality flags). For all these reasons, we limited our analysis of photo- uncertainties to the Monte Carlo simulations described in Section 3.3, without trying to derive uncertainties also due to catastrophic failures. In Section 3.2 we also note that the different (far-IR) photo-z approach of Lapi et al. (2011) produces consistent LF results in the common part of parameter space.
The median redshift of the 70-m sample in GOODS-S is (70)0.67 (the mean is 0.86), while those of the 100- and 160-m samples are different in each of the fields, given the different flux density depths reached by PEP in each area. In Table 2 we report the median and the mean redshifts found for the different fields (and for the combined sample) at the different selection wavelengths. As expected, GOODS-S reaches the highest redshifts, while the surveys in the COSMOS and ECDFS fields are shallower and sample lower redshifts. On average, the 160-m selection favours higher redshifts than the 100-m one (see also Berta et al. 2011).
In Fig. 2 we show the redshift distribution of the PEP sources selected at 100 m and 160 m in the four different fields. The black solid histogram is the total redshift distribution in the field (one for each row of the plot), while the filled histograms in different colours represent the redshift distributions of the different populations (green, spiral; cyan, starburst; red, SF-AGN; magenta, AGN2; blue, AGN1). The line-filled dashed histograms shown in the spiral and starburst panels represent the redshift distributions of the SF-AGN(Spiral) and SF-AGN(SB) sub-classes, respectively. In addition to the principal redshift peak, in GOODS-S a secondary peak centred at 2 is clearly visible at both 100 and 160 m. A similar result has been shown and discussed also by Berta et al. (2011), while an extensive analysis of PACS GOODS-S large-scale structure at 2–3 and of a 2.2 filamentary overdensity have been presented by Magliocchetti et al. (2011).
## 3 The Luminosity Function
The sizes and depths of the PEP samples are such as to allow a direct and accurate determination of the far-IR LF in several redshift bins, from 0 up to 4. PEPHerMES is the unique Herschel survey to allow such analysis over such a wide redshift and luminosity range, sampling both the faint and bright ends of the far- and total IR LFs with sufficient statistics. Because of the redshift range covered by PEP, we would need to make significant extrapolations in wavelength when computing the rest-frame LFs at any chosen wavelengths. In order to apply the smallest extrapolations for the majority of our sources, we choose to derive the far-IR LFs at the rest-frame wavelengths corresponding to the median redshift of each sample. Given the median redshift of the 70-m sample in GOODS-S (0.67, see Table 2), we use that sample to derive the rest-frame luminosity function at 35 m. With the 100- and 160-m PEP samples (whose median redshifts are 0.64 and 0.73 respectively), we derived the rest-frame LFs at 60 and 90 m. Note that, given the excellent multi-wavelength coverage available for most of our sources (thanks also to the HerMES data available in all the PEP fields and providing reliable counterparts for most of our PEP sources), their SEDs are very well determined from the UV to the sub-mm. The extrapolations are therefore well constrained by accurately defined SEDs, even at high redshifts (i.e. at 3.5 the rest-frame 90-m luminosity corresponds to 400 m, which is still in the range covered by HerMES).
### 3.1 Method
The LFs are derived using the method (Schmidt 1968). This method is non-parametric and does not require any assumptions on the LF shape, but derives the LF directly from the data. We have first derived the LFs in each field separately, in order to check for consistency and to test the role of cosmic variance. Successively, we have made use of the whole data-sets to derive the monochromatic and total IR LFs, by means of the Avni & Bahcall (1980) method for coherent analysis of independent data-sets. We have divided the samples into different redshift bins, over the range 04, selected to be almost equally populated, at least up to 2.5. In each redshift bin we have computed the comoving volume available to each source belonging to that bin, defined as , where is the minimum between the maximum redshift at which a source would still be included in the sample given the limiting flux of the survey (different for each field) and the upper boundary of the considered redshift bin, while is just the lower boundary of the considered redshift bin.
When combining the four samples, we have constructed a complete sample over the whole GOODS-SGOODS-NECDFS(GOODS-S)COSMOS region, including all the observed objects (see details in Section 3.2). The depth of the sample is not constant throughout the region, but an object with a given flux density (included in the list of observed objects irrespective of the identity of its parent sample) can a priori be found in one (or more) region if its redshift is of that region (e.g. sources detected in the COSMOS area are detectable over the whole joint area, while the fainter sources detected in GOODS-S are detectable in GOODS-S only). The maximum volume of space which is available to such an object to be included in the joint sample is then defined by
Vzmax,i = ΩGS4πVGSzmax+ΩGN4πVGNzmax+ΩE4πVEzmax+ +ΩC4πVCzmax (if zmax,i≤zCmax) = ΩGS4πVGSzmax+ΩGN4πVGNzmax+ΩE4πVEzmax (if zCmax
where V (with =, , , corresponding to GOODS-S, GOODS-N, ECDFS and COSMOS, respectively) is the comoving volume available to each source in that field, in a given redshift bin, while is the solid angle subtended by that field sample on the sky.
For each luminosity and redshift bin, the LF is given by:
where is the comoving volume over which the - galaxy could be observed, is the size of the luminosity bin, and is the completeness correction factor of the - galaxy. These completeness correction factors are a combination of the completeness corrections given by Berta et al. (2010) and Berta et al. (2011), derived as described in Lutz et al. (2011), to be applied to each source as function of its flux density, together with a correction for redshift incompleteness (for the ECDFS and COSMOS only, see Section 2.3). Since, as mentioned in 2.3, the redshift incompleteness in the COSMOS and ECDFS areas is independent on PACS flux density, in these fields we have applied the corrections regardless of the source luminosity and redshift (i.e. by multiplying by 1.07 and 1.14 in COSMOS and ECDFS, respectively). However, the redshift incompleteness does not affect our conclusions, since 95 per cent of all our sources do have a redshift.
Uncertainties in the infrared LF values depend on photometric redshift uncertainties. To quantify the effects of the uncertainties in photometric redshifts on our luminosity functions, we performed a set of Monte Carlo simulations, as described in Section 3.3.
### 3.2 The Rest-Frame 35-, 60- and 90-μm Luminosity Function
By following the method described above, we have derived the 35-m, 60-m and 90-m rest-frame LFs from the 70-m (in GOODS-S only), 100-m and 160-m samples, respectively. In order to check the consistency between the catalogues and the effects of cosmic variance, we have first derived the monochromatic LFs in each field separately. Note that, since the 70-m data are available in the GOODS-S field only, to have a better sampling of the LF especially at the bright-end, we have also computed the rest-frame 35-m LF from the 100-m samples in the four fields and compared them with that obtained from the 70-m sample (see Fig 3; Table 3). The agreement between the two derivations is very good, implying correct extrapolations in wavelength due to the good and complete SED coverage.
We have divided the samples into seven redshift bins: 0.00.4; 0.40.8; 0.81.2; 1.21.8; 1.82.5; 2.53.5; and 3.54.5. The results of the computation of our 35-m (reported in Table 3), 60-m and 90-m LFs are shown in Figs. 3, 4 and 5, respectively. The LFs in the four different fields appear to be consistent with each other within the error bars (1) in most of the common luminosity bins. The COSMOS and GOODS-S Surveys are complementary, with the faint end of the LFs being mostly determined by data in GOODS-S, and the bright end by COSMOS data. After having checked the field-to-field consistency, we have combined the 100- and 160-m samples in all fields, obtaining the global rest-frame 60- and 90-m LFs (reported in Tables 4 and 5, respectively).
The data from each field in each -bin have been plotted (and considered in the combination) only in the luminosity bins where we expect our sample to be complete, given that at fainter luminosities not all galaxy types are observable (depending on their SEDs; Ilbert et al. 2004). For comparison, we overplot the LFs at 35 m from Magnelli et al. (2009) and Magnelli et al. (2011), the local LFs at 60 m from Saunders et al. (1990) and those at 90 m from Serjeant et al. (2004) and Sedgwick et al. (2011) and at 100 m from Lapi et al. (2011), respectively. The comparison between the 35-m LF, derived from the 70-m PEP sample in GOODS-S, and the results of Magnelli et al. (2009) and Magnelli et al. (2011), based on a 24-m prior extraction and stacking analysis on Spitzer maps, shows very good agreement, both with the data and with the double power-law fit. The 1.82.5 PEP LF is consistent within 1 with the Magnelli et al. (2011) data points, though the power-law fit at bright (10 L) seems to be slightly lower than our data. At 2.5 no comparison data from Spitzer are available, while our LF derivation can provide hints of evolution at the bright end of the LF. In the common redshift intervals (between 1 and 3.5), our 90-m LF is in very good agreement with the 100-m Lapi et al. (2011) derivation from the H-ATLAS survey (although their redshift bins are somewhat different than ours: 1.21.6; 1.62.0; 2.02.4; and 2.44.0) and with the previous PEP-SDP derivation (Gruppioni et al. 2010). The consistency between our 90-m LFs and the Lapi et al. (2011) ones (derived from a different sample, using a different template SED to fit the data and a rest-frame far-IR based method to derive photometric redshifts) gives us confidence that, at least up to 3.5, our computation is not significantly affected by incompleteness or by photo- uncertainties.
### 3.3 The Total Infrared Luminosity Function
We integrate the best-fit SED of each source over 81000 m to derive the total IR luminosities ([8–1000 m]) in 11 redshift bins (0.0–0.3; 0.3–0.45; 0.45–0.6; 0.6–0.8; 0.8–1.0; 1.0–1.2; 1.2–1.7; 1.7–2.0; 2.0–2.5; 2.5–3.0; and 3.0–4.2), selected to be almost equally populated, at least up to 2.5. Our approach is similar to that of other studies based on mid-IR selected galaxy samples (e.g. Le Floc’h et al. 2005; Rodighiero et al. 2010a; Magnelli et al. 2011), but this is the first time that the SEDs have been accurately constrained by sufficiently deep data in the far-IR/sub-mm domain and not simply extrapolated from the mid-IR to the longer wavelengths or from average flux density ratios.
As mentioned in Section 2.3, we have studied the impact of redshift uncertainties on the total IR LF by performing Monte Carlo simulations. As test cases, we used the COSMOS and GOODS-S samples (which are basically defining the bright and faint ends of the LF in an almost complementary way), and we checked the effect on the total IR luminosity function. We iterated the computation of the total IR LF by each time varying the photometric redshift of each source (assigning a randomly selected value, according to a Gaussian distribution, within the 68 per cent confidence interval). Each time, we then recomputed the monochromatic and total IR luminosities, as well as the value, but we did not perform the SED-fitting again, keeping the previously found best-fit template for each object (the effect on the k-correction is not relevant in the far-IR wavelength range). The results of this Monte Carlo simulation are reported in Fig. 6, where we show the total IR LFs derived in each of the PEP fields independently: the red and blue filled circles represent the estimates of the GOODS-S and COSMOS LFs, with their range of values derived with 20 iterations by allowing a change in photo- represented by the pink and sky-blue shaded areas, respectively. The comparison shows that the effect of the uncertainty of the photometric redshifts on the error bars is slightly larger than the simple Poissonian value (1/), and affects mainly the lower and the higher redshift bins (especially at low and high luminosities). Using these Monte Carlo simulations, we find no evident systematic offsets caused by the photometric redshift uncertainties (see Fig. 6). This is due to the very accurate photometric redshifts available in these fields. For the total uncertainty in each luminosity bin in GOODS-S and COSMOS, we have therefore assumed the dispersion given by the Monte Carlo simulations (as shown in Fig. 7). We note that at the higher (2.5), where we must rely on a majority of photometric redshifts, the true uncertainties (taking into account also catastrophic failures or incompleteness effects) might be larger than derived with simulations. The unavailability of a “true” comparison sample (i.e. a large comparison sample with accurate spectroscopic redshifts and fully representative for the PACS flux selection) at high z does not allow to properly quantify this statement.
In Fig. 7 the total IR LFs obtained by combining the 160-m selected samples with the Avni & Bahcall (1980) technique is plotted and compared with other derivations available in the literature. The total IR LF of Sanders et al. (2003) is plotted as a local reference, in addition to the LFs of Le Floc’h et al. (2005), Rodighiero et al. (2010a), Caputi et al. (2007), Magnelli et al. (2009) and Magnelli et al. (2011) in various redshift intervals. Globally, data from surveys at different wavelengths agree relatively well over the common -range. No data for comparison are available at 2.5, apart from the IR LF of sub-mm galaxies from Chapman et al. (2005) and Wall et al. (2008) at 2.5, which represent reasonably well just the very bright end of the total IR LF. Our derivation is the first at such high redshifts, especially in the 34.2 range. Note the good agreement between our PEP-based total IR LF and the HerMES-based one derived by Vaccari et al. (in preparation), shown by the red open squares in Fig. 7. Though consistent within the error-bars, in the highest redshift bin the HerMES LF is slightly higher than ours. Since the 250-m HerMES selection favours the detection of higher redshift sources than the PEP one, it is more likely that the PEP LF in the higher-z bin is affected by some flux/redshift incompleteness rather than by the presence of low- sources erroneously placed at high- by incorrect photometric redshift assignment (catastrophic failures). The values of our total IR LF for each redshift and luminosity bin are reported in Table 6.
### 3.4 Evolution
In order to study the evolution of the total IR LF, we derive a parametric estimate of the luminosity function at different redshifts. For the shape of the LF we assume a modified-Schechter function (i.e. Saunders et al. 1990), where is given by
Φ(L)dlogL=Φ⋆(LL⋆)1−αexp[−12σ2log210(1+LL⋆)]dlogL, (3)
behaving as a power law for and as a Gaussian in for . The adopted LF parametric shape depends on 4 parameters (, , and ), whose best fitting values and uncertainties have been found using a non-linear least squares fitting procedure. In detail, while in the first -bin all the parameters have been estimated, starting from the second -bin, the values of and have been frozen at the values found at lower redshift, leaving only and free to vary (see Table 7). Note that, in the highest redshift bin (3.04.2) we are not able to constrain the LF break, while we are up to 3. Therefore, the results found at this redshift are affected by larger uncertainties than the 3 ones. However, although there is some degeneracy in the values of and at 3.04.2, the range of allowed value combinations still giving a reasonable fit to the three observed data points (constraining the bright-end of the LF) is limited and do not significantly affect our results.
In Fig. 8 we plot the total IR LFs at all redshifts with the 1 Poissonian uncertainty regions (different colours for different -intervals). There is a clear luminosity evolution with redshift, at least up to 3. The apparent “fall” of the 3 LF with respect to those at lower redshifts, if real, might imply a global negative evolution of the IR galaxies and/or AGN starting at 3. However, as mentioned above, we must point out that in the highest redshift bin PACS data could be affected by incompleteness, since with increasing redshift (and for intrinsically “cold” sources) the true PACS fluxes might fall below the detection limit (i.e. faint sources should be missed even if completeness corrections are perfect), while, if luminous enough, they can still be detectable by SPIRE. This effect is expected to be more relevant in COSMOS, where SPIRE data are quite deep relatively to PACS, while it should not happen in GOODS-S, where PACS data are very deep compared to the SPIRE ones (which are limited by confusion). The high- incompleteness of PACS surveys might also be emphasised by the redshift incompleteness of the COSMOS sample, that could affect the highest redshift bins more than the lower ones (although the redshift incompleteness seems to be independent of redshift; see Berta et al. 2011). However, we must point out that a decrease at 2.7–3 similar to that observed in our data, is also observed in the space density of X-ray (see Brusa et al. 2009; Civano et al. 2011) and optically selected AGN (Richards et al. 2006), and of sub-mm galaxies (Wall et al. 2008), as well as in the HerMES total IR LF – though less evident – (from 250-m data; Vaccari et al., in preparation: see Fig. 7). Moreover, our result is in agreement with the recent finding of Smit et al. (2012), that the characteristic value of the galaxy SFR exhibits a substantial, linear decrease as a function of redshift from 2 to 8.
In Fig. 9 we show the values of and at different redshifts, with the best curve () fit to the data points. The values of the curve slopes and of the redshifts corresponding to evolutionary breaks have been chosen to be those which minimise the of the fit with two power-laws. We find a significant variation of L with , which increases as up to 1.85, and as between 1.85 and 4. The variation of with redshift starts with a slow decrease as up to 1.1, followed by a rapid decrease (1) at 1.1 and up to 4.
Previous estimations of the evolution of and (i.e. Caputi et al. 2007, Bethermin et al. 2011 and Marsden et al. 2011) discussed a decrease in the density of far-IR sources between 1 and 2. In particular, Bethermin et al. (2011) and Marsden et al. (2011), by evolving a parameterised far-IR LF, explored the evolution required by the source counts in the parameter space. The results of these works (especially those of Bethermin et al. 2011, showing a decrease of at 1 and a flatter trend on the evolution of at 2), are close to ours, though with the source counts only it was not possible to constrain the evolution of IR sources at 2.
### 3.5 Evolution of the Different IR Populations
The PEP survey, given its size and its coverage in luminosity and redshift, allows us to go further in investigating the evolution of the total IR LF: it gives us the opportunity to study the evolution of the different galaxy classes that compose the global IR population. To investigate the different evolutionary paths of the various IR populations, we have computed the 1/V LFs separately for the five galaxy classes defined by the SED-fitting analysis. In Fig. 10 we show the total IR LFs derived from the combined PEP samples for the different SED classes (coloured filled areas). The results of the fit to a modified Schechter function (see equation 3) for each population are overplotted on the data. In fact, by following the same procedure adopted for the global luminosity function, a parametric fit to the LFs at different redshifts has been performed also for the single populations. The and parameters, for each population, have been estimated at the redshift where the corresponding LF is best sampled (not necessarily at the lowest -bin as for the global LF). Subsequently, the values of and have been frozen at the values found in the “optimal” redshift bin, leaving only and free to vary. In Fig. 11, analogously to Fig. 9, we show the values of and at different redshifts for the different populations, with the best least square fitting curves () overplotted. The best-fitting values of , , and in the first redshift bin, together with the parameters describing the luminosity (, and : (1) to , (1) at ) and density evolution (, and : (1) out to , (1) at ) are reported in Table 8.
A clear result of our analysis is that the evolution derived for the global IR LF is indeed a combination of different evolutionary paths: the far-IR population does not evolve all together “as a whole”, as it is often assumed in the literature, but is composed by different galaxy classes evolving differently and independently. As shown in Fig. 10, the normal spiral galaxy population dominates the luminosity function at low-, from the local Universe up to 0.5. Moving to higher redshifts, the number density of galaxies with spiral galaxy SEDs sharply decreases, while their luminosity continues to increase, at least up to 1 (see the and parameter trends shown in Fig. 11). We note that what we observe between 0 and 1 for the spiral SED galaxies is an increase of by a factor of 5, and a decrease of by a factor of 10. Since the two evolutions are not independent, the “total” evolutionary effect results from the combination of the two (as can be observed in the total IR luminosity density, see Section 4). A way to derive the ”total” effect of evolution on a LF is to fix at a given volume density value and see how the luminosity corresponding to that value changes: indeed we find an increases by a factor of 2.5 between 0 and 1 for the spiral LF, in good agreement with previous results, either empirical (for morphologically classified disky galaxies; Scarlata et al. 2007) or theoretical (from chemical evolution models of Milky Way-like galaxies; Colavitti, Matteucci & Murante 2008).
Over the whole redshift range 0.53, the “totall” luminosity function is dominated by the SF-AGN population. The number density of SF-AGN is nearly constant from the local Universe up to 1–1.5, showing a slight decrease at higher redshifts, while their luminosities show positive evolution up to the highest redshifts (3.5–4). From Fig. 10 we note a sort of bimodality in the SF-AGN LFs (at 0.45, where we are able to cover a larger luminosity range). This bimodality is indeed to be ascribed to the crossing of two contributions: that of the SF-AGN(Spiral) population, responsible for the faint-end steepness of the LFs, and that of the SF-AGN(SB) population, dominating the bright-end of the SF-AGN LFs and declining at low (not reported in the figure).
The starburst galaxy population never dominates. The redshift range where we observe the highest contribution from the starburst galaxies is at 1–2, while in the local Universe their contribution is almost negligible (i.e. their parameter shows an opposite trend with respect to that of spiral galaxies see Fig. 11).
The AGN1 and AGN2 populations show a very similar evolutionary trend as a function of , both in and . These powerful AGN populations dominate only the very bright end of the total IR LF, although their number densities and luminosities keep increasing from the local Universe up to the higher redshifts. At 2.5 the AGN1 and AGN2 populations become as important as the SF-AGN one, with the total IR LF of PACS-selected sources in the redshift range 2.5–4 being totally dominated by objects containing an AGN.
### 3.6 Total IR LF in Mass and Specific Star-Formation Rate bins
#### 3.6.1 Stellar masses and SFR from SED fitting
The wealth of multi-wavelength data available in the cosmological fields included in our work allow us to perform a detailed SED fitting of all sources, in order to derive their most relevant physical parameters (e.g. stellar masses). To derive stellar masses we have fitted the broad-band SEDs of our sources using a modified version of MAGPHYS (Da Cunha et al. 2008), which is a code describing the SEDs using a combination of stellar light and emission from dust heated by stellar populations. In particular, the MAGPHYS software simultaneously fits the broad-band UV-to-far IR observed SED of each object, ensuring an energy balance between the absorbed UV light and that re-emitted in the far-IR regime. The main assumptions are that the energy re-radiated by dust is equal to that absorbed, and that starlight is the only significant source of dust heating. We refer to Da Cunha et al. (2008) for a thorough formal description of how galaxy SEDs are build. At each source’s redshift, the code chooses among different combinations of star formation histories, metallicities and dust contents, associating a wide range of optical models to a wide range of infrared spectra and comparing to observed photometry, seeking for minimization. Each star formation (SF) history is parameterised in terms of an underlying continuous model with exponentially declining star formation rate (SFR), on top of which are superimposed random bursts (see Da Cunha et al. 2008, Da Cunha et al. 2010). We note that, although the MAGPHYS assumption of exponentially declining SFR might not be the best to reproduce the SFR history of 1.5 star-forming galaxies (i.e. exponentially increasing or increasing SFR would be better choices, as widely discussed by Maraston et al. 2010 and Reddy et al. 2012), in our specific case it does not affect the results. In fact, we do not use the MAGPHYS derived SFRs, but we compute them by integrating the best-fitting SED (resulting from Le Phare). The models are distributed uniformly in metallicity between 0.2 and 2 times solar. Since the MAGPHYS code assumes that starlight is the only significant source of dust heating, thus ignoring the presence of a possible AGN component, Berta et al. (2013) have developed a modified version of the MAGPHYS code by adding a torus component to the modelled SED emission, combining the Da Cunha et al. (2008) original code with the Fritz et al. (2006) AGN torus library (see also Feltre et al. 2012). The spectral fitting is performed by comparing the observed SED of our galaxies to every model in the generated library, at the corresponding redshift. A minimisation provides the quality of each fit. We must point out that the mass derivation for unobscured AGN (i.e. AGN1) is a problematic issue, therefore the masses estimated for that class of objects are the most uncertain ones. One source of uncertainty in the mass measurement for AGN1 is due to the fact that, in these objects, the UV part of the SED is likely dominated by the AGN rather than by the host galaxy. This may produce an underestimate of the mass, since, if the AGN contribution is not taken into account, the data can be fit by a bluer, younger and smaller mass object. On the other hand, the mid-IR part of the AGN SED is dominated by dust emission from the dusty torus heated by the central black hole. If a proper decomposition into a stellar and a torus component is not performed, the use of a pure stellar template for estimating the mass from SED-fitting tends to reproduce the mid-IR emission with an older, redder and larger mass object (mass sometimes larger by a factor of 2 than those derived through a decomposition procedure; Santini et al. 2012). These two effects lead to an increase in the uncertainty of the mass derivation, although they might somewhat compensate their effects for a large sample of objects. For this reason, we obtained measurements of the stellar masses of our objects containing an AGN by means of the specific decomposition technique developed by Berta et al. (2013), to separate stellar and nuclear emission components. Examples of the results of this decomposition applied to SF-AGN, AGN2 and AGN1 are shown in Fig. 12. Masses of AGN estimated with the original MAGPHYS and with the Berta et al. (2013) code have been compared, showing very good agreement and small dispersion around the 1–1 relation. Similarly, as further check, we have also computed stellar masses with different code (Hyperz; Bolzonella et al. 2000) and stellar library (BC03, Bruzual & Charlot 2003, instead of the CB07 used as default by MAGPHYS), finding good agreement and no sistematics, too.
We have derived the SFRs from the total IR luminosities (estimated from the SED fitting described in Section 2.1) with the standard Kennicutt (1998) relation (converted to Chabrier IMF), after subtracting the AGN contribution to . We note that the total IR luminosity in PEP sources is usually dominated by star formation, even in objects for which an AGN dominates the optical/near-IR/mid-IR part of the spectrum.
#### 3.6.2 LFs in different mass bins
We compute the total IR LF for galaxies of different stellar masses: 8.5log(/M)10, 10log(/M)11, and 11log(/M)12, by means of the standard 1/V formalism, and we show the results in Figs. 13 (total IR LF in different -bins) and 14 (ratio between the LF in the mass intervals and the total IR LF).
We have compared our results with the SFR function (SFR converted to IR luminosity using the Kennicutt (1998) relation) of massive (log(/M10) galaxies derived by Fontanot et al. (2012) from the GOODS-MUSIC sample at redshift 0.41.8. In the common redshift and luminosity range we find an excellent agreement with our total IR LF, which is dominated by sources with 10log(/M)11. At lower luminosities, not sampled by our data, the Fontanot et al. (2012) LFs are characterized by a double-peaked structure, interpreted in terms of the well-known bimodality in the colour(SFR)-Mass diagram. As expected from the SFR-Mass relation, the knee () of our IR LFs in different mass bins moves to higher luminosities with increasing masses (i.e. at 0.00.3, log(/L) changes from 9.5 for log(/M)8.5–10 sources, to 11.3 for sources with log(/M)11).
The slope of the total IR LF in each mass bin is always similar to (or flatter than) the “global” LF (total, including all the masses). The lower mass galaxies dominate at lower luminosities (log(/L)9), while the most massive galaxies (log(/M)11) contribute only at higher luminosities, even if they never dominate the LF. At all masses, the LF evolves with redshift, following the evolution of the “global” LF. Fig. 14 shows that the main contribution (50 per cent) to the total IR LF is due to intermediate-mass objects (log(/M)10–11) at all redshifts and luminosities, with their fraction remaining almost the same from 0 to 4, simply shifting to higher luminosities. Lower mass objects (log(/M)8.5–10) contribute significantly only at log(/L)10, with their fraction just shifting to higher luminosities with redshift, but always being below 30 per cent at 0.45 and log(/L)11. The contribution of the most massive objects (log(/M)11–12) increases with IR luminosity and redshift, becoming significant (50 per cent) only at 1.7 and log(/L)12.5. Thus the bulk of the IR luminosity is produced by star-forming galaxies of mass around the characteristic mass M of the Schecter mass function.
### 3.7 Specific-SFR and the main sequence of star-forming galaxies
Having computed stellar masses and SFRs for each source, we can check how the PACS selected sources populate the SFR–stellar mass plane and the so called main sequence (MS) of star-forming galaxies, as a function of redshift. This relation (i.e. SFR versus stellar mass) has been shown to be quite tight in the local Universe (Peng et al. 2010, 2011) and well established at redshift 1 (Elbaz et al. 2007) and up to 2 (Daddi et al. 2007, Rodighiero et al. 2011) and 3 (Magdis et al. 2010), with normalisation scaling as (1) out to 2, as shown by Sargent et al. (2012) (see also Elbaz et al. 2007, Rodighiero et al. 2010b, Pannella et al. 2009, Karim et al. 2011). At 2 and 1.5 we assume a slope of 0.79 for the MS in the SFR versus stellar-mass plane (according to Rodighiero et al. 2011 and Sargent et al. 2012), while at 1 we assume a slope of 0.9, as found by Elbaz et al. (2007), and we limit our investigation to the redshift range 0.82.2.
By combining UV and far-IR data, Rodighiero et al. (2011) re-evaluated the locus of the MS at 2, showing that objects lying a factor of 4 above the MS (in SFR) can be considered as outliers with respect to the average locus where smoothly star-forming galaxies spend most of their lives in a secular and steady regime. In that work, off-sequence sources (characterized by very high specific-SFRs) are assumed to be in a starburst mode, and are found to contribute only 2 per cent of mass-selected star-forming galaxies and to account for only 10 per cent of the cosmic SFR density at 2 (Rodighiero et al. 2011).
In order to check what kind of objects we could classify as on- and off-MS sources in our IR sample compared to previous findings, based either on IR or on optical surveys (e.g. Rodighiero et al. 2011; Sargent et al. 2012), we have splitted our sample into off-MS and on-MS. For consistency with previous studies we have applied the same criterion as Rodighiero et al. (2011) (0.6 dex above the MS) over the whole 0.82.2 redshift range, by using as a reference MS the relation found by Rodighiero et al. (2011) at 2, scaled as described above at 1.5, and the relation found by Elbaz et al. (2007) at 1. In Fig. 15, we show the SFR versus stellar mass distributions in three redshift bins (0.81.25, 1.251.8 and 1.82.2), for the PACS sources included in the computation of the luminosity functions presented in this work. The colour code marks the different SED-classes to which each source belongs. We also report the typical loci of the MS at the various redshifts (scaling as (1), as mentioned above). Details are given in the caption of the Figure. The typical far-IR selection bias (PACS-Herschel in this case) appears as an approximate horizontal SFR cut (Rodighiero et al. 2011, Wuyts et al. 2011), shown as thin dotted line in Fig. 15. We note that the trends of mid/far-IR SEDs with offset from the main sequence observed in Fig.15 (and widely discussed in Section 5) are in good agreement with the results of Elbaz et al. (2011) and Nordon et al. (2012).
With the selection based on Rodighiero et al. (2011) and overplotted in Fig. 15 (sources qualify as “off-MS” if they lie more than 0.6 dex above the observed SFR-stellar mass relation) , we can compute the contribution of off-MS (also called “starburst” in the literature) and on-MS (“steady star-formers” in the literature) sources to the total IR LFs. This is presented in Fig. 16, where the total IR LFs of on- and off-MS sources have been computed independently for the three redshift bins. The pink and yellow filled areas correspond to the 1 uncertainty regions of the total IR LFs estimated for the on-MS and off-MS populations, respectively, while the black filled area marks the global population. Our results are compared with the recent estimates by Sargent et al. (2012), a non-parametric approach that is based on three basic observables: the redshift evolution of the stellar mass function for star-forming galaxies; the evolution of the sSFR of MS galaxies; and a double-Gaussian decomposition of the sSFR distribution at fixed stellar mass into a contribution (assumed redshift- and mass-invariant) from MS and off-MS (i.e. starburst) activity. The evolution of the two populations found both in our data and the Sargent et al. (2012) model are very similar. Both data and estimates indicate that the bright-end of the total IR LF is dominated by off-MS sources. However, although consistent within the uncertainties, the relative contribution of off-MS sources seems to be stronger in the Sargent et al. (2012) model than observed in the present computation, where we find that the bright-ends of the PEP IR LFs are more similarly populated by MS and off-MS sources (especially at 2). This difference can be at least partly ascribed to the sharp cut we apply to separate MS from off-MS sources, while Sargent et al. (2012) model the off- and on-MS populations with two continuous log-normal distributions centred at 0.6 dex above the MS and exactly on the MS respectively, of which the one describing the “starburst” population has wings that extend into our on-MS selection region (hence attributing more sources to the starburst category than are selected in our off-MS class). A better agreement between our data and the estimates of Sargent et al. (2012) is found at the faint-end of the LFs that appear to be completely dominated by the “normal” MS galaxies at all redshifts (although the total and relative contributions at log(/L)12 are slightly lower in the data than predicted by Sargent et al. 2012). Good agreement is also found with respect to the evolution of the cross-over luminosity (i.e. where the contributions from on- and off-MS sources are equal); in the model of Sargent et al. (2012) the cross-over luminosity simply shifts to higher luminosities and lower densities (according to the luminosity and density evolution considered). Note that the model assumptions rely on results from different surveys, selected at different wavelengths, complete in mass and with good sampling of the MS. On the other hand, our selection is in SFR and our sources do not follow any clear sequence in stellar mass–SFR plane, because, except at the highest masses, the data are not deep enough to reach well into the main sequence. These different selection effects are likely to lead to some differences between our LFs and the estimates of Sargent et al. (2012).
To quantify the relative contribution of the two populations, our observed data have been fitted with a modified Schechter function, in order to integrate them and compute their comoving number and luminosity densities as functions of redshift (see next Section).
## 4 Number Density and IR Luminosity Density
We derive the evolution of the comoving number and luminosity density (total, see Fig 18) of the PEP sources, either belonging to the different SED classes (Fig 18, ), to the on- and off-MS categories () and to the different mass intervals (), by integrating the total IR LF in the different redshift bins from 0 to 4. To compute the number (and IR luminosity) density, we integrate the Schechter functions that best reproduce the different populations/mass/sSFR-classes, down to log(/L)8. We note that here we consider lower limits the number and luminosity densities at 3.04.2, since our LF estimate in that redshift bin is likely to be incomplete, as discussed in Section 3.3. We find that the number density of the whole IR population is nearly constant in the 0–1.2 redshift range (slightly increasing from 0 to 0.5), decreasing at 1.2 (see panels of Fig. 18). When decomposing the number density according to the different SED classes, we observe that normal spiral galaxies dominate the local density, with a smaller contribution also from the SF-AGN population and a negligible one from starburst, AGN1 and AGN2. The space density of spiral galaxies decreases rapidly at 0.5, while that of SF-AGN stays nearly constant at 0.52.5, largely dominating in that redshift range. Starburst galaxies never dominate, while the number density of the bright AGN (both AGN1 and AGN2) increases with redshift, from 10 Mpc at 0 to 1–210 Mpc at 3. At higher redshifts the AGN population largely dominates the number density.
If the overall contribution to the IR luminosity density () from the AGN components of galaxies is small, can be considered as a proxy of the SFR density (). As a further check, we have therefore studied the evolution of the SF-AGN population (which dominates the distribution of sources) by dividing this class into SF-AGN(SB) and SF-AGN(Spiral) sub-classes and studying their evolution separately. Indeed, we have found different evolutionary paths for the two populations, the former dominating at higher redshifts and showing a behaviour similar to that of AGN-dominated sources (e.g. AGN1 and AGN2), the latter dominating at intermediate redshifts (between 1 and 2), rising sharply from 2 toward the lower redshifts and decreasing, while the spiral population rises at 1. These evolutionary trends, in terms of number and luminosity density, have been reported in Fig. 18 as orange dot-dot-dot-dashed (SF-AGN(SB)) and dark-green dashed (SF-AGN(Spiral)) curves.
Galaxies following the SFR–mass relation are always dominant over the off-MS population, at all redshifts (although their space density decreases with increasing , as well as the “global” number density), while the number density of the latter population remains nearly constant between 0.8 and 2.2.
In all the mass bins, the trends with redshift of the galaxy number densities are similar to the “global” one, decreasing at higher redshifts, although with slightly different slopes for the different mass intervals. The number densities of low mass galaxies (8.5log(/M)10), reported in the top right panel of Fig. 18, have been computed by integrating the best-fitting modified Schechter function only to 2, since data were not enough to derive reliable fits at higher redshifts. To this redshift, these sources outnumber the higher mass ones, although they fall steeply above 1, when they reach about the same volume density of higher mass galaxies (10log(/M)11). Massive objects (log(/M)11) never dominate (always below 5 per cent) the total number density.
The total IR LF allows a direct estimate of the total comoving IR luminosity density () as a function of , which is a crucial tool for understanding galaxy formation and evolution. Although can be converted to a SFR density () under the assumption that the SFR and quantities are connected by the Kennicutt (1998) relation, before doing that we must be sure that the total IR luminosity is produced uniquely by star-formation, without contamination from an AGN. The SED decomposition and separation into AGN and SF contributions show a negligible contribution to (10 per cent) from the AGN in most of the SF-AGN, and a SF component dominating the far-IR even in the majority of more powerful AGN (AGN1 and AGN2). Here we prefer to speak in terms of rather than of , since, especially at high redshift – where the AGN-dominated sources are more numerous – the conversion of could represent only an upper limit to . Note, however, that since this population is never dominant in our IR survey, we do not expect that contamination related to accretion activity occurring in these objects (mainly at high-) can significantly affect the results in terms of . | {"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": 0.9388276934623718, "perplexity": 2262.5881895892926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499890.39/warc/CC-MAIN-20230131190543-20230131220543-00742.warc.gz"} |
http://engineerexperiences.com/author/ismail-sarwar/page/2 | ## Special Types of Energy Meters
Special Categories of Energy Meters: In the previous article, we discussed basics types of energy meters. In this article I will continue the discussion on other special types of energy meters. Let’s take this hierarchy diagram again. We already understood about the types of energy meters with respect to technology. Now, let’s talk about category […]
## Basic Types of Energy Meters
Basic Categories of Energy Meters There are many basic types of Energy Meters that it is not an easy task to categorize them. Still I will try my best to sort out different types of energy meters in some simple manner. To Study about the Basics of Energy Meters, click here. First, look at this […]
## Objectives of Energy Meters
Objectives of Energy Meters With the advancement of technology, objectives of energy meters are not merely to measure energy units. Objectives went far behind from just giving energy readings of some specific consumer. Also, these objectives are not constant, they are increasing day by day with the increase in advancement in energy sector. Let’s discuss […]
## Basics of Energy Meters
What is Energy Meter? A device which captures the “units of energy” is called as Energy Meter. This is the basic definition of an Energy Meters. The only difficult part in this definition is “units of energy”. Let’s describe this part in more detail. What are Units of Energy? To understand about the units […]
## Basics of GSM Module
Basics of GSM Module If you just started your work on GSM modules and you have no idea where to start then this article is right for you. I will give you step by step details on the basics of GSM modules. But first of all I should build your interest to work on GSM […]
## Smart Energy Meter using Atmel AVR Microcontrollers
What is Smart Energy Meter? A smart energy meter is device which can calculate the energy consumption and send it to energy provider through some communication medium. It must be compact, cost effective and able to communicate data by itself with the power control center. I designed a smart energy meter through an ATmega8 microcontroller […]
## 3 Phase Voltage Measurement using Atmel AVR Microcontrollers
3 Phase Voltage Measurement 3 Phase voltage measurement is a little bit tricky task. I will start from the basics and give you details step by step. Specification of this Project Measure voltage of each phase Measure angle of each phase Gives Line to Line voltage as well All results display on 16x2 LCD Less […] | {"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": 0.9032546281814575, "perplexity": 1136.0104261161848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999130.98/warc/CC-MAIN-20190620024754-20190620050754-00370.warc.gz"} |
https://byjus.com/angle-between-two-vectors-formula/ | # Angle between Two Vectors Formula
If the two vectors are assumed as $\vec{a}$ and $\vec{b}$ then the dot created is articulated as $\vec{a}. \vec{b}$. Let’s suppose these two vectors are separated by angle θ. To know what’s the angle measurement we solve with the below formula
we know that the dot product of two product is given as
$\vec{a}.\vec{b} =|\vec{a}||\vec{b}|cos\theta$
Thus, the angle between two vectors formula is given by
$\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$
where θ is the angle between $\vec{a}$ and $\vec{b}$
## Angle Between Two Vectors Examples
Let’s see some samples on the angle between two vectors:
Example 1:
Compute the angle between two vectors 3i + 4j – k and 2i – j + k.
solution:
Let
$\vec{a}$ = 3i + 4j – k and
$\vec{b}$ = 2i – j + k
The dot product is defined as
$\vec{a}. \vec{b}$ = (3i + 4j – k).(2i – j + k)
= (3)(2) + (4)(-1) + (-1)(1)
= 6-4-1
= 1
Thus, $\vec{a}. \vec{b}$ = 1
The Magnitude of vectors is given by
$|\vec{a}| =\sqrt{(3^{2}+4^{2}+(-1)^{2})} =\sqrt{26}= 5.09$
$|\vec{b}| =\sqrt{(2^{2}+(-1)^{2}+1^{2})} =\sqrt{6}= 2.45$
The angle between the two vectors is
$\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$
$\theta = cos^{-1}\frac{1}{(5.09)(2.45)}$
$\theta = cos^{-1}\frac{1}{12.47}$
$\theta = cos^{-1}(0.0802)$
$\theta = 85.39^{\circ}$
Example 2:
Find the angle between two vectors 5i – j + k and i + j – k.
Solution:
Let
$\vec{a}$ = 5i – j + k and
$\vec{b}$ = i + j – k
The dot product is defined as
$\vec{a}. \vec{b}$= (5i – j + k)(i + j – k)
$\vec{a}. \vec{b}$= (5)(1) + (-1)(1) + (1)(-1)
$\vec{a}. \vec{b}$= 5-1-1
$\vec{a}. \vec{b}$= 3
The Magnitude of vectors is given by
$|\vec{a}| =\sqrt{(5^{2}+(-1)^{2}+1^{2})} =\sqrt{27}= 5.19$
$|\vec{b}| =\sqrt{(1^{2}+1^{2}+(-1)^{2})} =\sqrt{3}= 1.73$
The angle between the two vectors is
$\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$
$\theta = cos^{-1}\frac{3}{(5.19)(1.73)}$
$\theta = cos^{-1}\frac{3}{8.97}$
$\theta = cos^{-1}(0.334)$
$\theta =70.48^{\circ}$ | {"extraction_info": {"found_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": 0.9976828694343567, "perplexity": 726.9823843757812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575541315293.87/warc/CC-MAIN-20191216013805-20191216041805-00017.warc.gz"} |
https://unapologetic.wordpress.com/2010/12/24/permutations-and-polytabloids/?like=1&source=post_flair&_wpnonce=eb8d9d939c | # The Unapologetic Mathematician
## Permutations and Polytabloids
We’ve defined a bunch of objects related to polytabloids. Let’s see how they relate to permutations.
First of all, I say that
$\displaystyle R_{\pi t}=\pi R_t\pi^{-1}$
Indeed, what does it mean to say that $\sigma\in R_{\pi t}$? It means that $\sigma$ preserves the rows of the tableau $\pi t$. And therefore it acts trivially on the tabloid $\{\pi t\}$. That is: $\sigma\{\pi t\}=\{\pi t\}$. But of course we know that $\{\pi t\}=\pi\{t\}$, and thus we rewrite $\sigma\pi\{t\}=\pi\{t\}$, or equivalently $\pi^{-1}\sigma\pi\{t\}=\{t\}$. This means that $\pi^{-1}\sigma\pi\in R_t$, and thus $\sigma\in\pi R_t\pi^{-1}$, as asserted.
Similarly, we can show that $C_{\pi t}=\pi C_t\pi^{-1}$. This is slightly more complicated, since the action of the column-stabilizer on a Young tabloid isn’t as straightforward as the action of the row-stabilizer. But for the moment we can imagine a column-oriented analogue of Young tabloids that lets the same proof go through. From here it should be clear that $\kappa_{\pi t}=\pi\kappa_t\pi^{-1}$.
Finally, I say that the polytabloid $e_{\pi t}$ is the same as the polytabloid $\pi e_t$. Indeed, we compute
$\displaystyle e_{\pi t}=\kappa_{\pi t}\{\pi t\}=\pi\kappa_t\pi^{-1}\pi\{t\}=\pi\kappa_t\{t\}=\pi e_t$
December 24, 2010 -
## 2 Comments »
1. […] polytabloids is a submodule, we must see that it’s invariant under the action of . We can use our relations to check this. Indeed, if is a polytabloid, then is another polytabloid, so the subspace spanned […]
Pingback by Specht Modules « The Unapologetic Mathematician | December 27, 2010 | Reply
2. […] use our relations to […]
Pingback by Examples of Specht Modules « The Unapologetic Mathematician | December 28, 2010 | Reply | {"extraction_info": {"found_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": 16, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9842222332954407, "perplexity": 526.2386739608728}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988717963.49/warc/CC-MAIN-20161020183837-00191-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/111466/upper-bound-on-joint-renyi-entropy | # Upper bound on joint Renyi entropy
Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by $$H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha.$$ Assume that $X$ and $Y$ have countably infinite alphabets, and observe the case $0\leq\alpha\leq 1$.
It is known that this functional is subadditive, i.e., that $H_\alpha(X,Y)\leq H_\alpha(X) + H_\alpha(Y)$, only for $\alpha=0$ (Hartley entropy) and $\alpha=1$ (Shannon entropy).
My question is the following: Does there exist some upper bound on the joint entropy for $0<\alpha<1$, in terms of the marginal entropies? In fact, I don't even need an explicit upper bound, answering the following simpler question would suffice: If $H_\alpha(X)<\infty$ and $H_\alpha(Y)<\infty$, does there exist a constant $h$ such that $H_\alpha(X,Y)\leq h$?
-
– Piotr Migdal May 14 '13 at 15:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9968237280845642, "perplexity": 173.32015390193556}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507452492.48/warc/CC-MAIN-20141017005732-00036-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://kerodon.net/tag/01Z1 | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
Remark 5.5.1.6. Let $X$ and $Y$ be Kan complexes. Then Remark 4.6.7.6 supplies a canonical homotopy equivalence of Kan complexes $\operatorname{Fun}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( X,Y)$. Beware that this homotopy equivalence is generally not an isomorphism. | {"extraction_info": {"found_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": 0.9630202651023865, "perplexity": 489.2744591609961}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610703514046.20/warc/CC-MAIN-20210117235743-20210118025743-00433.warc.gz"} |
https://samacheerkalvi.guide/samacheer-kalvi-12th-physics-guide-chapter-4/ | Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Physics Guide Pdf Chapter 4 Electromagnetic Induction and Alternating Current Text Book Back Questions and Answers, Notes.
## Tamilnadu Samacheer Kalvi 12th Physics Solutions Chapter 4 Electromagnetic Induction and Alternating Current
### 12th Physics Guide Electromagnetic Induction and Alternating Current Text Book Back Questions and Answers
Part – I
Text Book Evaluation:
I. Multiple choice questions:
Question 1.
An electron moves on a straight line path XY as shown in the figure. The coil abcd is adjacent to the path of the electron. What will be the direction of current, if any, induced in the coil?
a) The current will reverse its direction as the electron goes past the coil
b) No current will be induced
c) abcd
a) The current will reverse its direction as the electron goes past the coil
Solution:
First current develops in direction of abcd but when electron moves away, magnetic field inside loop decreases and current changes its direction.
Question 2.
A thin semi-circular conducting ring (PQR) of radius r is falling with its plane vertical in a horizontal magnetic field B, as shown in the figure. The potential difference developed across the ring when its speed v, is
a) Zero
b) $$\frac{\mathrm{B} v \pi \mathrm{r}^{2}}{2}$$ and P is at higher potential
c) πrBυ and k is at higher potential
d) 2rBυ and R is at higher potential
d) 2rBυ and R is at higher potential
Solution:
Motional emt induced in the semi circular ring PQR is equal to the motional emt induced in the imaginary conductor PR.
EPQR = EPR = BVl = BV (2r) (l = PR = 2r)
∴ Potential difference developed across the ring is 2r Br with R is at higher potential.
Question 3.
The flux linked with a coil at any instant t is given by φB= 10t2 – 50t + 250. The induced emf at t = 3s is
a) -190 V
b) -10 V
c) 10 V
d) 190 V
b) -10V
Solution:
e = – $$\frac{\mathrm{d} \phi}{\mathrm{dt}}$$ = $$\frac{d}{d t}$$ (10t2 – 50t + 250)
e = -20t + 50
e = -10V
Question 4.
When the current changes from +2A to -2A in 0.05 s, an emf of 8 V is induced in a coil. The coefficient of self-induction of the coil is
a) 0.2 H
b) 0.4 H
c) 0.8 H
d) 0.1 H
d) 0.1 H
Solution:
Question 5.
The current i flowing in a coil varies with time as shown in the figure. The variation of induced emf with time would be
Solution:
For $$\frac{3 \mathrm{~T}}{4}$$ to T i = 0, $$\frac{d i}{d t}$$ = 0, e = 0
Question 6.
A circular coil with a cross-sectional area of 4 cm2 has 10 turns. It is placed at the center of a long solenoid that has 15 turns/cm and a cross-sectional area of 10 cm2. The axis of the coil coincides with the axis of the solenoid. What is their mutual inductance?
a) 7.54 μH
b) 8.54 μH
c) 9.54 μH
d) 10.54 μH
a) 7.54 μH
Solution:
M = μ0 N1 N2 A2
=4π × 10-7 × 15 × 102 × 10 × 4 × 10-4
= 4π × 6 × 10-7
= 24 × 3.14 × 10-7
= 75.36 × 10-7
= 7.54 × 10-6
M = 7.54 μH
Question 7.
In a transformer the number of turns in the primary and the secondary are 410 and 123C respectively. If the current in primary is 6A, then that in the secondary coil is
a) 2A
b) 18A
c) 12A
d) 1A
a) 2 A
Solution:
Question 8.
A step – down transformer reduces the supply voltage from 220 V to 11 V and increase the current from 6 A to loo A. Then its efficiency is
a) 1.2
b) 0.83
c) 0.12
d) 0.9
b) 0.83
Solution:
P = VI
Input Power = 220 × 6 = 1320
Output Power = 11 × 100 = 1100
η = $$\frac{1100}{1320}$$ = 0.83
Question 9.
In an electrical circuit, R, L, C, and AC voltage sources are all connected in series. When L is removed from the circuit, the phase difference between the voltage and current in the circuit is π/3. Instead, if C is removed from the circuit, the phase difference is again π/3. The power factor of the circuit is
a) 1/2
b) $$\frac{1}{\sqrt{2}}$$
c) 1
d) $$\frac{\sqrt{3}}{2}$$
c) 1
Solution:
It is the condition for resonance therefore phase difference between v and i = 0
Power factor cos Φ = 1
Question 10.
In a series RL circuit, the resistance and inductive reactance are the same. Then the phase difference between the voltage and current in the circuit is
a) $$\frac{\pi}{4}$$
b) $$\frac{\pi}{2}$$
c) $$\frac{\pi}{6}$$
d) zero
a) $$\frac{\pi}{4}$$
Solution:
tan Φ = $$=\frac{X_{L}}{R}$$
χL = R tan Φ = 1, Φ = 45°
Φ = $$\frac{\pi}{4}$$
Question 11.
In a series resonant RLC circuit, the voltage across 100Ω resistor is 40 V. The resonant frequency is 250 rad/s. If the value of C is 4μF, then the voltage across L is
a) 600 V
b) 4000 V
c) 400 V
d) 1 V
c) 400 V
Solution:
χc = $$\frac{1}{C \omega}$$ = 1000Ω
At resonant χc = χL
∴ I = $$\frac{V}{R}$$ = 0.4 A
∴ VL = IXL = 400 V
Question 12.
An inductor 20 mH, a capacitor 50 µF, and a resistor 40Ω are connected in series across a source of emf V = 10 sin 340 t. The power loss in the AC circuit is
a) 0.76 W
b) 0.89 W
c) 0.46 W
d) 0.67 W
c) 0.46 W
Solution:
Power loss = $$\left(\frac{E_{\mathrm{rms}}}{Z}\right) \mathrm{R}$$
Erms = $$\frac{10}{\sqrt{2}}$$
Z = $$\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}$$
χL = 6.82Ω
χC = 58.8Ω
∴ Z = 65.6Ω
∴ Power Loss = 0.46 W
Question 13.
The instantaneous values of alternating current and voltage in a circuit are
i = $$\frac{1}{\sqrt{2}}$$ sin(100πt) A and
v = $$\frac{1}{\sqrt{2}}$$ sin(100π + π/3) V
The average power in watts consumed in the circuit is
a) $$\frac{1}{4}$$
b) $$\frac{\sqrt{3}}{4}$$
c) $$\frac{1}{2}$$
d) $$\frac{1}{8}$$
d) $$\frac{1}{8}$$
Solution:
Question 14.
In an oscillating IC circuit, the maximum charge on the capacitor is Q. The charge on, the capacitor when the energy is stored equally between the electric and magnetic fields is
a) $$\frac{Q}{2}$$
b) $$\frac{Q}{\sqrt{3}}$$
c) $$\frac{Q}{\sqrt{2}}$$
d) Q
c) $$\frac{Q}{\sqrt{2}}$$
Solution:
Question 15.
$$\frac{20}{\pi^{2}}$$ H inductor is connected to a capacitor of capacitance C. The value of C in order to
impart maximum power at 50 Hz is
a) 50 µF
b) 0.5 µF
c) 500 µF
d) 5 µF
d) 5 µF
Solution:
Maximum power at χL = χC
L 2πγ = $$\frac{1}{2 \pi \gamma}$$
C = 5 µF
Question 1.
What is meant by electromagnetic induction?
Whenever the magnetic flux linked with a closed coil changes, an emf (electromotive force) is induced and hence an electric current flows in the circuit.
Question 2.
State Faraday’s laws of electromagnetic induction.
First law:
Whenever magnetic flux linked with a closed circuit changes, an emf is induced in circuit.
Second law:
The magnitude of induced emf in a closed circuit is equal to the time rate of change of magnetic flux linked with the circuit
$$E=-\frac{\mathrm{d} \theta}{\mathrm{dt}}$$
Question 3.
State Lenz’s law.
Lenz’s law states that the direction of the induced current is such that it always opposes the cause responsible for its production.
Question 4.
State Fleming’s right-hand rule.
1. The Thumb, index finger, and middle finger of the right hand are stretched perpendicular to each other.
2. The index finger indicates the direction of the magnetic field.
3. Thumb indicates the direction of motion of the conductor.
4. The middle finger indicates the direction of induced current.
Question 5.
How is Eddy’s current produced? How do they flow in a conductor?
Even for a conductor in the form of a sheet or plate, an emf is induced when magnetic flux linked with it changes. But the difference is that there is no definite loop or path for induced current to flow away. As a result, the induced currents flow in concentric circular paths. As these electric currents resemble eddies of water, these are known as Eddy currents. They are also called Foucault currents.
Question 6.
Mention the ways of producing induced emf.
1. By changing the magnetic field B.
2. By changing the area A of the coil and
3. By changing the relative orientation of the coil.
Question 7.
What for an inductor is used? Give some examples.
Inductor is a device used to store energy in a magnetic field when an electric current flows through it. The typical examples are coils, solenoids, and toroids.
Question 8.
What do you mean by self-induction?
1. Its magnetic flux is changed by changing the current in the coil, and induced emf is induced in the same coil.
2. This is known as self-induction.
Question 9.
What is meant by mutual induction?
When an electric current passing through a coil changes with time, an emf is induced in the neighbouring coil. This phenomenon is known as mutual induction.
Question 10.
Give the principle of AC generator.
1. AC generator works on the principle of electromagnetic induction.
2. The relative motion between a conductor and a magnetic field changes the magnetic flux linked with the conductor which in turn, induces an emf.
Question 11.
List out the advantages of stationary armature -rotating field system of AC generator.
1. The current is drawn directly from fixed terminals on the stator without the use of brush contacts.
2. The insulation of stationary armature winding is easier.
3. The number of sliding contacts (slip rings) is reduced. Moreover, the sliding contacts are used for low-voltage DC Source.
4. Armature windings can be constructed more rigidly to prevent deformation due to any mechanical stress.
Question 12.
What are step-up and step-down transformers?
1. A transformer that increases the voltage by decreasing current is known as a step-up transformer.
2. A transformer that decreases the voltage by increasing the current is a step-down transformer.
Question 13.
Define the average value of an alternating current.
The average value of alternating current is defined as the average of all values of current over a positive half-cycle or negative half-cycle.
Question 14.
How will you define the RMS value of an alternating current?
The RMS value of an alternating current is defined as the square root of the mean of the squares of all currents over one cycle.
I rms = $$\frac{\mathrm{I}_{\mathrm{m}}}{\sqrt{2}}$$
Question 15.
What are phasors?
A sinusoidal alternating voltage (or current) can be represented by a vector which rotates about the origin in an anti-clockwise direction at a constant angular velocity ω. Such a rotating vector is called a phasor.
Question 16.
Define electric resonance.
Ans:
When frequency of applied alternating source (ωr) is equal to natural frequency
$$\left(\frac{1}{\sqrt{L C}}\right)$$
of RLC circuit, the current in the circuit reaches its maximum value. Then the circuit is said to be in electrical resonance.
Question 17.
What do you mean by resonant frequency?
When the frequency of the applied alternating source (ωr) is equal to the natural frequency $$\left[\frac{1}{\sqrt{L C}}\right]$$ of the RLC circuit, the current in the circuit reaches its maximum value. Then the circuit is said to be in electrical resonance. The frequency at which resonance takes place is called resonant frequency. Resonant angular frequency, ωr = $$\frac { 1 }{ \sqrt { LC } }$$
Question 18.
How will you define Q-factor?
1. Q factor is defined as the ratio of voltage across L or C to the applied voltage.
2. Q – factor = $$\frac{\text { Voltage across } L \text { or } C}{\text { Applied Voltage }}$$
3. Q factor = $$\frac{1}{R} \sqrt{\frac{L}{C}}$$
Question 19.
What is meant by wattles current?
The component of current (IRMS sin φ), which has a phase angle of $$\frac { π }{ 2 }$$ with the voltage is called reactive component. The power consumed is zero. So that it is also known as ‘Wattless’ current.
Question 20.
Give any one definition of power factor.
Power factor is defined as
1. Power factor = cos Φ = cosine of the angle of lead or leg.
2. Power factor = $$\frac{R}{Z}=\frac{\text { Resistance }}{\text { Impedance }}$$
3. Power factor = $$\frac{\text { True power }}{\text { Apparent power }}$$
Question 21.
What are LC oscillations?
Whenever energy is given to a LC circuit, the electrical oscillations of definite frequency are generated. These oscillations are called LC oscillations. During LC oscillations, the total energy remains constant. It means that LC oscillations take place in accordance with the law of conservation of energy.
Question 1.
Establish the fact that the relative motion between the coil and the magnet induces an emf in the coil of
a closed circuit.
1. In first experiment when bar magnet is placed close to the coil, magnetic lines pass through coil, the magnetic flux in coil increases and emf is induced hence electric current flows in the circuit.
2. At the same time when they recede away from one another magnetic flux decreases emf is induced in opposite direction. Current flows in opposite direction. So there is a deflection in the galvanometer.
3. In second experiment, when the primary circuit is open, no electric current flows in it, magnetic flux linked with the secondary coil is zero.
4. When primary circuit is closed, increasing current produce magnetic field. So magnetic flux linked with coil increases. This induced current in secondary coil.
5. When primary circuit is broken, decreasing primary current induces current in secondary coil but in opposite direction. So there is a deflection in Galvanorneter.
Question 2.
Give an illustration of determining direction of induced current by using Lenz’s law.
1. Move the bar magnet towards solenoid with north pole pointing solenoid.
2. When the Magnetic flux increases in the coil, induced current is produced, and the coil becomes magnetic dipole.
3. According to Lenz law, induced current opposes the movement of north pole towards coil.
4. It is possible if end nearer to magnet become north pole, then it repels the north pole of magnet and oppose the movement of magnet.
5. The direction of induced current is found by the right hand thumb rule.
6. When a bar magnet is withdrawn, nearer end becomes south pole which attracts north pole of the bar magnet, opposing the receding motion of magnet.
7. Direction of induced current can be found from Lenz law.
Question 3.
Show that Lenz’s law is in accordance with the law of conservation of energy.
Conservation of energy:
The truth of Lenz’s law can be established on the basis of the law of conservation of energy. According to Lenz’s law, when a magnet is moved either towards or away from a coil, the induced current produced opposes its motion. As a result, there will always be a resisting force on the moving magnet.
Work has to be done by some external agency to move the magnet against this resisting force. Here the mechanical energy of the moving magnet is converted into electrical energy which in turn, gets converted into Joule heat in the coil i.e., energy is converted from one form to another.
Question 4.
Obtain an expression for motional emf from Lorentz force.
1. Consider a straight rod AB of length l in uniform magnetic field perpendicularly to plane of paper.
2. Let rod move with constant velocity $$\overrightarrow{\mathrm{v}}$$ towards right side free electrons present in it also move with same $$\overrightarrow{\mathrm{v}}$$ in $$\overrightarrow{\mathrm{B}}$$
3. The Lorentz force is $$\overrightarrow{\mathrm{F}}_{\mathrm{B}}$$ = -e ($$\overrightarrow{\mathrm{v}}$$ × $$\overrightarrow{\mathrm{B}}$$ )
4. Due to electric field E , the coulomb force starts acting on free electrons along AB.
$$\overrightarrow{\mathrm{F}}_{\mathrm{B}}$$ = -e$$\overrightarrow{\mathrm{E}}$$
5. Magnitude of $$\overrightarrow{\mathrm{E}}$$ increasing as long as accumulation of electrons at the end A continues $$\overrightarrow{\mathrm{F}}_{\mathrm{E}}$$ increases until equilibrium is reached.
6. At equilibrium,
$$\left|\overrightarrow{\mathrm{F}}_{\mathrm{B}}\right|=\left|\overrightarrow{\mathrm{F}}_{\mathrm{E}}\right|$$
$$|-e(\vec{v} \times \vec{B})|=|-e \quad \overrightarrow{\mathrm{E}}|$$
vB sin 90° = E
vB = E
7. The Potential difference is V = El
V = vBl
So, ε = Blv
8. An emf is produced due to movement of rod, it is called as motional emf.
9. If A and B are connected by external circuit of resistance R, then the current $$i=\frac{\varepsilon}{R}=\frac{B l V}{R}$$ flows in it.
10. Direction of current is found from right-hand thumb rule.
Question 5.
Give the uses of Foucault current.
Though the production of eddy current is undesirable in some cases, it is useful in some other cases. A few of them are
1. Induction stove
2. Eddy current brake
3. Eddy current testing
4. Electromagnetic damping
1. Induction stove:
An induction stove is used to cook the food quickly and safely with less energy consumption. Below the cooking zone, there is a tightly wound coil of insulated wire. The cooking pan made of a suitable material is placed over the cooking zone.
When the stove is switched on, an alternating current flowing in the coil produces a high frequency alternating magnetic field which induces very strong eddy currents in the cooking pan. The eddy currents in the pan produce so much of heat due to Joule heating which is used to cook the food.
2. Eddy current brake:
This eddy current braking system is generally used in high-speed trains and roller coasters. Strong electromagnets are fixed just above the rails. To stop the train, electromagnets are switched on. The magnetic field of these magnets induces eddy currents in the rails which oppose or resist the movement of the train. This is Eddy’s current linear brake.
In some cases, the circular disc, connected to the wheel of the train through a common shaft, is made to rotate in between the poles of an electromagnet. When there is a relative motion between the disc and the magnet, eddy currents are induced in the disc which stop the train. This is Eddy current circular brake.
3. Eddy current testing:
It is one of the simple non-destructive testing methods to find defects like surface cracks and air bubbles present in a specimen. A coil of insulated wire is given an alternating electric current so that it produces an alternating magnetic field.
When this coil is brought near the test surface, eddy current is induced in the test surface. The presence of defects causes the change in phase and amplitude of the eddy current that can be detected by some other means. In this way, the defects present in the specimen are identified.
4. Electromagnetic damping:
The armature of the galvanometer coil is wound on a soft iron cylinder. Once the armature is deflected, the relative motion between the soft iron cylinder and the radial magnetic field induces eddy current in the cylinder. The damping force due to the flow of eddy current brings the armature to rest immediately and then the galvanometer shows a steady deflection. This is called electromagnetic damping.
Question 6.
Define self – inductance of a coil in terms of
(i) magnetic flux and
(ii) induced emf.
1. Let be the magnetic flux linked each turn of the coil of N turns, then the total flux linked with the coil NPB is proportional to the current i in the coil
B α i
B = Li ……………(1)
2. Self-inductance of a coil is defined as the flux linkage of the coil when 1 A current flows through it.
L = $$\frac{\mathrm{N} \Phi_{\mathrm{B}}}{i}$$ If i = 1A then L = NφB
3. When the current i changes with time, an emf is induced in it. From Faradys law of electro magnetic induction, this self induced emf is given
ε = $$-d \frac{\left(\mathrm{N} \varphi_{\mathrm{B}}\right)}{d t}=-d \frac{(\mathrm{Li})}{d t}$$
Using equation (1)
L = $$\frac{-\varepsilon}{d i / d t}$$ If di/dt = 1A.S-1 then L = -ε
4. Self inductance of coil is defined as opposing emf induced in the coil when the rate of change of current through the coil is 1As-1
Question 7.
How will you define the unit of inductance?
Unit of inductance: Inductance is a scalar and its unit is Wb A-1 or V s A-1. It is also measured in henry (H).
1 H = 1 Wb A-1 = 1 V s A-1
The dimensional formula of inductance is M L2 T-2A-2.
If i = 1 A and NΦB = 1 Wb turns, then L = 1 H.
Therefore, the inductance of the coil is said to be one henry if a current of 1 A produces unit flux linkage in the coil.
If $$\frac { di}{ dt }$$ = 1 As-1 and ε = -1 V, then L = 1 H.
Therefore, the inductance of the coil is one henry if a current changing at the rate of 1 A s-1 induces an opposing emf of 1 V in it.
Question 8.
What do you understand by the self-inductance of a coil? Give its physical significance.
Self Inductance of coil:
1. Inductance or simply inductance of a coil is defined as the flux linkage of the coil when I A current flows through it.
2. Inductance of a coil is also defined as opposing emf induced in the coil when the rate of change of current through the is 1 As-1.
Physical Significance:
3. In translational motion, mass is measure of inertia, for rotational motion, moment of inertia is a measure of rotational inertia.
4. The inductance plays the same role in circuit as mass and moment of inertia play in mechanical motion.
5. When a circuit is switched on, the increasing current induces on emf which opposes the growth of current in circuit (Figure (a))
6. Induced emf e opposes the changing current i When circuit is broken, the decreasing current induces an emf in the reverse direction. This emf now opposes the decay of current (Figure (b))
7. Thus, inductance of the coil opposes any change in current and tries to maintain the original state.
Question 9.
Assuming that the length of the solenoid is large when compared to its diameter, find the equation for its inductance.
1. Consider a long solenoid of length l and cross-sectional area A. Let n be the number of turns per unit length (1> n).
2. When the current i pass through solenoid, uniform magnetic field B is produced.
B = µ0ni
3. The magnetic field passes through each turn is
ΦB = $$\int_{A} \vec{B} \cdot d \vec{A}$$ = BA cosθ = BA, Since θ = 0°
ΦB = (µ0 ni)A
Self inductance of a long solenoid
4. Total magnetic flux of solenoid with N turns (N = nl)
B = (nl)(µ0 ni) AB
B = (µ0 n2 Al) i
5. we have L = µ0n2Al
6. Inductance depends on geometry of solenoid and medium present inside the solenoid.
7. For dielectric medium of µr
L = µn2Al (or)
L = µ0 µr n2 Al
Question 10.
An inductor of inductance L carries an electric current i. How much energy is stored while establishing the current in it?
When current is established in circuit, inductance opposes the growth of current. So, work is done against this opposition by external agency. This work done is stored as magnetic potential energy.
Induced emf at any instant is
ε = -L$$\frac{d i}{d t}$$
Work done dW = -ε dq = -εidt
∴ dq = idt
Substituting for ε from first equation
ε = -L $$\frac{d i}{d t}$$
dw = -(-L $$\frac{d i}{d t}$$) i dt
dw = Li di
Total work done
W = ∫ dw = $$\int_{0}^{i}$$Li di = L$$\left[\frac{i^{2}}{2}\right]$$
W = $$\frac{1}{2}$$ Li2
This work done is stored as magnetic potential energy
∴ UB = $$\frac{1}{2}$$ Li2
Energy density is the energy stored per unit volume of space
UB = $$\frac{\mathrm{U}_{\mathrm{B}}}{\mathrm{Al}}$$
∴ Volume of solenoid = Al
Question 11.
Show that the mutual inductance between a pair of coils is same (M12 = M21)
Consider two coils placed close to each other, i1 is the electric current sent through coil 1 linked with coil 2.
Let Φ21 be the magnetic flux linked with coil 2 of N2 due to coil 1
N2 Φ21 α i1
N2 Φ21 = M21 i1
M21 = $$\frac{\mathrm{N}_{2} \Phi_{21}}{i_{1}}$$
M21 is called mutual inductance of coil 2 with respect to coil 1. It is also called coefficient of mutual inductance.
If i1 = 1A then M21 = N2 Φ21
Mutual Inductance M21 is defined as flux linkage of coil 2 when 1 A current flows through coil 1.
When i1 changes with time emf ε2 is induced in coil 2. From Faraday’s law of electromagnetic
induction
Mutual inductance M21 is defined as emf induced in coil 2 when rate of change of current through the coil I is 1 As-1
Similarly,
M12 = $$\frac{\mathrm{N}_{1} \Phi_{12}}{i_{2}}$$ and
M12 = $$\frac{-\varepsilon_{2}}{d i_{2} / d t}$$
The mutual inductance is same ie M21 = M12 = M
The mutual inductance between two coils depends on size, shape, number of turns of the coils, relative orientation and permeability of the medium.
Question 12.
How will you induce an emf by changing the area enclosed by the coil?
1. Consider a conducting rod of length l moving with velocity v towards left.
2. The whole arrangement is in uniform magnetic field $$\overrightarrow{\mathrm{B}}$$, magnetic lines perpendicular to plane of the paper.
3. As the rod moves from AB to CD in time dt area enclosed by the loop and the magnetic flux through the loop decreases.
Induction of emf by changing the area enclosed by the loop
Change in magnetic flux in time dt is
B = B × change in area
= B × Area ABCD
= Blv dt
(or) $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$ = Blv
The magnitude of induced emf.
ε = $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$
ε = Blv
The emf is called motional Emf,
Direction of induced current is clockwise from Flemings Right hand rule.
Question 13.
Show mathematically that the rotation of a coil in a magnetic field over one rotation induces an alternating emf of one cycle.
1. Consider a rectangular coil of N turns in a uniform magnetic field $$\overrightarrow{\mathrm{B}}$$
2. Coil rotates with angular velocity ω about an axis in an anticlockwise direction perpendicular to the field.
3. At t = 0 plane of coil is perpendicular to the field so flux linked with the coil is Φm = BA
4. At t seconds coil is rotated through angle Φ flux is Φm cos ωt, component of Φm normal to the coil.
Variation of induced emf as a function ωt
According to Faraday’s law the ernf at any instant is
ε = –$$\frac{\mathrm{d}}{\mathrm{dt}}$$ (N ΦB)
= –$$\frac{\mathrm{d}}{\mathrm{dt}}$$ (N Φm cos ωt)
= -NΦm(-sin ωt) ω
= NΦm ω sin ωt
5. When the coil is rotated through 90°, sin ωt = 1.
6. The maximum induced emf is
εm = NΦm ω
εm = NBAω since Φm = BA
∴ Value of induced emf at any instant is
ε = εm sin ωt
The alternating current is given by i = Im sin ωt
Im = maximum value of induced current.
Question 14.
Elaborate the standard construction details of AC generator.
Alternator consists of two major parts namely stator and rotor.
i) Stator:
1. The stationary part which has armature windings mounted in it, is called stator.
2. This is outer frame used for holding stator core and armature windings in proper position. It provides best ventilation.
Stator core:
It is made up of iron or steel alloy. It is hollow cylinder and is laminated to minimize eddy current.
Armature winding:
1. It is the coil, wound on slots, provided in the armature core.
2. Two types are single layer winding and double layer winding.
ii) Rotor:
1. Rotor contains magnetic field windings. Magnetic poles are magnetized by DC source.
2. Ends of field wings are connected to slip rings attached to rotor rotates.
3. Slip rings rotate along with the rotor.
4. Brushes are used which continuously slide over slip rings to maintain connection between DC source and windings.
5. Two types of rotors are
i) salient pole rotor.
ii) cylindrical pole rotor.
i) Salient pole rotator:
It has a number of projecting poles having their bases riveted to the rotor. It is used in low-speed alternators.
ii) Cylindrical pole rotor:
It consists of a smooth solid cylinder. The slots are cut on the outer surface of the cylinder along its length. It is suitable for very high-speed generator.
Question 15.
Explain the working of a single-phase AC generator with the necessary diagram.
1. The loop PQRS is stationary and perpendicular to plane, when field windings are excited, magnetic field is produced around it.
2. Let field magnet be rotated in clockwise direction. Assume initial position is horizontal, direction of magnetic field is perpendicular to plane of loop PQRS Induced emf is zero. It is represented by O in the graph.
3. When rotates to 90°, magnetic field is parallel to PQRS, an induced emf becomes maximum. Direction of induced emf is given by Flemming’s right hand rule.
The loop PQRS ad field magnet in its initial position
Variation of induced emf with respect to time angle
4. When rotates to 180°, field is perpendicular to PQRS, an induced emf is zero. It is represented by point B.
5. When rotates to 270°, field is parallel to PQRS, an induced emf is maximum but in reverse direction.
6. Current flows in SRQP. It is represented by point C.
7. On completion of 360°, field is perpendicular to PQRS, an induced emf is zero and noted by point D.
8. When field magenets complete one rotation, induced emf in PQRS finishes one cycle.
9. Frequency depends on speed of field magnet rotates.
Question 16.
How are the three different EMFs generated in a three-phase AC generator? Show the graphical representation of these three EMFs.
1. In three-phase AC generator, the armature has 6 slots cut on its inner rim. Each slot is 60° away from one another. Six armature conductors mounted in these slots.
2. The conductors 1 and 4 joined in series to form coil 1.
The conductors 3 and 6 form coil 2.
The conductors 5 and 2 form coil 3.
3. These coils are rectangular and 120° apart from one another.
4. The initial position of field magnets is horizontal and direction is perpendicular to plane of coil 1.
5. When rotates from clockwise direction alternating emf in coil 1 begins a cycle from 0.
6. The corresponding cycle for in coil 2 starts at A after field magnet rotated through 120°.
7. So phase difference between ε1 and ε2 is 120°.
8. Similarly ε3 in coil 3 would begin its cycle at B after 240° from initial position.
9. Thus emfs produced in three-phase AC generator have 120° phase difference between one another.
Construction of three – Phase AC generator
Variation of emfs ε1, ε2, and ε3 with time angle
Question 17.
Explain the construction and working of transformer.
1. The principle of transformer is the mutual induction between two coils.
2. Two coils of high mutual inductance wound over the transformer core.
3. The core is laminated and made up of silicon steel.
4. Coils are insulated but magnetically linked via transformer core.
5. The coil across which alternating voltage is applied is called primary coil P.
6. Core and coil are kept in containers filled with suitable medium for insulation and cooling.
Construction of Transformer
Working:
1. If primary coil is connected with a.c input voltage the production of alternative magnetic flux linked with primary coil is linked with secondary coil.
2. As a result of flux change emf is induced in both coils.
3. Emf induced in primary coil is
νp = εp = -Np $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$ ……………(1)
4. Emf induced in secondary coil is
εS = -NS $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$
5. Np, NS is a number of turns in primary and secondary coil.
6. If secondary circuit is open εS = νS
νS = εS = -NS $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$ ………………….(2)
7. From eqn (1) and (2)
$$\frac{v_{S}}{v_{p}}=\frac{N_{S}}{N_{P}}=K$$ ………………(3)
p is voltage transformer ratio
8. For ideal transformer,
Input power vpip = Output power vsis
(ip, is are currents in Primary and secondary coil)
$$\frac{\mathrm{V}_{\mathrm{S}}}{\mathrm{V}_{\mathrm{P}}}=\frac{\mathrm{N}_{\mathrm{S}}}{\mathrm{N}_{\mathrm{P}}}=\frac{\mathrm{i}_{\mathrm{P}}}{\mathrm{i}_{\mathrm{S}}}$$ ………..(4)
$$\frac{\mathrm{V}_{\mathrm{S}}}{\mathrm{V}_{\mathrm{P}}}=\frac{\mathrm{N}_{\mathrm{S}}}{\mathrm{N}_{\mathrm{P}}}=\frac{\mathrm{i}_{\mathrm{P}}}{\mathrm{i}_{\mathrm{S}}}=\mathrm{K}$$
Step up transformer:
(i) If Ns > Np or K > I, Vs > Vp and Is < Ip
Voltage is increased, the current is decreased.
(ii) Step down transformer
Ns < Np or K< l, Vs < Vp and Is > Ip
Voltage is decreased, the current is increased.
Effeciency of a transformer,
η = $$\frac{\text { Output power }}{\text { Input power }}$$ × 100%
Question 18.
Mention the various energy losses in a transformer.
i) Core loss or Iron loss:
1. Hysteresis loss and eddy current loss are known as core or Iron loss.
2. When transformer core is magnetized and demagnetized repeatedly by a.c voltage Hysteresis loss takes place. It is minimized by silicon steel core.
3. Alternating magnetic flux in core induce eddy current, due to flow of eddy current energy loss takes place. It is minimized by thin lamination of transformer core.
ii) Copper loss:
When current flows through core energy is dissipated due to Joule heating. It is a copper loss, it is minimized by using large-diameter wire.
iii) Flux leakage:
Flux leakage happens, when the magnetic lines of the primary coil are not completely linked with the secondary coil. Energy loss due to flux leakage is minimized by winding coil one over the other.
Question 19.
Give the advantage of AC in long-distance power transmission with an example.
1. When power is transmitting for long-distance a fraction of power is lost due to joule heating (I2R).
2. This power loss can be tackled either by reducing current i or by reducing resistance R of a transmission line.
3. At the transmitting point, voltage is increased and the current is decreased by a step-up transformer.
4. This reduced current at high voltage reaches the destination without applicable loss.
5. At the receiving point, voltage is decreased and current is increased by step down transformer.
Example 1:
2MW power is transmitted at 40Ω with 10 kV voltage
I = $$\frac{P}{V}=\frac{2 \times 10^{6}}{10 \times 10^{3}}$$ = 200 A
Power loss = I2 R = (200)2 × 40 = 1.6 × 106 W
% of Power loss = $$\frac{1.6 \times 10^{6}}{2 \times 10^{6}}$$ × 100% = 80%
Example 2:
When 2MW power at 40 Q is transmitted with 100 kV voltage.
I = $$\frac{P}{V}=\frac{2 \times 10^{3}}{100 \times 10^{3}}$$ = 20 A
Power loss = I2 R = (200)2 × 40 = 0.016 × 106 W
% of power loss = $$\frac{0.016 \times 10^{6}}{2 \times 10^{6}}$$ × 100%
= 0.8 × 100% = 80%
Question 20.
Find out the phase relationship between voltage and current in a pure inductive circuit.
1. Consider a circuit containing pure inductor L across a.c. voltage source
2. υ = Vm sin ωt
Back emf ε = – L $$\frac{d i}{d t}$$
3. By applying kirchoff’s loop rule
ν + ε = 0
Vm sin ωt = L $$\frac{d i}{d t}$$
di = $$\frac{V_{m}}{L}$$ sin ωt dt
Integrating both sides.
Phasor diagram and wave diagram for AC circuit with L.
i = $$\frac{V_{m}}{L}$$ ∫ sin ωt dt
i = $$\frac{V_{m}}{L}$$ (- cos ωt) + constant
i = $$\frac{V_{m}}{L}$$ sin (ωt – π/2) or
i = Im sin (ωt – π/2)
where, $$\frac{V_{m}}{\omega L}$$ = Im.
The peak value of a.c current lags behind voltage by π/2 in inductive circuit in phasor diagram. In wave diagram current lags voltage by 90°.
Im = $$\frac{V_{m}}{\omega L}$$
The quantity ωL is the resistance offered by inductor and called inductive reactance (χL)
χL = ωL
for ideal inductor χL = 0
Question 21.
Derive an expression for phase angle between the applied voltage and current in a series RLC circuit.
AC circuit containing R, L and C
Phasor diagram for a series RLC – circuit when VL > VC
L(VL) leads I by π/2 and voltage across C(VC) lags I by π/2
Phasor diagram is drawn with the current.
The length of these phasors are
OI = Im, OA = ImR, OB = ImχL;
OC = ImχC
Vm2 = VR2 + (VL – VC)2
Special cases:
i) If χL> χC, (χL – χC) is positive and Φ is also positive.
∴ υ = Vmsin ωt; i = Im(sin(ωt – Φ))
ii) if χL < χC, (χL – χC) is negative and Φ is negative.
∴ υ = Vmsin ωt; i = Im(sin(wt + Φ))
Question 22.
Define inductive and capacitive reactance. Give their units.
1. Resistance offered by inductor is called inductive reactance (χL). Its unit is the ohm (Ω)
χL = ωL
If f = 0. χL = 0.
2. Resistance offered by capacitor is called capacitive reactance (χC). Its unit is also ohm (Ω)
χC = $$\frac{1}{\omega C}$$
If f = 0 ;
χC = $$\frac{1}{\omega C}$$ = $$\frac{1}{2 \pi f C}=\frac{1}{0}$$ = ∞
Question 23.
Obtain an expression for the average power of AC over a cycle. Discuss its special cases.
In RLC circuit,
υ = Vm sin ωt and
i = Im sin (ωt + Φ)
where Φ is phase angle between υ and i
Instantaneous power is p = υi
= Vm Im sin ωt sin (ωt + Φ)
= Vm Im sin ωt (sin ωt cos Φ – cos ωt sin Φ)
P = Vm Im(cos Φ sin2ωt – sinωt cosωt sin Φ)
Average power over a cycle,
Pav = Vm Im cos Φ × $$\frac{1}{2}$$
= $$\frac{V_{m}}{\sqrt{2}} \frac{I_{m}}{\sqrt{2}}$$ cos Φ
∴ Pav = VRMS IRMS P cos Φ
VRMS IRMS is apparent power cos Φ is power factor Average power of AC circuit is known as the true power of circuit.
Special Cases:
i) For pure resistive circuit phase angle is zero, cos Φ = 1
∴ Pav = VRMS IRMS
ii) For pure inductive or capacitive circuit cos (± π/2) = 0
∴ Pav = 0
iii) For RLC circuit Φ = $$\tan ^{-1}\left(\frac{X_{L}-X_{L}}{R}\right)$$
∴ Pav = VRMS IRMS
iv) For RLC circuit at resonance cos Φ = 1
∴ Pav = VRMS IRMS
Question 24.
Explain the generation of LC oscillation in a circuit containing an inductor of inductance L and a capacitor of capacitance C.
LC Oscillations:
Whenever energy is given to a circuit containing a pure inductor of inductance L and a capacitor of capacitance C, the energy oscillates back and forth between the magnetic field of the inductor and the electric field of the capacitor. Thus the electrical oscillations of definite frequency are generated. These oscillations are called LC oscillations.
LC Oscillations
Generation of LC oscillations:
1. Let us assume that the capacitor is fully charged with maximum charge Qm at the initial stage. So that the energy store in the Qm capacitor is maximum and is given by UE = $$\frac{Q_{m}^{2}}{2 C}$$
2. As there is no current in the inductor, the energy stored in it is zero i.e., UE = 0. Therefore, the total energy is wholly electrical. (Figure (a)).
3. The capacitor now begins to discharge through the inductor that establishes current i in a clockwise direction. This current produces a magnetic field around the inductor and the energy stored in the inductor is given by
UE = $$\frac{\mathrm{Li}^{2}}{2}$$
As the charge in the capacitor decreases, the energy stored in it also decreases and is given by
UE = $$\frac{q^{2}}{2 C}$$ total energy is the sum of electrical and magnetic energies. (Figure (b)).
4. When the charges in the capacitor are exhausted, its energy becomes zero i.e., UE = 0.
The energy is fully transferred to the magnetic field of the inductor and its energy is maximum.
5. This maximum energy is given by UE = $$\frac{\mathrm{Li}^{2}}{2}$$ where Imis the maximum current flowing in the circuit. The total energy is wholly magnetic (Figure (c)).
6. Even though the charge in the capacitor is zero, the current will continue to flow in the same direction because the inductor will not allow it to stop immediately.
7. As a result of this, the capacitor begins to charge in the opposite direction. A part of the energy is transferred from the inductor back to the capacitor. The total energy is the sum of the electrical and magnetic energies. (Figure (d)).
8. (Fig. e) i = o the capacitor becomes fully charged in the opposite direction.
9. (Fig. f) The state of the circuit is similar to the initial state but the difference is that the capacitor is charged in opposite direction.
∴ Total energy = UE + UB.
10. As already explained, the process repeated in opposite direction (Fig. g and h). Finally alternating current flows in the circuit.
11. This process is repeated again and again to produce LC Oscillations.
Question 25.
Prove that the total energy is conserved during LC oscillations.
1. LC oscillation takes place in accordance with law of conservation of energy.
2. Total energy U = UE + UB = $$\frac{q^{2}}{2 C}+\frac{1}{2} L i^{2}$$
Case (i):
When q = Qm and i = O
Total energy is U = $$\frac{\mathrm{Q}_{\mathrm{m}}^{2}}{2 \mathrm{C}}$$ + o = $$\frac{\mathrm{Q}_{\mathrm{m}}^{2}}{2 \mathrm{C}}$$
Total energy is wholly electrical.
Case (ii):
When q = O i = Im total energy is
U = 0 + $$\frac{1}{2} L I^{2}$$ = $$\frac{1}{2} L I^{2}$$
= $$\frac{L}{2} \times\left(\frac{Q^{2} m}{L_{C}}\right)$$
since Im = Qm ω
= $$\frac{Q_{m}}{\sqrt{L M}}$$ = $$\frac{Q^{2} m}{2 C}$$
Total energy is wholly magnetic
Case (iii):
When charge = q, current = i
Question 26.
Compare the electromagnetic oscillations of the LC circuit with the mechanical oscillations of the block spring system qualitatively to find the expression for the angular frequency of LC oscillators mathematically.
i) Qualitative treatment:
• Two forms of energy involved in LC oscillations. The’ are
1. electrical energy of capacitor.
2. Magnetic energy of inductor.
• Angular frequency of oscillations of spring mass is
ω = $$\sqrt{\frac{k}{m}}$$ where k → 1/C and m → L
ω = $$\frac{1}{\sqrt{L C}}$$ …………..(1)
ii) Quantitative treatment:
Mechanical energy of spring mass system,
E = $$\frac{1}{2}$$ mv2 + $$\frac{1}{2}$$ kx2 …………(2)
m = $$\frac{d^{2} x}{d t^{2}}$$ + kx = 0 ………………..(3)
x(t) = Xm cos(ωt + Φ)
Electromagnetic energy of LC system is
$$\mathrm{U}=\frac{Q_{m}^{2} \cos ^{2} \omega \mathrm{t}}{2 \mathrm{C}}+\frac{L \omega^{2} Q_{m}^{2} \sin ^{2} \omega \mathrm{t}}{2}$$
U = $$\frac{1}{2} L i^{2}+\frac{1}{2}\left(\frac{1}{C}\right) q^{2}$$ = constant ……………(5)
Differentiate w.r.t to time
$$\frac{d U}{d t}=\frac{1}{2} \mathrm{~L}\left(2 \mathrm{i} \frac{d i}{d t}\right)+\frac{1}{2 C}\left(2 q \frac{d q}{d t}\right)=0$$
$$L \frac{d^{2} q}{d t^{2}}+\frac{1}{C} q=0$$ ………………(6)
q(t) = Qm cos(ωt + Φ) ……………..(7)
current i(t)
i(t) = – Im sin (ωt + Φ) ……………..(8)
Angular frequency of LC oscillations
$$\frac{d^{2} q}{d t^{2}}$$ = – Qm ω2 cos(ωt + Φ) ……………(9)
Substituting (7) and (9) in (6)
L[- Qm ω2 cos(ωt + ΦΠ)] + $$\frac{1}{C}$$ Qm cos(ωt + Φ) = 0
Rearranging the terms, angular frequency of LC oscillation is
ω = $$\frac{1}{\sqrt{L C}}$$
IV. Numeric Problems:
Question 1.
A square coil of side 30 cm with 500 turns is kept in a uniform magnetic field of 0.4 T. The plane of the coil is inclined at an angle of 30° to the field. Calculate the magnetic flux through the coil.
Given data:
Area A = 30 × 30 × 10-4 m2
n = 500
B = 0.4 T
Q = 90° – 30° = 60°
Φ = nBA cos Φ
=500 ×30 × 30 × 10-4 × 0.4 × cos60°
= 5 × 10 × 0.4× 9 × $$\frac{1}{2}$$
= 5 × 0.2 × 9 × 10
= 5 × 0.2 × 9 × 10-1
= 1.0 × 9 × 10-1
= 9 wb
Φ = 9 wb
Question 2.
A straight metal wire crosses a magnetic field of flux 4 mWb in a time of 0.4 s. Find the magnitude of the emf induced in the wire.
Given data:
dΦ = 4 mWb = 4 × 10-3
dt = 0.4 s
induced emf ε =?
Question 3.
The magnetic flux passing through a coil perpendicular to its plane is a function of time and is given by φB = (2t3 + 4t2 + 8t + 8) Wb. If the resistance of the coil is 5Ω, determine the induced current through the coil at a time t = 3 second.
Given data :
φB = (2t3 + 4t2 + 8t + 8) Wb
R = 5Ω
t = 3second
induced current i = ?
i = $$\frac{\varepsilon}{\mathrm{R}}$$
ε = $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$
ε = $$\frac{d}{d t}\left(2 t^{3}+4 t^{2}+8 t+8\right)$$
ε = 6t2 + 8t + 8
at t = 3second,
ε = 6 × 32 × 8 × 3 + 8
ε = 54 + 24 + 8
ε = 86 V
∴ i = $$\frac{\varepsilon}{R}=\frac{86}{5}$$ = 17.2 A
i = 17.2 A
Question 4.
A closely wound circular coil of radius 0.02 m is placed perpendicular to the magnetic field. When the magnetic field is changed from 8000 T to 2000 T in 6s, an emf of 44 V is induced in it. Calculate, the number of turns in the coil
Given data:
Q = 90° – 90° = 0°
B1 = 8000 T, B2 = 2000 T
dt = 68
ε = 68V
n = ?
ε = nA cos θ $$\frac{\mathrm{dB}}{\mathrm{dt}}$$
Question 5.
A rectangular coil of area 6 cm2 having 3500 turns is kept in a uniform magnetic field of 0.4 T. Initially, the plane of the coil is perpendicular to the field and is then rotated through an angle of 180°. If the resistance of the coil of 35Ω, find the amount of charge following through the coil.
Given data:
Area A = 6 × 10-4 m2
n = 3500
B = 0.4 T
Q1 = 180°
R = 35Ω
amount of charge Q =?
Q = di.dt = $$\frac{\varepsilon}{R}$$ dt
Question 6.
An induced current of 2.5 mA flows through a single conductor of resistance 1ooΩ. Find out the rate at which the magnetic flux is cut by the conductor.
Given data:
Induced current i = 2.5 mA
i= 2.5 × 10-3 A
Resistance R = 100Ω
ε = $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$
where ε = iR
∴ $$\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$ = iR = 2.5 × 10-3 × 100
= 250 × 10-3
= 250 mWbs-1
Question 7.
A fan of metal blades of length 0.4 m rotates normal to a magnetic field of 4 × 10-3 T. If the induced emf between the center and edge of the blade is 0.02 V, determine the rate of rotation of the blade.
Given data:
length = 0.4 m
B = 4 × 10-3 T
ε = 0.02 V
The rate of rotation υ =?
Question 8.
A bicycle wheel with metal spokes of 1m long rotates in Earth’s magnetic field. The plane of the wheel is perpendicular to the horizontal component of Earth’s field of 4 × 10-5 T. If the emf induced across the spokes is 31.4 mV, calculate the rate of revolution of the wheel.
length l = 1 m
B = 4 × 10-5 T
E = 3.14 mV
= 3.14 × 10-3 V
The rate of revolution = ?
Question 9.
Determine the self-inductance of 4000 turn air-core solenoid of length 2 m and diameter 0.04 m.
Given data:
n = 4000
l = 2m
diameter d = 0.04 m
Self inductance L =?
L = µnA2 l
L = 4π × 10-7 × 4000 × π × 0.02 × 0.02 × 2
L = 16π × 10-1 × π × 8 × 10-4
L = 1262 × 10-5
L = 12.62 × 10-3 H
L = 12.62 mH
Question 10.
A coil of 200 turns carries a current of 4 A. If the magnetic flux through the coil is 6 x 10-5 Wb, find the magnetic energy stored in the medium surrounding the coil.
Given data:
N = 200 turns
i = 4A
ΦB = 6 × 10-5 Wb
Magnetic energy UB =?
Question 11.
A 50 cm long solenoid has 400 turns per cm. The diameter of the solenoid is 0.04m. Find the, magnetic flux linked turn when it carries a current of IA.
Given data:
Length of the solenoid, l = 50 cm = 50 × 10-2 m
No. of turns / cm = 400
For 50 cm, No. of turns N = 400 × 50 = 20,000
Diameter of the solenoid = 0.04 m
∴ Radius of the solenoid = 0.02 m
Current passing through the solenoid = 1 A
Area of the solenoid = πr²
= 3.14 × 0.02 × 0.02 m2
Formula :-
Magnetic flux, φ = µ0 n2 AIl
Question 12.
A coil of 200 turns carries a current of 0.4 A. If the magnetic flux of 4m Wb is linked with each turn of the coil, find the inductance of the coil.
Given data:
N = 200
i = 0.4 A
ΦB = 4 mWb = 4 × 10-3 Wb
L = ?
Bi = Li
Question 13.
Two air core solenoids have the same length of 80 cm and same cross-sectional area 5 cm2. Find the mutual inductance between them if the number of turns in the first coil is 1200 turns and that in the second coil is 400 turns.
Given data:
l = 80cm
A = 52 = 5 × 10-4 m2
= 80 × 10-2 m
turns = 1500 turns
n1 = $$\frac{1200}{0.8}$$;
n1 = 1500turns
n2 = $$\frac{400}{0.8}$$;
n2 = 500 turns
M = ?
Mn = μ0 n1 n2 A2 l
M = 4π × 10-7 × 1500 × 1500 × 5 × 10-4 × 0.8
M = 4π × 10-7 × 75 × 5 × 0.8
M = 3768 × 10-7 H
= 0.3768 × 10-3 H
M = 0.38 mH
Question 14.
A long solenoid having 400 turns per cm carries a current 2A. A 100 turn coil of cross-sectional area 4cm2 is placed co-axially inside the solenoid so that the coil is in the field produced by the solenoid. Find the emf induced in the coil if the current through the solenoid reverses its direction in 0.04 sec.
Given data:
n1 = 400 turns/cm
∴ n1 = 4000 turns/m
n2 = 100
A2 = 4 × 10-4 m2
i = 1 m
di = 2-(-2) = 48
dt = 0.04 sec
Question 15.
A 200 turn circular coil of radius 2 cm is placed co-axially within a long solenoid of 3 cm radius. If the turn density of the solenoid is 90 turns per cm, then calculate mutual inductance of the coil and solenoid.
Given data:
r2 = 2 cm, n2 = 200
r1 = 3 cm
n1 = 90 turns/cm
n1 = 9000 turns/m
M =?
M = μ0 n1 n2 A2 l
M = 4π × 10-7 × 280 × 9000 × π × 4 × 10-4 × 1
M = 4π × 10-6 × 18 × 4π
M = 2.839.56 × 10-6 H
M = 2.839 × 10-3 H
M = 2.84 mH
Question 16.
The solenoids S1 and S2 are wound on an iron-core of relative permeability 900. The area of their cross-section and their lengths are the same and are 4 cm2 and 0.04 m respectively. If the number of turns in S1 is 200 and that in S2 is 800, calculate the mutual inductance between the solenoids. If the current in solenoid 1 is increased from 2A to 8A in 0.04 second. Calculate the induced emf in solenoid 2.
Given data:
μr = 900
A2 = 4 × 10-4 m2
l = 0.04 m
n1 = 5000
n2 = 20,000
i1 =2A to i2 = 8A
dt = 0.048
M = ?
ε2 = ?
Solution:
M = μ0 μr n1 n2 A2 l
M = 4π × 10-7 × 900 × 5000 × 20000 × 4 × 10-4 ×0.04
M = 4π × 90 × 4 × 4 × 10-4
M = 18086 × 10-4H
M = 1.81 H
induced emf,
ε2 = M $$\frac{d i}{d t}$$
ε2 = 1.81 $$\left(\frac{(8-2)}{0.04}\right)$$
ε2 = $$\frac{1.81 \times 6}{4 \times 10^{-2}}$$
= 2.715 × 10-2 V
ε2 = 271.5 V
Question 17.
A step-down transformer connected to main supply of 220 V is used to operate 11V, 88W lamp. Calculate
(i) Transformation ratio and
(ii) current in the primary
Given data:
Vp = 220 V, Ps = 88 W, Vs = 11 V
Question 18.
A 200 V/ 120 V step down transformer of 90% efficiency is connected to an induction stove of resistance 40Ω. Find the current drawn by the primary of the transformer.
Given data:
Vp = 200 V
Vs = 120 V
η = 90% = $$\frac{90}{100}$$ = 0.9
η = $$\frac{\text { output power }}{\text { input power }}$$
Question 19.
The 300 turn primary of a transformer has resistance 0.82 Q and the resistance of its secondary of 1200 turns is 6.2 Q. Find the voltage across the primary if the power output from the secondary at 1600 V is 32 kW. Calculate the power losses in both coils when the transformer efficiency is 80%.
Given data:
NP = 300
NS = 1200
VS = 1600 V,
RP = 0.82Ω
RS = 6.2Ω
PS = 32 KW
iv) Power loss in primary = (IP)2 × RP
= 100 × 1000 × 0.82
= 0.82 × 1000
= 8.2000 = 8.2 KW
v) Power loss in secondary = (IS)2 × RS
= 20 × 20 × 6.2
= 400 × 6.2
= 2480W = 2.48 KW
Question 20.
Calculate the instantaneous value at 60°, average value and RMS value of an alternating current whose peak vaue is 20A.
Given data:
Im = 20 A Q = 60°
i) Instantaneous value of current
i = Im sin θ
= 20 × Sin 60°
= 20 × $$\frac{\sqrt{3}}{2}$$
= 10 × √3
= 10 × 1.732
i = 17.32 A
ii) Average value
Iav = 0.637 Im
Iav = 0.637 × 20
= 6.37 × 2
Iav = 12.74 A
iii) Irms = 0.707 Im
= 0.707 × 20
= 7.07 × 2
Irms = 14.14 A
V. Conceptual Questions:
Question 1.
A Graph between the magnitude of the magnetic flux linked with a closed-loop and time is given in the figure. Arrange the regions of the graph in ascending order of the magnitude of induced emf in the loop.
induced emf ε = $$\frac{-\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}}$$
i) Φ – t graph at ab is a straight line
∴ $$\frac{\mathrm{d} \phi}{\mathrm{dt}}$$ = constant
∴ ε = -ve and constant
ii) Φ – t graph at bc, Φ is constant
∴ ε = 0
iii) Φ – t graph at cd is a straight line
$$\frac{\mathrm{d} \phi}{\mathrm{dt}}$$ = constant
∴ ε = true and constant
iv) ab, bc, cd are ascending order of the magnitude of induced emf.
Question 2.
Using Lenz’s law, predict the direction of induced current in conducting rings 1 and 2 when the current in the wire is steadily decreasing.
Current in the wire is steadily decreasing, so the induced current in rings 1 and 2 will flow in such a way that it oppose the decrease of current
• ring 1 is clockwise
• ring 2 is anti-clockwise
Question 3.
A flexible metallic loop abcd in the shape of a square is kept in a magnetic field with its plane perpendicular to the field. The magnetic field is directed into the paper normally. Find the direction of the induced current when the square loop is crushed into an irregular shape as shown in the figure.
1. If a wire of irregular shape turns into a square loop then its area increases, so that the magnetic flux linked also increases
2. The induced current is in an anticlockwise direction, along abcd.
Question 4.
Predict the polarity of the capacitor in a closed circular loop when two bar magnets are moved as shown in the figure.
1. From Lenz’s law, the direction of the induced current is in a clockwise sense.
2. This implies that plate A of the capacitor is at a higher potential than plate B
3. B will be a negative plate while A will be a positive plate.
Question 5.
In a series LC circuit, the voltages across L and C are 180° out of phase. Is it correct? Explain.
1. Yes, it is correct
2. In a pure inductor, the voltage leads the current by a phase angle of 90°.
3. In a perfect capacitor, the voltage lags behind the current by a phase angle of 90°
4. So, In series LC circuit, the voltage across L and C are 180° out of phase.
Question 6.
When does the power factor of a series RLC circuit become maximum?
1. Cos Φ is the power factor
2. An RLC series circuit, the phase difference between voltage and current is zero when the circuit is at resonance.
3. The power factor becomes maximum.
Part II:
### Physics Guide Electromagnetic Induction and Alternating Current Additional Questions and Answers
Question 1.
A coil of area of cross section 0.5 m2 with 10 turns is in a plane which is perpendicular to an uniform magnetic field of 0.2 Wb/m2. The flux through the coil is –
(a) 100 Wb
(b) 10 Wb
(c) 1 Wb
(d) zero
(c) 1 Wb
Hint:
Φ = NBA cos θ
= 10 x 0.2 x 0.5 x cos 0° = 1 Wb
Question 2.
What happens to the current in a coil while alternating a magnet inside it?
a) Increase
b) decreases
c) Remains constant
d) Reverse
a) Increases
Question 3.
A wire of length 1 m moves with a speed of 10 ms-1 perpendiculars to a magnetic field. If the emf induced in the wire is 1 V, the magnitude of the field is-
(a) 0.01 T
(b) 0.1 T
(c) 0.2 T
(d) 0.02 T
(b) 0.1 T
Hint:
ε = Blv
⇒ B = $$\frac { ε }{ lv }$$ = $$\frac { 1 }{ 1 × 10 }$$ = 0.02 T
Question 4.
If a conductor 0.2m long moves with a velocity of 0.3m/s in the magnetic field of 5T. Calculate the emf induced
a) 0.3 V
b) 0.03 V
c) 30 V
d) 3 V
a) 0.3 V
Solution:
emf = Blv = 5 × 0.2 × 0.3 = 0.3 V
Question 5.
A coil of cross-sectional area 400 cm2 having 30 turns is making 1800 rev/min in a magnetic field of IT. The peak value of the induced emf is-
(a) 113 V
(b) 226 V
(c) 339 V
(d) 452 V
(b) 226 V
Hint:
εm = NBA ω = 30 x 1 x 400 x 10-4 x 30 x 2π = 226 V
Question 6.
An electric generator consists of a 10-turn square wire loop of a side 50cm. The loop is turned so as to produce 50 Hz A.C. How strong must the magnetic field be for the peak output voltage to be 300V?
a) 2.4 T
b) 0.417 T
c) 2.62 T
d) 0.382 T
d) 0.382 T
Question 7.
By accelerating magnet inside the coil current in it _____
a) increases
b) decreases
c) remains constant
d) reverse
a) increases
Question 8.
When a direct current ‘i’ is passed through an inductance L, the energy stored is-
(a) Zero
(b) Li
(c) $$\frac { 1 }{ 2 }$$ Li2
(d) $$\frac {{ L }^{ 2 }}{2i}$$
(c) $$\frac { 1 }{ 2 }$$ Li2
Question 9.
In an LCR series circuit, the voltage across each component L, C, and R is 50V. The voltage across the LC combination will be
a) 50 V
b) 0 V
c) 50 V
d) 100 V
b) 0 V
Solution:
V = (ωL – $$\frac{1}{\omega_{C}}$$) Irms
= ωLIrms – $$\frac{1}{\omega_{C}}$$Irms
= 50 – 50 = 0
Question 10.
In AC circuit with inductance and capacitance are joined in series, current is found to be maximum when the value of inductance is 0.5 H and capacitance is 8 μF. The angular frequency of applied alternating voltage will be______
a) 400 Hz
b) 5000 Hz
c) 2 × 105 Hz
d) 300 Hz
d) 500 Hz
Solution:
ω = $$\frac{1}{\sqrt{\mathrm{LC}}}$$ = $$1 / \sqrt{0.5 \times 8 \times 10^{-6}}$$
ω = 500 Hz
Question 11.
AC supply gives 30V which passes through a 10ω resistance. The power dissipated in it is _______.
a) 90√2 W
b) 90 W
c) 45√2 W
d) 45 W
b) 90 W
Solution:
P = $$\frac{V^{2}}{R}=\frac{30^{2}}{10}$$ = 90 W
Question 12.
In an a.c circuit, an alternating voltage e = 200√2 sin 100t volts is connected to a capacitor of capacity 1 μF. The RMS value of the current in the circuit is
a) 10 mA
b) 100 mA
C) 200 mA
d) 20 mA
d) 20 mA
Solution:
Question 13.
In an a.c circuit, the emf and current at any instant are e = E0 sin ωt, i = I0 sin ωt – Ø Average power is given by
a) E0 I0/2
b) (E0 I0/2) sin Φ
c) (E0 I0/2) cos Φ
d) E0 I0
c) (E0 I0/2) cos Φ
Solution:
Question 14.
AC power is transmitted from a powerhouse at a high voltage as-
(a) the rate of transmission is faster at high voltages
(b) it is more economical due to less power loss
(c) power cannot be transmitted at low voltages
(d) a precaution against theft of transmission lines
(b) it is more economical due to less power loss
Question 15.
What is the maximum value of inductance L for which the current is maximum in LCR circuit with C = 10 μF and ω = 1000 s-1
a) 1 mH
b) cannot calculate
c) 10 mH
d) 100 mH
d) 100 mH
Solution:
Question 16.
The primary winding of a transformer has 500 turns, whereas its secondary has 5000 turns. Primary is connected to a.c supply of 20V, 50Hz. The secondary will have an output of
a) 2 V, 5 Hz
b) 200 V, 500 Hz
c) 2 V, 50 Hz
d) 200 V, 50 Hz
d) 200 V, 50 Hz
Solution:
Question 17.
A step-up transformer operates on a 230 V line and supplied a load of 2A. The ratio of primary and secondary wings is 1:25. The current in primary is_______
a) 25 A
b) 50 A
c) 15 A
d) 12.5 A
b) 50 A
Solution:
Question 18.
If N is the number of turns in a coil, the value of self-inductance varies as-
(a) N°
(b) N
(c) N2
(d) N-2
(c) N2
Hint:
According to self-inductance of the long solenoid
L = $$\frac{\mu_{0} \mathrm{N}^{2} \mathrm{A}}{l}$$
⇒ L ∝ N2
Question 19.
A solenoid has n turns. Its coefficient of an inductance L varies with n as
a) L α n
b) L α n2
c) L α n-1
d) L α n
b) L α n2
Question 20.
What is the coefficient of mutual inductance when magnetic flux charges by 2 × 10-2 Wb, one’s change in current is 0.01 A
a) 2 H
b) 3 H
c) 1/2 H
d) zero
a) 2 H
Solution:
E = M$$\frac{d I}{d t}$$ = 2 × 10-2 × 0.01 = 2
Question 21.
The phase difference between VL and VC in series RLC circuit.
a) 2π
b) $$\frac{\pi}{2}$$
c) $$\frac{2 \pi}{3}$$
d) π
d) π
Question 22.
The quantity that remains unchanged in a transformer is-
(a) voltage
(b) current
(c) frequency
(d) none of these
(c) frequency
Question 23.
The direction of the induced current is such that opposes the very cause that has produced it. This is the law of
a) Lenz
c) Kirchoff
d) Fleming
a) Lenz
Question 24.
Which one is correct?
a) Self-inductance is directly proportional to the current flowing through the coil
b) Self-inductance is directly proportional to the length
c) Self-inductance is directly proportional to its area of cross-section
d) Self-inductance is inversely proportional to the area of cross-section
c) Self-inductance is directly proportional to its area of cross-section
Question 25.
A coil has a self-inductance of 0.04 H. The energy required to establish a steady-state current of 5 A in it is-
(a) 0.5 J
(b) 1.0 J
(c) 0.8 J
(d) 0.2 J
(a) 0.5 J
Question 26.
A 50 mH coil carries a current of 4 amp. The energy stored in joule is_______
a) 0.4 J
b) 4.0 J
c) 0.8 J
d) 0.04 J
a) 0.4J
Solution:
U = $$\frac{1}{2}$$ Li2
= $$\frac{1}{2}$$ × 50 × 10-3 × 45 × (4)2
Question 27.
If the angular speed of rotation of an armature of AC generator is doubled, the induced emf will be _______
a) same
b) doubled
c) halved
b) doubled
Question 28.
Faraday’s law of electromagnetic induction is related to _______
a) law of conservation of charge
b) law of conservation of energy
c) Third law of Newton
d) law of conservation of angular momentum
b) law of conservation of energy
Question 29.
The rms value of an alternating current, which when passed through a resistor produces heat three times of that produced by a direct current of 2 A in the same resistor, is-
(a) 6 A
(b) 3 A
(c) 2 A
(d) 2√3 A
(d) 2√3 A
Hint:
$${ I }_{ rms }^{ 2 }$$R = 3(22R) (or) Irms = 2√3 A
Question 30.
RMS voltage and frequency of V = 230 sin (314t) A.C. source.
a) 162.6 V, 50 Hz
b) 230 V, 50 Hz
c) 230 V, 60 Hz
d) 162.6 V, 25 Hz
d) 162.6 V, 25 Hz
Question 31.
From the reactance and frequency graph value of the inductance of given above the inductor is
a) 3.18 × 10-14 H
b) 1/100π H
c) 50π H
d) 6.37 × 10-3 H
d) 6.37 × 10-3 H
Question 32.
The impedance of a circuit consists of 3 Ω resistance and 4 Ω resistance. The power factor of the circuit is
(a) 0.4
(b) 0.6
(c) 0.8
(d) 1.0
(b) 0.6
Hint:
tan Φ = $$\frac { 4 }{ 3 }$$.
Power factor = cos Φ = $$\frac { 3 }{ 5 }$$ = 0.06
Question 33.
Calculate the Q factor of RLC circuit if L = 80 µH, C = 2000 pF and R = 50Ω
a) 40
b) 400
c) 4
d) 0.4
c) 4
Solution:
Q = $$\frac{1}{R} \sqrt{\frac{L}{C}}=\frac{1}{50} \sqrt{\frac{80 \times 10^{-6}}{200 \times 10}}$$ = 4
Question 34.
The average power dissipation in pure (Ideal) inductor is
a) $$\frac{1}{2}$$ Li2
b) 2 Li2
c) $$\frac{1}{2}$$ Li2
d) zero
d) zero
Question 35.
In an AC circuit, Resonance is obtained when
a) Z = R
b) Z = ωL – (1/ωC)
c) L=R
d) None
a) Z = R
Question 36.
Change of current of 1 As-1 causes emf of 1V to be equal to______
a) 1 H
b) 1 V m-1
c) 1 Am
d) 1 J
d) 1 H
Question 37.
In an AC circuit with voltage V and current I, the power dissipated is-
(a) VI
(b) $$\frac { 1 }{ 2 }$$ VI
(c) $$\frac { 1 }{ √2 }$$ VI
(d) depends on the phase difference between I and V
(d) depends on the phase difference between I and V
Question 38.
A metallic ring is attached to wall of room. When north pole of magnet is brought near ring induced current in ring is
a) zero
b) clockwise
c) anticlockwise
d) infinite
c) anticlockwise
Question 39.
In a series LCR circuit, R = 10 Ω and the impedance Z = 20 Ω. Then the phase difference between the current and the voltage is-
(a) 60°
(b) 30°
(c) 45°
(d) 90°
(c) 60°
Hint:
cos Φ = $$\frac { R }{ Z }$$ = $$\frac { 10 }{ 20 }$$ = $$\frac { 1 }{ 2 }$$
⇒ Φ = 60°
Question 40.
Which of the following effects does an alternating current shows
a) Chemical effect
b) magnetic effect
c) heating effect
d) all the above
c) heating effect
Question 41.
In the AC circuit the rms value of Irms is related to peak current I0 by the reaction
a) Irms = 1 I0
b) Irms = (1/√2) I0
c) Irms = 2 I0
d) Irms = π2I0
b) Irms = (1/√2) I0
Question 42.
Match the following:
i. Faraday’s law a. conservation of energy ii. Lenz law b. Electromagnetic Induction iii. Transformer c. Right-hand rule iv. Induced current d. ε = – dΦ / dt
a) i – a ii – d iii – c iv – b
b) i – c ii – a iii – d iv – b
c) i – d ii – a iii – b iv – c
d) i – a ii – d iii – b iv – c
c) i-d, ii-a, iii-b, iv-c
Question 43.
The wattless circuit is obtained when the phase difference between virtual voltage and virtual current is
a) 90°
b) 45°
c) 80°
d) 60°
a) 90°
II. Choose the Wrong statement:
Question 1.
Tick out the wrong statement:
a) An emf can be induced between the ends of a straight conductor by moving it through a uniform magnetic field.
b) The self-induced emf produced by changing the current in a coil always tends to decrease the current.
c) inserting an iron core in a coil increases the coefficient of self-inductance.
d) According to Lenz’s law direction of induced current oppose the flux change that causes it.
Ans :
b) The self-induced emf produced by changing the current in a coil always tends to decrease the current.
Question 2.
Tick out the wrong statement:
a) Inductor is used to store energy in a magnetic field when electric current flows through it,
b) According to Faraday’s law, an emf is induced in a conductor when magnetic flux passing through it.
c) Self-inductance is that current passing through coil changes with time emf is induced in neighbouring coil
d) The work done is stored as magnetic potential energy in the inductor.
c) Self-inductance is that current passing through coil changes with time emf is induced in neighbouring coil.
Question 3.
Tick out the wrong one:
a) Changing the magnetic field B
b) Changing area A
c) Changing the orientation of the coil Q
d) Changing mass of the rod
d) Changing mass of the rod
III. Assertion and Reason:
Question 1.
Assertion (A):
Emf is produced due to the movement of the rod it is often called motional emf.
Reason (R):
When the rod moves, free electrons in it also moves with the same velocity in B
a) A and R are correct R is the correct explanation of A
b) A and R arc correct But R is not the correct explanation of A
c) A is correct R is wrong
d) A is the wrong R is correct
a) A and R are correct R is the correct explanation of A.
Question 2.
Assertion (A):
Three-phase generator produce higher output than single-phase generator
Reason (R):
Three-phase transmission is very cheaper.
a) A and R are correct R is the correct explanation of A
b) A and R are correct But R is not the correct explanation of A
c) A is correct R is wrong
d) A is the wrong R is correct
b) A and R are correct But R is not the correct explanation of A
Question 3.
Assertion (A):
Alternating voltage and current in inductor system is V = Vm sin ωt and i = 1m sin(ωt – π/ 2)
Reason (R):
Current lags behind applied voltage by π/2
a) A and R are correct R is the correct explanation of A
b) A and R are correct But R is not the correct explanation of A
c) A is correct R is wrong
d) A is the wrong R is correct
a) A and R are correct R is the correct explanation of A
IV. Fill in the blanks:
Question 1.
Condition for Resonance in RLC circuit is ______.
χL = χC
Question 2.
Efficiency range of transformer is _______.
96 – 99%
Question 3.
Energy stored in capacitor is ________.
VE = ((Qm)2 / 2C)
Question 4.
Energy stored in the inductor is ______.
VB = Li2/2
Question 5.
The magnification of vo1tae at series resonance is _______
Q factor
V. Two Mark Questions:
Question 1.
Define magnetic flux
The magnetic flux through an area A in a magnetic field is defined as the number of magnetic field lines passing through that area.
ΦB = ∫A $$\overrightarrow{\mathrm{B}}$$ . d $$\overrightarrow{\mathrm{A}}$$ = BA cos θ
Question 2.
Write down the drawbacks of Eddy currents.
When eddy currents flow in the conductor, a large amount of energy is dissipated in the form of heat. The energy loss due to the flow of eddy current is inevitable but it can be reduced to a greater extent with suitable measures. The design of the transformer core and electric motor armature is crucial in order to minimise the eddy current loss.
To reduce these losses, the core of the transformer is made up of thin laminas insulated from one another while for electric motor the winding is made up of a group of wires insulated from one another. The insulation used does not allow huge eddy currents to flow and hence losses are minimized.
Question 3.
What is an inductor?
Inductor is a device used to store energy in a magnetic field when an electric current flows through it.
Question 4.
Define self-inductance in terms of flux.
Self-inductance of a coil is defined as the flux linkage of the coil when 1 A current flows through it
L = $$\frac{\mathrm{N} \Phi_{\mathrm{B}}}{i}$$
Question 5.
Define mutual inductance in terms of emf and current.
Mutual inductance M21 is also defined as the opposing emf induced in coil 2 when the rate of change of current through coil 1 is 1 As-1.
M12 = $$\frac{-\varepsilon_{1}}{d i_{2} / d t}$$
Question 6.
Define Mutual Inductance.
Mutual inductance is defined as opposing emf induced in the coil when the rating of change of current through another coil.
Question 7.
Define the unit of mutual Inductance.
The mutual inductance between two coils is said to be one henry if a current of 1A in coil 1 produces unit flux linkage in coil 2.
Question 8.
Define the principle of Transformer.
The principle of transformer is mutual induction between two coils. When electric current passing through a coil changes with time, an emf is induced in neighboring coil.
Question 9.
Define the Efficiency of Transformer.
The ratio of output power to input power is the efficiency of the transformer.
η = $$\frac{\text { Output power }}{\text { Input power }}$$
The efficiency range is 96 – 99%
Question 10.
What is alternating voltage?
An alternating voltage is a voltage which changes polarity at regular interval of time and the direction of the resulting alternating current also changes accordingly.
Question 11.
What is sinusoidal alternating voltage?
If the waveform of the alternating voltage is a sine wave, then it is known as sinusoidal alternating voltage.
ν = Vm sin ωt.
Question 12.
Using Lenz’s law, predict the direction of induced current in conducting rings 1 and 2 when the current in the wire is
(i) 1f the current is steadily decreasing inducing current will be in a direction so as to oppose the decreasing magnetic flux according to Lenz law. Hence the direction of current is anticlockwise in the ring I and clockwise ring 2.
(ii) If the current is steadily increasing the induced current with the flow in such a way the direction of current is clockwise in ring 1 and anticlockwise in ring 2.
Question 13.
What is meant by sinusoidal alternating voltage?
If the waveform of the alternating voltage is a sine wave, then it is known as sinusoidal alternating voltage, which is given by the relation.
υ = Vm sin ωt
Question 14.
Define power in AC circuit
Power of a circuit is defined as the rate of consumption of electric energy in that circuit.
Question 15.
What is ‘wattfull current’?
1. Current component IRMS cos φ is the active component.
2. Power consumed by current is VRMS IRMS cos φ. It is known as a wattful current.
Question 16.
Write any three definitions of power factor.
1. Power factor = cos φ is = cosine of angle of lead or lag
2. Power factor = $$\frac{R}{Z}=\frac{\text { Resistance }}{\text { Impedance }}$$
3. Power factor = $$\frac{\mathrm{VI} \cos \phi}{\mathrm{VI}}=\frac{\text { True power }}{\text { Apparent power }}$$
VI. Three Mark Questions:
Question 1.
List out the importance of Electromagnetic induction.
1. The application of electromagnetic induction is almost everywhere in the present life.
2. Right from home appliances to huge factories, satellite communication all need electricity for their operation.
3. So electric generators and transformers function on electromagnetic induction.
4. Sophisticated human life would not be possible without the discovery of electromagnetic induction.
Question 2.
What are the drawbacks of eddy current?
1. When an eddy current flows in the conductor, a large amount of heat is dissipated in the form of heat.
2. This can be reduced by suitable measures.
3. The design of the transformer core and the electric motor armature is crucial in order to minimize the eddy current loss.
4. To reduce this loss, the core of the transformer is made up of laminas insulated from one another.
5. Electric motor winding is made up of a group of wires insulated from one another. The insulation does not allow eddy current to flow.
Question 3.
Write down the applications of the series RLC resonant circuit.
RLC circuits have many applications like filter circuits, oscillators, and voltage multipliers, etc. An important use of series RLC resonant circuits is in the tuning circuits of radio and TV systems. The signals from many broadcasting stations at different frequencies are available in the air. To receive the signal of a particular station, tuning is done.
Question 4.
What are the advantages of the three-phase generator?
1. Three-phase produce a high power output
2. For the same capacity, the three-phase alternator is small in size.
3. It is cheaper. A relatively thinner wire is sufficient for transmission.
Question 5.
What is the mean or Average value of AC? Derive it.
The average value of alternating current is defined as the average of all values of current over the positive or negative half cycle.
$$I_{a v}=\frac{\text { Area of }+\text { +ve half cycle }(\text { or }) \text { -ve half cycle }}{\text { Base length of half cycle }}$$
Let i be mid ordinate of that strip
Area of elementary strip = i dθ
Area of +ve half – cycle = $$\int_{0}^{\pi}$$ idθ = $$\int_{0}^{\pi}$$ Im sinθ dθ
= Im $$[-\cos \theta]_{\mathrm{o}}^{\pi}$$ = Im [cos π – cosθ]
= 2 Im
Average value of AC, Iav = $$\frac{2 \mathrm{I}_{\mathrm{m}}}{\pi}$$
Iav = 0.637 Im
For negative half -cycle Iav = – 0.637 Im
Question 6.
How to draw phasor diagram in AC
1. Length of line segment equals to peak value
2. Vm or Im of alternating voltage.
3. Its angular velocity ω is equal to the angular frequency.
4. The projection of phasor on any vertical axis gives the instantaneous value of alternating voltage or current.
5. The angle between phasor and axis of reference indicates the phase of alternating voltage.
Question 7.
Write three examples of power factor.
1. Power factor = cos 0° = 1 for pure resistive circuit because the phase angle p between voltage and current is zero.
2. Power factor = cos(±π/2) = 0 for the purely inductive or capacitive circuit, because the phase angle between voltage and current is ±π/2
3. Power factor lies between 0 and I for a circuit having R, L, and C in varying proportions.
Question 8.
Write the analogies between electrical and mechanical quantities.
VII. Five Mark Questions:
Question 1.
Explain the first illustration of Lenz’s law and find the direction of induced current.
1. Consider a uniform magnetic field, Field lines represented by (x).
2. A rectangular metallic frame ABCD is placed in this magnetic field its plane ⊥ to B.
3. If arm AB slides to our right side, a number of lines passing through ABCD increases and current is induced.
4. By lenz’s law induced current oppose this flux increases and try to reduce it by producing another magnetic field pointing outwards.
5. Magnetic loops of the induced field are represented by circle. Direction of the induced current is found to be anti-clockwise by using right-hand thumb rule.
6. The leftward motion of arm AB decreases magnetic flux. The induced current, this time produce a magnetic field inward direction. Flux decrease is opposed by flow of induced current, it flows in a clockwise direction.
Question 2.
Explain the simple demonstration of the production of eddy current.
1. Consider a pendulum oscillate between poles of a powerful electromagnet.
2. the First electromagnet is switched off, the pendulum is slightly displaced and released. It begins to oscillate and execute a large number of oscillations before stopping Friction is only damping force.
3. When the electromagnet is switched on, the disc of the pendulum is made to oscillate, eddy currents are produced in it which oppose the oscillation.
4. A heavy damping force of eddy currents will bring the pendulum to rest within few oscillations.
5. If some slots are cut in the disc, eddy currents are reduced. The pendulum now execute several oscillations before coming to rest.
6. This clearly demonstrates the production of eddy current in the disc of the pendulum.
Question 3.
Derive the equation for mutual inductance between two long co-axial solenoids.
Consider two long co-axial solenoids of the same length l, A1 and A2 be an area of two solenoids n1 and n2 be the turn density i1 is current in solenoid 1.
B1 = µ0 n1 i1
Magnetic flux Φ21 = $$\int_{A_{2}} \overrightarrow{B_{1}} \cdot \overrightarrow{d A}$$ = B1 A2(0 = 0°)
Flux with total turns N2
N2 φ21 = (n2 l) (µ0 n1 i1) A
N2 φ21 = (µ0 n1 n2 A2 l) i1 ………………(1)
N2 φ21 = M21 i1
Comparing (1) and (2)
M21 = µ0 n1 n2 A2 l …………………..(3)
Magnetic field produced by solenoid 2 when carrying current i2
B2 = µ0 n2 i2
Φ12 = ∫(A2) B2 → . dA → = B2 A2
= (µ0 n1 n2) A2
N1 Φ12 = (n1 l) (µ0 n2 i2) A2
N1 Φ12 = (µ0 n1 n2 A2 l) i2
N1 Φ12 = M12 i2
M12 i2 = (µ0 n1 n2 A2 l) i2
M12 = µ0 n1 n2 A2 l ………………..(4)
Mutual inductance between two long co axial solenoid
M = µ0 n1 n2 A2 l ……………(5)
For relative permittivity µ
M = µn1 n2 A2 l
M = µ0 µr n1 n2 A2 l.
Question 4.
Find the instantaneous current when the AC circuit containing only the capacitor and find the capacitive reactance.
1. Consider a circuit containing a capacitor.
2. The alternating voltage is ν = Vm sin ωt ……………(1)
emf a cross capacitor is q/c
According to Kirchoff’s loop rule
ν = q/C = 0
q = C Vm sin ωt
i = $$\frac{\mathrm{dq}}{\mathrm{dt}}$$ = $$\frac{d}{d t}$$ (C Vm sin ωt)
= C Vm $$\frac{d}{d t}$$ sin ωt
= C Vm ω cos ωt
i = $$\frac{\mathrm{V}_{\mathrm{m}}}{\mathrm{I} / \mathrm{C} \omega}$$ sin (ωt + π/2)
Instantaneous Current i = I m sin (ωt + π/2)
I m = $$\frac{\mathrm{V}_{\mathrm{m}}}{1 / \omega \mathrm{c}}$$ peak value of alternating current.
Current leads the voltage by π/2 (or) 90°
Capacitive Reactance χC:
Resistance offered by capacitor is known as capacitive reactance χC = $$\frac{1}{\mathrm{C} \omega}$$
It varies inversely with frequency,
χC = $$\frac{1}{\mathrm{C} = [latex]\frac{1}{2 \pi \mathrm{fC}}=\frac{1}{0}$$ = ∞
Capacitive circuit offers infinite resistance to the steady current.
Question 5.
Explain the effect of series resonance and draw the Resonance curve.
1. When Series resonance occurs, the impedance of the circuit is minimum and equal to the resistance of the circuit, current in the circuit becomes maximum.
2. This is shown in the Resonance curve drawn between current and frequency. At resonance impedance is
Question 6.
1. Generation of AC is cheaper than DC.
2. At high voltage transmission loss is small.
3. We can easily convert AC to DC by rectifiers.
1. We cannot use AC for certain applications like charging of batteries, electroplating, electric traction, etc.
2. At high voltage, it is more dangerous to work with AC than DC.
Question 7.
Explain energy conservation of loop in the magnetic field and prove that mechanical work done in moving loop appears as thermal energy in the loop.
1. In order to move loop with constant velocity constant $$\overrightarrow{\mathrm{V}}$$ force is equal and opposite to magnetic force applied. Rate of doing work or power is
P = $$\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{V}}$$ = F V cos θ, θ = 0° = F V
2. Three deflecting forces $$\overrightarrow{\mathrm{F}}_{1}$$, $$\overrightarrow{\mathrm{F}}_{2}$$, $$\overrightarrow{\mathrm{F}}_{3}$$ acting on three segments of loop, the general equation is
$$\overrightarrow{\mathrm{F}_{\mathrm{d}}}=\overrightarrow{\mathrm{il}} \quad \times \overrightarrow{\mathrm{B}}$$
3. $$\overrightarrow{\mathrm{F}}_{1}$$, $$\overrightarrow{\mathrm{F}}_{2}$$ equal and opposute so cancel each other.
4. When induced current flows in loop Joule heating takes plan Thermal energy is dissipated in the loop.
P = i2 R
P = $$\left(\frac{B l V}{R}\right)^{2}$$ R ……………..(2)
(1) = (2)
5. Thus mechanical work done is moving the loop appears as thermal energy in the loop.
Question 1.
A series LCR circuit with R = 20Ω L = 1.5 H and c = 35 μF is connected to a variable frequency 200 V ac supply when the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
R = 20Ω, L = 1.5 H
C = 35 μF = 35 × 10-6 F, V = 200 V
Impedence of the circuit is $$Z=\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}$$
At resonance χL = χC
Z = R = 20Ω
I = $$\frac{V}{Z}=\frac{200}{20}$$ = 10A
Average power transferred to the one complete cycle.
P = I2 R
P = 10 × 10 × 20 = 2000W.
Question 2.
An ideal transformer has 460 and 40,000 turns in the primary and secondary coils respectively. Find the voltage developed per turn of the secondary coil if the transformer is connected to a 230 V AC main.
Np= 460 turns,
Ns = 40,000 turns
Vp = 230 V
Question 3.
At a hydroelectric power plant, the water pressure head is at a height of 300m and the water flow available is 100 m3s-1. If the turbine generate efficiency is 60% estimate the electric power available from the plant (g = 9.8 ms-2)
Height of water pressure head h = 300 m
Volume of water flow per second V = 100 m3 s-1
Efficiency of turbine generator η = 60% = 0.6
Acceleration due to gravity g = 9.8 ms-2
Density of water ρ = 103 kg/m3
Electric Power available from plant = η × hρg
= 0.6 × 300 × 103 × 9.8 × 100
= 176.4 × 106 W
= 176.4 MW
Question 4.
A long solenoid with 15 turns per cm has small loops of area 2.0 cm2 placed inside the solenoid normal to its axis. If the current carried by the solenoid change steadily from 2.0 A to 4.0 A is 0.1 s. What is the induced emf in the loop while the current is changing?
Number of turns in solenoid =15 turns/cm = 1500 turns/m
No. of turns per unit length n = 1500 turns
Small loop Area A = 2.0 cm2 = 2 × 10-4 m2
Change in current di = 4 – 2 = 2 A
Change in time dt = 0.1 s
induced emf in solenoid e = $$\frac{\mathrm{d} \varphi}{\mathrm{d} t}$$
e = $$\frac{d}{d t} (BA)$$
= Aµ0n $$\frac{d i}{d t}$$
= 2 × 10-4 × 4 × 10-7 × 1500 × $$\frac{2}{0.1}$$
= 7.54 × 10-6 V
Induced emf in the loop = 7.54 × 10-6 V.
Question 5.
A transmitter consists of an LC circuit with an inductance of 1 µH and capacitance of 1 µF. What is the wavelength of the electromagnetic waves it emits? | {"extraction_info": {"found_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": 0.8700396418571472, "perplexity": 2104.7885147166257}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056856.4/warc/CC-MAIN-20210919095911-20210919125911-00033.warc.gz"} |
http://math.stackexchange.com/questions/799519/algebraic-solution-for-the-intersection-points-of-two-parabolas | # Algebraic solution for the intersection point(s) of two parabolas
I recently ran through an algebraic solution for the intersection point(s) of two parabolas $ax^2 + bx + c$ and $dx^2 + ex + f$ so that I could write a program that solved for them. The math goes like this:
$$ax^2 - dx^2 + bx - ex + c - f = 0 \\ x^2(a - d) + x(b - e) = f - c \\ x^2(a - d) + x(b - e) + \frac{(b - e)^2}{4(a - d)} = f - c + \frac{(b - e)^2}{4(a - d)} \\ (x\sqrt{a - d} + \frac{b - e}{2\sqrt{a - d}})^2 = f - c + \frac{(b - e)^2}{4(a - d)} \\ (a - d)(x + \frac{b - e}{2(a - d)})^2 = f - c + \frac{(b - e)^2}{4(a - d)} \\ x + \frac{b - e}{2(a - d)} = \sqrt{\frac{f - c + \frac{(b - e)^2}{a - d}}{a - d}} \\ x = \pm\sqrt{\frac{f - c + \frac{(b - e)^2}{a - d}}{a - d}} - \frac{b - e}{2(a - d)} \\$$
Then solving for $y$ is as simple as plugging $x$ into one of the equations.
$$y = ax^2 + bx + c$$
Is my solution for $x$ and $y$ correct? Is there a better way to solve for the intersection points?
-
You lost a factor $4$ somewhere. You can simply rewrite your problem as
$$(a-d)x^2+(b-e)x+(c-f)=0$$
and use the standard formula for a quadratic equation, i.e.
$$x=-\frac{b-e}{2(a-d)}\pm\sqrt{\frac{(b-e)^2}{4(a-d)^2}-\frac{c-f}{a-d}}$$
Before evaluating this equation, you need to check if $a-d=0$, in which case
$$x=\frac{f-c}{b-e}$$
In this case you of course need to check if $b-e=0$.
-
Excellent to spot the exceptions. – Mark Bennet May 17 '14 at 20:58
You should recognise a form of the quadratic formula:$$(a-d)x^2+(b-e)x+(c-f)=0$$
which gives $$x=\frac {-(b-e)\pm \sqrt {(b-e)^2-4(a-d)(c-f)}}{2(a-d)}$$
This is the same as yours except for a missing factor of $\frac 14$ under your square root, which you lost when you took the square root near the end.
-
All the other answers are fine from a mathematical point of view, but they ignore the fact that using the quadratic formula is a very bad way to solve quadratic equations in computer code (using floating point arithmetic).
The problem arises when one of the roots is near zero. In this case, either the "$+$" or the "$-$" formula of the $\pm$ will cause you to subtract two numbers that are nearly equal, and this will result in catastrophic cancellation error. The problem is illustrated here, and the well-known solutions are described.
For other interesting examples, see George Forsythe's famous paper "Why a Math Book is not Enough".
-
Well, you somehow lost the factor of 4, otherwise seems valid. As you are going to write a program for this, you should also write a solution gfor the case $a=d$ and deal with the case the radical is zero or does not exist.
Actually, you did not have to write all this, as people usually know the solution of quadratic equation (isn't it school material?). For $ax^2 + bx +c = 0$ usually _the discriminant _$D=b^2 - 4ac$ is calculated, and then $x_{1,2}$ are given as $\frac {-b ±\sqrt D}{2a}$. If D=0 then the only sulution $-\frac b {2a}$ exists, and if $D<0$ there are no real solutions (as $\sqrt D$ is not a real number then or does not exist if we only work with real numbers). Then just use $a-d,b-e$ and $c-f$ as coefficients in the general solution formula. A separate solution for the case $a=d$.
- | {"extraction_info": {"found_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": 0.9134387373924255, "perplexity": 230.66033435571623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701162648.4/warc/CC-MAIN-20160205193922-00116-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://efinancemanagement.com/financial-analysis/loan-to-value-ratio | # Loan to Value Ratio
## Concept of Loan to Value Ratio
A mortgage is a legal document wherein the lender lends money in exchange for the title of the debtor’s property. Until the payment is received for the entire debt, the lender is the title holder of the property which is mortgaged.
There are various types of mortgages. However, despite the type, the lender has to bear significant risk upon lending such large amounts of money. In order to assess the risks, Loan to Value Ratio is used.
Let us understand what this ratio does.
## What is Loan to Value Ratio?
Loan to Value Ratio is a risk assessment ratio which compares the value of the asset to the amount of the loan given. Based on the ratio, the cost of taking the loan is calculated. It is expressed as a percentage of the loan to the cost or value of the asset.
## Loan to Value Ratio Definition
Thus, Loan to Value ratio can be defined as the amount of the mortgage lien divided by the appraised value of the property and expressed as a percentage. Its various uses are as given below.
## Uses of Loan to Value Ratio
The Loan to Value ratio is mainly used to determine the value of mortgage costs. The higher the percentage of Loan to Value ratio, the higher is the credit risk. Therefore, the cost of taking the mortgage goes up in terms of interest rates.A high percentage also requires the need to take insurance which in turn increases the cost of borrowing. A lower Loan to Value ratio percentage for the lenders means that they can sell the property at a rate equal to or more than the loan amount in case of a default in repayment.
The Loan to Value ratio is also used primarily for the purpose of mortgage underwriting. This is done while purchasing residential property, refinancing a current mortgage into new loans and borrowing against the real value of the property. The real value of the property would be the value minus the mortgages and loans on it. The loan to value ratio is an indicator of the feasibility of such additional lendings. It has a simple formula as shown below.
## Loan to Value Ratio Formula
The formula used by the lenders to calculate this ratio is as follows.
Loan to Value Ratio = Loan Amount / Appraised Value of Property
## What is a good Loan to Value Ratio?
Although there is no set optimal LTV ratio, all lenders prefer a low percentage as it decreases the credit risk. Around 80% is preferred in most of the cases as there is a good chance of recovery.
This can be explained with the help of a basic example.
## Loan to Value Ratio Example
A borrower wants a loan for purchasing a house worth \$100,000. The amount of the mortgage given by the lender is \$93,500.
The Loan to Value Ratio will be \$93,500 / \$100,000 or 93.5%.
Thus, this is not a viable ratio for the lender as the percentage is too high. The lender then runs the risk of a loss upon selling the property in case of a default in the repayment. At 93.5% the lender may not even recover the initial principal amount.
Conclusion
Loan to Value Ratio thus makes it easier for lenders to determine the risk and approve the mortgage. However, it must be noted that this is only one determinant and the approval of a mortgage depends on various other subjective factors and criteria.1,2
Share Knowledge if you liked | {"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": 0.837161123752594, "perplexity": 1217.2667493178874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243991562.85/warc/CC-MAIN-20210519012635-20210519042635-00447.warc.gz"} |
https://tex.stackexchange.com/questions/245143/how-to-draw-a-number-line-with-arrows-inside-it | # How to draw a number line with arrows inside it
I am having trouble using TiKz to draw a correct picture. Here is how I started it out:
\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows}
\begin{document}
\tikzset{->-/.style={decoration={
markings,
mark=at position #1 with {\arrow{<}}},postaction={decorate}}}
\begin{tikzpicture}
% a straight line segment
\draw[latex-] (-2,0) -- (0,0);
\draw[->-=.3] (0,0) -- (1,0);
\draw[->-=.6] (0,0) -- (1,0);
\draw[-latex] (1,0) -- (2,0);
% the ticks and their labels
\foreach \x in {-1,...,1}
\draw[xshift=\x cm] (0pt,2pt) -- (0pt,-1pt) node[below,fill=white] {\the\numexpr\x +1\relax};
\end{tikzpicture}
\end{document}
But I want my picture to look like this:
How do I make the numbers appear correctly, and also make the curved arrows?
Like this?
\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows}
\begin{document}
\tikzset{->-/.style={decoration={
markings,
mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
\begin{tikzpicture}
% a straight line segment
\draw[latex-] (-2,0) -- (0,0);
\draw[->-=.3] (0,0) -- (1,0);
\draw[->-=.6] (0,0) -- (1,0);
\draw[-latex] (1,0) -- (2,0);
% the ticks and their labels
\foreach \x [count=\i start from 0] in {-1,...,1}
\draw[xshift=\x cm] (0pt,2pt) -- (0pt,-1pt) node[below,fill=white] (\i) {\x};
\draw[->] (2) to [out=-95, in=-85] (0);
\draw[<-] (.25mm,1mm) arc(-85:265:3mm);
\end{tikzpicture}
\end{document}
• @ᴇʏᴇs I've updated the answer. You just need {\x} as node contents. \x cannot be used as node names because they are considered real values, and I've used \i which is an integer. But there is no problem using in nodes contents. – Ignasi May 15 '15 at 15:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9280164837837219, "perplexity": 3367.5804498419707}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1566027330750.45/warc/CC-MAIN-20190825151521-20190825173521-00051.warc.gz"} |
https://en.wikipedia.org/wiki/Cardinal_utility | # Cardinal utility
A simple example of two cardinal utility functions u (first column) and v (second column) whose values in all circumstances are related by v=2u+3
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations.[1][2] Two utility indices are related by an affine transformation if for the value $u(x_i)$ of one index u, occurring at any quantity $x_i$ of the goods bundle being evaluated, the corresponding value $v(x_i)$ of the other index v satisfies a relationship of the form
$v(x_i) = au(x_i) + b\!$,
for fixed constants a and b. Thus the utility functions themselves are related by
$v(x) = au(x) + b.$
The two indices differ only with respect to scale and origin.[1] Thus if one is concave, so is the other, in which case there is said to be diminishing marginal utility.
Thus the use of cardinal utility imposes the assumption that levels of absolute satisfaction exist, so that the magnitudes of increments to satisfaction can be compared across different situations. This contrasts with ordinal utility, in which concavity or convexity of the utility function has no economic relevance.
The idea of cardinal utility is considered outdated except for specific contexts such as decision making under risk, utilitarian welfare evaluations, and discounted utilities for intertemporal evaluations where it is still applied.[3] Elsewhere, such as in general consumer theory, ordinal utility with its weaker assumptions Is preferred because results that are just as strong can be derived.
## History
The first one to theorize about the marginal value of money was Apurva Chaurasia in 1738. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns. Since Bernoulli tacitly assumed that an interpersonal measure for the utility reaction of different persons can be discovered, he was then inadvertedly using an early conception of cardinality.[4]
Bernoulli's imaginary logarithmic utility function and Gabriel Cramer's U=W1/2 function were conceived at the time not for a theory of demand but to solve the St. Petersburg's game. Bernoulli assumed that "a poor man generally obtains more utility than a rich man from an equal gain"[5] an approach that is more profound that the simple mathematical expectation of money as it involves a law of moral expectation.
Early theorists of utility considered that it had physically quantifiable attributes. They thought that utility behaved like the magnitudes of distance or time, in which the simple use of a ruler or stopwatch resulted in a distinguishable measure. "Utils" was the name actually given to the units in a utility scale.
In the Victorian era many aspects of life were succumbing to quantification.[6] The theory of utility soon began to be applied to moral-philosophy discussions. The essential idea in utilitarianism is to judge people's decisions by looking at their change in utils and measure whether they are better off. The main forerunner of the utilitarian principles since the end of the 18th century was Jeremy Bentham, who believed utility could be measured by some complex introspective examination and that it should guide the design of social policies and laws. For Bentham a scale of pleasure has as a unit of intensity "the degree of intensity possessed by that pleasure which is the faintest of any that can be distinguished to be pleasure";[7] he also stated that, as these pleasures increase in intensity higher and higher numbers could represent them.[7]
In the 18th and 19th centuries utility's measurability received plenty of attention from European schools of political economy, most notably through the work of marginalists (e.g. William Stanley Jevons,[8] Léon Walras, Alfred Marshall). However, neither of them offered solid arguments to backup up the assumption of measurability. In Jevon's case he added to the later editions of his work a note on the difficulty of estimating utility with accuracy.[7] Walras, too, struggled for many years before he could even attempt to formalize the assumption of measurability.[9] Marshall was ambiguous about the measurability of hedonism because he adhered to its psychological-hedonistic properties but he also argued that it was "unrealistical" to do so.[10]
Supporters of cardinal utility theory in the 19th century suggested market prices reflected utility, although they did not say much about them being incompatible (i.e. prices are objective measures but utility is subjective). Accurately measuring subjective pleasure (or pain) seemed awkward, as the thinkers of the time were surely aware. They renamed utility in imaginative ways such as subjective wealth, overall happiness, moral worth, psychic satisfaction, or ophélimité. During the second half of the 19th century, many studies related to this fictional magnitude -utility- were conducted, but the conclusion was always the same: it proved impossible to definitively say whether a good is worth 50, 75, or 125 utils to a person, or to two different people. Moreover, the mere dependence of utility on notions of hedonism, led academic circles to be skeptical of this theory.[11]
Francis Edgeworth was also aware of the need to ground the theory of utility into the real world. He discussed the quantitative estimates that a person can make of his own pleasure or the pleasure of others, borrowing methods developed in psychology to study hedonic measurement: psychophysics. This field of psychology was built on work by Ernst H. Weber, but around the time of World War I, psychologists grew discouraged of it.[12][13]
In the late 19th century, Carl Menger and his followers from the Austrian school of economics undertook the first successful departure from measurable utility, in the clever form of a theory of ranked uses. Despite abandoning the thought of quantifiable utility (i.e. psychological satisfaction mapped into the set of real numbers) Menger managed to establish a body of hypothesis about decision-making, resting solely on a few axioms of ranked preferences over the possible uses of goods and services. His numerical examples are "illustrative of ordinal, not cardinal, relationships".[14]
Around the turn of the 19th century neoclassical economists started to embrace alternative ways to deal with the measurability issue. By 1900, Pareto was hesitant about accurately measuring pleasure or pain because he thought that such a self-reported subjective magnitude lacked scientific validity. He wanted to find an alternative way to treat utility that did not rely on erratic perceptions of the senses.[15] Pareto's main contribution to ordinal utility was to assume that higher indifference curves have greater utility, but how much greater does not need to be specified to obtain the result of increasing marginal rates of substitution.
The works and manuals of Vilfredo Pareto, Francis Edgeworth, Irving Fischer, and Eugene Slutsky departed from cardinal utility and served as pivots for others to continue the trend on ordinality. According to Viner,[16] these economic thinkers came up with a theory that explained the negative slopes of demand curves. Their method avoided the measurability of utility by constructing some abstract indifference curve map.
During the first three decades of the 20th century, economists from Italy and Russia became familiar with the Paretian idea that utility does not need to be cardinal. According to Schultz,[17] by 1931 the idea of ordinal utility was not yet embraced by American economists. The breakthrough occurred when a theory of ordinal utility was put together by John Hicks and Roy Allen in 1934.[18] In fact pages 54–55 from this paper contain the first use ever of the term 'cardinal utility'.[19] The first treatment of a class of utility functions preserved by affine transformations, though, was made in 1934 by Oskar Lange.[20]
In 1944 Frank Knight argued extensively for cardinal utility. In the decade of 1960 Parducci studied human judgements of magnitudes and suggested a range-frequency theory.[21] Since the late 20th century economists are having a renewed interest in the measurement issues of happiness.[22][23] This field has been developing methods, surveys and indices to measure happiness.
Several properties of Cardinal utility functions can be derived using tools from measure theory and set theory.
### Measurability
A utility function is considered to be measurable, if the strength of preference or intensity of liking of a good or service is determined with precision by the use of some objective criteria. For example, suppose that eating an apple gives to a person exactly half the pleasure of that of eating an orange. This would be a measurable utility if and only if the test employed for its direct measurement is based on an objective criterion that could let any external observer repeat the results accurately.[24] One hypothetical way to achieve this would be by the use of an hedonometer, which was the instrument suggested by Edgeworth to be capable of registering the height of pleasure experienced by people, diverging according to a law of errors.[12]
Before the 1930s, the measurability of utility functions was erroneously labeled as cardinality by economists. A different meaning of cardinality was used by economists who followed the formulation of Hicks-Allen. Under this usage, the cardinality of a utility function is simply the mathematical property of uniqueness up to a linear transformation. Around the end of the 1940s, some economists even rushed to argue that von Neumann-Morgenstern axiomatization of expected utility had resurrected measurability.[15]
The confusion between cardinality and measurability was not to be solved until the works of Armen Alchian,[25] William Baumol,[26] and John Chipman.[27] The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature at the time.
It is helpful to consider the same problem as it appears in the construction of scales of measurement in the natural sciences.[28] In the case of temperature there are two degrees of freedom for its measurement - the choice of unit and the zero. Different temperature scales map its intensity in different ways. In the celsius scale the zero is chosen to be the point where water freezes, and likewise, in cardinal utility theory one would be tempted to think that the choice of zero would correspond to a good or service that brings exactly 0 utils. However this is not necessarily true. The mathematical index remains cardinal, even if the zero gets moved arbitrarily to another point, or if the choice of scale is changed, or if both the scale and the zero are changed. Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity. The point of this example was used to prove that (as with temperature) it is still possible to predict something about the combination of two values of some utility function, even if the utils get transformed into entirely different numbers, as long as it remains a linear transformation.
Von Neumann and Morgenstern stated that the question of measurability of physical quantities was dynamic. For instance, temperature was originally a number only up to any monotone transformation, but the development of the ideal-gas-thermometry led to transformations in which the absolute zero and absolute unit were missing. Subsequent developments of thermodynamics even fixed the absolute zero so that the transformation system in thermodynamics consists only of the multiplication by constants. According to Von Neumann and Morgenstern (1944, p. 23) "For utility the situation seems to be of a similar nature [to temperature]".
The following quote from Alchian served to clarify once and for all the real nature of utility functions, emphasizing that they no longer need to be measurable:
Can we assign a set of numbers (measures) to the various entities and predict that the entity with the largest assigned number (measure) will be chosen? If so, we could christen this measure "utility" and then assert that choices are made so as to maximize utility. It is an easy step to the statement that "you are maximizing your utility", which says no more than that your choice is predictable according to the size of some assigned numbers. For analytical convenience it is customary to postulate that an individual seeks to maximize something subject to some constraints. The thing -or numerical measure of the "thing"- which he seeks to maximize is called "utility". Whether or not utility is of some kind glow or warmth, or happiness, is here irrelevant; all that counts is that we can assign numbers to entities or conditions which a person can strive to realize. Then we say the individual seeks to maximize some function of those numbers. Unfortunately, the term "utility" has by now acquired so many connotations, that it is difficult to realize that for present purposes utility has no more meaning than this.
Armen Alchian, The meaning of utility measurement[25]
### Order of preference
For more details on this topic, see Preference (economics).
In 1955 Patrick Suppes and Muriel Winet solved the issue of the representability of preferences by a cardinal utility function, and derived the set of axioms and primitive characteristics required for this utility index to work.[29]
Suppose an agent is asked to rank his preferences of A relative to B and his preferences of B relative to C. If he finds that he can state, for example, that his degree of preference of A to B exceeds his degree of preference of B to C, we could summarize this information by any triplet of numbers satisfying the two inequalities: UA > UB > UC and UA - UB > UB - UC.
If A and B were sums of money, the agent could vary the sum of money represented by B until he could tell us that he found his degree of preference of A over the revised amount B' equal to his degree of preference of B' over C. If he finds such a B', then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships: (a) UA > UB' > UC , and (b) UA - UB' = UB' - UC. Any two triplets obeying these relationships must be related by a linear transformation; they represent utility indices differing only by scale and origin. In this case, "cardinality" means nothing more being able to give consistent answers to these particular questions. Note that this experiment does not require measurability of utility. Itzhak Gilboa gives a sound explanation of why measurability can never be attained solely by introspection:
It might have happened to you that you were carrying a pile of papers, or clothes, and didn't notice that you dropped a few. The decrease in the total weight you were carrying was probably not large enough for you to notice. Two objects may be too close in terms of weight for us to notice the difference between them. This problem is common to perception in all our senses. If I ask whether two rods are of the same length or not, there are differences that will be too small for you to notice. The same would apply to your perception of sound (volume, pitch), light, temperature, and so forth...
—Itzhak Gilboa, Theory of decision under uncertainty[30]
According to this view, those situations where a person just cannot tell the difference between A and B will lead to indifference not because of a consistency of preferences, but because of a misperception of the senses. Moreover, human senses adapt to a given level of stimulation and then register changes from that baseline.[31]
## Construction
Suppose a certain agent has a preference ordering over random outcomes (lotteries). If the agent can be queried about his preferences, it is possible to construct a cardinal utility function that represents these preferences. This is the core of the Von Neumann–Morgenstern utility theorem.
## Applications
### Welfare economics
Among welfare economists of the utilitarist school it has been the general tendency to take satisfaction (in some cases, pleasure) as the unit of welfare. If the function of welfare economics is to contribute data which will serve the social philosopher or the statesman in the making of welfare judgements, this tendency leads perhaps, to a hedonistic ethics.[32]
Under this framework, actions (including production of goods and provision of services) are judged by their contributions to the subjective wealth of people. In other words, it provides a way of judging the "greatest good to the greatest number of persons". An act that reduces one person's utility by 75 utils while increasing two others' by 50 utils each has increased overall utility by 25 utils and is thus a positive contribution; one that costs the first person 125 utils while giving the same 50 each to two other people has resulted in a net loss of 25 utils.
If a class of utility functions is cardinal, intrapersonal comparisons of utility differences are allowed. If, in addition, some comparisons of utility are meaningful interpersonally, the linear transformations used to produce the class of utility functions must be restricted across people. An example is cardinal unit comparability. In that information environment, admissible transformations are increasing affine functions and, in addition, the scaling factor must be the same for everyone. This information assumption allows for interpersonal comparisons of utility differences, but utility levels cannot be compared interpersonally because the intercept of the affine transformations may differ across people.[33]
### Marginalism
For more details on this topic, see Marginal utility.
• Under cardinal utility theory, the sign of the marginal utility of a good is the same for all the numerical representations of a particular preference structure.
• The magnitude of the marginal utility is not the same for all cardinal utility indices representing the same specific preference structure.
• The sign of the second derivative of a differentiable utility function that is cardinal, is the same for all the numerical representations of a particular preference structure. Given that this is usually a negative sign, there is room for a law of diminishing marginal utility in cardinal utility theory.
• The magnitude of the second derivative of a differentiable utility function is not the same for all cardinal utility indices representing the same specific preference structure.
### Expected utility theory
For more details on this topic, see Expected utility theory.
This type of indices involves choices under risk. In this case, A, B, and C, are lotteries associated with outcomes. Unlike cardinal utility theory under certainty, in which the possibility of moving from preferences to quantified utility was almost trivial, here it is paramount to be able to map preferences into the set of real numbers, so that the operation of mathematical expectation can be executed. Once the mapping is done, the introduction of additional assumptions would result in a consistent behavior of people regarding fair bets. But fair bets are, by definition, the result of comparing a gamble with an expected value of zero to some other gamble. Although it is impossible to model attitudes toward risk if one doesn't quantify utility, the theory should not be interpreted as measuring strength of preference under certainty.[34]
### Construction of the utility function
Suppose that certain outcomes are associated with three states of nature, so that x3 is preferred over x2 which in turn is preferred over x1; this set of outcomes, X, can be assumed to be a calculable money-prize in a controlled game of chance, unique up to one positive proportionality factor depending on the currency unit.
Let L1 and L2 be two lotteries with probabilities p1, p2, and p3 of x1, x2, and x3 respectively being
$L_1 =(0.6, 0, 0.4),$
$L_2 =(0,1,0)\ .$
Assume that someone has the following preference structure under risk:
$L_{1} \succ L_{2},$
meaning that L1 is preferred over L2. By modifying the values of p1 and p3 in L1, eventually there will be some appropriate values (L1') for which she is found to be indifferent between it and L2—for example
$L_{1}' =(0.5, 0, 0.5).$
Expected utility theory tells us that
$EU(L_{1}') = EU(L_2)\!$
and so
$(0.5)*u(x_1)+(0.5)*u(x_{3})=1*u(x_{2}).$
In this example from Majumdar[35] fixing the zero value of the utility index such that the utility of x1 is 0, and by choosing the scale so that the utility of x2 equals 1, gives
$(0.5)*u(x_{3})=1.$
$u(x_{3}) = 2.$
### Intertemporal utility
For more details on this topic, see Intertemporal choice.
Models of utility with several periods, in which people discount future values of utility, need to employ cardinalism in order to have well-behaved utility functions. According to Paul Samuelson the maximization of the discounted sum of future utilities implies that a person can rank utility differences.[19]
## Controversies
Some authors have commented on the misleading nature of the terms "cardinal utility" and "ordinal utility", as used in economic jargon:
These terms, which seem to have been introduced by Hicks and Allen (1934), bear scant if any relation to the mathematicians' concept of ordinal and cardinal numbers; rather they are euphemisms for the concepts of order-homomorphism to the real numbers and group-homomorphism to the real numbers
—John Chipman, The foundations of utility[27]
There remain economists who believe that utility, if it cannot be measured, at least can be approximated somewhat to provide some form of measurement, similar to how prices, which have no uniform unit to provide an actual price level, could still be indexed to provide an "inflation rate" (which is actually a level of change in the prices of weighted indexed products). These measures are not perfect but can act as a proxy for the utility. Lancaster's[36] characteristics approach to consumer demand illustrates this point.
## Comparison between ordinal and cardinal utility functions
The following table compares the two types of utility functions common in economics:
Level of measurement Represents preferences on - Unique up to - Existence proved by- Mostly used in -
Ordinal utility Ordinal scale - sure outcomes - Increasing monotone transformation - Debreu (1954) Consumer theory
Cardinal utility Interval scale - random outcomes (lotteries) - Increasing monotone linear transformation - Von Neumann-Morgenstern (1947) Game theory
## References
1. ^ a b Ellsberg, Daniel (1954). "Classic and current notions of 'Measurable utility'". Economic Journal 64 (255): 528–556. doi:10.2307/2227744.
2. ^ Strotz, Robert. (1953). "Cardinal utility". American economic review Vol. 43, No. 2, pp. 384–397
3. ^ Köbberling, Veronika. (2006). "Strength of preference and cardinal utility". Economic theory, No. 27, p. 375
4. ^ Kauder, Emil (1953). "Genesis of the Marginal Utility Theory: From Aristotle to the End of the Eightteenth Century". Economic Journal 63 (251): 648. Retrieved 29 April 2015.
5. ^ Samuelson, Paul (1977). "St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described". Journal of Economic Literature 15 (1): 38. Retrieved 11 May 2015.
6. ^ Bernstein, Peter. (1996). Against the gods. The remarkable story of risk. New York: John Wiley and Sons, p. 191
7. ^ a b c Stigler, George. (1950). "The development of utility theory. I". Journal of political economy Vol. 58, No. 4, pp. 307–327
8. ^ Jevons, William Stanley (1862). "Brief account of a general mathematical theory of political economy". Journal of the Royal Statistical Society 29: 282–287.
9. ^ Jaffé, William (1977). "The Walras-Poincaré Correspondence on the Cardinal Measurability of Utility". Canadian Journal of Economics 10 (2): 300. Retrieved 19 May 2015.
10. ^ Martinoia, Rozenn (2003). "That which is desired, which pleases, and which satisfies: Utility according to Alfred Marshall" (PDF). Journal of the History of Economic Thought 25 (3): 350. Retrieved 21 May 2015.
11. ^ Stigler, George. (1950). "The development of utility theory. II". Journal of political economy Vol. 58, No. 5, pp. 373–396
12. ^ a b Colander, David. (2007). "Retrospectives: Edgeworth's hedonimeter and the quest to measure utility". Journal of economic perspectives, Vol. 21, No. 2, pp. 215–226.
13. ^ McCloskey, Deirdre N. "Happyism". New Republic. Retrieved 11 March 2013.
14. ^ Stigler, George (1937). "The Economics of Carl Menger". Journal of Political Economy 45 (2): 240. Retrieved 21 May 2015.
15. ^ a b Lewin, Shira. (1996). "Economics and psychology: lessons for our own day from the early twentieth century". Journal of economic literature 34 (3), 1293–1323.
16. ^ Viner, Jacob. (1925a). "The utility concept in value theory and its critics". Journal of political economy Vol. 33, No. 4, pp. 369–387
17. ^ Schultz, Henry (1931). "The Italian School of Mathematical Economics". Journal of Political Economy 39 (1): 77. Retrieved 3 June 2015.
18. ^ Hicks, John and Roy Allen. (1934). "A reconsideration of the theory of value". Economica Vol. 1, No. 1, pp. 52–76
19. ^ a b Moscati, Ivan. "How cardinal utility entered economic analysis during the Ordinal Revolution" (PDF). Working Paper. Universita Dell'Insubria Facolta di Economia. Retrieved 9 February 2013.
20. ^ Lange, Oskar (1934). "The Determinateness of the Utility Function". Review of Economic Studies 1 (3): 218–225. Retrieved 10 June 2015.
21. ^ Kornienko, T. (2004). A cognitive bases for cardinal utility. Retrieved November 4, 2012 from the website of the University of Stirling: http://staff.stir.ac.uk/tatiana.kornienko/tape.pdf, p. 3
22. ^ Kahneman, Daniel., Peter Wakker, and Rakesh Sarin. (1997). "Back to Bentham? Explorations of experienced utility?". Quarterly journal of economics Vol. 112, No. 2, pp. 375–405.
23. ^ Kahneman, Daniel., Ed Diener and Norbert Schwarz. (1999). Well-being: the foundations of hedonic psychology. New York: Rusell Sage Foundation
24. ^ Bernadelli, H. (1938). "The end of the marginal utility theory". Economica, Vol. 5, No. 18, p. 196
25. ^ a b Alchian, Armen. (1953). "The meaning of utility measurement". American economic review Vol. 43, No. 1, pp. 26–50.
26. ^ Baumol, William. (1958). "The cardinal utility which is ordinal". Economic journal Vol. 68, No. 272, pp. 665–672
27. ^ a b Chipman, John. (1960). "The foundations of utility". Econometrica Vol. 28, No. 2, pp. 215–216
28. ^ Allen, Roy. (1935). "A note on the determinateness of the utility function". Review of economic studies, Vol. 2, No. 2, pp. 155–158
29. ^ Suppes, Patrick and Muriel Winet. (1955). "An axiomatization of utility based on the notion of utility differences". Management science No. 1, pp. 259–270
30. ^ Gilboa, Itzhak. (2008). Theory of Decision under uncertainty. Cambridge University Press.
31. ^ Poundstone, William. (2010). Priceless. The myth of fair value (and how to take advantage of it). New York: Hill and Wang, p. 39
32. ^ Viner, Jacob. (1925b). "The utility concept in value theory and its critics. II. The utility concept in welfare economics". Journal of political economy Vol. 33, No. 6, pp. 638–659.
33. ^ Blackorby, Charles; Bossert, Walter; Donaldson, David (2002). "Utilitarianism and the theory of justice". In Arrow, Kenneth; Sen, Amartya; Suzumura, Kotaru. Handbook of social choice and welfare. Elsevier. p. 552. ISBN 978-0-444-82914-6.
34. ^ Shoemaker, Paul. (1982). "The expected utility model: its variants, purposes, evidence and limitations". Journal of economic literature, Vol. 20, No. 2, pp. 529–563
35. ^ Majumdar, Tapas. (1958). "Behaviourist cardinalism in utility theory". Economica, Vol. 25, No. 97, pp. 26–33
36. ^ Lancaster, Kelvin. (1966). "A new approach to consumer theory". Journal of political economy Vol. 74, No. 2, pp. 132–157 | {"extraction_info": {"found_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": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8525060415267944, "perplexity": 1714.5622195409658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1443738095178.99/warc/CC-MAIN-20151001222135-00195-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/179775-continuous-metric-spaces.html | # Math Help - continuous metric spaces
1. ## continuous metric spaces
Show that
1. the projection map p : R^2 → R given by p(x; y) := x is continuous.
.
My proofs:
We want to show that $\forall$ $\epsilon$ > 0, $\exists$ $\delta$>0 such that:
d(x,y)< $\delta$ $\Rightarrow$ d(f(x),f(y)) < $\epsilon$
(x1,y1) (x2,y2) are points in R^2
1) d(x1,x2) < $\epsilon$
|x1-x2| < $\epsilon$
We can define $\delta$ = the square root of ( $\epsilon$^2 - (y2-y1)^2)
Since there exists a delta for any epsilon we choose, implies that p is continuous.
Is this correct...
2. Looks quite correct to me. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9871405959129333, "perplexity": 2738.9904881174793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1432207927104.48/warc/CC-MAIN-20150521113207-00187-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.sarthaks.com/9826/define-1-j-of-work | # Define 1 J of work.
43 views
in Physics
Define 1 J of work.
by (127k points)
1 J is the amount of work done by a force of 1 N on an object that displaces it through a distance of 1 m in the direction of the applied force. 80 | {"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": 0.9464606642723083, "perplexity": 521.5902584842074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141745780.85/warc/CC-MAIN-20201204223450-20201205013450-00190.warc.gz"} |
http://math.stackexchange.com/questions/375299/calculating-circle-properties | # Calculating circle properties.
How can I incrementally calculate the angle from angle 0 and the point (x, y) in a circumference path if I have the center of the circle coordinates and the radius of the circle.
I have 127 segments each with 5 points in a segment. This is 635 in circumference. The closest radius to this is 102.
-
The question isn't terribly clear. Could you try to clarify? What, e.g. is the center (it's coordinates), and what is the radius? I'm not sure what you mean when you say you have the radius, but end your post with "the closest radius to 'this' [what this] is 102"? – amWhy Apr 28 '13 at 15:34
I have 127 segments that I need to render points for, each segment has 5 points. This gives me 635. So I need a circle with the circumference of 635, each point in the circumference can be mapped by a point in a segment. Calculating the radius of this causes a flooring of a floating point number to an integer of 102. I would like to calculate each point in the circumference, each point in each segment. So e.g. for each angle in the circle, 360 is used, incremented by 1 each time, how can I determine the (x, y) coordinate of each point in the circumference path, (going clockwise)? – Helium3 Apr 28 '13 at 15:42
If you have 635 points, and want them evenly dispersed along a circle, you can increment the angle by $\dfrac{360}{635}^\circ$. – amWhy Apr 28 '13 at 15:59
$$x = r\cos\theta, \quad y = r\sin\theta, \quad\text{for circle radius r centered at origin}$$
With an arbitrary center $(x_c, y_c)$, then
$$(x - x_c) = r\cos\theta, \quad (y - y_c) = r\sin\theta$$
$$x = r\cos\theta + x_c, \quad y = r\sin\theta + y_c$$
$$\tan\theta = \dfrac{y - y_c}{x - x_c} \iff \theta = \arctan\left(\frac{y - y_c}{x - x_c}\right)$$
-
Thank you for your time on helping me. :) – Helium3 Apr 28 '13 at 17:22
You're welcome, Helium3! – amWhy Apr 28 '13 at 17:25
Nice, clean answer +1 – Amzoti Apr 29 '13 at 0:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8752407431602478, "perplexity": 706.3327907496237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042986625.58/warc/CC-MAIN-20150728002306-00256-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://practice-questions.wizako.com/gmat/quant/permutation-combination-probability/specific-sequence-probability-rearrangement-3.shtml | # GMAT® Quant Practice : Probability
Concept: Rearranging letters of a word. Basics of probability.
This GMAT quant practice question is a problem solving question in probability. The concept tested in this question is to find the probability that a specific sequence occurs when letters of a word are rearranged.
#### Question: What is the probability that the position in which the consonants appear remain unchanged when the letters of the word Math are re-arranged?
1. $\frac{1}{4}\\$
2. $\frac{1}{6}\\$
3. $\frac{1}{3}\\$
4. $\frac{1}{24}\\$
5. $\frac{1}{12}\\$
Video explanation will be added soon
#### Compute denominator
In any probability question, the denominator represents the total number of outcomes for an event. The numerator represents the number of favorable outcomes.
The total number of ways in which the letters of the word MATH can be re-arranged = 4! = 4*3*2*1 = 24 ways.
#### Compute numerator and the probability
If the positions in which the consonants appear do not change, the first, third and the fourth positions are reserved for consonants and the vowel A remains at the second position.
The consonants M, T and H can be re-arranged in the first, third and fourth positions in 3! = 6 ways so that the positions in which the consonants appear remain unchanged.
Therefore, the required probability $\frac{6}{24} = \frac{1}{4}\\$
Choice A is the correct answer.
### Are you targeting Q-51 in GMAT Quant? Make it a reality!
Comprehensive Online classes for GMAT Math. 20 topics.
Focused preparation for the hard-to-crack eggs in the GMAT basket! | {"extraction_info": {"found_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": 0.9508455991744995, "perplexity": 711.6996946023011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934806736.55/warc/CC-MAIN-20171123050243-20171123070243-00767.warc.gz"} |
https://galoisrepresentations.wordpress.com/2015/10/17/hilbert-modular-forms-of-partial-weight-one-part-iii/ | Hilbert Modular Forms of Partial Weight One, Part III
My student Richard Moy is graduating!
Richard’s work has already appeared on this blog before, where we discussed his joint work with Joel Specter showing that there existed non-CM Hilbert modular forms of partial weight one. Today I want to discuss a sequel of sorts to that paper, which also forms part of Richard’s thesis (I should note that he already has five publications and will have 7 or 8 papers by the time he graduates.) The starting observation is as follows. Fix a real quadratic field F. From the perspective of Galois representations, the Hilbert modular forms of partial weight one fall under the case $\ell_0 = 1$ in the notation of my paper with David Geraghty (this is in the context of coherent cohomology). To orient the reader, let us discuss three classes of such forms:
1. Hilbert modular forms of weight $[2k+1,1]$ for a real quadratic field $F.$
2. Regular algebraic cuspidal automorphic forms for $\mathrm{GL}(3)/\mathbf{Q}.$
3. Regular algebraic cuspidal automorphic forms for $\mathrm{GL}(2)/F$ for an imaginary quadratic field $F.$
Suppose one fixes a tame level $N$ and then looks at the space of such forms as the weights vary. In both of the latter cases, the problem has been raised (or even conjectured, for $N = 1$ and $\mathrm{GL}(3)$ by Ash and Pollack here), of whether all but finitely many such forms arise via functoriality from a smaller group. More explicitly, one can ask whether:
1. If $G = \mathrm{GL}(2)/F,$ then all but finitely many cuspidal regular algebraic forms of conductor $N$ either arise (up to twist) via base change from $\mathrm{GL}(2)/\mathbf{Q},$ or are induced from a quadratic CM extension $E/F.$
2. If $G = \mathrm{GL}(3)/\mathbf{Q},$ then all but finitely many cuspidal regular algebraic forms of conductor $N$ arise up to twist as the symmetric square of a form from $\mathrm{GL}(2)/\mathbf{Q}.$
Naturally enough, one can make the same conjecture whenever $\ell_0 > 0,$ appropriately formulated. There does not seem to be any case of this conjecture which is known, although there are analogous results (where one fixes the weight and varies the level) in both weight one (where it is almost trivial) and for imaginary quadratic fields (in the work of Calegari-Dunfield and Boston-Ellenberg). Still, the conjectures in varying weight seem pretty hard even for $N = 1.$ In that context, Richard proves the following nice complementary pair of theorems below. Let $F = \mathbf{Q}(\sqrt{7}).$ The field $F$ has narrow class number $2$ and there is a unique odd everywhere unramified quadratic character $\chi$ of $G_F$ with fixed field $E = F(\sqrt{-1}).$
Theorem I (Moy) Let $F$ and $\chi$ be as above. Every Hilbert modular form over $F$ of weight $[2k+1,1]$ and level $N = 1$ is CM, and in particular is induced from $E.$
Theorem II (Moy) Let $F$ and $\chi$ be as above. Let $M$ be a strongly compatible family of two dimensional Galois representations of $F$ with determinant $\chi,$ level $N = 1,$ and Hodge–Tate weights $[0,0]$ and $[k,-k].$ Then $M$ is induced from $E.$
Theorem I is almost an immediate consequence of Theorem II, with the caveat that one doesn’t quite have complete local-global compatibility for partial weight one modular forms (though results and methods of Luu, Jorza, and Newton get close). Theorem II on the other hand is a consequence of the following:
Theorem III (Moy) Let $F$ and $\chi$ be as above. Let
$\rho: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_3)$
be a continuous irreducible representation with determinant $\chi$ that is unramified at all finite places except for one prime $v|3.$ Then $\rho$ is induced from a character of $G_E.$
The argument in this case is (roughly) the following. Using a Tate-style argument (with discriminant bounds), one proves that the residual representation $\overline{\rho}$ must have semi-simplification $\chi \oplus 1.$ The restriction of $\rho$ to $G_E$ then has the property that its image is pro-3 and unramified outside the fixed prime $v|3.$ Yet one shows by a class field theory computation that the largest abelian 3-extension unramified outside $v|3$ is cyclic, which (by consideration of the Frattini quotient) immediately implies that the image of $\rho$ restricted to $G_E$ factors through a cyclic quotient as well, and one is done.
Note that to deduce Theorem I, one first has to prove (using a congruence argument) that at the other prime $w|3,$ either:
1. The representation $\rho$ is unramified at $w,$
2. The representation $\rho$ restricted to $D_w$ has unramified semi-simplification. In particular, the generalized eigenvalues of $\mathrm{Frob}_w$ for $\overline{\rho}$ are both the same.
To finish, one rules out the second possibility by computing all the modular residual representations explicitly by doing computations in low weight (this can ultimately be reduced to a computation on the definite quaternion side, although Richard had to write his own programs to do this since the current magma implementation required trivial character for non-parallel weight.)
It is true that these arguments will not suffice for the more general conjecture, but then, I haven’t seen a viable strategy to prove those conjectures either!
This entry was posted in Mathematics and tagged , , , , , , , , , , . Bookmark the permalink.
One Response to Hilbert Modular Forms of Partial Weight One, Part III
1. Pingback: Graduation Day | Persiflage | {"extraction_info": {"found_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": 62, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9681989550590515, "perplexity": 297.16800071784}, "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-2018-13/segments/1521257647545.84/warc/CC-MAIN-20180320224824-20180321004824-00068.warc.gz"} |
https://onepetro.org/SNAMETOS/proceedings-abstract/TOS15/1-TOS15/D013S003R002/3712 | The determining of aerodynamic coefficients is one of the essential steps in the design of an offshore structure such as an FPSO (Floating Production Storage and Offloading). It is extremely important as they are one of the dimensioning criteria for the mooring design. Nowadays, these loads are mainly assessed through wind tunnel tests performed at model scale. Estimating realistic wind loads however, remains a big challenge. The complexity and associated simplification level of FPSO topside structures, the scale effects and the establishment of the atmospheric boundary layer imply that many simplifications are to be made. Nowadays with the evolution of CFD (Computational Fluid Dynamics) software, and the increase of the meshing capacity, new scopes open to CFD. Aerodynamic simulations on complex FPSO structures are therefore now possible, but need specific developments and validations that are presented in this paper. The first steps for setting up a procedure for computing these coefficients are thus presented.
The main objective of the work presented here is to investigate the ability of CFD for evaluating wind loads on complex FPSOs topsides. In a first stage, the first elements of the numerical model used as a numerical wind tunnel are presented and studied. Then, some specific effects such as the blockage effects, the atmospheric boundary layer and their numerical modelling are studied. The geometric model used corresponds to the one used in wind tunnel. The same Atmospheric Boundary Layer is simulated and a thorough effort is performed to ensure the mesh convergence. Then, the accuracy of the blockage effect correction is evaluated by performing computations with and without blockage, and results are compared with classical corrections applied in wind tunnel tests.
In the last step, a relevant comparison between wind tunnel tests and simulations on different geometries with grids density up to 50 million cells is presented. The ability for CFD to evaluate aerodynamic coefficients is then discussed.
This content is only available via PDF. | {"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": 0.9250544309616089, "perplexity": 956.5110210097383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780061350.42/warc/CC-MAIN-20210929004757-20210929034757-00263.warc.gz"} |
https://www.arxiv-vanity.com/papers/cond-mat/0109042/ | # Statistical properties of contact vectors
A. Kabakçıoğlu, I. Kanter, M. Vendruscolo, and E. Domany Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
Department of Physics, Bar Ilan University, 52900 Ramat Gan,Israel
Oxford Centre for Molecular Sciences, Central Chemistry Laboratory
University of Oxford, South Parks Road, OX1 3QH Oxford UK
###### Abstract
We study the statistical properties of contact vectors, a construct to characterize a protein’s structure. The contact vector of an -residue protein is a list of integers , representing the number of residues in contact with residue . We study analytically (at mean-field level) and numerically the amount of structural information contained in a contact vector. Analytical calculations reveal that a large variance in the contact numbers reduces the degeneracy of the mapping between contact vectors and structures. Exact enumeration for lengths up to on the three dimensional cubic lattice indicates that the growth rate of number of contact vectors as a function of is only less than that for contact maps. In particular, for compact structures we present numerical evidence that, practically, each contact vector corresponds to only a handful of structures. We discuss how this information can be used for better structure prediction.
I. Introduction
The protein folding problem has been the subject of extensive research in the last decade and although much has been learned a satisfactory understanding of the phenomenon has not been reached yet [1, 2, 3]. The physical approach to the problem is to consider the native state of a protein as the ground-state of a Hamiltonian which acts on sequence space and summarizes the inter-residue and residue-solvent interactions [1, 2, 4]. Recently it was shown that there are several cases for which there is no possible choice of pairwise contact interactions between residues that suffices to pin down the native state even for a single protein [5, 6]. This conclusion is supported by molecular dynamics studies [7] and lattice models [8] on residue-solvent interactions, where many-body forces are shown, or can be deduced to be, as relevant as two-body forces. To get around this failure of the two-body Hamiltonian approach while retaining a coarse-grained description (as opposed to, say, an all-atom one, including water [9]), we need to introduce new terms at the residue level, to bias the optimization procedure towards the true minima.
It is widely accepted that hydrophobicity is the force driving the folding process [10]. At the individual residue level, hydrophobicity is correlated with the solvent-exposed surface area in the native state [11]. In addition, as reported below, a statistical analysis of the native structures deposited in the Protein Data Bank (PDB) [12]) reveals a good correlation (coefficient of correlation 0.8) between the solvent-hidden surface area per residue and the number of inter-residue contacts per residue in the native state. We therefore propose the following two-step procedure for predicting the native state of a protein. First, a reasonably accurate prediction of the exposed surface area in the native fold is made on the basis of sequence information [11]. Second, this information is translated into a prediction of the number of native contacts of each residue, e.g. to a predicted native contact vector. Even if this scheme will turn out to be insufficient to perform a successful prediction, it opens the possibility to confine the search for the native fold to a small portion of the conformational space. The question then becomes “How many folded configurations are there, consistent with a given set of contact numbers ?” And, for that matter, “Is such a contact-number representation of the protein structure degenerate at all ?” The rest of this paper addresses these questions.
II. Contact maps versus contact vectors
The contact map (CM) [13] of a protein of amino-acids is a symmetric binary matrix of size , such that when the and the amino-acids of the sequence are neighbors, with some suitable definition of “neighborhoodness” (e.g., a common construct is to threshold the pairwise distance matrix for the C atoms [5]). The CM has proven to be a convenient encoding of the 3-dimensional native fold: 1. The native backbone conformation can be reproduced to within 1.5 Å average uncertainty (the same as most X-ray data) [14], 2. It allows for an efficient search of the configuration space, since large conformational changes can be obtained by minor modifications of the CM [15]. Within such minimalistic framework one hopes to gain new insight to the protein-folding problem since it is amenable to different physical and mathematical tools. For instance, the following Hamiltonian acting on the contact map space has been extensively used in the past [16, 17, 18, 19] :
H=∑ijw(ai,aj)Cij , (1)
where is one of the 210 energy parameters representing the contact energy between the amino-acid types and . Unfortunately, this formulation has limited predictive power. For example, given a large enough set of sequences and decoys (obtained by threading) from the PDB, no set of exists, for which has its ground-states at the native-folds [6]. This is in accordance with the recent studies on the nature of the hydrophobic interaction [7, 8], whose conclusion is that many-body interactions are of the same order of magnitude as two-body interactions.
One possible way to improve the Hamiltonian in Eq. (1) is to include an energy penalty for deviations from the native-contacts:
H=(1−λ)∑ijw(ai,aj)Cij+λ∑i(ni−nnati)2 (2)
where we define a “contact vector” (CV) of rank , which is the sum of the entries of the CM on each row (or column) (see Fig.(1)) :
ni=∑jCij . (3)
Contact vectors have already been studied in the context of protein folding [20, 21, 22, 23, 24, 25, 26, 27, 28]. We note in particular that the second term in Eq. (2) resembles a hydrophobic term introduced previously [29] and studied in Ref. [30], with the difference that there the desired number of contacts of residue is determined by its species. Here instead we assume the knowledge of , the correct number of contacts of residue in the native structure. Hence the second term in Eq. (2) carries the same spirit as the Go model [31]. In this work we are interested in studying the statistical properties of contact vectors. For our more general purpose, it would seem inconsistent to use Eq. (2) to predict the native structure of a protein, as we bias the Hamiltonian towards the minimum by using information which is not accessible to us before we actually solve the problem. However, unlike in the Go model, the information required here about the native state (the number of contacts for each residue) is modest, and, most crucially, can be predicted. Learning algorithms have been recently developed, which are trained on known structures to predict the surface exposure of the amino-acids in the native fold [11, 32]. Since the hydrophobic effect is driving the folding process [10], it is natural to expect that an accurate prediction of the solvent exposed surface of each residue in the folded state may lead to prediction of the correct native structure. To bridge the gap between the exposed-surface information and the CV defined above, we performed an analysis on a representative set of proteins from the PDB database. We found a linear correlation with a coefficient of correlation of 0.8 between the solvent hidden surface area of a residue and the number of amino-acids it is in contact with (see Fig.(2)). Therefore, in future work we expect to replace the term in Eq. (2) by , thereby breaking the causality loop which is a characteristic of Go-like models. Another reason to study the model of Eq. (2) is that a related kind of Hamiltonian has been recently proved to be useful to determine the structure of nearly-native protein conformations [33]. In that study, represent the number of native contacts formed by residue in the contact map . Also in that case, it was found that a large variance in (see below) implies a low degeneracy in mapping between contact vectors and three-dimensional conformations.
In studying Eq. (2), first we tried to use a set of contact energy parameters , found earlier by an optimization process, using Eq. (2) with . This attempt failed to assign the minimal energy to the native state for any choice of . However, an optimization of over the known structures by using the Hamiltonian in Eq. (2) with may, perhaps, successfully identify the native state. We will investigate this possibility in the future.
The Hamiltonian (2), with , fails to identify the native fold. This statement means that it is possible to find conformations which on the one hand are very different from the native one and, on the other, each amino-acid has exactly the correct number of neighbors, that is the same number of neighbors as in the native state. This result was first found by Ejtehadi et al. [22] by exact enumeration of all the compact conformations on a cubic lattice. For actual proteins, an example is given in Fig. 3 in the case of protein CI2 (PDB code 2ci2), where the CMs of the native fold and of another conformation are superimposed. These two conformations have identical CVs. At first glance, it would seem unlikely to find two compact configurations where each residue has exactly the same number of neighboring residues in contact. On the other hand, the cautious reader will attribute this degeneracy to the loss of information (from binary variables to integers of size ) associated with going from a given CM to its corresponding CV via Eq. (3). Quantifying the resulting degeneracy is a non-trivial problem. The next section is an analytical attempt in this direction.
III. An analytical approach
We ask the following question: “How many contact maps exist for a given contact vector ?”. In fact, we should be counting, for a given , only the physical CM that are consistent with it. A physical CM [29, 14] is one for which a perfectly matching chain configuration can be found. There is, however, no known analytical selection rule for the physical CMs among all symmetric and traceless matrices; therefore in our analytic study we will consider all binary symmetric matrices. This is essentially the mean-field treatment of the problem, since in the limit of infinite dimensions, all the constraints on the CM, except being symmetric with zero trace, will be relaxed. For any finite dimension we overestimate the degeneracy - the number of physical CMs scales exponentially, as , whereas the number of possible CMs scales as [13].
The formal expression for the number of symmetric, traceless binary matrices consistent with a given vector, , is
d(→n)=i>j∑{xij} N∏i=1δ(∑jxij),ni . (4)
The sum over represents a trace over all binary matrices, and the constraint ensures symmetry and zero trace. In order to perform the summation, we rewrite the Kronecker as a discrete Fourier sum:
d(→n) = i>j∑{cij} N∏i=1[1NN−1∑k=0ei2πkN(∑jxij−ni)] (5) = 1NNN−1∑k1=0N−1∑k2=0⋯N−1∑kN=0⎛⎜⎝i>j∑{xij}ei2πN∑iki(∑jxij−ni)⎞⎟⎠.
Scaling by , approximate the sums by integrals. Then, evaluate the trace over the matrix elements, paying special attention to and :
d(→n) = ∫10dk1dk2...dkN⎛⎜⎝i>j∑{xij}ei2π∑iki(∑jxij−ni)⎞⎟⎠ (6) = 2N(N−1)/2∫10dk1dk2...dkNe−i2π∑iki[(N−1)/2−ni]∏i>jcos[π(ki+kj)] .
The integral can now be evaluated around its saddle points, and , which contribute equally. After we set and assume is divisible by 4, we obtain
≃ 2N(N−1)/22∫1/2−1/2dq1dq2...dqNe−i2π∑iqi[(N−1)/2−ni]−(π2/2)[N∑iq2i+(∑iqi)2] . (7)
The last square term in the exponent can be eliminated by a Hubbard- transformation after rescaling by and defining :
d(→n) ≃ 2N(N−1)/2√2π∫1/2−1/2dq1dq2...dqN∫∞−∞dye−y2/2+iπy∑iqi+i2π∑iηiqi−Nπ2/2∑iq2i ,
which finally simplifies to yield
d(→n) ≃ 2N2/2(N/π)N/2√Ne−2σ2η−¯η2 , (8)
where and are the average and the standard deviation of . This is a mean-field estimate of how the degeneracy of a CV scales with respect to the statistical properties of the CV. The leading behavior is clearly far from being realistic, since the degeneracy should scale at most as for some ( is in 2d [13] and 1.32 in 3d as calculated here). Eq. (8) further suggests that the maximally degenerate CV with a fixed average number of contacts has , i.e., all the amino-acids have equal number of contacts, whereas an unbiased sample of CVs will be dominated by those vectors with a typical standard deviation of . The mean-field message is that the degeneracy is a decreasing function of , i.e., variation in contact number is desirable for low degeneracy. In the next section, we argue that this is true away from the saddle point as well.
IV. Finite connectivity : Graph counting
In the previous section, we allowed for the number of contacts to take any value between and . In reality, and also in lattice models, the number of contacts is of order unity. Therefore, it is desirable to have an estimate of the degeneracy of such CVs. Once again, we consider all traceless, symmetric, binary matrices. We first observe that every such matrix encodes a unique graph with N vertices, a vertex pair being connected if the corresponding matrix element is 1. Symmetry ensures that the graph is undirected. We can ensure chain connectivity (but not the graph being physical!) by freezing connections on the first off-diagonal; if we choose to relax these “backbone connections”, the remaining graph need not be connected.
The degeneracy of a CV, can then be approximated by the number of graphs with N vertices and given connectivities. We imagine the vertices from 1 to N with corresponding number of legs sticking out of each and we ask in how many ways these legs can be connected such that none will be left out (the total number of legs is an even number). Eq. (9) follows immediately if one imagines connecting pairs of legs sequentially (the numerator) and remembering that legs coming out of the same vertex are interchangeable (denominator).
Let’s assume we allow the entries of the CV to be one of , , and the composition given by , being the number of amino acids with contacts. The average number of contacts is . The corresponding number of graphs reads
d(N,{pi})=(cN−1)!!(0!)N0(1!)N1...(n!)Nn . (9)
(The only difference with the usual Feynman diagram counting is the missing in the denominator: our vertices are distinguishable since they correspond to the amino-acids labelled by their sequence number.)
Note that this expression is an approximation to the number of symmetric traceless CMs, since diagrams with small loops involving one vertex, as well as with more than one line connecting the same two vertices are counted in Eq. (9), even though they do not correspond to any CMs. However, corrections due to excluding such diagrams do not change the scaling with . Applying Stirling’s formula to Eq. (9),
d(N,{pi})≃exp[cN2lnN+N(c2lnc−1−∑pnln(n!))] . (10)
The leading order is now with . Better estimations require taking into consideration the spatial correlations in the contact numbers due to the underlying one-dimensional chain. Our next task is to find the compositions with the minimum and maximum degeneracy. The leading order in Eq. (10) depends only on the total number of contacts, so it is sensible to confine the search into the subspace of CVs with a fixed average connectivity. We then extremize the next order term with respect to , subject to the constraints and to find which distribution of contacts allows for the better “designability” (i.e., less degeneracy). Fig.(4) shows the choice of with maximum/minimum degeneracy obtained numerically, as a function of the average contact number, (maximum number of non-backbone contacts, , is chosen to be 4 as for the cubic lattice). As read from the graphs, the highly degenerate scenario is when the number of contacts for each residue is minimally away from the average, and vice versa for the low degeneracy. Even though here we deal with low connectivity , whereas in the previous section we had , the result obtained here is the same as there - low degeneracy goes in parallel with maximal variation in contact number.
One application of this principle is an order of magnitude estimation for the “optimal” length for a protein. Consider a necklace model of the protein, each residue represented by a sphere of fixed radius, and the necklace itself folded into a large compact sphere, where compactness is imposed as a necessary condition for stability. Then, maximal contact number fluctuation is attained when the number of buried residues equals the number of residues on the surface. From this purely geometric construction, one can estimate an “ideal” chain size: Let the radii of the individual residues be unity, and the radius of the protein be . Assuming hcp-like packing, each residue occupies a volume of , and those on the surface cover, roughly, of surface area. Then, if is the number of residues, we have and , which yields .
V. Numerical results
To compare the analytical findings presented above with numerical simulations, we performed several exact enumeration studies on the square and the cubic lattices. Therefore, in this section, we deal with physical CMs and contact vectors, i.e., those generated by self-avoiding walks in two and three dimensions. In each analysis, we kept a record of the distinct CMs we encountered and the corresponding CVs. Our first observation is that, the number of distinct CVs for a given size scales exponentially with :
Ncv∼eacvN .
For given , the number of CVs, CMs, and self-avoiding walks increase in the given order. Yet, it is interesting that the growth rate is only about less than the corresponding rate for the CMs in three dimensions (see Fig.(5), and also [13]). The discrepancy between the mean-field analytical calculations and the exact enumeration results points to the fact that the finite dimensionality and the correlations between contacts due to the underlying one-dimensional chain (i.e. working with physical CMs and CVs) are crucial.
The almost identical growth rates is in accordance with our next analysis on the compact configurations on a square lattice (see [34]): Considering all the Hamiltonian walks inside a square, we identified the number of walks that correspond to each CV and found that the number of CVs with degeneracy drops more or less exponentially with (see Fig.(6)). In fact, more than of all the contact vectors have degeneracy , although it is possible to find a vector with 69 Hamiltonian walks mapped on it (not shown in Fig.(6)). The degeneracy gets even smaller in the case of a compact but less than perfect packing , in our case when the square is mostly filled with a 32-residue chain: introduced vacancies (especially when in the core) “label” some of the residues with otherwise identical contact number. Rearrangement of the core, where all the residues have identical number of contacts, is the dominant mechanism of degeneracy. Hence, it gets more difficult to find conformations with the same CV, once this degeneracy is lifted by the vacancies. In this case, for practically all the CVs we have . In three dimensions, for the system sizes within reach, practically all the residues are on the surface. Therefore, we have not extended this analysis to such case.
VI. Conclusion
Existing and future prediction methods for the accessible surface area of individual residues can be adopted to predict the number of native contacts of each amino acid of a given protein. This prediction can then be used for an efficient search of the native contact map (and the corresponding conformation) in a dramatically reduced configuration space. The prerequisite of such a program is to be able to identify different folds consistent with a given set of contact numbers for each residue. We investigated at the mean-field level the partition of the configuration space (or rather the contact map space) into degeneracy classes labelled by the CVs. The average degeneracy predicted by the analytical calculations disagrees with the numerical findings, indicating that the finite dimensionality and the correlations induced by the underlying one-dimensional chain are crucial even for a qualitatively satisfactory result. We did find, already at the mean-field level, that the increasing the fluctuations in the native contact-numbers reduces the contact vectors’ degeneracy. This finding is also supported by another analytical calculation, valid in a different regime, where the average contact number is .
We further investigated by exact enumeration the degeneracy spectrum of CVs for self-avoiding walks on the square and the cubic lattice. We found that for compact self-avoiding walks the CM and the CV representations carry nearly the same amount of information. This is an encouraging result, for an accurate enough prediction of solvent exposed surface areas in the native state may then be used to reduce the search space sufficiently, so that within the limited set of remaining candidate CMs a simple pairwise interaction potential may suffice to single out the native fold of the protein. In addition, we performed exact enumeration over all SAWs of steps in three dimension, and found that the number of CVs grows exponentially with the protein length, with a prefactor only a few percent smaller than that for the CMs. The slow exponential growth of the average degeneracy of the CVs is largely overestimated by our mean-field calculations. Further analytical and numerical research is certainly called for. We also observed that for compact configurations, CV CM mapping is practically one-to-few. The Hamiltonian in Eq. (1), therefore, may still be promising if the pairwise interactions are optimized within the context of a (even roughly) predicted CV.
A. K. acknowledges many useful discussions with G. Getz and A. Punnose and is also grateful to the Bilkent University Physics Department for their hospitality during his visit. This work was partially supported by grants from the US-Israel Binational Science Foundation (BSF) and the Minerva Foundation. I. Kanter thanks the Einstein Center for Theoretical Physics for partial support.
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8937209844589233, "perplexity": 563.4028896935762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1614178381989.92/warc/CC-MAIN-20210308052217-20210308082217-00462.warc.gz"} |
http://mathoverflow.net/revisions/37381/list | 5 added 46 characters in body
Let me start with a question for which I know the answer. Consider a symmetric integral $n\times n$ matrix $A=(a_{ij})$ such that $a_{ii}=2$, and for $i\ne j$ one has $a_{ij}=0$ or $-1$. One can encode such a matrix by a graph $G$: it has $n$ vertices, and two vertices $i,j$ are connected by an edge iff $a_{ij}=-1$.
Question 1: Which graphs correspond to positive definite $A$?
Answer 1: (Classical, well-known, and easy) $G$ is a disjoint union of $A_n$, $D_n$, $E_n$. (http://en.wikipedia.org/wiki/Root_system)
Now let me put a little twist to this. Let us also allow $a_{ij}=+1$ for $i\ne j$, and encode this by putting a broken edge between $i$ and $j$ (or an edge of different color, if you prefer).
Real question: Which of these graphs correspond to positive definite $A$?
Let me add some partial considerations which do not quite go far enough. (Some of these were in the helpful Gjergji's answer.)
(1) Consider the set $R$ of shortest vectors in $\mathbb Z^n$; they have square 2. Reflections in elements $r\in R$ send $R$ to itself, and $R$ spans $\mathbb Z^n$ since it contains the standard basis vectors $e_i$. By the standard result about root lattices, $\mathbb Z^n$ is then a direct sum of the $A_n$, $D_n$, $E_n$ root lattices, and one can restrict to the case of a single direct summand.
Hence, the question equivalent to the following: what are the graphs corresponding to all possible bases of $\mathbb R^n$ in which the basis vectors are roots?
The case of $R=A_n$ is relatively easy. The roots are of the form $f_a-f_b$, with $a,b \in (1,\dots,n+1)$. Every collection $e_i$ corresponds to an oriented spanning tree on the set $(1,\dots,n+1)$. The 2-colored graph is computed from that tree. I don't see a clean description of the graphs obtained via this procedure, but it is something.
For $D_n$, similarly, a basis is described by an auxiliary connected graph $S$ on $n$ vertices with $n$ edges whose ends are labeled $+$ or $-$. The graph $G$ is computed from $S$.
And for $E_6,E_7,E_8$ there are of course only finitely many cases, but for me the emphasis is on MANY, very many.
So has anybody done this? Is there a table in some paper or book which contains all the 2-colored graphs obtained this way, or -- better still -- a clean characterization of such graphs?
(2) There is a notion of "weakly positive quadratic forms" used in the cluster algebra theory (see for example the first pages of LNM 1099 (Tame Algebras and Integral Quadratic Forms) by Ringel. And there is some kind of classification theory for them. Maybe I am mistaken, but this seems to be quite different: a quadratic form $q$ is "weakly positive" if $q\ge 0$ on the first quadrant $\mathbb Z_{\ge0}^n$. So there is no direct relation to my question, it seems.
4 added 145 characters in body; edited tags; added 10 characters in body; added 9 characters in body
Let me add some partial considerations which do not quite go far enough. (Some of these were in the helpful Gjergji's answer, which I appreciated, which he unfortunately removed.answer.)
(1) Clearly, Consider the standard basis set $R$ shortest vectors in $e_i$ are roots (\mathbb Z^n$; they have square 2)2. Reflections in elements$r\in R$send$R$to itself, and generate the lattice$R$spans$\mathbb Z^n$. since it contains the standard basis vectors$e_i$. By the standard result about root lattices,$\mathbb Z^n$is then a direct sum of the$A_n$,$D_n$,$E_n$root lattices, and one can restrict to the case of a single direct summand. Then$e_i$'s are some of Hence, the roots in$R$, and we askquestion equivalent to the following: what are the collections graphs corresponding to all possible bases of such roots spanning the vector space, and with the property that the pairwise dot products$(e_i,e_j)$\mathbb R^n$ in which the basis vectors are 0,-1,+1. roots?
The case of $R=A_n$ is relatively easy. The roots are of the form $f_a-f_b$, with $a,b \in (1,\dots,n+1)$. Every collection $e_i$ corresponds to a an oriented spanning tree on the set $(1,\dots,n+1)$. The 2-colored graph is computed from that tree. I don't see a clean description of the graphs obtained via this procedure, but it is something.
For $D_n$, the procedure may be similar but messier. similarly, a basis is described by an auxiliary connected graph $S$ on $n$ vertices with $n$ edges whose ends are labeled $+$ or $-$. The graph $G$ is computed from $S$.
(2) There is a notion of "weakly positive quadratic forms" used in the cluster algebra theory (see for example the first pages of LMN1099 LNM 1099 (Tame Algebras and Integral Quadratic Forms) by Ringel. And there is some kind of classification theory for them. Maybe I am mistaken, but this seems to be quite different: a quadratic form $q$ is "weakly positive" if $q\ge 0$ on the first quadrant $\mathbb Z_{\ge0}^n$. So there is no direct relation to my question, it seems.
3 added 1850 characters in body
Let me add some partial considerations which do not quite go far enough. (Some of these were in Gjergji's answer, which I appreciated, which he unfortunately removed.)
(1) Clearly, the standard basis vectors $e_i$ are roots (they have square 2), and generate the lattice $\mathbb Z^n$. By the standard result about root lattices, $\mathbb Z^n$ is then a direct sum of the $A_n$, $D_n$, $E_n$ root lattices, and one can restrict to the case of a single direct summand.
Then $e_i$'s are some of the roots in $R$, and we ask: what are the collections of such roots spanning the vector space, and with the property that the pairwise dot products $(e_i,e_j)$ are 0,-1,+1.
The case of $R=A_n$ is relatively easy. The roots are of the form $f_a-f_b$, with $a,b \in (1,\dots,n+1)$. Every collection $e_i$ corresponds to a spanning tree on the set $(1,\dots,n+1)$. The 2-colored graph is computed from that tree. I don't see a clean description of the graphs obtained via this procedure, but it is something.
For $D_n$, the procedure may be similar but messier. And for $E_6,E_7,E_8$ there are of course only finitely many cases, but for me the emphasis is on MANY, very many.
So has anybody done this? Is there a table in some paper or book which contains all the 2-colored graphs obtained this way, or -- better still -- a clean characterization of such graphs?
(2) There is a notion of "weakly positive quadratic forms" used in the cluster theory (see for example the first pages of LMN1099 (Tame Algebras and Integral Quadratic Forms) by Ringel. And there is some kind of classification theory for them. Maybe I am mistaken, but this seems to be quite different: a quadratic form $q$ is "weakly positive" if $q\ge 0$ on the first quadrant $\mathbb Z_{\ge0}^n$. So there is no direct relation to my question, it seems.
2 edited tags; edited title; deleted 198 characters in body
1 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9401609897613525, "perplexity": 246.52181838487033}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368697420704/warc/CC-MAIN-20130516094340-00041-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/121977/residue-of-a-pole-of-order-6 | # Residue of a pole of order 6
I am in the process of computing an integral using the Cauchy residue theorem, and I am having a hard time computing the residue of a pole of high order.
Concretely, how would one compute the residue of the function $$f(z)=\frac{(z^6+1)^2}{az^6(z-a)(z-\frac{1}{a})}$$ at $z=0$?
Although it is not needed here, $a$ is a complex number with $|a|<1$.
Thanks in advance for any insight.
-
Can you compute the Taylor series at $0$? – Mariano Suárez-Alvarez Mar 19 '12 at 6:50
You can write $$f(z) = \frac{1}{az^6} (z^{12} + 2z^6 + 1) \left(\sum_{k=0}^\infty \frac1{a^k} z^k \right) \left( \sum_{k=0}^\infty a^k z^k \right).$$ You want to extract the coefficient of $z^5$ in the product of the two series.
$$g(z)=\frac{1}{(z-a)(z-\frac{1}{a})}=\frac{\frac{1}{a-\frac{1}{a}}}{z-a}+\frac{\frac{-1}{a-\frac{1}{a}}}{z-\frac{1}{a}}$$ we know: $$(a+b)^n =a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+...+b^n$$ $$\text{As regards }: |a|<1$$ Taylor series of f(z) is: $$g(z)=\frac{\frac{1}{a-\frac{1}{a}}}{z-a}+\frac{\frac{1}{a-\frac{1}{a}}}{\frac{1}{a}-z}=(\frac{1}{a-\frac{1}{a}}) \left[ \frac{-\frac{1}{a}}{1-\frac{z}{a}}+\frac{a}{1-az} \right]$$ $$g(z)=(\frac{1}{a-\frac{1}{a}}) \left[ \frac{-1}{a} \sum_{n=0}^{\infty}(\frac{z}{a})^n+a \sum_{n=0}^{\infty} (az)^n \right]$$ $$f(z)=\frac{(z^6+1)^2}{az^6}g(z)=\frac{z^{12}+2z^2+1}{az^6}g(z)=\left( \frac{z^6}{a} + \frac{2}{az^4} + \frac{1}{az^6} \right)g(z)$$ $$f(z)= \left( \frac{z^6}{a} + \frac{2}{az^4} + \frac{1}{az^6} \right) \left(\frac{1}{a-\frac{1}{a}}\right) \left[ \frac{-1}{a} \sum_{n=0}^{\infty}(\frac{z}{a})^n+a \sum_{n=0}^{\infty} (az)^n \right]$$ $$\text{ so residue is coefficient of term }z^{-1}$$ $$f(z)=\frac{1}{a(a-\frac{1}{a})} \left[ \frac{-1}{a}\left( \sum_{n=0}^{\infty}\frac{z^{n+6}}{a^n} +2\sum_{n=0}^{\infty}\frac{z^{n-4}}{a^n} +\sum_{n=0}^{\infty} \frac{z^{n-6}}{a^n}\right) +a \left( \sum_{n=0}^{\infty} a^nz^{n+6}+2\sum_{n=0}^{\infty} a^nz^{n-4} +\sum_{n=0}^{\infty} a^nz^{n-6} \right) \right]$$
$$\text{residue of function at z=0 is :}$$ $$\frac{1}{a(a-\frac{1}{a})} \left[ \frac{-1}{a}\left( 0 +2\frac{1}{a^3} +\frac{1}{a^5}\right) +a \left( 0+2a^3 +a^5 \right) \right]$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9925040006637573, "perplexity": 103.90719329299023}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042988511.77/warc/CC-MAIN-20150728002308-00254-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/connection-and-tensor-issue-with-the-proof.715032/ | Connection and tensor-issue with the proof
1. Oct 7, 2013
1. The problem statement, all variables and given/known data
I am tying to prove the following:
$\Gamma^{a}_{bc}$ T$^{bc}$ =0
2. Relevant equations
3. The attempt at a solution
I approached this problem as follows:
$dx_{b}/dx^{c} * e^{a} (e^{b} . e^{c})$ but it did not yield anything.
Then I expanded the christoeffel symbols into g s and again I am not sure what to do next.So any hints please
2. Oct 7, 2013
WannabeNewton
Your post is incomplete to say the least. What is $\tau^{bc}$ to start with?
3. Oct 7, 2013
That's a tensor.
4. Oct 7, 2013
a symmetric tensor.
5. Oct 7, 2013
George Jones
Staff Emeritus
In a coordinate basis? A non-holonomic basis?
6. Oct 7, 2013
WannabeNewton
Again you are not being specific enough. $\Gamma^{a}_{bc}\tau^{bc} = 0$ is certainly not true in general for any symmetric tensor $\tau^{bc}$ if $\Gamma^{a}_{bc}$ are the coefficients of the Levi-Civita connection. It is only true in general if $\tau^{bc}$ is antisymmetric so you must specify what exactly the tensor $\tau^{bc}$ is.
7. Oct 7, 2013
Γabc are the christoffel symbols/connection and T^(bc) = (e^b,e^c)
8. Oct 7, 2013
George Jones
Staff Emeritus
Do you mean $T^{bc} = T \left( e^b , e^c \right)$?
9. Oct 7, 2013
yes.
10. Oct 7, 2013
George Jones
Staff Emeritus
And $\left\{ e_a \right\}$ is an orthonormal basis?
11. Oct 7, 2013
Yes, it is,
12. Oct 7, 2013
George Jones
Staff Emeritus
First, In don't think that you should write $\Gamma^a{}_{bc}$ instead of $\Gamma^a_{bc}$.
Second, what anti-symmetry property does the Levi-Civita connection have with respect to an orthonormal basis? | {"extraction_info": {"found_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": 0.8860675692558289, "perplexity": 2499.4809243758914}, "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-2018-09/segments/1518891815318.53/warc/CC-MAIN-20180224033332-20180224053332-00065.warc.gz"} |
http://math.stackexchange.com/questions/68154/numerical-software-to-solve-partial-differential-equations-in-spherical-coordina | # Numerical software to solve partial differential equations in spherical coordinates?
Which numerical libraries / math software can allow me to solve partial differential equations in spherical coordinates? (my system consists of N degrees of freedom, each degree lives in $\mathbb{C}^{2}$ and the whole system is $(\mathbb{C}^{2})^{N}$).
- | {"extraction_info": {"found_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": 0.9740782976150513, "perplexity": 354.9011245233093}, "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-2016-22/segments/1464051113990.53/warc/CC-MAIN-20160524005153-00146-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.sarthaks.com/1518738/long-capillary-radius-placed-vertically-inside-beaker-water-surface-tension-water-2xx10 | # A long capillary tube of radius 0.2 mm is placed vertically inside a beaker of water. If the surface tension of water is 7.2xx10^(-2)N//m the angl
140 views
in Physics
closed
A long capillary tube of radius 0.2 mm is placed vertically inside a beaker of water.
If the surface tension of water is 7.2xx10^(-2)N//m the angle of contact between glass and water is zero, then determine the height of the water column in the tube.
A. 3cm
B. 9cm
C. 7cm
D. 5cm
by (87.1k points)
selected
Here s=7.0x10^(-2) N//m,r=0.2xx10^(-3)m,r=10^(3)kg//m^(3)
we know that the hieght of liquid in the tube is
h=2sigmacostheta//rhoRg
where R is the radius of the menisus, and theta is the angle of contact.
Since theta=0 (given), radius of the meniscus is equal to the radius of the capillary tube i.e. R=r
:. h=(2(7.0xx10^(-2)))/((10^(3))(0.2xx10^(-3))(10))=0.07m
or h=7.0cm
When the length of the capillary tube above the free surface of the liquid is less than the height of liquid that rise in the tube, radius R of the free surface is not equal to the radius of the tube. It is greater than r as the surface tennis to be flatter.
According to the equation p_(1)-p_(2)=(4sigma//R), the pressure difference across te surface is given by
/_p=(2sigmacostheta)/R
If p_(1) and p_(2) are the pressure just above and below the mensisus, respectively then p_(1)-p_(2)=rhogh_(0)
:. rhogh_(0)=(2sigma)/r...........i
In part a we have seen than when h_(0)=h,theta=0, R=r, and rhogh=(2sigma)/r.......ii
Therefore dividing eqn i by eqn ii we have r/R=(h_(0))/h
From the figure, it is clear that costheta=r/R
Therefore the angle of contact is
theta=(cos^(-1)h_(0))/h=cos^(-1)(5/7)~~44^@ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9421091675758362, "perplexity": 1036.0657686572497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950383.8/warc/CC-MAIN-20230402043600-20230402073600-00798.warc.gz"} |
https://cs.stackexchange.com/questions/13286/interactive-proofs-for-conp | # Interactive Proofs for coNP
I am trying to understand interactive proof systems and tried the following problem as an exercise. We know that $PH \subseteq PSPACE$ and $IP=PSPACE$, so come up with (easy to understand) interactive proof systems for $PH$?
An interactive proof system for $NP$ is trivial, but I failed to get an interactive proof system even for $coNP$. Do you know of an explicit interactive proof system (by explicit I mean without going through the $IP=PSPACE$ route) for $coNP$?
• Could you clarify what you mean by interactive proof system? For those who aren't familiar with the term. – jmite Jul 15 '13 at 17:28
• Even the inclusion $coNP \subseteq IP$ requires nonrelativizing techniques; the only known way to show it is via algebrization, as in Yuval's answer. Showing $IP=PSPACE$ is merely a slight technical modification of this proof. – sdcvvc Jul 16 '13 at 14:09
• @sdcvvc, I think your comment worths being posted as an answer. It explains why there are not any examples as simple as those for NP. – Kaveh Jul 17 '13 at 8:05
Wikipedia outlines such an example. Consider the coNP-complete problem UNSAT: given a CNF $\varphi$ on $n$ variables, we want to convince the verifier that $\varphi$ is not satisfiable. We arithmetize $\varphi$ to a polynomial $p$ and choose some large prime $q$. Let $$p(x_1,\ldots,x_k) = \sum_{x_{k+1}=0}^1 \cdots \sum_{x_n=0}^1 p(x_1,\ldots,x_n).$$ The protocol proceeds as follows:
1. Prover sends verifier a prime $q \in (2^n,2^{n+1})$, and the latter verifies that $q$ is prime.
2. Prover sends verifier $p(z) \in \mathbb{Z}_q[z]$. Verifier verifies that $p(0) + p(1) = 0$, and sends prover a random $r_1$.
3. Prover sends verifier $p(r_1,z) \in \mathbb{Z}_q[z]$. Verifier verifies that $p(r_1,0) + p(r_1,1) = p(r_1)$, and sends prover a random $r_2$.
4. Eventually, verifier gets $p(r_1,\ldots,r_n) \in \mathbb{Z}_q$, and verifies that it has the correct value by evaluating $p$ directly.
Because the degree of $p$ is small compared to $q$, if the prover is cheating then the verifier will probably catch her (see Wikipedia for the proof, or work it out yourself using the Schwartz-Zippel lemma).
Graph non-isomorphism at Proofs that Yield Nothing But their Validity or All Languages in NP have Zero-Knowledge Proofs, Goldreich, Micali and Wigderson, JACM, 1991.
Common input is a pair of graphs: $G_1, G_2$. At the start of each round, verifying party chooses an index $i \in \{1,2\}$ at random and sends a random permutation of graph $G_i$. Proving party responds with an index $b \in \{1,2\}$.
Completeness property: for non-isomorphic graphs, prover always give correct response $b=i$.
Soundness: for isomorphic graphs, prover give correct response with probability $\frac{1}{2}$.
• Please give a proper reference to a peer-reviewed article and a short summary of the content. Links like the one you provide tend to break, and then your answer contains zero information. – Raphael Jan 31 '15 at 9:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8555546998977661, "perplexity": 674.8381316169259}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655933254.67/warc/CC-MAIN-20200711130351-20200711160351-00403.warc.gz"} |
http://demonstrations.wolfram.com/GeodesicsInTheMorrisThorneWormholeSpacetime/ | # Geodesics in the Morris-Thorne Wormhole Spacetime
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.
Requires a Wolfram Notebook System
Edit on desktop, mobile and cloud with any Wolfram Language product.
The simplest wormhole geometry is given by the line element , see [2]. The parameter defines the size of the throat of the wormhole, and represents the proper length radius.
[more]
Light rays and objects in free motion in four-dimensional spacetimes follow lightlike or timelike geodesics. In general, these geodesics must be computed numerically. However, in the Ellis wormhole spacetime, there is an analytic solution of the geodesic equation in terms of elliptic integral functions. Because of the spherical symmetry and staticity of the metric, it suffices to consider geodesics in the hypersurface . This two-dimensional surface can be embedded in the three-dimensional Euclidean space. The corresponding embedding function reads with .
In this application, you can change the throat size , the initial position of the observer, and the initial angle of the geodesic with respect to the local reference frame of the observer.
[less]
Contributed by: Thomas Müller (July 2010)
Open content licensed under CC BY-NC-SA
## Details
A detailed discussion about analytic geodesics in the Morris–Thorne wormhole spacetime can be found in [1].
The metric described in [2] was first mentioned in [3]. Hence, it should be called Ellis wormhole instead. See also the apology in [4], Ref. 14.
References
[1] T. Müller, "Exact Geometric Optics in a Morris–Thorne Wormhole Spacetime," Physical Review D, 77(4) 2008. doi: 10.1103/PhysRevD.77.044043.
[2] M. S. Morris and K. S. Thorne, "Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity," American Journal of Physics, 56(5), 1988 pp. 395–412.
[3] H. G. Ellis, "Ether Flow through a Drainhole: A Particle Model in General Relativity," Journal of Mathematical Physics, 14, 1973 pp. 104–118; 1974 Errata: 15, p. 520.
[4] O. James, E. von Tunzelmann, P. Franklin, and K. S. Thorne, "Visualizing Interstellar's Wormhole," American Journal of Physics 83, 2015 pp. 483–499.
## Permanent Citation
Thomas Müller
Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send | {"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": 0.8124597072601318, "perplexity": 1671.6002410695764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376824180.12/warc/CC-MAIN-20181212225044-20181213010544-00102.warc.gz"} |
http://planetmath.org/KantorovitchsTheorem | # Kantorovitch’s theorem
Let $\mathbf{a}_{0}$ be a point in $\mathbb{R}^{n},U$ an open neighborhood of $\mathbf{a}_{0}$ in $\mathbb{R}^{n}$ and $\mathbf{f}\colon U\rightarrow\mathbb{R}^{n}$ a differentiable mapping, with its derivative $[\mathbf{D}\mathbf{f}(\mathbf{a}_{0})]$ invertible. Define
$\mathbf{h}_{0}=-[\mathbf{D}\mathbf{f}(\mathbf{a}_{0})]^{-1}\mathbf{f}(\mathbf{% a}_{0})\>,\>\mathbf{a}_{1}=\mathbf{a}_{0}+\mathbf{h}_{0}\>,\>U_{0}=\{\mathbf{x% }|\>|\mathbf{x}-\mathbf{a}_{1}|\leq|\mathbf{h}_{0}|\}.$
If $U_{0}\subset U$ and the derivative $[\mathbf{D}\mathbf{f}(\mathbf{x})]$ satisfies the http://planetmath.org/node/765Lipschitz condition
$|[\mathbf{D}\mathbf{f}(\mathbf{u}_{1})]-[\mathbf{D}\mathbf{f}(\mathbf{u}_{2})]% |\leq M|\mathbf{u}_{1}-\mathbf{u}_{2}|$
for all points $\mathbf{u}_{1},\mathbf{u}_{2}\in U_{0}$, and if the inequality
$\left|\mathbf{f}(\mathbf{a_{0}})\right|\left|[\mathbf{D}\mathbf{f}(\mathbf{a_{% 0}})]^{-1}\right|^{2}M\leq\frac{1}{2}$
is satisfied, the equation $\mathbf{f}(\mathbf{x})=\mathbf{0}$ has a unique solution in $U_{0}$, and Newton’s method with initial guess $\mathbf{a}_{0}$ converges to it. If we replace $\leq$ with $<$, then it can be shown that Newton’s method http://planetmath.org/node/793superconverges! If you want an even stronger version, one can replace $|...|$ with the norm $||...||$.
## Logic behind the theorem:
Let’s look at the useful part of the theorem:
$\left|\mathbf{f}(\mathbf{a_{0}})\right|\left|[\mathbf{D}\mathbf{f}(\mathbf{a_{% 0}})]^{-1}\right|^{2}M\leq\frac{1}{2}.$
It is a product of three distinct properties of your function such that the product is less than or equal to a certain number, or bound. If we call the product $R$, then it says that $\mathbf{a}_{0}$ must be within a ball of radius $R$. It also says that the solution $\mathbf{x}$ is within this same ball. How was this ball defined?
The first term, $|\mathbf{f}(\mathbf{a_{0}})|$, is a measure of how far the function is from the domain; in the Cartesian plane, it would be how far the function is from the x-axis. Of course, if we’re solving for $\mathbf{f}(\mathbf{x})=\mathbf{0}$, we want this value to be small, because it means we’re closer to the axis. However a function can be annoyingly close to the axis, and yet just happily curve away from the axis. Thus we need more.
The second term, $|[\mathbf{D}\mathbf{f}(\mathbf{a_{0}})]^{-1}|^{2}$ is a little more difficult. This is obviously a measure of how fast the function is changing with respect to the domain (x-axis in the plane). The larger the derivative, the faster it’s approaching wherever it’s going (hopefully the axis). Thus, we take the inverse of it, since we want this product to be less than a number. Why it’s squared though, is because it is the denominator where a product of two terms of like units is the numerator. Thus to conserve units with the numerator, it is multiplied by itself. Combined with the first term, this also seems to be enough, but what if the derivative changes sharply, but it changes the wrong way?
The third term is the Lipschitz ratio $M$. This measures sharp changes in the first derivative, so we can be sure that if this is small, that the function won’t try to curve away from our goal on us too sharply.
By the way, the number $\frac{1}{2}$ is unitless, so all the units on the left side cancel. Checking units is essential in applications, such as physics and engineering, where Newton’s method is used.
Title Kantorovitch’s theorem KantorovitchsTheorem 2013-03-22 11:58:09 2013-03-22 11:58:09 stevecheng (10074) stevecheng (10074) 25 stevecheng (10074) Theorem msc 49K10 Kantorovitch inequality LipschitzCondition NewtonsMethod Superconvergence | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 29, "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": 0.9636784791946411, "perplexity": 255.77947946478164}, "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-2018-13/segments/1521257645824.5/warc/CC-MAIN-20180318145821-20180318165821-00167.warc.gz"} |
http://www.researchgate.net/publication/45918625_Less_space_for_a_new_family_of_fermions | Article
# Less space for a new family of fermions
(Impact Factor: 4.86). 05/2010; DOI: 10.1103/PhysRevD.82.095006
Source: arXiv
ABSTRACT We investigate the experimentally allowed parameter space of an extension of the standard model (SM3) by one additional family of fermions. Therefore we extend our previous study of the CKM like mixing constraints of a fourth generation of quarks. In addition to the bounds from tree-level determinations of the 3$\times$3 CKM elements and FCNC processes ($K$-, $D$-, $B_d$-, $B_s$-mixing and the decay $b \to s \gamma$) we also investigate the electroweak $S$, $T$, $U$ parameters, the angle $\gamma$ of the unitarity triangle and the rare decay $B_s \to \mu^+ \mu^-$. Moreover we improve our treatment of the QCD corrections compared to our previous analysis. We also take leptonic contributions into account, but we neglect the mixing among leptons. As a result we find that typically small mixing with the fourth family is favored, but still some sizeable deviations from the SM3 results are not yet excluded. The minimal possible value of $V_{tb}$ is 0.93. Also very large CP-violating effects in $B_s$ mixing seem to be impossible within an extension of the SM3 that consists of an additional fermion family alone. We find a delicate interplay of electroweak and flavor observables, which strongly suggests that a separate treatment of the two sectors is not feasible. In particular we show that the inclusion of the full CKM dependence of the $S$ and $T$ parameters in principle allows the existence of a degenerate fourth generation of quarks. Comment: 38 pages, 26 figures; references added
### Full-text
Available from: Alexander Lenz, Apr 26, 2015
0 Followers
·
86 Views
• Source
##### Article: On new physics in $\Delta \Gamma_d$
[Hide abstract]
ABSTRACT: Motivated by the recent measurement of the dimuon asymmetry by the D{\O} collaboration, which could be interpreted as an enhanced decay rate difference in the neutral $B_d$-meson system, we investigate the possible size of new-physics contributions to $\Delta \Gamma_d$. In particular, we perform model-independent studies of non-standard effects associated to the dimension-six current-current operators $(\bar{d} p)(\bar p^{\hspace{0.25mm}\prime} b)$ with $p,p^\prime= u,c$ as well as $(\bar{d}b) (\bar\tau\tau)$. In both cases we find that for certain flavour or Lorentz structures of the operators sizable deviations of $\Delta \Gamma_d$ away from the Standard Model expectation cannot be excluded in a model-independent fashion.
Journal of High Energy Physics 04/2014; 06(2014):040. DOI:10.1007/JHEP06(2014)040 · 5.62 Impact Factor
• Source
• Source
##### Article: Search for pair produced fourth-generation up-type quarks in $pp$ collisions at $sqrts=7$ TeV with a lepton in the final state
[Hide abstract]
ABSTRACT: The results of a search for the pair production of a fourth-generation up-type quark (t') in proton-proton collisions at sqrt(s) = 7 TeV are presented, using data corresponding to an integrated luminosity of about 5.0 inverse femtobarns collected by the Compact Muon Solenoid experiment at the LHC. The t' quark is assumed to decay exclusively to a W boson and a b quark. Events with a single isolated electron or muon, missing transverse momentum, and at least four hadronic jets, of which at least one must be identified as a b jet, are selected. No significant excess of events over standard model expectations is observed. Upper limits for the t' anti-t' production cross section at 95% confidence level are set as a function of t' mass, and t'-quark production for masses below 570 GeV is excluded. The search is equally sensitive to nonchiral heavy quarks decaying to Wb. In this case, the results can be interpreted as upper limits on the production cross section times the branching fraction to Wb. | {"extraction_info": {"found_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": 0.9768912196159363, "perplexity": 970.2051809562627}, "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-2015-18/segments/1430457788926.49/warc/CC-MAIN-20150501052308-00060-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://crypto.stackexchange.com/questions/1422/demonstrating-the-insecurity-of-an-rsa-signature-encoding-scheme?answertab=votes | # Demonstrating the insecurity of an RSA signature encoding scheme
I'm working on problem 12.4 from Katz-Lindell. The problem is as follows:
Given a public encoding function $\newcommand{\enc}{\operatorname{enc}}\enc$ and a textbook RSA signature scheme where signing occurs by finding $\enc(m)$ and raising it to the private key $d \bmod N$, how can we demonstrate the scheme's insecurity for $\enc(m) = 0||m||0^{L/10}$, where $L = |N|$ and $|m| = 9L/10 - 1$ and m is not the message of all zeroes?
-
Okay, and where do you need help here? – Paŭlo Ebermann Dec 9 '11 at 21:52
I need to know how to find a forgery on an m not in Q, where Q is the set of queries to the adversary's signing oracle. Is there something dead simple that I'm missing about this? – pg1989 Dec 9 '11 at 21:56
Have a look at the corresponding verifying scheme. Can you find a number, which, when taken to the power $e$ (the public exponent), gives something in this encoding? (This depends on the public key, but assume it it something like $3$.) – Paŭlo Ebermann Dec 9 '11 at 22:08
Well, here's a hint: remember that for textbook RSA $enc(X) \cdot enc(Y) = enc( X \cdot Y) \mod N$ -- how can we find two messages $X$ and $Y$ such that $X \cdot Y \mod N$ is also a valid message? – poncho Dec 9 '11 at 22:10
@pg1989: Presumably that should go "and $m$ is not ...". $\;\;$ – Ricky Demer Dec 9 '11 at 22:20
An RSA signature scheme with public key $(n,e)$, private exponent $d$, and encoding function $enc$ (including but not limited to the question's $enc$), signs message $m$ as $$Sign(m) = enc(m)^d\bmod n$$
Such scheme is insecure if an adversary can figure out $k>0$ distinct messages $m_i$, and integers $u_i$, $r$, $s$ verifying $$s^e \cdot enc(m_0) \cdot \prod_{0\lt i\lt k} enc(m_i)^{u_i} \equiv r^e \pmod n$$ because this implies (by raising to the power $d$) $$Sign(m_0) \equiv r \cdot s^{-1} \cdot\prod_{0\lt i\lt k}Sign(m_i)^{-u_i} \pmod n$$ which allows computing the signature of $m_0$ (if $k\gt 1$, it is also necessary that the attacker obtain the signatures of the other messages $m_i$; that becomes an existential forgery, or chosen-message attack). Although dated, Jean-Francois Misarsky's How (Not) to Design RSA Signature Schemes is an interesting and relatively easy reading on that topic.
In fact, every known attack on an RSA signature scheme is either of the above kind (with more or less involved computations to exhibit $m_i$, $u_i$, $r$, $s$); or amounts to factorization of $n$ (which includes anything recovering $d$, perhaps by side-channel attack); or is some implementation error, perhaps widespread.
In order to mount an attack of the above kind, a relation of the form $enc(m_0)=r^e$ is ideal. It gives the signature of $m_0$ without any consideration on $n$ or known signature. When $e$ is 3, 5 or 7, this can be done with the encoding $enc$ in the question, by considering $r=2^t$ for some appropriate $t$, and extended to $r=v\cdot2^t$ for some small $v$.
Similarly, $enc(m_0) = r^e\cdot enc(m_1)$ gives the signature of one message from the signature of the other, without any consideration on $n$. This can be done with the encoding $enc$ in the question, for a wider choice of $e$.
Similarly, $enc(m_0) \cdot enc(m_1) = enc(m_2) \cdot enc(m_3)$ gives the signature of one message from the signature of the other three, for any public key $(n,e)$. With the encoding $enc$ in the question, there is ample choice (the equation simplifies to $m_0\cdot m_1=m_2\cdot m_3$, and all messages which left bit is 0 or which integer representation is composite are vulnerable). The ISO/IEC 9796:1991 signature encoding scheme (section 11.3.5 of the Handbook of Applied Cryptography), now withdrawn, turned out to be vulnerable to that, of course if the adversary can obtain the signature of three chosen messages, and is content with the signature of the fourth.
Even the hash-based ISO/IEC 9796-2:1997 (now known as ISO/IEC 9796-2:2010 scheme 1), still in wide use, is vulnerable if the adversary can obtain the signature of many weird chosen messages and is content with the signature of another, which fortunately is seldom the case in practice.
Some require $e>2^{16}$ (FIPS 186-3 appendix B3.1, RGS Annex B1 section 2.2.1.1, and I have seen suggestions for much wider random $e$), because some attacks on weak encoding schemes or implementations of RSA signature/encryption have been easiest for $e=3$ or other small $e$, as is the case for the scheme in the question. I will not condone a course of action that will lead us to loose the main appeal of RSA (or Rabin) signature schemes: fast and simple verification with modest hardware.
- | {"extraction_info": {"found_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": 0.9672331213951111, "perplexity": 801.0910166460479}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510267876.49/warc/CC-MAIN-20140728011747-00328-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://mathmaine.com/2010/04/01/sigma-and-pi-notation/?replytocom=1507 | # Sigma and Pi Notation (Summation and Product Notation)
### Sigma (Summation) Notation
The Sigma symbol, $\sum$, is a capital letter in the Greek alphabet. It corresponds to “S” in our alphabet, and is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum).
The Sigma symbol can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:
$\displaystyle\sum a_i~\\*\\*=~a_1+a_2+a_3+...$
But this is not something that can be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term.
A more typical use of Sigma notation will include an integer below the Sigma (the “starting term number”), and an integer above the Sigma (the “ending term number”). In the example below, the exact starting and ending numbers don’t matter much since we are being asked to add the same value, two, repeatedly. All that matters in this case is the difference between the starting and ending term numbers… that will determine how many twos we are being asked to add, one two for each term number.
$\displaystyle\sum_{1}^{5}2~\\*\\*=~2+2+2+2+2$
Sigma notation, or as it is also called, summation notation is not usually worth the extra ink to describe simple sums such as the one above… multiplication could do that more simply.
Sigma notation is most useful when the “term number” can be used in some way to calculate each term. To facilitate this, a variable is usually listed below the Sigma with an equal sign between it and the starting term number. If this variable appears in the expression being summed, then the current term number should be substituted for the variable:
$\displaystyle\sum_{i=1}^{5}i~\\*\\*=~1+2+3+4+5$
Note that it is possible to have a variable below the Sigma, but never use it. In such cases, just as in the example that resulted in a bunch of twos above, the term being added never changes:
$\displaystyle\sum_{n=1}^{5}x~\\*\\*=~x+x+x+x+x$
The “starting term number” need not be 1. It can be any value, including 0. For example:
$\displaystyle\sum_{k=3}^{7}k~\\*\\*=~3+4+5+6+7$
That covers what you need to know to begin working with Sigma notation. However, since Sigma notation will usually have more complex expressions after the Sigma symbol, here are some further examples to give you a sense of what is possible:
$\displaystyle\sum_{i=2}^{5}2i\\*~\\*=2(2)+2(3)+2(4)+2(5)\\*~\\*=4+6+8+10$
$\displaystyle\sum_{j=1}^{4}jx\\*~\\*=1x+2x+3x+4x$
$\displaystyle\sum_{k=2}^{4}(k^2-3kx+1)\\*~\\*=(2^2-3(2)x+1)+(3^2-3(3)x+1)+(4^2-3(4)x+1)\\*~\\*=(4-6x+1)+(9-9x+1)+(16-12x+1)$
$\displaystyle\sum_{n=0}^{3}(n+x)\\*~\\*=(0+x)+(1+x)+(2+x)+(3+x)\\*~\\*=0+1+2+3+x+x+x+x$
Note that the last example above illustrates that, using the commutative property of addition, a sum of multiple terms can be broken up into multiple sums:
$\displaystyle\sum_{i=0}^{3}(i+x)\\*~\\*=\displaystyle\sum_{i=0}^{3}i+\displaystyle\sum_{i=0}^{3}x$
And lastly, this notation can be nested:
$\displaystyle\sum_{i=1}^{2}\displaystyle\sum_{j=4}^{6}(3ij)\\*~\\*=\displaystyle\sum_{i=1}^{2}(3i\cdot4+3i\cdot5+3i\cdot6)\\*~\\*=(3\cdot1\cdot4+3\cdot1\cdot5+3\cdot1\cdot6)+ (3\cdot2\cdot4+3\cdot2\cdot5+3\cdot2\cdot6)$
The rightmost sigma (similar to the innermost function when working with composed functions) above should be evaluated first. Once that has been evaluated, you can evaluate the next sigma to the left. Parentheses can also be used to make the order of evaluation clear.
### Pi (Product) Notation
The Pi symbol, $\prod$, is a capital letter in the Greek alphabet call “Pi”, and corresponds to “P” in our alphabet. It is used in mathematics to represent the product of a bunch of terms (think of the starting sound of the word “product”: Pppi = Ppproduct). It is used in the same way as the Sigma symbol described above, except that succeeding terms are multiplied instead of added:
$\displaystyle\prod_{k=3}^{7}k\\*~\\*=(3)(4)(5)(6)(7)$
$\displaystyle\prod_{n=0}^{3}(n+x)\\*~\\*=(0+x)(1+x)(2+x)(3+x)$
$\displaystyle\prod_{i=1}^{2}\displaystyle\prod_{j=4}^{6}(3ij)\\*~\\*=\displaystyle\prod_{i=1}^{2}((3i\cdot4)(3i\cdot5)(3i\cdot6))\\*~\\*=((3\cdot1\cdot4)(3\cdot1\cdot5)(3\cdot1\cdot6)) ((3\cdot2\cdot4)(3\cdot2\cdot5)(3\cdot2\cdot6))$
### Summary
Sigma (summation) and Pi (product) notation are used in mathematics to indicate repeated addition or multiplication. Sigma notation provides a compact way to represent many sums, and is used extensively when working with Arithmetic or Geometric Series. Pi notation provides a compact way to represent many products.
To make use of them you will need a “closed form” expression (one that allows you to describe each term’s value using the term number) that describes all terms in the sum or product (just as you often do when working with sequences and series). Sigma and Pi notation save much paper and ink, as do other math notations, and allow fairly complex ideas to be described in a relatively compact notation.
### Whit Ford
Math teacher, substitute teacher, and tutor (along with other avocations)
## 49 thoughts on “Sigma and Pi Notation (Summation and Product Notation)”
1. Douglas maindo says:
am very thankful 2 the information above.it is very helpful to me
2. I always see these equations on in technical papers but I never knew how to decode them. This was so helpful! It’s basically a for loop in scripting, makes so much sense. Also your blog is awesome, thank you for sharing!
1. Thank you! And yes, a little programming experience with loops makes Sigma and Pi Notation much easier to understand…
3. Sundaram says:
Very very useful. Thanks a lot – Sundaram
4. Anonymous says:
Thank you, this was very helpful. I was finding how to use Sigma notation, and finally found such a good one.
5. Samama Fahim says:
Indeed a very lucid exposition of Sigma and Pi notations! Thanks 🙂
6. Very useful post. But what if the Pi notation is not in closed form, such as
$\displaystyle\prod^{n}_{k=2}(1-\dfrac{1}{k^2})$
1. If the index limit above the Pi symbol is a variable, as in the example you gave:
$\displaystyle\prod^{n}_{k=2}(1-\dfrac{1}{k^2})$
then there are an indeterminate number of factors in the product until such time as “n” is specified.
I suppose a problem could be posed this way if you are being asked to come up with an expression for such a product that does not involve Pi notation: is there some closed form expression involving “n” that represents this product?
So, if n=3, then
$\displaystyle\prod^{n}_{k=2}(1-\dfrac{1}{k^2})=(1-\dfrac{1}{4})\cdot(1-\dfrac{1}{9})$
and if n=4, then
$\displaystyle\prod^{n}_{k=2}(1-\dfrac{1}{k^2})=(1-\dfrac{1}{4})\cdot(1-\dfrac{1}{9})\cdot(1-\dfrac{1}{16})$
and if you leave the final index as “n” becomes:
$=(1-\dfrac{1}{4})\cdot(1-\dfrac{1}{9})\cdot(1-\dfrac{1}{16})\cdot ... \cdot (1-\dfrac{1}{n^2})$
$=\dfrac{3}{4}\cdot\dfrac{8}{9}\cdot\dfrac{15}{16}\cdot ... \cdot\dfrac{n^2-1}{n^2}$
Is there some closed form expression that represents this product?
1. Stephen Kazoullis says:
Shouldn’t the k be squared ?
2. Which “k” are you referring to? There are several in the posting… Ooops – I just realized you were asking about my reply to the comment. You are correct. I will modify my response shortly.
3. Jason Z Okoro says:
Actually n SHOULD be squared in his reply since he’s saying that that’s the LAST TERM in the product. Basically this is where k = n. It’s important to emphasize that.
7. what is the relation between the two when they are logarithmic differentiated?
1. It would help if you could provide an example of what you are asking about.
If you need to differentiate a sum, I would not expect logarithmic differentiation to be very useful, as the laws of logarithms do not allow us to do anything with something like
$ln(y)=ln(\displaystyle\sum _i x^{ix})$
Differentiating this would turn the right side into the reciprocal of the original sum times its derivative = a mess.
However, if you need to differentiate a product, logarithmic differentiation could make life simpler by converting a long succession of product rule applications into a sum of logs. Since
$ln(xy)=ln(x)+ln(y)$
we can rewrite the log of a product as a sum of logs:
$ln(y)=ln(\displaystyle\prod _i x^{ix})\\*\\ln(y)=\displaystyle\sum_i ln(x^{ix})$
which, in many cases, could simplify the differentiation process.
If sigma is for summation, and pi is for multiplication, are there any notations for division and subtraction? Just out of curiosity?
1. Good question!
Subtraction can be rewritten as the addition of a negative. So Sigma notation describes repeated subtraction when its argument is a negative quantity.
Division can be rewritten as multiplication by the reciprocal. So Pi notation describes repeated division when its argument has a denominator other than 1.
Therefore, additional notations are not needed to describe repeated subtraction or division… Which is quite convenient.
9. john walter says:
Sir, how about expressing thing one 1x2x3 + 2x3x4 + 3x4x5 + ….will it be a combination of
sigma and pi? If you can illustrate it please.
1. You are correct – this can be represented using a combination of Sigma and Pi notation:
$\displaystyle\sum_{i=1}^{N} \displaystyle\prod_{j=0}^{2}(i+j)$
In the above notation, i is the index variable for the Sum, and provides the starting number for each product. By having the Product index variable start at zero, the expression to generate each value is a bit simpler. If j went from one to three each time, the expression on the right would have to be (i + j – 1).
1. gargi says:
hello sir,
thank you for the amazing and very helpful post.
I was just practicing the question wanted to know can 30….. n(n+1)(n+2) be the ans to the above sigma and product equation given by you.
2. Gargi,
The Sigma and Pi expression I used to answer the previous question did not have a value specified for “N”, so any value given for the expression will have to be in terms of “N”… as your question is. However, your expression leaves me uncertain as to whether you are analyzing the situation correctly or not.
Let’s list the first few terms of this sequence individually to get a sense of how this series behaves:
$6+24+60+120+...+(N)(N+1)(N+2)$
So, the sum of the first two terms would indeed be 30. But if you are trying to give a general answer, you should show each term individually so that the person reading your answer can see any pattern that is developing, and understand how to fill in the “…” used to represent all the terms that are not shown.
3. Yucel says:
Hi Mr. Ford,
any hint for the solution of following infinite series will appretiated.
Thanks…
$\displaystyle\sum_{i=0}^\infty \displaystyle\prod_{j=0}^i \dfrac{j+7}{6(j+1)}$
4. Yucel,
Evaluating the first few terms, just to get a sense of its behavior, produces the following (after converting all fractions to have a common denominator so that they are easier to compare quickly:
i=0: (14/12)
i=1: (14/12)(8/12)
i=2: (14/12)(8/12)(6/12)
i=3: (14/12)(8/12)(6/12)(5/12)
It seems that successive terms are growing smaller, since each is the previous term multiplied by a factor that is less than 1 and shrinking, so the series will converge (it shrinks faster than a geometric sequence with a common ratio that is less than one). But to what value?
I don’t know what context this problem arises in for you, and therefore what tools you are expected to use to analyze the problem (assuming it is a problem from a class). Plus I have not worked with infinite series in a while – off the top of my head, I might try to “squeeze” this between two series for which I know the sum, to at least provide upper and lower bounds for the sum. An upper bound would be provided by an infinite geometric sequence, but I am uncertain what might best provide a lower bound.
Does that help?
10. Anonymous says:
$\displaystyle\sum_{n=0}^{3}(n+x)\\*~\\*=(0+x)+(1+x)+(2+x)+(3+x)\\*~\\*=0+1+2+3+x+x+x+x$
what can be the correct answer this equation?
1. The example was an expression, not an equation, therefore it cannot be “solved”. However, this particular example can be “simplified” by collecting like terms to become
$6+4x$
which would raise the question: “why write it using Sigma Notation when you could just as easily write $4x+6$?”. My answer to that would be: I probably would not use Sigma Notation to write such a simple expression. This example was intended show how to interpret Sigma Notation in some of the many ways that it can be used.
11. Jonty says:
I have a student asking whether there is a symbol for exponentiation of a sequence? So there’s SIGMA for summation of a sequence, PI for multiplication of a sequence and perhaps something else for exponentiation of a sequence?
So like E(x+n) for n=1 to 3 would produce (x+1)^(x+2)^(x+3)… Or maybe((x+1)^(x+2))^(x+3)
Thanks,
Jonty
1. Jonty,
Good question! I am not aware of such notation, and furthermore, I am not aware of situations where such notation would be needed. Do you know of situations that require repeating exponentiation to model them? I suppose that some multi-dimensional models (perhaps like String Theory) could require some repeated exponentiation, but even there I doubt they would need to get beyond several levels of exponentiation (the result would grow really fast…).
I’ll research this a bit to see if I can find anything, and if I do I’ll post another reply.
1. Jonty says:
Thanks Whit. The student in question is actually only 11 years old and somehow I don’t think that he will accept the “not needed” reason! I’ll challenge him to find a need for it and maybe he can create his own notation. He said it had something to do with his investigation into combination formulae… He’s currently using a backwards SIGMA symbol!
2. Jonty,
One other thought… if
$(x^a)^b=x^{ab}$
and therefore
$((x^a)^b)^c=x^{abc}$
etc…
then there is no need for a notation to represent repeated exponentiation, since exponents that are products already represent repeated exponentiation. Using Pi notation in the exponent achieves the desired purpose.
3. Jonty says:
Good point, however, x^a^b is not the same as x^ab. For repeated exponentiation I would assume that form rather than (x^a)^b. So maybe we do still need something?
4. Ooops – didn’t think of
$x^{a^{b^c}}$
I still cannot think of either an application for such an expression or a notation for it. Perhaps this is a good question for a forum like http://math.stackexchange.com/questions
12. Jeremy says:
If the sum of a bunch of terms in known as a “summation of a series”, then what is the product of a bunch of terms known as in mathematics?
1. The exact vocabulary used is likely to differ from one person to another, and I doubt that everyone will care that much about the words used, but I will be picky about the words used in an effort to clarify the situation and answer your question.
A “sequence” is an ordered set of terms which are NOT added together. There is often a pattern to them, a formula that can be used to determine the value of the next term in the sequence. Sequence definitions usually have no need for summation notation.
A “series” is the sum of the first N terms of a sequence. Series definitions almost always rely on summation notation.
The phrase you wrote, “summation of a series”, is either redundant (they could have just said “a series”), or indicated that they wish to sum the first N terms of a series (the sum of terms, each which is a sum, something that might have a use, but I have not seen used).
A polynomial (such as a quadratic) can be called “a sum of terms”.
And finally to your question. A “product” is the result of multiplying two or more “factors”. The entire product is a single “term”.
So when using pi notation, the expression after the pi describes each “factor” (not “term”), and the final result after the pi notation has been evaluated is a “product”. No new vocabulary is needed.
13. Nikson says:
Does Multiplication operator always increment? does $\displaystyle\prod_{i~=~n-1}^{0}$ work?
i.e. can there be a bigger value at the base and smaller value at top of the PI operator? I want to do that to signify that the matrices do not commute.
1. Interesting question! Notation is a convention, a commonly shared interpretation of some symbols. So, even if it is not commonly used in a particular way, there is no strong reason I can think of why you couldn’t use it that way (if necessary, including a note or example describing how you intend the notation to be interpreted).
Loops in programming languages can be written to decrease the index each time just as easily as they can increase it. The convention is to increase it, just like with Sigma and Pi notation, but they also support decreasing indeces.
So, my opinion would be: sure! Why not? If I were to see an upper index value that is smaller than the lower one, my first assumption would be that I would need to decrease the index by 1 for each iteration – which seems to be what you intend.
I do not follow your thinking though when you say you wish to use a descending index value to indicate that matrices do not commute… I would not perceive a descending index value, or an ascending one, to indicate anything about the commutative property’s applicability to the resulting expression. After expanding the Pi notation into the full expression that it represents, the person working with that expression must follow the rules of algebra (or matrix algebra), and the index number of each factor would not have any effect on such rules. But, perhaps I do not understand the situation you seek to describe.
14. Shri Krishan Baghel says:
I like it … And I hope it will help other students too to acheive their goals …
15. Appiah Godfred says:
How can u write this using summation notation: 3 -5+7 -9+11-13+15?
1. Appiah,
I notice three things when I look at this sequence:
1) The values alternate sign, so we need a factor that changes sign for each value of “n”. (-1)^n will change sign every time “n” grows by one, but when n=1 it is negative – which is the wrong sign for the first time. Adding or subtracting 1 from “n” will make the factor positive when n=1 (since a negative raised to the zero, or an even, power is positive). So $(-1)^{n-1}$ will provide the correct sign for the nth term.
2) The values grow in magnitude linearly by 2 each time. A factor of (2n) will produce such numbers, but when n=1 this will have a value of 2, not 3… so I need to add 1 to each value: $(2n+1)$.
3) There are seven terms, so n will need a starting value of 1, and an ending value of 7.
Putting the three thoughts above together, I get:
$\displaystyle\sum_{n=1}^{7}{(-1)^{n-1}(2n+1)}$
16. 18mn6@cis.dk says:
What if I want to write the sequence:
$2^n-1 + 2(2^n-2 + 2(2^n-3 + 2(2^n-4 + ... )))$
using Sigma or Pi notation, or possibly both. Furthermore is there a way of simplifying the notation and finding a result that is a function of n?
1. If I have interpreted the expression you show correctly, it is neither an arithmetic nor a geometric sequence. Futhermore, it appears to me as though it will always have an infinite number of sub-expressions that need to be evaluated, regardless of the value of “n”.
It is not
– a sum of consistent terms (the third term contains all the of the remaining “terms”)
– a product of consistent factors (the first two terms are not multiplied by what follows)
so I do not see a way of representing it using either Sigma or Pi notation.
You may be able to simplify this expression by expanding the values a bit to see if there is a pattern, but the result will probably vary a great deal depending on the value of “n”. For example, if n=1, then the expression would be:
$2-1+2(2-2+2(2-3+2(2-4+2(...))))$
$=~1+2(0+2(-1+2(-2+2(...)))$
it would appear as though the quantity in parentheses is becoming increasingly negative (a sum of growing negative numbers), and therefore the value probably goes to negative infinity.
If n=2, then
$4-1+2(4-2+2(4-3+2(4-4+2(...))))$
$=~3+2(2+2(1+2(0+2(...)))$
once again it would appear as though the quantity in parentheses is going to become increasingly negative (a sum of growing negative numbers), and therefore the value propably goes to negative infinity again, even though it starts out a bit larger.
As n grows, the constant power of 2 in the expression will dominate the initial results a lot more, but the infinite number of subtractions from it will eventually catch up to its value, no matter how large it is.
17. Thanks for your clear explanations. It helps me to understands the notation means and how to use it.
18. Hello,
How would I derive the polynomial for the following expression:
n = 112
expression is
$x-y^{11n}$
multiply until n reaches 143 (i.e. n=112, n=113 etc.)
I’m interested in simplifying the polynomial to 32 terms and determine the exponents of y
Thank you.
1. Alin,
Using Pi notation, I interpret your question to be
$\displaystyle\prod_{n=112}^{143}(x-y^{11n})$
$~=~(x-y^{1232})(x-y^{1243})(x-y^{1254})...(x-y^{1573}))$
Using a binomial expansion, the terms will be
$(_{32}C_i)(x^{32-i})(y^{\sum_{k=112}^{112+i} 11k})$
for i = 0 to 32. Terms with odd values of i will be negative.
The coefficients will be “32 Choose i”, or
$\dfrac{32!}{(32-i)!~i!}$
Does that help?
Hello, I am trying to utilize the Pi notation to represent a repeating multiplication, but one that rounds up to the nearest whole after each time there is a multiplication(or division). Before I continue please forgive my mathematical illiteracy, I am taking an amateur interest in this.
What I am wondering about is this. If I wanted to take, let’s say “I”, and multiply “I” by a repeating multiple, let’s say “1/(1-r)”.
I might write it as: I×(1÷(1-r))×(1÷(1-r))×(1÷(1-r))… or I÷(1-r)÷(1-r)÷(1-r)…
Reading this post it seems like this would be easy to use the big Pi Π notation.
Ex: I × Π(1÷(1-r))….. something like that.
If I wanted to represent something being rounded up I think I could use ceiling function brackets: ⌈⌉. So for instance, if I wanted to round the above to the nearest whole after each division (or multiplication) step I think I could write: ⌈⌈⌈I÷(1-r)⌉÷(1-r)⌉÷(1-r)⌉…. In my mind, this rounds up each time the value is divided by (1-r). I simply cannot figure out how to represent that using big Pi Π.
I hope that makes some sense.
Any insights would be very appreciate.
Thanks,
The notation that follows a capital Pi describes only the term that is to be multiplied. The difficulty you describe is that you wish to specify what happens to the result of that product, and capital Pi notation does not provide any means to do that.
Two ways to resolve the problem come to mind:
1) your expansion of the problem using square brackets
2) using a programming language to describe a loop in which each product is then rounded, before repeating the loop until the specified number of multiplications have been carried out.
20. Anonymous says:
Hello,
Sir,If I have equation like this :
X1=(1-P1)(1-P2)P3+(1-P2)(1-P3)P1+(1-P3)(1-P1)P2
X2=(1-P1)P2.P3+(1-P2)P2P3+(1-P3)P2P3
X3=P1.P2.P3
For example, X1 means we have One term say P3 and rest two are (1-P) and summation of such product terms for 3 values(P1,P2 and P3).
How should I proceed if I want to get it for n instead of 3.
Equation for Xn in terms of P1,P2,……Pn.
1. Summation notation does not provide an easy way that I can think of to do what you describe. While it can add a bunch of terms very nicely, the challenge is describing each of the terms you show as a function of the term number. This would be easy to do in a computer program, but not so much using summation notation.
21. Dharmendra paswan says:
limit,n–>infinity {tan(p/2n)tan(2p/2n)tan(3p/2n)……}^(1/n) . find the value where p=pi. option (a) 1 (b) 2-log2 (c) 3 (d) 3 -log4. please reply in my email ( dpaswan309@gmail.com). thanks
1. Dharmendra,
This problem is not strictly a Pi Notation problem, as it involves a limit and a power outside of any Pi Notation.
Also, I am not certain where the product you describe is supposed to end. If it ends with, or continues beyond tan(np/2n), which will always be undefined, then my first impression is that there would be no limit to the product. However, I have never worked with infinite products. Your answer options suggest that there is some expansion of a a logarithm that results in an infinite product of tangent functions, however I am not familiar with that.
Sorry!
22. Mr. Unknown says:
How to find the derivative of the pi notation
1. If each factor described by the pi notation contains an instance of a the variable, you would need to use the product rule… potentially many times. However, if each factor does not contain the variable (or a function of the variable) that you are differentiating with respect to, then the whole product would be a constant. So, depending on the number of factors in the product, it could be a very long process, or a very short one.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"extraction_info": {"found_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": 48, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8834459781646729, "perplexity": 486.3146050602524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195529737.79/warc/CC-MAIN-20190723215340-20190724001340-00225.warc.gz"} |
http://physics.stackexchange.com/questions/106457/expectation-values-and-derivation-of-heisenberg-equation | # Expectation Values and Derivation of Heisenberg Equation?
Consider a system of particles with wave function $\psi$(x) (x can be understood to stand for all degrees of freedom of the system; so, if we have a system of two particles then x should represent {$x_1; y_1; z_1; x_2; y_2; z_2$}). The expectation value of an operator $\hat{A}$ that operates on is defined by : $$\langle\hat{A}\rangle = \int\psi^{*}\hat{A}\psi dx$$
Yup this makes sense to me and there's nothing new here.
If $\psi$ is an eigenfunction of $\hat{A}$ with eigenvalue $a$, then, assuming the wave function to be normalized, we have : $$⟨ \hat{A} ⟩ = a$$
This is where I want to confirm something.
$$\hat{A}\psi = a\psi$$
Hence, $$⟨ \hat{A} ⟩ =\int\psi^{*} a \psi dx$$
Since $a$ is a constant I can take it out :
$$\langle\hat{A}\rangle = a \int\psi^{*} \psi dx$$
We assumed that the wave function was normalized hence $$\int\psi^{*} \psi dx = 1$$
Leaving $$\langle\hat{A}\rangle = a$$
Now consider the rate of change of the expectation value of $\langle\hat{A}\rangle$:
$$\frac{d\langle\hat{A}\rangle}{dt} = \int{\frac{\partial}{\partial t}}(\psi^{*}\hat{A}\psi)dx$$
$$=\int{\frac{\partial \psi^{*}}{\partial t}\hat{A}\psi+\psi^{}\frac{\partial\hat{A}}{\partial t}\psi^{*}}+\frac{\partial \psi}{\partial t}\hat{A}\psi^{*} dx$$
$$=\int{\langle\frac{\partial\hat{A}}{\partial t}\rangle} +\frac{i}{\hbar}\int{[(\hat{H}\psi)^{*}\hat{A}\psi-\psi^{*}\hat{A}\hat{H}\psi]}dx$$
where we have used the Schrodinger equation :
$$i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi$$
The second line is easily obtained via differentiation. The second term in the second line corresponds to the first term in the third line, correct ?
I do not see how this term was obtained. In particular where the $\frac{i}{\hbar}$ originates from :
$$\frac{i}{\hbar}\int{[(\hat{H}\psi)^{*}\hat{A}\psi-\psi^{*}\hat{A}\hat{H}\psi]}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9849386811256409, "perplexity": 145.3085272722131}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1432207927824.26/warc/CC-MAIN-20150521113207-00343-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Kluyver%27s_Formula_for_Ramanujan%27s_Sum | # Kluyver's Formula for Ramanujan's Sum
## Theorem
Let $q \in \N_{>0}$.
Let $n \in \N$.
Let $\map {c_q} n$ be Ramanujan's sum.
Let $\mu$ denote the Möbius function.
Then:
$\displaystyle \map {c_q} n = \sum_{d \mathop \divides \gcd \set {q, n} } d \map \mu {\frac q d}$
where $\divides$ denotes divisibility.
## Proof
Let $\alpha \in \R$.
Let $e: \R \to \C$ be the mapping defined as:
$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let $\zeta_q$ be a primitive $q$th root of unity.
Let:
$\displaystyle \map {\eta_q} n := \sum_{1 \mathop \le a \mathop \le q} \map e {\frac {a n} q}$
By Complex Roots of Unity in Exponential Form this is the sum of all $q$th roots of unity.
Therefore:
$\displaystyle \map {\eta_q} n = \sum_{d \mathop \divides q} \map {c_d} n$
By the Möbius Inversion Formula, this gives:
$\displaystyle \map {c_q} n = \sum_{d \mathop \divides q} \map {\eta_d} n \map \mu {\frac q d}$
Now by Sum of Roots of Unity, we have:
$\displaystyle \map {c_q} n = \sum_{d \mathop \divides q} d \map \mu {\frac q d}$
as required.
$\blacksquare$
## Source of Name
This entry was named for Jan Cornelis Kluyver. | {"extraction_info": {"found_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": 0.9986546635627747, "perplexity": 1203.558287065147}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400198942.13/warc/CC-MAIN-20200921050331-20200921080331-00401.warc.gz"} |
https://www.physicsforums.com/threads/physics-laws-and-expansion-of-universe.150396/ | # Physics laws and expansion of universe?
• Start date
682
1
## Main Question or Discussion Point
my interpretation of the expansion of the universe might be naive... correct me if I make any mistake.
So, space in general is expanding, or being created all over the universe. If you have a photon traveling in free space, the frequency gets shifted. But Energy=h*frequency; the energy of the photon decreases as it travels. Where did the energy of that wave go?
Similarly, what about momentum? p=E/c, is that not conserved as well?
I think it has something to do with reference frame. How can one resolve the paradox?
Last edited:
Related Special and General Relativity News on Phys.org
1,222
2
This question is not uncommon, and there is no good answer at this point.
Conservation of energy and general relativity have not been reconciled.
305
1
Yes, this is quite disturbing to me.
Energy conservation has to do with time symmetry. Since there is an expansion, there is an arrow of time, so the symmetry is broken.
If energy cannot be defined, then how about entropy?
If energy and entropy cannot be defined, we cannot do much thermodynamics with the universe. There will be no 1st law and 2nd law of universal thermodynamics. And yet in thermodynamics, we often need to consider the entropy of the universe and things like that. Really strange and mind boggling.
Yet we can define a temperature of the universe, flatness of the universe. So it seems at least we can come out with an equation of state of the universe.
488
0
I would say perhaps that energy goes into the expansion? Though cause and effect kind of go out the window with that one
Hey the apparent frequency of the photon changes.I don't think the real frequency of the photon changes.Please correct me if im wrong
what i believe is that mass is a form of energy.Mass occurs between a specific range of energy density but the range is not clearly defined.Even if energy density is slightly different characteristics of mass may occur.All energy forms differ due to energy density.And when an object attains relative velocity the energy density tends to increase which gives an effect of mass
305
1
The photon frequency does change really, not apparently.
The universe expansion actually stretches the photon lengthening its wavelength.
682
1
and since the rate of expansion of the universe is increasing, a photon's frequency will eventually be red-shifted to near zero... so its energy is gradually being drained...
Also, if the space is expanding, then distance stars will move farther and farther away from us (accelerated frame of reference). yet, in that distance stars' frame, there will be no fictitious force due to that apparent acceleration... does it mean the equivalence principle is broken?
• Last Post
Replies
4
Views
2K
• Last Post
Replies
12
Views
3K
• Last Post
Replies
1
Views
2K
• Last Post
Replies
23
Views
8K
• Last Post
Replies
2
Views
2K
• Last Post
Replies
5
Views
718
• Last Post
Replies
3
Views
1K
• Last Post
Replies
1
Views
4K | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9724065065383911, "perplexity": 689.4942017500414}, "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/1575540527205.81/warc/CC-MAIN-20191210095118-20191210123118-00142.warc.gz"} |
http://math.stackexchange.com/questions/298137/counting-monic-polynomials-over-mathbb-z-p-mathbb-z | # Counting monic polynomials over $\mathbb Z / p\mathbb Z$.
Let $p$ be prime. Consider monic polynomials of degree $d$ over $\mathbb Z / p \mathbb Z$. Denote the number of such polynomials with degree less than $p$, which are not zero for all $x \in \mathbb Z / p \mathbb Z$, with $m_d$. Let $d \geq p$. Then I have to show that there are $m_p p^{d-p}$ of such polynomials which are not zero for all $x \in \mathbb Z / p \mathbb Z$.
-
I am confused. You appear to define $m_{d}$ as the number of monic polynomials of degree $d$ with coefficients in $\mathbb Z / p \mathbb Z$ which have no roots in $\mathbb Z / p \mathbb Z$. But I don't understand which polynomials you want to count then. – Andreas Caranti Feb 8 '13 at 16:51
It seems you want to prove $m_d=m_pp^{d-p}$? If so, I think it would be clearer if you put it that way. It's confusing that you say "such polynomials" but then only repeat one of the conditions on these polynomials from the previous sentence, so it's not clear whether these are the same kind of polynomials as in the other sentence or not. Also it's not clear whether the polynomials should be non-zero for all $x$ or whether it shouldn't be the case that they're not zero for all $x$. – joriki Feb 8 '13 at 18:23
Hint: The polynomials that are zero for all $x \in \mathbb F_p=\mathbb Z/p\mathbb Z$ are precisely those in the ideal generated by $x^p-x$. | {"extraction_info": {"found_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": 0.9642418622970581, "perplexity": 83.39380267877792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042988650.53/warc/CC-MAIN-20150728002308-00027-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://clumath343s15s2.wikidot.com/clumath2015s2ch6s2-definitions | Definitions
An set of vectors is an orthogonal set if each pair of vectors is orthogonal to eachother such that
(1)
\begin{align} \begin{Bmatrix} u_1,...,u_n \end{Bmatrix}\\ u_1 \cdot u_2 = 0, u_1 \cdot u_2 = 0, ... , u_{n-1\text{}} \cdot u_n = 0 \end{align}
In addition, if there is a subspace S formed by an orthogonal set, that set is linearly independent and is thus a basis for S. Therefore, if there exists a vector $\vec{y}$ is S, then $\vec{y}$ can be formed by the orthogonal set
(2)
\begin{align} \vec{y} = c_1\vec{u_1}+c_2\vec{u_2}+...+c_n\vec{u_n} \end{align}
We can therefore multiply each term by $\vec{u_1}$ for instance, and since all other u's are perpendicular to $u_1$, all other terms drop out except $\vec{y} \cdot \vec{u_1} = c_1 \vec{u_1}^2$. This can be simplified and $c_1$ can be found using
(3)
\begin{align} \frac{\vec{y} \cdot \vec{u_1}}{\vec{u_1}^2} = c_1 \end{align}
This is true for all c's so we now have the tools to find any vector in S by a linear combination of the orthogonal basis.
## Orthogonal Projection
Now, if we're looking to express $\vec{y}$ as a sum of two vectors, one which is a multiple of $\vec{u}$ and one perpendicular to $\vec{u}$, we must apply some of what we just learned. We know the scalar $c_1$ to create a vector in S that is a component of $\vec{y}$ is $\frac{\vec{y} \cdot \vec{u_1}}{\vec{u_1}^2}$. Using this to scale $\vec{u_1}$ which we'll call $\hat{y}$, we find
(4)
\begin{align} \vec{y} = \hat{y} + \vec{v} \end{align}
where $\vec{v}$ is the perpendicular vector such that
(5)
\begin{align} \vec{y} - \hat{y} = \vec{v} \\ \vec{v} \cdot \hat{y} = 0 \end{align}
Orthonormal Sets
An Orthonormal set is a set of orthogonal vectors whose magnitudes are all 1.
To normalize a vector, just divide each component by the magnitude of the vector so…
(6)
\begin{align} \hat{x} = \frac{\vec{x}}{|\vec{x}|} \end{align}
Then the orthonormal set has the property $U^TU = I$. Thus, $U^T = U^{-1}$.
page revision: 0, last edited: 02 May 2015 17:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 6, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9963857531547546, "perplexity": 376.26605061619466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1563195524679.39/warc/CC-MAIN-20190716160315-20190716182315-00008.warc.gz"} |
http://code7700.com/aero_properties_of_the_atmosphere.htm | # Properties of the Atmosphere
## Aeronautical
#### Eddie sez:
"Any final questions?" the aero professor would ask at the end of each class. "Why is there air?" somebody would invariably ask.
"Without air," he would say, "basketballs wouldn't bounce." The class would laugh and we would leave. We did this every single class. I can't explain it.
So too it is with most pilots when asked "How do airplane's fly?" They answer "lift." Then comes, "what is lift?"
If you want the answer to "What is Lift?" you need to understand the atmosphere and how we, as pilots, measure it.
Everything here is from the references shown below, with a few comments in an alternate color.
Wright Flyer, first flight, from Wikimedia Commons.
20120926
### Static Pressure
[Dole, pg. 15] The static pressure of the air P is simply the weight per unit area of the air above the level under consideration. For instance, the weight of a column of air, with a cross sectional area of 1 ft2 and extending upward from sea level through the atmosphere is 2116 lb. The sea level static pressure is, therefore, 2116 psf (or 14.7 psi). Static pressure is reduced as altitude is increased because there is less air weight above. At 18,000 ft altitude the static pressure is about half that at sea level.
Another commonly used measure of static pressure is inches of mercury. On a standard sea level day the air's static pressure will support a column of mercury (Hg) that is 29.92 in. high. Weather reports use a third method of measuring static pressure called millibars. In addition to these rather confusing systems, there are the metric measurements in use throughout most of the world.
In aerodynamics is is convenient to use pressure ratios, rather than actual pressures. Thus the units of measurement are canceled out.
Pressure Ratio: δ = p / p0
Where P0 is the sea level standard pressure (2116 psf or 29.92 in. Hg).
### Temperature
[Dole, pg. 16] The commonly used measures of temperature are the Fahrenheit (F) and Celsius (C) scales. Neither degrees F nor C are based upon absolute zero and cannot be used in calculations. Absolute temperature must be used instead. Absolute zero in the Fahrenheit system is -460 and in the Celsius system is -273°.
The symbol for absolute temperature is T, and the symbol for sea level standard temperature is T0. Using temperature ratios rather than actual temperatures cancels out the units and simplifies things. The symbol for temperature ratio is θ (theta).
Temperature Ratio: θ = T / T0
### Density property of the atmosphere
[Dole, pg. 16] The density of the air is the most important property of the air in the study of aerodynamics. It is defined as the mass of the air per unit volume. The symbol for density is ρ (rho).
$ρ = mass unit volume$
Standard sea level density ρ0 = 0.002377 slugs/ft3. Density decreases with altitude. At about 22,000 ft. the density is about one-half of ρ0. It is desirable in aerodynamics to use density ratios, rather than the actual values of density. The symbol for density ratio is σ (sigma).
Density Ratio: σ = ρ / ρ0
Density is directly proportional to pressure and inversely proportional to absolute temperature, as shown by the universal gas law:
σ = P / RT
and so
$ρ ρ0 = P/RT P0 / RT0$
where R is the gas constant. Therefore:
Density: σ = δ / θ
### Viscosity
[Dole, pg. 17] Viscosity can be simply defined as the internal friction of a fluid caused by molecular attraction which makes it resist a tendency to flow. The viscosity of air is important when discussing airflow in the region very close to the surface of the aircraft. This region is called the boundary layer. The viscosity of gases is unlike that of liquids, in that with gases an increase in temperature causes an increase in viscosity. The coefficient of absolute viscosity has been assigned the symbol μ (mu). Since aerodynamics often involves considerations f both viscosity and density, ma more usual form of viscosity measurement, known as kinematic viscosity, is often used. It is denoted by the symbol ν (nu).
Kinematic Viscosity: ν = μ / ρ
### International Standard Atmosphere (ISA)
Figure: Standard Altitude Table, from [Hurt, pg. 5].
As pilots we never deal directly with σ, ρ, θ, or just about any of the other Greek symbols. But we deal with the air mass and much of what we do is constrained by what has become known as the "International Standard Atmosphere."
Scientists, engineers, and people who write aviation textbooks can't seem to agree on what exactly constitutes a standard atmosphere or even where one layer ends and the next begins. Most agree that temperature is key, so that's where we will begin. Since we are international pilots, we'll use the ICAO's definitions.
### Layers of the Atmosphere
#### ICAO Standard Atmosphere Temperature Model
Figure: Temperature and Vertical Temperature Gradients Table, from ICAO Doc 7488/3, Table D.
Converting km to feet and °Kelvin to °Celsius, we come up with an ICAO altitude vs. temperature model we can use:
Altitude (ft) Temperature (°C) Gradient (°C)
0 15
-1.98
36,089 -56.5
0
65,617 -56.5
+0.3
104,987
Note:
• °C = °K - 273.15
• Feet = (km) 3280.8399
From this we surmise the troposphere starts at the surface and ends when the temperature no longer loses 2°C every 1,000 feet of altitude, at about 36,000' where the temperature will be -56.5°C. At that point the temperature remains constant for the remainder of most our flight envelopes. Of course all this is based on that so-called "standard" day.
Pilots should be concerned with the height of the tropopause because it determines where most of the weather is, where fuel economy increases plateau, and where aircraft components may be subject to limiting temperatures. Newer data suggests the tropopause is a bit lower than quoted by the advisory circular, typically around 17 km near the equator, above FL 550. At the poles, the tropopause dips as low as 8 km, around FL 260. For most mid-latitudes, we will be spending our time right at or just above the transition layer, the tropopause.
### Tropopause Height and ISA
Figure: Tropopause Height, from Geerts and Linacre.
• The height of the tropopause depends on the location, notably the latitude, as shown in the figure on the right (which shows annual mean conditions). It also depends on the season.
• At latitudes above 60°, the tropopause is less than 9-10 km above sea level; the lowest is less than 8 km high, above Antarctica and above Siberia and northern Canada in winter. The highest average tropopause is over the oceanic warm pool of the western equatorial Pacific, about 17.5 km high, and over Southeast Asia, during the summer monsoon, the tropopause occasionally peaks above 18 km. In other words, cold conditions lead to a lower tropopause, obviously because of less convection.
• Deep convection (thunderstorms) in the Intertropical Convergence Zone, or over mid-latitude continents in summer, continuously push the tropopause upwards and as such deepen the troposphere.
• On the other hand, colder regions have a lower tropopause, obviously because convective overturning is limited there, due to the negative radiation balance at the surface. In fact, convection is very rare in polar regions; most of the tropospheric mixing at middle and high latitudes is forced by frontal systems in which uplift is forced rather than spontaneous (convective). This explains the paradox that tropopause temperatures are lowest where the surface temperatures are highest.
The tropopause is actually quite lower than the ICAO model predicts, especially at the poles. G450 pilots, for example, spend most of their cruise flight in the tropopause. From a stick and rudder perspective, once the temperature stops decreasing you are not necessarily improving fuel mileage with altitude. (Winds and other weather concerns should determine altitude selection once the lapse rate nears zero.)
### Altitude Measurement
[FAA-H-8083-15, pg. 3-1] Flight instruments depend upon accurate sampling of the ambient atmospheric pressure to determine the height and speed of movement of the aircraft through the air, both horizontally and vertically. This pressure is sampled at two or more locations outside the aircraft by the pitot-static system. The pressure of the static, or still air, is measured at a flush port where the air is not disturbed. On some aircraft, this air is sampled by static ports on the side of the electrically heated pitot-static head, such as the one in [the figure shown]. Other aircraft pick up the static pressure through flush ports on the side of the fuselage or the vertical fin. These ports are in locations proven by flight tests to be in undisturbed air, and they are normally paired, one on either side of the aircraft. This dual location prevents lateral movement of the aircraft from giving erroneous static pressure indications. The areas around the static ports may be heated with electric heater elements to prevent ice forming over the port and blocking the entry of the static air.
[FAA-H-8083-15, pg. 3-3] A sensitive altimeter is an aneroid barometer that measures the absolute pressure of the ambient air and displays it in terms of feet or meters above a selected pressure level. The sensitive element in a sensitive altimeter is a stack of evacuated, corrugated bronze aneroid capsules like those shown in [the figure]. The air pressure acting on these aneroids tries to compress them against their natural springiness, which tries to expand them. The result is that their thickness changes as the air pressure changes. Stacking several aneroids increases the dimension change as the pressure varies over the usable range of the instrument.
### Airspeed Measurement
[FAA-H-8083-15, pg. 3-7] An airspeed indicator is a differential pressure gauge that measures the dynamic pressure of the air through which the aircraft is flying. Dynamic pressure is the difference in the ambient static air pressure and the total, or ram, pressure caused by the motion of the aircraft through the air. These two pressures are taken from the pitot-static system. The mechanism of the airspeed indicator in [the figure] consists of a thin, corrugated phosphor-bronze aneroid, or diaphragm, that receives its pressure from the pitot tube. The instrument case is sealed and connected to the static ports. As the pitot pressure increases, or the static pressure decreases, the diaphragm expands, and this dimensional change is measured by a rocking shaft and a set of gears that drives a pointer across the instrument dial. Most airspeed indicators are calibrated in knots, or nautical miles per hour; some instruments show statute miles per hour, and some instruments show both.
There are many types of airspeed: indicated airspeed (IAS), calibrated airspeed (CAS), equivalent airspeed (EAS), and true airspeed (TAS).
• IAS. Indicated airspeed is shown on the dial of the instrument, uncorrected for instrument or system errors.
• CAS. Calibrated airspeed is the speed the aircraft is moving through the air, which is found by correcting IAS for instrument and position errors. The POH/AFM has a chart or graph to correct IAS for these errors and provide the correct CAS for the various flap and landing gear configurations.
• EAS. Equivalent airspeed is CAS corrected for compression of the air inside the pitot tube. Equivalent airspeed is the same as CAS in standard atmosphere at sea level. As the airspeed and pressure altitude increase, the CAS becomes higher than it should be and a correction for compression must be subtracted from the CAS.
• TAS. True airspeed is CAS corrected for nonstandard pressure and temperature. True airspeed and CAS are the same in standard atmosphere at sea level. But under nonstandard conditions, TAS is found by applying a correction for pressure altitude and temperature to the CAS.
### Indicated Airspeed (IAS)
An airspeed indicator is simply a pressure meter which uses static port pressure as a reference. In terms of Bernoulli, the meter reads q which can be found by subtract P from H; because H = P + q.
The airspeed indicator reads this speed which is termed, plainly enough, indicated airspeed.
### Calibrated Airspeed (CAS)
Figure: Typical position error correction, from [Hurt, pg. 12].
The speed indicated by the pitot tube might be in error because of the placement of the pitot tube relative to the relative wind. This is known as position error and when applicable, aircraft manuals will include a correction factor.
Calibrated airspeed is indicated airspeed adjusted for position error. Some aircraft, the G450 for example, make these corrections for the pilot so that the speed indicated on cockpit instruments is actually calibrated airspeed.
### Equivalent Airspeed (EAS)
Figure: Compressibility correction, from [Hurt, pg. 12].
Another error occurs when the aircraft travels fast enough to compress the air entering the pitot tube. A compressibility correction chart is used by some aircraft to factor this error..
### True Airspeed (TAS)
[Dole, pg. 25] True airspeed, coupled with ambient density ratio, produces the same dynamic pressure as EAS, coupled with standard sea level density ratio. That is
TAS2 σ = EAS2 σ0
But σ0 = 1, so:
TAS = EAS √ ( 1/ σ)
It can also be found using a standard table, shown here.
Figure: Density altitude correction, from [Hurt, pg. 13].
### Indicated to Calibrated to Equivalent to True Airspeed Conversion
As fledgling Air Force pilots or freshman aeronautical engineering students, the ability to convert from one type of airspeed to another was important. These days computers do it all for us. In the G450, for example, the pilot's primary flight instruments show CAS and TAS. In some aircraft, like the early T-37B, all you got was IAS and you couldn't fly outside the local area without TAS for risk of running out of fuel. For more about that, take a look at my Day One with Ice-T.
#### References:
14 CFR 25, Title 14: Aeronautics and Space, Federal Aviation Administration, Department of Transportation
Advisory Circular 61-107B, Aircraft Operations at Altitudes Above 25,000 Feet Mean Sea Level or Mach Numbers Greater than .75, 3/29/13, U.S. Department of Transportation
Air Training Command Manual 51-3, Aerodynamics for Pilots, 15 November 1963
Connolly, Thomas F., Dommasch, Daniel 0., and Sheryby, Sydney S., Airplane Aerodynamics, Pitman Publishing Corporation, New York, NY, 1951.
Davies, D. P., Handling the Big Jets, Civil Aviation Authority, Kingsway, London, 1985.
Dole, Charles E., Flight Theory and Aerodynamics, 1981, John Wiley & Sons, Inc, New York, NY, 1981.
FAA-H-8083-15, Instrument Flying Handbook, U.S. Department of Transportation, Flight Standards Service, 2001.
Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.
Hage, Robert E. and Perkins, Courtland D., Airplane Peformance Stability and Control, John Wiley & Sons, Inc., 1949.
Hurt, H. H., Jr., Aerodynamics for Naval Aviators, Skyhorse Publishing, Inc., New York NY, 2012.
ICAO Doc 7488/3 - Manual of the ICAO Standard Atmosphere, International Civil Aviation Organization, 1993
Technical Order 1T-38A-1, T-38A/B Flight Manual, USAF Series, 1 July 1978.
The Height of the Tropopause, B. Geerts and E. Linacre, University of Wyoming, Atmospheric Science, 11/97. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.8358920812606812, "perplexity": 2869.2658323288256}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00463.warc.gz"} |
https://socratic.org/questions/what-is-the-domain-of-f-x-5x-2-2x-1 | Precalculus
Topics
# What is the domain of f(x)=5x^2+2x-1?
So the domain is $\left(- \infty , + \infty\right)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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": 0.8493256568908691, "perplexity": 1012.5331265589193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370490497.6/warc/CC-MAIN-20200328074047-20200328104047-00413.warc.gz"} |
http://www.gbhatnagar.com/2018/11/ | Friday, November 16, 2018
A bibasic Heine transformation formula
While studying chapter 1 of Andrews and Berndt's Lost Notebook, Part II, I stumbled upon a bibasic Heine's transformation. A special case is Heine's 1847 transformation. Other special cases include an identity of Ramanujan (c. 1919), and a 1966 transformation formula of Andrews. Eventually, I realized that it follows from a Fundamental Lemma given by Andrews in 1966. Still, I'm happy to have rediscovered it. Using this formula one can find many identities proximal to Ramanujan's own $_2\phi_1$ transformations.
And of course, the multiple series extensions (some in this paper, and others appearing in another paper) are all new.
Here is a preprint.
Here is a video of a talk I presented at the Alladi 60 Conference. March 17-21, 2016.
Update (November 10, 2018). The multi-variable version has been accepted for publication in the Ramanujan Journal. This has been made open access. It is now available online, even though the volume and page number has not been decided yet. The title is: Heine's method and $A_n$ to $A_m$ transformation formulas.
Here is a reprint.
--
UPDATE (Feb 11, 2016). This has been published. Reference (perhaps to be modified later): A bibasic Heine transformation formula and Ramanujan's $_2\phi_1$ transformations, in Analytic Number Theory, Modular Forms and q-Hypergeometric Series, In honor of Krishna Alladi's 60th Birthday, University of Florida, Gainesville, Mar 2016, G. E. Andrews and F. G. Garvan (eds.), 99-122 (2017)
The book is available here. The front matter from the Springer site.
--
UPDATE (June 16, 2016). The paper has been accepted to appear in: Proceedings of the Alladi 60 conference held in Gainesville, FL. (Mar 2016), K. Alladi, G. E. Andrews and F. G. Garvan (eds.) | {"extraction_info": {"found_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": 0.8182755708694458, "perplexity": 2162.011622061692}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500251.38/warc/CC-MAIN-20230205094841-20230205124841-00068.warc.gz"} |
http://mca.ignougroup.com/2017/04/answers-kripke-models-evaluating.html | World's most popular travel blog for travel bloggers.
# [Answers] Kripke Models - evaluating the meaning of $\Box\Box p$
, ,
Problem Detail:
In Kripke models the evaluation of $x \vdash \Box p$ would be that every world reachable from $x$ satisfies $p$.
But how would the truth of $\Box\Box p$ be evaluated in Kripke models?
#### Answered By : David Richerby
This is an unfortunate use of the word “reachable”, in that Kripke structures are graphs but “reachable” in a Kripke structure is not the same thing as “reachable” in a graph. Let us avoid this confusion by saying that a state $y$ is a successor of state $x$ if there is a (directed) edge $(x,y)$ in the structure.
Now, the semantics of modal logic says that $\Box\varphi$ is true at state $x$ if, and only if, $\varphi$ is true at every successor of $x$. Informally, $\Box\varphi$ means, “$\varphi$ is true everywhere I can get to from here in one step.” So, to understand the meaning of $\Box\Box p$, just substitute $\Box p$ for $\varphi$:
• $\Box p$ is true at every successor
• $p$ is true at every successor of every successor.
In other words, $p$ is true everywhere I can get in two steps. Note that this is not, in general, the same thing as $\Box p$, which means that $p$ is true after one step. Indeed, one can show that $(\Box \varphi)\rightarrow (\Box\Box\varphi)$ is a tautology in a particular Kripke frame if, and only if, the successor relation is transitive.
Note that, without assuming transitivity, basic modal logic with only $\Box$ and $\Diamond$ has no way of expressing “$\varphi$ is true everywhere that can be reached from here, in any number of steps” (i.e., the usual graph-theoretic meaning of “reachable”). | {"extraction_info": {"found_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": 0.939094603061676, "perplexity": 438.65873326179786}, "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-2019-18/segments/1555578527135.18/warc/CC-MAIN-20190419041415-20190419063415-00382.warc.gz"} |
https://atcoder.jp/contests/abc216/tasks/abc216_h | Contest Duration: - (local time) (100 minutes) Back to Home
H - Random Robots /
Time Limit: 2 sec / Memory Limit: 1024 MB
Score : 600 points
### Problem Statement
There are K robots on a number line. The i-th robot (1 \leq i \leq K) is initially at the coordinate x_i.
The following procedure is going to take place exactly N times.
• Each robot chooses to move or not with probability \frac{1}{2} each. The robots that move will simultaneously go the distance of 1 in the positive direction, and the other robots will remain still.
Here, all probabilistic decisions are independent.
Find the probability that no two robots meet, that is, there are never two or more robots at the same coordinate at the same time throughout the procedures, modulo 998244353 (see Notes).
### Notes
It can be proved that the probability in question is always a rational number. Additionally, under the Constraints in this problem, when that value is represented as \frac{P}{Q} using two coprime integers P and Q, it can be proved that there uniquely exists an integer R such that R \times Q \equiv P\pmod{998244353} and 0 \leq R \lt 998244353. Find this R.
### Constraints
• 2 \leq K \leq 10
• 1 \leq N \leq 1000
• 0 \leq x_1 \lt x_2 \lt \cdots \lt x_K \leq 1000
• All values in input are integers.
### Input
Input is given from Standard Input in the following format:
K N
x_1 x_2 \ldots x_K
### Sample Input 1
2 2
1 2
### Sample Output 1
374341633
The probability in question is \frac{5}{8}.
We have 374341633 \times 8 \equiv 5\pmod{998244353}, so you should print 374341633.
### Sample Input 2
2 2
10 100
### Sample Output 2
1
The probability in question may be 1.
### Sample Input 3
10 832
73 160 221 340 447 574 720 742 782 970
### Sample Output 3
553220346
### 問題文
これから以下の操作をちょうど N 回行います。
• K 個のロボットそれぞれについて、「進む」か「止まる」かを確率 \frac{1}{2} で決める。「進む」と決めたロボットたちは同時に正の方向に 1 進み、「止まる」と決めたロボットたちはその場から動かない。
ただし、すべての確率的な決定は独立であるとします。
### 制約
• 2 \leq K \leq 10
• 1 \leq N \leq 1000
• 0 \leq x_1 \lt x_2 \lt \cdots \lt x_K \leq 1000
• 入力は全て整数
### 入力
K N
x_1 x_2 \ldots x_K
### 入力例 1
2 2
1 2
### 出力例 1
374341633
374341633 \times 8 \equiv 5\pmod{998244353} ですので、374341633 を出力します。
### 入力例 2
2 2
10 100
### 出力例 2
1
### 入力例 3
10 832
73 160 221 340 447 574 720 742 782 970
### 出力例 3
553220346 | {"extraction_info": {"found_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": 0.9983696341514587, "perplexity": 3295.3057389970104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00742.warc.gz"} |
https://proceedings.neurips.cc/paper/2018/hash/03c6b06952c750899bb03d998e631860-Abstract.html | #### Authors
Simon S. Du, Yining Wang, Xiyu Zhai, Sivaraman Balakrishnan, Russ R. Salakhutdinov, Aarti Singh
#### Abstract
A widespread folklore for explaining the success of Convolutional Neural Networks (CNNs) is that CNNs use a more compact representation than the Fully-connected Neural Network (FNN) and thus require fewer training samples to accurately estimate their parameters. We initiate the study of rigorously characterizing the sample complexity of estimating CNNs. We show that for an $m$-dimensional convolutional filter with linear activation acting on a $d$-dimensional input, the sample complexity of achieving population prediction error of $\epsilon$ is $\widetilde{O(m/\epsilon^2)$, whereas the sample-complexity for its FNN counterpart is lower bounded by $\Omega(d/\epsilon^2)$ samples. Since, in typical settings $m \ll d$, this result demonstrates the advantage of using a CNN. We further consider the sample complexity of estimating a one-hidden-layer CNN with linear activation where both the $m$-dimensional convolutional filter and the $r$-dimensional output weights are unknown. For this model, we show that the sample complexity is $\widetilde{O}\left((m+r)/\epsilon^2\right)$ when the ratio between the stride size and the filter size is a constant. For both models, we also present lower bounds showing our sample complexities are tight up to logarithmic factors. Our main tools for deriving these results are a localized empirical process analysis and a new lemma characterizing the convolutional structure. We believe that these tools may inspire further developments in understanding CNNs. | {"extraction_info": {"found_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": 0.9265298843383789, "perplexity": 645.2511465936991}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046150266.65/warc/CC-MAIN-20210724125655-20210724155655-00565.warc.gz"} |
https://www.physicsforums.com/threads/field-theory.213796/ | # Field Theory
1. Feb 7, 2008
### johnson123
1. The problem statement, all variables and given/known data
Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x],
and g has degree n.
Its clear that F[x]/( g(x) ) is a vector space and that
B= (1,$$x^{2}$$,.....,$$x^{n-1}$$) spans F[x]/( g(x) ),
but im having trouble showing that B is linearly independent
I realize this is pretty much a HW problem and it should be in the HW section, but I
read a post from one of the pf mentors noting that for gradlevel/seniorlevel problems
you might have a chance at a response from the non hw sections. thanks for any suggestions.
2. Feb 7, 2008
### Hurkyl
Staff Emeritus
Well, what happens if they are linearly dependent, so that a nontrivial linear combination of them is equal to zero in F[x] / (g(x))?
3. Feb 7, 2008
### ejungkurth
It's not clear that you have tied B to either F[x] or g(x). First relate B to F and g. Assume for the moment that I am not the person who doesn't have the answer. | {"extraction_info": {"found_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": 0.893610954284668, "perplexity": 1346.3973066243452}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988720941.32/warc/CC-MAIN-20161020183840-00381-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.delorie.com/gnu/docs/avl/libavl_70.html | www.delorie.com/gnu/docs/avl/libavl_70.html search
GNU libavl 2.0.1
[ < ] [ > ] [ << ] [ Up ] [ >> ] [Top] [Contents] [Index] [ ? ]
## 5.12 Balance
Sometimes binary trees can grow to become much taller than their optimum height. For example, the following binary tree was one of the tallest from a sample of 100 15-node trees built by inserting nodes in random order:
The average number of comparisons required to find a random node in this tree is \ASCII\{(1 + 2 + (3 \times 2) + (4 \times 4) + (5 \times 4) + 6 + 7 + 8) / 15 = 4.4, (1 + 2 + (3 * 2) + (4 * 4) + (5 * 4) + 6 + 7 + 8) / 15 = 4.4} comparisons. In contrast, the corresponding optimal binary tree, shown below, requires only \ASCII\{(1 + (2 \times 2) + (3 \times 4) + (4 \times 8))/15 = 3.3, (1 + (2 * 2) + (3 * 4) + (4 * 8))/15 = 3.3} comparisons, on average. Moreover, the optimal tree requires a maximum of 4, as opposed to 8, comparisons for any search:
Besides this inefficiency in time, trees that grow too tall can cause inefficiency in space, leading to an overflow of the stack in bst_t_next(), bst_copy(), or other functions. For both reasons, it is helpful to have a routine to rearrange a tree to its minimum possible height, that is, to balance (see balance) the tree.
The algorithm we will use for balancing proceeds in two stages. In the first stage, the binary tree is "flattened" into a pathological, linear binary tree, called a "vine." In the second stage, binary tree structure is restored by repeatedly "compressing" the vine into a minimal-height binary tree.
Here's a top-level view of the balancing function:
<@xref{\NODE\, , BST to vine function.>,89} <@xref{\NODE\, , Vine to balanced BST function.>,90} void bst_balance (struct bst_table *tree) { assert (tree != NULL); tree_to_vine (tree); vine_to_tree (tree); tree->bst_generation++; }
This code is included in @refalso{29
/* Special BST functions. */ void bst_balance (struct bst_table *tree);
This code is included in @refalso{24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9052618145942688, "perplexity": 1785.9523985452315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934806720.32/warc/CC-MAIN-20171123031247-20171123051247-00004.warc.gz"} |
https://en.wikipedia.org/wiki/Pythagorean_theorem_(baseball) | # Pythagorean expectation
(Redirected from Pythagorean theorem (baseball))
Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team "should" have won based on the number of runs they scored and allowed. Comparing a team's actual and Pythagorean winning percentage can be used to evaluate how lucky that team was (by examining the variation between the two winning percentages). The name comes from the formula's resemblance to the Pythagorean theorem.[1]
The basic formula is:
${\displaystyle \mathrm {Win\ Ratio} ={\frac {{\text{runs scored}}^{2}}{{\text{runs scored}}^{2}+{\text{runs allowed}}^{2}}}={\frac {1}{1+({\text{runs allowed}}/{\text{runs scored}})^{2}}}}$
where Win Ratio is the winning ratio generated by the formula. The expected number of wins would be the expected winning ratio multiplied by the number of games played.
## Empirical origin
Empirically, this formula correlates fairly well with how baseball teams actually perform. However, statisticians since the invention of this formula found it to have a fairly routine error, generally about three games off. For example, in 2002, the New York Yankees scored 897 runs and allowed 697 runs. According to James' original formula, the Yankees should have won 62.35% of their games.
${\displaystyle \mathrm {Win} ={\frac {{\text{897}}^{2}}{{\text{897}}^{2}+{\text{697}}^{2}}}=0.623525865}$
Based on a 162-game season, the Yankees should have won 101.01 games. The 2002 Yankees actually went 103–58.[2]
In efforts to fix this error, statisticians have performed numerous searches to find the ideal exponent.
If using a single-number exponent, 1.83 is the most accurate, and the one used by baseball-reference.com.[3] The updated formula therefore reads as follows:
${\displaystyle \mathrm {Win} ={\frac {{\text{runs scored}}^{1.83}}{{\text{runs scored}}^{1.83}+{\text{runs allowed}}^{1.83}}}={\frac {1}{1+({\text{runs allowed}}/{\text{runs scored}})^{1.83}}}}$
The most widely known is the Pythagenport formula[4] developed by Clay Davenport of Baseball Prospectus:
${\displaystyle \mathrm {Exponent} =1.50\cdot \log \left({\frac {R+RA}{G}}\right)+0.45}$
He concluded that the exponent should be calculated from a given team based on the team's runs scored (R), runs allowed (RA), and games (G). By not reducing the exponent to a single number for teams in any season, Davenport was able to report a 3.9911 root-mean-square error as opposed to a 4.126 root-mean-square error for an exponent of 2.[4]
Less well known but equally (if not more) effective is the Pythagenpat formula, developed by David Smyth.[5]
${\displaystyle \mathrm {Exponent} =\left({\frac {R+RA}{G}}\right)^{.287}}$
Davenport expressed his support for this formula, saying:
After further review, I (Clay) have come to the conclusion that the so-called Smyth/Patriot method, aka Pythagenpat, is a better fit. In that, X = ((rs + ra)/g)0.285, although there is some wiggle room for disagreement in the exponent. Anyway, that equation is simpler, more elegant, and gets the better answer over a wider range of runs scored than Pythagenport, including the mandatory value of 1 at 1 rpg.[6]
These formulas are only necessary when dealing with extreme situations in which the average number of runs scored per game is either very high or very low. For most situations, simply squaring each variable yields accurate results.
There are some systematic statistical deviations between actual winning percentage and expected winning percentage, which include bullpen quality and luck. In addition, the formula tends to regress toward the mean, as teams that win a lot of games tend to be underrepresented by the formula (meaning they "should" have won fewer games), and teams that lose a lot of games tend to be overrepresented (they "should" have won more).
## "Second-order" and "third-order" wins
In their Adjusted Standings Report,[7] Baseball Prospectus refers to different "orders" of wins for a team. The basic order of wins is simply the number of games they have won. However, because a team's record may not reflect its true talent due to luck, different measures of a team's talent were developed.
First-order wins, based on pure run differential, are the number of expected wins generated by the "pythagenport" formula (see above). In addition, to further filter out the distortions of luck, Sabermetricians can also calculate a team's expected runs scored and allowed via a runs created-type equation (the most accurate at the team level being Base Runs). These formulas result in the team's expected number of runs given their offensive and defensive stats (total singles, doubles, walks, etc.), which helps to eliminate the luck factor of the order in which the team's hits and walks came within an inning. Using these stats, sabermetricians can calculate how many runs a team "should" have scored or allowed.
By plugging these expected runs scored and allowed into the pythagorean formula, one can generate second-order wins, the number of wins a team deserves based on the number of runs they should have scored and allowed given their component offensive and defensive statistics. Third-order wins are second-order wins that have been adjusted for strength of schedule (the quality of the opponent's pitching and hitting). Second- and third-order winning percentage has been shown[according to whom?] to predict future actual team winning percentage better than both actual winning percentage and first-order winning percentage.
## Theoretical explanation
Initially the correlation between the formula and actual winning percentage was simply an experimental observation. In 2003, Hein Hundal provided an inexact derivation of the formula and showed that the Pythagorean exponent was approximately 2/(σπ) where σ was the standard deviation of runs scored by all teams divided by the average number of runs scored.[8] In 2006, Professor Steven J. Miller provided a statistical derivation of the formula[9] under some assumptions about baseball games: if runs for each team follow a Weibull distribution and the runs scored and allowed per game are statistically independent, then the formula gives the probability of winning.[9]
More simply, the Pythagorean formula with exponent 2 follows immediately from two assumptions: that baseball teams win in proportion to their "quality", and that their "quality" is measured by the ratio of their runs scored to their runs allowed. For example, if Team A has scored 50 runs and allowed 40, its quality measure would be 50/40 or 1.25. The quality measure for its (collective) opponent team B, in the games played against A, would be 40/50 (since runs scored by A are runs allowed by B, and vice versa), or 0.8. If each team wins in proportion to its quality, A's probability of winning would be 1.25 / (1.25 + 0.8), which equals 50^2 / (50^2 + 40^2), the Pythagorean formula. The same relationship is true for any number of runs scored and allowed, as can be seen by writing the "quality" probability as [50/40] / [ 50/40 + 40/50], and clearing fractions.
The assumption that one measure of the quality of a team is given by the ratio of its runs scored to allowed is both natural and plausible; this is the formula by which individual victories (games) are determined. [There are other natural and plausible candidates for team quality measures, which, assuming a "quality" model, lead to corresponding winning percentage expectation formulas that are roughly as accurate as the Pythagorean ones.] The assumption that baseball teams win in proportion to their quality is not natural, but is plausible. It is not natural because the degree to which sports contestants win in proportion to their quality is dependent on the role that chance plays in the sport. If chance plays a very large role, then even a team with much higher quality than its opponents will win only a little more often than it loses. If chance plays very little role, then a team with only slightly higher quality than its opponents will win much more often than it loses. The latter is more the case in basketball, for various reasons, including that many more points are scored than in baseball (giving the team with higher quality more opportunities to demonstrate that quality, with correspondingly fewer opportunities for chance or luck to allow the lower-quality team to win.)
Baseball has just the right amount of chance in it to enable teams to win roughly in proportion to their quality, i.e. to produce a roughly Pythagorean result with exponent two. Basketball's higher exponent of around 14 (see below) is due to the smaller role that chance plays in basketball. And the fact that the most accurate (constant) Pythagorean exponent for baseball is around 1.83, slightly less than 2, can be explained by the fact that there is (apparently) slightly more chance in baseball than would allow teams to win in precise proportion to their quality. Bill James realized this long ago when noting that an improvement in accuracy on his original Pythagorean formula with exponent two could be realized by simply adding some constant number to the numerator, and twice the constant to the denominator. This moves the result slightly closer to .500, which is what a slightly larger role for chance would do, and what using the exponent of 1.83 (or any positive exponent less than two) does as well. Various candidates for that constant can be tried to see what gives a "best fit" to real life data.
The fact that the most accurate exponent for baseball Pythagorean formulas is a variable that is dependent on the total runs per game is also explainable by the role of chance, since the more total runs scored, the less likely it is that the result will be due to chance, rather than to the higher quality of the winning team having been manifested during the scoring opportunities. The larger the exponent, the farther away from a .500 winning percentage is the result of the corresponding Pythagorean formula, which is the same effect that a decreased role of chance creates. The fact that accurate formulas for variable exponents yield larger exponents as the total runs per game increases is thus in agreement with an understanding of the role that chance plays in sports.
In his 1981 Baseball Abstract, James explicitly developed another of his formulas, called the log5 formula (which has since proven to be empirically accurate), using the notion of 2 teams having a face-to-face winning percentage against each other in proportion to a "quality" measure. His quality measure was half the team's "wins ratio" (or "odds of winning"). The wins ratio or odds of winning is the ratio of the team's wins against the league to its losses against the league. [James did not seem aware at the time that his quality measure was expressible in terms of the wins ratio. Since in the quality model any constant factor in a quality measure eventually cancels, the quality measure is today better taken as simply the wins ratio itself, rather than half of it.] He then stated that the Pythagorean formula, which he had earlier developed empirically, for predicting winning percentage from runs, was "the same thing" as the log5 formula, though without a convincing demonstration or proof. His purported demonstration that they were the same boiled down to showing that the two different formulas simplified to the same expression in a special case, which is itself treated vaguely, and there is no recognition that the special case is not the general one. Nor did he subsequently promulgate to the public any explicit, quality-based model for the Pythagorean formula. As of 2013, there is still little public awareness in the sabermetric community that a simple "teams win in proportion to quality" model, using the runs ratio as the quality measure, leads directly to James's original Pythagorean formula.
In the 1981 Abstract, James also says that he had first tried to create a "log5" formula by simply using the winning percentages of the teams in place of the runs in the Pythagorean formula, but that it did not give valid results. The reason, unknown to James at the time, is that his attempted formulation implies that the relative quality of teams is given by the ratio of their winning percentages. Yet this cannot be true if teams win in proportion to their quality, since a .900 team wins against its opponents, whose overall winning percentage is roughly .500, in a 9 to 1 ratio, rather than the 9 to 5 ratio of their .900 to .500 winning percentages. The empirical failure of his attempt led to his eventual, more circuitous (and ingenious) and successful approach to log5, which still used quality considerations, though without a full appreciation of the ultimate simplicity of the model and of its more general applicability and true structural similarity to his Pythagorean formula.
American sports executive Daryl Morey was the first to adapt James' Pythagorean expectation to professional basketball while a researcher at STATS, Inc.. He found that using 13.91 for the exponents provided an acceptable model for predicting won-lost percentages:
${\displaystyle \mathrm {Win} ={\frac {{\text{points for}}^{13.91}}{{\text{points for}}^{13.91}+{\text{points against}}^{13.91}}}.}$
Daryl's "Modified Pythagorean Theorem" was first published in STATS Basketball Scoreboard, 1993-94.[10]
Noted basketball analyst Dean Oliver also applied James' Pythagorean theory to professional basketball. The result was similar.
Another noted basketball statistician, John Hollinger, uses a similar Pythagorean formula, except with 16.5 as the exponent.
## Use in pro football
The formula has also been used in pro football by football stat website and publisher Football Outsiders, where it is known as Pythagorean projection. The formula is used with an exponent of 2.37 and gives a projected winning percentage. That winning percentage is then multiplied by 16 (for the number of games played in an NFL season), to give a projected number of wins. This projected number given by the equation is referred to as Pythagorean wins.
${\displaystyle {\text{Pythagorean wins}}={\frac {{\text{Points For}}^{2.37}}{{\text{Points For}}^{2.37}+{\text{Points Against}}^{2.37}}}\times 16.}$
The 2011 edition of Football Outsiders Almanac[11] states, "From 1988 through 2004, 11 of 16 Super Bowls were won by the team that led the NFL in Pythagorean wins, while only seven were won by the team with the most actual victories. Super Bowl champions that led the league in Pythagorean wins but not actual wins include the 2004 Patriots, 2000 Ravens, 1999 Rams and 1997 Broncos."
Although Football Outsiders Almanac acknowledges that the formula had been less-successful in picking Super Bowl participants from 2005–2008, it reasserted itself in 2009 and 2010. Furthermore, "[t]he Pythagorean projection is also still a valuable predictor of year-to-year improvement. Teams that win a minimum of one full game more than their Pythagorean projection tend to regress the following year; teams that win a minimum of one full game less than their Pythagoerean projection tend to improve the following year, particularly if they were at or above .500 despite their underachieving. For example, the 2008 New Orleans Saints went 8-8 despite 9.5 Pythagorean wins, hinting at the improvement that came with the next year's championship season."
## Use in ice hockey
In 2013, statistician Kevin Dayaratna and mathematician Steven J. Miller provided theoretical justification for applying the Pythagorean Expectation to ice hockey. In particular, they found that by making the same assumptions that Miller made in his 2007 study about baseball, specifically that goals scored and goals allowed follow statistically independent Weibull distributions, that the Pythagorean Expectation works just as well for ice hockey as it does for baseball. The Dayaratna and Miller study verified the statistical legitimacy of making these assumptions and estimated the Pythagorean exponent for ice hockey to be slightly above 2.[12]
## Notes
1. ^ "The Game Designer: Pythagoras Explained". Retrieved 7 May 2016.
2. ^ "2002 New York Yankees". Baseball-Reference.com. Retrieved 7 May 2016.
3. ^ "Frequently Asked Questions". Baseball-Reference.com. Retrieved 7 May 2016.
4. ^ a b "Baseball Prospectus - Revisiting the Pythagorean Theorem". Baseball Prospectus. Retrieved 7 May 2016.
5. ^ "W% Estimators". Retrieved 7 May 2016.
6. ^ "Baseball Prospectus - Glossary". Retrieved 7 May 2016.
7. ^ "Baseball Prospectus - Adjusted Standings". Retrieved 7 May 2016.
8. ^ Hundal, Hein. "Derivation of James Pythagorean Formula (Long)".
9. ^ a b Miller (2007). "A Derivation of the Pythagorean Won-Loss Formula in Baseball". Chance. 20: 40–48. arXiv:. Bibcode:2005math......9698M. doi:10.1080/09332480.2007.10722831.
10. ^ Dewan, John; Zminda, Don; STATS, Inc. Staff (October 1993). STATS Basketball Scoreboard, 1993-94. STATS, Inc. p. 17. ISBN 0-06-273035-5.
11. ^ Football Outsiders Almanac 2011 (ISBN 978-1-4662-4613-3), p.xviii
12. ^ Dayaratna, Kevin; Miller, Steven J. (2013). "The Pythagorean Won-Loss Formula and Hockey: A Statistical Justification for Using the Classic Baseball Formula as an Evaluative Tool in Hockey" (PDF). The Hockey Research Journal 2012/13. XVI: 193–209. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "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": 0.8058832883834839, "perplexity": 1909.8848655074144}, "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-2017-47/segments/1510934804610.37/warc/CC-MAIN-20171118040756-20171118060756-00015.warc.gz"} |
https://www.lessonplanet.com/teachers/using-electricity-4th-5th | # Using Electricity
In this electricity instructional activity, students study the circuit diagram color the light bulb yellow and tick the box if they think there is electricity for that example. Students cross out the boxes with no electricity current.
Concepts
Resource Details | {"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": 0.9108841419219971, "perplexity": 3911.972403664053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1500549424884.51/warc/CC-MAIN-20170724142232-20170724162232-00134.warc.gz"} |
http://tex.stackexchange.com/questions/175746/how-to-force-a-label-to-be-a-given-string | # How to force a label to be a given string?
Normally I issue a command like \label{foo}, and this means that foo gets entered into a symbol table somewhere that says that it equals 14, or xiv, or aa, or whatever. These symbols are generated and incremented automatically, and to change the numbering system, I can use packages like alphalph.
Suppose I want to force foo to evaluate to some arbitrary string, such as "Socrates." E.g., I want something like \mynameis{Socrates}\label{foo}, and then when I say \ref{foo}, the result will not be "14" but "Socrates."
I want to completely bypass the normal system involving counters. If Socrates is the name of a chapter, then I don't expect LaTeX to be smart enough to automatically name the next chapter Plato. So, e.g.:
\chapter{The life of Socrates}\mynameis{Socrates}\label{foo}
...
\chapter{The life of Plato}\label{bar}
...
This is an error on my part. It's OK with me if bar is set to garbage, or if LaTeX chokes, or if bar is set to some arbitrary string such as Socrates2.
Is there some way to do this?
-
## 1 Answer
This is fairly easy:
\documentclass{article}
\makeatletter
\newcommand{\mynameis}[1]{#1\renewcommand{\@currentlabel}{#1}}
\makeatother
\begin{document}
\mynameis{Socrates}\label{foo}%
Do you know about \ref{foo}?
\end{document}
\@currentlabel is the macro that is stored when you use \label (in addition to \thepage). So, all we do is update this to our liking before calling \label.
If you wish for this to be compatible with hyperref, you could issue an additional \phantomsection so the hyperlink is correct (and perhaps also update \@currentlabelname)
\documentclass{article}
\usepackage{hyperref}
\makeatletter
\newcommand{\mynameis}[1]{%
\phantomsection#1% Mark hyperlink
\renewcommand{\@currentlabel}{#1}%
\renewcommand{\@currentlabelname}{#1}}
\makeatother
\begin{document}
\mynameis{Socrates}\label{foo}%
Do you know about \ref{foo} or \nameref{foo}?
\end{document}
For more on cross-referencing, see Understanding how references and labels work.
- | {"extraction_info": {"found_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": 0.8509626984596252, "perplexity": 2257.114812524598}, "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-2015-35/segments/1440645265792.74/warc/CC-MAIN-20150827031425-00205-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/integral-of-e-constant-x-2.9061/ | # Integral of e^-(constant)x^2
1. Nov 16, 2003
### Kristen
Need the integral of e^-(constant)x^2.........don't want to use guass integral trick
2. Nov 16, 2003
### Ambitwistor
3. Nov 17, 2003
### mathman
The indefinite integral cannot be expressed in simple form. Usually it is given in terms of a function called "erf", which is simply a standard form (constant=1/2) integral. | {"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": 0.9998171925544739, "perplexity": 4256.676461308775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743968.63/warc/CC-MAIN-20181118052443-20181118074443-00256.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10B_Problems/Problem_18&direction=prev&oldid=141353 | # 2004 AMC 10B Problems/Problem 18
## Problem
In the right triangle , we have , , and . Points , , and are located on , , and , respectively, so that , , and . What is the ratio of the area of to that of ?
## Solution 1
Let . Because is divided into four triangles, .
Because of triangle area, .
and , so .
, so .
## Solution 2
First of all, note that , and therefore .
Draw the height from onto as in the picture below:
Now consider the area of . Clearly the triangles and are similar, as they have all angles equal. Their ratio is , hence . Now the area of can be computed as = .
Similarly we can find that as well.
Hence , and the answer is .
## Solution 3 (Coordinate Geometry)
We will put triangle ACE on a xy-coordinate plane with C being the origin. The area of triangle ACE is 96. To find the area of triangle DBF, let D be (4, 0), let B be (0, 9), and let F be (12, 3). You can then use the shoelace theorem to find the area of DBF, which is 42.
## Solution 4
You can also place a point on such that is , creating trapezoid . Then, you can find the area of the trapezoid, subtract the area of the two right triangles and , divide by the area of , and get the ratio of .
## Solution 5
It is well known that for when two triangles share an angle, the two sides around the shared angle is proportional to the areas of each of the two triangles.
We can find all the ratios of the triangles except for and then subtract from
In this case, we have sharing with .
Therefore, we have
Also note that shares with .
Therefore, we have
Lastly, note that shares with .
Therefore, we have
Thus, the ratio of to is
~mathboy282
## Solution 6 (Wooga Looga Theorem)
We know that , so by the Wooga Looga Theorem we have . | {"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": 0.9909345507621765, "perplexity": 698.8075182999422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039594808.94/warc/CC-MAIN-20210423131042-20210423161042-00384.warc.gz"} |
https://www.clutchprep.com/chemistry/practice-problems/90229/indicate-the-number-of-significant-figures-in-each-of-the-following-measured-qua | # Problem: Indicate the number of significant figures in each of the following measured quantities.3.774 km
###### FREE Expert Solution
We are asked to identify the number of significant figures.
significant figures → digits that carry meaningful contributions to its measurement resolution
84% (139 ratings)
###### Problem Details
Indicate the number of significant figures in each of the following measured quantities.
3.774 km | {"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": 0.8834533095359802, "perplexity": 2408.9589421070477}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00738.warc.gz"} |
https://hal.inria.fr/hal-01283619 | # On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition
2 EDP - Equations aux dérivées partielles
IECL - Institut Élie Cartan de Lorraine
Abstract : We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resol-vent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.
Document type :
Journal articles
Domain :
https://hal.inria.fr/hal-01283619
Contributor : Renata BUNOIU Connect in order to contact the contributor
Submitted on : Saturday, March 5, 2016 - 7:18:40 PM
Last modification on : Saturday, October 16, 2021 - 11:18:03 AM
### Citation
Denis Borisov, Renata Bunoiu, Giuseppe Cardone. On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition. Annales de l'Institut Henri Poincaré, 2010, 11 (8), pp.1591-1627. ⟨10.1007/s00023-010-0065-0⟩. ⟨hal-01283619⟩
Record views | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8742558360099792, "perplexity": 1316.6995057326822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103033816.0/warc/CC-MAIN-20220624213908-20220625003908-00379.warc.gz"} |
http://mathhelpforum.com/calculus/142916-please-help-ap-calc-test-tomorrow-print.html | • May 3rd 2010, 07:53 PM
helpmeee
is there a formula for integrating fractions??? sorry if this is hard to understand but i need to find the INTEGRAL of x/(2+e^x).
• May 3rd 2010, 08:10 PM
surjective
integrating fractions
Hello,
When facing such problems you should realize that:
$\int \frac{x}{2+e^{x}}dx=\int \left(\frac{1}{2+e^{x}}\cdot x \right)dx$
If you define $f(x)=\frac{1}{2+e^{x}}$ and $g(x)=x$ then you have:
$\int f(x)g(x) dx$
From here you can simply apply the rule of integration by parts.
• May 3rd 2010, 08:22 PM
helpmeee
• May 3rd 2010, 08:27 PM
CalculusCrazed
• May 3rd 2010, 08:34 PM
helpmeee
That made no sense to me.
Please show me with my problem
• May 3rd 2010, 08:49 PM
lovek323
I don't think integration by parts should be used here.
Are you sure this was the question? This integral is rather difficult to evaluate. Cf. Wolfram Alpha
• May 3rd 2010, 08:51 PM
Debsta
Quote:
Originally Posted by lovek323
I don't think integration by parts should be used here.
Are you sure this was the question? This integral is rather difficult to evaluate. Cf. Wolfram Alpha
Yes state the exact question you are having problems with.
• May 3rd 2010, 09:01 PM
CalculusCrazed
Yeah, because I was trying to do this by parts. It is not easy by any means.
• May 3rd 2010, 09:20 PM
There's always an integral involving Li(x) on the AP exam, right you guys? The AP board loves those almost as much as Si(x).
• May 3rd 2010, 09:36 PM
CalculusCrazed
Quote:
There's always an integral involving Li(x) on the AP exam, right you guys? The AP board loves those almost as much as Si(x).
I never took AP and we didn't learn that in calc 1 or 2. What is Li(x) and Si(x)?
• May 4th 2010, 04:10 AM
mr fantastic
Quote:
Originally Posted by helpmeee
is there a formula for integrating fractions??? sorry if this is hard to understand but i need to find the INTEGRAL of x/(2+e^x).
Are there integral terminals, that is, is it a definite integral?
• May 4th 2010, 04:48 AM
skeeter
Quote:
Originally Posted by helpmeee
is there a formula for integrating fractions??? sorry if this is hard to understand but i need to find the INTEGRAL of x/(2+e^x).
no such integration is required on either the AB or BC exam unless it is a definite integral on the calculator part of the exam. | {"extraction_info": {"found_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": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.99075847864151, "perplexity": 1586.409832300411}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187824618.72/warc/CC-MAIN-20171021062002-20171021082002-00374.warc.gz"} |
http://amj.math.stonybrook.edu/html-articles/Files/15-12/ | Arnold Mathematical Journal Research Contribution
Received: 17 December 2014 / Accepted: 4 April 2015
# On an Equivariant Version of the Zeta Function of a Transformation
S. M. Gusein-Zade Faculty of Mathematics and Mechanics, GSP-1 Moscow State University Moscow 119991 Russia I. Luengo Present address: ICMAT-Institute of Mathematical Sciences Madrid Spain
Faculty of Mathematical Sciences Complutense University of Madrid 28040 Madrid Spain
A. Melle-Hernández Present address: ICMAT-Institute of Mathematical Sciences Madrid Spain
Faculty of Mathematical Sciences Complutense University of Madrid 28040 Madrid Spain
### Abstract
Earlier the authors offered an equivariant version of the classical monodromy zeta function of a $G$-invariant function germ with a finite group $G$ as a power series with the coefficients from the Burnside ring of the group $G$ tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a $G$-equivariant transformation given by W. Lück and J. Rosenberg. Here we use another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring itself. We give an A’Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones.
#### Keywords
Equivariant Lefschetz numbers, Zeta functions, Burnside ring
#### Mathematics Subject Classification
32S05, 32S50, 57R91, 58K10
## 1 Introduction
Many topological invariants have equivariant versions for spaces with actions of a group $G$, say, a finite one. For example, in [Verdier1973], an equivariant version of the Euler characteristic is an element of the Grothendieck ring of ${\mathbb{Z}}[G]$- or ${\mathbb{Q}}[G]$-modules. In tom Dieck ( [tom Dieck1979] , Section 5.4) it is defined as an element of the Burnside ring of the group $G$ (that is of the Grothendieck ring $K_{0}({\mbox{f.$G$-s.}})$ of finite $G$-sets). Applying these concepts to the Milnor fibre, one gets an equivariant version of the Milnor number of a $G$-invariant function-germ. For example, in [Wall1980] it is an element of the ring of virtual representations of the group $G$.
An important invariant of a germ of a holomorphic function (on $({\mathbb{C}}^{n},0)$ or on a germ of a complex analytic variety) is its monodromy and its corresponding zeta function, see e.g. [Arnold et al.1988]. It is defined as the zeta function of the classical monodromy transformation on the Milnor fibre. A number of statements have natural formulations in terms of monodromy zeta functions. As an example one can indicate the well-known monodromy conjecture: see, e.g., [Denef and Loeser1992]. The monodromy zeta function is connected with a number of other invariants, topological and analytic ones. For example, in [Gusein-Zade et al.1999], it was shown that, for an irreducible plane curve singularity, the monodromy zeta function of the corresponding function-germ coincides with the Poincaré series of the natural filtration on the local ring defined by the curve valuation. There are generalizations of this fact to some other situations [(see, e.g., a survey in [Gusein-Zade2010]]. In all these cases one has no intrinsic explanation of the relation. The relation is obtained by independent computation of the right and left hand sides of it in the same terms and comparison of the obtained results.
Generalizations of relations of this sort to equivariant settings could help to understand the general framework. For example, in [Ebeling and Gusein-Zade2012b] an equivariant version of a relation obtained earlier gave a better understanding of the role of the Saito duality in it. This leads to the desire to define equivariant analogues of monodromy zeta functions and of the Poincaré series of filtrations. These problem is not trivial and equivariant analogues are not unique. Moreover, up to now there were no definitions of equivariant analogues of monodromy zeta functions and of the Poincaré series which were elements of the same rings. For example, in [Campillo et al.2007]; [Campillo et al.2013], there were offered different approaches to equivariant Poincaré series. In [Campillo et al.2007] it is a power series with coefficients from the ring of one-dimensional representation of a group. In [Campillo et al.2013] it is an element of the Grothendieck ring of “locally finite” $G$-sets with an additional structure. In [Gusein-Zade et al.2008], there was given an equivariant version of the monodromy zeta function as a power series with the coefficients from $K_{0}({\mbox{f.$G$-s.}})\otimes{\mathbb{Q}}$. The fact that it was defined only after tensoring by the field ${\mathbb{Q}}$ of rational numbers makes it less reasonable, in particular, to compare it with the equivariant versions of the Poincaré series which were defined over integers. In [Gusein-Zade2013] an equivariant version of the monodromy zeta function was defined as an element of a generalization of the Burnside ring different from that in [Campillo et al.2013]. Just recently one gave a definition of an equivariant version of the Poincaré series as a power series with the coefficients from the Burnside ring of the group: [Campillo et al.2014]. (In that paper initially the Poincaré series is defined as a power series with the coefficients from a certain modification of the Burnside ring. A simple reductions sends this modification to the usual Burnside ring.)
One of the main ingredients of the definition of the equivariant version of the monodromy zeta function in [Gusein-Zade et al.2008] was the definition of the equivariant Lefschetz number of a transformation from [Lück and Rosenberg2003]. The definition from [Lück and Rosenberg2003] is rather natural. Moreover, one can say that it is the only possible definition possessing some reasonable properties. However the fact that it leads to a “non-integer” definition of the (monodromy) zeta function gives a hint that this definition is not absolutely adequate to this purpose.
There is certain freedom in a definition of an equivariant version of the Lefschetz number of a transformation connected with the question whether it should count the fixed points of the transformation or the fixed $G$-orbits of it. Here we use the second approach to the definition of the equivariant version of the Lefschetz number. This definition was introduced in [Dzedzej2001]. We describe the corresponding equivariant version of the zeta function of a transformation. This zeta-function is a power series with the coefficients from the ring $K_{0}({\mbox{f.$G$-s.}})$. We give an A’Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution.
Difficulties to compare equivariant versions of monodromy zeta functions and of the Poincaré series being elements of different nature (e.g. those described above) leads to the idea to compare their “integer valued reductions”. These reductions can be made with the help of the usual Euler characteristic and also with the help of the orbifold Euler characteristic. In the light of this, we also discuss possible orbifold versions of the zeta function of a transformation.
###### Remark.
The defined equivariant version of the zeta function is not a new invariant in the sense that it cannot distinguish more transformations than existing ones. In particular, it is expressed in terms of equivariant Lefschetz numbers of iterates defined earlier. The same holds for the usual (non-equivariant) zeta function of a transformation. However, it appears to be better adapted to a number of problems. (One more example: the well-known monodromy conjecture on poles of the topological Igusa zeta function (see, e.g., [Denef and Loeser1992]) is also formulated in terms of the monodromy zeta function.) To find equivariant analogues of these problems, one would need to have an equivariant generalization of the usual zeta function of a transformation. Moreover, there are other indices of equivariant transformations which are more fine invariants than the monodromy zeta function. For example, equivariant generalizations of Dold’s indices of iterates defined in [Crabb2007] are of this sort: our equivariant version of the zeta function can be expressed through them (private communication by M. C. Crabb).
## 2 Burnside Ring and the Equivariant Euler Characteristic
A finite $G$-set is a finite set with an action (say a left one) of the group $G$. Isomorphism classes of irreducible $G$-sets (i.e. those which consist of exactly one orbit) are in one-to-one correspondence with the set $\mbox{Consub}(G)$ of conjugacy classes of subgroups of $G$: to the conjugacy class containing a subgroup $H\subset G$ one associates the isomorphism class $[G/H]$ of the $G$-set $G/H$. The Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ of finite $G$-sets (also called the Burnside ring of $G$) is the group generated by isomorphism classes of finite $G$-sets with the relation $[A\coprod B]=[A]+[B]$ and with the multiplication defined by the cartesian product. As an abelian group $K_{0}({\mbox{ f.$G$-s.}})$ is freely generated by the isomorphism classes $[G/H]$ of irreducible $G$-sets. The element 1 in the ring $K_{0}({\mbox{ f.$G$-s.}})$ is represented by the $G$-set consisting of one point (with the trivial $G$-action).
Recall that given a subgroup $H$ of $G$ there are two natural maps $\mbox{Res}_{H}^{G}:K_{0}(\mbox{f.}{G}\mbox{-sets})\to K_{0}(\mbox{f.}{H}% \mbox{-sets})$ and $\mbox{Ind}_{H}^{G}:K_{0}(\mbox{f.}{H}\mbox{-sets})\to K_{0}(\mbox{f.}{G}% \mbox{-sets})$. The restriction map $\mbox{Res}_{H}^{G}$ sends a $G$-set X to the same set considered with the $H$-action. The induction map $\mbox{Ind}_{H}^{G}$ sends an $H$-set $X$ to the product $G\times X$ factorized by the natural equivalence: $(g_{1},x_{1})\sim(g_{2},x_{2})$ if there exists $g\in H$ such that $g_{2}=g_{1}g$, $x_{2}=g^{-1}x_{1}$ with the natural (left) $G$-action. The induction map $\mbox{Ind}_{H}^{G}$ sends the class $[H/H^{\prime}]$ ($H^{\prime}$ is a subgroup of $H$) to the class $[G/H^{\prime}]$. Both maps are group homomorphisms, however the induction map $\mbox{Ind}_{H}^{G}$ is not a ring homomorphism.
In some places, say, in [tom Dieck1979], [Lück and Rosenberg2003] and [Gusein-Zade et al.2008], the equivariant Euler characteristic of a $G$-space is considered as an element of the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$. For a relatively good $G$-space $X$ (say, for a quasiprojective variety) the equivariant Euler characteristic $\chi^{G}(X)\in K_{0}({\mbox{ f.$G$-s.}})$ can be defined in the following way. For a point $x\in X$, let $G_{x}=\{g\in G:\,g\cdot x=x\}$ be the isotropy subgroup of the point $x$. For a conjugacy class ${\mathcal{H}}\in\mbox{ Consub}(G)$, set $X^{{\mathcal{H}}}=\{x\in X:x\mbox{ is a fixed point of a subgroup }H\in{\mathcal{H}}\}$ and let $X^{({\mathcal{H}})}=\{x\in X:G_{x}\in{\mathcal{H}}\}$ be the set of points with the isotropy subgroups from ${\mathcal{H}}$. One can see that in the natural sense $X^{({\mathcal{H}})}=X^{{\mathcal{H}}}{\setminus}X^{>{\mathcal{H}}}$, where $X^{>{\mathcal{H}}}=\bigcup\nolimits_{{\mathcal{H}}^{\prime}>{\mathcal{H}}}X^{{% \mathcal{H}}^{\prime}}$.
Then the equivariant Euler characteristic of the $G$-space $X$ is defined as
$\chi^{G}(X):=\sum_{{\mathcal{H}}\in{ Consub}(G)}\frac{\chi(X^{({\mathcal{H}% })})\,|H|}{|G|}[G/H]=\sum_{{\mathcal{H}}\in{ Consub}(G)}\chi(X^{({\mathcal{% H}})}/G)[G/H],$ (1)
where $H$ is a representative of the conjugacy class ${\mathcal{H}}$.
###### Remark.
Here we use the additive Euler characteristic $\chi(\cdot)$, i.e. the alternating sum of the ranks of the cohomology groups with compact support. For a complex analytic variety this Euler characteristic is equal to the alternating sum of the ranks of the usual cohomology groups.
###### Definition.
A pre-$\lambda$ ring structure on a commutative ring $R$ is an additive to multiplicative group homomorphism $\lambda_{T}:R\to 1+T\cdot R[[T]]$, that is $\lambda_{T}(m+n)=\lambda_{T}(m)\lambda_{T}(n)$, such that $\lambda_{T}(m)=1+mT$ (mod $T^{2}$).
A pre-$\lambda$ ring homomorphism is a ring homomorphism between pre-$\lambda$ rings which commutes with the pre-$\lambda$ ring structures.
The Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ has a natural pre-$\lambda$-ring structure defined by the series
$\sigma_{X}(t)=1+[X]\,t+[S^{2}X]\,t^{2}+[S^{3}X]\,t^{3}+\cdots,$
where $S^{k}X=X^{k}/S_{k}$ is the $k$-th symmetric power of the $G$-set $X$ with the natural $G$-action. This pre-$\lambda$-ring structure induces a power structure over the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$: see [Gusein-Zade et al.2006]. This means that for a power series $A(t)\in 1+t\cdot K_{0}({\mbox{ f.$G$-s.}})[[t]]$ and $m\in K_{0}({\mbox{ f.$G$-s.}})$ there is defined a series $\left(A(t)\right)^{m}\in 1+t\cdot K_{0}({\mbox{ f.$G$-s.}})[[t]]$ so that all the properties of the exponential function hold. In these notations $\sigma_{X}(t)=(1-t)^{-[X]}$. The geometric description of the natural power structure over the Grothendieck ring of quasiprojective varieties given in [Gusein-Zade et al.2006] using graded spaces is also valid for the power structure over $K_{0}({\mbox{ f.$G$-s.}})$ as well.
Some examples of computation of the series $(1-t)^{-[G/H]}$ for $G$ being the cyclic group ${\mathbb{Z}}_{6}$ of order 6 and the group ${\mathcal{S}}_{3}$ of permutations on three elements can be found in [Gusein-Zade et al.2008] (with some misprints). For example
\renewcommand\Z{\mathbb Z}\begin{align} (1-t)^{-[{\mathcal S}_3/\langle e\rangle]}=&\frac{1}{1-t^6}[1]+\frac{t^3}{(1-t^3)(1-t^6)}[{\mathcal S}_3/\Z_3]\\ &+\frac{3t^2}{(1-t^2)^2(1-t^6)}[{\mathcal S}_3/\Z_2]\\ &+\frac{t(1+4t^2+t^3+4t^4-2t^5+3t^6+t^7)}{(1-t^2)^2(1-t^3)(1-t^6)(1-t)^2}[{\mathcal S}_3/\langle e\rangle].\end{align}
There is a natural homomorphism from the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ to the ring $R(G)$ of virtual representations of the group $G$ which sends the class $[G/H]\in K_{0}({\mbox{ f.$G$-s.}})$ to the representation $i^{G}_{H}[1_{H}]$ induced from the trivial one-dimensional representation $1_{H}$ of the subgroup $H$. (A virtual representation of the group $G$ is an element of the Grothendieck ring of representations, i.e. a formal difference of two representations.) This homomorphism is a homomorphism of pre-$\lambda$-rings ([Knutson1973]).
Let us show that for any subgroup $H$ of $G$ the series $(1-t)^{-[G/H]}$ represents a rational function with the denominator equal to a product of the binomials of the form $(1-t^{m})$, $m\in{\mathbb{Z}}_{\geq 1}$. Since irreducible $G$-sets are in one-to-one correspondence with the set $\mbox{ Consub}(G)$ of conjugacy classes of subgroups of $G$ and $K_{0}({\mbox{ f.$G$-s.}})$ is freely generated by isomorphism classes $[G/H]$ of irreducible $G$-sets then
$(1-t)^{-[G/H]}=\sum_{{\mathcal{F}}\in{ Consub}(G)}{\mathcal{A}}_{H,{% \mathcal{F}}}(t)[G/F]$ (2)
where $F$ is a representative of the conjugacy class ${\mathcal{F}}$ and ${\mathcal{A}}_{H,{\mathcal{F}}}(t)\in{\mathbb{Z}}[[t]]$.
Let ${\mathcal{F}}$ be a conjugacy class of subgroups of $G$ and let $F$ be a representative of it. The subgroup $F$ acts on the $G$-space $G/H$. Let $F\backslash G/H$ be the quotient of $G/H$ by this action and let $p:G/H\to F\backslash G/H$ be the quotient map. For $m=1,2,\ldots,$ let $Y_{m}$ be the set of points of $F\backslash G/H$ with $m$ preimages in $G/H$ and let ${\ell}^{{\mathcal{F}}}_{m}=|Y_{m}|$. (The numbers ${\ell}^{{\mathcal{F}}}_{m}$ depend only on the conjugacy class ${\mathcal{F}}$.) For an abelian $G$, ${\ell}^{{\mathcal{F}}}_{m}$ is different from zero if and only if $m=\frac{|F|}{|F\cap H|}$ and in this case ${\ell}^{{\mathcal{F}}}_{m}=|G|/|F+H|$.
For conjugacy classes ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime}$ from ${ Consub}(G)$, let $F$ and $F^{\prime}$ be their representatives, and let $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ be the number of fixed points of the group $F$ on $G/F^{\prime}$. The integer $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ is different from zero if and only if ${\mathcal{F}}^{\prime}\geq{\mathcal{F}}$ (i.e. there exist representatives $F^{\prime}$ of $F$ of them such that $F^{\prime}\supset F$. For an abelian $G$ and for ${\mathcal{F}}^{\prime}\geq{\mathcal{F}}$, one has $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}=|G/F^{\prime}|$. For a non-abelian group the equation is more involved and $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ depends on ${\mathcal{F}}$ as well.
###### Lemma 1.
For ${\mathcal{F}}\in{ Consub}(G)$ one has
$\prod_{m\geq 1}(1-t^{m})^{-{\ell}^{{\mathcal{F}}}_{m}}=\sum_{{\mathcal{F}}^{% \prime}\in{ Consub}(G)}r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}\,\,{% \mathcal{A}}_{H,{\mathcal{F}}^{\prime}}(t).$ (3)
###### Proof.
Let $F$ be a representative of ${\mathcal{F}}$ and let us count fixed points of the subgroup $F$ in the left hand side and right hand side of (2) .
For a finite set $X$ an element of $\coprod_{k\geq 0}S^{k}X$ can be identified with an integer valued function on $X$ with non-negative values. The corresponding element belongs to $S^{k}X$ if and only if the sum of all the values of the function is equal to $k$. An element of $\coprod_{k\geq 0}S^{k}[G/H]$ is fixed with respect to $F$ if and only if the corresponding function is invariant with respect to the $F$-action on $G/H$. Such a function can be identified with a function on $F\backslash G/H$. A function on $F\backslash G/H$ can be also considered as the direct sum of functions on the subset $Y_{s}$ defined above. The generating series for the number of functions on $Y_{s}$ (i.e. the series $\sum_{k\geq 0}\arrowvert S^{k}Y_{s}\arrowvert\,t^{k}$) is $(1-t)^{-|Y_{s}|}=(1-t)^{-{\ell}^{{\mathcal{F}}}_{s}}$. Each function on $Y_{s}$ with the sum of values equal to $k$ lifts to an $H$ invariant function on $G/H$ with the sum of the values equal to $ks$. Therefore the generating series for $F$-invariants functions on $p^{-1}(Y_{s})$ is $(1-t^{s})^{-{\ell}^{{\mathcal{F}}}_{s}}$. The generating series for all $F$-invariants functions on $G/H$ is the product of those for $p^{-1}(Y_{s})$. This is the left hand side of (3) .
The right hand side of (3) is obviously the set of fixed points of $F$ on the right hand side of (2) . $\square$
Since $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ is different from zero if and only if ${\mathcal{F}}\leq{\mathcal{F}}^{\prime}$ and $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}^{\prime}}$ is different from zero, the system of Eq. (3) is a triangular one (with respect to the partial order on the set of conjugacy classes of subgroups of $G$). Together with the fact that the denominators of the left hand side of the Eq. (3) are product of the binomials of the form $(1-t^{m})$ this implies the following statement.
###### Proposition 1.
For any subgroup $H$ of $G$ the series $(1-t)^{-[G/H]}$ belongs to the localization $K_{0}({\mbox{ f.$G$-s.}})[t]_{(\{1-t^{m}\})}$ of the polynomial ring $K_{0}({\mbox{ f.$G$-s.}})[t]$ at all the elements of the form $(1-t^{m})$, $m\geq 1$.
The natural homomorphism from the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ to the ring $R(G)$ of virtual representations of the group $G$ sends the equivariant Euler characteristic $\chi^{G}(X)$ to the one used in [Wall1980]. Since this homomorphism is, generally speaking, neither injective, no surjective, the equivariant Euler characteristic as an element in $K_{0}({\mbox{ f.$G$-s.}})$ is a somewhat finer invariant than the one as an element of the ring $R(G)$.
## 3 An Alternative Version of the Equivariant Lefschetz Number of a Map
Let $X$ be a relatively good topological space (say, a quasiprojective complex or real variety) with a $G$-action and let $\varphi:X\to X$ be a $G$-equivariant proper map. The usual (“non-equivariant”) Lefschetz number $L(\varphi)$ counts the fixed points of $\varphi$ (or rather of its generic perturbation). The equivariant version $L^{G}(\varphi)$ of the Lefschetz number from [Lück and Rosenberg2003] counts the fixed points of $\varphi$ as a (finite) $G$-set. This leads to the following equation for the equivariant Lefschetz number
$L^{G}(\varphi)=\sum\limits_{{\mathcal{H}}\in\mbox{Consub}(G)}\frac{L(\varphi_{% |(X^{{\mathcal{H}}},X^{>{\mathcal{H}}})})|H|}{|G|}[G/H],$ (4)
where $H$ is a representative of the class ${\mathcal{H}}$. If $\varphi$ is a $G$-homeomorphism (like the monodromy transformation, see Sect. , (5) and (6) , (7) with the two parts of the equation (1) .]
###### Example.
For some simplicity, let the group $G$ be abelian, let $X=(G/H)\times{\mathbb{Z}}_{k}=\{0,1,\ldots,k-1\}$, $k>0$, with the natural action of the group $G$ on the first factor, and let the map $\varphi:X\to X$ be defined by
$\varphi(a,i)=\begin{cases}(a,i+1)&\mbox{ for }0\leq i If$k>1$, then$L^{G}(\varphi)={\widetilde{L}}^{G}(\varphi)=0$since$\varphi$has neither fixed points, no fixed orbits. The smallest$i$for which${\widetilde{L}}^{G}(\varphi^{i})\neq 0$is$i=k$. In this case all the$G$-orbits in$X$are fixed by$\varphi^{k}$and therefore${\widetilde{L}}^{G}(\varphi^{k})=k[G/H]$. On the other hand, if$g\notin H$, the map$\varphi^{k}$has no fixed points and thus$L^{G}(\varphi^{k})=0$. The smallest$i$for which$L^{G}(\varphi^{i})\neq 0$is$i=\ell k$, where$\ell$is the order of the element$g$in the group$G/H$. In this case all the points of$X$are fixed by$\varphi^{\ell k}$and therefore${\widetilde{L}}^{G}(\varphi^{\ell k})=k[G/H]$. Just in the same way as in [Lück and Rosenberg2003] one can formulate the equivariant version of the Lefschetz fixed point theorem for${\widetilde{L}}^{G}(\varphi)$(an analogue of Theorem 2.1 in [Lück and Rosenberg2003]). ## 4 The Zeta Function of a Transformation Let$\varphi:X\to X$be as above. The usual (non-equivariant) zeta function of$\varphi$is defined in terms of the action of$\varphi$in the (co)homology groups of$X$(in a way somewhat similar to the definition of the Lefschetz number). This definition is not convenient for a direct generalization to the equivariant case. It is more convenient to use the definition of the zeta function of the transformation$\varphi$in terms of the Lefschetz numbers of the iterates of$\varphi$. One defines integers$s_{i}$,$i=1,2\ldots,$recursively by the equation $L(\varphi^{m})=\sum_{i|m}s_{i}.$(8) The number$s_{m}$counts (with integer multiplicities) the points$x\in X$with the$\varphi$-order equal to$m$(i.e.$\varphi^{m}(x)=x$,$\varphi^{i}(x)\neq x$for$0<i<m$). Together with each such point all its images under the iterates of$\varphi$(there are exactly$m$different ones) are of this sort. Therefore$s_{m}$is divisible by$m$. One defines the zeta function$\zeta_{\varphi}(t)$to be $\zeta_{\varphi}(t):=\prod_{m\geq 1}(1-t^{m})^{-{s_{m}}/{m}}.$(9) ###### Remark. There are two traditions to define the zeta function of a transformation. The other one does not contain the minus sign in the exponent and therefore is the inverse to this one. Here we follow the definition from [A’Campo1975]. In the equivariant version, let$s_{m}^{{{G}}}(\varphi)$and${\widetilde{s}}_{m}^{{{G}}}(\varphi)$be defined through$L^{G}(\varphi^{i})$and${\widetilde{L}}^{G}(\varphi^{i})$respectively by the analogues of the Eq. (8) $L^{G}(\varphi^{m})=\sum_{i|m}s_{i}^{G}(\varphi),\quad{\widetilde{L}}^{G}(% \varphi^{m})=\sum_{i|m}{\widetilde{s}}_{i}^{G}(\varphi).$(10) The elements$s_{m}^{G}(\varphi)$and${\widetilde{s}}_{m}^{G}(\varphi)$count the points in$X$the$\varphi$-order of which is equal to$m$in$X$and in$X/G$respectively. ###### Example. In the Example from Sect. 2 with Proposition 1 gives $\begin{align}{\widetilde{\zeta}}^{{\mathcal{S}}_{3}}_{f}(t)&(1-t^{6k})^{-1}\cdot(1-t^{6k}% )^{(6k-1)[{\mathcal{S}}_{3}/{\mathbb{Z}}_{2}]}\cdot(1-t^{3k})^{-[{\mathcal{S}}% _{3}/\langle e\rangle]}\\ &\cdot(1-t^{2k})^{-[{\mathcal{S}}_{3}/\langle e\rangle]}\cdot(1-t^{6k})^{(1-6% k^{2})[{\mathcal{S}}_{3}/\langle e\rangle]}.\end{align}$## 5 On Orbifold Versions of the Equivariant Monodromy Zeta Function For a$G$-variety$X$, its orbifold Euler characteristic$\chi^{orb}(X,G)\in{\mathbb{Z}}$is defined, e.g., in [Atiyah and Segal1989] or [Hirzebruch and Höfer1990]. For a subgroup$H$of$G$, let$X^{H}=\{x\in X:Hx=x\}$be the fixed point set of$H$. The orbifold Euler characteristic$\chi^{orb}(X,G)$of the$G$-space$X$is defined, e.g., in [Atiyah and Segal1989] and [Hirzebruch and Höfer1990]: $\chi^{orb}(X,G)=\sum_{[g]\in{ Consub}(G)}\chi(X^{\langle g\rangle}/C_{G}(g)),$(13) where$C_{G}(g)=\{h\in G:h^{-1}gh=g\}$is the centralizer of$g$, and$\langle g\rangle$the subgroups generated by$g$. There is a natural homomorphism of abelian groups$\Phi:K_{0}({\mbox{ f.$G$-s.}})\to{\mathbb{Z}}$which sends the generator$[G/H]$of$K_{0}({\mbox{ f.$G$-s.}})$to$\chi^{orb}(G/H,G)$and therefore the equivariant Euler characteristic$\chi^{G}(X)\in K_{0}({\mbox{ f.$G$-s.}})$to the orbifold Euler characteristic$\chi^{orb}(X,G)$. For an abelian$G$,$\Phi([G/H])=|H|$and$\Phi$is a ring homomorphism, but this is not the case in general. The Lefschetz number is a sort of generalization of the Euler characteristic: the Euler characteristic is the Lefschetz number of the identity map. The definition of the orbifold Euler characteristic gives the hint that there can be the corresponding definition(s) of the orbifold Lefschetz number of a$G$-equivariant transformation. It can be expressed through an equivariant version of the Lefschetz number with values in the Burnside ring: the image of the Lefschetz number by the homomorphism$\Phi$. The two versions$L^{G}(\varphi)$and${\widetilde{L}}^{G}(\varphi)$of the equivariant Lefschetz number [see (4) and (6) ] give two versions $L^{orb}(\varphi)=\Phi(L^{G}(\varphi))\,\,\mbox{ and }\,\,{\widetilde{L}}^{orb}(\varphi)=\Phi({\widetilde{L}}^{G}(\varphi))$of orbifold Lefschetz numbers. The usual definition of the zeta function of a transformation [e.g. Eqs. (8) , (9) , (10) and (11) ] gives two orbifold versions of the zeta function of a$G$-equivariant transformation$\varphi:X\to X$: ${\zeta}_{\varphi}^{orb}(t)=\prod_{m\geq 1}(1-t^{m})^{-{{s}_{m}^{{orb}}}/{m}},% \,\,\mbox{ and }\,\,{\widetilde{\zeta}}_{\varphi}^{orb}(t)=\prod_{m\geq 1}(1-t% ^{m})^{-{{\widetilde{s}}_{m}^{{orb}}}/{m}},$(14) where$L^{orb}(\varphi^{m})=\sum_{i|m}s^{orb}_{i}(\varphi)$and${\widetilde{L}}^{orb}(\varphi^{m})=\sum_{i|m}{\widetilde{s}}^{orb}_{i}(\varphi)$. The exponents$-{{\widetilde{s}}_{m}^{{orb}}}/{m}$are integers and therefore the orbifold monodromy zeta function${\widetilde{\zeta}}_{\varphi}^{orb}(t)$is a rational function in$t$. The exponents$-{{s}_{m}^{{orb}}}/{m}$are in general rational numbers. For instance, for$f$from Example 2 in Sect. 5 one has ${\widetilde{\zeta}}_{f}^{orb}(t)=(1-t^{6k})^{-1+2(6k-1)+(1-6k^{2})}\cdot(1-t^{% 3k})^{-1}\cdot(1-t^{2k})^{-1}.$This follows from the fact that, for$G={\mathcal{S}}_{3}$and for a subgroup$H$of${\mathcal{S}}_{3}$,$\chi^{orb}({\mathcal{S}}_{3}/H,{\mathcal{S}}_{3})=|H|$. #### Acknowledgements S. M. Gusein-Zade partially supported by the Grants RFBR-13-01-00755 and NSh-5138.2014.1. I. Luengo and A. Melle-Hernández partially supported by the Spanish Grant MTM2013-45710-C2-2-P. ### References • [A’Campo1975] A’Campo, N.: La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233–248 (1975) • [Arnold et al.1988] Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. vol. II. Monodromy and Asymptotics of Integrals, Monographical Mathematics, vol. 83. Birkhäuser, Boston (1988) • [Atiyah and Segal1989] Atiyah, M., Segal, G.: On equivariant Euler characteristics. J. Geom. Phys. 6(4), 671–677 (1989) • [Gusein-Zade et al.1999] Gusein-Zade, S.M., Delgado, F., Campillo, A.: On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions. Funktsional. Anal. i Prilozhen. 33(1), 66–68 (1999); translation in Funct. Anal. Appl. 33(1), 56–57 (1999) • [Campillo et al.2007] Campillo, A., Delgado, F., Gusein-Zade, S.M.: On Poincaré series of filtrations on equivariant functions of two variables. Mosc. Math. J. 7(2), 243–255 (2007) • [Campillo et al.2013] Campillo, A., Delgado, F., Gusein-Zade, S.M.: Equivariant Poincaré series of filtrations. Rev. Mat. Complut. 26(1), 241–251 (2013) • [Campillo et al.2014] Campillo, A., Delgado, F., Gusein-Zade, S.M.: An equivariant Poincaré series of filtrations and monodromy zeta functions. Rev. Mat. Complut. (2014). doi:10.1007/s13163-014-0160-8 • [Clemens1969] Clemens, C.H.: Picard–Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities. Trans. Am. Math. Soc. 136, 93–108 (1969) • [Crabb2007] Crabb, M.C.: Equivariant fixed-point indices of iterated maps. J. Fixed Point Theory Appl. 2(2), 171–193 (2007) • [Denef and Loeser1992] Denef, J., Loeser, F.: Caractéristiques d’Euler–Poincaré, fonctions zêta locales et modifications analytiques. J. Am. Math. Soc. 5(4), 705–720 (1992) • [Dzedzej2001] Dzedzej, Z.: Fixed orbit index for equivariant maps. In: Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000). Nonlinear Analytics, vol. 47(4), pp. 2835–2840 (2001) • [Ebeling and Gusein-Zade2012a] Ebeling, W., Gusein-Zade, S.M.: Saito duality between Burnside rings for invertible polynomials. Bull. Lond. Math. Soc. 44, 814–822 (2012) • [Ebeling and Gusein-Zade2012b] Ebeling, W., Gusein-Zade, S.M.: Equivariant Poincaré series and monodromy zeta functions of quasihomogeneous polynomials. Publ. Res. Inst. Math. Sci. 48(3), 653–660 (2012) • [Gusein-Zade2010] Gusein-Zade, S.M.:Integration with respect to the Euler characteristic and its applications. Uspekhi Mat. Nauk 65(393), 5–42 (2010) (no. 3); translation in Russ. Math. Surv. 65(3), 399–432 (2010) • [Gusein-Zade2013] Gusein-Zade, S.M.: On an equivariant analogue of the monodromy zeta function. Funktsional. Anal. i Prilozhen. 47(1), 17–25 (2013); translation in Funct. Anal. Appl. 47(1), 14–20 (2013) • [Gusein-Zade et al.2006] Gusein-Zade, S.M., Luengo, I., Melle-Hernández, A.: Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points. Mich. Math. J. 54(2), 353–359 (2006) • [Gusein-Zade et al.2008] Gusein-Zade, S.M., Luengo, I., Melle-Hernández, A.: An equivariant version of the monodromy zeta function. In: Geometry, Topology, and Mathematical Physics, pp. 139–146. American Mathematical Society Translation Series 2, 224, American Mathematical Society, Providence (2008) • [Hirzebruch and Höfer1990] Hirzebruch, F.: Höfer, Th: On the Euler number of an orbifold. Math. Ann. 286(1–3), 255–260 (1990) • [Knutson1973] Knutson, D.:$\lambda$-Rings and the Representation Theory of the Symmetric Group. Lecture Notes in Mathematics, vol. 308. Springer, Berlin, New York (1973) • [Lück and Rosenberg2003] Lück, W., Rosenberg, J.: The equivariant Lefschetz fixed point theorem for proper cocompact$G\$-manifolds. In: Farrell, F.T., Lück, W. (eds.) High-Dimensional Manifold Topology, pp. 322–361. World Science Publishing, River Edge (2003)
• [tom Dieck1979] tom Dieck, T.: Transformation Groups and Representation Theory. Lecture Notes in Mathematics, vol. 766. Springer, Berlin (1979)
• [Verdier1973] Verdier, J.-L.: Caractéristique d’Euler–Poincaré. Bull. Soc. Math. France 101, 441–445 (1973)
• [Wall1980] Wall, C.T.C.: A note on symmetry of singularities. Bull. Lond. Math. Soc. 12(3), 169–175 (1980) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9507799744606018, "perplexity": 239.29601257293334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689775.56/warc/CC-MAIN-20170923175524-20170923195524-00525.warc.gz"} |
https://link.springer.com/article/10.1007%2Fs12220-017-9760-0 | The Journal of Geometric Analysis
, Volume 27, Issue 3, pp 2269–2277
# The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary
• Stephen McCormick
Article
## Abstract
We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension $$n\ge 3$$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric g, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with g on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: given a function f on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to g whose boundary mean curvature is given by f. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.
## Keywords
Yamabe problem Asymptotically flat manifold Scalar curvature
## Mathematics Subject Classification
58J05 53C21 53A30 35B33
## Notes
### Acknowledgements
The author would like to thank the Institut Henri Poincaré, for their hospitality while part of this work was completed, and gratefully acknowledge that this work was supported by a UNE Research Seed Grant.
## References
1. 1.
Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure. Appl. Math. 39, 661–693 (1986)
2. 2.
Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62(20), 845–885 (1989)
3. 3.
Bartnik, R., Isenberg, J.: The constraint equations. In: Chruściel, P., Friedrich, H. (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 1–38. Birkhäuser, Basel (2004)Google Scholar
4. 4.
Brendle, S., Marques, F.C.: Recent progress on the Yamabe problem. arXiv:1010.4960 (2010)
5. 5.
Cantor, M.: A necessary and sufficient condition for York data to specify an asymptotically flat spacetime. J. Math. Phys. 20(8), 1741–1744 (1979)
6. 6.
Cantor, M.: Some problems of global analysis on asymptotically simple manifolds. Compos. Math. 38(1), 3–35 (1979)
7. 7.
Cantor, M., Brill, D.: The laplacian on asymptotically flat manifolds and the specification of scalar curvature. Compos. Math. 43(3), 317–330 (1981)
8. 8.
Escobar, J.F.: Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Ann. Math. 136, 1–50 (1992)
9. 9.
Escobar, J.F.: The Yamabe problem on manifolds with boundary. J. Differ. Geom. 35(1), 21–84 (1992)
10. 10.
Escobar, J.F.: Conformal metrics with prescribed mean curvature on the boundary. Calc. Var. Partial Differ. Equ. 4(6), 559–592 (1996)
11. 11.
Kazdan, J.L., Warner, F.W.: Scalar curvature and conformal deformation of riemannian structure. J. Differ. Geom. 10(1), 113–134 (1975)
12. 12.
Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. 17(1), 37–91 (1987)
13. 13.
Maxwell, D.: Solutions of the Einstein constraint equations with apparent horizon boundaries. Commun. Math. Phys. 253(3), 561–583 (2005)
14. 14.
McCormick, S.: The hilbert manifold of asymptotically flat metric extensions. arXiv:1512.02331 (2015)
15. 15.
Schwartz, F.: The zero scalar curvature Yamabe problem on noncompact manifolds with boundary. Indiana Univ. Math. J. 55(4), 1449–1459 (2006)
16. 16.
Szabados, L.B.: Quasi-local energy-momentum and angular momentum in general relativity: a review article. Living Rev. Relativ. 7, 4 (2004)
17. 17.
Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Della Sc. Norm. Super. Pisa-Classe Sci. 22(2), 265–274 (1968)
18. 18.
Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12(1), 21–37 (1960)
## Authors and Affiliations
• Stephen McCormick
• 1
• 2
1. 1.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia
2. 2.Institutionen för MatematikKungliga Tekniska HögskolanStockholmSweden | {"extraction_info": {"found_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": 0.9258082509040833, "perplexity": 2300.6671801024054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823657.20/warc/CC-MAIN-20181211151237-20181211172737-00104.warc.gz"} |
http://mathhelpboards.com/differential-equations-17/method-integrating-factor-20250.html?s=937dcc8a83392d4f10f5259d4891b7f7 | # Thread: method of integrating factor
1. $\tiny{206.q3.2}\\$
$\textsf{3. use the method of integrating factor}\\$
$\textsf{to find the general solution to the first order linear differential equation}\\$
\begin{align}
\displaystyle
\frac{dy}{dx}+5y=10x
\end{align}
$\textit{clueless !!!}$
2. Given a first order linear ODE of the form:
$\displaystyle \d{y}{x}+f(x)y=g(x)$
We can use an integrating factor $\mu(x)$ to make the LHS of the ODE into the derivative of a product, using the special properties of the exponential function with regard to differentiation. Consider what happens if we multiply though by:
$\displaystyle \mu(x)=\exp\left(\int f(x)\,dx\right)$
We get:
$\displaystyle \exp\left(\int f(x)\,dx\right)\d{y}{x}+\exp\left(\int f(x)\,dx\right)f(x)y=g(x)\exp\left(\int f(x)\,dx\right)$
Now, let's use:
$\displaystyle F(x)=\int f(x)\,dx\implies F'(x)=f(x)$
And we now have:
$\displaystyle \exp\left(F(x)\right)\d{y}{x}+\exp\left(F(x)\right)F'(x)y=g(x)\exp\left(F(x)\right)$
Now, if we observe that, via the product rule, we have:
$\displaystyle \frac{d}{dx}\left(\exp(F(x))y\right)=\exp\left(F(x)\right)\d{y}{x}+\exp\left(F(x)\right)F'(x)y$
Then, we may now write our ODE as:
$\displaystyle \frac{d}{dx}\left(\exp(F(x))y\right)=g(x)\exp\left(F(x)\right)$
Now, we may integrate both sides w.r.t $x$.
So, in the given ODE:
$\displaystyle \d{y}{x}+5y=10x$
We identify: $\displaystyle f(x)=5$
And so we compute the integrating factor as:
$\displaystyle \mu(x)=\exp\left(5\int\,dx\right)=$? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9931864142417908, "perplexity": 389.9794483820826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818690203.42/warc/CC-MAIN-20170924190521-20170924210521-00329.warc.gz"} |
http://mathhelpforum.com/calculus/65917-differential-geometry.html | differential geometry?
hello there.
i was reading through some work on curves of pursuit and the author stated a certain equivalence without any comment as to where it came from.
Say i have a differentiable curve $C$ in the plane. Given a point $X \in C$ let the distance of the tangent line from the origin at $X$ be $p$, the angle the tangent line makes with the x-axis be $\omega$, and the length of the curve be $s$. Then we have the relation:
$p + \frac{d^2p}{d\omega^2} = \frac{ds}{d\omega}$
---
this sort of thing looks like it should be intuitive but i can't see why it is at all, or at least like it should follow from some other nice results. i've been able to show it by letting $C = (x_1(t), x_2(t))$:
$p = \frac{x_1\dot{x}_2-\dot{x}_1x_2}{\sqrt{\dot{x}_1^2+\dot{x}_2^2}}$
$\omega = \text{arctan}\left(\frac{\dot{x}_2}{\dot{x}_1}\rig ht)$
$\frac{ds}{dt} = \sqrt{\dot{x}_1^2+\dot{x}_2^2}$
and then working out all the derivatives manually but that doesn't seem to help me understand it either.
i hope that someone recognizes it and can point me in the direction for a nicer explanation of it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9021983742713928, "perplexity": 108.4099851360109}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738659753.31/warc/CC-MAIN-20160924173739-00150-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://en.wikiludia.com/wiki/Expected_value | Expected value
In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the same experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity (see § Examples for details). In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.
More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. The same principle applies to an absolutely continuous random variable, except that an integral of the variable with respect to its probability density replaces the sum. The formal definition subsumes both of these and also works for distributions which are neither discrete nor absolutely continuous; the expected value of a random variable is the integral of the random variable with respect to its probability measure.[1][2]
The expected value does not exist for random variables having some distributions with large "tails", such as the Cauchy distribution.[3] For random variables such as these, the long-tails of the distribution prevent the sum or integral from converging.
The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations: it is the expected value of the squared deviation of the variable's value from the variable's expected value (var(X) = E[(X - E[X])2] = E(X2) - [E(X)]2).
The expected value plays important roles in a variety of contexts. In regression analysis, one desires a formula in terms of observed data that will give a "good" estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, so the estimate it gives is itself a random variable. A formula is typically considered good in this context if it is an unbiased estimator— that is if the expected value of the estimate (the average value it would give over an arbitrarily large number of separate samples) can be shown to equal the true value of the desired parameter.
In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information. For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for risk averse agents it involves maximizing the expected value of some objective function such as a von Neumann–Morgenstern utility function. One example of using expected value in reaching optimal decisions is the Gordon–Loeb model of information security investment. According to the model, one can conclude that the amount a firm spends to protect information should generally be only a small fraction of the expected loss (i.e., the expected value of the loss resulting from a cyber or information security breach).[4]
Definition
Finite case
Let ${\displaystyle X}$ be a random variable with a finite number of finite outcomes ${\displaystyle x_{1},x_{2},\ldots ,x_{k}}$ occurring with probabilities ${\displaystyle p_{1},p_{2},\ldots ,p_{k},}$ respectively. The expectation of ${\displaystyle X}$ is defined as
${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{k}x_{i}\,p_{i}=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.}$
Since all probabilities ${\displaystyle p_{i}}$ add up to 1 (${\displaystyle p_{1}+p_{2}+\cdots +p_{k}=1}$), the expected value is the weighted average, with ${\displaystyle p_{i}}$’s being the weights.
If all outcomes ${\displaystyle x_{i}}$ are equiprobable (that is, ${\displaystyle p_{1}=p_{2}=\cdots =p_{k}}$), then the weighted average turns into the simple average. This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average. If the outcomes ${\displaystyle x_{i}}$ are not equiprobable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the others. The intuition however remains the same: the expected value of ${\displaystyle X}$ is what one expects to happen on average.
An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3.5 as the number of rolls (trials) grows.
Examples
• Let ${\displaystyle X}$ represent the outcome of a roll of a fair six-sided die. More specifically, ${\displaystyle X}$ will be the number of pips showing on the top face of the die after the toss. The possible values for ${\displaystyle X}$ are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of 1/6. The expectation of ${\displaystyle X}$ is
${\displaystyle \operatorname {E} [X]=1\cdot {\frac {1}{6}}+2\cdot {\frac {1}{6}}+3\cdot {\frac {1}{6}}+4\cdot {\frac {1}{6}}+5\cdot {\frac {1}{6}}+6\cdot {\frac {1}{6}}=3.5.}$
If one rolls the die ${\displaystyle n}$ times and computes the average (arithmetic mean) of the results, then as ${\displaystyle n}$ grows, the average will almost surely converge to the expected value, a fact known as the strong law of large numbers. One example sequence of ten rolls of the die is 2, 3, 1, 2, 5, 6, 2, 2, 2, 6, which has the average of 3.1, with the distance of 0.4 from the expected value of 3.5. The convergence is relatively slow: the probability that the average falls within the range 3.5 ± 0.1 is 21.6% for ten rolls, 46.1% for a hundred rolls and 93.7% for a thousand rolls. See the figure for an illustration of the averages of longer sequences of rolls of the die and how they converge to the expected value of 3.5. More generally, the rate of convergence can be roughly quantified by e.g. Chebyshev's inequality and the Berry–Esseen theorem.
• The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable ${\displaystyle X}$ represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability 1/38 in American roulette), the payoff is$35; otherwise the player loses the bet. The expected profit from such a bet will be
${\displaystyle \operatorname {E} [\,{\text{gain from }}\1{\text{ bet}}\,]=-\1\cdot {\frac {37}{38}}+\35\cdot {\frac {1}{38}}=-\0.0526.}$
That is, the bet of $1 stands to lose$0.0526, so its expected value is -\$0.0526.
Countably infinite case
Let ${\displaystyle X}$ be a random variable with a countable set of outcomes ${\displaystyle x_{1},x_{2},\ldots ,}$ occurring with probabilities ${\displaystyle p_{1},p_{2},\ldots ,}$ respectively, such that the infinite sum ${\displaystyle \textstyle \sum _{i=1}^{\infty }|x_{i}|\,p_{i}}$ converges. The expected value of ${\displaystyle X}$ is defined as the series
${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i}.}$
Remark 1. Observe that ${\displaystyle \textstyle {\Bigl |}\operatorname {E} [X]{\Bigr |}\leq \sum _{i=1}^{\infty }|x_{i}|\,p_{i}<\infty .}$
Remark 2. Due to absolute convergence, the expected value does not depend on the order in which the outcomes are presented. By contrast, a conditionally convergent series can be made to converge or diverge arbitrarily, via the Riemann rearrangement theorem.
Example
• Suppose ${\displaystyle x_{i}=i}$ and ${\displaystyle p_{i}={\frac {k}{i2^{i}}},}$ for ${\displaystyle i=1,2,3,\ldots }$, where ${\displaystyle k={\frac {1}{\ln 2}}}$ (with ${\displaystyle \ln }$ being the natural logarithm) is the scale factor such that the probabilities sum to 1. Then
${\displaystyle \operatorname {E} [X]=1\left({\frac {k}{2}}\right)+2\left({\frac {k}{8}}\right)+3\left({\frac {k}{24}}\right)+\dots ={\frac {k}{2}}+{\frac {k}{4}}+{\frac {k}{8}}+\dots =k.}$
Since this series converges absolutely, the expected value of ${\displaystyle X}$ is ${\displaystyle k}$.
• For an example that is not absolutely convergent, suppose random variable ${\displaystyle X}$ takes values 1, −2, 3, −4, ..., with respective probabilities ${\displaystyle {\frac {c}{1^{2}}},{\frac {c}{2^{2}}},{\frac {c}{3^{2}}},{\frac {c}{4^{2}}}}$, ..., where ${\displaystyle c={\frac {6}{\pi ^{2}}}}$ is a normalizing constant that ensures the probabilities sum up to one. Then the infinite sum
${\displaystyle \sum _{i=1}^{\infty }x_{i}\,p_{i}=c\,{\bigg (}1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\dotsb {\bigg )}}$
converges and its sum is equal to ${\displaystyle {\frac {6\ln 2}{\pi ^{2}}}\approx 0.421383}$. However it would be incorrect to claim that the expected value of ${\displaystyle X}$ is equal to this number—in fact ${\displaystyle \operatorname {E} [X]}$ does not exist (finite or infinite), as this series does not converge absolutely (see Alternating harmonic series).
• An example that diverges arises in the context of the St. Petersburg paradox. Let ${\displaystyle x_{i}=2^{i}}$ and ${\displaystyle p_{i}={\frac {1}{2^{i}}}}$ for ${\displaystyle i=1,2,3,\ldots }$. The expected value calculation gives
${\displaystyle \sum _{i=1}^{\infty }x_{i}\,p_{i}=2\cdot {\frac {1}{2}}+4\cdot {\frac {1}{4}}+8\cdot {\frac {1}{8}}+16\cdot {\frac {1}{16}}+\cdots =1+1+1+1+\cdots \,.}$
Since this does not converge but instead keeps growing, the expected value is infinite.
Absolutely continuous case
If ${\displaystyle X}$ is a random variable whose cumulative distribution function admits a density ${\displaystyle f(x)}$, then the expected value is defined as the following Lebesgue integral:
${\displaystyle \operatorname {E} [X]=\int _{\mathbb {R} }xf(x)\,dx.}$
Remark. From computational perspective, the integral in the definition of ${\displaystyle \operatorname {E} [X]}$ may often be treated as an improper Riemann integral ${\displaystyle \textstyle \int _{-\infty }^{+\infty }xf(x)\,dx.}$ Specifically, if the function ${\displaystyle xf(x)}$ is Riemann-integrable on every finite interval ${\displaystyle [a,b]}$, and
${\displaystyle \min \left((-1)\cdot {\hbox{(R)}}\int _{-\infty }^{0}xf(x)\,dx,\ {\hbox{(R)}}\int _{0}^{+\infty }xf(x)\,dx\right)<\infty ,}$
then the values (whether finite or infinite) of both integrals agree.
General case
In general, if ${\displaystyle X}$ is a random variable defined on a probability space ${\displaystyle (\Omega ,\Sigma ,\operatorname {P} )}$, then the expected value of ${\displaystyle X}$, denoted by ${\displaystyle \operatorname {E} [X]}$, ${\displaystyle \langle X\rangle }$, or ${\displaystyle {\bar {X}}}$, is defined as the Lebesgue integral
${\displaystyle \operatorname {E} [X]=\int _{\Omega }X(\omega )\,d\operatorname {P} (\omega ).}$
Remark 1. If ${\displaystyle X_{+}(\omega )=\max(X(\omega ),0)}$ and ${\displaystyle X_{-}(\omega )=-\min(X(\omega ),0)}$, then ${\displaystyle X=X_{+}-X_{-}.}$ The functions ${\displaystyle X_{+}}$ and ${\displaystyle X_{-}}$ can be shown to be measurable (hence, random variables), and, by definition of Lebesgue integral,
{\displaystyle {\begin{aligned}\operatorname {E} [X]&=\int _{\Omega }X(\omega )\,d\operatorname {P} (\omega )\\&=\int _{\Omega }X_{+}(\omega )\,d\operatorname {P} (\omega )-\int _{\Omega }X_{-}(\omega )\,d\operatorname {P} (\omega )\\&=\operatorname {E} [X_{+}]-\operatorname {E} [X_{-}],\end{aligned}}}
where ${\displaystyle \operatorname {E} [X_{+}]}$ and ${\displaystyle \operatorname {E} [X_{-}]}$ are non-negative and possibly infinite.
The following scenarios are possible:
• ${\displaystyle \operatorname {E} [X]}$ is finite, i.e. ${\displaystyle \max(\operatorname {E} [X_{+}],\operatorname {E} [X_{-}])<\infty ;}$
• ${\displaystyle \operatorname {E} [X]}$ is infinite, i.e. ${\displaystyle \max(\operatorname {E} [X_{+}],\operatorname {E} [X_{-}])=\infty }$ and ${\displaystyle \min(\operatorname {E} [X_{+}],\operatorname {E} [X_{-}])<\infty ;}$
• ${\displaystyle \operatorname {E} [X]}$ is neither finite nor infinite, i.e. ${\displaystyle \operatorname {E} [X_{+}]=\operatorname {E} [X_{-}]=\infty .}$
Remark 2. If ${\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}$ is the cumulative distribution function of ${\displaystyle X}$, then
${\displaystyle \operatorname {E} [X]=\int _{-\infty }^{+\infty }x\,dF_{X}(x),}$
where the integral is interpreted in the sense of Lebesgue–Stieltjes.
Remark 3. An example of a distribution for which there is no expected value is Cauchy distribution.
Remark 4. For multidimensional random variables, their expected value is defined per component, i.e.
${\displaystyle \operatorname {E} [(X_{1},\ldots ,X_{n})]=(\operatorname {E} [X_{1}],\ldots ,\operatorname {E} [X_{n}])}$
and, for a random matrix ${\displaystyle X}$ with elements ${\displaystyle X_{ij}}$,
${\displaystyle (\operatorname {E} [X])_{ij}=\operatorname {E} [X_{ij}].}$
Basic properties
The properties below replicate or follow immediately from those of Lebesgue integral.
${\displaystyle \operatorname {E} [{\mathbf {1} }_{A}]=\operatorname {P} (A)}$
If ${\displaystyle A}$ is an event, then ${\displaystyle \operatorname {E} [{\mathbf {1} }_{A}]=\operatorname {P} (A),}$ where ${\displaystyle {\mathbf {1} }_{A}}$ is the indicator function of the set ${\displaystyle A}$.
Proof. By definition of Lebesgue integral of the simple function ${\displaystyle {\mathbf {1} }_{A}={\mathbf {1} }_{A}(\omega )}$,
${\displaystyle \operatorname {E} [{\mathbf {1} }_{A}]=1\cdot \operatorname {P} (A)+0\cdot \operatorname {P} (\Omega \setminus A)=\operatorname {P} (A).}$
If X = Y (a.s.) then E[X] = E[Y]
The statement follows from the definition of Lebesgue integral (${\displaystyle X_{+}=Y_{+}}$ (a.s.), ${\displaystyle X_{-}=Y_{-}}$ (a.s.)), and that changing a simple random variable on a set of probability zero does not alter the expected value.
Expected value of a constant
If ${\displaystyle X}$ is a random variable, and ${\displaystyle X=c}$ (a.s.), where ${\displaystyle c\in [-\infty ,+\infty ]}$, then ${\displaystyle \operatorname {E} [X]=c}$. In particular, for an arbitrary random variable ${\displaystyle X}$, ${\displaystyle \operatorname {E} [\operatorname {E} [X]]=\operatorname {E} [X]}$.
Linearity
The expected value operator (or expectation operator) ${\displaystyle \operatorname {E} [\cdot ]}$ is linear in the sense that
{\displaystyle {\begin{aligned}\operatorname {E} [X+Y]&=\operatorname {E} [X]+\operatorname {E} [Y],\\[6pt]\operatorname {E} [aX]&=a\operatorname {E} [X],\end{aligned}}}
where ${\displaystyle X}$ and ${\displaystyle Y}$ are arbitrary random variables, and ${\displaystyle a}$ is a constant.
More rigorously, let ${\displaystyle X}$ and ${\displaystyle Y}$ be random variables whose expected values are defined (different from ${\displaystyle \infty -\infty }$).
• If ${\displaystyle \operatorname {E} [X]+\operatorname {E} [Y]}$ is also defined (i.e. differs from ${\displaystyle \infty -\infty }$), then
${\displaystyle \operatorname {E} [X+Y]=\operatorname {E} [X]+\operatorname {E} [Y].}$
• Let ${\displaystyle \operatorname {E} [X]}$ be finite, and ${\displaystyle a\in \mathbb {R} }$ be a finite scalar. Then ${\displaystyle \operatorname {E} [aX]=a\operatorname {E} [X].}$
E[X] exists and is finite if and only if E[|X|] is finite
The following statements regarding a random variable ${\displaystyle X}$ are equivalent:
• ${\displaystyle \operatorname {E} [X]}$ exists and is finite.
• Both ${\displaystyle \operatorname {E} [X_{+}]}$ and ${\displaystyle \operatorname {E} [X_{-}]}$ are finite.
• ${\displaystyle \operatorname {E} [|X|]}$ is finite.
Sketch of proof. Indeed, ${\displaystyle |X|=X_{+}+X_{-}}$. By linearity, ${\displaystyle \operatorname {E} [|X|]=\operatorname {E} [X_{+}]+\operatorname {E} [X_{-}]}$. The above equivalency relies on the definition of Lebesgue integral and measurability of ${\displaystyle X}$.
Remark. For the reasons above, the expressions "${\displaystyle X}$ is integrable" and "the expected value of ${\displaystyle X}$ is finite" are used interchangeably when speaking of a random variable throughout this article.
Monotonicity
If ${\displaystyle X\leq Y}$ (a.s.), and both ${\displaystyle \operatorname {E} [X]}$ and ${\displaystyle \operatorname {E} [Y]}$ exist, then ${\displaystyle \operatorname {E} [X]\leq \operatorname {E} [Y]}$.
Remark. ${\displaystyle \operatorname {E} [X]}$ and ${\displaystyle \operatorname {E} [Y]}$ exist in the sense that ${\displaystyle \min(\operatorname {E} [X_{+}],\operatorname {E} [X_{-}])<\infty }$ and ${\displaystyle \min(\operatorname {E} [Y_{+}],\operatorname {E} [Y_{-}])<\infty .}$
Proof follows from the linearity and the previous property for ${\displaystyle Z=Y-X}$, since ${\displaystyle Z\geq 0}$ (a.s.).
If ${\displaystyle |X|\leq Y}$ (a.s.) and ${\displaystyle \operatorname {E} [Y]}$ is finite then so is ${\displaystyle \operatorname {E} [X]}$
Let ${\displaystyle X}$ and ${\displaystyle Y}$ be random variables such that ${\displaystyle |X|\leq Y}$ (a.s.) and ${\displaystyle \operatorname {E} [Y]<\infty }$. Then ${\displaystyle \operatorname {E} [X]\neq \pm \infty }$.
Proof. Due to non-negativity of ${\displaystyle |X|}$, ${\displaystyle \operatorname {E} |X|}$ exists, finite or infinite. By monotonicity, ${\displaystyle \operatorname {E} |X|\leq \operatorname {E} [Y]<\infty }$, so ${\displaystyle \operatorname {E} |X|}$ is finite which, as we saw earlier, is equivalent to ${\displaystyle \operatorname {E} [X]}$ being finite.
If ${\displaystyle \operatorname {E} |X^{\beta }|<\infty }$ and ${\displaystyle 0<\alpha <\beta }$ then ${\displaystyle \operatorname {E} |X^{\alpha }|<\infty }$
The proposition below will be used to prove the extremal property of ${\displaystyle \operatorname {E} [X]}$ later on.
Proposition. If ${\displaystyle X}$ is a random variable, then so is ${\displaystyle X^{\alpha }}$, for every ${\displaystyle \alpha >0}$. If, in addition, ${\displaystyle \operatorname {E} |X^{\beta }|<\infty }$ and ${\displaystyle 0<\alpha <\beta }$, then ${\displaystyle \operatorname {E} |X^{\alpha }|<\infty }$.
Counterexample for infinite measure
The requirement that ${\displaystyle \operatorname {P} (\Omega )<\infty }$ is essential. By way of counterexample, consider the measurable space
${\displaystyle ([1,+\infty ),{\mathcal {B}}_{\mathbb {R} _{[1,+\infty )}},\lambda ),}$
where ${\displaystyle {\mathcal {B}}_{\mathbb {R} _{[1,+\infty )}}}$ is the Borel ${\displaystyle \sigma }$-algebra on the interval ${\displaystyle [1,+\infty ),}$ and ${\displaystyle \lambda }$ is the linear Lebesgue measure. The reader can prove that ${\displaystyle \textstyle \int _{[1,+\infty )}{\frac {1}{x}}\,dx=\infty ,}$ even though ${\displaystyle \textstyle \int _{[1,+\infty )}{\frac {1}{x^{2}}}\,dx=1.}$ (Sketch of proof: ${\displaystyle \textstyle \int _{S}{\frac {1}{x}}\,dx}$ and ${\displaystyle \textstyle \int _{S}{\frac {1}{x^{2}}}\,dx}$ define a measure ${\displaystyle \mu }$ on ${\displaystyle \textstyle [1,+\infty )=\cup _{n=1}^{\infty }[1,n].}$ Use "continuity from below" w.r. to ${\displaystyle \mu }$ and reduce to Riemann integral on each finite subinterval ${\displaystyle [1,n]}$).
Extremal property
Recall, as we proved early on, that if ${\displaystyle X}$ is a random variable, then so is ${\displaystyle X^{2}}$.
Proposition (extremal property of ${\displaystyle \operatorname {E} [X])}$). Let ${\displaystyle X}$ be a random variable, and ${\displaystyle \operatorname {E} [X^{2}]<\infty }$. Then ${\displaystyle \operatorname {E} [X]}$ and ${\displaystyle \operatorname {Var} [X]}$ are finite, and ${\displaystyle \operatorname {E} [X]}$ is the best least squares approximation for ${\displaystyle X}$ among constants. Specifically,
• for every ${\displaystyle c\in \mathbb {R} }$, ${\displaystyle \textstyle \operatorname {E} [X-c]^{2}\geq \operatorname {Var} [X];}$
• equality holds if and only if ${\displaystyle c=\operatorname {E} [X].}$
(${\displaystyle \operatorname {Var} [X]}$ denotes the variance of ${\displaystyle X}$).
Remark (intuitive interpretation of extremal property). In intuitive terms, the extremal property says that if one is asked to predict the outcome of a trial of a random variable ${\displaystyle X}$, then ${\displaystyle \operatorname {E} [X]}$, in some practically useful sense, is one's best bet if no advance information about the outcome is available. If, on the other hand, one does have some advance knowledge ${\displaystyle {\mathcal {F}}}$ regarding the outcome, then — again, in some practically useful sense — one's bet may be improved upon by using conditional expectations ${\displaystyle \operatorname {E} [X\mid {\mathcal {F}}]}$ (of which ${\displaystyle \operatorname {E} [X]}$ is a special case) rather than ${\displaystyle \operatorname {E} [X]}$.
Proof of proposition. By the above properties, both ${\displaystyle \operatorname {E} [X]}$ and ${\displaystyle \operatorname {Var} [X]=\operatorname {E} [X^{2}]-\operatorname {E} ^{2}[X]}$ are finite, and
{\displaystyle {\begin{aligned}\operatorname {E} [X-c]^{2}&=\operatorname {E} [X^{2}-2cX+c^{2}]\\[6pt]&=\operatorname {E} [X^{2}]-2c\operatorname {E} [X]+c^{2}\\[6pt]&=(c-\operatorname {E} [X])^{2}+\operatorname {E} [X^{2}]-\operatorname {E} ^{2}[X]\\[6pt]&=(c-\operatorname {E} [X])^{2}+\operatorname {Var} [X],\end{aligned}}}
whence the extremal property follows.
Non-degeneracy
If ${\displaystyle \operatorname {E} |X|=0}$, then ${\displaystyle X=0}$ (a.s.).
${\displaystyle |\operatorname {E} [X]|\leq \operatorname {E} |X|}$
For an arbitrary random variable ${\displaystyle X}$, ${\displaystyle |\operatorname {E} [X]|\leq \operatorname {E} |X|}$.
Proof. By definition of Lebesgue integral,
{\displaystyle {\begin{aligned}|\operatorname {E} [X]|&={\Bigl |}\operatorname {E} [X_{+}]-\operatorname {E} [X_{-}]{\Bigr |}\leq {\Bigl |}\operatorname {E} [X_{+}]{\Bigr |}+{\Bigl |}\operatorname {E} [X_{-}]{\Bigr |}\\[5pt]&=\operatorname {E} [X_{+}]+\operatorname {E} [X_{-}]=\operatorname {E} [X_{+}+X_{-}]\\[5pt]&=\operatorname {E} |X|.\end{aligned}}}
This result can also be proved based on Jensen's inequality.
Non-multiplicativity
In general, the expected value operator is not multiplicative, i.e. ${\displaystyle \operatorname {E} [XY]}$ is not necessarily equal to ${\displaystyle \operatorname {E} [X]\cdot \operatorname {E} [Y]}$. Indeed, let ${\displaystyle X}$ assume the values of 1 and -1 with probability 0.5 each. Then
${\displaystyle \operatorname {E^{2}} [X]=\left({\frac {1}{2}}\cdot (-1)+{\frac {1}{2}}\cdot 1\right)^{2}=0,}$
and
${\displaystyle \operatorname {E} [X^{2}]={\frac {1}{2}}\cdot (-1)^{2}+{\frac {1}{2}}\cdot 1^{2}=1,{\text{ so }}\operatorname {E} [X^{2}]\neq \operatorname {E^{2}} [X].}$
The amount by which the multiplicativity fails is called the covariance:
${\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y].}$
However, if ${\displaystyle X}$ and ${\displaystyle Y}$ are independent, then ${\displaystyle \operatorname {E} [XY]=\operatorname {E} [X]\operatorname {E} [Y]}$, and ${\displaystyle \operatorname {Cov} (X,Y)=0}$.
Counterexample: ${\displaystyle \operatorname {E} [X_{i}]\not \to \operatorname {E} [X]}$ despite ${\displaystyle X_{i}\to X}$ pointwise
Let ${\displaystyle \left([0,1],{\mathcal {B}}_{[0,1]},{\mathrm {P} }\right)}$ be the probability space, where ${\displaystyle {\mathcal {B}}_{[0,1]}}$ is the Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle [0,1]}$ and ${\displaystyle {\mathrm {P} }}$ the linear Lebesgue measure. For ${\displaystyle i\geq 1,}$ define a sequence of random variables
${\displaystyle X_{i}=i\cdot {\mathbf {1} }_{\left[0,{\frac {1}{i}}\right]}}$
and a random variable
${\displaystyle X={\begin{cases}+\infty &{\text{if}}\ x=0\\0&{\text{otherwise.}}\end{cases}}}$
on ${\displaystyle [0,1]}$, with ${\displaystyle {\mathbf {1} }_{S}}$ being the indicator function of the set ${\displaystyle S\subseteq [0,1]}$.
For every ${\displaystyle x\in [0,1],}$ as ${\displaystyle i\to +\infty ,}$ ${\displaystyle X_{i}(x)\to X(x),}$ and
${\displaystyle \operatorname {E} [X_{i}]=i\cdot {\mathrm {P} }\left(\left[0,{\frac {1}{i}}\right]\right)=i\cdot {\dfrac {1}{i}}=1,}$
so ${\displaystyle \lim _{i\to \infty }\operatorname {E} [X_{i}]=1.}$ On the other hand, ${\displaystyle \mathop {\mathrm {P} } (\{0\})=0,}$ and hence ${\displaystyle \operatorname {E} \left[X\right]=0.}$
In general, the expected value operator is not ${\displaystyle \sigma }$-additive, i.e.
${\displaystyle \operatorname {E} \left[\sum _{i=0}^{\infty }X_{i}\right]\neq \sum _{i=0}^{\infty }\operatorname {E} [X_{i}].}$
By way of counterexample, let ${\displaystyle \left([0,1],{\mathcal {B}}_{[0,1]},{\mathrm {P} }\right)}$ be the probability space, where ${\displaystyle {\mathcal {B}}_{[0,1]}}$ is the Borel ${\displaystyle \sigma }$-algebra on ${\displaystyle [0,1]}$ and ${\displaystyle {\mathrm {P} }}$ the linear Lebesgue measure. Define a sequence of random variables ${\displaystyle \textstyle X_{i}=(i+1)\cdot {\mathbf {1} }_{\left[0,{\frac {1}{i+1}}\right]}-i\cdot {\mathbf {1} }_{\left[0,{\frac {1}{i}}\right]}}$ on ${\displaystyle [0,1]}$, with ${\displaystyle {\mathbf {1} }_{S}}$ being the indicator function of the set ${\displaystyle S\subseteq [0,1]}$. For the pointwise sums, we have
${\displaystyle \sum _{i=0}^{n}X_{i}=(n+1)\cdot {\mathbf {1} }_{\left[0,{\frac {1}{n+1}}\right]},}$
${\displaystyle \sum _{i=0}^{\infty }X_{i}(x)={\begin{cases}+\infty &{\text{if}}\ x=0\\0&{\text{otherwise.}}\end{cases}}}$
${\displaystyle \sum _{i=0}^{\infty }\operatorname {E} [X_{i}]=\lim _{n\to \infty }\sum _{i=0}^{n}\operatorname {E} [X_{i}]=\lim _{n\to \infty }\operatorname {E} \left[\sum _{i=0}^{n}X_{i}\right]=1.}$
On the other hand, ${\displaystyle \mathop {\mathrm {P} } (\{0\})=0,}$ and hence
${\displaystyle \operatorname {E} \left[\sum _{i=0}^{\infty }X_{i}\right]=0\neq 1=\sum _{i=0}^{\infty }\operatorname {E} [X_{i}].}$
Countable additivity for non-negative random variables
Let ${\displaystyle \{X_{i}\}_{i=0}^{\infty }}$ be non-negative random variables. It follows from monotone convergence theorem that
${\displaystyle \operatorname {E} \left[\sum _{i=0}^{\infty }X_{i}\right]=\sum _{i=0}^{\infty }\operatorname {E} [X_{i}].}$
Inequalities
Cauchy–Bunyakovsky–Schwarz inequality
The Cauchy–Bunyakovsky–Schwarz inequality states that
${\displaystyle (\operatorname {E} [XY])^{2}\leq \operatorname {E} [X^{2}]\cdot \operatorname {E} [Y^{2}].}$
Markov's inequality
For a nonnegative random variable ${\displaystyle X}$ and ${\displaystyle a>0}$, Markov's inequality states that
${\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} [X]}{a}}.}$
Bienaymé-Chebyshev inequality
Let ${\displaystyle X}$ be an arbitrary random variable with finite expected value ${\displaystyle \operatorname {E} [X]}$ and finite variance ${\displaystyle \operatorname {Var} [X]\neq 0}$. The Bienaymé-Chebyshev inequality states that, for any real number ${\displaystyle k>0}$,
${\displaystyle \operatorname {P} {\Bigl (}{\Bigl |}X-\operatorname {E} [X]{\Bigr |}\geq k{\sqrt {\operatorname {Var} [X]}}{\Bigr )}\leq {\frac {1}{k^{2}}}.}$
Jensen's inequality
Let ${\displaystyle f:{\mathbb {R} }\to {\mathbb {R} }}$ be a Borel convex function and ${\displaystyle X}$ a random variable such that ${\displaystyle \operatorname {E} |X|<\infty }$. Jensen's inequality states that
${\displaystyle f(\operatorname {E} (X))\leq \operatorname {E} (f(X)).}$
Remark 1. The expected value ${\displaystyle \operatorname {E} (f(X))}$ is well-defined even if ${\displaystyle X}$ is allowed to assume infinite values. Indeed, ${\displaystyle \operatorname {E} |X|<\infty }$ implies that ${\displaystyle X\neq \pm \infty }$ (a.s.), so the random variable ${\displaystyle f(X(\omega ))}$ is defined almost sure, and therefore there is enough information to compute ${\displaystyle \operatorname {E} (f(X)).}$
Remark 2. Jensen's inequality implies that ${\displaystyle |\operatorname {E} [X]|\leq \operatorname {E} |X|}$ since the absolute value function is convex.
Lyapunov's inequality
Let ${\displaystyle 0. Lyapunov's inequality states that
${\displaystyle {\Bigl (}\operatorname {E} |X|^{s}{\Bigr )}^{1/s}\leq \left(\operatorname {E} |X|^{t}\right)^{1/t}.}$
Proof. Applying Jensen's inequality to ${\displaystyle |X|^{s}}$ and ${\displaystyle g(x)=|x|^{t/s}}$, obtain ${\displaystyle {\Bigl |}\operatorname {E} |X^{s}|{\Bigr |}^{t/s}\leq \operatorname {E} |X^{s}|^{t/s}=\operatorname {E} |X|^{t}}$. Taking the ${\displaystyle t}$th root of each side completes the proof.
Corollary.
${\displaystyle \operatorname {E} |X|\leq {\Bigl (}\operatorname {E} |X|^{2}{\Bigr )}^{1/2}\leq \cdots \leq {\Bigl (}\operatorname {E} |X|^{n}{\Bigr )}^{1/n}\leq \cdots }$
Hölder's inequality
Let ${\displaystyle p}$ and ${\displaystyle q}$ satisfy ${\displaystyle 1\leq p\leq \infty }$, ${\displaystyle 1\leq q\leq \infty }$, and ${\displaystyle 1/p+1/q=1}$. The Hölder's inequality states that
${\displaystyle \operatorname {E} |XY|\leq (\operatorname {E} |X|^{p})^{1/p}(\operatorname {E} |Y|^{q})^{1/q}.}$
Minkowski inequality
Let ${\displaystyle p}$ be an integer satisfying ${\displaystyle 1\leq p\leq \infty }$. Let, in addition, ${\displaystyle \operatorname {E} |X|^{p}<\infty }$ and ${\displaystyle \operatorname {E} |Y|^{p}<\infty }$. Then, according to the Minkowski inequality, ${\displaystyle \operatorname {E} |X+Y|^{p}<\infty }$ and
${\displaystyle {\Bigl (}\operatorname {E} |X+Y|^{p}{\Bigr )}^{1/p}\leq {\Bigl (}\operatorname {E} |X|^{p}{\Bigr )}^{1/p}+{\Bigl (}\operatorname {E} |Y|^{p}{\Bigr )}^{1/p}.}$
Taking limits under the ${\displaystyle \operatorname {E} }$ sign
Monotone convergence theorem
Let the sequence of random variables ${\displaystyle \{X_{n}\}}$ and the random variables ${\displaystyle X}$ and ${\displaystyle Y}$ be defined on the same probability space ${\displaystyle (\Omega ,\Sigma ,\operatorname {P} ).}$ Suppose that
• all the expected values ${\displaystyle \operatorname {E} [X_{n}],}$ ${\displaystyle \operatorname {E} [X],}$ and ${\displaystyle \operatorname {E} [Y]}$ are defined (differ from ${\displaystyle \infty -\infty }$);
• ${\displaystyle \operatorname {E} [Y]>-\infty ;}$
• for every ${\displaystyle n,}$
${\displaystyle -\infty \leq Y\leq X_{n}\leq X_{n+1}\leq +\infty \quad {\hbox{(a.s.)}};}$
• ${\displaystyle X}$ is the pointwise limit of ${\displaystyle \{X_{n}\}}$ (a.s.), i.e. ${\displaystyle X(\omega )=\lim \nolimits _{n}X_{n}(\omega )}$ (a.s.).
The monotone convergence theorem states that
${\displaystyle \lim _{n}\operatorname {E} [X_{n}]=\operatorname {E} [X].}$
Fatou's lemma
Let the sequence of random variables ${\displaystyle \{X_{n}\}}$ and the random variable ${\displaystyle Y}$ be defined on the same probability space ${\displaystyle (\Omega ,\Sigma ,\operatorname {P} ).}$ Suppose that
• all the expected values ${\displaystyle \operatorname {E} [X_{n}],}$ ${\displaystyle \textstyle \operatorname {E} [\liminf _{n}X_{n}],}$ and ${\displaystyle \operatorname {E} [Y]}$ are defined (differ from ${\displaystyle \infty -\infty }$);
• ${\displaystyle \operatorname {E} [Y]>-\infty ;}$
• ${\displaystyle -\infty \leq Y\leq X_{n}\leq +\infty }$ (a.s.), for every ${\displaystyle n.}$
Fatou's lemma states that
${\displaystyle \operatorname {E} [\liminf _{n}X_{n}]\leq \liminf _{n}\operatorname {E} [X_{n}].}$
(${\displaystyle \textstyle \liminf _{n}X_{n}}$ is a random variable, for every ${\displaystyle n,}$ by the properties of limit inferior).
Corollary. Let
• ${\displaystyle X_{n}\to X}$ pointwise (a.s.);
• ${\displaystyle \operatorname {E} [X_{n}]\leq C,}$ for some constant ${\displaystyle C}$ (independent from ${\displaystyle n}$);
• ${\displaystyle \operatorname {E} [Y]>-\infty ;}$
• ${\displaystyle -\infty \leq Y\leq X_{n}\leq +\infty }$ (a.s.), for every ${\displaystyle n.}$
Then ${\displaystyle \operatorname {E} [X]\leq C.}$
Proof is by observing that ${\displaystyle \textstyle X=\liminf _{n}X_{n}}$ (a.s.) and applying Fatou's lemma.
Dominated convergence theorem
Let ${\displaystyle \{X_{n}\}_{n}}$ be a sequence of random variables. If ${\displaystyle X_{n}\to X}$ pointwise (a.s.), ${\displaystyle |X_{n}|\leq Y\leq +\infty }$ (a.s.), and ${\displaystyle \operatorname {E} [Y]<\infty }$. Then, according to the dominated convergence theorem,
• the function ${\displaystyle X}$ is measurable (hence a random variable);
• ${\displaystyle \operatorname {E} |X|<\infty }$;
• all the expected values ${\displaystyle \operatorname {E} [X_{n}]}$ and ${\displaystyle \operatorname {E} [X]}$ are defined (do not have the form ${\displaystyle \infty -\infty }$);
• ${\displaystyle \lim _{n}\operatorname {E} [X_{n}]=\operatorname {E} [X]}$ (both sides may be infinite);
• ${\displaystyle \lim _{n}\operatorname {E} |X_{n}-X|=0.}$
Uniform integrability
In some cases, the equality ${\displaystyle \displaystyle \lim _{n}\operatorname {E} [X_{n}]=\operatorname {E} [\lim _{n}X_{n}]}$ holds when the sequence ${\displaystyle \{X_{n}\}}$ is uniformly integrable.
Relationship with characteristic function
The probability density function ${\displaystyle f_{X}}$ of a scalar random variable ${\displaystyle X}$ is related to its characteristic function ${\displaystyle \varphi _{X}}$ by the inversion formula:
${\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.}$
For the expected value of ${\displaystyle g(X)}$ (where ${\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}$ is a Borel function), we can use this inversion formula to obtain
${\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\left[\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt\right]\,dx.}$
If ${\displaystyle \operatorname {E} [g(X)]}$ is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem,
${\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,}$
where
${\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}$
is the Fourier transform of ${\displaystyle g(x).}$ The expression for ${\displaystyle \operatorname {E} [g(X)]}$ also follows directly from Plancherel theorem.
Uses and applications
It is possible to construct an expected value equal to the probability of an event by taking the expectation of an indicator function that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the law of large numbers to justify estimating probabilities by frequencies.
The expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of X − E[X]. The moments of some random variables can be used to specify their distributions, via their moment generating functions.
To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller.
This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate (probabilistic) quantities of interest via Monte Carlo methods, since most quantities of interest can be written in terms of expectation, e.g. ${\displaystyle \operatorname {P} ({X\in {\mathcal {A}}})=\operatorname {E} [{\mathbf {1} }_{\mathcal {A}}]}$, where ${\displaystyle {\mathbf {1} }_{\mathcal {A}}}$ is the indicator function of the set ${\displaystyle {\mathcal {A}}}$.
The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β).
In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose X is a discrete random variable with values xi and corresponding probabilities pi. Now consider a weightless rod on which are placed weights, at locations xi along the rod and having masses pi (whose sum is one). The point at which the rod balances is E[X].
Expected values can also be used to compute the variance, by means of the computational formula for the variance
${\displaystyle \operatorname {Var} (X)=\operatorname {E} [X^{2}]-(\operatorname {E} [X])^{2}.}$
A very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator ${\displaystyle {\hat {A}}}$ operating on a quantum state vector ${\displaystyle |\psi \rangle }$ is written as ${\displaystyle \langle {\hat {A}}\rangle =\langle \psi |A|\psi \rangle }$. The uncertainty in ${\displaystyle {\hat {A}}}$ can be calculated using the formula ${\displaystyle (\Delta A)^{2}=\langle {\hat {A}}^{2}\rangle -\langle {\hat {A}}\rangle ^{2}}$.
The law of the unconscious statistician
The expected value of a measurable function of ${\displaystyle X}$, ${\displaystyle g(X)}$, given that ${\displaystyle X}$ has a probability density function ${\displaystyle f(x)}$, is given by the inner product of ${\displaystyle f}$ and ${\displaystyle g}$:
${\displaystyle \operatorname {E} [g(X)]=\int _{\mathbb {R} }g(x)f(x)\,dx.}$
This formula also holds in multidimensional case, when ${\displaystyle g}$ is a function of several random variables, and ${\displaystyle f}$ is their joint density.[5][6]
Alternative formula for expected value
Formula for non-negative random variables
Finite and countably infinite case
For a non-negative integer-valued random variable ${\displaystyle X:\Omega \to \{0,1,2,3,\ldots \}\cup \{+\infty \},}$
${\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }\operatorname {P} (X\geq i).}$
General case
If ${\displaystyle X:\Omega \to [0,+\infty ]}$ is a non-negative random variable, then
${\displaystyle \operatorname {E} [X]=\int \limits _{[0,+\infty )}\operatorname {P} (X\geq x)\,dx=\int \limits _{[0,+\infty )}\operatorname {P} (X>x)\,dx,}$
and
${\displaystyle \operatorname {E} [X]={\hbox{(R)}}\int \limits _{0}^{\infty }\operatorname {P} (X\geq x)\,dx={\hbox{(R)}}\int \limits _{0}^{\infty }\operatorname {P} (X>x)\,dx,}$
where ${\displaystyle {\hbox{(R)}}\textstyle \int _{0}^{\infty }}$ denotes improper Riemann integral.
Formula for non-positive random variables
If ${\displaystyle X:\Omega \to [-\infty ,0]}$ is a non-positive random variable, then
${\displaystyle \operatorname {E} [X]=-\int \limits _{(-\infty ,0]}\operatorname {P} (X\leq x)\,dx=-\int \limits _{(-\infty ,0]}\operatorname {P} (X
and
${\displaystyle \operatorname {E} [X]=-{\hbox{(R)}}\int \limits _{-\infty }^{0}\operatorname {P} (X\leq x)\,dx=-{\hbox{(R)}}\int \limits _{-\infty }^{0}\operatorname {P} (X
where ${\displaystyle {\hbox{(R)}}\textstyle \int _{-\infty }^{0}}$ denotes improper Riemann integral.
This formula follows from that for the non-negative case applied to ${\displaystyle -X.}$
If, in addition, ${\displaystyle X}$ is integer-valued, i.e. ${\displaystyle X:\Omega \to \{\ldots ,-3,-2,-1,0\}\cup \{-\infty \}}$, then
${\displaystyle \operatorname {E} [X]=-\sum _{i=-1}^{-\infty }\operatorname {P} (X\leq i).}$
General case
If ${\displaystyle X}$ can be both positive and negative, then ${\displaystyle \operatorname {E} [X]=\operatorname {E} [X_{+}]-\operatorname {E} [X_{-}]}$, and the above results may be applied to ${\displaystyle X_{+}}$ and ${\displaystyle X_{-}}$ separately.
History
The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players who have to end their game before it's properly finished. This problem had been debated for centuries, and many conflicting proposals and solutions had been suggested over the years, when it was posed in 1654 to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. He began to discuss the problem in a now famous series of letters to Pierre de Fermat. Soon enough they both independently came up with a solution. They solved the problem in different computational ways but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution and this in turn made them absolutely convinced they had solved the problem conclusively. However, they did not publish their findings. They only informed a small circle of mutual scientific friends in Paris about it.[7]
Three years later, in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise (see Huygens (1657)) "De ratiociniis in ludo aleæ" on probability theory. In this book he considered the problem of points and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens also extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players). In this sense this book can be seen as the first successful attempt at laying down the foundations of the theory of probability.
In the foreword to his book, Huygens wrote: "It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from the same principle. But finally I have found that my answers in many cases do not differ from theirs." (cited by Edwards (2002)). Thus, Huygens learned about de Méré's Problem in 1655 during his visit to France; later on in 1656 from his correspondence with Carcavi he learned that his method was essentially the same as Pascal's; so that before his book went to press in 1657 he knew about Pascal's priority in this subject.
Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: "That my Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure me in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal Chance of gaining them, my Expectation is worth a+b/2." More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract "Théorie analytique des probabilités", where the concept of expected value was defined explicitly:
… this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for the sum hoped for. We will call this advantage mathematical hope.
The use of the letter E to denote expected value goes back to W.A. Whitworth in 1901,[8] who used a script E. The symbol has become popular since for English writers it meant "Expectation", for Germans "Erwartungswert", for Spanish "Esperanza matemática" and for French "Espérance mathématique".[9]
References
1. ^ Sheldon M Ross (2007). "§2.4 Expectation of a random variable". Introduction to probability models (9th ed.). Academic Press. p. 38 ff. ISBN 0-12-598062-0.
2. ^ Richard W Hamming (1991). "§2.5 Random variables, mean and the expected value". The art of probability for scientists and engineers. Addison–Wesley. p. 64 ff. ISBN 0-201-40686-1.
3. ^ Richard W Hamming (1991). "Example 8.7–1 The Cauchy distribution". The art of probability for scientists and engineers. Addison-Wesley. p. 290 ff. ISBN 0-201-40686-1. Sampling from the Cauchy distribution and averaging gets you nowhere — one sample has the same distribution as the average of 1000 samples!
4. ^ Gordon, Lawrence; Loeb, Martin (November 2002). "The Economics of Information Security Investment". ACM Transactions on Information and System Security. 5 (4): 438–457. doi:10.1145/581271.581274.
5. ^ Expectation Value, retrieved August 8, 2017
6. ^ Papoulis, A. (1984), Probability, Random Variables, and Stochastic Processes, New York: McGraw–Hill, pp. 139–152
7. ^ "Ore, Pascal and the Invention of Probability Theory". The American Mathematical Monthly. 67 (5): 409–419. 1960. doi:10.2307/2309286.
8. ^ Whitworth, W.A. (1901) Choice and Chance with One Thousand Exercises. Fifth edition. Deighton Bell, Cambridge. [Reprinted by Hafner Publishing Co., New York, 1959.]
9. ^
Literature
• Edwards, A.W.F (2002). Pascal's arithmetical triangle: the story of a mathematical idea (2nd ed.). JHU Press. ISBN 0-8018-6946-3.
• Huygens, Christiaan (1657). De ratiociniis in ludo aleæ (English translation, published in 1714:). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 611, "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": 0.9870748519897461, "perplexity": 978.2881045520115}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319470.94/warc/CC-MAIN-20190824020840-20190824042840-00363.warc.gz"} |
https://mathoverflow.net/questions/301527/biased-vs-unbiased-lax-monoidal-categories | # Biased vs unbiased lax monoidal categories
There are two principal ways to define a monoidal category:
• The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes B)\otimes C\cong A\otimes (B\otimes C)$ and unit isomorphisms satisfying appropriate axioms.
• The unbiased definition includes an $n$-ary tensor product $(A_1\otimes\cdots \otimes A_n)$ for all $n\ge 0$ (where $n=0$ gives the unit $I = ()$), with associativity isomorphisms such as $((A\otimes B) \otimes () \otimes (C)) \cong (A\otimes B\otimes C)$ satisfying appropriate axioms.
The two definitions are equivalent in an appropriate sense (though this is a nontrivial coherence theorem). However, this is no longer true for "lax" kinds of monoidal category, where the associativity and unit isomorphisms are replaced by not-necessarily-invertible transformations. In the lax case, the unbiased definition seems to be more-studied, and is usually what people mean by a "lax monoidal category". There are good reasons for this, but "biased-lax" monoidal categories, and more general biased-lax structures, do occasionally pop up.
In the unbiased case, there are only two consistent choices of direction for the transformations: $((A\otimes B) \otimes () \otimes (C)) \to (A\otimes B\otimes C)$ gives a lax monoidal category, while the opposite direction gives a colax one. In the biased case, there are more choices: in addition to choosing $(A\otimes B)\otimes C\to A\otimes (B\otimes C)$ or the opposite, we can choose how to orient the two unit morphisms: either $A \otimes I \to A$ or $A \to A\otimes I$, and also either $I\otimes A \to A$ or $A\to I\otimes A$. For instance, a skew monoidal category pairs $A\to I\otimes A$ with $A\otimes I\to A$. (Thanks Maxime for pointing this out in the comments.)
In this question I am interested in biased-lax monoidal categories where the unit transformations go in the same direction, say $A\otimes I\to A$ and $I\otimes A\to A$. It seems that it should be possible to identify a biased-lax monoidal category of this sort with a particular kind of unbiased-lax one, by defining the $n$-ary tensor product in terms of the binary one by right-associativity: $(A_1\otimes\cdots \otimes A_n) = (A_1 \otimes (A_2 \otimes \cdots \otimes (A_{n-1}\otimes A_n)\cdots ))$ (or perhaps left associativity, depending on which direction the biased-lax associativity map goes). I have seen this claimed in print, and have even claimed it myself, but I have not seen a proof written out. So my questions are:
1. Has anyone studied biased-lax monoidal categories of this sort (or related structures like biased-lax bicategories, biased-lax monoids in a monoidal bicategory, etc.) in detail?
2. In particular, is there a better name for them? (The only reference I know of is the paper "$T$-categories" by Albert Burroni, who called "biased-colax" bicategories of this sort "pseudo-categories" — clearly not a good name in light of modern terminological conventions.)
3. (The main question) Has anyone written out a proof that biased-lax monoidal categories of this sort can be identified with certain unbiased-lax ones?
4. What is an intrinsic characterization of the unbiased-lax monoidal categories that arise in this way? (I expect they should be the ones such that certain of the associativity maps happen to be isomorphisms.)
• Re 2: Skew monoidal categories are an example of biased-monoidal categories which seem to 'ping' a bit more when searching google, but it depends on how you want to orient the unit morphisms. – Maxime Lucas May 30 '18 at 10:05
• @MaximeLucas Thanks! I should have mentioned skew-monoidal categories and specified the direction of unit morphisms; I'll edit the question. I think both unit morphisms have to go in the same direction for my question (3) to be true, which is not the case for skew-monoidal categories. (The actual direction is then arbitrary, up to passage to opposite categories.) – Mike Shulman May 30 '18 at 14:45
• There seems to be an answer to questions 3 and 4 for skew monoidal categories in arxiv.org/abs/1708.06087: in that case, unbiased-(co)lax monoidal categories have to be replaced by (co)lax algebras over a slightly fancier Cat-operad, and there is a "certain of the associativity maps are identities" condition called being an "LBC-algebra" that characterizes the lax algebras arising from skew-monoidal categories in this way. – Mike Shulman May 31 '18 at 20:33
• That is right. You can restrict this to obtain an equivalence between left normal skew monoidal categories (those for which $l:ia \to a$ is invertible) and normal colax monoidal categories satisfying the LBC condition (when viewed as colax $L$-algebras). (The corresponding multicategory result was described in Theorem 6.3 of arxiv.org/abs/1708.06088) – john Jun 1 '18 at 7:47
• This special case about left normal skew monoidal categories would, I suspect, also be a special case of the result you are after about biased (co)lax monoidal categories. – john Jun 1 '18 at 7:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9453216791152954, "perplexity": 615.412869346913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998879.63/warc/CC-MAIN-20190619003600-20190619025600-00518.warc.gz"} |
https://academic.tips/question/describe-stellar-parallax-and-explain-how-one-would-mathematically-measure-and-calculate-the-distance-to-a-star-using-this-method/ | # Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method.
A
Stellar Parallax describes the movement of a star against other more distant stars during the revolution of the Earth around the Sun. The concept can be used to determine the distance of a few stars that are close to the Sun. Mathematically, the distance to a star in parsecs will be given by: d= 1/p where p is in seconds of arc while d is in parsecs.
An answer to this question is provided by one of our experts who specializes in physics. Let us know how much you liked it and give it a rating.
Select a citation style:
References
Academic.Tips. (2021) 'Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method'. 1 July.
Reference
Academic.Tips. (2021, July 1). Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method. Retrieved from https://academic.tips/question/describe-stellar-parallax-and-explain-how-one-would-mathematically-measure-and-calculate-the-distance-to-a-star-using-this-method/
References
Academic.Tips. 2021. "Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method." July 1, 2021. https://academic.tips/question/describe-stellar-parallax-and-explain-how-one-would-mathematically-measure-and-calculate-the-distance-to-a-star-using-this-method/.
1. Academic.Tips. "Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method." July 1, 2021. https://academic.tips/question/describe-stellar-parallax-and-explain-how-one-would-mathematically-measure-and-calculate-the-distance-to-a-star-using-this-method/.
Bibliography
Academic.Tips. "Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method." July 1, 2021. https://academic.tips/question/describe-stellar-parallax-and-explain-how-one-would-mathematically-measure-and-calculate-the-distance-to-a-star-using-this-method/.
Work Cited
"Describe stellar parallax and explain how one would mathematically measure and calculate the distance to a star using this method." Academic.Tips, 1 July 2021, academic.tips/question/describe-stellar-parallax-and-explain-how-one-would-mathematically-measure-and-calculate-the-distance-to-a-star-using-this-method/.
Copy | {"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": 0.9799016118049622, "perplexity": 736.6721204369765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056892.13/warc/CC-MAIN-20210919160038-20210919190038-00266.warc.gz"} |
https://latex.org/forum/viewtopic.php?f=44&t=30819 | ## LaTeX forum ⇒ Text Formatting ⇒ Pagenumbering in textblock Topic is solved
Information and discussion about LaTeX's general text formatting features (e.g. bold, italic, enumerations, ...)
hirscsil
Posts: 2
Joined: Sun Jan 07, 2018 12:42 pm
### Pagenumbering in textblock
Dear Members
As this is my first post in this forum, I don't know if I post it in the correct category. So if not, I'm sorry
Right now I'm writing summaries for the final exams in latex, and since I have three columns and almost no margin, I can't and don't want to use headers or footers.
Because of that I wanted to print the pagenumber in a textblock which works actually quite nice. But the problem is, that the page always starts at page 2 with number one, but of course in the table of content the fist page is actually the one where also the table of content is contained.
I tried many things like putting
\setcounter{page}{1}
on different places but it still didn't work.
This is the code I use right now (shortened). I include my single chapters with the
\input{document}
command, but the error is also visible with the blinddocument.
What i actually want: starting the pagenumbering at the very first page, that the pagenumbers in the table of content are consistant with the pages containing the section.
\documentclass[landscape,a4paper,fontsize=8pt]{scrartcl}\usepackage[dvipsnames]{xcolor}\usepackage[ngerman]{babel} \usepackage{amsmath,color}\usepackage{amssymb}\usepackage{helvet}\usepackage{fancyhdr}\usepackage{blindtext} % To generate the box with the pagenumter\usepackage{atbegshi}\usepackage[absolute,overlay]{textpos} \usepackage[a4paper, left=0.5cm, right=0.5cm, top=0.3cm, bottom=0.3cm]{geometry}\usepackage{flowfram} % Three columns\Ncolumninarea{3}{\textwidth}{\textheight}{0pt}{0pt}\insertvrule{flow}{1}{flow}{2}\insertvrule{flow}{2}{flow}{3} % Generate the textblock with the pagenumber\TPGrid{8}{11}\AtBeginShipout{ \begin{textblock}{3}(7.85,10.7) \footnotesize \texttt{\colorbox{Salmon}{\textbf{\thepage}}} \end{textblock}%} \begin{document} \fontsize{8pt}{3pt}\selectfont \begin{center} \noindent {\scshape\Large Topic \\} {\scshape\large Name, Class \\} \end{center} \tableofcontents %\input{doc1} %\input{doc2} %\input{doc3} \Blinddocument \end{document}
Kind regards and thank you in advance
hirscsil
Tags:
Stefan Kottwitz
Posts: 8953
Joined: Mon Mar 10, 2008 9:44 pm
Hi hirscsil,
welcome to the forum!
Quick fix:
\AtBeginShipout{% \begin{textblock}{3}(7.85,10.7)% \footnotesize\stepcounter{page}% \texttt{\colorbox{Salmon}{\textbf{\thepage}}}% \addtocounter{page}{-1}% \end{textblock}%}
I guess at "begin shipout time" the page counter was not yet incremented.
Stefan
hirscsil
Posts: 2
Joined: Sun Jan 07, 2018 12:42 pm
Stefan Kottwitz wrote:Hi hirscsil,
welcome to the forum!
Quick fix:
\AtBeginShipout{% \begin{textblock}{3}(7.85,10.7)% \footnotesize\stepcounter{page}% \texttt{\colorbox{Salmon}{\textbf{\thepage}}}% \addtocounter{page}{-1}% \end{textblock}%}
I guess at "begin shipout time" the page counter was not yet incremented.
Stefan
Hi Stefan
Oh wow, that was a quick response and works perfectly. Thank you very much.
hirscsil | {"extraction_info": {"found_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": 0.8597541451454163, "perplexity": 4517.9470319734255}, "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-2018-30/segments/1531676591718.31/warc/CC-MAIN-20180720154756-20180720174756-00091.warc.gz"} |
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.em/1175789760 | ## Experimental Mathematics
### Computing Varieties of Representations of Hyperbolic $3$-Manifolds into $\SLfR$
#### Abstract
The geometric structure on a closed orientable hyperbolic 3-manifold determines a discrete faithful representation $\rho$ of its fundamental group into $\mathrm{SO^{+}(3,1)}$, unique up to conjugacy. Although Mostow rigidity prohibits us from deforming $\rho$, we can try to deform the composition of $\rho$ with inclusion of $\mathrm{SO^{+}(3,1)}$ into a larger group. In this sense, we have found by exact computation a small number of closed manifolds in the Hodgson-Weeks census for which $\rho$ deforms into $\mathrm{SL(4,\mathbb R)}$, thus showing that the hyperbolic structure can be deformed in these cases to a real projective structure. In this paper we describe the method for computing these deformations, particular attention being given to the manifold Vol3.
#### Article information
Source
Experiment. Math. Volume 15, Issue 3 (2006), 291-306.
Dates
First available in Project Euclid: 5 April 2007
Cooper, Daryl; Long, Darren; Thistlethwaite, Morwen. Computing Varieties of Representations of Hyperbolic $3$-Manifolds into $\SLfR$. Experiment. Math. 15 (2006), no. 3, 291--306. http://projecteuclid.org/euclid.em/1175789760. | {"extraction_info": {"found_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": 0.9252007007598877, "perplexity": 522.7843334273232}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1429246638820.85/warc/CC-MAIN-20150417045718-00222-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://colvertgroup.com/standard-error/in-statistics-and-estimation-error.php | Home > Standard Error > In Statistics And Estimation Error
# In Statistics And Estimation Error
## Contents
Because of random variation in sampling, the proportion or mean calculated using the sample will usually differ from the true proportion or mean in the entire population. A statistics is a consistent estimator of a parameter if its probability that it will be close to the parameter's true value approaches 1 with increasing sample size. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Stat Trek Teach yourself statistics Skip to main content Home Tutorials AP Statistics Stat Tables Stat Tools Calculators Books A margin of error. his comment is here
The graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16. All rights Reserved.EnglishfrançaisDeutschportuguêsespañol日本語한국어中文（简体）By using this site you agree to the use of cookies for analytics and personalized content.Read our policyOK Home ResearchResearch Methods Experiments Design Statistics Reasoning Philosophy Ethics History AcademicAcademic A natural way to describe the variation of these sample means around the true population mean is the standard deviation of the distribution of the sample means. We might describe this interval estimate as a 95% confidence interval.
## Standard Error Formula
I have chosen an extreme sample size to just make this clear. I. A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. Gurland and Tripathi (1971)[6] provide a correction and equation for this effect.
1. A standard deviation is the spread of the scores around the average in a single sample.
2. May 3, 2015 Matthew Clare · Lancaster University Jochen, I do apologise for providing an answer.
3. Relative standard error See also: Relative standard deviation The relative standard error of a sample mean is the standard error divided by the mean and expressed as a percentage.
4. National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more
They may be used to calculate confidence intervals. May 8, 2015 Can you help by adding an answer? The survey with the lower relative standard error can be said to have a more precise measurement, since it has proportionately less sampling variation around the mean. Standard Error Definition The data set is ageAtMar, also from the R package openintro from the textbook by Dietz et al.[4] For the purpose of this example, the 5,534 women are the entire population
Add to my courses 1 Frequency Distribution 2 Normal Distribution 2.1 Assumptions 3 F-Distribution 4 Central Tendency 4.1 Mean 4.1.1 Arithmetic Mean 4.1.2 Geometric Mean 4.1.3 Calculate Median 4.2 Statistical Mode Point Estimate Formula If one survey has a standard error of $10,000 and the other has a standard error of$5,000, then the relative standard errors are 20% and 10% respectively. II. http://stattrek.com/estimation/standard-error.aspx?Tutorial=AP Follow us!
Trochim, All Rights Reserved Purchase a printed copy of the Research Methods Knowledge Base Last Revised: 10/20/2006 HomeTable of ContentsNavigatingFoundationsSamplingExternal ValiditySampling TerminologyStatistical Terms in SamplingProbability SamplingNonprobability SamplingMeasurementDesignAnalysisWrite-UpAppendicesSearch menuMinitab® 17 SupportWhat is How To Find Point Estimate The standard error is a measure of central tendency. (A) I only (B) II only (C) III only (D) All of the above. (E) None of the above. Because the age of the runners have a larger standard deviation (9.27 years) than does the age at first marriage (4.72 years), the standard error of the mean is larger for Greek letters indicate that these are population values.
## Point Estimate Formula
In this scenario, the 400 patients are a sample of all patients who may be treated with the drug. Comments View the discussion thread. . Standard Error Formula And isn't that why we sampled in the first place? Standard Error Calculator The proportion or the mean is calculated using the sample.
Blackwell Publishing. 81 (1): 75–81. Never did I suggest that it is perfect, so don't ridicule like that. In this scenario, the 2000 voters are a sample from all the actual voters. The margin of error and the confidence interval are based on a quantitative measure of uncertainty: the standard error. Standard Error Vs Standard Deviation
The standard error estimated using the sample standard deviation is 2.56. Confidence Level The probability part of a confidence interval is called a confidence level. The distribution of the mean age in all possible samples is called the sampling distribution of the mean. weblink Sign up today to join our community of over 11+ million scientific professionals.
And, of course, we don't actually know the population parameter value -- we're trying to find that out -- but we can use our best estimate for that -- the sample Standard Error Excel All Rights Reserved. For the runners, the population mean age is 33.87, and the population standard deviation is 9.27.
## And it has been proven that it is an underestimation several times, and the underestimation of your 2 point sample is roughly 25% whereas in 6 data point, the SEM only
From previous experience we know that the population standard deviation is \$5,000 Using alpha = 1 - 0.99 = 0.01, we find the z-values for the endpoints of the CI when Your last post actually clarifies that not the SE itself is indicating the representativeness of the data but rather the sample size (given an appropriate sampling procedure, for sure). Hyattsville, MD: U.S. Standard Error Regression n is the size (number of observations) of the sample.
When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9] The ages in one such sample are 23, 27, 28, 29, 31, 31, 32, 33, 34, 38, 40, 40, 48, 53, 54, and 55. For the purpose of hypothesis testing or estimating confidence intervals, the standard error is primarily of use when the sampling distribution is normally distributed, or approximately normally distributed. The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years.
For the age at first marriage, the population mean age is 23.44, and the population standard deviation is 4.72. This gives 9.27/sqrt(16) = 2.32. The z-score for the normal variable statistics is used to help determine the interval endpoints that correspond to the probability of degree of certainty one which to use for the interval The standard error of the mean now refers to the change in mean with different experiments conducted each time.
A confidence interval is a type of interval estimate, not a type of point estimate. An important aspect of statistical inference is using estimates to approximate the value of an unknown population parameter. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE. The standard error of the mean now refers to the change in mean with different experiments conducted each time.Mathematically, the standard error of the mean formula is given by: σM =
No problem, save it as a course and come back to it later. For example, the sample mean is the usual estimator of a population mean. This gives 9.27/sqrt(16) = 2.32. I leave to you to figure out the other ranges.
Consider the following scenarios. Since the maximum margin of error, E is given by the formula: then solving for n, the sample size for some expected level of error, E. If σ is known, the standard error is calculated using the formula σ x ¯ = σ n {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} where σ is the Standard error of the mean Further information: Variance §Sum of uncorrelated variables (Bienaymé formula) The standard error of the mean (SEM) is the standard deviation of the sample-mean's estimate of a | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9220634698867798, "perplexity": 718.8231335993544}, "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-2018-26/segments/1529267867055.95/warc/CC-MAIN-20180624195735-20180624215735-00226.warc.gz"} |