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https://math.stackexchange.com/questions/3328299/beth-cardinals-and-inacceesible-cardinals/3328318 | # Beth cardinals and inacceesible cardinals
Since my last question here about the Alephs was too imprecise and thus went over like a lead balloon, I am trying a new and simpler question which asks what I probably should have asked before.
Under "Beth Numbers" in Wikipedia I read:
"In ZF, for any cardinals $$\kappa$$ and $$\mu$$, there is an ordinal $$\alpha$$ such that:
$$\kappa \leq \beth_\alpha(\mu)$$."
But under "Inaccessible Cardinals" I read:
"a cardinal $$\kappa$$ is strongly inaccessible if it is uncountable, it is not a sum of fewer than $$\kappa$$ cardinals that are less than $$\kappa$$, and $$\alpha < \kappa$$ implies $$2^\alpha < \kappa$$."
These two passages are troubling to me since they seem to be contradictory. The first one seems to imply that for ANY cardinal one can always find a Beth number which exceeds it. While the second one clearly seems to imply that the first inaccesible and any cardinal larger than it, of which there are uncountably many, of course, are all vastly larger than any Beth cardinal generated by even $$\omega$$ applications of the Power Set operation could ever be.
I assume that I am simply missing something important here, and that both statements from the Wikipedia are actually true. But what exactly am I missing??
• Why are you limiting yourself to only $\omega$ applications of the power set operation? – Eric Wofsey Aug 19 at 19:54
• OK, I thought of that and almost said "any number of Power Set operations," but I don't understand what it would be to apply the operation an uncountable number of times, if such is possible at all. – Wd Fusroy Aug 19 at 20:02
• @WdFusroy "I don't understand what it would be to apply the operation an uncountable number of times, if such is possible at all." This is exactly what's going on here. Are you familiar with transfinite recursion? – Noah Schweber Aug 19 at 22:21
• @WdFusroy "no such restriction is placed on the equation in the Beth numbers article." Right, that's what I said. That's why there's no contradiction. If $\kappa$ is strongly inaccessible, then $\beth_\alpha\lt\kappa$ for $\alpha\lt\kappa$, but $\beth_\alpha\ge\kappa$ for $\alpha\ge\kappa$. What's the problem? – bof Aug 19 at 23:15
• An inaccessible cardinal is still an ordinal. – Asaf Karagila Aug 20 at 6:37
It seems like the issue you're having is with understanding what $$\beth_\alpha$$ means when $$\alpha \geq \omega$$. The definition is by recursion: $$\beth_0 = \aleph_0$$, for any ordinal $$\alpha$$ we define $$\beth_{\alpha+1} = 2^{\beth_\alpha}$$, and when $$\alpha$$ is a limit ordinal we have $$\beth_\alpha = \sup\{\beth_\beta : \beta < \alpha\}$$. This allows us to continue the powerset operation transfinitely.
Notice that, for every $$\alpha$$, $$\aleph_\alpha \leq \beth_\alpha$$. Thus, if $$\kappa = \aleph_\alpha$$, then $$\beth_\alpha \geq \aleph_\alpha = \kappa$$, so it is indeed true that there is a $$\beth$$ number larger than $$\kappa$$ (if you want strictly larger, go for $$\beth_{\alpha+1}$$).
The reason the previous paragraph doesn't contradict the definition of $$\kappa$$ being inaccessible is that if $$\kappa$$ is inaccessible then the $$\alpha$$ for which $$\kappa = \aleph_\alpha$$ is $$\kappa$$ itself, i.e., $$\kappa = \aleph_\kappa$$. Thus, unlike in the definition of inaccessibility, we're not in the position of calculating $$2^{\alpha}$$ with $$\alpha < \kappa$$.
• Well, that all sounds very sensible. But I am still troubled by your claim that:"if κ is inaccessible then the α for which κ=ℵα is κ itself, i.e., κ=ℵκ." I have encountered that claim many times before but never been fully comfortable with it since it isn't obvious to me that there could not exist some big cardinal "j" such that there is NO "a," not even j itself, for which j = Aleph[j], or in the case mentioned here, for which j = Beth[j]. More simply put, how can we know, a priori, that many cardinals do not exceed even uncountably infinite recursive Power Set taking altogether? – Wd Fusroy Aug 19 at 21:02
• I am also still quite troubled by the idea that it would ever be possible to take an uncountable number of recursive Power Set applications. Since these recursions must always be taken in precise sequence, each one after, and building upon, the previous one, they should always remain countable by some huge, but still denumerable, ordinal index, I think. – Wd Fusroy Aug 19 at 21:07
• I should probably also say that I am interested in all of these matters primarily because it has always seemed to me that the Power Set operation is far too "slow" to serve as something like an index of the comparative size of various cardinals, which, I assume, was the general idea behind inventing the Beth Numbers in the first place. There are many other operations, like, say, the tetration operation, i.e. 2 Tet [Super]Beth[n] = [Super]Beth[n+1], which would certainly grow vastly faster than the usual 2 exp Beth[n] = Beth[n+1] But could even that recursion get to any of the inaccessibles?? – Wd Fusroy Aug 19 at 21:20
• There are a few things going on here. The existence of an inaccessible cardinal is something you could reasonably not believe (in the sense that we don't even know that it is consistent, and indeed there are significant barriers to us knowing that, because ZFC+"there is an inaccessible cardinal" implies Con(ZFC)). – Chris Eagle Aug 19 at 22:11
• @WdFusroy "Since these recursions must always be taken in precise sequence, each one after, and building upon, the previous one, they should always remain countable by some huge, but still denumerable, ordinal index, I think." No, that's not necessary. Definition by transfinite recursion makes sense along all the ordinals. – Noah Schweber Aug 19 at 22:32
Your comments indicate that you are dubious about iterating operations on ordinals more than finitely many times, and very dubious about iterating them uncountably many times. This is a fundamental point in set theory. The danger is in thinking of recursive definitions as processes which need to be carried out, in which case our "finitary biases" get in the way. Instead, you should think of a recursive definition as "happening all at once." Essentially, we can show that every recursive description corresponds to a unique function, and what lets us do this is transfinite induction. (It shouldn't be surprising that "we can do recursion for as long as we can do induction.")
Specifically, suppose that $$F:Ord\rightarrow Ord$$ is a function on ordinals (or rather, class function; for simplicity I'm assuming we're working in a theory like NBG that makes all this much simpler to say). For $$\theta>0$$ an ordinal, say that a function $$G$$ iterates $$F$$ along $$\theta$$ starting at $$\alpha$$ iff
• The domain of $$G$$ is $$\theta$$,
• $$G(0)=\alpha$$,
• for $$\beta+1<\theta$$ we have $$G(\beta+1)=F(G(\beta))$$, and
• for $$\lambda<\theta$$ a limit we have $$G(\lambda)=\sup\{G(\beta): \beta<\lambda\}$$.
Incidentally, this last condition is really only a natural thing to do if $$F$$ is nondecreasing, but strictly speaking this works for any $$F$$.
In principle, there could be many $$G$$ with this property, or none at all. However, it turns out that there is only ever exactly one:
For every $$F:Ord\rightarrow Ord$$ (= the function to be iterated), $$\theta>0$$ (= the iteration length), and $$\alpha$$ (= the starting value), there is exactly one $$G$$ which iterates $$F$$ along $$\theta$$ starting at $$\alpha$$.
Moreover, the $$G$$s "cohere" in the sense that if $$G$$ iterates $$F$$ along $$\theta$$ starting at $$\alpha$$ and $$G'$$ iterates $$F$$ along $$\theta'$$ starting at $$\alpha$$, with $$\theta<\theta'$$, then for each $$\eta<\theta$$ we have $$G(\eta)=G'(\eta)$$. So in some sense there is a unique way to iterate $$F$$ along $$Ord$$.
The proof is by transfinite induction: fixing an arbitrary $$F$$ and $$\alpha$$, consider some $$\theta$$ such that the claim holds for all iteration lengths $$<\theta$$. Intuitively, if $$\theta=\gamma+1$$ we just take the $$G$$ for $$\gamma$$ and "stick one more value onto it," and if $$\theta$$ is a limit we "glue the earlier $$G$$s together." It's a good exercise to turn this vague hint into an actual proof.
The sequence of $$\beth$$ numbers can be constructed in this way:
• $$F$$ is the map sending an ordinal $$\alpha$$ to the cardinality of the powerset of $$\alpha$$ (which, remember, is itself an ordinal - cardinals are just initial ordinals).
• The starting value $$\alpha$$ is $$\omega$$: this amounts to setting $$\beth_0=\omega$$.
• To determine what $$\beth_\eta$$ should be, we set $$\theta=\eta+1$$ - or really we pick any $$\theta>\eta$$, by the "coherence" point above it doesn't affect the answer.
• Thanks so much for that very complete explanation, Noah. I can't quite grasp everything you have written yet but will work hard on it in the next few days. I find "Essentially, we can show that every recursive description corresponds to a unique function, and what lets us do this is transfinite induction," the crux of the matter, and it strikes me as funny that I have trouble with transfinite recursion of the Beths, when I've never had any problem with the similarly "transfinite" definition of simple functions from R to R! [Still the Beths seem "sequential" in a way the Reals do not.] – Wd Fusroy Aug 20 at 15:54
Say that a natural number $$n$$ is "large" if for any number $$k$$ you'd think of in the next 24 hours, $$n>10^k$$. Let's go bigger, say $$n>k\uparrow^k k$$, using the Knuth notation.
Pretty much by definition, $$n$$ is unimaginably large. So large that for the next 24 hours there is no way you can even imagine it. But tomorrow evening, you'd be sitting with a beer and realize that you can imagine an even larger number. Why is that even possible? Because even though this $$n$$ is large, almost all the other natural numbers are larger.
Inaccessible cardinals are ordinals. They are incredibly large, yes. But at the end of the day, most ordinals are in fact larger. More cardinals are larger.
If $$\kappa$$ is inaccessible and $$\mu<\kappa$$, then we can prove that the smallest $$\alpha$$ for which $$\kappa\leq\beth_\alpha(\mu)$$ is in fact $$\kappa$$ itself. Namely, $$\kappa\leq\beth_\kappa(\mu)$$, and they are in fact equal, and if $$\alpha<\kappa$$, then $$\beth_\alpha(\mu)<\kappa$$ as well. But since $$\kappa$$ is an ordinal, taking $$\alpha=\kappa$$ is perfectly valid.
So there is no contradiction there. You can move to $$\aleph$$ fixed points, whose existence is provable in $$\sf ZFC$$, and replace the $$\beth_\alpha(\mu)$$ by $$\mu^{+\alpha}$$, the $$\alpha$$th successor of $$\mu$$. If $$\kappa=\aleph_\kappa$$, and $$\mu,\alpha<\kappa$$, then $$\mu^{+\alpha}<\kappa$$. Nevertheless, there is some $$\beta$$ such that $$\mu^{+\beta}\geq\kappa$$, in fact a proper class of those $$\beta$$s. And specifically, $$\kappa$$ itself works for that.
• Thanks so much Asaf! I always prize your answers and try to learn from them. Here I love the way you start out with an analogy from the finite case. I've always thought that the study of the "Large Finites" has been relatively neglected by mathematicians, although in recent years it is starting to develop. I've always found it amazing that if one asks someone to pick any natural number of their choice, however cleverly large, it will still fall in the first INFINTESIMAL segment of the whole set of nat. numbers. And so too with the infinite cardinals, even if it sometimes seems less obvious. – Wd Fusroy Aug 20 at 16:09
• The meat of your answer is however this: "If κ is inaccessible and μ<κ, then we can prove that the smallest α for which κ≤ℶα(μ) is in fact κ itself" I have certainly learned that basic fact, but as I do not know the proof it still seems less than apodeictic to me. At the risk of making a fool of myself again, I still don't see why k can't be GREATER than even Beth[k] [mu] in some cases. That would just mean that k is so large there is NO "a," not even k, that will put Beth[a] [mu] beyond what we might call, say, the "recursively inaccessible" k. Why is such an odd thing impossible?? – Wd Fusroy Aug 20 at 16:20
• Perhaps my problem here is as simple as not knowing for sure whether Beth[omega] is a limit cardinal. I had always assumed it must be, -- since it doesn't get to the inaccessibles -- but now it seems to me like the limit cardinal for the whole class of Beths must come only much farther out -- whatever exactly that means -- if in fact there IS anything like a limit cardinal for the whole sequence, which the answers here seem strongly to indicate there is not. – Wd Fusroy Aug 20 at 16:35
• I think it would be very good for you to do some basic exercises about cardinals, ordinals, and recursive definitions. Most of the questions you raise in the comments are very easy to answer once you have a firm grasp on the basics. – Asaf Karagila Aug 20 at 16:59
• Do you mean $\beth_{\omega+\omega}$? Or just generally limit cardinals? For the latter case, that is undecided in ZFC, since it is consistent (i.e., GCH) that $\aleph_\alpha=\beth_\alpha$, but it's also consistent that $\beth_{\alpha+2}$ is a limit cardinal for any fixed $\alpha$. – Asaf Karagila Sep 19 at 20:48 | {"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": 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": 117, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8842326998710632, "perplexity": 402.07355721518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987751039.81/warc/CC-MAIN-20191021020335-20191021043835-00542.warc.gz"} |
https://www.lektorium.tv/node/34173 | # Коллоквиум. Automorphisms of Weyl Algebra
Лекция
Предмет:
Дата записи:
18.04.19
Дата публикации:
24.04.19
Код для блога:
The famous Jacobian Conjecture states that locally invertible polynomial mapping over is globally invertible. Dixmier conjecture says that any endomorphism of the ring of differential operators (Weyl algebra ) is an automorphism. In the paper A. Ya. Kanel-Belov and M. L. Kontsevich, «The Jacobian conjecture is stably equivalent to the Dixmier conjecture», Mosc. Math. J., 7:2 (2007), 209-218, arXiv:math/0512171, we constructed a homomorphism between the automorphism semigroup of and polynomial symplectomorphisms of Kontsevich Conjecture states that this homomorphism is an isomorphism of invertible mappings. In fact, Kontsevich’s Conjecture states that deformation quantization of affine space preserves the group of symplectic polynomial automorphisms, i.e. the group of polynomial symplectomorphisms in dimension 2n is canonically isomorphic to the group of automorphisms of the corresponding n-th Weyl algebra. The conjecture is confirmed for n=1 and open for n>1. We play with Plank constants and use singularity trick to confirm the general case of the conjecture, see Alexei Kanel-Belov, Andrey Elishev, Jie-Tai Yu, Augmented Polynomial Symplectomorphisms and Quantization, 2018, 21 pp., arXiv:1812.02859.
17
## Комментарии
Хорошо,мне нравится это видео. | {"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": 0.9672409296035767, "perplexity": 1689.0686729654028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575540545146.75/warc/CC-MAIN-20191212181310-20191212205310-00099.warc.gz"} |
http://mathhelpforum.com/math-challenge-problems/111337-integral-basic.html | # Thread: integral , basic ..
1. ## integral , basic ..
Use this formula only $\int x^n ~dx = \frac{ x^{n+1}}{n+1} + c$
Find the integral $\int \frac{dx}{\sqrt{ x + \sqrt{ x^2 + 1}}}$
Of course , little Algebra is allowed . Try to answer this without applying substitution , integration by parts , etc
2. Edit
3. What is the solution
i lost 2 hours from my life
and BTW, you should said that n dont equal -1
4. ## intermediate step
I'm considering squaring.
5. It seems impossible to solve without using substitution
6. Below is the solution to this problem ,
Spoiler:
To avoid appearing the imaginary number $i$ in the solution ,
Consider
$\int \frac{dx}{\sqrt{ x + \sqrt{ x^2 - a^2 }}}$
$= \int \frac{dx}{\sqrt{ x + \sqrt{ x^2 + 1}}} \cdot \frac{ \sqrt{ x - \sqrt{x^2 - a^2 }} }{\sqrt{ x - \sqrt{x^2 - a^2 }}}$
$= \int \frac{ \sqrt{ x - \sqrt{x^2 - a^2 }} }{ \sqrt{ x^2 - (x^2 - a^2)}}~dx$
$= \frac{1}{a} \int \sqrt{ x - \sqrt{x^2 - a^2 }} ~dx$
Now , the integral looks much more simplier but that's not enough , we have to do something special in the following steps :
$= \frac{1}{a} \int \sqrt{\frac{2( x - \sqrt{x^2 - a^2})}{2}} ~dx$
$= \frac{1}{a \sqrt{2}} \int \sqrt{2x - 2\sqrt{(x-a)(x+a)}}~dx$
$= \frac{1}{a \sqrt{2}} \int \sqrt{ ( x+a) - 2\sqrt{(x-a)(x+a)} + (x-a)}~dx$
$= \frac{1}{a \sqrt{2}} \int \sqrt{ \sqrt{x+a}^2 - 2\sqrt{(x-a)(x+a)} + \sqrt{x-a}^2}~dx$
$= \frac{1}{a \sqrt{2}} \int \sqrt{ ( \sqrt{x+a} - \sqrt{x-a})^2}~dx$
$= \frac{1}{a \sqrt{2}} \int [\sqrt{x+a} - \sqrt{x-a} ]~dx$
$= \frac{\sqrt{2}}{3a} \left[ \sqrt{x+a}^3 - \sqrt{x-a}^3 \right ] + C$
$= \frac{\sqrt{2}}{3a} ( \sqrt{x+a} - \sqrt{x-a})( (x+a) + \sqrt{x^2 - a^2} + (x-a) ) + C$
$= \frac{2}{3a} \sqrt{ x - \sqrt{x^2 - a^2}} ( 2x + \sqrt{ x^2 - a^2}) + C$
$= \frac{2}{3} \frac{1}{\sqrt{x+ \sqrt{x^2 - a^2}}} ( 2x + \sqrt{x^2 - a^2} ) + C$
Sub. $-a^2 = +1$
the integral
$= \frac{2}{3} \frac{2x + \sqrt{x^2 +1} }{\sqrt{x+ \sqrt{x^2 +1}}} + C$
7. Originally Posted by TWiX
What is the solution
i lost 2 hours from my life
and BTW, you should said that n dont equal -1
I just forgot to add this point but actually it is also true when $n = -1$
Although , from the formula , it gives us something like this :
$\frac{ x^0}{0} + C$ or $\lim_{a\to 0} \frac{x^a}{a} + C$
the numerator tends to $1$ while the denominator tends to zero .
,if we rewrite the constant $C$ as $-\frac{1}{a} + C'$ , and make $a$ very small $\to 0$
so we have
$\int \frac{dx}{x} = \lim_{a\to 0} \frac{x^a -1 }{a} + C'$
Then use L rule to evaluate the limit , we will find that
$I = \ln(x) + C'$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 29, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9935799837112427, "perplexity": 918.6315992421328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125881.93/warc/CC-MAIN-20170423031205-00460-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/105260/clustering-numbers-by-factors-count | # Clustering numbers by factors count
Is there a formula (or an efficient approach) for counting amount of positive numbers in range up to $N$ which have exactly $K$ divisors?
P.S.
Initial problem was to cluster number in range [1..N] according to the number of divisors. Then find multiplication of clusters' sizes factorials. So we just need compute the answer according to this scheme. The final result is a amount of sequencies formed such that first group has 1 divisor, second one has 2 and so on.
-
While I don't know of an explicit formula giving what you want, there are good ways to simplify the problem.
First, let $\tau(n)$ be the number of divisors of $n$. If $n=ab$ with $\operatorname{GCD}(a,b)=1$, then every divisor of $n$ can be uniquely factored as a divisor of $a$ and a divisor of $b$. Because of this, $\tau(ab)=\tau(a)\tau(b)$. By induction, if we factor $n$ as a product of powers of primes $n=\prod p_i^{a_i}$, then
$$\tau(n)=\prod \tau(p_i^{a_i})=\prod (a_i+1)$$ because $p^k$ has divisors $1,p,p^2,\ldots p^k$. Thus, the number of divisors a number has is determined by the exponents of its prime factorization (and not the individual factors).
With this in mind, we can solve the original problem as follows.
1. Find all possible factorizations of $K$. Note that there will be $\tau(K)$ of them.
2. For each factorization of $K$, subtract $1$ from each of the factors to write $K=\prod (a_i+1)$ for some collection of positive integers $a_i$.
3. For each collection of $a_i$, look at the possible assignment of primes such that $\prod p_i^{a_i}$ is less than $N$.
This is dependent on having a good way to generate all the primes, and the third step probably needs to be fleshed out a little, especially if one wants to be efficient, but overall, this is probably the best approach to the problem.
Note that (as mentioned in the other comments), if you let $K=2$, the problem simplifies to counting the number of primes less than $N$, and so a general formula is going to be more complicated than computing $\pi(N)$. I don't think anything less involved than the procedure above is likely to be fruitful (although you might be able to get asymptotics for specific values of $K$).
-
No, there's no such formula. If there were, then finding new primes would not be an interesting problem.
-
It rather depends on what other functions you have to hand.
If $K$ is prime then you want to count the primes up to the $K-1^{\text{th}}$ root of $N$, i.e. $$\pi(\sqrt[K-1]{N})$$ where $\pi(x)$ is the prime counting function.
But for $K=4$ you also want to count the numbers up to $N$ which are the product of distinct primes to add to $\pi(\sqrt[3]{N})$ and I am not aware of an explicit function notation for that.
-
Initial problem was to cluster number in range $[1..N]$ according to the number of divisors. Then find multiplication of clusters' sizes factorials. – Jane Feb 3 '12 at 9:11
@Jane Was the problem a computation problem? Or something else? It's possible that calculating the size of individual clusters is hard, but some functions of all the clusters (e.g., the product of the factorials) has some other interpretation/characterization that makes it possible to calculate without calculating the individual cluster sizes. – Aaron Feb 3 '12 at 9:21
@Aaron Yes, we just need compute the answer according to the scheme. So the final result is a amount of sequencies formed such that first group has 1 divisor, second one has 2 and so on. – Jane Feb 3 '12 at 9:41
@Jane You should edit your question to include the added context so that people can consider whether there are indirect solutions that don't require computing individual cluster sizes. Unless this was a homework problem and you explicitly needed to compute the cluster sizes? However, this sounds more like a project euler type problem, which generally uses math to go from intractable to solvable-but-only-with-significant-computer-assistance. – Aaron Feb 3 '12 at 10:05
There is another generalization of the problem, according to using of factorials of clusters sizes.
One wants to find amount of permutations of $1..N$ which have the following property -$\tau(A_i)=\tau(i)$, where $A_i$ are the elements of permutation.
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http://mathhelpforum.com/discrete-math/193473-question-about-cardinality-empty-set.html | # Math Help - question about cardinality of empty set
1. ## question about cardinality of empty set
Hi
I am doing the chapter "Equinemerous sets" from Velleman's "How to prove it" and
I have some doubts. At one point he says that "for each natural number n , let
$I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$. A set $A$
is called $\mathit{finite}$ if there is a natural number n such that
$I_n \sim A$ . Otherwise A is $\mathit{infinite}$. "
Further down he says that "it makes sense to define the number of elements of
a finite set A to be the unique n such that $I_n \sim A$. This number
is also sometimes called the cardinality of A and its denoted
$\lvert A \rvert$. Note that according to this definition,
$\varnothing$ is finite and $\lvert \varnothing \rvert =0$."
So that will mean that we will need to choose n=0 for an empty set, so that
$I_0 \sim \varnothing$. Now according to the author's defnition of
$I_n$ , $I_0=\varnothing$. So
$I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing$
which is true since for any set A, we have $A \sim A$.
Do you think its correct understanding ?
Thanks
2. ## Re: question about cardinality of empty set
Originally Posted by issacnewton
At one point he says that "for each natural number n , let
$I_n=\{i \in \mathbb{Z^+}\lvert i \le n \}$. A set $A$
is called $\mathit{finite}$ if there is a natural number n such that
$I_n \sim A$ . Otherwise A is $\mathit{infinite}$. "
Further down he says that "it makes sense to define the number of elements of
a finite set A to be the unique n such that $I_n \sim A$. This number
is also sometimes called the cardinality of A and its denoted
$\lvert A \rvert$. Note that according to this definition,
$\varnothing$ is finite and $\lvert \varnothing \rvert =0$."
So that will mean that we will need to choose n=0 for an empty set, so that
$I_0 \sim \varnothing$. Now according to the author's definition of
$I_n$ , $I_0=\varnothing$. So $I_0 \sim \varnothing \Rightarrow \varnothing \sim \varnothing$ which is true since for any set A, we have $A \sim A$.
Do you think its correct understanding ?
That is consistent with his definition because $S_0$ must be $\emptyset$.
3. ## Re: question about cardinality of empty set
What is $S_0$ ?
4. ## Re: question about cardinality of empty set
Originally Posted by issacnewton
What is $S_0$ ?
Well look at Velleman's own definition: $S_n=\{i\in\mathbb{Z}^+:i\le n\}$.
Now what positive integer is less than or equal to zero?
Note that Velleman did not specify the nature of n in that definition.
So $S_{0.5}=S_0$
5. ## Re: question about cardinality of empty set
Oh you meant $I_0$. Ya I get that. Does your book have a different symbol there ? I think you have first edition. anyway...
Velleman's definitions are too formal. Even wikipedia doesn't give such definitions and I have not seen other maths books giving
the definitions so formally. I think he is a set theorist/logician, that's why he is too formal. Or since he is teaching "how to prove it" so
it makes sense to give definitions as rigorous as possible
6. ## Re: question about cardinality of empty set
Originally Posted by issacnewton
Oh you meant $I_0$. Ya I get that. Does your book have a different symbol there ? I think you have first edition. anyway...
Velleman's definitions are too formal. Even wikipedia doesn't give such definitions and I have not seen other maths books giving the definitions so formally. I think he is a set theorist/logician, that's why he is too formal. Or since he is teaching "how to prove it" so
it makes sense to give definitions as rigorous as possible
This may surprise you but I have never seen that textbook.
But if you look at Velleman's pedigree you see that one of his advisers was Mary Ellen Rudin (Walter Rudin's wife) and an R L Moore's PhD student. That should at once tell you that her students should be a stickler for precise definitions. There you have some of the best mathematicians of the last century . So if I were you I would rethink that remark.
7. ## Re: question about cardinality of empty set
Well, thanks for the information. Not being from US, didn't know that.
Its strange that even the real analysis books I have seen, don't use such precise
definitions. So that style is not followed everywhere. I was checking Moore's wikipedia page and there is mention of Moore's method. May be this style comes from there.... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 36, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.942555844783783, "perplexity": 622.3725704644766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394023864908/warc/CC-MAIN-20140305125104-00073-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/percent-difference-and-uncertainty.431036/ | # Homework Help: Percent difference and uncertainty
1. Sep 21, 2010
### mike2601
1. The problem statement, all variables and given/known data
How do we solve this problem?
% Difference = (l 0.282 +/- 0.0019 - 0.288 +/- 0.04 l) / 0.285 +/- 0.02
2. Relevant equations
Percent Difference = (l value - average value l) / average value x 100%
3. The attempt at a solution | {"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": 0.8530508279800415, "perplexity": 3853.0630814569035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221210615.10/warc/CC-MAIN-20180816093631-20180816113631-00144.warc.gz"} |
https://www.groundai.com/project/astrophysical-configurations-with-background-cosmology-probing-dark-energy-at-astrophysical-scales/ | Probing Dark Energy at Astrophysical Scales
# Astrophysical Configurations with Background Cosmology: Probing Dark Energy at Astrophysical Scales
A. Balaguera-Antolínez, D. F. Mota and M. Nowakowski
Max Planck Institute für Extraterrestrische Physik, D-85748, Garching, Germany
Institute for Theoretical Physics, University of Heidelberg, 69120 Heidelberg, Germany
Departamento de Física, Universidad de los Andes, A.A. 4976, Bogotá, D.C., Colombia
E-mail: [email protected]: [email protected]: [email protected]
###### Abstract
We explore the effects of a positive cosmological constant on astrophysical and cosmological configurations described by a polytropic equation of state. We derive the conditions for equilibrium and stability of such configurations and consider some astrophysical examples where our analysis may be relevant. We show that in the presence of the cosmological constant the isothermal sphere is not a viable astrophysical model since the density in this model does not go asymptotically to zero. The cosmological constant implies that, for polytropic index smaller than five, the central density has to exceed a certain minimal value in terms of the vacuum density in order to guarantee the existence of a finite size object. We examine such configurations together with effects of in other exotic possibilities, such as neutrino and boson stars, and we compare our results to N-body simulations. The astrophysical properties and configurations found in this article are specific features resulting from the existence of a dark energy component. Hence, if found in nature would be an independent probe of a cosmological constant, complementary to other observations.
###### keywords:
Cosmology – Theory – Dark Energy – Structure Formation
pagerange: Astrophysical Configurations with Background Cosmology: Probing Dark Energy at Astrophysical ScalesReferencespubyear: 2007
## 1 Introduction
Some of the most relevant properties of the universe have been established through astronomical data associated to light curves of distant Supernova Ia riess (), the temperature anisotropies in the cosmic microwave background radiation wmap3 () and the matter power spectrum of large scale structures tegmark (). Such observations give a strong evidence that the geometry of the universe is flat and that our Universe is undertaking an accelerated expansion at the present epoch. This acceleration is attributed to a dominant dark energy component, whose most popular candidate is the cosmological constant, .
The present days dominance of dark energy make us wonder if this component may affect the formation and stability of large astrophysical structures, whose physics is basically Newtonian. This is in fact an old question put forward already by Einstein einstein () and pursued by many other authors noer (); chernin2 (); manera (); nunes (); baryshev () and kagramanova (); jetzer1 (). In general, the problem is rooted in the question whether the expansion of the universe, which in the Newtonian sense could be understood as a repulsive force, affects local astrophysical properties of large objects. The answer is certainly affirmative if part of the terms responsible for the Universe expansion survives the Newtonian limit of the Einstein equations. This is indeed the case of which is part of the Einstein tensor. In fact, as explained in the main text below, all the effects of the universe expansion can be taken into account, regardless of the model, by generalizing the Newtonian limit. This approach allow us to calculate the impact of a given cosmological model on astrophysical structures.
Although, there are several candidates to dark energy which have their own cosmological signature, e.g. Koivisto1 (); daly (); wang2 (); koivisto (); koivisto2 (); shaw2 () and seo (); brook (); daly2 (); Koivisto3 (); shaw3 (), in this paper we will investigate the CDM model only. Such consideration is in fact not restrictive and our results will be common to most dark energy models. At astrophysical scales and within the Newtonian limit one does not expect to find important differences among the different dark energy models. This, however should not be interpreted as if has no effect at smaller astrophysical scales. In fact, the effects of a cosmological constant on the equilibrium and stability of astrophysical structures is not negligible, and can be of relevance to describe features of astrophysical systems such as globular clusters, galaxy clusters or even galaxies cher (); iorio (); bala1 (); nowakowski1 (); cardoso (). Motivated by this, we investigate the effects of a dark energy component on the Newtonian limit of Einstein gravity and its consequences at astrophysical scales.
In this article we investigate how the cosmological constant changes certain aspects of astrophysical hydrostatic equilibrium. In particular we search for specific imprints which are unique to the existence of a dark energy fluid. For instance, the instability of previously viable astrophysical models when is included. We explore such possibility using spherical configurations described by a polytropic equation of state (e.o.s) . The polytropic equation of state derives its importance from its success and consistency, and it is widely used in determining the properties of gravitational structures ranging from stars chan () to galaxies binney (). It leads to an acceptable description of the behavior of astrophysical objects in accordance with observations and numerical simulations kennedy1 (); gruzinov (); kaniadakis (); sadeth (); pinzon1 (); ruffet (). The description of such configurations can be verified in the general relativistic framework herrera () and applications of these models to the dark energy problem have in fact been explored mukhop ().
The effect of a positive cosmological constant can be best visualized as a repulsive non local force acting on the matter distribution. It is clear that this extra force will result into a minimum density (either central or average) which is possible for the distribution to be in equilibrium. This minimum density is a crucial crossing point: below this value no matter can be in equilibrium, above this value low density objects exist lahav91 (). Both effects are novel features due to . We will demonstrate such inequalities, which are generalizations of corresponding inequalities found in nowakowski1 () and bala2 (), for every polytropic index . However, the most drastic effect can be found in the limiting case of the polytropic equation of state, i.e, the isothermal sphere where the polytropic index goes to infinity. This case captures, as far as the effects of are concerned, many features also for higher, but finite . The model of the isothermal sphere is often used to model galaxies and galactic clusters natarajan () and used in describing effects of gravitational lensing kawano (); sereno (); maccio (). Herein lies the importance of the model. Regarding the isothermal sphere we will show that renders the model unacceptable on general grounds. This essentially means that the model does not even have an appealing asymptotic behavior for large radii and any attempt to definite a physically acceptable radius has its severe drawbacks.
The positive cosmological constant offers, however, yet another unique opportunity, namely the possible existence of young low density virialized objects, understood as configurations that have reached virial equilibrium just at the vacuum dominated epoch (in contrast to the structures forming during the matter dominated era, where the criteria for virialization is roughly ). This low density hydrostatic/virialized objects can be explained again due to which now partly plays the role of the outward pressure.
The applicability of fluid models, virial theorem and hydrostatic equilibrium to large astrophysical bodies has been discussed many times in the literature. For a small survey on this topic we refer the reader to (jackson, ; bala5, ) where one can also find the relevant references.
It is interesting to notice that dark matter halos represent a constant density background which, in the Newtonian limit, objects embedded in them feel the analog to a negative cosmological constant. The equilibrium analysis for such configurations has been performed in Umemura (); Horedt (). A negative will just enhance the attractive gravity effect, whereas a positive one opposes this attraction. As a result the case reveals different physical concepts as discussed in this paper.
The article is organized as follows. In the next section we introduce the equations relevant for astrophysical systems as a result of the weak field limit and the non-relativistic limit of Einstein field equations taking into account a cosmological constant. There we derive such limit taking into account the background expansion independently of the dark energy model. In section 3 we derive the equations governing polytropic configurations, the equilibrium conditions and stability criteria. In section 4 we describe the isothermal sphere and investigate its applicability in the presence of . In section 5 we explore some examples of astrophysical configurations where the cosmological constant may play a relevant role. In particular, we probe into low density objects, fermion (neutrino) stars and boson stars. Finally we perform an important comparison between polytropic configurations with and parameterized density of Dark Matter Halos. We end with conclusions. We use units except in section 5.4 where we restore and use natural units .
## 2 Local dynamics in the cosmology background
The dynamics of the isotropic and homogeneous cosmological background is determined by the evolution of the (dimensionless) scale factor given through the Friedman-Robertson-Walker line element as solution of Einstein field equations,
(1)
corresponding to the Raychaudhury equation and Friedman equation, respectively. The total energy density is a contribution from a matter component - baryonic plus dark matter - (), radiation () and a dark energy component ( with ). The function is given as
f(a)≡3lna∫a1ωx(a′)+1a′da′, (2)
where the term represents the equation of state for the dark energy component. The case corresponds to the cosmological constant . The effects of the background on virialized structures can be explored through the Newtonian limit of field equations from which one can derive a modified Poisson’s equation (see e.g. noer (); nowakowski2 ()). Recalling that pressure is also a source for gravity, the gravitational potential produced by an overdensity is given by
∇2Φ=4π(ρt+3Pt), (3)
where and . Where is the local overdensity with respect to the background density . Notice that equation (3) reduces to the usual Poisson equation, , when non relativistic matter dominates the Universe, and . However, at present times, when dark energy dominates, the pressure is non-negligible and might even be non zero, such as in the case of quintessence models ml (); wang (); mota1 (). In this work, however, we will focus in the case of an homogeneous dark energy component where . With this in mind, one can then write the modified Poisson equation as
∇2Φ=4πδρ−3¨a(t)a(t), (4)
Note that this equation allows one to probe local effects of different Dark Energy models through the term given in Eq.(1).
Since we will be investigating the configuration and stability of astrophysical objects nowadays, when dark energy dominates, it is more instructive to write the above equations in terms of an effective vacuum density i.e.
∇2Φ=4πδρ−8πρeffvac(a), (5)
where by using (1) has been defined as
ρeffvac(a)≡−12[(ΩcdmΩvac)a−3+(1+3ωx)a−f(a)]ρvac, (6)
which reduces to for and negligible contribution from the cold dark matter component with respect to the over-density . With being the solution associated to the pure gravitational interaction, the full solution for the potential can be simply written as
Φ(r,a)=Φgrav(r)−43πρeffvac(a)r2,Φgrav(r)=−∫V′δρ(r′)|r−r′|d3r′, (7)
which defines the Newton-Hooke space-time for a scale factor close to the present time (vacuum dominated epoch), and gibbons (); aldro (). For a CDM universe with and we get : that is, the positive density of matter which has an attractive effect opposing the repulsive one of reduces effectively the strength of the ’external force’ in (5). Note that, although in the text we will use the notation which would be vaild in the case of a Newton-Hook space time, it must be understood that we can replace by for a CDM background.
Given the potential , we can write the Euler’s equation for a self-gravitating configuration as
ρd⟨vi⟩dt+∂ip+ρ∂iΦ=0
, where is the (statistical) mean velocity and is the total energy density in the system. We can go beyond Euler’s equation and write down the tensor virial equation which reduces to its scalar version for spherical configurations. The (scalar) virial equation with the background contribution reads as bala3 (); caimmi ()
d2Idt2=2T+Wgrav+3Π+83πρeffvac(a)I−∫∂Vp(→r⋅^n)dA, (8)
where is the gravitational potential energy defined by
Wgrav=12∫Vρ(r)Φgrav(r)d3r, (9)
and is the contribution of ordered motions to the kinetic energy. Also, is the moment of inertia about the center of the configuration and is the trace of the dispersion tensor. The full description of a self gravitating configuration is completed with an equation for mass conservation, energy conservation and an equation of state . If we assume equilibrium via , we obtain the known virial theorem jackson (); bala3 ()
|Wgrav|=2T+3Π+83πρeffvac(a)I, (10)
where we have neglected the surface term in (8), which is valid in the case when we define the boundary of the configuration where .
With given in (6), one must be aware that an equilibrium configuration is at the most a dynamical one. This is to say that the ’external repulsive force’ in (6) is time dependent through the inclusion of the background expansion and so are the terms in (8). This leads to a violation of energy conservation, which also occurs in the virialization process wang3 (); ml (); caimmi (); mota1 (); shaw ()). Traced over cosmological times, this implies that if we insist on the second derivative of the inertial tensor to be zero, then the internal properties of the object like angular velocity or the internal mean velocity of the components will change with time. Even if in the simplest case, one can assume that the objects shape and its density remain constant. Hence, equilibrium here can be thought of as represented by long time averages in which case the second derivative also vanishes, not because of constant volume and density, but because of stability bala4 (); bala5 ().
The expressions derived in the last section, especially (5) and (10) can be used for testing dark energy models on configurations in a dynamical state of equilibrium. However, in these cases, one should point out that, in this approach there is no energy conservation within the overdensity: dark energy flows in and out of the overdensity. Such feature is a consequence of the assumption that dark energy does not cluster at small scales (homogeneity of dark energy). This is in fact the most common assumption in the literaturewang (); chen (); hore (), with a few exceptions investigated in wang3 (); ml (); caimmi (); mota1 ().
In this paper, we will concentrate on the possible effects of a background dominated by a dark energy component represented by the cosmological constant at late times (1). It implies that the total density involved in the definitions of the integral quantities appearing in the virial equation can be approximated to . In that case the Poisson equation reduces to the form . As mentioned before, the symbol has no multiplicative factors in the case of a Newton-Hooke space time, while for a CDM model it must be understood as . As the reader will see, the most relevant quantities derived here come in forms of ratio of a characterizing density and , and hence the extra factor appearing in the CDM can be re-introduced in the characterizing density. For general consideration of equilibrium in the spherical case see boehmer1 () and boehmer (), while the quasi spherical collapse with cosmological constant has been discussed in debnath ().
## 3 Polytropic configurations and the Λ-Lane-Emden equation
We can determine the relevant features of astrophysical systems by solving the dynamical equations describing a self gravitating configuration (Euler’s equation, Poisson’s equation, continuity equations). In order to achieve this goal we must first know the potential to be able to calculate the gravitational potential energy. To obtain , one must supply the density profile and solve Poisson’s equation. In certain cases, the potential is given and we therefore can solve for the density profile in a simple way. Here we face the situation where no information on the potential (aside from its boundary conditions) is available and we also do not have an apriori information about the density profile (see for instance binney () for related examples). In order to determine both, the potential and the density profile, a complete description of astrophysical systems required. This means we need to know an equation of state (here we change notation and we call the proper density of the system). The equation of state can take several forms and the most widely used one is the so-called polytropic equation of state, expressed as
p=κργ,γ≡1+1n, (11)
where is the polytropic index and is a parameter that depends on the polytropic index, central density, the mass and the radius of the system. The exponent is defined as and is associated with processes with constant (non-zero) specific heat . It reduces to the adiabatic exponent if . The polytropic equation of state was introduced to model fully convective configurations. From a statistical point of view, Eq. (11) represents a collisionless system whose distribution function can be written in the form , with being the relative energy and being the relative potential (where is a constant chosen such that )binney (). In astrophysical contexts, the polytropic equation of state is widely used to describe astrophysical systems such as the sun, compact objects, galaxies and galaxy clusters chan (); binney (); kennedy1 (). We now derive the well known Lane-Emden equation. We start from Poisson equation and Euler equation for spherically symmetric configurations, written as
d2Φdr2+2rdΦdr=4πρ−8πρvac,dpdr=−ρdΦdr. (12)
This set of equations together with Eq. (11) can be integrated in order to solve for the density in terms of the potential as
ρ(r)=ρc[1−(γ−1κγ)ρ1−γc(Φ(r)−Φ(0))]1γ−1, (13)
where is the central density. In view of eq. (13) in conjunction with eq. (7) it is clear that will have the effect to increase the value of . Therefore the boundary of the configuration will be located in a greater as compared to the case . In order to determine the behavior of the density profile, we again combine Eq(12) and Eq.(11) in order to eliminate the potential . We obtain
1n(∇ρρ)2+∇2lnρ=−4πnρ1−1nκ(n+1)(1−ζ), (14)
where we defined the function
ζ=ζ(r)≡2(ρvacρ(r)). (15)
We can rewrite Eq (14) by introducing the variable defined by , where is the central density. We also introduce the variable where
a≡ ⎷κ(n+1)4πρ1−1nc (16)
is a length scale. Eq.(14) is finally written as bala2 (); chan ()
1ξ2ddξ(ξ2dψdξ)=ζc−ψn,ζc≡2(ρvacρc). (17)
This is the -Lane-Emden equation (). Note that for constant density, we recover as the first non trivial solution of equation. This is consistent with the results from virial theorem for constant density spherical objects which tell us that nowakowski1 (). Note that using Eq. (13) we can write the solution with the explicit contribution of as
ψ(ξ=r/a)=1−(4πa2ρc)−1(Φgrav(r)−Φgrav(0))+6ζcξ2, (18)
so that for a given smaller than the radius we will obtain
ψ(r/a)>ψ(r/a)Λ=0, (19)
as already pointed out before. Then the differential equation (17) must be solved with the initial conditions , satisfied by (18). Numerical solutions were obtained for the first time in bala3 (). The solutions presented in Fig. 1 are given in terms of the ratio for and . This choice of variables are useful also since sets a fundamental scale of density (the choice will be explored for the isothermal sphere, where figure 1 will be helpful for discussions). The radius of a polytropic configuration is determined as the value of when the density of matter with the e.o.s (11) vanishes. This happens at a radius located at
R=aξ1suchthatψ(ξ1)=0. (20)
Note that equation (17) yields a transcendental equation to determine . Also one notes from Fig. 1 that not all values of yield allowed configurations in the sense that we cannot find a value of such that . This might not be surprising since for and we find the situation where the asymptotic behavior is as (we consider this still as an acceptable behavior). There is, however, one crucial difference when we switch on a non-zero . For not only we cannot reach a definite radius but the derivative of the density changes sign and hence becomes non-physical. The situation for the cases is somewhat similar to the extreme case of (isothermal sphere). Clearly, these features are responsible of the last term in Eq.(18), which for high values of may become dominant over the remaining (gravitational) terms. We will discuss this case in section four where we will attempt another definition of a finite radius with the constraint . For now it is sufficient to mention that, as expected, the radius of the allowed configurations are larger than the corresponding radius when .
### 3.1 Equilibrium and stability for polytropes
In this section we will derive the equilibrium conditions for polytropic configurations in the presence of a positive cosmological constant. We will use the results of last section in order to write down the virial theorem. The total mass of the configuration can be determined as usual with together with Eq.(17). One then has a relation between the mass, the radius and the central density:
R=M1/3ρ−1/3cf0(ζc;n)=(Mr2Λ)1/3(4π)1/3ζ1/3cf0(ζc;n) (21)
where
f0(ζc;n)≡(ξ314π)13(∫ξ10ξ2ψn(ξ)dξ)−13, (22)
Note that we have introduced the cosmological constant in the equation for the radius, leading to the appearance of the astrophysical length scale (with Mpc Mpc for the concordance values and ). This scale has been already found in the context of Schwarzschild - de Sitter metric where it is the maximum allowed radius for bound orbits. At the same time it is the scale of the maximum radius for a self gravitating spherical and homogeneous configuration in the presence of a positive bala3 (). This also let us relate the mean density of the configuration with its central density and/or cosmological parameters as .
Similarly we can determine the other relevant quantities appearing in the equations for the energy and the scalar virial theorem (10). For the traces of the moment of inertia tensor and the dispersion tensor we can write
I=MR2f1,Π=κρ1ncMf2, (23)
where the functions have been defined as
f1(ζc;n)≡∫ξ10ξ4ψndξξ21∫ξ10ξ2ψndξ,f2(ζc;n)≡∫ξ10ξ2ψn+1dξ∫ξ10ξ2ψndξ, (24)
using (21). These functions are numerically determined in the sequence , such that for a given mass we obtain the radius as .
Let us consider the virial theorem (10) for a polytropic configuration. The gravitational potential energy can be obtained following the same arguments shown in chan (). The method consist in integrating Euler’s equation and solve for , then using Eq.(9) one obtains . The final result is written as
Wgrav=−M22R−12(n+1)Π+23πρvac(I−MR2), (25)
To show the behavior of with respect to the index , we can solve the the virial theorem (10) for and replace it in Eq.(25). We obtain
Wgrav=−35−n[1−ρvac¯ρ(13(5+2n)f1−1)]M2R. (26)
This expression shows the typical behavior of a polytrope (even if ): the configuration has an infinite potential energy, due to the fact that the matter is distributed in a infinite volume. The energy of the configuration in terms of the polytropic index can be easily obtained by using (10), (25) and (26)
E=Wgrav+83πρvacI+nΠ=−(3−n5−n)M2R[1−ρvac¯ρ(5f(n)1−1)]. (27)
One is tempted to use as the condition to be fulfilled for a gravitationally bounded system. For we recover the condition () for gravitationally bounded configurations in equilibrium. On the other hand, for this condition might not be completely true due to the following reasoning: The two-body effective potential in the presence of a positive cosmological constant does not go asymptotically to zero for large distances, which is to say that is not stringent enough to guarantee a bound system. Therefore, we rather rely on the numerical solutions from which, for every , we infer the value of such that
ρc≥Anρvac. (28)
This gives us the lowest possible central density in terms of . The behavior of the , the functions , the solution and the values of are shown in fig 2. Note that this inequality can be understood as a generalization of the equilibrium condition , which, when applied for a spherical homogeneous configurations yields with constant bala3 (); bala2 ().
Note that, at the radius of the configuration becomes undefined, as well as the energy. A polytrope is highly concentrated at the center chan (). No criteria can be written since even for there is not a finite radius. But it is this high concentration at the center and a smooth asymptotic behavior which makes this case still a viable phenomenological model if is zero. On the contrary for non-zero, positive the solutions start oscillating around which makes the definition of the radius more problematic. For the polytropic e.o.s describes an ideal gas (isothermal sphere). Since in this limit the expressions derived before are not well definite, this case will be explored in more detail in the next section. In spite of the mathematical differences, the isothermal sphere bears many similarities to the cases and our conclusions regarding the definition of a radius in the case equally apply to finite bigger than .
### 3.2 Effects with generalized dark energy equation of state
In the last section we have explored the effects of a dark energy-dominated background with the equation of state . Other dark energy models are often used with and or even , in the so-called phantom regime, or even a time dependent dark energy model (quintessence). A simple generalization to such models can be easily done by making the following replacement in our equations
ζ→ζ(a)eff≡−12ζη(a)a−f(a), (29)
where , and where the function is defined in Eq.(2). Note that for this generalization to be coherent with the derivation of LE in (17), one needs to consider that the equation of state is close to , so that there is almost no time-dependence, and the energy density for dark energy is almost constant. This is indeed the case for most popular candidates of dark energy specially at low redshifts . Also, notice once again, that we are still assuming an homogeneous dark energy component which flows freely to and from the overdensity. Hence, violating energy conservation inside it. Clearly, other models of dark energy will posses dynamical properties that the cosmological constant does not have. For instance, we could allow some fraction of dark energy to take part in the collapse and virialization ml (); caimmi (); mota1 (), which would lead to the presence of self and cross interaction terms for dark energy and the (polytropic like) matter in Euler equation, which at the end modifies the Lane Emden equation. With this simplistic approach, we see that the effects with a general equation of state are smaller than those associated to the cosmological constant. In particular, the equation of state displays a null effects since it implies (note that this equation of state can also resemble the curvature term in evolution equation for the background). On the other hand, phantom models of dark energy, which are associated to equations of state cald (); nojiri (), have quite a strong effect. In Fig. 3 we show numerical solutions of Lane-Emden equations for a background dominated with dark energy with and a phantom dark energy with , with . These curves are to be compared with those at fig.1. Clearly equations of state with will generate larger radius than the case described in the main text. Furthermore, the asymptotic behavior of the ratio between the density and is .
### 3.3 Stability criteria with cosmological constant
Stability criteria for polytropic configurations can be derived from the virial equation. Using equations (23), (24) and (25) we can write the virial theorem (10) in terms of the radius and the mass :
−M22R+12(5−n)κρ1ncMf2+23πρvacMR2(5f1−1)=0, (30)
where we have assumed that the only contribution to the kinetic energy comes from the pressure in the form of . Note that for and finite mass, one obtains while for we would obtain a cubic equation for the radius. Instead of solving for the virial radius, we solve for the mass as a function of central density with the help of Eq (21). We have
M=Gρ3−n2nc,G=G(ζc;n)≡⎡⎣κf0f2(5−n)1−23πζcf30(5f1−1)⎤⎦32. (31)
The explicit dependence of the mass with respect to the central density splits into two parts: on one hand it has the same form as the usual case with , that is, ; on the other hand the function has a complicated dependence on the central density because of the term . With the help of (21) and (31) we can write a mass-radius relation and the radius-central density relation
M=(G23(nn−1)f3−nn−10)R3−n1−n,R=(G13f0)ρ12(1−nn)c. (32)
Following the stability theorem (see for instance in weinberg ()), the stability criteria can be determined from the variations of the mass in equilibrium with respect to the central density. We derive from Eq. (32):
∂M∂ρc=[32(γ−43)ρ−1cG+∂G∂ρc]ρ32(γ−43)c. (33)
Stability (instability) stands for (). This yields a critical value of the polytropic exponent when given by
γcrit=γcrit(ζc)≡43+23∂lnG∂lnρc, (34)
in the sense that polytropic configurations are stable under small radial perturbations if . It is clear that the second term in (34) also depends on the polytropic index and therefore this equation is essentially a transcendental expression for .
It is worth mentioning that by including the corrections due to general relativity, the critical value for is also modified as shapiro () and hence for compact objects the correction to the critical polytropic index is stronger from the effects of general relativity than from the effects of the background. This is as we would expect it. Stability of relativistic configurations with non-zero cosmological constant has been explored in boehmer (); boehmer1 ().
Going back to equation (31), we can write the mass of the configuration as , where is the mass when and is the enhancement factor. Both quantities can be calculated to give
M(0)≡(κ(5−n)f(n)0f(n)2)32ρ3−n2nc,αM≡⎡⎢ ⎢⎣f0f2f(n)0f(n)2(1−23πζcf30(5f1−1))⎤⎥ ⎥⎦32, (35)
where are numerical factors (tabulated in table 1) that can be determined in a straightforward way. Similarly, by using Eq (21), the radius can be written as , where
R(0)=(κ(5−n)f(n)0f(n)2)12f(n)0ρ1−n2nc,αR≡⎛⎝f0f(n)0⎞⎠⎡⎢ ⎢⎣f0f2f(n)0f(n)2(1−23πζcf30(5f1−1))⎤⎥ ⎥⎦12. (36)
In table 1 we show the values of the enhancement factors and for different values of and different polytropic index . We also show the values of the critical ratio which separates the configurations with definite ratio such that a zero exist provided that . We will show some examples where the enhancement factors may be relevant in section 5.
## 4 The isothermal sphere
The isothermal sphere is a popular model in astrophysics, either to model large astrophysical and cosmological objects (galaxies, galaxy clusters) lynden (); penston (); yabu (); sommer (); chavanis1 (); more (), to examine the so-called gravothermal catastrophe binney (); natarajan (); lombardi () and finally to compare observations with model predictions rines (); more (). In the limit in the polytropic equation of state one obtains the description for an isothermal sphere, (ideal gas configuration) with
p=σ2ρ (37)
where is the velocity dispersion (). The pattern we found in section 3 for gets confirmed here: no finite radius of the configurations is found with , the asymptotic behavior is not as (but rather ) and, as we will show below, other attempts to define a proper finite radius are not satisfactory.
The results for a finite value of the index are defined in the limit only asymptotically in the case . We consider this as an acceptable behavior of the density. Because of the limiting case the analysis for the isothermal sphere must be done in a slightly different way. As was done in Eq. (13), we can integrate the equilibrium equations (Euler and Poisson’s equations) and obtain an explicit dependence of the density with as
ρ(r)=ρcexp[−1σ2(Φgrav(r)−Φgrav(0))]exp[83σ2πρvacr2], (38)
The resulting differential equation for the density with cosmological constant can be written as
σ2r2ddr(r2dlnρdr)=−4πρ+8πρvac,
This differential equation could be treated in the same way as we did for the polytropic equation of state, i.e, by defining a new function , but here we can already use the fact that the cosmological constant introduces scales of density, length and time bala1 (). Let us then define the function and , with the associated length scale. Since we are now scaling the density with , the associated length scale should be also scaled by the length scale imposed by :
r0=σrΛ=13.34(σ103km/s)Mpc. (39)
For an hydrogen cloud with kms we have kpc which is approximately the radius of an elliptical (E0) galaxy. In terms of the function , the differential equation governing the density profile is then written as 111Compare with Eq. 374 of chan () or Eq.1 of natarajan () where the density is scaled by the central density. The factor on the r.h.s of (40) is due to .
1ξ2ddξ(ξ2dψdξ)=1−12eψ, (40)
so that according to Eq. (38) we may write
ψ(r/r0)=ln(ρcρvac)−σ−2(Φgrav(r)−83πρvacr2). (41)
From this we can derive different solutions depending on the initial condition and at . In Fig. 4 we show numerical results for the solutions of equation (40) using different values values of . As it is the case for , the radius cannot be defined by searching the first zero of the density i.e. the value such that (including ). In this case, the behavior of the derivative of the density profile changes as compared with the the case since with increasing the density starts oscillating around the value such that for one has a solution . This can be checked from (40) which corresponds to the first non trivial solution for . This behavior implies that there exist a value of where the derivative changes sign and hence the validity of the physical condition required for any realistic model i.e. should be given up unless we define the size of the configuration as the radius at the value of the first local minimum. We will come back to this option below to show that it is not acceptable. A second option would be to set the radius at the position where the density acquires for the first time its asymptotic value . We could motivate such a definition by demanding that the density at the boundary goes smoothly to the background density. This is for tow reasons, however, not justified. First, we recall that a positive cosmological constant leads to a repulsive ’force’ as it accelerates the expansion. A negative cosmological constant could be modeled in a Newtonian sense by a constant positive density which, however, is strictly speaking still not a background density. Secondly, if we include the background density we would have started with (with a dynamical equation for being constant) in which case the boundary condition would again be to define the extension of the body (or at least, as ). Hence, this second option can be excluded on general grounds. In any case as can be seen from Figure 4 both definitions would yield two different values of radius. Since the first candidate to define a radius is based upon a physical condition of the configuration, we could expect this definition as the more suitable one. However such a definition must be in agreement with the observed values for masses and radius of specific configurations and the validity of this definition can be put to test by the total mass of the configuration, given as
M=1.55×1015(σ103km/s)3f(ξ1)M⊙,f(ξ1) ≡ ∫ξ10ξ2eψdξ. (42)
Combining (39) and (42) we can write
M=653.55(Rkpc)3ξ−31f(ξ1)M⊙. (43)
If we define the radius at the first minimum (see Fig. 4), we find . Although this might set the right order of magnitude for the mass of a E0 galaxy if we insist on realistic values for the respective radius, say kpc, the picture changes again as the radius is fixed by (39) which gives . In order to get a radius of the order of kpc with masses of the order of we would require . This differs by almost eight orders of magnitude with the measured values for the velocity dispersion in elliptical galaxies ( kms) or with the Faber-Jackson Law for velocity dispersion padma (). We conclude that defining the radius by the position of the first minimum is not a realistic solution. As the last option to get realistic values for the parameters of the configuration we consider the brute force method to simply fix the value of . It is understood, however, that this method is not acceptable if we insist that the model under consideration has some appealing features (without such features almost any model would be phenomenologically viable). Therefore we discuss this option only for completeness. For a configuration with , say an elliptical galaxy, we fix the radius at kpc in Eq.(39) and using a typical value for the velocity dispersion kms we get which implies . The mass in Eq.(42) is then given as while the density at the boundary is , that is, . In table 2 we perform the same exercise for other radii. The resulting mean density is in accordance with the observed values of the mean density of astrophysical objects ranging from an small elliptical galaxy to a galaxy cluster. However, as mentioned above, the model introduces an arbitrary cut-off and cannot be considered as a consistent model of hydrostatic equilibrium.
In summary, the attempts to define a finite radius for the isothermal sphere fail in the presence of a cosmological constant either because such a model fails to reproduce certain phenomenological values (if the definition of the radius is fixed by the first minimum) or because the definition is technically speaking quite artificial to the extent of introducing arbitrary cut-offs. Note that this conclusion is valid almost for any object as the density of the isothermal sphere with has a minimum whatever the central density we choose.
## 5 Exotic astrophysical configurations
In this section we will probe into the possibilities of exotic, low density configurations. The global interest in such structures is twofold. First, the cosmological constant will affect the properties of low density objects. Secondly, plays effectively the role of an external, repulsive force. Hence a relevant issue that arises in this context is to see whether the vacuum energy density can partly replace the pressure which essentially is encoded in the parameter (). By the word replace we mean that we want to explore the possibility of a finite radius as long as the pressure effects are small in the presence of .
### 5.1 Minimal density configurations
As mentioned above the effect of a positive cosmological constant on matter is best understood as an external repulsive force. In previous sections we have probed into one extreme which describes the situation where a relatively low density object is pulled apart by this force (to an extent that we concluded that the isothermal sphere is not a viable model in the presence of ). Limiting conditions when this happens were derived. On the other hand, approaching with our parameters these limiting conditions, but remaining still on the side of equilibrium, means that relatively low density objects can be still in equilibrium thanks to the positive cosmological constant. The best way to investigate low density structures is to to use the lowest possible central density. As explained in section 3, for every there exist a such that which defines the lowest central density. Certainly, a question of interest is to see what such objects would look like. We start with the parameters of the configuration. The radius at the critical value is given by Eq.(21) after taking the limit (see (28)). It is given by
Rcrit=2.175(M1012M⊙)1/3f0(ζcrit;n)ζ1/3critMpc. (44)
From fig. 5 we can see that the product changes a little round the value as we change the index , so that these polytropic configurations will have roughly the same radius (for a given mass). This implies that such configurations have approximately the same average density . That is, such configurations have a mean density of the order of the of the critical density of the universe. Given such a density we would, at the first glance, suspect that the object described by this density cannot be in equilibrium. However, our result follows strictly from hydrostatic equilibrium and therefore there is no doubt that such object can theoretically exist. Furthermore, satisfies the inequalities derived in nowakowski1 () and bala2 () from virial equations and Buchdahl inequalities which guarantee that the object is in equilibrium (). Interestingly, the central density for such objects has to be much higher than as, e.g., for we have and therefore .
Note that these values have been given from the solution of the Lane-Emden equation, which is a consequence of dynamical equations reduced to describe our system in a steady state. However, we have not tried to solve explicitly quantities from the virial theorem. This makes sense as for Dark Matter Halos (DMH) the parametrized density profiles go often only asymptotically to zero and the radius of DMH is defined as a virial radius where the density is approximately two hundred times over the critical one. Therefore, an analysis using virial equations seems to be adequate here. For constant density Eq.(30) can be expressed as a cubic equation bala1 () for the radius at which the virial theorem is satisfied (let us ignore for these analysis any surface terms coming from the tensor virial equation). However, if the density is not constant, this expression becomes a transcendental equation for the dimensionless radius . This equation is
ξ2vir=6(5−n)f30(n+1)[3−2πζc(5f1−1)f30], (45)
understanding the functions now as integrals up to the value . Once we fix for a given index , we use as a first guess for the iteration process the value . In fig 5 we show the behavior of the functions and the solutions of Eq.(45). For these values, Eq. (44) gives for a radius
Rvir=25.8(MM⊙)1/3pc, (46)
which can be compared with the radius-mass relation derived in the top-hat sphericall collapse padma ()
Rvir=21.5h−2/3(1+zvir)−1(MM⊙)1/3pc, (47)
where is the dimensionless Hubble parameter and is the redshift of virialization. The resulting average density is then of the order of the value predicted by the top-hat sphericall model:
(48)
Note with the help of Eq.(16) and (44) that the mass can be written as proportional to the parameter (introduced in the polytropic equation of state Eq.(11)). Therefore is equivalent to choosing a small pressure and, at the same time, a small mass which, in case of a relatively small radius, amounts to a diluted configuration with small density and pressure. Without such configurations would be hardly in equilibrium. Hence, for the configuration which has the extension of pc, Eq.(46), with one solar mass we conclude that the equilibrium is not fully due to the pressure, but partially maintained also by . This is possible, as exerts an outwardly directed non local force on the body. Other mean densities, also independent of are for and , respectively:
¯ρ=15.3ρcrit,¯ρ=2.6×103ρcrit, (49)
The first value is close to and therefore also to . Certainly, if in this example we choose a small mass, equivalent to choosing a negligible pressure, part of the equilibrium is maintained by the repulsive force of . In bala2 () we found a simple solution of the hydrostatic equation which has a constant density of the order of . The above is a non-constant and non-trivial generalization of this solution.
### 5.2 Cold white dwarfs
The neutrino stars which we will discuss in the subsequent subsection are modeled in close analogy to white dwarfs. Therefore it makes sense to recall some part of the physics of white dwarfs. In addition we can contrast the example of white dwarfs to the low density cases affected by .
In the limit where the thermal energy of a (Newtonian) white dwarf is much smaller than the energy at rest of the electrons (, these configurations can be treated as polytropic configuration with . This is the ultra-relativistic limit where the mass of electrons is much smaller than Fermi’s momentum . In the opposite case we obtain a polytrope or configuration with . weinberg (); shapiro (). In both cases, the parameter from the polytropic equation of state is given as
κ3=112π2(3π2mnμ)43,κ3/2=115meπ2(3π2mnμ)53, (50)
where is the nucleon mass, is the electron mass and is the number of nucleons per electron. Using the Newtonian limit with cosmological constant, we can derive the mass and radius of these configurations in equilibrium. In the first case, for the mass is written using (31) as which corresponds approximately to the Chandrasekhar’s limit (strictly speaking a configuration would have the critical mass, i.e, the Chandrasekhar’s limit, if its polytropic index is such that ). For this situation one has and . On the other hand, for one obtains and the radius is given by where is the mean density of the sun. Since for these configurations the ratio is much smaller than we see from Fig. 2 that the effects of are almost negligible. The critical value of the ratio gives for the inequality and for the same limit reads . Central densities of white dwarfs are of the order of which corresponds to a deviation of nearly thirty orders of magnitude of .
### 5.3 Neutrino stars
An interesting possibility is to determine the effects of on configurations formed by light fermions. Such configurations can be used, for instance, to model galactic halos dolgov (); lattanzi (); jetzer (); boerner (). While discussing the phenomenological interest of fermion stars below, we intend to describe such a halo. Clearly, these kind of systems will maintain equilibrium by counterbalancing gravity with the degeneracy pressure as in a white dwarf. For stable configurations, i.e, , one must replace the mass of the electron and nucleon by the mass of the considered fermion and set in (50) and (31). We then get for the mass and the radius:
M0 = 3.28×1028(ρc¯ρ⊙)12(eVmf)4M⊙=3.64×1014ζ−1/2c(eVmf)4M⊙ | {"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": 0.9759220480918884, "perplexity": 456.92762887349005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00231.warc.gz"} |
http://en.wikipedia.org/wiki/Euler's_four-square_identity | # Euler's four-square identity
In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. Specifically:
$(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,$
$(a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4)^2 +\,$
$(a_1 b_2 - a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +\,$
$(a_1 b_3 - a_2 b_4 - a_3 b_1 + a_4 b_2)^2 +\,$
$(a_1 b_4 + a_2 b_3 - a_3 b_2 - a_4 b_1)^2.\,$
Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach[1][2] (but he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring. If the $a_k$ and $b_k$ are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers.
The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any $a_k$ to $-a_k$, $b_k$ to $-b_k$, or by changing the signs inside any of the squared terms on the right hand side.
Hurwitz's theorem states that an identity of form,
$(a_1^2+a_2^2+a_3^2+...+a_n^2)(b_1^2+b_2^2+b_3^2+...+b_n^2) = c_1^2+c_2^2+c_3^2+...+c_n^2\,$
where the $c_i$ are bilinear functions of the $a_i$ and $b_i$ is possible only for n = {1, 2, 4, 8}. However, the more general Pfister's theorem allows that if the $c_i$ are just rational functions of one set of variables, hence has a denominator, then it is possible for all $n = 2^m$.[3] Thus, a different kind of four-square identity can be given as,
$(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,$
$(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1)^2 +\,$
$(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2)^2 +\,$
$\left(a_1 b_2 + a_2 b_1 + \frac{a_3 u_1}{b_1^2+b_2^2} - \frac{a_4 u_2}{b_1^2+b_2^2}\right)^2+\,$
$\left(a_1 b_1 - a_2 b_2 - \frac{a_4 u_1}{b_1^2+b_2^2} - \frac{a_3 u_2}{b_1^2+b_2^2}\right)^2\,$
where,
$u_1 = b_1^2b_4-2b_1b_2b_3-b_2^2b_4$
$u_2 = b_1^2b_3+2b_1b_2b_4-b_2^2b_3$
Note also the incidental fact that,
$u_1^2+u_2^2 = (b_1^2+b_2^2)^2(b_3^2+b_4^2)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 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": 0.9209285974502563, "perplexity": 127.71772417179001}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430455962510.77/warc/CC-MAIN-20150501045242-00084-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://matheducators.stackexchange.com/questions/15214/why-do-standard-geometry-textbooks-not-start-with-trigonometry | # Why do standard geometry textbooks not start with trigonometry?
Throughout my geometry course, I was given many theorems and postulates, which I was were expected to memorize and apply. At the time, I sorta went along with it, but I couldn’t help but wonder where these came from. Probably 90% of these questions were answered when I took trigonometry.
Take, for example, proving two triangles congruent. As we all know, one can prove them congruent by side-side-side, side-angle-side, angle-side-angle, angle-angle-side, and hypotenuse-leg. My teacher, I distinctly remember, made sure to emphasize to “stay away from the bad word, forward and backward” (angle-side-side).
But wait! I wondered at the time. Why doesn’t angle-side-side work, but hypotenuse-leg does? Isn’t hypotenuse-leg just a special case of angle-side-side, where the angle is a right angle? To this day I cannot come up with a rigorous proof that justifies these that doesn’t rely on the fact that $$\sin\theta=\sin(\pi-\theta)$$, meaning that, unless $$\theta=\frac\pi2$$, there are two possible side lengths corresponding to the same angle, and thus two triangles that contain the same angle-side-side but differing third sides.
In general, I think, explaining the basic trigonometry functions and the unit circle as the very first thing in the course would make pretty much all of geometry make so much more sense. Granted, even the unit circle requires certain geometric knowledge (ex. definition of a circle, that all radii are congruent, etc.), but not too much that would seem to preclude some form of this working?
1. Is what I’ve described a typical geometry course worldwide, or at least US-wide, or did I happen to end up with a poor geometry course?
2. If this is the normal way of teaching geometry, why? Why is the course focused more on memorizing theorems rather than understanding where they come from — something which, in my experience, seems to be nigh-impossible without the tools learned after Geometry?
• The traditional study of geometry has followed Euclid's Elements for the past 1900 years or so. Postulates, etc. and no trigonometry. Feb 13 '19 at 13:47
• @Gerald Followed to the extent that it was my father's school geometry textbook in the 1930s, in fact! Feb 13 '19 at 14:08
• @GeraldEdgar Just because that’s how it’s always been done doesn’t necessarily mean that’s the best way to do it. Feb 13 '19 at 14:26
• The non-answer to "Is what I’ve described a typical [secondary education] geometry course worldwide" is mu. Secondary education in the UK has one mathematics course until age 16: mathematics. If you want the question to be globally applicable then it needs to be reworded as something like "How much geometry is it typical to teach before trigonometry?" Feb 13 '19 at 15:08
• Geometry is not trigonometry. The latter is the study of two interrelated functions of real numbers. The former is the study of certain sets satisfying certain properties. Your suggestion would be like trying to start algebra books by a detailed study of the exponential functions. Just not it. Feb 14 '19 at 18:02
If you really want to understand why the curriculum is structured the way it is, and how it got that way, you might want to read a brief history of the discourse around geometry education for the past 150 years or so; I recommend Chapter 1 of The Learning and Teaching of Geometry in Secondary Schools: A Modeling Perspective, by Herbst, Halverscheid, Fujita, and me.
However, I think the question actually relies on a (pardon the pun) circular argument: Certainly one can introduce $$\sin, \cos, \tan$$ etc as "circular function" defined via the coordinates of points on a unit circle. But in order to make use of those functions in triangles, you have to know how to adapt them to the case when the hypotenuse is not $$1$$. How do you know that the circular functions are scale-invariant?
The answer, of course, is that you rely on similarity properties: specifically, if two angles of one triangle are congruent to two angles of a second triangle, then the ratios of corresponding sides are equal ("Angle-Angle Similarity"). To solve any complicated problems involving triangles, you also need to know that it works in the other order: if the legs of one right triangle are in the same ratio as the legs of a second right triangle, then the corresponding angles are congruent (a special case of "Side-Angle-Side Similarity"). So if you really want to start with trigonometry, you have to preface it with a unit on similarity.
But similarity properties (and their proofs) depend in an essential way on the properties of parallels, so you can't really cover similarity without first covering parallels. And at that point you have basically reproduced about 2/3 of the first few chapters of a traditional geometry course anyway, so why not do the whole thing?
I cannot speak to curricula outside of the US, and I don't really buy your argument that geometry courses are "focused more on memorizing theorems rather than understanding where they come from." It is a poor geometry instructor indeed who admonishes students "Don't be an ASS" before first taking the time to discuss the failure of the angle-side-side congruence relation. Because I don't find much in your question that I can directly respond to, let me start by questioning the framing of your question, and instead answer the question of why geometry is taught the way that it is (which is, after all, the essence of the title of your question).
My impression is that geometry is usually taught (in the modern world, at least) because it is a relatively concrete and hands-on way of getting students to engage in mathematically rigorous argumentation. Very few people actually care if two triangles are congruent or not, or how to construct a regular hexagon. However, the process of proving statements about triangles and hexagons introduces students to the idea of mathematical logic.
In the example you give, I would hope that the pedagogy follows something like the following outline (modulo some permutation of ideas, depending on the instructor, the author, or where the students lead you):
1. We observe that a triangle has three sides and three angles. To a mathematician, it is (I think) natural to ask how many of these data are actually necessary to uniquely specify a triangle, up to congruence.
2. So, we get out our compass and straight-edge, and start doodling. Experiment a little. If we are clever, we should quickly discover that if two angles are specified, then the third angle is completely determined. This gives us the Angle Sum Theorem, which we can prove satisfactorily in the Euclidean setting.
3. However, we might note that three (or really, two) angles are insufficient to uniquely specify a triangle. Two triangles can have the same set of angle measures, but be of radically different sizes. However, such triangles will be similar (i.e. there is a constant ratio between the lengths of corresponding sides). We can prove this result, and, perhaps, call it the AAA Similarity Theorem (for angle-angle-angle).
4. At this point, we know that if we have determined two angles, we need at least one side length in order to uniquely determine the triangle. The natural question here should be "is this sufficient?" That is, if I know the measures of two angles and the length of one side, can I determine everything else about the triangle? The answer is yes, giving us an AAS Congruence Theorem (or, in your question SAA, for side-angle-angle).
5. Now we might want to know if we can lose even more information and still make progress. For example, is there an AS theorem? That is, if we know an angle and a side, is that enough to determine a triangle. Very quickly, we should see that the answer is "No"–counterexamples are fairly easy to construct. Fair enough. If we only know one angle, knowing the length of only one side is insufficient.
6. Here the real fun starts: if we are clever (or have good guidance), we should start to realize that if we know an angle and two sides, then the arrangement of those objects matters: if the angle is "between" the two sides, then a triangle is uniquely determined, a result which we can prove as the SAS Congruence Theorem. On the other hand, if the angle is not "between" the two sides, we get into trouble.
7. The mathematically inclined might ask to classify the ways in which we can get into trouble. After some investigation, we should determine that there are essentially three things which can go wrong (assume that $$AB$$ is a side with a specified angle at $$A$$; we wish to construct the side $$BC$$ of specified length):
• We construct a circle of the specified radius centered at $$B$$, but the radius is too small, and so we cannot construct any triangle with the specified angle and sides.
• We construct a circle of the specified radius centered at $$B$$, but the circle intersects the the opposite side of the angle at $$A$$ at two different points, leading to two possible triangles.
• We construct a circle of the specified radius centered at $$B$$, and the circle intersects the opposite side of the angle at $$A$$ at exactly one point, giving exactly one triangle. There are actually a couple of ways that this can happen: either the specified length is longer than $$AB$$ (in which case the circle will meet the line at only one point on the "correct" side of the the angle), or the circle will meet the line at a point of tangency (in which case, the side $$BC$$ will be perpendicular to the side $$AC$$).
8. Because there is ambiguity in the "ASS" case, we don't get a nice theorem, and we demand more information in such a case. However, do note that the "ASS" case is not always indeterminate: for example, we get the Hypotenuse-Leg Congruence Theorem as a special case.
This topic could be further expanded (it might be nice to carefully consider exactly when ASS fails and how and what extra data may be used to patch things up), but I've already hit the main points alluded to in the question, so I'll stop here.
Having laid out an outline for a possible series of lectures or activities for students, let me make the following points:
• There are (by my count) five theorems here which require proof. Real mathematics is about rigorously proving statements, so there are a number of good exercises here for students to work on. As an added bonus, each of the theorems to be proved should be more-or-less intuitively obviously true, and good constructions will help the students to see how to make the arguments. In higher mathematics, it is sometimes hard to visualize an argument, and it is often difficult to tell a priori whether or not a result is true (there are a lot of "obvious" algebraic statements that are a pain to prove in an abstract ring, and a lot of wicked estimates in analysis which are hard to visualize or otherwise understand intuitively), hence Euclidean geometry is a good sandbox for learning to pose and prove theorems.
• There are a number of cases where things don't work. This can be used to emphasize the power of a counter-example. Once we show that there is an example in which ASS fails to produce a unique triangle, then we know that no ASS congruence theorem can possibly exist. Counter-examples are amazing! Indeed, two of my favorite books are slim Dover volumes full of counter-examples in analysis and topology. However, these require a lot more technical knowledge than a geometric construction. Again, Euclidean geometry is a good sandbox for learning about counter-examples.
• Finally, to reiterate, no one really cares about most of the results in a high school geometry class. By suggesting that we skip straight to trigonometry, which is its own nightmare of abstract notation and endless confusion for students, you are suggesting that the goal of such a class is to learn computational techniques for determining angles and sides (or whatever). This completely misses the point, which is to explore a topic in a mathematically rigorous manner. The results are not at all important (for the most part); it is the process by which those results are obtained that is important.
My mental picture of why angle-side-side doesn't work as a congruence condition doesn't involve trigonometry and seems to me much simpler than trigonometry: Imagine that you're given a line segment $$AB$$ to serve as one of the two specified sides of the triangle and you're given the angle at $$A$$, so you can draw a line $$l$$ through $$A$$ in the desired direction. The third vertex $$C$$ of the desired triangle should be on $$l$$ and should be at a specified distance from $$B$$. That gives you a circle $$m$$ of possible locations for $$C$$, centered at $$B$$, and $$C$$ must be at an intersection of $$l$$ and $$m$$. In general, a circle will intersect a line twice; for the two choices to produce congruent triangles, you need that they're equidistant from $$A$$. In other words, the chord of $$m$$ joining the two intersection points should be bisected at $$A$$ by the radius of $$m$$ that passes through $$A$$. But a chord of a circle is bisected by a radius if and only if it's perpendicular to that radius, and that's why the angle-side-side condition works iff the angle is a right angle. (Strictly speaking, one should pay attention to the case where $$l$$ meets $$m$$ only once, i.e., where $$l$$ is tangent to $$m$$, but there again you have a right angle.)
So this example doesn't need (or, in my opinion, even benefit from) trigonometry. Offhand, I can't think of other examples either that would justify putting trigonometry earlier in the curriculum. Trigonometry is a useful tool, but the essential ideas of geometry don't involve it.
• Ultimately this is the same as the trigonometry proof, just explained without actually relying on trig and isn’t as formal. I was only using that as an example anyway, and this doesn’t address my main question. Feb 13 '19 at 14:40
• @DonielF You're right about my not addressing the question. I added a few sentences to clarify that. On the other hand, I"m not at all convinced that this is the same as the trigonometry proof. My argument looks like one that Euclid could have used. If it's really the same as trig, that would suggest that trig is just making things look more complicated. Feb 13 '19 at 15:33
• Strong objection to "isn't as formal". Just because there are no equations does not imply non-formalness. Most proofs in geometry look like this and are as formal as needed. Feb 13 '19 at 16:12
Anecdotal evidence from Germany: congruence is taught way before trigonometry (~7th grade vs ~9th grade for trig functions)
Why is the course focused more on memorizing theorems rather than understanding where they come from.
The fact that two triangles are congruent if their sides are pairwise of equal length has nothing to do with trigonometry in the first place; one can give a constructive proof (that also shows why the triangle inequality must hold) without trigonometry.
To see the same fact using trigonometry one would have to apply the law of cosines twice which in my eyes is much harder. This law will also most likely be memorized without "understanding where it comes from" because the proof is again much harder than the constructive ones for congruence using compass, straightedge, and protractor (In Germany, we call this a Geodreieck).
• Thanks, edited. Feb 13 '19 at 17:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 37, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.819175124168396, "perplexity": 349.10443386324573}, "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-2022-05/segments/1642320301341.12/warc/CC-MAIN-20220119125003-20220119155003-00192.warc.gz"} |
https://brilliant.org/problems/subtle-integral/ | # Subtle integral
Calculus Level pending
The differentiate function y= f(x) has a property that a secant joining any two points ( a, f( a)) and (b,f(b)) always intersect the y axis at(0,3ab) given that f(1)= 2 then. Integral from 0 to 1 f(x)dx is ?
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https://onepetro.org/snameicetech/proceedings-abstract/ICETECH14/3-ICETECH14/D031S015R001/487881 | In this paper, the quantitative influence of aspect ratio (B/h), and dimensionless velocity or thickness Froude number [TFN = u/√(gh)] on dimensionless ice-induced pressures (pe/ρiu2) is briefly reviewed and discussed. Since material properties of ice (E, σf, K1c) have not been reported for many data-sets, a strategy for generating appropriate material properties for ice is proposed. Two dimensionless terms for material properties of ice, {(E/σf)×[K1c/(σf√h)]} and {[K1c/(σfu)]×√[E/(ρih)]} were identified and their influence on pe/ρiu2 is discussed. It was found that (1) pe/ρiu2 on rigid vertical structures decreases with (a) increasing B/h at a rate of about 0.42, when u/√(gh) and {[K1c/(σfu)]×√[E/(ρih)]} remain constant; (b) pe/ρiu2 decreases with increasing u/√(gh) at a rate of about 1.80 when u/√(gh) is < about 6.0×10-3and at a rate of about 1.93 when u/√(gh) is > about 6.0×10-3 when B/h and {[K1c/(σfu)]×√[E/(ρih)]} remain constant. (2) Preliminary analyses of the datasets shows that pe/ρiu2 decreases with increasing {(E/σf)× [K1c/(σf√h)]} at a rate of 0.335 and 0.469 and that pe/ρiu2 decreases with increasing {[K1c/(σfu)]×√[E/(ρih)]} at a rate of 0.729 and 0.808. (3) It was also found that shapes of structures do not influence dimensionless ice-induced pressures on structures.
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http://mathoverflow.net/questions/108498/cohomological-criterion-for-g-equivariant-maps?answertab=votes | # Cohomological criterion for $G$-equivariant maps
Let $f:M\rightarrow N$ be a surjective map from a fourmanifold $M$ to a surface $N$, with connected fibers (each fiber is connected). Assume that $f$ admits a multiple section $s:N\rightarrow M$.
Suppose now that we are given a finite group $G$ acting freely on $M$ in such a way that the fiber class and the multiple section class in $H^2(M,\mathbb{Z})$ are invariant or multiplied by $-1$. Can we conclude that the map $f$ is $G$-equivariant and descends to $f':M/G\rightarrow N/G$ as topological spaces (as $G$ action may not be free on $N$)?
A point I am confused by is that the fiber class and the section class are preserved (invariant or multiplied by -1) as classes, not as cycles.
Edit My question turns out to be non-sense in its original form above. So let me change my question. I am reading this paper by M. Gross and P.H.M. Wilson, where they study SYZ conjecture for certain CY3s. I now try to understand the proof of Theorem 2.1. where certain holomorphic map $f$ is claimed to be anti-holomorphic $C_2$-action equivariant. Here the assumption in my question is satisfied. I now understand the (anti-)holomorphicity of the map $f$ ($C_2$-action) is crucial. But it is not yet clear to me why $f$ is $C_2$-action equivariant.
-
Koopa -- what is the action of $G$ on $N$? – algori Sep 30 '12 at 22:41
@algori We don't know whether or not the $G$-action descends to $N$. Rather that's exactly what I am asking; if $G$ preserves the fiber class and the multiple section class, can one conclude that $f$ preserves the fiber and thus the $G$-action on $M$ induces that of $N$? – Koopa Oct 1 '12 at 0:24
Koopa -- what do you mean by "$f$ preserves the fiber"? – algori Oct 1 '12 at 0:55
.. if you meant to say that $G$ takes fibers to fibres, than this is certainly false, at least in this generality (at least when $G\neq\{e\}$): e.g., take a map $f:M\to N$ for which everything is fine and perturb it slightly to make sure the image of some fiber under some element of $G$ is not a fiber. – algori Oct 1 '12 at 1:03
What is a "multiple section" or "multisection" of a surjective map? (My guess is it assigns to each point in the base a subset of the fibre, but then I have several guesses for what continuity means in this context.) – Mark Grant Oct 1 '12 at 15:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9105653762817383, "perplexity": 267.56327707761125}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454702039825.90/warc/CC-MAIN-20160205195359-00088-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/q-f-x-ln-12x-5-9x-2-so-by-using-the-chain-rule-i-can.254914/ | Q. f(x)=ln (12x-5/9x-2)So by using the chain rule, i can
1. Sep 9, 2008
fr33pl4gu3
Q. f(x)=ln (12x-5/9x-2)
So by using the chain rule, i can get:
(-4/3)((9x-2)2/(12x-5)2)
and by using the quotient rule, i can get the final answer, which is:
(2(-36x-8)(-36x-15)2-2(-36x-15)(-36x-8)2)
------------------------------------------------------------------
(-36x-15)4
The answer that i got here is wrong, but i don't know, why?
2. Sep 9, 2008
NoMoreExams
Re: Differentiation4
First of all do you mean: $$f(x) = ln \left( \frac{12x-5}{9x-2} \right)$$? In which case yes you would use chain rule and quotient rule. Chain rule applies to cases like f(g(x)), what is your f? what is your g? let's start there.
3. Sep 9, 2008
fr33pl4gu3
Re: Differentiation4
yes, and f is ln (12x-5/9x-2), where g is (12x-5/9x-2), correct??
4. Sep 9, 2008
NoMoreExams
Re: Differentiation4
Perhaps I used too many f's, let's say we have
$$f(x) = g(h(x)) = ln \left( \frac{12x-5}{9x-2} \right)$$
What is g? what is h?
5. Sep 10, 2008
fr33pl4gu3
Re: Differentiation4
Thanks, but i solve the problem, the f(x) will be the nominator of the log and the g(x) will be the denominator of the log.
6. Sep 10, 2008
NoMoreExams
Re: Differentiation4
No, I asked about the chain rule, not the quotient rule. Actually f is the whole function, h(x) would be the argument of ln( ) and g would be ln itself. In general you should know this rule:
If
$$f(x) = ln( g(x) )$$
Then
$$f'(x) = \frac{g'(x)}{g(x)}$$
So in this case the derivative of
$$ln \left( \frac{12x-5}{9x-2} \right)$$
is
$$\frac{ \left(\frac{12x-5}{9x-2} \right)'} {\left( \frac{12x-5}{9x-2} \right) }$$
Where ' denotes differentiation.
7. Sep 11, 2008
HallsofIvy
Staff Emeritus
Re: Differentiation4
Since this doesn't really have anything to do with differential equations, I am moving it to Calculus and Analysis.
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http://www.physicsforums.com/showthread.php?p=4155861 | # Astrophysics and formulas
by Gliese123
Tags: astrophysics, extraterrestial, math and physics, method, planet
P: 134 Hello there! I'm currently working on an essay regarding astrophysics. The level of this essay is of high school standards but I do like math and physics so advanced calculations doesn't frighten me. Anyway so the essay's is about the astrophysics behind our search for extraterrestrial planets and I'm keen to conclude this. However, I seem to get stuck at the math part since I don't know what kind of formulas I should use for this. For instance the transit method or the Doppler method and more. To get this essay as complete as possible, I would like to demonstrate my own examples in this by having mathematical formulas and calculations for my "own solar system" Are there any persons here who are familiar with the very physics behind extraterrestrial methods? Not the literal part but the mathematical part. If you know any about anything then you're the best!
Mentor P: 10,511 You might be interested in the Kepler laws and the Doppler effect. Transits are simple geometry - if the luminosity of the star goes down by 0.01%, the planetary disk has 0.01% of the star's visible area, which corresponds to 1% of its radius, for example.
P: 134
Quote by mfb You might be interested in the Kepler laws and the Doppler effect. Transits are simple geometry - if the luminosity of the star goes down by 0.01%, the planetary disk has 0.01% of the star's visible area, which corresponds to 1% of its radius, for example.
Thank you! Well I've looked over the Kepler laws a bit.. But it seems to be many ways of finding planets.
Mentor
P: 10,511
## Astrophysics and formulas
There are many ways, but most of them measure the orbital period in some way and calculate the orbital radius based on the period and the estimated stellar mass.
Astrometry (not successful yet, but Gaia should change this), direct imaging (rare) and microlensing (rare) are the only examples, I think. They can get a direct measurement related to the distance.
P: 204 If you want to get a bit more technical, read up on single-lined spectroscopic binaries. Incase you aren't already aware, single lined just means that only one object in the system is visible to us (as opposed to double lined, where we can see two distinct spectra). Of course, as mfb has already alluded to, it is usually the case that a star-planet system is a single-lined spectroscopic binary. Here is a link to some excellent material from J. B. Tatum's Celestial Mechanics notes: http://www.astro.uvic.ca/~tatum/celmechs.html Double and single lined spectroscopic binary systems are covered in chapter 18, though chapter 17 on visual binary stars introduces the important orbital elements as they pertain to binaries. Of particular interest for star-planet(s) systems is the mass function when M>>m. (see section 18.4) It is also well worth mentioning, however, that Tatum is largely skeptical of concluding that the unseen object is a planet without additional supporting evidence. His reasons for this stance are particularly interesting and noteworthy, especially if you're writing a paper on methods used in planetary searches.
P: 134
Quote by bossman27 If you want to get a bit more technical, read up on single-lined spectroscopic binaries. Incase you aren't already aware, single lined just means that only one object in the system is visible to us (as opposed to double lined, where we can see two distinct spectra). Of course, as mfb has already alluded to, it is usually the case that a star-planet system is a single-lined spectroscopic binary. Here is a link to some excellent material from J. B. Tatum's Celestial Mechanics notes: http://www.astro.uvic.ca/~tatum/celmechs.html Double and single lined spectroscopic binary systems are covered in chapter 18, though chapter 17 on visual binary stars introduces the important orbital elements as they pertain to binaries. Of particular interest for star-planet(s) systems is the mass function when M>>m. (see section 18.4) It is also well worth mentioning, however, that Tatum is largely skeptical of concluding that the unseen object is a planet without additional supporting evidence. His reasons for this stance are particularly interesting and noteworthy, especially if you're writing a paper on methods used in planetary searches.
Great facts and Great link! Thanks a lot :D
Related Discussions Astrophysics 0 Career Guidance 1 Introductory Physics Homework 3 Introductory Physics Homework 2 Astrophysics 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": 0.8249734044075012, "perplexity": 1050.0979822205334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999674031/warc/CC-MAIN-20140305060754-00030-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://crypto.stackexchange.com/questions/59449/proving-one-time-pad-is-perfectly-secret | # Proving one time pad is perfectly secret
I'm reading about one-time pad in "Introduction to Modern Cryptography" by Katz and Lindell. I can understand the definition of perfect secrecy. However, how is OTP proven to be perfectly secure ? I'm quite rusty on probability, so please explain the steps.
The proof in the book goes like this,
$M$ is the set of messages (all possible bit strings of length $\ell$), the message space. $K$ (the key space) is equal to the message space, and the cipher is computer as $c=k\oplus m$, where $k$ belongs to $K$ (chosen uniformly at random) and m belongs to $M$, \begin{align*} \Pr&[C=c\mathrel|M=m] \\ &= \Pr[M \oplus K = c \mathrel| M = m] \\&\quad\text{(How did the random variable $C$ become $M\oplus K$ here?)} \\ &= \Pr[m \oplus K = c] \\&\quad\text{(where did the conditional probability go ? what rule should I apply here ?)} \\ &= \Pr[K = m \oplus c] \\&\quad\text{(what is this ?)} \\ &= 1/2^\ell \\&\quad\text{(how should I arrive at this finally ?)} \end{align*} I did not understand any of these four steps. Please explain the reasoning behind these steps. This result is being used to show that it satisfies the definition of perfect secrecy. Please help me understand this result.
• The uppercase letters denote random variables in some probability space.
• The lowercase letters denote particular values in the domains of the corresponding random variables.
For example, if the probability space is $\Omega$, then $C\colon \Omega \to \{0,1\}^\ell$ is a random variable for the ciphertext in the one-time pad system, and $c \in \{0,1\}^\ell$ is a particular value of a ciphertext, which at some sample $\omega \in \Omega$ might be the value of $C(\omega) = c$.
The random variable $C$ is related to the random variables $M$ and $K$ by $C = M \oplus K$, which is the standard mild abuse of notation for random variables meaning that $C(\omega) = M(\omega) \oplus K(\omega)$ in any sample $\omega \in \Omega$. Any fixed values $c$, $m$, and $k$ might or might not be related by $c = m \oplus k$, although if in some sample $\omega$ we have $C(\omega) = c$ and $M(\omega) = m$ and $K(\omega) = k$ then $c$, $m$, and $k$ are indeed related by $c = m \oplus k$.
For any fixed message $m \in \{0,1\}^\ell$, in the event $E = \{\omega \in \Omega \mathrel: M(\omega) = m\}$ where the message $M$ takes on the value $m \in \{0,1\}^\ell$, what can you conclude about $C$? In other words, if for any $\omega$ we have $M(\omega) = m$, what can you say about $C(\omega)$? Can you use this to eliminate $C$ from the ellipsis in $\Pr[\;\cdots \mathrel| M = m]$?
Next, if you have an equation about the random variable $K$ without reference to $M$, given an equation about $M$, recall the definition of independence of random variables and how it relates to conditional probabilities. Finally, use the probability mass function for $K$, and you'll have an answer.
• Thanks for taking time to help. Is it possible that for some single value omega in {0,1}^l be that C(omega) = c and M(omega) = m and K(omega) = k such that c = m exor k ? I understand that you want me to think about it and arrive at the solution myself but would it be possible for you explain in much simpler terms ? I'm very bad at probability. To be honest, I still did not understand. Sorry, I'm not able to write in LaTeX here in comments. – rranjik May 22 '18 at 15:49
• @user3222 $\omega$ is an abstract sample in a sample space, not a bit string. We could make it concrete by defining $\Omega$ to be the cartesian product $\{0,1\}^\ell\times\{0,1\}^\ell$, and then defining defining $M(\omega)=x_0$ and $K(\omega)=x_1$ when $\omega = (x_0, x_1)$, but the language of random variables is convenient because it is not necessary to write out such concrete details. If you're not familiar with this language, I suggest finding an introductory text on probability theory first discussing random variables; then you can return to this. (For LaTeX, use \$dollar signs\$.) – Squeamish Ossifrage May 22 '18 at 15:55
• Okay. I understand that I should learn probability before I start reading this. Thanks for being nice to me so far. But, can you please give me a hint of what I can conclude about $C(\omega)$ here ? Only thing I can conclude is that it is take up a value within $\{0,1\}^\ell$, and that is all I can guarantee. What should I do knowing that $M(\omega) = m$ ? And, please give a hint on how to go from step 2 to 3 ($\Pr[m \oplus K = c] = \Pr[K = m \oplus c]$). I will not pester you more. Thanks for teaching me this thing LaTeX thing. I'm picking up. – rranjik May 22 '18 at 16:36
• @user3222 If $C(\omega) = M(\omega) \oplus K(\omega)$ for all $\omega$, and you know $M(\omega) = m$, then…(fill in the blank). For the other question, look up or try to figure out properties of xor. (What's $a \oplus a$ for any $a$?) – Squeamish Ossifrage May 22 '18 at 20:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9913065433502197, "perplexity": 227.7257596739759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550249550830.96/warc/CC-MAIN-20190223203317-20190223225317-00625.warc.gz"} |
http://ndl.iitkgp.ac.in/document/K2M4U0VmUUU5Tnp4NHFiTEhqMWxwYUVBVnRNUFl1WVcybmcwZWxGZndUQT0 | ### Tensor form factor for the D → π(K) transitions with Twisted Mass fermions.Tensor form factor for the D → π(K) transitions with Twisted Mass fermions.
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Author Lubicz, Vittorio ♦ Riggio, Lorenzo ♦ Salerno, Giorgio ♦ Simula, Silvano ♦ Tarantino, Cecilia Source Paperity Content type Text File Format PDF ♦ HTM / HTML Copyright Year ©2018
Abstract We present a preliminary lattice calculation of the D → π and D → K tensor form factors fT (q2) as a function of the squared 4-momentum transfer q2. ETMC recently computed the vector and scalar form factors f+(q2) and f0(q2) describing D → π(K)lv semileptonic decays analyzing the vector current and the scalar density. The study of the weak tensor current, which is directly related to the tensor form factor, completes the set of hadronic matrix element regulating the transition between these two pseudoscalar mesons within and beyond the Standard Model where a non-zero tensor coupling is possible. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 + 1 + 1 flavors of dynamical quarks. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV and with the valence heavy quark in the mass range from ≃ 0.7 mc to ≃ 1.2mc. The matrix element of the tensor current are determined for a plethora of kinematical conditions in which parent and child mesons are either moving or at rest. As for the vector and scalar form factors, Lorentz symmetry breaking due to hypercubic effects is clearly observed in the data. We will present preliminary results on the removal of such hypercubic lattice effects. Learning Resource Type Article Publisher Date 2018-01-01 Journal EPJ Web of Conferences Issue Number 175 | {"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": 0.8900005221366882, "perplexity": 2377.9581499612545}, "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-2020-40/segments/1600400201601.26/warc/CC-MAIN-20200921081428-20200921111428-00167.warc.gz"} |
https://www.physicsforums.com/threads/why-is-c-even-in-the-equation.16754/ | # Why is c even in the equation?
1. Mar 21, 2004
### donkeyhide
why is "c" even in the equation?
i was hoping someone could explain why light speed even enters the e=mc2 equation. energy equals mass times the speed of light squared. i get that, and what the formula predicts. but why the speed of light? i was never one to go into the mathematics of all of this.
its just one of those things that you read and never really fully answer for yourself, i guess. i have read many books on relativity, and understand some things about it as a whole. but why is it that light-speed squared just happens to work out as the number that gives the right figure to multiply mass against? does it have something to do with the units that are used to calculate energy?
2. Mar 21, 2004
### Janus
Staff Emeritus
Re: why is "c" even in the equation?
It all goes back to the fact that c (the speed of light) is a constant for all observers. becuase of this, measurements of length and time between moving systems have to be adjusted by formulas that contain c (the Lorentz transformations).
Now if you take these formulas to calculate the energy of a moving object (which depends on these measurements) you get a formula that looks like this:
$$E = \frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$
Note that if v =0, the formula reduces to
$$E=mc^{2}$$
which is the energy equivalence of the object's rest mass.
3. Mar 24, 2004
### yogi
There are many books that give a derivation of the E=mc^2 forumla all based upon c as the velocity of light. There is another approach - a global derivation based upon expansion, where c is the expansion velocity of the Hubble sphere (which has the same value as the velocity of light)... accelerations of mass leads to forces, and force times distance is energy - when you calculate the energy required to remove a mass from the universe against this force - vola - you also get E = mc^2
4. Mar 24, 2004
### marcus
Re: why is "c" even in the equation?
It seems to me that it does have something to do with the units.
Maybe the units do not explain everything but they might help give some understanding.
In any consistent system of units the unit of energy is
always equal to the unit of mass multiplied by the square of
the unit speed
take for example the normal metric system where the energy unit is the joule and the unit mass is the kilogram
the joule is equal to one kilogram multiplied by the square of the unit speed
that is one kilogram x (meter/second)2
So if you have a mass and you want to get an energy quantity you are algebraically required (by the units system) to multiply by the square of some speed.
So suppose Einstein had figured out that the energy content of an object is proportional to the object's inertia but he had not yet discovered that the ratio was c2
he would think: if E is proportional to m, then there must be SOME speed that I can square and multiply m by, to get E. What could that speed be? What speed could give the constant of proportionality?
If this proportion between E and m is a universal proportion then the speed I need must be some universal fundamental speed constant. Well the only universal constant speed I know is c, could it maybe be that?
----------------
I guess the basic help that units give here is to show that if E and m are going to be proportional to each other there must be SOME speed whose square is the constant proportion between them.
This could have been deduced 100 years before Einstein, people have known for as long as they have had energy units that energy units are always equal to the mass unit multiplied by the square of a speed.
Newton could have told you that. Energy (work) = F x d = force x dist
force = mass x accel
force = mass x dist/time2
force x dist = mass x dist2/time2
force x dist = mass x speed2
Last edited: Mar 24, 2004
5. Mar 24, 2004
### da_willem
Einstein used thought experiments involving photons, so from studying these, one can understand where c, the speed of light, enters the story.
There is one experimental fact you will have to acknowledge; photons carry besides energy (E) also an amount of momentum p=E/c.
Imagine a box with mass M and length L on which no forces work, so the center of gravity will remain fixed. One the left side a photon is emitted and will cause the box to move to the other side. Because of the conservation of momentum the box will move with a velocity v wich obeys: Mv=p=E/c -> v=E/Mc. After a time t=L/c the photon will reach the oher side and the box will be moved a distance d=vt.
The center of gravity will be shifted by an amount mL-Md. But wait, because the are no external forces at work the center of gravity must be fixed. So: mL=Md=Mvt=EL/c^2 -> m=E/c^2 or E=mc^2.
So this shows that a photon can be assigned a 'mass' of m=E/c^2...
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http://math.stackexchange.com/questions/21060/notation-what-does-fx-x-omega1-mean | # Notation: What does $f(x) = x^{-\omega(1)}$ mean?
I am reading a cryptography paper, and the authors introduce a function $f(x) = x^{-\omega(1)}$ and call it a negligible function in $x$. What is the possible meaning of this?
-
This is a relative of big-O notation. We say that $f(x) = \omega(g(x))$ if $f(x) \ge k g(x)$ for all positive constants $k$, at least asymptotically.
So the notation should really be read $- \log f(x) = \omega(\log x)$, or $- \log f(x) \ge k \log x$ for every positive constant $k$ asymptotically. In other words, $f(x) \le x^{-k}$ for every positive constant $k$ asymptotically, hence $f$ vanishes faster than polynomially as $x \to \infty$. | {"extraction_info": {"found_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": 0.9721286296844482, "perplexity": 185.91902660295148}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1464049278244.7/warc/CC-MAIN-20160524002118-00150-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://rdrr.io/github/ramhiser/clusteval/man/fowlkes_mallows.html | fowlkes_mallows: Computes the Fowlkes-Mallows similarity index of two... In ramhiser/clusteval: Evaluation of Clustering Algorithms
Description
For two clusterings of the same data set, this function calculates the Fowlkes-Mallows similarity coefficient of the clusterings from the comemberships of the observations. Basically, the comembership is defined as the pairs of observations that are clustered together.
Usage
1 fowlkes_mallows(labels1, labels2)
Arguments
labels1 a vector of n clustering labels labels2 a vector of n clustering labels
Details
To calculate the Fowlkes-Mallows index, we compute the 2x2 contingency table, consisting of the following four cells:
n_11:
the number of observation pairs where both observations are comembers in both clusterings
n_10:
the number of observation pairs where the observations are comembers in the first clustering but not the second
n_01:
the number of observation pairs where the observations are comembers in the second clustering but not the first
n_00:
the number of observation pairs where neither pair are comembers in either clustering
The Fowlkes-Mallows similarity index is defined as:
\frac{n_{11}}{√{(n_{11} + n_{10})(n_{11} + n_{01})}}.
To compute the contingency table, we use the comembership_table function.
Value
the Fowlkes-Mallows index for the two sets of cluster labels
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ## Not run: # We generate K = 3 labels for each of n = 10 observations and compute the # Fowlkes-Mallows similarity index between the two clusterings. set.seed(42) K <- 3 n <- 10 labels1 <- sample.int(K, n, replace = TRUE) labels2 <- sample.int(K, n, replace = TRUE) fowlkes_mallows(labels1, labels2) # Here, we cluster the \code{\link{iris}} data set with the K-means and # hierarchical algorithms using the true number of clusters, K = 3. # Then, we compute the Fowlkes-Mallows similarity index between the two # clusterings. iris_kmeans <- kmeans(iris[, -5], centers = 3)\$cluster iris_hclust <- cutree(hclust(dist(iris[, -5])), k = 3) fowlkes_mallows(iris_kmeans, iris_hclust) ## End(Not run)
ramhiser/clusteval documentation built on Oct. 17, 2017, 12:26 p.m. | {"extraction_info": {"found_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": 0.9191973209381104, "perplexity": 3761.056029558481}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583657557.2/warc/CC-MAIN-20190116175238-20190116201238-00032.warc.gz"} |
http://www.transtutors.com/questions/rank-difference-method-of-correlation-27275.htm | # Rank Difference method of correlation
Using the rank-difference method of correlation, calculate the correlation coeffcient for the scores in the 100 meter dash and the average length of steps in the dash collected from twelve athletes. X y 1 10.41 2.29 2 9.98 2.23 3 10.22 2.15 4 10.11 2.13 5 9.90 2.13 6 10.25 2.11 7 10.30 2.04 8 10.30 2.04 9 10.00 2.00 10 10.30 2.00 11 10.70 1.98 12 10.30 1.95
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http://mathhelpforum.com/discrete-math/83413-using-pmi-help.html | 1. ## Using PMI help!
Hi all,
Well I went to my teacher to get help, but his method just did not work (no idea why). I'm trying to solve this:
For all n in N (where N is the set of natural numbers) $, 1/2(5) + 1/5(8) + ... + 1/(3n-1)(3n+2) = n/6n+4$
I understand the steps to PMI, but when I get to the Inductive Step $P(k+1)$ and try to solve the equality I get completely stuck.
I don't know what I am doing wrong, whether it be the PMI steps or otherwise. Thanks guys.
2. Originally Posted by teacast
Hi all,
Well I went to my teacher to get help, but his method just did not work (no idea why). I'm trying to solve this:
For all n in N (where N is the set of natural numbers) $, 1/2(5) + 1/5(8) + ... + 1/(3n-1)(3n+2) = n/6n+4$
I understand the steps to PMI, but when I get to the Inductive Step $P(k+1)$ and try to solve the equality I get completely stuck.
I don't know what I am doing wrong, whether it be the PMI steps or otherwise. Thanks guys.
for the P(k+1) you must have:
1/2(5) +1/5(8)+.......................+1/(3n-1)(3n+2) + 1/(3n+2)(3n+5) =
=(n+1)/6[(n+1)+4]= (n+1)/2(3n+5)........................................... ............................1
IF (1) is satisfied then the P(k+1) step is satisfied.
Since you already have : 1/2(5) +....................+1/(3n-1)(3n+2)= n/(6n+4) ,substitute that into (1) and you get:
n/(6n+4) + 1/(3n+2)(3n+5) = (n+1)/2(3n+5) or:
n/2(3n+2) + 1/(3n+2)(3n+5) = (n+1)/2(3n+5).
DO THE calculations and you will see that the equation is satisfied | {"extraction_info": {"found_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": 0.9158493876457214, "perplexity": 676.3782048429565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889733.57/warc/CC-MAIN-20180120201828-20180120221828-00500.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/210534-limiting-distribution.html | 1. Limiting Distribution
Let $X_i,i=1,2,3,...$ be independent Bernoulli( $\frac{1}{2}$) random variables and let $Y_n=\frac{\sum_{1=1}^nX_i}{n}$. Find the limitiing distribution of $W_n=n(Y_n(1-Y_n)-\frac{1}{4})$
2. Re: Limiting Distribution
First, let's find the distribution of $Y_n$, this can be done by computing the characteristic function $\phi_{Y_n}(t) = \displaystyle \phi_{\frac{1}{n}\sum_{i=1}^{n}}(t)=\phi_{\sum_{i= 1}^{n }X_i}\left(\frac{t}{n}\right)$ and because $\forall i \in {1,\ldots,n\}$ we have given that $X_i \sim B\left(\frac{1}{2}\right)$ are independent random variables we obtain $\phi_{Y_n}(t) = \prod_{i=1}^{n} \phi_{X_i}\left(\frac{t}{n}\right)$. Now, the only thing you have to do is calculating the characteristic function of a random variable with a Bernoulli distribution. Can you do that?
Second, we want the distribution of $W_n$, in fact we want to find the distribution function $F_{W_n}(x) = P(W_n \leq x)$. Can you compute this? | {"extraction_info": {"found_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": 0.9931529760360718, "perplexity": 99.82201151964603}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607649.26/warc/CC-MAIN-20170523183134-20170523203134-00597.warc.gz"} |
https://www.degruyter.com/view/j/math.2016.14.issue-1/math-2016-0078/math-2016-0078.xml | Show Summary Details
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The homogeneous balance of undetermined coefficients method and its application
Yi Wei
• Corresponding author
• Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, Shaanxi, China
• Email:
/ Xin-Dang He
• School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an, 710072, Shaanxi, China
/ Xiao-Feng Yang
• College of Science, Northwest A&F University, Yangling 712100, Shaanxi, China
Published Online: 2016-10-29 | DOI: https://doi.org/10.1515/math-2016-0078
Abstract
The homogeneous balance of undetermined coefficients method is firstly proposed to solve such nonlinear partial differential equations (PDEs), the balance numbers of which are not positive integers. The proposed method can also be used to derive more general bilinear equation of nonlinear PDEs. The Eckhaus equation, the KdV equation and the generalized Boussinesq equation are chosen to illustrate the validity of our method. The proposed method is also a standard and computable method, which can be generalized to deal with some types of nonlinear PDEs.
MSC 2010: 35Q55; 35Q80; 35G25
1 Introduction
Nonlinear PDEs are known to describe a wide variety of phenomena not only in physics, but also in biology, chemistry and several other fields. In recent years, many powerful methods to construct the exact solutions of nonlinear PDEs have been established and developed, which are one of the most excited advances of nonlinear science and theoretical physics. The exact solutions of nonlinear PDEs play an important role since they can provide much physical information and more insight into the physica1 aspects of the problems and thus lead to further applications. Seeking the exact solutions of nonlinear PDEs has long been an interesting topic in the nonlinear mathematical physics. With the development of soliton theory, various methods for obtaining the exact solutions of nonlinear PDEs have been presented, such as the inverse scattering method [1], the Bäcklund and Darboux transformation method [2], the homotopy perturbation method [3], the first integral method [4], the variational iteration method [5], the Riccati-Bernoulli sub-ODE method [6], the Jacobi elliptic function method [7], the tanhsech method [8], the (G'/G)-expansion method [9, 10], the Hirota’s method [11], the homogeneous balance method (HBM) [12, 13], the differential transform method (DTM) [1417] and so on.
In these traditional methods, the HBM and the DTM are straightforward and popular tools for handling many types of functional equations. Recently, the Adomian’s decomposition method (ADM) for solving differential and integral equations, linear or nonlinear, has been the subject of extensive analytical and numerical studies because the ADM provides the solution in a rapid convergent series with elegantly computable components [1823]. The Jacobi pseudo spectral approximation method [24], the fully spectral collocation approximation method [25, 26], the Jacobi tau approximation method [27], and the homotopy perturbation Sumudu transform method [28,29] are powerful and effective tools for solving nonlinear PDEs.
As a direct method, the HBM provides a convenient analytical technique to construct the exact solutions of nonlinear PDEs and has been generalized to obtain multiple soliton (or multiple solitary-wave) solutions [12, 13]. Fan improved this method to investigate the Bäcklund transformation, Lax pairs, symmetries and exact solutions for some nonlinear PDEs [30]. He also showed that there are many links among the HBM, Weiss-Tabor-Carnevale method and Clarkson-Kruskal method [31].
However, the HBM usually encounters complicated and tedious algebraic calculation. The exact solutions are fixed and single types when we deal with nonlinear PDEs by using the HBM. Moreover, the balance numbers of all nonlinear PDEs dealt with by this method usually are limited to positive integers.
Based on these problems, the homogeneous balance of undetermined coefficients method is proposed to improve the HBM, to derive more general bilinear equation of the KdV equation and the generalized Boussinesq equation, and to the exact solution of the Eckhaus equation, the balance numbers of which is not a positive integer.
The remainder of this paper is organized as follows: the homogeneous balance of undetermined coefficients method is described in Section 2. In Section 3, the exact solutions of the Eckhaus equation are obtained by using the homogeneous balance of undetermined coefficients method. In Section 4, the bilinear equation of the KdV equation and the generalized Boussinesq equation are derived respectively. A brief conclusion is given in Section 5.
2 Description of the homogeneous balance of undetermined coefficients method
Let us consider a general nonlinear PDE, say, in two variables, $P(u,ut,ux,uxx,uxt,⋯)=0,$(1) where P is a polynomial function of its arguments, the subscripts denote the partial derivatives. The homogeneous balance of undetermined coefficients method consists of three steps.
Step 1 Suppose that the solution of Eq. (1) is of the form $u=amn((Inw)m,n+∑k=m,j=nk,j=0k+j≠0,m+nakj(Inw)k,j+a00)b,$(2) where u = (u(x,t), w = w(x,t), ${\left(\text{In\hspace{0.17em}}w\right)}_{k,j}=\frac{{\mathrm{\partial }}^{k+j}\left(\text{In\hspace{0.17em}}w\left(x,t\right)\right)}{\mathrm{\partial }{x}^{k}\mathrm{\partial }{t}^{j}}$, and m, n, b (balance numbers) and akj (k = 0, 1, ...,m, j = 0, 1, ...,n) (balance coefficients) are constants to be determined later.
By balancing the highest nonlinear terms and the highest order partial derivative terms, m, n and b can be determined.
Step 2 Substituting Eq. (2) into Eq. (1) and arranging it at each order of w yield an equation as follows: $∑l=0flwl=0,$(3) where fl (l = 0, 1, ...) are differential-algebraic expressions of w and akj. Setting fl = 0 and using compatible condition (wxt = wxt) yield a set of differential-algebraic equations (DAEs).
Step 3 Solving the set of DAEs, w and akj (k = 0, 1, ...,m, j = 0, 1, ...,n) can be determined. By substituting w, m, n, b and akj into Eq. (2), the exact solutions of Eq. (1) can be obtained.
In the following section, the exact solutions of Eckhaus equation can be obtained by using the homogeneous balance of undetermined coefficients method.
3 Application to the Eckhaus equation
Let us consider the Eckhaus equation in the form $iψt+ψxx+2(|ψ|2)xψ+|ψ|4ψ=0,$(4) where $i=\sqrt{-1}$ and Ψ = Ψ(x, t) is a complex-valued function of two real variables x, t.
The Eckhaus equation was found as an asymptotic multiscale reduction of certain classes of nonlinear Schrödinger type equations [32]. In [33], a lot of the properties of the Eckhaus equation were obtained. In [34], the Eckhaus equation is linearized by making some transformations of dependent and independent variables. Exact traveling wave solutions of the Eckhaus equation can be obtained by the (G'/G)-expansion method [10] and the first integral method [4].
In this section, by applying the homogeneous balance of undetermined coefficients method to the Eckhaus equation, new exact solutions of the Eckhaus equation can be obtained.
Suppose that the solution of Eq. (4) is of the form $ψ=am n((Inw)m,n+∑k=m,j=n k,j=0k+j≠0,m+nakj(Inw)k,j+a00)bei(c x+dt) ,$(5) where m, n, b, c, d and akj (k = 0, 1, ...,m, j = 0, 1, ...,n) are constants to be determined later.
Balancing |Ψ|4Ψ and Ψxx in Eq. (4), it is required that 3mb + 1 = 5mb, 3nb = 5nb. Solving this set of equations, we get m = 1, n = 0, $b=\frac{1}{2}$. Then Eq. (5) can be written as $ψ=a((Inw)x+b)12ei(cx+dt),$(6) where a, b, c(a10 = a, a00 = b) and d are real constants to be determined later.
Substituting Eq. (6) into Eq. (4), we get $f0w0+f1w1+f2w2+f3w3+f4w4+i(g1w1+g2w2+g3w3)=0,$(7) where $f0=(2a2−1)(2a2−3)wx4,$(7a) $f1=4(2a2−1)(b(2a2−1)wx3+wx2wxx),$(7b) $f2=(24a2b2−4c2−8a2b2−4d)wx2+((16a2b−6b)wxx+2wxxx)wx−wxx2,$(7c) $f3=(16a4b3−8bd−8bc2)wx+8a2b2wxx+2bwxxx,$(7d) $f4=4b2(a4b2−c2−d),$(7e) $g1=−2(wt+2cwx)wx2,$(7f) $g2=−4bcwx2+(2wxt+4cwxx−2bwt)wx,$(7g) $g3=2b(wxt+2cwxx).$(7h)
Obviously, Ψ is a solution of Eq. (4) provided that fk = 0 (k = 0, 1, 2, 3, 4) and gj = 0 (j = 1, 2, 3). Firstly, suppose that wx = 0, from Eqs. (6) and (7), we get an exact solution of Eq. (4) as follows: $ψ1=C0ei(cx+(C04−c2)t),$(8) where C0 and c are arbitrary real constants.
Secondly, suppose that wx ≠ 0, setting f0 = 0 and g1 = 0, we get $a=±\frac{\sqrt{2}}{2}$, or $a=±\frac{\sqrt{6}}{2}$ $wt=−2cwx.$(9)
Substituting Eq. (9) into Eqs. (7g) and (7h), we get g2 = g3 = 0.
Case A When $a=±\frac{\sqrt{6}}{2}$, substituting Eq. (9) and $a=±\frac{\sqrt{6}}{2}$ into Eqs. (7b), (7c), (7d), and (7e), we get $f1=8(wxx+2bwx)wx2,$(10a) $f2=(42b2−4c2−4d)wx2+(18bwxx+2wxxx)wx−wxx2,$(10b) $f3=2b((18b2−4d−4c2)wx+6bwxx+wxxx),$(10c) $f4=b2(9b2−4d−4c2).$(10d)
Case A-1 When b = 0, substituting b = 0 into Eqs. (10), we get $f3=f4=0,f1=8wxxwx2,f2=−4(c2+d)wx2+2wxxxwx−wxx2.$(11)
Setting f1 = f2 = 0 and noting that wx ≠ 0, we get $wxx=0,d=−c2.$(12)
From Eqs. (9) and (12), we get $w=C1(x−2ct)+C0,$(13) where C0, C1 and c are arbitrary real constants.
Substituting $a=±\frac{\sqrt{6}}{2}$, b = 0, Eqs. (12) and (13) into Eq. (6), we get an exact solution of Eq. (4) as follows: $ψ2=±2C13(C1(x−2ct)+C0ei(cx−c2t),$(14) where C0, C1 and c are arbitrary real constants.
Case A-2 When b ≠ 0, setting f1 = f4 = 0 and noting that wx ≠ 0, we get $wx=−wxx2b,d=9b24−c2.$(15)
Substituting Eqs. (15) into Eqs. (10b) and (10c), we get $f2=b2wx2,f3=2b3wx.$(16)
Setting f2 = f3 = 0, we get bwx = 0 from Eqs. (16), which contradicts with b ≠ 0 and wx ≠ 0. Therefore, Eq. (4) has no solution in this case.
Case B When $a=±\frac{\sqrt{2}}{2}$, substituting $a=±\frac{\sqrt{2}}{2}$ and Eq. (9) into Eqs. (7b), (7c), (7d) and (7e), we get $f1=0,$(17a) $f2=(2b2−4c2−4d)wx2+(2bwxx+2wxxx)wx−wxx2,$(17b) $f3=2b((2b2−4c2−4d)wx+2bwxx+wxxx),$(17c) $f4=b2(b2−4c2−4d).$(17d)
Case B-1 When b = 0, substituting b = 0 into Eqs. (17), we get $f1=f3=f4=0,f2=−4(c2+d)wx2+2wxwxxx−wxx2.$(18)
Setting f2 = 0 yields an equation as follows: $−αwx2+2wxwxxx−wxx2=0,$(19) where α = 4(c2 + d).
Using transformations Z = wx and $Y=\frac{{w}_{xx}}{{w}_{x}}$, Eq. (19) is reduced to $2YdYY2−α=−dZZ.$(20)
Solving Eq. (20), we get $dZdx=±αZ2+βZ,$ namely $wxx=±αwx2+β(t)wx,$(21) where β(t) is an arbitrary function of t.
(1) When α = β(t) = 0, from Eq. (21) we get $w=C1(t)x+C2(t),$ where C1(t) and C2(t) are arbitrary functions of t.
Substituting the above equation into Eq. (9), we get $xdC1(t)dt+dC2(t)dt=−2cC1(t).$
Setting the coefficients of xj (j = 0, 1) to zero in the above equation, we get $dC1(t)dt=0,dC2(t)dt=2cC1(t).$
Solving the above equations, we get $C1(t)=C1,C2(t)=C0−2cC1(t),$ where Ck (k = 0, 1) are arbitrary real constants.
Then we get $w=C1(x−2ct)+C0.$(22)
Substituting $a=±\frac{\sqrt{2}}{2}$, d = −c2 and Eq. (22) into Eq. (6), we get an exact solution of Eq. (4) as follows: $ψ3=±C12(C1(x−2ct)+C0)ei(cx−c2t),$(23) where Ck (k = 0, 1) and c are arbitrary real constants.
(2) When β(t) = 0, α > 0, similar to (1), we get $w=C0+C1e2εc2+d(x−2ct),$ and an exact solution of Eq. (4) as follows: $ψ4=±εC1(c2+d)e2εc2+d(x−2ct)C0+C1e2εc2+d(x−2ct)ei(cx+dt),$(24) where ε = ±1, Ck (k = 0, 1), c and d are arbitrary real constants.
Especially, if C0C1 > 0, then Eq. (24) can be reduced to $ψ5=±(c2+d)142(1+tanh(c2+d(x−2ct)+ξ0))12ei(cx+dt),$(25) $ψ6=±(c2+d)142(−1+tanh(c2+d(x−2ct)+ξ0))12ei(cx+dt),$(26) where Ck (k = 0, 1), c, d and ξ0 are arbitrary real constants.
If C0C1 < 0, then Eq. (24) can be reduced to $ψ7=±(c2+d)142(1+coth(c2+d(x−2ct)+ξ0))12ei(cx+dt),$(27) $ψ8=±(c2+d)142(−1+coth(c2+d(x−2ct)+ξ0))12ei(cx+dt),$(28) where Ck (k = 0, 1), c, d and ξ0 are arbitrary real constants.
(3) When β(t) ≠ 0, α = 0, similar to (1), we get $w=13((x−2ct+C1)3+C0),$ and an exact solution of Eq. (4) as follows: $ψ9=±6(x−2ct+C1)2(x−2ct+C1)3+C0ei(cx−c2t),$(29) where Ck (k = 0, 1) and c are arbitrary constants.
(4) When β(t) ≠ 0, α > 0, similar to (1), we get $w=C1eC4(x−2ct)+C2e−C4(x−2ct)+C3(x−2ct)+C0,$ and an exact solution of Eq. (4) as follows: $ψ10=±(C4(C1eC4(x−2ct)−C2e−C4(x−2ct))+C32(C1eC4(x−2ct)+C2e−C4(x−2ct)+C3(x−2ct)+C0))12ei(cx+(C424−c2)t),$(30) where ${C}_{3}^{2}-4\left({c}^{2}+d\right)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}4{C}_{1}{C}_{4}^{2}{C}_{2}+{C}_{3}^{2}=0,\left(k=0,1,2,3,4\right)$, c and d are arbitrary real constants.
(5) When β(t) ≠ 0, α < 0, similar to (1), we get $w=εC1(x−2ct)+cos(C1(x−2ct)+C2)+C0,$ and an exact solution of Eq. (4) as follows: $ψ11 = ± (C1(ε − sin (C1 (x − 2ct) + C2))2(εC1 (x − 2ct) + cos (C1 (x − 2ct) + C2) + C0))12ei(c x − (c124 + c2) t),$(31) where ${C}_{1}^{2}+4\left({c}^{2}+d\right)=0,\phantom{\rule{thinmathspace}{0ex}}\epsilon =±1,\phantom{\rule{thinmathspace}{0ex}}{C}_{k}\left(k=0,1,2\right)$, c and d are arbitrary real constants.
Case B-2 When b ≠ 0, setting f2 = f3 = f4 = 0 in Eqs. (17b), (17c) and (17d), we get $b=±2c2+d,wx=−wxxb.$(32)
Solving Eqs. (32) and noting that Eq. (9), we get $w=C0+C1e±2c2+d(x−2ct).$(33)
Substituting $a=±\frac{\sqrt{2}}{2}$, $b=±2\sqrt{{c}^{2}+d}$ and Eq. (33) into Eq. (6), we find that this case is identical to Case B-1-(2).
In this section, by applying the homogeneous balance of undetermined coefficients method to the Eckhaus equation, new exact solutions of the Eckhaus equation are obtained. Among the solutions of the Eckhaus equation, Ψj (j = 3, 4, 5, 6, 7, 8) are the same as the results of [4, 10]. To our knowledge, other solutions of Eckhaus equation, Ψj (j = 2, 9, 10, 11) have not been reported in any literature.
4 Application to derive the bilinear equation of nonlinear PDEs
In this section, firstly, we modify the homogeneous balance of undetermined coefficients method to derive the bilinear equation of nonlinear PDEs. Then, more general bilinear equation of the KdV equation and the generalized Boussinesq equation are obtained by using the proposed method.
Let us still consider Eq. (1). Suppose that the solution of Eq. (1) is Eq. (2). By balancing the highest nonlinear terms and the highest order partial derivative terms, m, n and b can be determined. Substituting Eq. (2) into Eq. (1) and balancing the terms with ${\left(\frac{{w}_{x}}{w}\right)}^{i}{\left(\frac{{w}_{t}}{w}\right)}^{j}$ yield a set of algebraic equations for aij (i = 1, ...,m, j = 1, ...,n). Solving the set of algebraic equations and simplifying Eq. (1), we can get the bilinear equation of Eq. (1) directly or after integrating some times (generally, integrating times equal to the orders of lowest partial derivative of Eq. (1).) with respect to x, t.
Next, the KdV equation and the generalized Boussinesq equation are chosen as examples to illustrate our method.
Example 4.1: Let us consider the celebrated KdV equation [9] in the form $ut+uux+ δ uxxx=0,$(34) where δ is a constant.
Suppose that the solution of Eq. (34) is Eq. (2). Balancing uxxx and uux in Eq. (34), it is required that mb + 3 = 2mb + 1, nb = 2nb. Choosing m = 2, n = 0 and b = 1, Eq. (2) can be written as $u=a20(Inw)xx+a10(Inw)x+a00,$(35) where aj0(j = 0, 1, 2) are constants to be determined later.
From Eq. (35), one can calculate the following derivatives: $u=a20(wxxw−wx2w2)+a10wxw+a00,$(36a) $ux=a20(wxxxw−3wxxwxw2+2wx3w3)+a10(wxxw−wx2w2),$(36b) $ut=a20(wxxtw−wxxwt+2wxwxtw2+2wx2wtw3)+a10(wxtw−wxwtw2),$(36c) $uxxx = a20 (wxxxxxw − 5wxxxxwx + 10wxxxwxxw2 + 20wxxxwx2 + 30wxx2wxw3 − 60wxxwx3w4 + 24wx5w5) +a10(wxxxxw − 4wxxxwx + 3wxx2w2 + 12wxxwx2w3 − 6wx4w4).$(36d)
Equating the coefficients of ${\left(\frac{{w}_{x}}{w}\right)}^{5}$ and ${\left(\frac{{w}_{x}}{w}\right)}^{4}$ on the left-hand side of Eq. (34) to zero yields a set of algebraic equations for a20 and a10 as follows: $−2a202+24 δ a20=0,3a20a10−6 δ a10=0,$
Solving the above algebraic equations, we get a20 = 12δ, a10 = 0. Substituting a20 and a10 back into Eq. (35), we get $u=12 δ (Inw)xx+a00,$(37) where a00 is an arbitrary constant.
Substituting Eq. (37) into Eq. (34) and simplifying it, we get $12 δ (K1+K2+K3)=0,$(38) where $K1=wxxtw−2wxwxt+wxxwtw2+2wx2wtw3,K2=a00(wxxxw−3wxxwxw2+2wx3w3),K3= δ (wxxxxxw+2wxxxwxx−5wxxxxxwxw2+16wxxxwx2−6wxwxx2w3).$
Simplifying Eq. (38) and integrating once with respect to x, we get $∂∂x((wxtw−wxwt)+ δ (wxxxxw−4wxwxxx+3wxx2)+a00(wxxw−wx2)w2)=0.$(39)
Eq. (39) is identical to $(wxtw−wxwt)+ δ (wxxxxw−4wxwxxx+3wxx2)+a00(wxxw−wx2)−C(t)w2=0,$(40) where C(t) is an arbitrary function of t, and a00 is an arbitrary constant.
Especially, taking C(t) as zero in Eq. (40), we get the bilinear equation of Eq. (34) as follows: $(wxtw−wxwt)+ δ (wxxxxw−4wxwxxx+3wxx2)+a00(wxxw−wx2)=0.$(41)
Eq. (41) can be written concisely in terms of D-operator as $(DxDt+ δ Dx4+a00Dx2)w⋅w=0,$(42) where $DxmDtna⋅b=(∂x−∂x′)m(∂t−∂x′)na(x,t)b(x′,t′)|x′=x,t′=t.$
Remark 4.2: Applying the Hirota’s method [11], the bilinear equation of Eq. (34) can be written as $(DxDt+ δ Dx4)w⋅w=0.$(43)Eq. (43) is obtained by setting a00 = 0 in Eq. (42). Obviously, Eq. (43) is a special case of Eq. (42). Therefore, more general bilinear equation of the KdV equation is obtained by using our method.
Remark 4.3: Applying the perturbation method [11] to Eq. (42), we can get 1-soliton solution and 2-soliton solution of Eq. (42) as follows: $w1=1+eη1,w2=1+eη1+eη2+(P1−P2)2(P1+P2)2eη1+η2,$(44) where $\eta j={{P}_{j}}_{{}^{x}}-\left({a}_{00}{P}_{j}+\delta {P}_{j}^{3}\right)t+{\xi }_{j}^{0}$, and Pj ${\xi }_{j}^{0}\left(j=\phantom{\rule{thinmathspace}{0ex}}1,2\right)$ and a00 are arbitrary constants.Substituting Eqs. (44) into Eq. (37), 1-soliton solution and 2-soliton solution of Eq. (34) can be obtained. Similarly, N -soliton solution of Eq. (34) can be obtained.
Example 4.4: The generalized Boussinesq equation reads $utt+2 α uxt+( α 2+β)uxx+γuuxx+ δ uxxxx=0,$(45) where α, β, γ and δ are known constants.
Suppose that the solution of Eq. (45) is Eq. (2). In order to balance uxxxx and uux in Eq. (45), it is required that mb + 4 = 2mb + 2,nb = 2nb. Choosing m = 2, n = 0 and b = 1, Eq. (2) can be written as $u=a20(Inw)xx+a10(Inw)x+a00,$(46) where aj0 (j = 0, 1, 2) are constants to be determined later.
Substituting Eq. (46) into Eq. (45) and equating the coefficients of${\left(\frac{{w}_{x}}{w}\right)}^{6}$ and ${\left(\frac{{w}_{x}}{w}\right)}^{6}$ on the left-hand side of Eq. (45) to zero yield a set of algebraic equations for a20 and a10. Solving the algebraic equations, we get ${a}_{20}=\frac{6\delta }{\gamma }$, a10 = 0. Substituting a20 and a10 back into Eq. (46), we get $u=6 δ γ(Inw)xx+a00,$(47) where a00 is an arbitrary constant.
Substituting Eq. (47) into Eq. (45), we get $6 δ γ(K1+K2+K3)=0,$(48) where $K1= wx x t tw−2wxwx t t+wx x wt t+2wtwx xt+2wx t2w2+2wx2w t t+2wx x wt2+8wxwtwxtw2−6wx2wt2w4+β(wx x x xw−3wxx2+4wxwx x xw2+12wx xwx2w3−6wx4w4)+α2(wx x x xw−3wxx2+4wxwx x xw2+12wx xwx2w3−6wx4w4)+α(2wx x x tw−6wx xwx t+6wx x t wx +2wtwxxxw2+12wxwtwxx+12wx2wxtw3−12wtwx3w4),k2=a00(2γwx x x xw−8γwxwx x x +6γwxx2w2+24γwx xwx2w3−12γwx4w4),k3=δ(wx x x x x xw+2wx x x2−6wxwx x x x−3wxxwx x x xw2+18wx2wx x x x −6wxx3w3+18wx2wxx2−24wxx x3w4).$
Simplifying Eq. (48) and integrating twice with respect to x, we get $∂2∂x2 (( α 2 + β +2γa00) (wwxx − wx2w2) + 2 α (wwxt − wxwtw2) + (wwtt − wt2w2) + δ (wwxxxx − 4wxwxxx + 3wxx2w2)) = 0.$(49)
Eq. (49) is identical to $( α 2 + β +2γa00) (wwxx − wx2) + 2 α (wwxt − wxwt) + (wwtt − wt2) + δ (wwxxxx − 4wxwxxx + 3wxx2) − (C1 (t) x + C1 (t)) w2 = 0,$(50) where C1 (t) and C2 (t) are arbitrary functions of t, and a00 is an arbitrary constant.
Especially, letting C1 (t) = C2 (t) = 0 in Eq. (50), we get the bilinear equation of Eq. (45) as follows: $( α 2 + β + 2γa00) (wwxx − wx2) + 2 α (wwxt − wxwt)+ (wwtt − wt2) + δ (wwxxxx − 4wxwxxx + 3wxx2) =0.$(51)
Eq. (51) can be written concisely in terms of D-operator as $(2 α DxDt+ δ Dx4+( α 2+β+2γa00)Dx2+Dt2)w⋅w=0,$(52) where a00 is an arbitrary constant.
So far, the proposed method is successfully used to establish the bilinear equation of the KdV equation and the generalized Boussinesq equation. Our method can be used to derive the bilinear equations of some nonlinear PDEs.
5 Conclusions
The homogeneous balance of undetermined coefficients method is successfully used to solve such nonlinear PDEs, the balance numbers of which are not positive integers. By applying our method to the Eckhaus equation, more exact solutions are obtained. The proposed method can also be used to derive the bilinear equation of nonlinear PDEs. More general bilinear equation of the KdV equation and the generalized Boussinesq equation can be obtained by applying our method.
Once the bilinear equation of nonlinear PDE is obtained, the perturbation method can be employed to obtain the N -soliton solution for the nonlinear PDE. Many exact solutions of the nonlinear PDE are obtained by using the three-wave method and the homoclinic test approach. Generally, some nonlinear PDE can be linearized or homogenized by using the homogeneous balance of undetermined coefficients method.
Many well-known nonlinear PDEs can be handled by our method. The performance of our method is found to be simple and efficient. The availability of computer systems like Maple facilitates the tedious algebraic calculations. Our method is also a standard and computable method, which allows us to solve complicated and tedious algebraic calculations.
Acknowledgement
The research is supported by the National Natural Science Foundation of China (61402375), the Fundamental Research Funds for the Central Universities (G2015KY0103), the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX201517).
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Accepted: 2016-01-27
Published Online: 2016-10-29
Published in Print: 2016-01-01
Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 107, "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": 0.8698402643203735, "perplexity": 1210.1310996598138}, "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-2017-22/segments/1495463608416.96/warc/CC-MAIN-20170525195400-20170525215400-00138.warc.gz"} |
https://cstheory.stackexchange.com/questions/36834/expected-kolmogorov-complexity-under-kolmogorov-complexity-distribution?noredirect=1 | # Expected Kolmogorov complexity under Kolmogorov complexity distribution
If $K(w)$ is the Kolmogorov complexity of a string $w$, where programs are prefix-encoded so $\sum_{w} 2^{-K(w)} \leq 1$, what is
$$\lim_{n\to\infty} \frac{\sum_{|w|=n}2^{-K(w)} K(w)}{\sum_{|w|=n} 2^{-K(w)}}?$$
Also, what is
$$\lim_{n\to\infty} \frac{\sum_{|w|=n}2^{-K(w)} \frac{K(w)}{n}}{\sum_{|w|=n} 2^{-K(w)}}?$$
The distribution here is closely related to the "universal distribution". The second limit would say whether $\Theta(|w|)$ is an approximation for $K(w)$.
• Possibly related: cstheory.stackexchange.com/questions/7993/… – Andrew Oct 25 '16 at 23:31
• I could just ask many similar questions and don't see at all why you have asked these. Why prefix-free? Why is the denominator summed for only $|w|=n$? Don't we know that this denominator is practically the slowest convergent series, i.e., $1/\sum_{|w|=n} 2^{-K(w)}=O(n\log n \log^2\log n)$? I think that instead of $n$, you can divide by practically anything that tends to $\infty$ to get a convergent sequence. – domotorp Oct 27 '16 at 7:37
• @domotorp The denominator is there to make it an expectation over all strings of length $n$ – Bjørn Kjos-Hanssen Oct 28 '16 at 19:10
• @B I know, but why not take all strings? Anyhow, I agree that this is not as far-fetched as the other parameters. – domotorp Oct 28 '16 at 19:31
• @domotorp, perhaps Andrew means proportional for fixed $n$. E.g., if you take $n$ into account, proportional to $f(n)2^{j}$ for some $f$ (e.g. $f(n)=2^{-n}$). If my calculations are right, any such $f$ also gives the limit of the ratio value 1/2. – Neal Young Nov 2 '16 at 20:22
If $\alpha$ is the answer to the 1st question then $\alpha=\infty$. Namely, for any $c$ there is an $n$ such that all strings $w$ of length at least $n$ have $K (w) \ge c$. In particular the expectation of $K (w)$ with respect to any distribution on strings of length $n$ is $\ge c$.
Similarly if $\beta$ is the answer to the 2nd question then $0\le\beta\le 1$, since $$(\exists c)(\forall w)(K(w)\le |w|+2\log |w| + c),$$ but I don't know exactly what $\beta$ is.
• OK the first one is easy; in other words, minimum $K$ goes to infinity so $E(K)$ goes to infinity. – Andrew Nov 1 '16 at 16:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9719306230545044, "perplexity": 309.3408835335438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573415.58/warc/CC-MAIN-20190919015534-20190919041534-00501.warc.gz"} |
http://mathhelpforum.com/pre-calculus/147002-imaginary-numbers-trig.html | # Thread: Imaginary numbers with trig
1. ## Imaginary numbers with trig
Hi, I wasn't too sure what category to put this in, so i'm sorry if pre-calculus is wrong.
I came across this question but can't seem to prove it:
If $z=Cos\theta + \iota Sin\theta$, show that:
(i) $|1+z|=2Cos\frac{\theta}{2}$, for $0^{\circ} \leq \theta \leq 180^{\circ}$
(ii) $\frac{2}{1+z}=1-\iota tan\frac{\theta}{2}$, $z\neq -1$
My workings:
(i)
$|1+z|$
$Cos0+\iota Sin0+Cos\theta +\iota Sin\theta$
$2Cos\frac{\theta}{2}Cos\frac{-\theta}{2}+2\iota Sin\frac{\theta}{2}Cos\frac{\theta}{2}$
$Cos^2\frac{\theta}{2}+\iota Sin\frac{\theta}{2}Cos\frac{\theta}{2}=Cos\frac{\t heta}{2}$
After this I couldn't find a way to prove it and I just kept going round in circles.
(ii)
$\frac{2}{1+z}$
$\frac{1}{Cos\frac{\theta}{2}} = 1-\iota tan\frac{\theta}{2}$ ... from part one
$Sec^2\frac{\theta}{2} = 1-tan^2\frac{\theta}{2}-2\iota tan\frac{\theta}{2}$
$Sec^2A = 1-(Sec^2A-1)-2\iota tanA$
$Sec^2A = 1-\iota tanA$
$tan^2A = -\iota tanA$
$tanA = -\iota$
Which I don't think's possible.
Any help here would be appreciated.
Thanks.
2. Originally Posted by FreeT
Hi, I wasn't too sure what category to put this in, so i'm sorry if pre-calculus is wrong.
I came across this question but can't seem to prove it:
If $z=Cos\theta + \iota Sin\theta$, show that:
(i) $|1+z|=2Cos\frac{\theta}{2}$, for $0^{\circ} \leq \theta \leq 180^{\circ}$
(ii) $\frac{2}{1+z}=1-\iota tan\frac{\theta}{2}$, $z\neq -1$
My workings:
(i)
$|1+z|$ Don't put the absolute value signs around the 1+z. At this point, you are finding an expression for 1+z, not |1+z|.
$Cos0+\iota Sin0+Cos\theta +\iota Sin\theta$
$2Cos\frac{\theta}{2}Cos\frac{-\theta}{2}+2\iota Sin\frac{\theta}{2}Cos\frac{\theta}{2}$
That's correct up to that point. Also, $\cos(-x) = \cos x$, so you can write that last line as $1+z = 2\cos^2\tfrac\theta2 + 2i\sin\tfrac\theta2\cos\tfrac\theta2 = 2\cos\tfrac\theta2(\cos\tfrac\theta2 + i\sin\tfrac\theta2)$.
Now remember that the absolute value of a complex number is given by $|x+iy| = \sqrt{x^2+y^2}$. Therefore $|1+z| = \sqrt{(2\cos\tfrac\theta2)^2(\cos^2\tfrac\theta2 + \sin^2\tfrac\theta2)}
$
.
3. Thanks. I got it out.
4. Originally Posted by FreeT
(ii) $\frac{2}{1+z}=1-\iota tan\frac{\theta}{2}$, $z\neq -1$
(ii)
$\frac{2}{1+z}$
$\frac{1}{Cos\frac{\theta}{2}} = 1-\iota tan\frac{\theta}{2}$ ... from part one
$Sec^2\frac{\theta}{2} = 1-tan^2\frac{\theta}{2}-2\iota tan\frac{\theta}{2}$
$Sec^2A = 1-(Sec^2A-1)-2\iota tanA$
$Sec^2A = 1-\iota tanA$
$tan^2A = -\iota tanA$
$tanA = -\iota$
Which I don't think's possible.
Any help here would be appreciated.
Thanks.
$\frac{2}{1 + z} = \frac{2}{1 + \cos{\theta} + i\sin{\theta}}$
$= \frac{2(1 + \cos{\theta} - i\sin{\theta})}{(1 + \cos{\theta} + i\sin{\theta})(1 + \cos{\theta} - i\sin{\theta})}$
$= \frac{2(1 + \cos{\theta} - i\sin{\theta})}{(1 + \cos{\theta})^2 + \sin^2{\theta}}$
$= \frac{2 + 2\cos{\theta} - 2i\sin{\theta}}{1 + 2\cos{\theta} + \cos^2{\theta} + \sin^2{\theta}}$
$= \frac{2 + 2\cos{\theta} - 2i\sin{\theta}}{1 + 2\cos{\theta} + 1}$
$= \frac{2 + 2\cos{\theta} - 2i\sin{\theta}}{2 + 2\cos{\theta}}$
$= 1 - \frac{2i\sin{\theta}}{2 + 2\cos{\theta}}$
$= 1 - \frac{i\sin{\theta}}{1 + \cos{\theta}}$
$= 1 - i\tan{\frac{\theta}{2}}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 46, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9711135625839233, "perplexity": 650.0406686398701}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1490218189495.77/warc/CC-MAIN-20170322212949-00562-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/78805/the-covering-lemma-for-lu/79026 | MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
## The Covering Lemma for L[U]
Hi,
The covering lemma for L[U] (minimal k-model) is stated in Mitchel's handbook chapter:
"Assuming zero-dagger does not exist then covering holds for L[U] or there is a sequence C ⊆ κ, which is Prikry generic over L[U], such that for all sets x of ordinals there is a set y ∈ L[U,C] such that y ⊇ x and |y| = |x| + ℵ1."
And so a Prikry sequence is the only counter example to covering. Kanamori (19.18) proves a theorem of Solovay saying that the critical points of an iteration define a Prikry sequence. So in the core model L[U] (under zero-dagger) we must have prikry sequences.
What are the main differences between the proof of the (more familiar) covering lemma for L and the theorem above for L[U]? I assume that since the mice of L[U] are not simple $L_\alpha$'s we get indiscernibles when trying to cover a set X in a collapsed model. But is the proof the same as before "modulu" Prikry sequences, or is it more complicated than that and more cases should be handled regarding these sequences?
-
I don't know enough about these proofs to really answer the questions, but the following seems relevant to Question b. One of the key facts about constructibility is that, if you have two transitive models of an appropriate finite part of ZFC plus V=L, then one of these is an initial segment of the other (because both are of the form $L_\alpha$). The corresponding fact for transitive models of $(\exists U)\ V=L[U]$ (where the notation $U$ means a normal ultrafilter on a measurable cardinal) is that, given any two such models, you can form iterated ultrapowers of both to arrange that the measurable cardinal is the same in both, and then one of the two iterated ultrapowers will be an initial segment of the other. The need to take iterated ultrapowers in such comparison arguments is a major difference between working with L and working with inner models and core models for large cardinals. The larger the cardinals, the more complicated the iterations get, including non-linearly ordered "tree iterations" once you get to Woodin cardinals. (I urge the people here who really know this part of set theory to do whatever editing is needed to make what I wrote here true.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9513901472091675, "perplexity": 592.1645631596805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368709135115/warc/CC-MAIN-20130516125855-00057-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://asmedigitalcollection.asme.org/heattransfer/article-abstract/105/4/895/414373/Study-on-Properties-and-Growth-Rate-of-Frost?redirectedFrom=fulltext | An experimental study was carried out on the properties and growth rate of the frost layer which developed on a cooled vertical plate in free convective flow. Dimensionless parameters introduced by dimensional analysis were found to be effective in predicting frost densities, its thermal conductivities and growth rates. It was also found that the frost formation process can be divided into two periods if the frost growth data are correlated with the dimensionless parameters presented.
This content is only available via PDF.
You do not currently have access to this content. | {"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": 0.8275166153907776, "perplexity": 675.5223692837966}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987828425.99/warc/CC-MAIN-20191023015841-20191023043341-00361.warc.gz"} |
https://www.physicsforums.com/threads/restate-the-question.831323/ | # Restate the question ...
1. Sep 7, 2015
### Antonius
1. The problem statement, all variables and given/known data
2. Relevant equations
E = kq1q2/r2
3. The attempt at a solution
Last line -- When it says find the charge, does it mean find the net charge or...? Can you please restate the question? Thanks
Last edited by a moderator: Sep 7, 2015
2. Sep 7, 2015
### ehild
The question is: " Find the charge on the outer surface of sphere 2. "
Have something to add?
Draft saved Draft deleted
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https://www.physicsforums.com/threads/existence-of-a-two-variable-function-limit.639920/ | # Existence of a two variable function limit
• Start date
• #1
Jalo
120
0
## Homework Statement
In this exercice I'm asked to find out if the limif of the function f(x,y) exists.
lim (x,y)→(0,0) sqrt(x*y) / (x^2 - y^2)
## The Attempt at a Solution
I've tried to approach it from different coordinates (y=x, y=0, x=0, y=sqrt(x),...) but I didn't found any limit...
I also tried to prove it didn't exist with the iteration property without luck...
Any hint would be highly appreciated.
Thanks.
• #2
Homework Helper
Gold Member
9,568
774
## Homework Statement
In this exercice I'm asked to find out if the limif of the function f(x,y) exists.
lim (x,y)→(0,0) sqrt(x*y) / (x^2 - y^2)
## The Attempt at a Solution
I've tried to approach it from different coordinates (y=x, y=0, x=0, y=sqrt(x),...) but I didn't found any limit...
I also tried to prove it didn't exist with the iteration property without luck...
Any hint would be highly appreciated.
Thanks.
Try along lines like ##y=mx## for various ##m##.
• #3
Staff Emeritus
Homework Helper
Gold Member
11,647
1,223
## Homework Statement
In this exercise I'm asked to find out if the limit of the function f(x,y) exists.
lim (x,y)→(0,0) sqrt(x*y) / (x^2 - y^2)
## The Attempt at a Solution
I've tried to approach it from different coordinates (y=x, y=0, x=0, y=sqrt(x),...) but I didn't found any limit...
I also tried to prove it didn't exist with the iteration property without luck...
Any hint would be highly appreciated.
Thanks.
Along x=0 and y=0, the limit definitely exists.
Along the line y=x, what is true of the denominator, x-y2 ?
See what happens along the curve, x=y3.
• #4
Jalo
120
0
Along x=0 and y=0, the limit definitely exists.
Along the line y=x, what is true of the denominator, x-y2 ?
See what happens along the curve, x=y3.
If I substitute y for x I'll get
sqrt(y*y) / (y² - y²) = y / (y² - y²) = 1 / (y-y)
Which is 1/0, an indetermination. I don't get a physical value therefore I can't conclude nothing from this limit.. ( I think? correct me if I'm wrong)
As to x=y³
sqrt(y³*y) / ( (y³)² - y²) = y² / (y⁶ - y²) = 1 / (y⁴ - 1) = -1
I decided to compare it to
x=4*y³
sqrt(4*y³*y) / ( (4*y³)² - y²) = 2*y² / (16*y⁶ - y²) = 2 / (16*y⁴ - 1) = -2
I can now conclude that the limit doesn't exist.
Could someone just comment on the result of the x=y substitution? Have I done it incorrectly?
Thank you very much.
D.
EDIT: Also, can I substitute x for y+n, where n is some random number belonging |R² ?
Last edited:
• #5
Homework Helper
43,021
970
If I substitute y for x I'll get
sqrt(y*y) / (y² - y²) = y / (y² - y²) = 1 / (y-y)
Which is 1/0, an indetermination. I don't get a physical value therefore I can't conclude nothing from this limit.. ( I think? correct me if I'm wrong)
No, "1/0" is NOT an "indetermination" (better English would be "indeterminate"). "0/0" is an indeterminate. Getting "1/0" is itself enough to conclude that the limit does not exist.
As to x=y³
sqrt(y³*y) / ( (y³)² - y²) = y² / (y⁶ - y²) = 1 / (y⁴ - 1) = -1
I decided to compare it to
x=4*y³
sqrt(4*y³*y) / ( (4*y³)² - y²) = 2*y² / (16*y⁶ - y²) = 2 / (16*y⁴ - 1) = -2
I can now conclude that the limit doesn't exist.
Could someone just comment on the result of the x=y substitution? Have I done it incorrectly?
Thank you very much.
D.
EDIT: Also, can I substitute x for y+n, where n is some random number belonging |R² ?
• #6
Staff Emeritus
Homework Helper
Gold Member
11,647
1,223
If I substitute y for x I'll get
sqrt(y*y) / (y² - y²) = y / (y² - y²) = 1 / (y-y)
Which is 1/0, an indetermination. I don't get a physical value therefore I can't conclude nothing from this limit.. ( I think? correct me if I'm wrong)
You're right in that you have division by zero. You have |y|/0 all along the line y=x. But, that's not an indeterminate form. When taking the limit as y→0 you get a result of +∞.
As to x=y³
sqrt(y³*y) / ( (y³)² - y²) = y² / (y⁶ - y²) = 1 / (y⁴ - 1) = -1
So, there you have it. The limit is +∞ along one path, and -1 along another.
I decided to compare it to
x=4*y³
sqrt(4*y³*y) / ( (4*y³)² - y²) = 2*y² / (16*y⁶ - y²) = 2 / (16*y⁴ - 1) = -2
I can now conclude that the limit doesn't exist.
Could someone just comment on the result of the x=y substitution? Have I done it incorrectly?
Thank you very much.
D.
EDIT: Also, can I substitute x for y+n, where n is some random number belonging |R² ?
For x = y+n, for n a non-zero number, that does not pass through the origin.
If you literally mean x=y+(a, b), that's not any defined operation as far as I know.
You originally implied that the limit doesn't exist along x=0 and y=0. Those limits do exist, and they are zero. So, your original paths, along y=x, and y=0 or x=0, are enoough to show that the limit does not exist, but using a path such as x = ay3 or y = ax3 does give nice results.
• #7
Homework Helper
Gold Member
9,568
774
Could someone just comment on the result of the x=y substitution? Have I done it incorrectly?
Others have commented about your calculation of the x=y substitution. But I would comment about its appropriateness. None of the points on y=x are in the domain of the function so it doesn't make sense to even consider that path when studying the limit in question. But, as you have seen, other paths will do.
• #8
Homework Helper
Gold Member
9,568
774
So, there you have it. The limit is +∞ along one path,
I would disagree with that. See my post #7.
• #9
Staff Emeritus
Homework Helper
Gold Member
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I would disagree with that. See my post #7.
Yes, of course you are correct !
Unfortunately, I had a typo in post #3, but that should have said that along the line y=x, what does the denominator x2 - y2 become? Of course, the answer is that the denominator is zero, so that the overall expression is undefined along y = x.
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407 | {"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": 0.842162549495697, "perplexity": 2304.7002394929013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1669446710690.85/warc/CC-MAIN-20221129064123-20221129094123-00069.warc.gz"} |
http://www.cfd-online.com/W/index.php?title=Hydrodynamic/acoustic_splitting&diff=prev&oldid=9197 | # Hydrodynamic/acoustic splitting
(Difference between revisions)
Revision as of 21:30, 30 July 2008 (view source)Jhseo (Talk | contribs)← Older edit Revision as of 21:35, 30 July 2008 (view source)Jhseo (Talk | contribs) Newer edit → Line 21: Line 21: $\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec [itex]\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)$ u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)[/itex] + + where $D/Dt = \partial /\partial t + (\vec U \cdot \nabla )$, $\vec f'_{vis}$ is the perturbed viscous force + vector, $\Phi$ and $\vec q$ represent thermal viscous dissipation and heat flux vector, respectively. + The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic + fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called + 'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of + hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, + in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited.
## Revision as of 21:35, 30 July 2008
In the hydrodynamic/acoustic splitting method, the total flow variables are decomposed into the incompressible variables and the perturbed compressible ones as,
$\rho(\vec{x},t)=\rho_0+\rho'(\vec{x},t)$
$\vec{u}(\vec{x},t)=\vec{U}(\vec{x},t)+\vec{u'}(\vec{x},t)$
$p(\vec{x},t)=P(\vec{x},t)+p'(\vec{x},t)$
The incompressible variables represent the hydrodynamic flow field, while the acoustic fluctuations and other compressibility effects are resolved by the perturbed quantities denoted by ('). The original full perturbed compressible equations (PCE) to calculate the perturbed quantities can be derived by subtracting the incompressible Navier-Stokes equations from the compressible ones. The PCE recently revised by Seo and Moon are written in a vector form as,
$\frac{\partial\rho'}{\partial t}+(\vec{u}\cdot\nabla)\rho'+\rho(\nabla\cdot\vec{u'})=0$
$\frac{\partial\vec{u'}}{\partial t}+(\vec{u}\cdot\nabla)\vec{u'}+(\vec{u'}\cdot\nabla)\vec{U}+\frac{1}{\rho}\nabla p'+\frac{\rho'}{\rho}\frac{D\vec{U}}{Dt}=\frac{1}{\rho}\vec{f'_{vis}}$
$\frac{{\partial p'}}{{\partial t}}\!+\!(\vec u \cdot \nabla )p'\!+\!\gamma p(\nabla \cdot \vec u') + (\vec u' \cdot \nabla )P = - \frac{{DP}}{{Dt}} + (\gamma - 1)\left( {\Phi - \nabla \cdot \vec q} \right)$
where $D/Dt = \partial /\partial t + (\vec U \cdot \nabla )$, $\vec f'_{vis}$ is the perturbed viscous force vector, $\Phi$ and $\vec q$ represent thermal viscous dissipation and heat flux vector, respectively. The PCE are the mixed-scales, non-linear equations, in which terms are involved with coupling effects between the acoustic fluctuations and the incompressible flow field. Through the coupling effects, a non-radiating vortical component, so-called 'perturbed vorticity', is generated in the perturbed system. The perturbed vorticity may be regarded as the modification of hydrodynamic vorticity due to acoustic fluctuations, but it just reside in the perturbed system by this 'one-way' splitting method, in which backscattering of acoustic fluctuations onto the incompressible flow field is prohibited. | {"extraction_info": {"found_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": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.939153790473938, "perplexity": 2425.3421989628464}, "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-2015-32/segments/1438042982745.46/warc/CC-MAIN-20150728002302-00016-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://ablconnect.harvard.edu/book/students-explaining-math-problems | # Students Explaining Math Problems
Alexander Isakov, a graduate student in physics, has his students explain in class how to do crucial steps of math problems for Math 21a.
In order to do the activity, students need only to have looked at the homework and know the lecture material. During section, the instructor starts working out a problem, and then feigns ignorance about how to do crucial steps. He makes sure that multiple people have the chance to give him some ideas and explain why they think their idea is better. Then, he proceeds a bit more with the calculations/proof (as the case may be), and calls on other students to fill in details.
Alexander finds that this activity works best in smaller sections; if you have a larger section, make sure that everyone is involved. Even if someone clearly does not know what to say, give them a small hint or ask them to take a guess. Don't rush it. | {"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": 0.8435516953468323, "perplexity": 792.750467951859}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171163.39/warc/CC-MAIN-20170219104611-00061-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://worldwidescience.org/topicpages/s/super+yang-mills+theories.html | #### Sample records for super yang-mills theories
1. Higher derivative super Yang-Mills theories
International Nuclear Information System (INIS)
Bergshoeff, E.; Rakowski, M.; Sezgin, E.
1986-11-01
The most general higher derivative Yang-Mills actions of the type (F 2 +α2F 4 ) which are globally supersymmetric up to order α 2 in six and ten dimensional spacetimes are given. The F 4 -terms turn out to occur in the combination α 2 (tr F 4 - 1/4(tr F 2 ) 2 ), where the trace is over the Lorentz indices. This result agrees with the low energy limit of the open superstring in ten dimensions, where α is the string tension. Surprisingly, the transformation rules of the Yang-Mills multiplet receive order α 2 corrections even in the off-shell formulation. For the case of Abelian Yang-Mills group, the action is expressed in Born-Infeld form with a metric generically given by (1+α 2 F 2 +...). (author)
2. Two-dimensional N = 2 Super-Yang-Mills Theory
Science.gov (United States)
August, Daniel; Wellegehausen, Björn; Wipf, Andreas
2018-03-01
Supersymmetry is one of the possible scenarios for physics beyond the standard model. The building blocks of this scenario are supersymmetric gauge theories. In our work we study the N = 1 Super-Yang-Mills (SYM) theory with gauge group SU(2) dimensionally reduced to two-dimensional N = 2 SYM theory. In our lattice formulation we break supersymmetry and chiral symmetry explicitly while preserving R symmetry. By fine tuning the bar-mass of the fermions in the Lagrangian we construct a supersymmetric continuum theory. To this aim we carefully investigate mass spectra and Ward identities, which both show a clear signal of supersymmetry restoration in the continuum limit.
3. Non commutative geometry and super Yang-Mills theory
International Nuclear Information System (INIS)
Bigatti, D.
1999-01-01
We aim to connect the non commutative geometry 'quotient space' viewpoint with the standard super Yang Mills theory approach in the spirit of Connes-Douglas-Schwartz and Douglas-Hull description of application of noncommutative geometry to matrix theory. This will result in a relation between the parameters of a rational foliation of the torus and the dimension of the group U(N). Namely, we will be provided with a prescription which allows to study a noncommutative geometry with rational parameter p/N by means of a U(N) gauge theory on a torus of size Σ/N with the boundary conditions given by a system with p units of magnetic flux. The transition to irrational parameter can be obtained by letting N and p tend to infinity with fixed ratio. The precise meaning of the limiting process will presumably allow better clarification. (Copyright (c) 1999 Elsevier Science B.V., Amsterdam. All rights reserved.)
4. One-dimensional structures behind twisted and untwisted superYang-Mills theory
CERN Document Server
Baulieu, Laurent
2011-01-01
We give a one-dimensional interpretation of the four-dimensional twisted N=1 superYang-Mills theory on a Kaehler manifold by performing an appropriate dimensional reduction. We prove the existence of a 6-generator superalgebra, which does not possess any invariant Lagrangian but contains two different subalgebras that determine the twisted and untwisted formulations of the N=1 superYang-Mills theory.
5. One-dimensional structures behind twisted and untwisted super Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Baulieu, Laurent [CERN, Geneve (Switzerland). Theoretical Div.; Toppan, Francesco, E-mail: [email protected], E-mail: [email protected] [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)
2010-07-01
We give a one-dimensional interpretation of the four-dimensional twisted N = 1 super Yang-Mills theory on a Kaehler manifold by performing an appropriate dimensional reduction. We prove the existence of a 6-generator superalgebra, which does not possess any invariant Lagrangian but contains two different subalgebras that determine the twisted and untwisted formulations of the N = 1 super Yang-Mills theory. (author)
6. One-dimensional structures behind twisted and untwisted super Yang-Mills theory
International Nuclear Information System (INIS)
Baulieu, Laurent
2010-01-01
We give a one-dimensional interpretation of the four-dimensional twisted N = 1 super Yang-Mills theory on a Kaehler manifold by performing an appropriate dimensional reduction. We prove the existence of a 6-generator superalgebra, which does not possess any invariant Lagrangian but contains two different subalgebras that determine the twisted and untwisted formulations of the N = 1 super Yang-Mills theory. (author)
7. Boundary effects in super-Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Shah, Mushtaq B.; Ganai, Prince A. [National Institute of Technology, Department of Physics, Srinagar, Kashmir (India); Faizal, Mir [University of British Columbia-Okanagan, Irving K. Barber School of Arts and Sciences, Kelowna, BC (Canada); University of Lethbridge, Department of Physics and Astronomy, Alberta (Canada); Zaz, Zaid [University of Kashmir, Department of Electronics and Communication Engineering, Srinagar, Kashmir (India); Bhat, Anha [National Institute of Technology, Department of Metallurgical and Materials Engineering, Srinagar, Kashmir (India); Masood, Syed [International Islamic University, Department of Physics, Islamabad (Pakistan)
2017-05-15
In this paper, we shall analyze a three dimensional supersymmetry theory with N = 2 supersymmetry. We will analyze the quantization of this theory, in the presence of a boundary. The effective Lagrangian used in the path integral quantization of this theory, will be given by the sum of the gauge fixing term and the ghost term with the original classical Lagrangian. Even though the supersymmetry of this effective Lagrangian will also be broken due to the presence of a boundary, it will be demonstrated that half of the supersymmetry of this theory can be preserved by adding a boundary Lagrangian to the effective bulk Lagrangian. The supersymmetric transformation of this new boundary Lagrangian will exactly cancel the boundary term generated from the supersymmetric transformation of the effective bulk Lagrangian. We will analyze the Slavnov-Taylor identity for this N = 2 Yang-Mills theory with a boundary. (orig.)
8. Boundary effects in super-Yang-Mills theory
International Nuclear Information System (INIS)
Shah, Mushtaq B.; Ganai, Prince A.; Faizal, Mir; Zaz, Zaid; Bhat, Anha; Masood, Syed
2017-01-01
In this paper, we shall analyze a three dimensional supersymmetry theory with N = 2 supersymmetry. We will analyze the quantization of this theory, in the presence of a boundary. The effective Lagrangian used in the path integral quantization of this theory, will be given by the sum of the gauge fixing term and the ghost term with the original classical Lagrangian. Even though the supersymmetry of this effective Lagrangian will also be broken due to the presence of a boundary, it will be demonstrated that half of the supersymmetry of this theory can be preserved by adding a boundary Lagrangian to the effective bulk Lagrangian. The supersymmetric transformation of this new boundary Lagrangian will exactly cancel the boundary term generated from the supersymmetric transformation of the effective bulk Lagrangian. We will analyze the Slavnov-Taylor identity for this N = 2 Yang-Mills theory with a boundary. (orig.)
9. Super Yang-Mills theory with impurity walls and instanton moduli spaces
Science.gov (United States)
Cherkis, Sergey A.; O'Hara, Clare; Sämann, Christian
2011-06-01
We explore maximally supersymmetric Yang-Mills theory with walls of impurities respecting half of the supersymmetries. The walls carry fundamental or bifundamental matter multiplets. We employ three-dimensional N=2 superspace language to identify the Higgs branch of this theory. We find that the vacuum conditions determining the Higgs branch are exactly the bow equations yielding Yang-Mills instantons on a multi-Taub-NUT space. Under electric-magnetic duality, the super Yang-Mills theory describing the bulk is mapped to itself, while the fundamental- and bifundamental-carrying impurity walls are interchanged. We perform a one-loop computation on the Coulomb branch of the dual theory to find the asymptotic metric on the original Higgs branch.
10. Supersymmetric renormalization prescription in N=4 super-Yang-Mills theory
International Nuclear Information System (INIS)
Baulieu, Laurent; Bossard, Guillaume
2006-01-01
Using the shadow dependent decoupled Slavnov-Taylor identities associated to gauge invariance and supersymmetry, we discuss the renormalization of the N=4 super-Yang-Mills theory and of its coupling to gauge-invariant operators. We specify the method for the determination of non-supersymmetric counterterms that are needed to maintain supersymmetry
11. Wess-Zumino and super Yang-Mills theories in D=4 integral superspace
Science.gov (United States)
Castellani, L.; Catenacci, R.; Grassi, P. A.
2018-05-01
We reconstruct the action of N = 1 , D = 4 Wess-Zumino and N = 1 , 2 , D = 4 super-Yang-Mills theories, using integral top forms on the supermanifold M^{(.4|4)} . Choosing different Picture Changing Operators, we show the equivalence of their rheonomic and superspace actions. The corresponding supergeometry and integration theory are discussed in detail. This formalism is an efficient tool for building supersymmetric models in a geometrical framework.
12. An off-shell superspace reformulation of D = 4, N = 4 super-Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Cederwall, Martin [Division for Theoretical Physics, Department of Physics, Chalmers University of Technology, Gothenburg (Sweden)
2018-01-15
D = 4, N = 4 super-Yang-Mills theory has an off-shell superspace formulation in terms of pure spinor superfields, which is directly inherited from the D = 10 theory. That superspace, in particular the choice of pure spinor variables, is less suitable for dealing with fields that are inherently 4-dimensional, such as the superfields based on the scalars, which are gauge-covariant, and traces of powers of scalars, which are gauge-invariant. We give a reformulation of D = 4, N = 4 super-Yang-Mills theory in N = 4 superspace, using inherently 4-dimensional pure spinors. All local degrees of freedom reside in a superfield based on the physical scalars. The formalism should be suited for calculations of correlators of traces of scalar superfields. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
13. Simulations of N = 2 super Yang-Mills theory in two dimensions
International Nuclear Information System (INIS)
Catterall, Simon
2006-01-01
We present results from lattice simulations of N = 2 super Yang-Mills theory in two dimensions. The lattice formulation we use was developed and retains both gauge invariance and an exact (twisted) supersymmetry for any lattice spacing. Results for both U(2) and SU(2) gauge groups are given. We focus on supersymmetric Ward identities, the phase of the Pfaffian resulting from integration over the Grassmann fields and the nature of the quantum moduli space
14. Ward identities and mass spectrum of N=1 Super Yang-Mills theory on the lattice
International Nuclear Information System (INIS)
Kirchner, R.
2000-11-01
We study the lattice regularization of N=1 Super Yang-Mills theory. Projecting operators for the low-lying spectrum are discussed. We also consider a ''baryonic'' state consisting of three gluinos, and develop a numerical strategy to determine its mass in a Monte Carlo simulation. We present numerical results on the low-lying spectrum of SU(2) N=1 Super Yang-Mills theory with light dynamical gluinos. The lattice regularization of N=1 Super Yang-Mills theory breaks supersymmetry at any finite lattice spacing. We derive the form of the corresponding SUSY Ward identity and carry out renormalization. The ratios of the renormalization coefficients Z T /Z S and M R /Z S are determined non-perturbatively in a numerical simulation. The form of the renormalized SUSY Ward identity is confirmed numerically. We discuss how the SUSY Ward identity can be used to define a supersymmetric continuum limit, and how its approach can be monitored in numerical simulations. (orig.)
15. The three-loop form factor in N=4 super Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Gehrmann, Thomas [Universitaet Zuerich (Switzerland); Henn, Johannes [IAS Princeton (United States); Huber, Tobias [Universitaet Siegen (Germany)
2012-07-01
We present the calculation of the Sudakov form factor in N=4 super Yang-Mills theory to the three-loop order. At leading colour, the latter is expressed in terms of planar and non-planar loop integrals. We show that it is possible to choose a representation in which each loop integral has uniform transcendentality in the Riemann {zeta}-function. We comment on the expected exponentiation of the infrared divergences and the values of the three-loop cusp and collinear anomalous dimensions in dimensional regularisation. We also compare the form factor in N=4 super Yang-Mills to the leading transcendentality pieces of the quark and gluon form factor in QCD. Finally, we investigate the ultraviolet properties of the form factor in D>4 dimensions.
16. Finiteness preserving mass terms in N=4 super Yang-Mills theory
International Nuclear Information System (INIS)
Rajpoot, S.; Taylor, J.G.; Zaimi, M.
1983-01-01
It is shown using light cone gauge techniques that N = 4 super Yang-Mills theory is ultraviolet finite in the presence of a wide range of explicit symmetry breaking mass terms for (a) scalars and fermions (b) scalars alone. These mass terms satisfy sum rules that are part of the more general sum rule: μsub(s=0,) sub(1/2) (-1)sup(2S+1)(2s + 1)msub(S) 2 = 0, in which the mass of vector bosons is set to zero for reasons of gauge invariance. The resulting lagrangians offer the exciting possibility of realising explicit hierarchical descent of N = 4 super Yang-Mills through N = 2 and N = 1 supersymmetries. Tree level spontaneous symmetry breaking from the resulting scalar potentials are briefly discussed. (orig.)
17. Quantum metamorphosis of conformal symmetry in N=4 super Yang-Mills theory
International Nuclear Information System (INIS)
Kuzenko, S.M.; McArthur, I.N.
2002-01-01
In gauge theories, not all rigid symmetries of the classical action can be maintained manifestly in the quantization procedure, even in the absence of anomalies. If this occurs for an anomaly-free symmetry, the effective action is invariant under a transformation that differs from its classical counterpart by quantum corrections. As shown by Fradkin and Palchik years ago, such a phenomenon occurs for conformal symmetry in quantum Yang-Mills theories with vanishing beta function, such as the N=4 super Yang-Mills theory. More recently, Jevicki et al. demonstrated that the quantum metamorphosis of conformal symmetry sheds light on the nature of the AdS/CFT correspondence. In this paper, we derive the conformal Ward identity for the bosonic sector of the N=4 super Yang-Mills theory using the background field method. We then compute the leading quantum modification of the conformal transformation for a specific Abelian background which is of interest in the context of the AdS/CFT correspondence. In the case of scalar fields, our final result agrees with that of Jevicki et al. The resulting vector and scalar transformations coincide with those which are characteristic of a D3-brane embedded in AdS 5 xS 5 . (author)
18. Supersymmetric Adler-Bardeen anomaly in N=1 super-Yang-Mills theories
International Nuclear Information System (INIS)
Baulieu, Laurent; Martin, Alexis
2008-01-01
We provide a study of the supersymmetric Adler-Bardeen anomaly in the N=1, d=4,6,10 super-Yang-Mills theories. We work in the component formalism that includes shadow fields, for which Slavnov-Taylor identities can be independently set for both gauge invariance and supersymmetry. We find a method with improved descent equations for getting the solutions of the consistency conditions of both Slavnov-Taylor identities and finding the local field polynomials for the standard Adler-Bardeen anomaly and its supersymmetric counterpart. We give the explicit solution for the ten-dimensional case
19. Counting domain walls in N=1 super Yang-Mills theory
International Nuclear Information System (INIS)
2002-01-01
We study the multiplicity of BPS domain walls in N=1 super Yang-Mills theory, by passing to a weakly coupled Higgs phase through the addition of fundamental matter. The number of domain walls connecting two specified vacuum states is then determined via the Witten index of the induced world volume theory, which is invariant under the deformation to the Higgs phase. The world volume theory is a sigma model with a Grassmanian target space which arises as the coset associated with the global symmetries broken by the wall solution. Imposing a suitable infrared regulator, the result is found to agree with recent work of Acharya and Vafa in which the walls were realized as wrapped D4-branes in type IIA string theory
20. N=4 Super Yang-Mills: the harmonic oscillator of interacting quantum field theories
International Nuclear Information System (INIS)
Minahan, Joseph A.
2012-01-01
Full text: In this talk I discuss progress over the last ten years in solving N=4 Super Yang-Mills in the planar limit, where the number of colors is taken to infinity. The key to the solution is mapping the theory to an integrable one-dimensional spin chain. At the leading perturbative level the spin-chain in question is the Heisenberg chain which was solved by Bethe in 1931. We discuss how the analysis of spin-chains ultimately allows to compute the spectrum of observables in the theory for any value of the coupling constant. I then discuss ongoing work to find the so-called three-point functions, which when combined with the spectrum would completely solve the theory in the planar limit. (author)
1. Spectral parameters for scattering amplitudes in N=4 super Yang-Mills theory
International Nuclear Information System (INIS)
Ferro, Livia; Łukowski, Tomasz; Meneghelli, Carlo; Plefka, Jan; Staudacher, Matthias
2014-01-01
Planar N=4 Super Yang-Mills theory appears to be a quantum integrable four-dimensional conformal theory. This has been used to find equations believed to describe its exact spectrum of anomalous dimensions. Integrability seemingly also extends to the planar space-time scattering amplitudes of the N=4 model, which show strong signs of Yangian invariance. However, in contradistinction to the spectral problem, this has not yet led to equations determining the exact amplitudes. We propose that the missing element is the spectral parameter, ubiquitous in integrable models. We show that it may indeed be included into recent on-shell approaches to scattering amplitude integrands, providing a natural deformation of the latter. Under some constraints, Yangian symmetry is preserved. Finally we speculate that the spectral parameter might also be the regulator of choice for controlling the infrared divergences appearing when integrating the integrands in exactly four dimensions
2. N=4 super-Yang-Mills in LHC superspace part I: classical and quantum theory
Energy Technology Data Exchange (ETDEWEB)
Chicherin, Dmitry [LAPTH, Université de Savoie,CNRS, B.P. 110, F-74941 Annecy-le-Vieux (France); Sokatchev, Emery [LAPTH, Université de Savoie,CNRS, B.P. 110, F-74941 Annecy-le-Vieux (France); Theoretical Physics Department, CERN,CH -1211, Geneva 23 (Switzerland)
2017-02-10
We present a formulation of the maximally supersymmetric N=4 gauge theory in Lorentz harmonic chiral (LHC) superspace. It is closely related to the twistor formulation of the theory but employs the simpler notion of Lorentz harmonic variables. They parametrize a two-sphere and allow us to handle efficiently infinite towers of higher-spin auxiliary fields defined on ordinary space-time. In this approach the chiral half of N=4 supersymmetry is manifest. The other half is realized non-linearly and the algebra closes on shell. We give a straightforward derivation of the Feynman rules in coordinate space. We show that the LHC formulation of the N=4 super-Yang-Mills theory is remarkably similar to the harmonic superspace formulation of the N=2 gauge and hypermultiplet matter theories. In the twin paper https://arxiv.org/abs/1601.06804 we apply the LHC formalism to the study of the non-chiral multipoint correlation functions of the N=4 stress-tensor supermultiplet.
3. The Analytic Structure of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory
Science.gov (United States)
Litsey, Sean Christopher
We begin the dissertation in Chapter 1 with a discussion of tree-level amplitudes in Yang-. Mills theories. The DDM and BCJ decompositions of the amplitudes are described and. related to one another by the introduction of a transformation matrix. This is related to the. Kleiss-Kuijf and BCJ amplitude identities, and we conjecture a connection to the existence. of a BCJ representation via a condition on the generalized inverse of that matrix. Under. two widely-believed assumptions, this relationship is proved. Switching gears somewhat, we introduce the RSVW formulation of the amplitude, and the extension of BCJ-like features to residues of the RSVW integrand is proposed. Using the previously proven connection of BCJ representations to the generalized inverse condition, this extension is validated, including a version of gravitational double copy. The remainder of the dissertation involves an analysis of the analytic properties of loop. amplitudes in N = 4 super-Yang-Mills theory. Chapter 2 contains a review of the planar case, including an exposition of dual variables and momentum twistors, dual conformal symmetry, and their implications for the amplitude. After defining the integrand and on-shell diagrams, we explain the crucial properties that the amplitude has no poles at infinite momentum and that its leading singularities are dual-conformally-invariant cross ratios, and can therefore be normalized to unity. We define the concept of a dlog form, and show that it is a feature of the planar integrand as well. This leads to the definition of a pure integrand basis. The proceeding setup is connected to the amplituhedron formulation, and we put forward the hypothesis that the amplitude is determined by zero conditions. Chapter 3 contains the primary computations of the dissertation. This chapter treats. amplitudes in fully nonplanar N = 4 super-Yang-Mills, analyzing the conjecture that they. follow the pattern of having no poles at infinity, can be written in dlog
4. Galilean Yang-Mills theory
International Nuclear Information System (INIS)
Bagchi, Arjun; Basu, Rudranil; Kakkar, Ashish; Mehra, Aditya
2016-01-01
We investigate the symmetry structure of the non-relativistic limit of Yang-Mills theories. Generalising previous results in the Galilean limit of electrodynamics, we discover that for Yang-Mills theories there are a variety of limits inside the Galilean regime. We first explicitly work with the SU(2) theory and then generalise to SU(N) for all N, systematising our notation and analysis. We discover that the whole family of limits lead to different sectors of Galilean Yang-Mills theories and the equations of motion in each sector exhibit hitherto undiscovered infinite dimensional symmetries, viz. infinite Galilean Conformal symmetries in D=4. These provide the first examples of interacting Galilean Conformal Field Theories (GCFTs) in D>2.
5. Galilean Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Bagchi, Arjun [Center for Theoretical Physics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139 (United States); Basu, Rudranil [Saha Institute of Nuclear Physics,Block AF, Sector 1, Bidhannagar, Kolkata 700068 (India); Kakkar, Ashish [Indian Institute of Science Education and Research,Dr Homi Bhabha Road, Pashan. Pune 411008 (India); Mehra, Aditya [Indian Institute of Science Education and Research,Dr Homi Bhabha Road, Pashan. Pune 411008 (India); Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen (Netherlands)
2016-04-11
We investigate the symmetry structure of the non-relativistic limit of Yang-Mills theories. Generalising previous results in the Galilean limit of electrodynamics, we discover that for Yang-Mills theories there are a variety of limits inside the Galilean regime. We first explicitly work with the SU(2) theory and then generalise to SU(N) for all N, systematising our notation and analysis. We discover that the whole family of limits lead to different sectors of Galilean Yang-Mills theories and the equations of motion in each sector exhibit hitherto undiscovered infinite dimensional symmetries, viz. infinite Galilean Conformal symmetries in D=4. These provide the first examples of interacting Galilean Conformal Field Theories (GCFTs) in D>2.
6. Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra
International Nuclear Information System (INIS)
Thierry-Mieg, Jean
2006-01-01
In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space
7. BRST cohomology of N = 2 super-Yang-Mills theory in four dimensions
International Nuclear Information System (INIS)
Tanzini, A.; Ventura, O.S.; Vilar, L.C.Q.; Sorella, S.P.
2000-01-01
The BRST cohomology of the N = 2 supersymmetric Yang-Mills theory in four dimensions is discussed by making use of the twisted version of the N = 2 algebra. By the introduction of a set of suitable constant ghosts associated with the generators of N = 2, the quantization of the model can be done by taking into account both gauge invariance and supersymmetry. In particular, we show how the twisted N = 2 algebra can be used to obtain in a straightforward way the relevant cohomology classes. Moreover, we shall be able to establish a very useful relationship between the local gauge-invariant polynomial tr φ 2 and the complete N = 2 Yang-Mills action. This important relation can be considered as the first step towards a fully algebraic proof of the one-loop exactness of the N = 2 β-function.
8. The one-loop partition function of N=4 super-Yang-Mills theory on RxS3
International Nuclear Information System (INIS)
2005-01-01
We study weakly coupled SU(N)N=4 super-Yang-Mills theory on RxS 3 at infinite N, which has interesting thermodynamics, including a Hagedorn transition, even at zero Yang-Mills coupling. We calculate the exact one-loop partition function below the Hagedorn temperature. Our calculation employs the representation of the one-loop dilatation operator as a spin chain Hamiltonian acting on neighboring sites and a generalization of Polya's counting of necklaces (gauge-invariant operators) to include necklaces with a 'pendant' (an operator which acts on neighboring beads). We find that the one-loop correction to the Hagedorn temperature is δlnT H =+λ/8π 2
9. Integrability of N=3 super Yang-Mills equations
International Nuclear Information System (INIS)
Devchand, C.; Ogievetsky, V.
1993-10-01
We describe the harmonic superspace formulation of the Witten-Manin supertwistor correspondence for N=3 extended super Yang-Mills theories. The essence in that on being sufficiently supersymmetrised (up to the N=3 extension), the Yang-Mills equations of motion can be recast in the form of Cauchy-Riemann-like holomorphicity conditions for a pair of prepotentials in the appropriate harmonic superspace. This formulation makes the explicit construction of solutions a rather more tractable proposition than previous attempts. (orig.)
10. High-energy scattering in strongly coupled N=4 super Yang-Mills theory
International Nuclear Information System (INIS)
Sprenger, Martin
2014-11-01
This thesis concerns itself with the analytic structure of scattering amplitudes in strongly coupled N=4 super Yang-Mills theory (abbreviated N = 4 SYM) in the multi-Regge limit. Through the AdS/CFT-correspondence observables in strongly coupled N = 4 SYM are accessible via dual calculations in a weakly coupled string theory on an AdS 5 x S 5 -geometry, in which observables can be calculated using standard perturbation theory. In particular, the calculation of the leading order of the n-gluon amplitude in N = 4 SYM at strong coupling corresponds to the calculation of a minimal surface embedded into AdS 5 . This surface ends on the concatenation of the gluon momenta, which is a light-like curve. The calculation of the minimal surface area can be reduced to finding the solution of a set of non-linear, coupled integral equations, which have no analytic solution in arbitrary kinematics. In this thesis, we therefore specialise to the multi-Regge limit, the n-particle generalisation of the Regge limit. This limit is especially interesting as even in the description of scattering amplitudes in weakly coupled N = 4 SYM in this limit a certain set of Feynman diagrams has to be resummed. This description organises itself into orders of logarithms of the energy involved in the scattering process. In this expansion each order in logarithms includes terms from every order in the coupling constant and therefore contains information about the strong coupling sector of the theory, albeit in a very specific way. This raises the central question of this thesis, which is how much of the analytic structure of the scattering amplitudes in the multi-Regge limit is preserved as we go to the strong coupling regime. We show that the equations governing the area of the minimal surface simplify drastically in the multi-Regge limit, which allows us to obtain analytic results for the scattering amplitudes. We develop an algorithm for the calculation of scattering amplitudes in the multi
11. Collinear and Regge behavior of 2{yields}4 MHV amplitude in N=4 super Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Bartels, J.; Prygarin, A. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Lipatov, L.N. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; St. Petersburg Nuclear Physics Institute (Russian Federation)
2011-04-15
We investigate the collinear and Regge behavior of the 2{yields}4 MHV amplitude in N=4 super Yang-Mills theory in the BFKL approach. The expression for the remainder function in the collinear kinematics proposed by Alday, Gaiotto, Maldacena, Sever and Vieira is analytically continued to the Mandelstam region. The result of the continuation in the Regge kinematics shows an agreement with the BFKL approach up to to five-loop level. We present the Regge theory interpretation of the obtained results and discuss some issues related to a possible nonmultiplicative renormalization of the remainder function in the collinear limit. (orig.)
12. Holographic thermalization in N = 4 super Yang-Mills theory at finite coupling
Energy Technology Data Exchange (ETDEWEB)
Stricker, Stefan A. [Technische Universitaet Wien, Institut fuer Theoretische Physik, Vienna (Austria)
2014-02-15
We investigate the behavior of energy-momentum tensor correlators in holographic N = 4 super Yang-Mills plasma, taking finite coupling corrections into account. In the thermal limit we determine the flow of quasinormal modes as a function of the 't Hooft coupling. Then we use a specific model of holographic thermalization to study the deviation of the spectral densities from their thermal limit in an out-of-equilibrium situation. The main focus lies on the thermalization pattern with which the plasma constituents approach their thermal distribution as the coupling constant decreases from the infinite coupling limit. All obtained results point towards the weakening of the usual top-down thermalization pattern. (orig.)
13. Color superconductivity, ZN flux tubes and monopole confinement in deformed N=2* super Yang-Mills theories
International Nuclear Information System (INIS)
Kneipp, Marco A.C.
2003-11-01
We study the ZN flux tubes and monopole confinement in deformed N=2* super Yang-Mills theories. In order to do that we consider an N=4 super Yang-Mills theory with an arbitrary gauge group G and add some N=2, N=1 and N=0 deformation terms. We analyze some possible vacuum solutions and phases of the theory, depending on the deformation terms which are added. In the Coulomb phase for the N=2* theory, G is broken to U(1)r and the theory has monopole solutions. Then, by adding some deformation terms, the theory passes to the Higgs or color superconducting phase, in which G is broken to its center CG. In this phase we construct the ZN flux tubes Ansatz and obtain the BPS string tension. We show that the monopole magnetic fluxes are linear integer combinations of the string fluxes and therefore the monopoles can become confined. Then, we obtain a bound for the threshold length of the string-breaking. We also show the possible formation of a confining system with 3 different monopoles for the SU(3) gauge group. Finally we show that the BPS string tensions of the theory satisfy the Casimir scaling law. (author)
14. Form factors and the dilatation operator in N= 4 super Yang-Mills theory and its deformations
International Nuclear Information System (INIS)
Wilhelm, Matthias Oliver
2016-01-01
In the first part of this thesis, we study form factors of general gauge-invariant local composite operators in N=4 super Yang-Mills theory at various loop orders and for various numbers of external legs. We show how to use on-shell methods for their calculation and in particular extract the dilatation operator from the result. We also investigate the properties of the corresponding remainder functions. Moreover, we extend on-shell diagrams, a Grassmannian integral formulation and an integrability-based construction via R-operators to form factors, focussing on the chiral part of the stress-tensor supermultiplet as an example. In the second part, we study the β- and the γ i -deformation, which were respectively shown to be the most general supersymmetric and non-supersymmetric field-theory deformations of N=4 super Yang-Mills theory that are integrable at the level of the asymptotic Bethe ansatz. For these theories, a new kind of finite-size effect occurs, which we call prewrapping and which emerges from double-trace structures that are required in the deformed Lagrangians. While the β-deformation is conformal when the double-trace couplings are at their non-trivial IR fixed points, the γ i -deformation has running double-trace couplings without fixed points, which break conformal invariance even in the planar theory. Nevertheless, the γ i -deformation allows for highly non-trivial field-theoretic tests of integrability at arbitrarily high loop orders.
15. Calculating the anomalous supersymmetry breaking in super Yang-Mills theories with local coupling
International Nuclear Information System (INIS)
Kraus, E.
2002-01-01
Supersymmetric Yang-Mills theories with local gauge coupling have a new type of anomalous breaking, which appears as a breaking of supersymmetry in the Wess-Zumino gauge. The anomalous breaking generates the two-loop order of the gauge β function in terms of the one-loop β function and the anomaly coefficient. We determine the anomaly coefficient in the Wess-Zumino gauge by solving the relevant supersymmetry identities. For this purpose we use a background gauge and show that the anomaly coefficient is uniquely determined by convergent one-loop integrals. When evaluating the one-loop diagrams in the background gauge, it is seen that the anomaly coefficient is determined by the Feynman-gauge value of the one-loop vertex function to G μν G-tilde μν at vanishing momenta
16. Superlocalization formulas and supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Bruzzo, U.; Fucito, F.
2004-01-01
By using supermanifolds techniques we prove a generalization of the localization formula in equivariant cohomology which is suitable for studying supersymmetric Yang-Mills theories in terms of ADHM data. With these techniques one can compute the reduced partition functions of topological super-Yang-Mills theory with 4, 8 or 16 supercharges. More generally, the superlocalization formula can be applied to any topological field theory in any number of dimensions
17. Superspace gauge fixing of topological Yang-Mills theories
Energy Technology Data Exchange (ETDEWEB)
Constantinidis, Clisthenis P; Piguet, Olivier [Universidade Federal do Espirito Santo (UFES) (Brazil); Spalenza, Wesley [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro (Brazil)
2004-03-01
We revisit the construction of topological Yang-Mills theories of the Witten type with arbitrary space-time dimension and number of ''shift supersymmetry'' generators, using a superspace formalism. The super-BF structure of these theories is exploited in order to determine their actions uniquely, up to the ambiguities due to the fixing of the Yang-Mills and BF gauge invariance. UV finiteness to all orders of perturbation theory is proved in a gauge of the Landau type. (orig.)
18. Superspace gauge fixing of topological Yang-Mills theories
International Nuclear Information System (INIS)
Constantinidis, Clisthenis P.; Piguet, Olivier; Spalenza, Wesley
2004-01-01
We revisit the construction of topological Yang-Mills theories of the Witten type with arbitrary space-time dimension and number of ''shift supersymmetry'' generators, using a superspace formalism. The super-BF structure of these theories is exploited in order to determine their actions uniquely, up to the ambiguities due to the fixing of the Yang-Mills and BF gauge invariance. UV finiteness to all orders of perturbation theory is proved in a gauge of the Landau type. (orig.)
19. An extended topological Yang-Mills theory
International Nuclear Information System (INIS)
Deguchi, Shinichi
1992-01-01
Introducing infinite number of fields, we construct an extended version of the topological Yang-Mills theory. The properties of the extended topological Yang-Mills theory (ETYMT) are discussed from standpoint of the covariant canonical quantization. It is shown that the ETYMT becomes a cohomological topological field theory or a theory equivalent to a quantum Yang-Mills theory with anti-self-dual constraint according to subsidiary conditions imposed on state-vector space. On the basis of the ETYMT, we may understand a transition from an unbroken phase to a physical phase (broken phase). (author)
20. Numerical simulations of N=(1,1) 1+1-dimensional super Yang-Mills theory with large supersymmetry breaking
International Nuclear Information System (INIS)
Filippov, I.; Pinsky, S.
2002-01-01
We consider the N=(1,1) super Yang-Mills (SYM) theory that is obtained by dimensionally reducing SYM theory in 2+1 dimensions to 1+1 dimensions and discuss soft supersymmetry breaking. We discuss the numerical simulation of this theory using supersymmetric discrete light-cone quantization when either the boson or the fermion has a large mass. We compare our result to the pure adjoint fermion theory and pure adjoint boson discrete light-cone quantization calculations of Klebanov, Demeterfi, Bhanot and Kutasov. With a large boson mass we find that it is necessary to add additional operators to the theory to obtain sensible results. When a large fermion mass is added to the theory we find that it is not necessary to add operators to obtain a sensible theory. The theory of the adjoint boson is a theory that has stringy bound states similar to the full SYM theory. We also discuss another theory of adjoint bosons with a spectrum similar to that obtained by Klebanov, Demeterfi, and Bhanot
1. Infrared finiteness in Yang--Mills theories
International Nuclear Information System (INIS)
Appelquist, T.; Carazzone, J.; Kluberg-Stern, H.; Roth, M.
1976-01-01
The infrared divergences of renormalizable theories with coupled massless fields (in particular, the Yang--Mills theory) are shown to cancel for transition probabilities corresponding to finite-energy-resolution detectors, just as in quantum electrodynamics. This result is established through lowest nontrivial order in perturbation theory for the detection of massive muons in a quantum electrodynamic theory containing massless electrons or the detection of massive quarks in a Yang--Mills theory
2. Two loop diagrams in Yang Mills theory
International Nuclear Information System (INIS)
Jones, D.R.T.
1974-01-01
A calculation of the renormalization constants of the Yang Mills field to 0(g 4 ) is presented. The function β(g) is hence evaluated to 0(g 5 ) and possible implications for gauge theories of the strong interactions discussed
3. Generalized WDVV equations for F4 pure N=2 Super-Yang-Mills theory
NARCIS (Netherlands)
Hoevenaars, L.K.; Kersten, P.H.M.; Martini, Ruud
2001-01-01
An associative algebra of holomorphic differential forms is constructed associated with pure N=2 super-Yang–Mills theory for the Lie algebra F4. Existence and associativity of this algebra, combined with the general arguments in the work of Marshakov, Mironov and Morozov, proves that the
4. Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
CERN Document Server
Dixon, Lance J.; Henn, Johannes M.
2012-01-01
We provide a simple analytic formula for the two-loop six-point ratio function of planar N = 4 super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two function...
5. Light Dilaton at Fixed Points and Ultra Light Scale Super Yang Mills
DEFF Research Database (Denmark)
Antipin, Oleg; Mojaza, Matin; Sannino, Francesco
2012-01-01
of pure supersymmetric Yang-Mills. We can therefore determine the exact nonperturbative fermion condensate and deduce relevant properties of the nonperturbative spectrum of the theory. We also show that the intrinsic scale of super Yang-Mills is exponentially smaller than the scale associated...
6. Super Yang-Mills theory in 10+2 dimensions, The 2T-physics Source for N=4 SYM and M(atrix) Theory
CERN Document Server
Bars, Itzhak
2010-01-01
In this paper we construct super Yang-Mills theory in 10+2 dimensions, a number of dimensions that was not reached before in a unitary supersymmetric field theory, and show that this is the 2T-physics source of some cherished lower dimensional field theories. The much studied conformally exact N=4 Super Yang-Mills field theory in 3+1 dimensions is known to be a compactified version of N=1 SYM in 9+1 dimensions, while M(atrix) theory is obtained by compactifications of the 9+1 theory to 0 dimensions (also 0+1 and others). We show that there is a deeper origin of these theories in two higher dimensions as they emerge from the new theory with two times. Pursuing various alternatives of gauge choices, solving kinematic equations and/or dimensional reductions of the 10+2 theory, we suggest a web of connections that include those mentioned above and a host of new theories that relate 2T-physics and 1T-physics field theories, all of which have the 10+2 theory as the parent. In addition to establishing the higher spa...
7. Superspace gauge fixing of topological Yang-Mills theories
International Nuclear Information System (INIS)
Constantinidis, Clisthenis P.; Piguet, Olivier; Spalenza, Wesley
2003-10-01
We revisit the construction of topological Yang-Mills theories of the Witten type with arbitrary space-time dimension and number of 'shift supersymmetry' generators, using a superspace formalism. The super-B F structure of these theories is exploited in order to determine their actions uniquely, up to the ambiguities due to the fixing of the Yang-Mills and B F gauge invariance. UV finiteness to all orders of perturbation theory is proved in a gauge of the Landau type. (author)
8. Superspace gauge fixing of topological Yang-Mills theories
Energy Technology Data Exchange (ETDEWEB)
Constantinidis, Clisthenis P; Piguet, Olivier [Espirito Santo Univ. (UFES), Vitoria, ES (Brazil); Spalenza, Wesley
2003-10-15
We revisit the construction of topological Yang-Mills theories of the Witten type with arbitrary space-time dimension and number of 'shift supersymmetry' generators, using a superspace formalism. The super-B F structure of these theories is exploited in order to determine their actions uniquely, up to the ambiguities due to the fixing of the Yang-Mills and B F gauge invariance. UV finiteness to all orders of perturbation theory is proved in a gauge of the Landau type. (author)
9. Gauged supersymmetries in Yang-Mills theory
International Nuclear Information System (INIS)
Tissier, Matthieu; Wschebor, Nicolas
2009-01-01
In this paper we show that Yang-Mills theory in the Curci-Ferrari-Delbourgo-Jarvis gauge admits some up to now unknown local linear Ward identities. These identities imply some nonrenormalization theorems with practical simplifications for perturbation theory. We show, in particular, that all renormalization factors can be extracted from two-point functions. The Ward identities are shown to be related to supergauge transformations in the superfield formalism for Yang-Mills theory. The case of nonzero Curci-Ferrari mass is also addressed.
10. Nonperturbative Results for Yang-Mills Theories
DEFF Research Database (Denmark)
Sannino, Francesco; Schechter, Joseph
2010-01-01
Some non perturbative aspects of the pure SU(3) Yang-Mills theory are investigated assuming a specific form of the beta function, based on a recent modification by Ryttov and Sannino of the known one for supersymmetric gauge theories. The characteristic feature is a pole at a particular value....... Assuming the usual QCD value one finds it to be 1.67 GeV, which is in surprisingly good agreement with a quenched lattice calculation. A similar calculation is made for the supersymmetric Yang-Mills theory where the corresponding beta function is considered to be exact....
11. Ultraviolet divergences in higher dimensional supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Howe, P.S.; Stelle, K.S.
1984-01-01
We determine the loop orders for the onset of allowed ultra-violet divergences in higher dimensional supersymmetric Yang-Mills theories. Cancellations are controlled by the non-renormalization theorems for the linearly realizable supersymmetries and by the requirement that counterterms display the full non-linear supersymmetries when the classical equations of motion are imposed. The first allowed divergences in the maximal super Yang-Mills theories occur at four loops in five dimensions, three loops in six dimensions and two loops in seven dimensions. (orig.)
12. Solvable Relativistic Hydrogenlike System in Supersymmetric Yang-Mills Theory
DEFF Research Database (Denmark)
Caron-Huot, Simon; Henn, Johannes M.
2014-01-01
this symmetry? In this Letter we show that the answer is positive: in the nonrelativistic limit, we identify the dual conformal symmetry of planar N=4 super Yang-Mills theory with the well-known symmetries of the hydrogen atom. We point out that the dual conformal symmetry offers a novel way to compute...
13. A new double-scaling limit of N = 4 super-Yang-Mills theory and pp-wave strings
DEFF Research Database (Denmark)
Kristjansen, C.; Plefka, J.; Semenoff, G. W.
2002-01-01
. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double......The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS5 × S5. According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N = 4 supersymmetric Yang-Mills theory...
14. Auxiliary fields for super Yang-Mills from division algebras
CERN Document Server
Evans, Jonathan M.
1994-01-01
Division algebras are used to explain the existence and symmetries of various sets of auxiliary fields for super Yang-Mills in dimensions d=3,4,6,10. (Contribution to G\\"ursey Memorial Conference I: Strings and Symmetries)
15. Composite inflation from super Yang-Mills theory, orientifold, and one-flavor QCD
DEFF Research Database (Denmark)
Channuie, P.; Jorgensen, J. J.; Sannino, F.
2012-01-01
Recent investigations have shown that inflation can be driven by four-dimensional strongly interacting theories nonminimally coupled to gravity. We explore this paradigm further by considering composite inflation driven by orientifold field theories. The advantage of using these theories resides ...... nonminimally coupled QCD theory of inflation. The scale of composite inflation, for all the models presented here, is of the order of 10(16) GeV. Unitarity studies of the inflaton scattering suggest that the cutoff of the model is at the Planck scale. DOI: 10.1103/PhysRevD.86.125035...
16. The dilatation operator of conformal N=4 super-Yang-Mills theory
DEFF Research Database (Denmark)
Beisert, N.; Kristjansen, C.; Staudacher, M.
2003-01-01
We argue that existing methods for the perturbative computation of anomalous dimensions and the disentanglement of mixing in N=4 gauge theory can be considerably simplified, systematized and extended by focusing on the theory's dilatation operator. The efficiency of the method is first illustrate...
17. Random geometry and Yang-Mills theory
International Nuclear Information System (INIS)
Froehlich, J.
1981-01-01
The author states various problems and discusses a very few preliminary rigorous results in a branch of mathematics and mathematical physics which one might call random (or stochastic) geometry. Furthermore, he points out why random geometry is important in the quantization of Yang-Mills theory. (Auth.)
18. Gauge-string duality for superconformal deformations of N = 4 Super Yang-Mills theory
International Nuclear Information System (INIS)
2005-01-01
We analyze in detail the relation between an exactly marginal deformation of N = 4 SYM - the Leigh-Strassler or 'β-deformation' - and its string theory dual (recently constructed in hep-th/0502086) by comparing energies of semiclassical strings to anomalous dimensions of gauge-theory operators in the two-scalar sector. We stress the existence of integrable structures on the two sides of the duality. In particular, we argue that the integrability of strings in AdS 5 x S 5 implies the integrability of the deformed world sheet theory with real deformation parameter. We compare the fast string limit of the worldsheet action in the sector with two angular momenta with the continuum limit of the coherent state action of an anisotropic XXZ spin chain describing the one-loop anomalous dimensions of the corresponding operators and find a remarkable agreement for all values of the deformation parameter. We discuss some of the properties of the Bethe Ansatz for this spin chain, solve the Bethe equations for small number of excitations and comment on higher loop properties of the dilatation operator. With the goal of going beyond the leading order in the 't Hooft expansion we derive the analog of the Bethe equations on the string-theory side, and show that they coincide with the thermodynamic limit of the Bethe equations for the spin chain. We also compute the 1/J corrections to the anomalous dimensions of operators with large R-charge (corresponding to strings with angular momentum J) and match them to the 1-loop corrections to the fast string energies. Our results suggest that the impressive agreement between the gauge theory and semiclassical strings in AdS 5 x S 5 is part of a larger picture underlying the gauge/gravity duality
19. Instanton expansions for mass deformed N=4 super Yang-Mills theories
International Nuclear Information System (INIS)
Minahan, J.A.; Nemeschansky, D.; Warner, N.P.
1998-01-01
We derive modular anomaly equations from the Seiberg-Witten-Donagi curves for softly broken N=4 SU(n) gauge theories. From these equations we can derive recursion relations for the pre-potential in powers of m 2 , where m is the mass of the adjoint hypermultiplet. Given the perturbative contribution of the pre-potential and the presence of ''gaps'', we can easily generate the m 2 expansion in terms of polynomials of Eisenstein series, at least for relatively low rank groups. This enables us to determine efficiently the instanton expansion up to fairly high order for these gauge groups, e.g. eighth order for SU(3). We find that after taking a derivative, the instanton expansion of the pre-potential has integer coefficients. We also postulate the form of the modular anomaly equations, the recursion relations and the form of the instanton expansions for the SO(2n) and E n gauge groups, even though the corresponding Seiberg-Witten-Donagi curves are unknown at this time. (orig.)
20. Continuum regularized Yang-Mills theory
International Nuclear Information System (INIS)
1987-01-01
Using the machinery of stochastic quantization, Z. Bern, M. B. Halpern, C. Taubes and I recently proposed a continuum regularization technique for quantum field theory. This regularization may be implemented by applying a regulator to either the (d + 1)-dimensional Parisi-Wu Langevin equation or, equivalently, to the d-dimensional second order Schwinger-Dyson (SD) equations. This technique is non-perturbative, respects all gauge and Lorentz symmetries, and is consistent with a ghost-free gauge fixing (Zwanziger's). This thesis is a detailed study of this regulator, and of regularized Yang-Mills theory, using both perturbative and non-perturbative techniques. The perturbative analysis comes first. The mechanism of stochastic quantization is reviewed, and a perturbative expansion based on second-order SD equations is developed. A diagrammatic method (SD diagrams) for evaluating terms of this expansion is developed. We apply the continuum regulator to a scalar field theory. Using SD diagrams, we show that all Green functions can be rendered finite to all orders in perturbation theory. Even non-renormalizable theories can be regularized. The continuum regulator is then applied to Yang-Mills theory, in conjunction with Zwanziger's gauge fixing. A perturbative expansion of the regulator is incorporated into the diagrammatic method. It is hoped that the techniques discussed in this thesis will contribute to the construction of a renormalized Yang-Mills theory is 3 and 4 dimensions
1. Yang-Mills theory for non-semisimple groups
CERN Document Server
Nuyts, J; Nuyts, Jean; Wu, Tai Tsun
2003-01-01
For semisimple groups, possibly multiplied by U(1)'s, the number of Yang-Mills gauge fields is equal to the number of generators of the group. In this paper, it is shown that, for non-semisimple groups, the number of Yang-Mills fields can be larger. These additional Yang-Mills fields are not irrelevant because they appear in the gauge transformations of the original Yang-Mills fields. Such non-semisimple Yang-Mills theories may lead to physical consequences worth studying. The non-semisimple group with only two generators that do not commute is studied in detail.
2. Duality in supersymmetric Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Peskin, M.E.
1997-02-01
These lectures provide an introduction to the behavior of strongly-coupled supersymmetric gauge theories. After a discussion of the effective Lagrangian in nonsupersymmetric and supersymmetric field theories, the author analyzes the qualitative behavior of the simplest illustrative models. These include supersymmetric QCD for N{sub f} < N{sub c}, in which the superpotential is generated nonperturbatively, N = 2 SU(2) Yang-Mills theory (the Seiberg-Witten model), in which the nonperturbative behavior of the effect coupling is described geometrically, and supersymmetric QCD for N{sub f} large, in which the theory illustrates a non-Abelian generalization of electric-magnetic duality. 75 refs., 12 figs.
3. Duality in supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Peskin, M.E.
1997-02-01
These lectures provide an introduction to the behavior of strongly-coupled supersymmetric gauge theories. After a discussion of the effective Lagrangian in nonsupersymmetric and supersymmetric field theories, the author analyzes the qualitative behavior of the simplest illustrative models. These include supersymmetric QCD for N f c , in which the superpotential is generated nonperturbatively, N = 2 SU(2) Yang-Mills theory (the Seiberg-Witten model), in which the nonperturbative behavior of the effect coupling is described geometrically, and supersymmetric QCD for N f large, in which the theory illustrates a non-Abelian generalization of electric-magnetic duality. 75 refs., 12 figs
4. Perturbative spacetimes from Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Luna, Andrés [School of Physics and Astronomy, University of Glasgow,Glasgow G12 8QQ, Scotland (United Kingdom); Monteiro, Ricardo [Theoretical Physics Department, CERN,Geneva (Switzerland); Nicholson, Isobel; Ochirov, Alexander; O’Connell, Donal [Higgs Centre for Theoretical Physics,School of Physics and Astronomy, The University of Edinburgh,Edinburgh EH9 3JZ, Scotland (United Kingdom); Westerberg, Niclas [Institute of Photonics and Quantum Sciences,School of Engineering and Physical Sciences, Heriot-Watt University,Edinburgh (United Kingdom); Higgs Centre for Theoretical Physics,School of Physics and Astronomy, The University of Edinburgh,Edinburgh EH9 3JZ, Scotland (United Kingdom); White, Chris D. [Centre for Research in String Theory,School of Physics and Astronomy, Queen Mary University of London,327 Mile End Road, London E1 4NS (United Kingdom)
2017-04-12
The double copy relates scattering amplitudes in gauge and gravity theories. In this paper, we expand the scope of the double copy to construct spacetime metrics through a systematic perturbative expansion. The perturbative procedure is based on direct calculation in Yang-Mills theory, followed by squaring the numerator of certain perturbative diagrams as specified by the double-copy algorithm. The simplest spherically symmetric, stationary spacetime from the point of view of this procedure is a particular member of the Janis-Newman-Winicour family of naked singularities. Our work paves the way for applications of the double copy to physically interesting problems such as perturbative black-hole scattering.
5. Yang-Mills theory - a string theory in disguise
International Nuclear Information System (INIS)
Foerster, D.
1979-01-01
An examination of the Schwinger-Dyson equations of U(N) lattice Yang-Mills theory shows that this theory is exactly equivalent to a theory of strings that interact with one another only through their topology. (Auth.)
6. Semiclassical scattering in Yang-Mills theory
International Nuclear Information System (INIS)
Gould, T.M.; Poppitz, E.R.
1994-01-01
A classical solution to the Yang-Mills theory is given a semiclassical interpretation. The boundary value problem on a complex time contour which arises from the semiclassical approximation to multiparticle scattering amplitudes is reviewed and applied to the case of Yang-Mills theory. The solution describes a classically forbidden transition between states with a large average number of particles in the limit g→0. It dominates a transition probability with a semiclassical suppression factor equal to twice the action of the well-known BPST instanton. Hence, it is relevant to the problem of high-energy tunnelling. It describes transitions of unit topological charge for an appropriate time contour. Therefore, it may have a direct interpretation in terms of fermion-number violating processes in electroweak theory. The solution describes a transition between an initial state with parametrically fewer particles than the final state. Thus, it may be relevant to the study of semiclassical initial-state corrections in the limit of a small number of initial particles. The implications of these results for multiparticle production in electroweak theory are also discussed. (orig.)
7. Higgs Amplitudes from N=4 Supersymmetric Yang-Mills Theory.
Science.gov (United States)
Brandhuber, Andreas; Kostacińska, Martyna; Penante, Brenda; Travaglini, Gabriele
2017-10-20
Higgs plus multigluon amplitudes in QCD can be computed in an effective Lagrangian description. In the infinite top-mass limit, an amplitude with a Higgs boson and n gluons is computed by the form factor of the operator TrF^{2}. Up to two loops and for three gluons, its maximally transcendental part is captured entirely by the form factor of the protected stress tensor multiplet operator T_{2} in N=4 supersymmetric Yang-Mills theory. The next order correction involves the calculation of the form factor of the higher-dimensional, trilinear operator TrF^{3}. We present explicit results at two loops for three gluons, including the subleading transcendental terms derived from a particular descendant of the Konishi operator that contains TrF^{3}. These are expressed in terms of a few universal building blocks already identified in earlier calculations. We show that the maximally transcendental part of this quantity, computed in nonsupersymmetric Yang-Mills theory, is identical to the form factor of another protected operator, T_{3}, in the maximally supersymmetric theory. Our results suggest that the maximally transcendental part of Higgs amplitudes in QCD can be entirely computed through N=4 super Yang-Mills theory.
8. Yang-Mills theory in Coulomb gauge; Yang-Mills-theorie in Coulombeichung
Energy Technology Data Exchange (ETDEWEB)
Feuchter, C.
2006-07-01
In this thesis we study the Yang-Mills vacuum structure by using the functional Schroedinger picture in Coulomb gauge. In particular we discuss the scenario of colour confinement, which was originally formulated by Gribov. After a short introduction, we recall some basic aspects of Yang-Mills theories, its canonical quantization in the Weyl gauge and the functional Schroedinger picture. We then consider the minimal Coulomb gauge and the Gribov problem of the gauge theory. The gauge fixing of the Coulomb gauge is done by using the Faddeev-Popov method, which enables the resolution of the Gauss law - the constraint on physical states. In the third chapter, we variationally solve the stationary Yang-Mills Schroedinger equation in Coulomb gauge for the vacuum state. Therefor we use a vacuum wave functional, which is strongly peaked at the Gribov horizon. The vacuum energy functional is calculated and minimized resulting in a set of coupled Schwinger-Dyson equations for the gluon energy, the ghost and Coulomb form factors and the curvature in gauge orbit space. Using the angular approximation these integral equations have been solved analytically in both the infrared and the ultraviolet regime. The asymptotic analytic solutions in the infrared and ultraviolet regime are reasonably well reproduced by the full numerical solutions of the coupled Schwinger-Dyson equations. In the fourth chapter, we investigate the dependence of the Yang-Mills wave functional in Coulomb gauge on the Faddeev-Popov determinant. (orig.)
9. Nonperturbative aspects of Yang-Mills theory
International Nuclear Information System (INIS)
Schleifenbaum, Wolfgang
2008-01-01
The subject of this thesis is the theory of strong interactions of quarks and gluons, with particular emphasis on nonperturbative aspects of the gluon sector. Continuum methods are used to investigate in particular the confinement phenomenon. Confinement which states that the elementary quarks and gluons cannot be detected as free particles requires an understanding of large-scale correlations. In perturbation theory, only short-range correlations can be reliably described. A nonperturbative approach is given by the set of integral Dyson Schwinger equations involving all Green functions of the theory. A solution for the gluon propagator is obtained in the infrared and ultraviolet asymptotic limits. In chapter 1, redundant degrees of freedom of the Yang Mills gauge theory are removed by fixing the Weyl and Coulomb gauge prior to quantization. The constrained quantization in the Dirac bracket formalism is then performed explicitly to produce the quantized Yang Mills Hamiltonian. The asymptotic infrared limits of Coulomb gauge correlation functions are studied analytically in chapter 2 in the framework of the Gribov Zwanziger confinement scenario. The Coulomb potential between heavy quarks as part of the Yang Mills Hamiltonian is calculated in this limit. A connection between the infrared limits of Coulomb and Landau gauge is established. The Hamiltonian derived paves the way in chapter 3 for finding the Coulomb gauge vacuum wave functional by means of the variational principle. Numerical solutions for the propagators in this vacuum state are discussed and seen to reproduce the anticipated infrared limit. The discussion is extended to the vertex functions. The effect of the approximations on the results is examined. Chapter 4 is mainly devoted to the ultraviolet behavior of the propagators. The discussion is issued in both Coulomb and Landau gauge. A nonperturbative running coupling is defined and calculated. The ultraviolet tails of the variational solutions from
10. Nonperturbative aspects of Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Schleifenbaum, Wolfgang
2008-07-01
The subject of this thesis is the theory of strong interactions of quarks and gluons, with particular emphasis on nonperturbative aspects of the gluon sector. Continuum methods are used to investigate in particular the confinement phenomenon. Confinement which states that the elementary quarks and gluons cannot be detected as free particles requires an understanding of large-scale correlations. In perturbation theory, only short-range correlations can be reliably described. A nonperturbative approach is given by the set of integral Dyson Schwinger equations involving all Green functions of the theory. A solution for the gluon propagator is obtained in the infrared and ultraviolet asymptotic limits. In chapter 1, redundant degrees of freedom of the Yang Mills gauge theory are removed by fixing the Weyl and Coulomb gauge prior to quantization. The constrained quantization in the Dirac bracket formalism is then performed explicitly to produce the quantized Yang Mills Hamiltonian. The asymptotic infrared limits of Coulomb gauge correlation functions are studied analytically in chapter 2 in the framework of the Gribov Zwanziger confinement scenario. The Coulomb potential between heavy quarks as part of the Yang Mills Hamiltonian is calculated in this limit. A connection between the infrared limits of Coulomb and Landau gauge is established. The Hamiltonian derived paves the way in chapter 3 for finding the Coulomb gauge vacuum wave functional by means of the variational principle. Numerical solutions for the propagators in this vacuum state are discussed and seen to reproduce the anticipated infrared limit. The discussion is extended to the vertex functions. The effect of the approximations on the results is examined. Chapter 4 is mainly devoted to the ultraviolet behavior of the propagators. The discussion is issued in both Coulomb and Landau gauge. A nonperturbative running coupling is defined and calculated. The ultraviolet tails of the variational solutions from
11. A QCD Model Using Generalized Yang-Mills Theory
International Nuclear Information System (INIS)
Wang Dianfu; Song Heshan; Kou Lina
2007-01-01
Generalized Yang-Mills theory has a covariant derivative, which contains both vector and scalar gauge bosons. Based on this theory, we construct a strong interaction model by using the group U(4). By using this U(4) generalized Yang-Mills model, we also obtain a gauge potential solution, which can be used to explain the asymptotic behavior and color confinement.
12. N=1 supersymmetric Yang-Mills theory on the lattice
Energy Technology Data Exchange (ETDEWEB)
Piemonte, Stefano
2015-04-08
Supersymmetry (SUSY) relates two classes of particles of our universe, bosons and fermions. SUSY is considered nowadays a fundamental development to explain many open questions about high energy physics. The N=1 super Yang-Mills (SYM) theory is a SUSY model that describes the interaction between gluons and their fermion superpartners called ''gluinos''. Monte Carlo simulations on the lattice are a powerful tool to explore the non-perturbative dynamics of this theory and to understand how supersymmetry emerges at low energy. This thesis presents new results and new simulations about the properties of N=1 SYM, in particular about the phase diagram at finite temperature.
13. Representation dependence of k -strings in pure Yang-Mills theory via supersymmetry
Science.gov (United States)
Anber, Mohamed M.; Pellizzani, Vito
2017-12-01
We exploit a conjectured continuity between super Yang-Mills on R3×S1 and pure Yang-Mills to study k -strings in the latter theory. As expected, we find that Wilson-loop correlation functions depend on the N-ality of a representation R to the leading order. However, the next-to-leading order correction is not universal and is given by the group characters, in the representation R , of the permutation group. We also study W-bosons in super Yang-Mills. We show that they are deconfined on the string world sheet, and therefore, they can change neither the string N-ality nor its tension. This phenomenon mirrors the fact that soft gluons do not screen probe charges with nonzero N-ality in pure Yang-Mills. Finally, we comment on the scaling law of k -strings in super Yang-Mills and compare our findings with strings in Seiberg-Witten theory, deformed Yang-Mills theory, and holographic studies that were performed in the 't Hooft large-N limit.
14. The four-loop remainder function and multi-Regge behavior at NNLLA in planar N=4 super-Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Dixon, Lance J. [SLAC National Accelerator Laboratory,Stanford University, Stanford, CA 94309 (United States); Drummond, James M. [CERN,Geneva 23 (Switzerland); School of Physics and Astronomy, University of Southampton,Highfield, Southampton, SO17 1BJ (United Kingdom); LAPTH, CNRS et Université de Savoie,F-74941 Annecy-le-Vieux Cedex (France); Duhr, Claude [Institute for Particle Physics Phenomenology, University of Durham,Durham, DH1 3LE (United Kingdom); Pennington, Jeffrey [SLAC National Accelerator Laboratory,Stanford University, Stanford, CA 94309 (United States)
2014-06-19
We present the four-loop remainder function for six-gluon scattering with maximal helicity violation in planar N=4 super-Yang-Mills theory, as an analytic function of three dual-conformal cross ratios. The function is constructed entirely from its analytic properties, without ever inspecting any multi-loop integrand. We employ the same approach used at three loops, writing an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz using the multi-Regge limit and the operator product expansion in the near-collinear limit. We express the result in terms of multiple polylogarithms, and in terms of the coproduct for the associated Hopf algebra. From the remainder function, we extract the BFKL eigenvalue at next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at N{sup 3}LLA. We plot the remainder function along various lines and on one surface, studying ratios of successive loop orders. As seen previously through three loops, these ratios are surprisingly constant over large regions in the space of cross ratios, and they are not far from the value expected at asymptotically large orders of perturbation theory.
15. On maximally supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Movshev, M.; Schwarz, A.
2004-01-01
We consider ten-dimensional supersymmetric Yang-Mills theory (10D SUSY YM theory) and its dimensional reductions, in particular, BFSS and IKKT models. We formulate these theories using algebraic techniques based on application of differential graded Lie algebras and associative algebras as well as of more general objects, L ∞ - and A ∞ -algebras. We show that using pure spinor formulation of 10D SUSY YM theory equations of motion and isotwistor formalism one can interpret these equations as Maurer-Cartan equations for some differential Lie algebra. This statement can be used to write BV action functional of 10D SUSY YM theory in Chern-Simons form. The differential Lie algebra we constructed is closely related to differential associative algebra (Ω,∂) of (0,k)-forms on some supermanifold; the Lie algebra is tensor product of (Ω,) and matrix algebra. We construct several other algebras that are quasiisomorphic to (Ω,∂) and, therefore, also can be used to give BV formulation of 10D SUSY YM theory and its reductions. In particular, (Ω,∂) is quasiisomorphic to the algebra (B,d), constructed by Berkovits. The algebras (Ω 0 ,∂) and (B 0 ,d) obtained from (Ω,∂) and (B,d) by means of reduction to a point can be used to give a BV-formulation of IKKT model. We introduce associative algebra SYM as algebra where relations are defined as equations of motion of IKKT model and show that Koszul dual to the algebra (B 0 ,d) is quasiisomorphic to SYM
16. Green functions in a super self-dual Yang-Mills background
International Nuclear Information System (INIS)
McArthur, I.N.
1984-01-01
In euclidean supersymmetric theories of chiral superfields and vector superfields coupled to a super-self-dual Yang-Mills background, we define Green functions for the Laplace-type differential operators which are obtained from the quadratic parot the action. These Green functions are expressed in terms of the Green function on the space of right chiral superfields, and an explicit expression for the right chiral Green function in the fundamental representation of an SU(n) gauge group is presented using the supersymmetric version of the ADHM formalism. The superfield kernels associated with the Laplace-type operators are used to obtain the one-loop quantum corrections to the super-self-dual Yang-Mills action, and also to provide a superfield version of the super-index theorems for the components of chiral superfields in a self-dual background. (orig.)
17. On tree amplitudes of supersymmetric Einstein-Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Adamo, Tim; Casali, Eduardo; Roehrig, Kai A.; Skinner, David [Department of Applied Mathematics & Theoretical Physics, University of Cambridge,Wilberforce Road, Cambridge, CB3 0WA United Kingdom (United Kingdom)
2015-12-29
We present a new formula for all single trace tree amplitudes in four dimensional super Yang-Mills coupled to Einstein supergravity. Like the Cachazo-He-Yuan formula, our expression is supported on solutions of the scattering equations, but with momenta written in terms of spinor helicity variables. Supersymmetry and parity are both manifest. In the pure gravity and pure Yang-Mills sectors, it reduces to the known twistor-string formulae. We show that the formula behaves correctly under factorization and sketch how these amplitudes may be obtained from a four-dimensional (ambi)twistor string.
18. Reduction of 4-dim self dual super Yang-Mills onto super Riemann surfaces
International Nuclear Information System (INIS)
Mendoza, A.; Restuccia, A.; Martin, I.
1990-05-01
Recently self dual super Yang-Mills over a super Riemann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of 4-dim Euclidean self dual super Yang-Mills in terms of constraints on the supercurvature. By dimensional reduction we obtain the same set of superconformal field equations which define self dual connections on a super Riemann surface. (author). 10 refs
19. β-function in a noncovariant Yang-Mills theory
International Nuclear Information System (INIS)
Nielsen, H.B.; Ninomiya, M.
1978-05-01
The betafunction for a noncovariant pure Yang-Mills theory is calculated in perturbation theory to lowest order in the coupling constant and in the deviation from covariance. The authors use the methods developed by DeWitt, Hawking and Dowker. The β-function shows that Lorentz invariance becomes more and more accurate as one goes toward smaller mass scales. The relative deviation of the coupling constant set from covariance diminishes towards lower mass scales as αsub(s)sup(-7/11), where αsub(s) is the QCD 'fine structure constant', for a pure (noncovariant) Yang-Mills theory. (Auth.)
20. Yang-Mills theory on the mass shell
International Nuclear Information System (INIS)
Cvitanovic, P.
1976-01-01
Gauge-invariant mass-shell amplitudes for quantum electrodynamics (QED) and Yang-Mills theory are defined by dimensional regularization. Gauge invariance of the mass-shell renormalization constants is maintained through interplay of ultraviolet and infrared divergences. Quark renormalizations obey the same simple Ward identity as do the electron renormalizations in QED, while the gluon contributions to gluon renormalizations are identically zero. The simplest amplitude finite in QED, the magnetic moment, is gauge-invariant but divergent in Yang-Mills theory for both external gluon and external photon
1. Analysis of Ward identities in supersymmetric Yang-Mills theory
Science.gov (United States)
Ali, Sajid; Bergner, Georg; Gerber, Henning; Montvay, Istvan; Münster, Gernot; Piemonte, Stefano; Scior, Philipp
2018-05-01
In numerical investigations of supersymmetric Yang-Mills theory on a lattice, the supersymmetric Ward identities are valuable for finding the critical value of the hopping parameter and for examining the size of supersymmetry breaking by the lattice discretisation. In this article we present an improved method for the numerical analysis of supersymmetric Ward identities, which takes into account the correlations between the various observables involved. We present the first complete analysis of supersymmetric Ward identities in N=1 supersymmetric Yang-Mills theory with gauge group SU(3). The results indicate that lattice artefacts scale to zero as O(a^2) towards the continuum limit in agreement with theoretical expectations.
2. U(1) decoupling, Kleiss-Kuijf and Bern-Carrasco-Johansson relations in N=4 super Yang-Mills
International Nuclear Information System (INIS)
Jia Yin; Huang Rijun; Liu Changyong
2010-01-01
By using the Britto-Cachazo-Feng-Witten recursion relation of N=4 super Yang-Mills theory, we proved the color reflection, U(1) decoupling, Kleiss-Kuijf and Bern-Carrasco-Johansson relations for color-ordered amplitudes of N=4 super Yang-Mills theory. This proof verified the conjectured Bern-Carrasco-Johansson relations of matter fields. The proof depended only on general properties of superamplitudes. We showed also that the color reflection relation and U(1)-decoupling relation are special cases of Kleiss-Kuijf relations.
3. Full colour for loop amplitudes in Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Ochirov, Alexander [Higgs Centre for Theoretical Physics, School of Physics and Astronomy,The University of Edinburgh,Edinburgh EH9 3JZ, Scotland (United Kingdom); Page, Ben [Albert-Ludwigs-Universität Freiburg, Physikalisches Institut,D-79104 Freiburg (Germany)
2017-02-20
We present a general method to account for full colour dependence Yang-Mills amplitudes at loop level. The method fits most naturally into the framework of multi-loop integrand reduction and in a nutshell amounts to consistently retaining the colour structures of the unitarity cuts from which the integrand is gradually constructed. This technique has already been used in the recent calculation of the two-loop five-gluon amplitude in pure Yang-Mills theory with all positive helicities, (DOI: 10.1007/JHEP10(2015)064). In this note, we give a careful exposition of the method and discuss its connection to loop-level Kleiss-Kuijf relations. We also explore its implications for cancellation of nontrivial symmetry factors at two loops. As an example of its generality, we show how it applies to the three-loop case in supersymmetric Yang-Mills case.
4. Perturbation Theory of Massive Yang-Mills Fields
Science.gov (United States)
Veltman, M.
1968-08-01
Perturbation theory of massive Yang-Mills fields is investigated with the help of the Bell-Treiman transformation. Diagrams containing one closed loop are shown to be convergent if there are more than four external vector boson lines. The investigation presented does not exclude the possibility that the theory is renormalizable.
5. The zero mass limit in Yang-Mills theory
International Nuclear Information System (INIS)
Dombey, N.
1976-01-01
The zero mass limit of massive Yang-Mills theory is investigated and it is shown that there is a conflict between Lorentz invariance and the internal symmetry group in the theory. A necessary but not sufficient condition for the resolution of this conflict is the introduction of zero mass scalar fields. (author)
6. Einstein equation and Yang-Mills theory of gravitation
International Nuclear Information System (INIS)
Stedile, E.
1988-01-01
The possibility of Yang Mills theory of gravitation being a candidate as a gauge model for the Poincare group is pointed out. If the arguments favoring this theory are accepted then Einstein's equations can be derived by a different method in which they arise from a dynamical equation for the torsion field, in a particular case. (author) [pt
7. Supersymmetric sigma models and composite Yang-Mills theory
International Nuclear Information System (INIS)
Lukierski, J.
1980-04-01
We describe two types of supersymmetric sigma models: with field values in supercoset space and with superfields. The notion of Riemannian symmetric pair (H,G/H) is generalized to supergroups. Using the supercoset approach the superconformal-invariant model of composite U(n) Yang-Mills fields in introduced. In the framework of the superfield approach we present with some details two versions of the composite N=1 supersymmetric Yang-Mills theory in four dimensions with U(n) and U(m) x U(n) local invariance. We argue that especially the superfield sigma models can be used for the description of pre-QCD supersymmetric dynamics. (author)
8. Stochastic variables in N=1 supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Lechtenfeld, O.
1984-06-01
The stochastic structure of N=1 supersymmetric Yang-Mills theory is rederived by using a previously developed method for the construction of the (nonlocal) Nicolai map. The stochastic variables correspond to the fixed points of this mapping. The relations are derived in a light cone gauge and in general covariant gauges. (orig.)
9. Quantum Yang-Mills theory on arbitrary surfaces
International Nuclear Information System (INIS)
Blau, Matthias; Thompson, George
1991-05-01
Quantum Yang-Mills theory on 2-dimensional surfaces is studied. Using path integral methods general and explicit expressions are derived for the partition function and expectation values of homologically trivial and non-trivial Wilson loops on closed surfaces of any genus, as well as for the kernels on manifolds with handles and boundaries. (author). 15 refs
10. Bifurcation and stability in Yang-Mills theory with sources
International Nuclear Information System (INIS)
Jackiw, R.
1979-06-01
A lecture is presented in which some recent work on solutions to classical Yang-Mills theory is discussed. The investigations summarized include the field equations with static, external sources. A pattern allowing a comprehensive description of the solutions and stability in dynamical systems are covered. A list of open questions and problems for further research is given. 20 references
11. Quantization of the topological Yang-Mills theory
International Nuclear Information System (INIS)
Dahmen, H.D.; Marculescu, S.; Szymanowski, L.
1990-01-01
The structure of the conserved or partially conserved currents of the topological Yang-Mills theory is discussed. These currents are then used to show that at the one-loop level the β-function reproduces the N=2 supersymmetry value. The result holds even if N=2 supersymmetry is broken down to a singlet supersymmetry. (orig.)
12. Hamiltonian reduction of SU(2) Yang-Mills field theory
International Nuclear Information System (INIS)
Khvedelidze, A.M.; Pavel, H.-P.
1998-01-01
The unconstrained system equivalent to SU (2) Yang-Mills field theory is obtained in the framework of the generalized Hamiltonian formalism using the method of Hamiltonian reduction. The reduced system is expressed in terms of fields with 'nonrelativistic' spin-0 and spin-2
13. Massive Yang-Mills theory: Renormalizability versus unitarity
International Nuclear Information System (INIS)
Delbourgo, R.; Twisk, S.; Thompson, G.
1987-06-01
Various massive Yang-Mills theories not based on the Higgs mechanism are investigated. They are subject to conflicting demands in the twin requirements of unitarity and perturbative renormalizability. Either one or other of these requirements is violated. Unitarity is considered in some detail. (author). 18 refs, 5 figs
14. Topological susceptibility for the SU(3) Yang--Mills theory
DEFF Research Database (Denmark)
Del Debbio, Luigi; Giusti, Leonardo; Pica, Claudio
2004-01-01
We present the results of a computation of the topological susceptibility in the SU(3) Yang--Mills theory performed by employing the expression of the topological charge density operator suggested by Neuberger's fermions. In the continuum limit we find r_0^4 chi = 0.059(3), which corresponds to chi...
15. Quantum theory of massive Yang-Mills fields, 2
International Nuclear Information System (INIS)
Fukuda, Takashi; Monda, Minoru; Takeda, Minoru; Yokoyama, Kan-ichi.
1981-06-01
By generalization of a basic formulation presented in a preceding part of the same series, a massive Yang-Mills field theory with gauge covariance is formulated within one-parameter invariant gauge families. It is consequently concluded that all cases of different gauges belonging to the same gauge family are equivalent to one another in a rigorous field-theoretical sense. (author)
16. Quantum theory of massive Yang-Mills fields, 3
International Nuclear Information System (INIS)
Fukuda, Takashi; Matsuda, Hiroaki; Seki, Yoshinori; Yokoyama, Kan-ichi
1983-01-01
The renormalizable structure of a massive Yang-Mills field theory proposed previously is revealed in view of nonpolynomial Lagrangian theories. Analytic properties of several relevant superpropagators are elucidated in the sense of distributions. It is shown that these regularized superpropagators exhibit a strong infinity-suppression mechanism making the theory renormalizable. There appears a divergence-free model as a subcase of the present theory. (author)
17. Higher order BLG supersymmetry transformations from 10-dimensional super Yang Mills
Energy Technology Data Exchange (ETDEWEB)
Hall, John [Alumnus of Physics Department, Imperial College,South Kensington, London, SW7 2AZ (United Kingdom); Low, Andrew [Physics Department, Wimbledon High School,Mansel Road, London, SW19 4AB (United Kingdom)
2014-06-26
We study a Simple Route for constructing the higher order Bagger-Lambert-Gustavsson theory - both supersymmetry transformations and Lagrangian - starting from knowledge of only the 10-dimensional Super Yang Mills Fermion Supersymmetry transformation. We are able to uniquely determine the four-derivative order corrected supersymmetry transformations, to lowest non-trivial order in Fermions, for the most general three-algebra theory. For the special case of Euclidean three-algbera, we reproduce the result presented in arXiv:1207.1208, with significantly less labour. In addition, we apply our method to calculate the quadratic fermion terms in the higher order BLG fermion supersymmetry transformation.
18. Unconstrained N=2 matter, Yang-Mills and supergravity theories in harmonic superspace
International Nuclear Information System (INIS)
Galperin, A.; Kalitzin, S.; Sokatchev, E.
1984-04-01
A new approach to N=2 supersymmetry based on the concept of harmonic superspace is proposed and is used to give an unconstrained superfield geometric description of N=2 super Yang-Mills and supergravity theories as well as of matter N=2 hypermultiplets. The harmonic N=2 superspace has as independent coordinates, in addition to the usual ones, the isospinor harmonics Usub(i)sup(+-) on the sphere SU(2)/U(1). The role of Usub(i)sup(+-) is to relate the SU(2) group realized on the component fields to a U(1) group acting on the relevant superfields. Their introduction makes it possible to SU(2)-covariantize the notion of Grassmann analyticity. Crucial for our construction is the existence of an analytic subspace of the general harmonic N=2 superspace. The hypermultiplet superfields and the true prepotentials (pre-prepotentials) of N=2 super Yang-Mills and supergravity are unconstrained superfunctions over this analytic subspace. The pre-prepotentials have a clear geometric interpretation as gauge connections with respect to the internal SU(2)/U(1)-directions. A radically new feature arises: the number of gauge and auxiliary degrees of freedom becomes infinite while the number of physical degrees of freedom remains finite. Other new results are the massive N=2 Yang-Mills theory and various off-shell self-interactions of hypermultiplets. The propagators for matter and Yang-Mills superfields are given. (author)
19. Solving pure yang-mills theory in dimensions.
Science.gov (United States)
Leigh, Robert G; Minic, Djordje; Yelnikov, Alexandr
2006-06-09
We analytically compute the spectrum of the spin zero glueballs in the planar limit of pure Yang-Mills theory in 2 + 1 dimensions. The new ingredient is provided by our computation of a new nontrivial form of the ground state wave functional. The mass spectrum of the theory is determined by the zeroes of Bessel functions, and the agreement with large lattice data is excellent.
20. Note on dual superconformal symmetry of the N=4 super Yang-Mills S matrix
International Nuclear Information System (INIS)
Brandhuber, Andreas; Heslop, Paul; Travaglini, Gabriele
2008-01-01
We present a supersymmetric recursion relation for tree-level scattering amplitudes in N=4 super Yang-Mills. Using this recursion relation, we prove that the tree-level S matrix of the maximally supersymmetric theory is covariant under dual superconformal transformations. We further analyze the consequences that the transformation properties of the trees under this symmetry have on those of the loops. In particular, we show that the coefficients of the expansion of generic one-loop amplitudes in a basis of pseudoconformally invariant scalar box functions transform covariantly under dual superconformal symmetry, and in exactly the same way as the corresponding tree-level amplitudes.
1. Cut-and-join operators and N=4 super Yang-Mills
International Nuclear Information System (INIS)
Brown, T.W.
2010-02-01
We show which multi-trace structures are compatible with the symmetrisation of local operators in N=4 super Yang-Mills when they are organised into representations of the global symmetry group. Cut-and-join operators give the non-planar expansion of correlation functions of these operators in the free theory. Using these techniques we find the 1/N corrections to the quarter-BPS operators which remain protected at weak coupling. We also present a new way of counting these chiral ring operators using the Weyl group S N . (orig.)
2. Cut-and-join operators and N=4 super Yang-Mills
Energy Technology Data Exchange (ETDEWEB)
Brown, T W [DESY, Hamburg (Germany). Theory Group
2010-02-15
We show which multi-trace structures are compatible with the symmetrisation of local operators in N=4 super Yang-Mills when they are organised into representations of the global symmetry group. Cut-and-join operators give the non-planar expansion of correlation functions of these operators in the free theory. Using these techniques we find the 1/N corrections to the quarter-BPS operators which remain protected at weak coupling. We also present a new way of counting these chiral ring operators using the Weyl group S{sub N}. (orig.)
3. Saddle point solutions in Yang-Mills-dilaton theory
International Nuclear Information System (INIS)
Bizon, P.
1992-01-01
The coupling of a dilaton to the SU(2)-Yang-Mills field leads to interesting non-perturbative static spherically symmetric solutions which are studied by mixed analytical and numerical methods. In the abelian sector of the theory there are finite-energy magnetic and electric monopole solutions which saturate the Bogomol'nyi bound. In the nonabelian sector there exist a countable family of globally regular solutions which are purely magnetic but have zero Yang-Mills magnetic charge. Their discrete spectrum of energies is bounded from above by the energy of the abelian magnetic monopole with unit magnetic charge. The stability analysis demonstrates that the solutions are saddle points of the energy functional with increasing number of unstable modes. The existence and instability of these solutions are 'explained' by the Morse-theory argument recently proposed by Sudarsky and Wald. (author)
4. Introduction to instantons in Yang-Mills theory
International Nuclear Information System (INIS)
Pak, N.K.
1980-02-01
The Yang-Mills theory is outlined; the classical formalism is discussed first, and then the difficulties related to gauge invariance in the canonical quantization of the theory are taken up. Next, the task of finding and studying Euclidean gauge field configurations of finite action as solutions to the equation of motion is addressed. It is found that configurations which contribute the most in the semi-classical approximation are those which minimize the action. The question of a lower bound for the Euclidean action is considered. Properties of topological charge and the behavior of topological charge under gauge transformation are discussed. Then instanton solutions to the field equations are produced. Finally, the physical interpretation of the instanton is considered. It is found that the instanton, the Euclidean gauge field configuration which minimizes the action, induces tunneling among the infinitely degenerate vacua of the Yang-Mills theory by lifting the degeneracy and creating new distinct inequivalent (invariant under topologically nontrivial gauge transformations) vacua labelled by a superselection index theta. The angle theta is shown not to be a gauge artifact. In conclusion, the tunneling Hamiltonian and effective Lagrangian for the Yang-Mills theory are discussed
5. Real and virtual infrared behaviour in Yang-Mills theory
International Nuclear Information System (INIS)
Frenkel, J.; Meuldermans, R.; Mohammad, I.; Taylor, J.C.
1977-01-01
The leading infrared divergences, up to fourth order, from both virtual and from real soft gluon corrections to massive quark scattering have been calculated in an external colourless potential. In the combined corrections, the infrared divergences cancel, as expected. The form of the correction is very simple, and can be expressed in terms of the effective coupling constant of the pure Yang-Mills theory. (Auth.)
6. Ambiguities of the natural gauge in Yang-Mills theories
International Nuclear Information System (INIS)
Lazarides, G.
1978-01-01
We study the ambiguities of the natural gauge condition for the Euclidean SU(2) Yang-Mills theory in four dimensions. Then, we show that, in the stationary-phase approximation, these ambiguities do not affect the contribution of the sector with Pontryagin index q = 1 to the correlation functions of gauge-invariant operators. They affect only the higher-order corrections to this contribution
7. Quantum Yang-Mills theory of Riemann surfaces and conformal field theory
International Nuclear Information System (INIS)
Killingback, T.P.
1989-01-01
It is shown that Yang-Mills theory on a smooth surface, when suitably quantized, is a topological quantum field theory. This topological gauge theory is intimately related to two-dimensional conformal field theory. It is conjectured that all conformal field theories may be obtained from Yang-Mills theory on smooth surfaces. (orig.)
8. Loop groups and Yang-Mills theory in dimension two
DEFF Research Database (Denmark)
Gravesen, Jens
1990-01-01
Given a connection ω in a G-bundle over S2, then a process called radial trivialization from the poles gives a unique clutching function, i.e., an element γ of the loop group ΩG. Up to gauge equivalence, ω is completely determined by γ and a map f:S2 →g into the Lie algebra. Moreover, the Yang......-Mills function of ω is the sum of the energy of γ and the square of a certain norm of f. In particular, the Yang-Mills functional has the same Morse theory as the energy functional on ΩG. There is a similar description of connections in a G-bundle over an arbitrary Riemann surface, but so far not of the Yang...
9. YANG-MILLS Theory in, Beyond, and Behind Observed Reality
Science.gov (United States)
Wilczek, Frank
The primary interactions of Yang-Mills theory [1] are visibly embodied in hard processes, most directly in jets. The character of jets also reflects the deep structure of effective charge, which is dominated by the influence of intrinsically non-Abelian gauge dynamics. These proven insights into fundamental physics ramify in many directions, and are far from being exhausted. I will discuss three rewarding explorations from my own experience, whose point of departure is the hard Yang-Mills interaction, and whose end is not yet in sight. Given an insight so profound and fruitful as Yang and Mills brought us, it is in order to try to consider its broadest implications, which I attempt at the end.
10. Yangian Symmetry of Scattering Amplitudes and the Dilatation Operator in N=4 Supersymmetric Yang-Mills Theory.
Science.gov (United States)
Brandhuber, Andreas; Heslop, Paul; Travaglini, Gabriele; Young, Donovan
2015-10-02
It is known that the Yangian of PSU(2,2|4) is a symmetry of the tree-level S matrix of N=4 super Yang-Mills theory. On the other hand, the complete one-loop dilatation operator in the same theory commutes with the level-one Yangian generators only up to certain boundary terms found by Dolan, Nappi, and Witten. Using a result by Zwiebel, we show how the Yangian symmetry of the tree-level S matrix of N=4 super Yang-Mills theory implies precisely the Yangian invariance, up to boundary terms, of the one-loop dilatation operator.
11. One-loop effect of null-like cosmology's holographic dual super-Yang-Mills
International Nuclear Information System (INIS)
Lin, F.-L.; Tomino, Dan
2007-01-01
We calculate the 1-loop effect in super-Yang-Mills which preserves 1/4-supersymmetries and is holographically dual to the null-like cosmology with a big-bang singularity. Though the bosonic and fermionic spectra do not agree precisely, we do obtain vanishing 1-loop vacuum energy for generic warped plane-wave type backgrounds with a big-bang singularity. Moreover, we find that the cosmological 'constant' contributed either by bosons or fermions is time-dependent. The issues about the particle production of some background and about the UV structure are also commented. We argue that the effective higher derivative interactions are suppressed as long as the Fourier transform of the time-dependent coupling is UV-finite. Our result holds for scalar configurations that are BPS but with arbitrary time-dependence. This suggests the existence of non-renormalization theorem for such a new class of time-dependent theories. Altogether, it implies that such a super-Yang-Mills is scale-invariant, and that its dual bulk quantum gravity might behave regularly near the big bang
12. A gluon cluster solution of effective Yang-Mills theory
CERN Document Server
Pavlovsky, O V
2001-01-01
A classical solution of the effective Yang-Mills (YM) theory with a finite energy and nonstandard Lagrangian was obtained. Influence of vacuum polarization on gluon cluster formation was discussed. Appearance of cluster solutions in the theory of non-Abelian fields can take place only if the result goes beyond the framework of pure YM theory. It is shown that account of quantum effects of polarized vacuum in the presence of a classical gluon field can also result in formation of the solutions. Solutions with the finite intrinsic energy are provided. Besides, fields of colour groups SU(2) were studied
13. A less-constrained (2,0) super-Yang-Mills model: the coupling to non-linear σ-models
International Nuclear Information System (INIS)
Almeida, C.A.S.; Doria, R.M.
1990-01-01
Considering a class of (2,0) super-Yang-Mills multiplets characterised by the appearance of a pair of independent gauge potentials, we present here their coupling to non-linear σ-models in (2,0)-superspace. Contrary to the case of the coupling to (2,0) matter superfields, the extra gauge potential present in the Yang-Mills sector does not decouple from the theory in the case one gauges isometry groups of σ-models. (author)
14. Infrared divergence cancellation in pure Yang-Mills theory
International Nuclear Information System (INIS)
Alvarez, A.G.
1977-01-01
Virtual and real corrections to massless external lines in pure Yang-Mills theory are considered in order to look for general features of the infrared divergence cancellation. Use of the Ward identities and sums over transverse polarization states give rise to terms formally corresponding to real ghost emission, cancelling ghost loop singularities, and to a factorisation of the hard narrow single gauge boson emission. Other virtual corrections are examined in the soft region and a graph by graph cancellation is also found. An illustrative explicit calculation of scattering of a gauge particle in an external scalar potential, including hard narrow angle emission is presented. (Auth.)
15. Is the ground state of Yang-Mills theory Coulombic?
OpenAIRE
Heinzl, Thomas; Ilderton, Anton; Langfeld, Kurt; Lavelle, Martin; Lutz, Wolfgang; McMullan, David
2008-01-01
We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory. A state describing the flux tube between quarks as a thin string of glue is found to be a poor description of the continuum ground state; the infinitesimal thickness of the string leads to UV artifacts which suppress the overlap with the ground state. Contrastingly, a state which surrounds the quarks with non-abelian Coulomb fields is found to have a good overlap with the ground state for all ch...
16. Open spin chains in super Yang-Mills at higher loops: some potential problems with integrability
International Nuclear Information System (INIS)
Agarwal, Abhishek
2006-01-01
The super Yang-Mills duals of open strings attached to maximal giant gravitons are studied in perturbation theory. It is shown that non-BPS baryonic excitations of the gauge theory can be studied within the paradigm of open quantum spin chains even beyond the leading order in perturbation theory. The open spin chain describing the two loop mixing of non-BPS giant gravitons charged under an su(2) of the so(6) R symmetry group is explicitly constructed. It is also shown that although the corresponding open spin chain is integrable at the one loop order, there is a potential breakdown of integrability at two and higher loops. The study of integrability is performed using coordinate Bethe ansatz techniques
17. Confinement in a three-dimensional Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Frasca, Marco
2017-04-15
We show that, starting from known exact classical solutions of the Yang-Mills theory in three dimensions, the string tension is obtained and the potential is consistent with a marginally confining theory. The potential we obtain agrees fairly well with preceding findings in the literature but here we derive it analytically from the theory without further assumptions. The string tension is in strict agreement with lattice results and the well-known theoretical result by Karabali-Kim-Nair analysis. Classical solutions depend on a dimensionless numerical factor arising from integration. This factor enters into the determination of the spectrum and has been arbitrarily introduced in some theoretical models. We derive it directly from the solutions of the theory and is now fully justified. The agreement obtained with the lattice results for the ground state of the theory is well below 1% at any value of the degree of the group. (orig.)
18. Regge meets collinear in strongly-coupled N=4 super Yang-Mills
Energy Technology Data Exchange (ETDEWEB)
Sprenger, Martin [Institut für Theoretische Physik, Eidgenössische Technische Hochschule Zürich,Wolfgang-Pauli-Strasse 27, 8093 Zürich (Switzerland)
2017-01-10
We revisit the calculation of the six-gluon remainder function in planar N=4 super Yang-Mills theory from the strong coupling TBA in the multi-Regge limit and identify an infinite set of kinematically subleading terms. These new terms can be compared to the strong coupling limit of the finite-coupling expressions for the impact factor and the BFKL eigenvalue proposed by Basso et al. in https://www.doi.org/10.1007/JHEP01(2015)027, which were obtained from an analytic continuation of the Wilson loop OPE. After comparing the results order by order in those subleading terms, we show that it is possible to precisely map both formalisms onto each other. A similar calculation can be carried out for the seven-gluon amplitude, the result of which shows that the central emission vertex does not become trivial at strong coupling.
19. Scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills
International Nuclear Information System (INIS)
Henn, J.M.
2010-01-01
We discuss planar scattering amplitudes on the Coulomb branch of N=4 super Yang-Mills. The vacuum expectation values on the Coulomb branch can be used to regulate infrared divergences. We argue that this has a number of conceptual as well as practical advantages over dimensional regularisation.
20. Lectures on strings in flat space and plane waves from N = 4 super Yang Mills
International Nuclear Information System (INIS)
Maldacena, J.
2003-01-01
In these lecture notes we explain how the string spectrum in flat space and plane waves arises from the large N limit of U(N) N = 4 super Yang Mills. We reproduce the spectrum by summing a subset of the planar Feynman diagrams. We also describe some other aspects of string propagation on plane wave backgrounds. (author)
1. Gravitational catalysis of merons in Einstein-Yang-Mills theory
Science.gov (United States)
Canfora, Fabrizio; Oh, Seung Hun; Salgado-Rebolledo, Patricio
2017-10-01
We construct regular configurations of the Einstein-Yang-Mills theory in various dimensions. The gauge field is of meron-type: it is proportional to a pure gauge (with a suitable parameter λ determined by the field equations). The corresponding smooth gauge transformation cannot be deformed continuously to the identity. In the three-dimensional case we consider the inclusion of a Chern-Simons term into the analysis, allowing λ to be different from its usual value of 1 /2 . In four dimensions, the gravitating meron is a smooth Euclidean wormhole interpolating between different vacua of the theory. In five and higher dimensions smooth meron-like configurations can also be constructed by considering warped products of the three-sphere and lower-dimensional Einstein manifolds. In all cases merons (which on flat spaces would be singular) become regular due to the coupling with general relativity. This effect is named "gravitational catalysis of merons".
2. Some global charges in classical Yang-Mills theory
International Nuclear Information System (INIS)
Chrus'ciel, P.T.; Kondracki, W.
1987-01-01
Three classes of boundary conditions allowing the definition of a global field strength (''global color'') are presented. A definition of global color of the sources and of the Yang-Mills field is proposed. Some exact solutions of Yang-Mills equations with point sources and with ''topologically nontrivial electric color'' are presented
3. Exact and microscopic one-instanton calculations in N=2 supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Ito, K.; Sasakura, N.
1997-01-01
We study the low-energy effective theory in N=2 super Yang-Mills theories by microscopic and exact approaches. We calculate the one-instanton correction to the prepotential for any simple Lie group from the microscopic approach. We also study the Picard-Fuchs equations and their solutions in the semi-classical regime for classical gauge groups with rank r≤3. We find that for gauge groups G=A r , B r , C r (r≤3) the microscopic results agree with those from the exact solutions. (orig.)
4. Is the ground state of Yang-Mills theory Coulombic?
Science.gov (United States)
Heinzl, T.; Ilderton, A.; Langfeld, K.; Lavelle, M.; Lutz, W.; McMullan, D.
2008-08-01
We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory. A state describing the flux tube between quarks as a thin string of glue is found to be a poor description of the continuum ground state; the infinitesimal thickness of the string leads to UV artifacts which suppress the overlap with the ground state. Contrastingly, a state which surrounds the quarks with non-Abelian Coulomb fields is found to have a good overlap with the ground state for all charge separations. In fact, the overlap increases as the lattice regulator is removed. This opens up the possibility that the Coulomb state is the true ground state in the continuum limit.
5. Yang-Mills theory in null path space
International Nuclear Information System (INIS)
Kent, S.L.
1982-01-01
A reformulation of classical GL(n,c) Yang-Mills theory is presented. The reformulation is in terms of a single matrix-valued function G on a six-dimensional subspace of the space of paths in Minkowski space, M. This subspace is defined as the null paths beginning at each point, (X/sup a/), of M and ending at future null infinity. A convenient parametrization of these paths is to give the Minkowski coordinates x/sup a/ of the starting point and the (complex) stereographic coordinates (xi, antixi) on S 2 which label the light cone generators of x/sup a/. A path is thus labeled by (x/sup a/,xi, antixi). The function G(x/sup a/,xi, antixi) is defined by the parallel propagation (with a given connection) of n linearly independent fiber vectors from x/sup a/ to null infinity along the (xi, antixi) generator. From knowledge of G(x/sup a/,xi, antixi) the connection one-form γ/sub a/ at the point x/sup a/ can be obtained is shown. Furthermore how the vacuum Yang-Mills equations can be imposed on the G is shown. This results in a rather complicated integro-differential equation for G which involves the characteristic initial data (essentially the radiation field) acting as the driving term. Two simple special cases are immediately obtainable; in the case of self-dual (or anti-self dual) fields the author obtains a simple derivation of the Sparling equation, namely delta G = -GA, while for Abelian (Maxwell) theories obtained the equation delta anti delta log G = -anti delta A-anti delta A, where A and its conjugate anti A are the characteristic free data given on null infinity. The latter equation is equivalent to the vacuum Maxwell equations
6. On the exponentiation of leading infrared divergences in massless Yang-Mills theories
International Nuclear Information System (INIS)
Frenkel, J.; Garcia, R.L.
1977-01-01
We derive, in the axial gauge, the effective U-matrix which governs the behaviour of leading infrared singularities in the self-energy functions of Yang-Mills particles. We then show in a very simple manner, that these divergences, which determine the leading singularities in massless Yang-Mills theories, exponentiate [pt
7. 5D black hole solution in Einstein-Yang-Mills-Gauss-Bonnet theory
International Nuclear Information System (INIS)
Mazharimousavi, S. Habib; Halilsoy, M.
2007-01-01
By adopting the 5D version of the Wu-Yang ansatz we present in closed form a black hole solution in the Einstein-Yang-Mills-Gauss-Bonnet theory. In the Einstein-Yang-Mills limit, we recover the 5D black hole solution already known
8. A Yang-Mills structure for string field theory
International Nuclear Information System (INIS)
Tsousheung Tsun
1990-01-01
String theorists believe that one way to achieve a fully quantized theory of string is through string field theory. The other way is to study conformal field theory on Riemann surfaces of different genera, which is the subject of many of the talks at this Conference. In a way, string field theory is the more conservative approach, since it aims just to replace the spacetime points of conventional quantum field theory by string, which are extended objects. However, from this point of view string theory has one rather unsatisfactory aspect, in the sense that although it has been very well developed and minutely studied, we are still rather unclear about its basic structure. We can contrast this to both general relativity, which is based on the geometry of spacetime, and to gauge theory, which is about the structure of various natural bundles over spacetime. And yet string theory is supposed to embody both these two essentially geometric theories. To paraphrase Witten, in string theory we seem to have to work backwards to get at the still unknown basic structure. Some joint work with Chan Hong-Mo is reported in an attempt to gain some understanding in that general direction. It seems that one could in some sense consider string field theory as a generalized Yang-Mills theory. This idea is explored. (author)
9. Can large Nc equivalence between supersymmetric Yang-Mills theory and its orbifold projections be valid?
International Nuclear Information System (INIS)
Kovtun, Pavel; Uensal, Mithat; Yaffe, Laurence G.
2005-01-01
In previous work, we found that necessary and sufficient conditions for large N c equivalence between parent and daughter theories, for a wide class of orbifold projections of U(N c ) gauge theories, are just the natural requirements that the discrete symmetry used to define the projection not be spontaneously broken in the parent theory, and the discrete symmetry permuting equivalent gauge group factors not be spontaneously broken in the daughter theory. In this paper, we discuss the application of this result to Z k projections of N=1 supersymmetric Yang-Mills theory in four dimensions, as well as various multiflavor generalizations. Z k projections with k>2 yielding chiral gauge theories violate the symmetry realization conditions needed for large N c equivalence, due to the spontaneous symmetry breaking of discrete chiral symmetry in the parent super-Yang-Mills theory. But for Z 2 projections, we show that previous assertions of large N c inequivalence, in infinite volume, between the parent and daughter theories were based on incorrect mappings of vacuum energies, theta angles, or connected correlators between the two theories. With the correct identifications, there is no sign of any inconsistency. A subtle but essential feature of the connection between parent and daughter theories involves multivaluedness in the mapping of theta parameters from parent to daughter
10. Homotopy Lie superalgebra in Yang-Mills theory
International Nuclear Information System (INIS)
Zeitlin, Anton M.
2007-01-01
The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra
11. Electric vortex lines from the Yang-Mills theory
International Nuclear Information System (INIS)
Nielsen, N.K.; Olesen, P.
1978-08-01
The dynamics of an unstable Yang-Mills field mode previously found by the authors is developed. It is argued that this unstable mode corresponds to the transition to a state where electric vortex lines are created. (Auth.)
12. String tensions in deformed Yang-Mills theory
Science.gov (United States)
Poppitz, Erich; Shalchian T., M. Erfan
2018-01-01
We study k-strings in deformed Yang-Mills (dYM) with SU(N) gauge group in the semiclassically calculable regime on R^3× S^1 . Their tensions Tk are computed in two ways: numerically, for 2 ≤ N ≤ 10, and via an analytic approach using a re-summed perturbative expansion. The latter serves both as a consistency check on the numerical results and as a tool to analytically study the large-N limit. We find that dYM k-string ratios Tk/T1 do not obey the well-known sine- or Casimir-scaling laws. Instead, we show that the ratios Tk/T1 are bound above by a square root of Casimir scaling, previously found to hold for stringlike solutions of the MIT Bag Model. The reason behind this similarity is that dYM dynamically realizes, in a theoretically controlled setting, the main model assumptions of the Bag Model. We also compare confining strings in dYM and in other four-dimensional theories with abelian confinement, notably Seiberg-Witten theory, and show that the unbroken Z_N center symmetry in dYM leads to different properties of k-strings in the two theories; for example, a "baryon vertex" exists in dYM but not in softly-broken Seiberg-Witten theory. Our results also indicate that, at large values of N, k-strings in dYM do not become free.
13. Lattice super-Yang-Mills: a virial approach to operator dimensions
International Nuclear Information System (INIS)
Callan, Curtis G.; Heckman, Jonathan; McLoughlin, Tristan; Swanson, Ian
2004-01-01
The task of calculating operator dimensions in the planar limit of N=4 super-Yang-Mills theory can be vastly simplified by mapping the dilatation generator to the Hamiltonian of an integrable spin chain. The Bethe ansatz has been used in this context to compute the spectra of spin chains associated with various sectors of the theory which are known to decouple in the planar (large-Nc) limit. These techniques are powerful at leading order in perturbation theory but become increasingly complicated beyond one loop in the 't Hooft parameter λ=gYM2Nc, where spin chains typically acquire long-range (non-nearest-neighbor) interactions. In certain sectors of the theory, moreover, higher-loop Bethe ansatze do not even exist. We develop a virial expansion of the spin chain Hamiltonian as an alternative to the Bethe ansatz methodology, a method which simplifies the computation of dimensions of multi-impurity operators at higher loops in λ. We use these methods to extract previously reported numerical gauge theory predictions near the BMN limit for comparison with corresponding results on the string theory side of the AdS/CFT correspondence. For completeness, we compare our virial results with predictions that can be derived from current Bethe ansatz technology
14. Classical geometrical interpretation of ghost fields and anomalies in Yang-Mills theory and quantum gravity
International Nuclear Information System (INIS)
Thierry-Mieg, J.
1985-01-01
This paper discusses the reinterpretation of the BRS equations of Quantum Field Theory as the Maurer Cartan equation of a classical principal fiber bundle leads to a simple gauge invariant classification of the anomalies in Yang Mills theory and gravity
15. Classical geometrical interpretation of ghost fields and anomalies in Yang-Mills theory and quantum gravity
International Nuclear Information System (INIS)
Thierry-Mieg, J.
1985-01-01
The reinterpretation of the BRS equations of Quantum Field Theory as the Maurer Cartan equation of a classical principal fiber bundle leads to a simple gauge invariant classification of the anomalies in Yang Mills theory and gravity
16. Generating functional for Donaldson invariants and operator algebra in topological D=4 Yang-Mills theory
International Nuclear Information System (INIS)
Johansen, A.A.
1992-01-01
It is shown, that under the certain constraints the generating functional for the Donaldson invariants in the D=4 topological Yang-Mills theory can be interpreted as a partition function for the renormalizable theory. 20 refs
17. Finite Yang-Mills theories and the Bjorken--Johnson--Low limit
International Nuclear Information System (INIS)
Ali, A.; Bernstein, J.
1975-01-01
We consider the Bjorken-Johnson-Low limit for the propagator in massless Yang-Mills theories. The significance of our result in terms of imposing an eigenvalue on the theory so as to render it finite is discussed
18. Integrability in dipole-deformed \\boldsymbol{N=4} super Yang-Mills
Science.gov (United States)
Guica, Monica; Levkovich Maslyuk, Fedor; Zarembo, Konstantin
2017-09-01
We study the null dipole deformation of N=4 super Yang-Mills theory, which is an example of a potentially solvable ‘dipole CFT’: a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrödinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld-Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sl(2) sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrödinger holography. Dedicated to the memory of Petr Petrovich Kulish.
19. Analytic supersymmetric regularization for the pure N=1 super-Yang-Mills model
International Nuclear Information System (INIS)
Abdalla, E.; Jasinschi, R.S.
1987-01-01
We calculate for the pure N=1 super-Yang-Mills model the quantum correction to the background field strength up to two loops. In using background field method, analytic regularization and Seeley coefficient expansion we show how these corrections arise. Our method differs from the dimensional regularization via dimensional reduction scheme in various respects, in particular to the origin of the background field strength as appearing in the divergent expressions. (orig.)
20. 4-dimensional General Relativity from the instrinsic spatial geometry of SO(3) Yang-Mills theory
International Nuclear Information System (INIS)
Ita, Eyo Eyo
2011-01-01
In this paper we derive 4-dimensional General Relativity from three dimensions, using the intrinsic spatial geometry inherent in Yang-Mills theory which has been exposed by previous authors as well as some properties of the Ashtekar variables. We provide various interesting relations, including the fact that General Relativity can be written as a Yang-Mills theory where the antiself-dual Weyl curvature replaces the Yang-Mills coupling constant. We have generalized the results of some previous authors, covering Einstein's spaces, to include more general spacetime geometries.
1. N=1 supersymmetric yang-mills theory in Ito Calculus
International Nuclear Information System (INIS)
Nakazawa, Naohito
2003-01-01
The stochastic quantization method is applied to N = 1 supersymmetric Yang-Mills theory, in particular in 4 and 10 dimensions. In the 4 dimensional case, based on Ito calculus, the Langevin equation is formulated in terms of the superfield formalism. The stochastic process manifestly preserves both the global N = 1 supersymmetry and the local gauge symmetry. The expectation values of the local gauge invariant observables in SYM 4 are reproduced in the equilibrium limit. In the superfield formalism, it is impossible in SQM to choose the so-called Wess-Zumino gauge in such a way to gauge away the auxiliary component fields in the vector multiplet, while it is shown that the time development of the auxiliary component fields is determined by the Langevin equations for the physical component fields of the vector multiplet in an ''almost Wess-Zumino gauge''. The physical component expressions of the superfield Langevin equation are naturally extended to the 10 dimensional case, where the spinor field is Majorana-Weyl. By taking a naive zero volume limit of the SYM 10 , the IIB matrix model is studied in this context. (author)
2. Gluon scattering in N=4 super Yang-Mills at finite temperature
International Nuclear Information System (INIS)
Ito, Katsushi; Iwasaki, Koh; Nastase, Horatiu
2008-01-01
We extend the AdS/CFT prescription of Alday and Maldacena to finite temperature T, defining an amplitude for gluon scattering in N=4 Super Yang-Mills at strong coupling from string theory. It is defined by a lightlike 'Wilson loop' living at the horizon of the T-dual to the black hole in AdS space. Unlike the zero temperature case, this is different from the Wilson loop contour defined at the boundary of the AdS black hole metric. Thus at nonzero T there is no relation between gluon scattering amplitudes and the Wilson loop. We calculate a gauge theory observable that can be interpreted as the amplitude at strong coupling for forward scattering of a low energy gluon (E >T) in both cutoff and generalized dimensional regularization. The generalized dimensional regularization is defined in string theory as an IR modified dimensional reduction. For this calculation, the corresponding usual Wilson loop of the same boundary shape was argued to be related to the jet quenching parameter of the finite temperature N=4 SYM plasma, while the gluon scattering amplitude is related to the viscosity coefficient. (author)
3. On the renormalization of topological Yang-Mills field theory in N=1 superspace
Energy Technology Data Exchange (ETDEWEB)
Oliveira, M.W. de; Penna Firme, A.B.
1996-03-01
We discuss the renormalization aspects of topological super-Yang-Mills field theory in N=1 superspace. Our approach makes use of the regularization independent BRS algebraic technique adapted to the case of a N=1 supersymmetric model. We give the expression of the most general local counterterm to the classical action to all orders of the perturbative expansion. The counterterm is shown to be BRS-coboundary, implying that the co-homological properties of the super topological theory are not affected by quantum effects. We also demonstrate the vanishing of the Callan-Symanzik {beta}-function of the model by employing a recently discovered supersymmetric antighost Ward identity. (author). 30 refs.
4. On the renormalization of topological Yang-Mills field theory in N=1 superspace
International Nuclear Information System (INIS)
Oliveira, M.W. de; Penna Firme, A.B.
1996-03-01
We discuss the renormalization aspects of topological super-Yang-Mills field theory in N=1 superspace. Our approach makes use of the regularization independent BRS algebraic technique adapted to the case of a N=1 supersymmetric model. We give the expression of the most general local counterterm to the classical action to all orders of the perturbative expansion. The counterterm is shown to be BRS-coboundary, implying that the co-homological properties of the super topological theory are not affected by quantum effects. We also demonstrate the vanishing of the Callan-Symanzik β-function of the model by employing a recently discovered supersymmetric antighost Ward identity. (author). 30 refs
5. Masslessness of ghosts in equivariantly gauge-fixed Yang-Mills theories
International Nuclear Information System (INIS)
Golterman, Maarten; Zimmerman, Leah
2005-01-01
We show that the one-loop ghost self-energy in an equivariantly gauge-fixed Yang-Mills theory vanishes at zero momentum. A ghost mass is forbidden by equivariant BRST symmetry, and our calculation confirms this explicitly. The four-ghost self interaction which appears in the equivariantly gauge-fixed Yang-Mills theory is needed in order to obtain this result
6. Propagation of field disturbances in Yang-Mills theory
International Nuclear Information System (INIS)
De Lorenci, Vitorio A.; Li Shiyuan
2008-01-01
The propagation of field disturbances is examined in the context of the effective Yang-Mills Lagrangian, which is intended to be applied to QCD systems. It is shown that birefringence phenomena can occur in such systems provided some restrictive conditions, as causality, are fulfilled. Possible applications to phenomenology are addressed.
7. A Unified Field Theory of Gravity, Electromagnetism, and the Yang-Mills Gauge Field
Directory of Open Access Journals (Sweden)
Suhendro I.
2008-01-01
Full Text Available In this work, we attempt at constructing a comprehensive four-dimensional unified field theory of gravity, electromagnetism, and the non-Abelian Yang-Mills gauge field in which the gravitational, electromagnetic, and material spin fields are unified as intrinsic geometric objects of the space-time manifold S4 via the connection, with the general- ized non-Abelian Yang-Mills gauge field appearing in particular as a sub-field of the geometrized electromagnetic interaction.
8. Observables in topological Yang-Mills theories with extended shift supersymmetry
Energy Technology Data Exchange (ETDEWEB)
Constantinidis, Clisthenis P; Piguet, Olivier [Universidade Federal do Espirito Santo (UFES), Vitoria, ES (Brazil); Spalenza, Wesley [Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste (Italy); Centro Brasileiro de Pesquisas Fsicas (CBPF), Rio de Janeiro (Brazil)
2006-01-01
We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N-extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator. (orig.)
9. On the restoration of supersymmetry in twisted two-dimensional lattice Yang-Mills theory
International Nuclear Information System (INIS)
Catterall, Simon
2007-01-01
We study a discretization of N = 2 super Yang-Mills theory which possesses a single exact supersymmetry at non-zero lattice spacing. This supersymmetry arises after a reformulation of the theory in terms of twisted fields. In this paper we derive the action of the other twisted supersymmetries on the component fields and study, using Monte Carlo simulation, a series of corresponding Ward identities. Our results for SU(2) and SU(3) support a restoration of these additional supersymmetries without fine tuning in the infinite volume continuum limit. Additionally we present evidence supporting a restoration of (twisted) rotational invariance in the same limit. Finally we have examined the distribution of scalar field eigenvalues and find evidence for power law tails extending out to large eigenvalue. We argue that these tails indicate that the classical moduli space does not survive in the quantum theory
10. D-branes in a non-critical superstrings and minimal super Yang-Mills in various dimensions
International Nuclear Information System (INIS)
Ashok, S.K.; Murthy, S.; Troost, J.
2005-11-01
We construct and analyze D-branes in superstring theories in even dimensions less than ten. The backgrounds under study are supersymmetric R d-1,1 x SL(2,R) k /U(1) where the level of the supercoset is tuned such as to provide bona fide string theory backgrounds. We provide exact boundary states for D-branes that are localized at the tip of the cigar SL(2,R)/U(1) supercoset conformal field theory. We analyze the spectra of open strings on these D-branes and show explicitly that they are consistent with supersymmetry in d = 2,4 and 6. The low energy theory on the world-volume of the D-brane in each case is pure Yang-Mills theory with minimal supersymmetry. In the case with four macroscopic flat directions d = 4, we realize an N = 1 super Yang-Mills theory, and we interpret the backreaction for the dilaton as the running of the gauge coupling, and study the relation between R-symmetry breaking in the gauge theory and the backreaction on the Rr axion. (author)
11. The yang mills gravity dual
International Nuclear Information System (INIS)
Crooks, David E.; Evans, Nick
2003-01-01
We describe a ten dimensional supergravity geometry which is dual to a gauge theory that is non-supersymmetric Yang Mills in the infra-red but reverts to N=4 super Yang Mills in the ultra-violet. A brane probe of the geometry shows that the scalar potential of the gauge theory is stable. We discuss the infra-red behaviour of the solution. The geometry describes a Schroedinger equation potential that determines the glueball spectrum of the theory; there is a mass gap and a discrete spectrum. The glueball mass predictions match previous AdS/CFT Correspondence computations in the non-supersymmetric Yang Mills theory, and lattice data, at the 10% level. (author)
12. Pure N=2 super Yang-Mills and exact WKB
International Nuclear Information System (INIS)
Kashani-Poor, Amir-Kian; Troost, Jan
2015-01-01
We apply exact WKB methods to the study of the partition function of pure N=2ϵ i -deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in ϵ 2 /ϵ 1 (i.e. at large central charge) and in an expansion in ϵ 1 . We find corrections of the form ∼exp [−((/tiny SW periods)/(ϵ 1 ))] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of ϵ 1 when passing from the Nekrasov form of the partition function to the topological string theory inspired form. We conjecture that such corrections should be computable from a worldsheet perspective on the partition function. Our results follow upon the determination of the Stokes graphs associated to the Mathieu equation with complex parameters and the application of exact WKB techniques to compute the Mathieu characteristic exponent.
13. (2,0)-Super-Yang-Mills coupled to non-linear {sigma}-model
Energy Technology Data Exchange (ETDEWEB)
Goes-Negrao, M.S.; Penna-Firme, A.B. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil); Negrao, M.R. [Universidade Federal, Rio de Janeiro, RJ (Brazil). Inst. de Fisica
1999-07-01
Considering a class of (2,0)-super yang-Mills multiplets that accommodate a pair of independent gauge potentials in connection with a single symmetry group, we present here their coupling to ordinary matter to non-linear {sigma}-models in (2,0)-superspace. The dynamics and the coupling of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials once light-cone coordinates are chosen. (author)
14. (2,0)-Super-Yang-Mills coupled to non-linear σ-model
International Nuclear Information System (INIS)
Goes-Negrao, M.S.; Penna-Firme, A.B.; Negrao, M.R.
1999-07-01
Considering a class of (2,0)-super yang-Mills multiplets that accommodate a pair of independent gauge potentials in connection with a single symmetry group, we present here their coupling to ordinary matter to non-linear σ-models in (2,0)-superspace. The dynamics and the coupling of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials are discussed and the interesting feature that comes out is a sort of chirality for one of the gauge potentials once light-cone coordinates are chosen. (author)
15. Finite field equation of Yang--Mills theory
International Nuclear Information System (INIS)
Brandt, R.A.; Wing-Chiu, N.; Yeung, W.
1980-01-01
We consider the finite local field equation -][1+1/α (1+f 4 )]g/sup munu/D'Alembertian-partial/sup μ/partial/sup ν/]A/sup nua/ =-(1+f 3 ) g 2 N[A/sup c/νA/sup a/μA/sub ν//sup c/] +xxx+(1-s) 2 M 2 A/sup a/μ, introduced by Lowenstein to rigorously describe SU(2) Yang--Mills theory, which is written in terms of normal products. We also consider the operator product expansion A/sup c/ν(x+xi) A/sup a/μ(x) A/sup b/lambda(x-xi) approx.ΣM/sup c/abνμlambda/sub c/'a'b'ν'μ'lambda' (xi) N[A/sup nuprimec/'A/sup muprimea/'A/sup lambdaprimeb/'](x), and using asymptotic freedom, we compute the leading behavior of the Wilson coefficients M/sup ...//sub .../(xi) with the help of a computer, and express the normal products in the field equation in terms of products of the c-number Wilson coefficients and of operator products like A/sup c/ν(x+xi) A/sup a/μ(x) A/sup b/lambda(x-xi) at separated points. Our result is -][1+(1/α)(1+f 4 )]g/sup munu/D'Alembertian-partial/sup μ/partial/sup ν/]A/sup nua/ =-(1+f 3 ) g 2 lim/sub xiarrow-right0/] (lnxi)/sup -0.28/2b/[A/sup c/ν (x+xi) A/sup a/μ(x) A/sub ν//sup c/(x-xi) +epsilon/sup a/bcA/sup muc/(x+xi) partial/sup ν/A/sup b//sub ν/(x)+xxx] +xxx]+(1-s) 2 M 2 A/sup a/μ, where β (g) =-bg 3 , and so (lnxi)/sup -0.28/2b/ is the leading behavior of the c-number coefficient multiplying the operator products in the field equation
16. A generalized Yang-Mills Theory I: general aspects of the classical theory
International Nuclear Information System (INIS)
Galvao, C.A.P.
1987-01-01
A generalized Yang-Mills theory which is the non-Abelian version of the generalized eletrodinamics proposed by Podolsky is analysed both in the Lagrangian an Hamiltonian formulation. A simple class of solutions to the Euler-Lagrange equations is presented and the structure of the Hamiltonian constraints is studied in details. (Author) [pt
17. D0-D4 brane tachyon condensation to a BPS state and its excitation spectrum in noncommutative super Yang-Mills theory
International Nuclear Information System (INIS)
Wimmer, Robert
2005-01-01
We investigate the D0-D4-brane system for different B-field backgrounds including the small instanton singularity in noncommutative SYM theory. We discuss the excitation spectrum of the unstable state as well as for the BPS D0-D4 bound state. We compute the tachyon potential which reproduces the complete mass defect. The relevant degrees of freedom are the massless (4,4) strings. Both results are in contrast with existing string field theory calculations. The excitation spectrum of the small instanton is found to be equal to the excitation spectrum of the fluxon solution on R θ 2 x R which we trace back to T-duality. For the effective theory of the (0,0) string excitations we obtain a BFSS matrix model. The number of states in the instanton background changes significantly when the B-field becomes self-dual. This leads us to the proposal of the existence of a phase transition or cross over at self-dual B-field
18. Spin foam models of Yang-Mills theory coupled to gravity
International Nuclear Information System (INIS)
Mikovic, A
2003-01-01
We construct a spin foam model of Yang-Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barrett-Crane ansatz. In the Euclidean gravity case, we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidean gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang-Mills theory coupled to gravity
19. Higgs amplitudes from supersymmetric form factors Part II: $\\mathcal{N}<4$ super Yang-Mills arXiv
CERN Document Server
Brandhuber, Andreas; Penante, Brenda; Travaglini, Gabriele
The study of form factors has many phenomenologically interesting applications, one of which is Higgs plus gluon amplitudes in QCD. Through effective field theory techniques these are related to form factors of various operators of increasing classical dimension. In this paper we extend our analysis of the first finite top-mass correction, arising from the operator ${\\rm Tr} (F^3)$, from $\\mathcal{N}=4$ super Yang-Mills to theories with $\\mathcal{N}<4$, for the case of three gluons and up to two loops. We confirm our earlier result that the maximally transcendental part of the associated Catani remainder is universal and equal to that of the form factor of a protected trilinear operator in the maximally supersymmetric theory. The terms with lower transcendentality deviate from the $\\mathcal{N}=4$ answer by a surprisingly small set of terms involving for example $\\zeta_2$, $\\zeta_3$ and simple powers of logarithms, for which we provide explicit expressions.
20. The Hagedorn temperature and open QCD-string tachyons in pure N=1 super-Yang-Mills
International Nuclear Information System (INIS)
2008-01-01
We consider large-N confining gauge theories with a Hagedorn density of states. In such theories the potential between a pair of colour-singlet sources may diverge at a critical distance r c =1/T H . We consider, in particular, pure N=1 super-Yang-Mills theory and argue that when a domain wall and an anti-domain wall are brought to a distance near r c the interaction potential is better described by an 'open QCD-string channel'. We interpret the divergence of the potential in terms of a tachyonic mode and relate its mass to the Hagedorn temperature. Finally we relate our result to a theorem of Kutasov and Seiberg and argue that the presence of an open string tachyonic mode in the annulus amplitude implies an exponential density of states in the UV of the closed string channel
1. Singular Minkowski and Euclidean solutions for SU(2) Yang-Mills theory
International Nuclear Information System (INIS)
Singleton, D.
1996-01-01
In this paper it is examined a solution to the SU(2) Yang-Mills-Higgs system, which is a trivial mathematical extension of recently discovered Schwarzschild- like solutions (Singleton D., Phys. Rev. D, 51 (1955) 5911). Physically, however, this new solution has drastically different properties from the Schwarzschild-like solutions. It is also studied a new classical solution for Euclidean SU(2) Yang-Mills theory. Again this new solution is a mathematically trivial extension of the Belavin-Polyakov-Schwartz-Tyupkin (BPST) (Belavin A. A. et al., Phys. Lett. B, 59 (1975) 85) instanton, but is physically very different. Unlike the usual instanton solution, the present solution is singular on a sphere of arbitrary radius in Euclidean space. Both of these solutions are infinite-energy solutions, so their practical value is somewhat unclear. However, they may be useful in exploring some of the mathematical aspects of classical Yang-Mills theory
2. Proof of ultraviolet finiteness for a planar non-supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Ananth, Sudarshan; Kovacs, Stefano; Shimada, Hidehiko
2007-01-01
This paper focuses on a three-parameter deformation of N=4 Yang-Mills that breaks all the supersymmetry in the theory. We show that the resulting non-supersymmetric gauge theory is scale invariant, in the planar approximation, by proving that its Green functions are ultraviolet finite to all orders in light-cone perturbation theory
3. Center-stabilized Yang-Mills Theory:Confinement and Large N Volume Independence
International Nuclear Information System (INIS)
Unsal, Mithat; Yaffe, Laurence G.
2008-01-01
We examine a double trace deformation of SU(N) Yang-Mills theory which, for large N and large volume, is equivalent to unmodified Yang-Mills theory up to O(1/N 2 ) corrections. In contrast to the unmodified theory, large N volume independence is valid in the deformed theory down to arbitrarily small volumes. The double trace deformation prevents the spontaneous breaking of center symmetry which would otherwise disrupt large N volume independence in small volumes. For small values of N, if the theory is formulated on R 3 x S 1 with a sufficiently small compactification size L, then an analytic treatment of the non-perturbative dynamics of the deformed theory is possible. In this regime, we show that the deformed Yang-Mills theory has a mass gap and exhibits linear confinement. Increasing the circumference L or number of colors N decreases the separation of scales on which the analytic treatment relies. However, there are no order parameters which distinguish the small and large radius regimes. Consequently, for small N the deformed theory provides a novel example of a locally four-dimensional pure gauge theory in which one has analytic control over confinement, while for large N it provides a simple fully reduced model for Yang-Mills theory. The construction is easily generalized to QCD and other QCD-like theories
4. Center-stabilized Yang-Mills theory: Confinement and large N volume independence
International Nuclear Information System (INIS)
Uensal, Mithat; Yaffe, Laurence G.
2008-01-01
We examine a double trace deformation of SU(N) Yang-Mills theory which, for large N and large volume, is equivalent to unmodified Yang-Mills theory up to O(1/N 2 ) corrections. In contrast to the unmodified theory, large N volume independence is valid in the deformed theory down to arbitrarily small volumes. The double trace deformation prevents the spontaneous breaking of center symmetry which would otherwise disrupt large N volume independence in small volumes. For small values of N, if the theory is formulated on R 3 xS 1 with a sufficiently small compactification size L, then an analytic treatment of the nonperturbative dynamics of the deformed theory is possible. In this regime, we show that the deformed Yang-Mills theory has a mass gap and exhibits linear confinement. Increasing the circumference L or number of colors N decreases the separation of scales on which the analytic treatment relies. However, there are no order parameters which distinguish the small and large radius regimes. Consequently, for small N the deformed theory provides a novel example of a locally four-dimensional pure-gauge theory in which one has analytic control over confinement, while for large N it provides a simple fully reduced model for Yang-Mills theory. The construction is easily generalized to QCD and other QCD-like theories.
5. 1992 Trieste lectures on topological gauge theory and Yang-Mills theory
International Nuclear Information System (INIS)
Thompson, G.
1993-05-01
In these lecture notes we explain a connection between Yang-Mills theory on arbitrary Riemann surfaces and two types of topological field theory, the so called BF and cohomological theories. The quantum Yang-Mills theory is solved exactly using path integral techniques. Explicit expressions, in terms of group representation theory, are obtained for the partition function and various correlation functions. In a particular limit the Yang-Mills theory devolves to the topological models and the previously determined correlation functions give topological information about the moduli spaces of flat connections. In particular, the partition function yields the volume of the moduli space for which an explicit expression is derived. These notes are self contained, with a basic introduction to the various ideas underlying the topological field theories. This includes some relatively new work on handling problems that arise in the presence of reducible connections, which in turn, forms the bridge between the various models under consideration. These notes are identical to those made available to participants of the 1992 summer school in Trieste, except for one or two additions added circa January 1993. (author). 52 refs, 6 figs
6. Freedom and confinement in lattice Yang-Mills theories: a case for divorce
International Nuclear Information System (INIS)
Colangelo, P.; Cosmai, L.; Pellicoro, M.; Preparata, G.
1986-01-01
It is presented evidence that nonperturbative effects in lattice gauge theories do not obey at small coupling constant (large β) asymptotic scaling, but they rather behave as suggested by a recent result in continuum Yang-Mills theories. It is also discussed the possible impact of these results on our understanding of QCD
7. Infrared stability of the Yang-Mills theory in the zero-instanton sector
International Nuclear Information System (INIS)
Olesen, P.
1976-12-01
Abstracting the decoupling theorem of Appelquist and Carazzone from perturbation theory it is shown that the Yang-Mills theory is infrared stable in the zero-instanton sector. It is pointed out that the argument is not valid when instantons are present. (Auth.)
8. Classical and semi-classical solutions of the Yang--Mills theory
International Nuclear Information System (INIS)
Jackiw, R.; Nohl, C.; Rebbi, C.
1977-12-01
This review summarizes what is known at present about classical solutions to Yang-Mills theory both in Euclidean and Minkowski space. The quantal meaning of these solutions is also discussed. Solutions in Euclidean space expose multiple vacua and tunnelling of the quantum theory. Those in Minkowski space-time provide a semi-classical spectrum for a conformal generator
9. Off-shell superspace D=10 super Yang-Mills from covariantly quantized Green-Schwarz superstring
International Nuclear Information System (INIS)
Nissimov, E.; Pacheva, S.; Solomon, S.
1988-05-01
We construct a gauge invariant superspace action in terms of unconstrained off-shell superfields for the D=10 supersymmetric Yang-Mills (SYM) theory. We use to this effect the point particle limit of the BRST charge of the covariantly quantized harmonic Green-Schwarz superstring and a general covariant action principle for overdetermined systems of nonlinear field equations of motion. One obtains gauge and super Poincare invariant equations of motion equivalent to the Nilsson's constraints for D=10 SYM. In the previous approaches (light-cone-gauge, component-fields) one would have to sacrifice either explicit Lorenz invariance or explicit supersymmetry while in the present approach they are both manifest. (authors)
10. Regularization independent analysis of the origin of two loop contributions to N=1 Super Yang-Mills beta function
Energy Technology Data Exchange (ETDEWEB)
Fargnoli, H.G.; Sampaio, Marcos; Nemes, M.C. [Federal University of Minas Gerais, ICEx, Physics Department, P.O. Box 702, Belo Horizonte, MG (Brazil); Hiller, B. [Coimbra University, Faculty of Science and Technology, Physics Department, Center of Computational Physics, Coimbra (Portugal); Baeta Scarpelli, A.P. [Setor Tecnico-Cientifico, Departamento de Policia Federal, Lapa, Sao Paulo (Brazil)
2011-05-15
We present both an ultraviolet and an infrared regularization independent analysis in a symmetry preserving framework for the N=1 Super Yang-Mills beta function to two loop order. We show explicitly that off-shell infrared divergences as well as the overall two loop ultraviolet divergence cancel out, whilst the beta function receives contributions of infrared modes. (orig.)
11. Regularization independent analysis of the origin of two loop contributions to N=1 Super Yang-Mills beta function
International Nuclear Information System (INIS)
Fargnoli, H.G.; Sampaio, Marcos; Nemes, M.C.; Hiller, B.; Baeta Scarpelli, A.P.
2011-01-01
We present both an ultraviolet and an infrared regularization independent analysis in a symmetry preserving framework for the N=1 Super Yang-Mills beta function to two loop order. We show explicitly that off-shell infrared divergences as well as the overall two loop ultraviolet divergence cancel out, whilst the beta function receives contributions of infrared modes. (orig.)
12. Perturbative Yang-Mills theory without Faddeev-Popov ghost fields
Science.gov (United States)
Huffel, Helmuth; Markovic, Danijel
2018-05-01
A modified Faddeev-Popov path integral density for the quantization of Yang-Mills theory in the Feynman gauge is discussed, where contributions of the Faddeev-Popov ghost fields are replaced by multi-point gauge field interactions. An explicit calculation to O (g2) shows the equivalence of the usual Faddeev-Popov scheme and its modified version.
13. Ito calculus for σ-models and Yang-Mills theories
International Nuclear Information System (INIS)
Patrascioiu, A.; Richard, J.L.
1984-07-01
It is pointed out that the effective continuum action for σ-models and Yang-Mills theories may differ from the naive continuum action by terms of order g 2 or higher, which are non-symmetric. The modifications are produced by a generalization of the Ito calculus to dimensions higher than one
14. Two-dimensional Yang-Mills theory in the leading 1/N expansion
International Nuclear Information System (INIS)
Wu, T.T.
1977-01-01
Recent controversies about the gauge invariance of the two-dimensional SU(N) Yang-Mills theory in the 't Hooft limit of large N are resolved. The fermion (quark) propagator is found explicitly, and is qualitatively different from those in the previous literature. (Auth.)
15. On the quantization and asymptotic freedom of Yang-Mills theory in the temporal gauge
International Nuclear Information System (INIS)
Frenkel, J.
1979-01-01
The relevance of the longitudinal gluons in the derivation of the Feynman rules in the temporal gauge is studied, via the canonical approach. It is then shown that these gluons are also fundamental for the asymptotic freedom of the Yang-Mills theory [pt
16. q-trace for quantum groups and q-deformed Yang-Mills theory
International Nuclear Information System (INIS)
Isaev, A.P.; Popowicz, Z.
1992-01-01
The definitions of orbits and q-trace for the quantum groups are introduced. Then the q-trace is used to construct the invariants for the quantum group orbits and to formulate the q-deformed Yang-Mills theory. The amusing formal relation of the Weinberg type mixing angle with the quantum group deformation parameter is discussed. (orig.)
17. N = 1 quasi self-duality in the N = 2 Yang-Mills theory
International Nuclear Information System (INIS)
Pavlik, O.V.; Yatsun, V.A.
1998-01-01
The system of first-order equations-quasi self-duality equations-for component fields of the N = 2 SUSY Yang-Mills theory is suggested, which leads to equations of motion and reduces to self-duality equations if scalar fields vanish. The symmetries of quasi self-duality equations are studied
18. On a relation between massive Yang-Mills theories and dual string models
International Nuclear Information System (INIS)
Mickelsson, J.
1983-01-01
The relations between mass terms in Yang-Mills theories, projective representations of the group of gauge transformations, boundary conditions on vector potentials and Schwinger terms in local charge algebra commutation relations are discussed. The commutation relations (with Schwinger terms) are similar to the current algebra commutation relations of the SU(N) extended dual string model. (orig.)
19. Equations of motion for the new D=10 N=1 supergravity-Yang-Mills theory
International Nuclear Information System (INIS)
Vashakidze, Sh.I.
1988-01-01
An on-shell superfield formulation of the dual (type IB) ten-dimensional N=1 supergravity coupled to Yang-Mills theory is presented. The coupling is completely specified in superspace by A-tensor supercurrent which, at the same time, takes into account all superstring corrections in the slope parameter expansion. The complete set of equations of motion is derived
20. Spontaneous transition to a stochastic state in a four-dimensional Yang-Mills quantum theory
International Nuclear Information System (INIS)
Semikhatov, A.M.
1983-01-01
The quantum expectation values in a four-dimensional Yang-Mills theory are represented in each topological sector as expectation values over the diffusion which develops in the ''fourth'' Euclidean time. The Langevin equations of this diffusion are stochastic duality equations in the A 4 = 0 gauge
1. Bäcklund Transformations in 10D SUSY Yang-Mills Theories
Science.gov (United States)
Gervais, Jean-Loup
A Bäcklund transformation is derived for the Yang's type (super) equations previously derived (hep-th/9811108) by M. Saveliev and the author, from the ten-dimensional super-Yang-Mills field equations in an on-shell light cone gauge. It is shown to be based upon a particular gauge transformation satisfying nonlinear conditions which ensure that the equations retain the same form. These Yang's type field equations are shown to be precisely such that they automatically provide a solution of these conditions. This Bäcklund transformation is similar to the one proposed by A. Leznov for self-dual Yang-Mills in four dimensions. In the introduction a personal recollection on the birth of supersymmetry is given.
2. The thermodynamics of quantum Yang-Mills theory theory and applications
CERN Document Server
Hofmann, Ralf
2016-01-01
This latest edition enhances the material of the first edition with a derivation of the value of the action for each of the Harrington-Shepard calorons/anticalorons that are relevant for the emergence of the thermal ground state. Also included are discussions of the caloron center versus its periphery, the role of the thermal ground state in U(1) wave propagation, photonic particle-wave duality, and calculational intricacies and book-keeping related to one-loop scattering of massless modes in the deconfining phase of an SU(2) Yang-Mills theory. Moreover, a derivation of the temperature-redshift relation of the CMB in deconfining SU(2) Yang-Mills thermodynamics and its application to explaining an apparent early re-ionization of the Universe are given. Finally, a mechanism of mass generation for cosmic neutrinos is proposed.
3. A novel supersymmetry in 2-dimensional Yang-Mills theory on Riemann surfaces
International Nuclear Information System (INIS)
Soda, Jiro
1991-02-01
We find a novel supersymmetry in 2-dimensional Maxwell and Yang-Mills theories. Using this supersymmetry, it is shown that the 2-dimensional Euclidean pure gauge theory on a closed Riemann surface Σ can be reduced to a topological field theory which is the 3-dimensional Chern-Simons gauge theory in the special space-time topology Σ x R. Related problems are also discussed. (author)
4. Integrable open spin chain in Super Yang-Mills and the plane-wave/SYM duality
International Nuclear Information System (INIS)
Chen Bin; Wang Xiaojun; Wu Yongshi
2004-01-01
We investigate the integrable structures in an N = 2 superconfomal Sp(N) Yang-Mills theory with matter, which is dual to an open+closed string system. We restrict ourselves to the BMN operators that correspond to free string states. In the closed string sector, an integrable structure is inherited from its parent theory, N = 4 SYM. For the open string sector, the planar one-loop mixing matrix for gauge invariant holomorphic operators is identified with the Hamiltonian of an integrable SU(3) open spin chain. Using the K-matrix formalism we identify the integrable open-chain boundary conditions that correspond to string boundary conditions. The solutions to the algebraic Bethe ansatz equations (ABAE) with a few impurities are shown to recover the anomalous dimensions that exactly match the spectrum of free open string in the plane-wave background. We also discuss the properties of the solutions of ABAE beyond the BMN regime. (author)
5. Extended pure Yang-Mills gauge theories with scalar and tensor gauge fields
International Nuclear Information System (INIS)
Gabrielli, E.
1991-01-01
The usual abelian gauge theory is extended to an interacting Yang-Mills-like theory containing vector, scalar and tensor gauge fields. These gauge fields are seen as components along the Clifford algebra basis of a gauge vector-spinorial field. Scalar fields φ naturally coupled to vector and tensor fields have been found, leading to a natural φ 4 coupling in the lagrangian. The full expression of the lagrangian for the euclidean version of the theory is given. (orig.)
6. Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
Science.gov (United States)
Kökényesi, Zoltán; Sinkovics, Annamaria; Szabo, Richard J.
2016-11-01
We derive the analog of the large $N$ Gross-Taylor holomorphic string expansion for the refinement of $q$-deformed $U(N)$ Yang-Mills theory on a compact oriented Riemann surface. The derivation combines Schur-Weyl duality for quantum groups with the Etingof-Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of $q$-deformed Yang-Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit $q=1$, the expansion defines a new $\\beta$-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit $\\beta=1$ to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and $\\beta$-ensembles of matrix models arising in refined topological string theory.
7. BFKL approach and six-particle MHV amplitude in N=4 super Yang-Mills
International Nuclear Information System (INIS)
Lipatov, L.N.; Prygarin, A.
2010-12-01
We consider the planar MHV amplitude in N=4 supersymmetric Yang-Mills theory for 2→4 particle scattering at two and three loops in the Regge kinematics. We perform an analytic continuation of two-loop result for the remainder function found by Goncharov, Spradlin, Vergu and Volovich to the physical region, where the remainder function does not vanish in the Regge limit. After the continuation both the leading and the subleading in the logarithm of the energy terms are extracted and analyzed. Using this result we calculate the next-to-leading corrections to the impact factors required in the BFKL approach. The BFKL technique was used to find the leading imaginary and real parts of the remainder function at three loops. (orig.)
8. BFKL approach and six-particle MHV amplitude in N=4 super Yang-Mills
Energy Technology Data Exchange (ETDEWEB)
Lipatov, L.N. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; St. Petersburg Inst. of Nuclear Physics, Gatchina (Russian Federation); Prygarin, A. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik
2010-12-15
We consider the planar MHV amplitude in N=4 supersymmetric Yang-Mills theory for 2{yields}4 particle scattering at two and three loops in the Regge kinematics. We perform an analytic continuation of two-loop result for the remainder function found by Goncharov, Spradlin, Vergu and Volovich to the physical region, where the remainder function does not vanish in the Regge limit. After the continuation both the leading and the subleading in the logarithm of the energy terms are extracted and analyzed. Using this result we calculate the next-to-leading corrections to the impact factors required in the BFKL approach. The BFKL technique was used to find the leading imaginary and real parts of the remainder function at three loops. (orig.)
9. Extended Soliton Solutions in an Effective Action for SU(2 Yang-Mills Theory
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available The Skyrme-Faddeev-Niemi (SFN model which is an O(3 σ model in three dimensional space up to fourth-order in the first derivative is regarded as a low-energy effective theory of SU(2 Yang-Mills theory. One can show from the Wilsonian renormalization group argument that the effective action of Yang-Mills theory recovers the SFN in the infrared region. However, the theory contains an additional fourth-order term which destabilizes the soliton solution. We apply the perturbative treatment to the second derivative term in order to exclude (or reduce the ill behavior of the original action and show that the SFN model with the second derivative term possesses soliton solutions.
10. A Note on Supergravity Duals of Noncommutative Yang-Mills Theory
International Nuclear Information System (INIS)
Das, Sumit R.; Ghosh, Banhiman
2000-01-01
A class of supergravity backgrounds have been proposed as dual descriptions of strong coupling large-N noncommutative Yang-Mills (NCYM) theories in 3+1 dimensions. However calculations of correlation functions in supergravity from an evaluation of relevant classical actions appear ambiguous. We propose a resolution of this ambiguity. Assuming that holographic description exists - regardless of whether it is the NCYM theory - we argue that there should be operators in the holographic boundary theory which create normalized states of definite energy and momenta. An operator version of the dual correspondence then provides a calculation of correlators of these operators in terms of bulk Green's functions. We show that in the low energy limit the correlators reproduce expected answers of the ordinary Yang-Mills theory. (author)
11. The (confinement) structure of Yang-Mills-theories within a Bose-BCS-theory
International Nuclear Information System (INIS)
Schuette, D.
1984-01-01
It is the purpose of this talk to report on a first attempt to apply (non-perturbative) techniques of many-body theory to a field-theory of the Yang-Mills-type. The procedure is in principle analogous to lattice calculations: In order to make the field-theoretical hamiltonian a well-behaved operator in the Fock-space, a phasespace-cutoff is assumed for the definition of the field operators. The coupling constant g then becomes a function of this cutoff which is fixed by some physical property like a glue-ball mass. (orig./HSI)
12. On electromagnetic duality in locally supersymmetric N = 2 Yang-Mills theory
CERN Document Server
Ceresole, Anna; Ferrara, S.; Van Proeyen, Antoine; Ceresole, A; D'Auria, R; Ferrara, S; Van Proeyen, A
1995-01-01
We consider duality transformations in N=2 Yang--Mills theory coupled to N=2 supergravity, in a manifestly symplectic and coordinate covariant setting. We give the essential of the geometrical framework which allows one to discuss stringy classical and quantum monodromies, the form of the spectrum of BPS saturated states and the Picard--Fuchs identities encoded in the special geometry of N=2 supergravity theories.
13. Low energy dynamics of monopoles in supersymmetric Yang-Mills theories with hypermultiplets
International Nuclear Information System (INIS)
Kim, Chanju
2006-01-01
We derive the low energy dynamics of monopoles and dyons in N = 2 supersymmetric Yang-Mills theories with hypermultiplets in arbitrary representations by utilizing a collective coordinate expansion. We consider the most general case that Higgs fields both in the vector multiplet and in the hypermultiplets have nonzero vacuum expectation values. The resulting theory is a supersymmetric quantum mechanics which has been obtained by a nontrivial dimensional reduction of two-dimensional (4,0) supersymmetric sigma models with potentials
14. On possibility of the conformal infrared asymptotics in nonabelian Yang-Mills theories
International Nuclear Information System (INIS)
Vasil'ev, A.N.; Perekalin, M.M.; Pis'mak, Yu.M.
1983-01-01
A possibility of the conformal-invariant infrared asymptotics in nonabelian Yang-Mills theories is discussed. In the framework of the conformal bootstrap method it is shown that the hypothesis about the exact conformal invariance contradicts the transversality of the polarization operator i.e. the Ward identities. However, it is still possible to use the conformal theory as an approximate solution to the bootstrap equations
15. Amplitude relations in heterotic string theory and Einstein-Yang-Mills
Energy Technology Data Exchange (ETDEWEB)
Schlotterer, Oliver [Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Potsdam (Germany)
2016-11-11
We present all-multiplicity evidence that the tree-level S-matrix of gluons and gravitons in heterotic string theory can be reduced to color-ordered single-trace amplitudes of the gauge multiplet. Explicit amplitude relations are derived for up to three gravitons, up to two color traces and an arbitrary number of gluons in each case. The results are valid to all orders in the inverse string tension α{sup ′} and generalize to the ten-dimensional superamplitudes which preserve 16 supercharges. Their field-theory limit results in an alternative proof of the recently discovered relations between Einstein-Yang-Mills amplitudes and those of pure Yang-Mills theory. Similarities and differences between the integrands of the Cachazo-He-Yuan formulae and the heterotic string are investigated.
16. Supercurrent and the Adler-Bardeen theorem in coupled supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Ensign, P.W.
1987-01-01
By the Adler-Bardeen theorem, only one-loop Feynman diagrams contribute to the anomalous divergences of quantum axial currents. The anomalous nature of scale transformations is manifested by an anomalous trace of the energy-momentum tensor, T/sup μ//sub μ/. Renormalization group arguments show that the quantum T/sup μ//sub μ/ must be proportional to the β-function. Since the β-function receives contributions at all loop levels, the Adler-Bardeen theorem appears to conflict with supersymmetry. Recently Grisaru, Milewski and Zanon constructed a supersymmetric axial current for pure supersymmetric Yang-Mills theory which satisfies the Adler-Bardeen theorem to two-loops. They used supersymmetric background field theory and regularization by dimensional reduction to maintain manifest supersymmetry and gauge invariance. In this thesis, their construction is extended to supersymmetric Yang-Mills theory coupled to chiral matter fields. The Adler-Bardeen theorem is then proven to all orders in perturbation theory for both the pure and coupled theories. The extension to coupled supersymmetric Yang-Mills supports the general validity of these techniques, and adds considerable insight into the structure of the anomalies. The all orders proof demonstrates that there is no conflict between supersymmetry and the Adler-Bardeen theorem
17. Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model
International Nuclear Information System (INIS)
Szabo, Richard J; Tierz, Miguel
2010-01-01
We derive some new relationships between matrix models of Chern-Simons gauge theory and of two-dimensional Yang-Mills theory. We show that q-integration of the Stieltjes-Wigert matrix model is the discrete matrix model that describes q-deformed Yang-Mills theory on S 2 . We demonstrate that the semiclassical limit of the Chern-Simons matrix model is equivalent to the Gross-Witten model in the weak-coupling phase. We study the strong-coupling limit of the unitary Chern-Simons matrix model and show that it too induces the Gross-Witten model, but as a first-order deformation of Dyson's circular ensemble. We show that the Sutherland model is intimately related to Chern-Simons gauge theory on S 3 , and hence to q-deformed Yang-Mills theory on S 2 . In particular, the ground-state wavefunction of the Sutherland model in its classical equilibrium configuration describes the Chern-Simons free energy. The correspondence is extended to Wilson line observables and to arbitrary simply laced gauge groups.
18. Duality transformations in supersymmetric Yang-Mills theories coupled to supergravity
CERN Document Server
Ceresole, Anna T; Ferrara, Sergio; Van Proeyen, A; Ceresole, A; D'Auria, R; Ferrara, S; Van Proeyen, A
1995-01-01
We consider duality transformations in N=2, d=4 Yang-Mills theory coupled to N=2 supergravity. A symplectic and coordinate covariant framework is established, which allows one to discuss stringy classical and quantum duality symmetries' (monodromies), incorporating T and S dualities. In particular, we shall be able to study theories (like N=2 heterotic strings) which are formulated in symplectic basis where a holomorphic prepotential' F does not exist, and yet give general expressions for all relevant physical quantities. Duality transformations and symmetries for the N=1 matter coupled Yang--Mills supergravity system are also exhibited. The implications of duality symmetry on all N>2 extended supergravities are briefly mentioned. We finally give the general form of the central charge and the N=2 semiclassical spectrum of the dyonic BPS saturated states (as it comes by truncation of the N=4 spectrum).
19. BRST quantization of Yang-Mills theory: A purely Hamiltonian approach on Fock space
Science.gov (United States)
Öttinger, Hans Christian
2018-04-01
We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories in an explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stage of the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilation operators, thus enabling us to characterize the physical states on a well-defined Fock space. The Hamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimal invariant extension of the free theory. It is verified that the evolution equations implied by the resulting minimal Hamiltonian provide a quantum version of the classical Yang-Mills equations. The modifications and requirements for the inclusion of matter are discussed in detail.
20. Four-dimensional Yang-Mills theory, gauge invariant mass and fluctuating three-branes
International Nuclear Information System (INIS)
Niemi, Antti J; Slizovskiy, Sergey
2010-01-01
We are interested in a gauge invariant coupling between four-dimensional Yang-Mills field and a three-brane that can fluctuate into higher dimensions. For this we interpret the Yang-Mills theory as a higher dimensional bulk gravity theory with dynamics that is governed by the Einstein action, and with a metric tensor constructed from the gauge field in a manner that displays the original gauge symmetry as an isometry. The brane moves in this higher dimensional spacetime under the influence of its bulk gravity, with dynamics determined by the Nambu action. This introduces the desired interaction between the brane and the gauge field in a way that preserves the original gauge invariance as an isometry of the induced metric. After a prudent change of variables the result can be interpreted as a gauge invariant and massive vector field that propagates in the original spacetime R 4 . The presence of the brane becomes entirely invisible, expect for the mass.
1. Some Comments on the String Singularity of the Yang-Mills-Higgs Theory
International Nuclear Information System (INIS)
Lim, Kok-Geng; Teh, Rosy
2010-01-01
We are going to make use of the regulated polar angle which had been introduced by Boulware et al.. to show that in the SU(2) Yang-Mills-Higgs theory when the magnetic monopole is carried by the gauge field, the Higgs field does not carry the monopole and vice versa. In the Yang-Mills-Higgs theory, our solution shows that when the parameter ε ≠ 0, the monopole is carried by the gauge field and there is a string singularity in the gauge field. When the parameter ε → 0, the monopole is transferred from the gauge field to the Higgs field and the string singularity disappeared. The solution is only singular at the origin, that is at r = 0 as it becomes the Wu-Yang monopole.
2. Continuum strong-coupling expansion of Yang-Mills theory: quark confinement and infra-red slavery
International Nuclear Information System (INIS)
Mansfield, P.
1994-01-01
We solve Schroedinger's equation for the ground-state of four-dimensional Yang-Mills theory as an expansion in inverse powers of the coupling. Expectation values computed with the leading-order approximation are reduced to a calculation in two-dimensional Yang-Mills theory which is known to confine. Consequently the Wilson loop in the four-dimensional theory obeys an area law to leading order and the coupling becomes infinite as the mass scale goes to zero. (orig.)
3. On generating functional of vertex functions in the Yang-Mills theories
International Nuclear Information System (INIS)
Lavrov, P.M.; Tyutin, I.V.
1981-01-01
It is shown that the generating functional GITA(kappa, PHI) in the Yang-Mills gauge theories for linear gauge conditions may be written as GITA(kappa, PHI)=GITA tilde(phi(kappa, PHI))-1/2t 2 , where t is a gauge function and GITA tilde(PHI) is a universal functional independent of kappa, parameters of the gauge condition [ru
4. Integrality of the monopole number in SU(2) Yang-Mills-Higgs theory on R3
International Nuclear Information System (INIS)
Groisser, D.
1984-01-01
We prove that in classical SU(2) Yang-Mills-Higgs theories on R 3 with a Higgs field in the adjoint representation, an integer-valued monopole number (magnetic charge) is canonically defined for any finite-action L 2 sub(1,loc) configuration. In particular the result is true for smooth configurations. The monopole number is shown to decompose the configuration space into path components. (orig.)
5. Higher conservation laws for ten-dimensional supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Abdalla, E.; Forger, M.; Freiburg Univ.; Jacques, M.
1988-01-01
It is shown that ten-dimensional supersymmetric Yang-Mills theories are integrable systems, in the (weak) sense of admitting a (superspace) Lax representation for their equations of motion. This is achieved by means of an explicit proof that the equations of motion are not only a consequence of but in fact fully equivalent to the superspace constraint F αβ =0. Moreover, a procedure for deriving infinite series of non-local conservation laws is outlined. (orig.)
6. The quantum cosmology of Einstein-Yang-Mills theory in Eight-dimensions
International Nuclear Information System (INIS)
Su Bing; Li Xinzhou
1991-01-01
The quantum cosmology of Einstein-Yang-Mills has been studied. The Hartle-Hawking's proposal for the boundary conditions of the universe is extended to Eight-dimensional Einstein-Yong-Mills theory. A miniuperspace wave function is calculated in the classical limit corresponding to a superposition of classical solutions in which four of the dimensions remain small while the other four behave like an inflationary universe
7. Monopole dynamics of yang-mills theory without gauge-fixing
International Nuclear Information System (INIS)
Jia Duojie; Li Xiguo
2003-01-01
A new off-shell decomposition of SU(2) gauge field without any gauge fixing is proposed. This decomposition yields, for an appropriate gauge-fixing, a Skyme-Faddeev-like Wilsonian action and confirms the presence of high-order derivatives of a color-unit-vector at the classical level. The 't Hooft's conjecture that 'monopole' dynamics of infrared Yang-Mills theory is projection independent is also independently demonstrated
8. Black Hole Solution of Einstein-Born-Infeld-Yang-Mills Theory
Directory of Open Access Journals (Sweden)
Kun Meng
2017-01-01
Full Text Available A new four-dimensional black hole solution of Einstein-Born-Infeld-Yang-Mills theory is constructed; several degenerated forms of the black hole solution are presented. The related thermodynamical quantities are calculated, with which the first law of thermodynamics is checked to be satisfied. Identifying the cosmological constant as pressure of the system, the phase transition behaviors of the black hole in the extended phase space are studied.
9. Correlation functions in topological Yang-Mills theory with two fermionic charges
International Nuclear Information System (INIS)
Marculescu, S.
1997-01-01
The solution of the Donaldson cohomology problem for the topological Yang-Mills theory with two fermionic symmetries needs besides the gauge field and its descendants additional fields, hereafter called ascendants of the gauge field. It is shown that the dependence of the ascendants disappears in the all the correlation functions. This property allows one for the usual interpretation of the Donaldson invariants as cocycles of the instanton moduli space. (orig.)
10. Algebraic renormalization of Yang-Mills theory with background field method
International Nuclear Information System (INIS)
Grassi, P.A.
1996-01-01
In this paper the renormalizability of Yang-Mills theory in the background gauge fixing is studied. By means of Ward identities of background gauge invariance and Slavnov-Taylor identities, in a regularization-independent way, the stability of the model under radiative corrections is proved and its renormalizability is verified. In particular, it is shown that the splitting between background and quantum field is stable under radiative corrections and this splitting does not introduce any new anomalies. (orig.)
11. Solitons in a six-dimensional super Yang-Mills-tensor system and non-critical strings
International Nuclear Information System (INIS)
Nair, V.P.; Randjbar-Daemi, S.
1997-11-01
In this letter we study a coupled system of six-dimensional N = 1 tensor and super Yang-Mills multiplets. We identify some of the solitonic states of this system which exhibit stringy behaviour in six dimensions. A discussion of the supercharges and energy for the tensor multiples as well as zero modes is also given. We speculate about the possible relationship between our solution and what is known as tensionless strings. (author)
12. Non-Abelian sigma models from Yang-Mills theory compactified on a circle
Science.gov (United States)
Ivanova, Tatiana A.; Lechtenfeld, Olaf; Popov, Alexander D.
2018-06-01
We consider SU(N) Yang-Mills theory on R 2 , 1 ×S1, where S1 is a spatial circle. In the infrared limit of a small-circle radius the Yang-Mills action reduces to the action of a sigma model on R 2 , 1 whose target space is a 2 (N - 1)-dimensional torus modulo the Weyl-group action. We argue that there is freedom in the choice of the framing of the gauge bundles, which leads to more general options. In particular, we show that this low-energy limit can give rise to a target space SU (N) ×SU (N) /ZN. The latter is the direct product of SU(N) and its Langlands dual SU (N) /ZN, and it contains the above-mentioned torus as its maximal Abelian subgroup. An analogous result is obtained for any non-Abelian gauge group.
13. Multiple direct exchange in a Yang-Mills theory at high energy
International Nuclear Information System (INIS)
McCoy, B.M.; Wu, T.T.
1976-01-01
For eighth and higher orders, we obtain the leading high-energy behavior of the sum of all one-layer Feynman diagrams in Yang-Mills theory. These are the contributions from Feynman diagrams where the two incident fast particles exchange directly Yang-Mills bosons that are much less energetic. The incident particles may be either bosons or fermions of arbitrary isospin, and the result is also generalized to include the case of the Higgs scalar. The scattering amplitudes in all these cases are closely related, and all behave as s ln/sub n/ -1 s in the 2n + 2 order. Furthermore, in this leading order for n > or = 2, the exchanged isospins are always 0 and 2, no matter how high the isospins of the incident particles are
14. Non-Douglas-Kazakov phase transition of two-dimensional generalized Yang-Mills theories
International Nuclear Information System (INIS)
2007-01-01
In two-dimensional Yang-Mills and generalized Yang-Mills theories for large gauge groups, there is a dominant representation determining the thermodynamic limit of the system. This representation is characterized by a density, the value of which should everywhere be between zero and one. This density itself is determined by means of a saddle-point analysis. For some values of the parameter space, this density exceeds one in some places. So one should modify it to obtain an acceptable density. This leads to the well-known Douglas-Kazakov phase transition. In generalized Yang-Mills theories, there are also regions in the parameter space where somewhere this density becomes negative. Here too, one should modify the density so that it remains nonnegative. This leads to another phase transition, different from the Douglas-Kazakov one. Here the general structure of this phase transition is studied, and it is shown that the order of this transition is typically three. Using carefully-chosen parameters, however, it is possible to construct models with the order of the phase transition not equal to three. A class of these non-typical models is also studied. (orig.)
15. Sigma-model formulation of the Yang-Mills theory on four-dimensional hypersphere
International Nuclear Information System (INIS)
Ivanov, E.A.; Krivonos, S.O.
1983-01-01
The bilocal sigma-model representation is constructed for Yang-Mills theory in the simplest conformally flat hyperspherical spases SO(1, 4)/SO(1, 3), SO(2, 3)/SO(1, 3) and SO(5)/SO(4) (for the Euclidean Yang-Mills). Like in the case of Minkowski and Euclidean spaces, Yang-Mills potential is defined as bsub(μ)(x)=dsub(μ)sup(y)b(x, y)sub(y=0), b(x, y) being a bilocal Goldstone field which takes values in the gauge group algebra and is subjected to certain covariant constraints. The minimal version of these constraints results in the ''string'' representation for b(x, y) through the P exponential of bsub(μ)(x) along the fixed paths coinciding with geodesics. Due to the presence of closed geodesics, the contour functional naturally appear in the theory, with contours being the circles with the hypersphere radius. The sigma-model representation is shown to be Weyl-covariant: its formulations in different conformally flat spaces are related by transformations of ysup(rho). The geometric meaning ysup(rho) and minimal constraints is explained, and the conformal group transofrmation of ysup(rho) is found
16. An introduction to topological Yang-Mills theory
International Nuclear Information System (INIS)
Baal, P. van; Rijksuniversiteit Utrecht
1990-01-01
In these lecture notes I give a ''historical'' introduction to topological gauge theories. My main aim is to clearly explain the origin of the Hamiltonian which forms the basis of Witten's construction of topological gauge theory. I show how this Hamiltonian arises from Witten's formulation of Morse theory as applied by Floer to the infinite dimensional space of gauge connections, with the Chern-Simons functional as the appriopriate Morse function(al). I therefore discuss the De Rham cohomology, Hodge theory, Morse theory, Floer homology, Witten's construction of the Lagrangian for topological gauge theory, the subsequent BRST formulation of topological quantum field theory and finally Witten's construction of the Donaldson polynomials. (author)
17. Center-symmetric effective theory for high-temperature SU(2) Yang-Mills theory
International Nuclear Information System (INIS)
Forcrand, Ph. de; Kurkela, A.; Vuorinen, A.
2008-01-01
We construct and study a dimensionally reduced effective theory for high-temperature SU(2) Yang-Mills theory that respects all the symmetries of the underlying theory. Our main motivation is to study whether the correct treatment of the center symmetry can help extend the applicability of the dimensional reduction procedure towards the confinement transition. After performing perturbative matching to the full theory at asymptotically high temperatures, we map the phase diagram of the effective theory using nonperturbative lattice simulations. We find that at lower temperature the theory undergoes a second-order confining phase transition, in complete analogy with the full theory, which is a direct consequence of having incorporated the center symmetry
18. Towards a 'pointless' generalisation of Yang-Mills theory
International Nuclear Information System (INIS)
Chan Hongmo; Tsou Sheungtsun
1989-05-01
We examine some generalisations in physical concepts of gauge theories, leading towards a scenario corresponding to non-commutative geometry, where the concept of locality loses its usual meaning of being associated with points on a base manifold and becomes intertwined with the concept of internal symmetry, suggesting thereby a gauge theory of extended objects. Examples are given where such generalised gauge structures can be realised, in particular that of string theory. (author)
19. Spinning superstrings at two loops: Strong-coupling corrections to dimensions of large-twist super Yang-Mills operators
International Nuclear Information System (INIS)
Roiban, R.; Tseytlin, A. A.
2008-01-01
We consider folded (S,J) spinning strings in AdS 5 xS 5 (with one spin component in AdS 5 and a one in S 5 ) corresponding to the Tr(D S Φ J ) operators in the sl(2) sector of the N=4 super Yang-Mills theory in the special scaling limit in which both the string mass ∼√(λ)lnS and J are sent to infinity with their ratio fixed. Expanding in the parameter l=(J/√(λ)lnS) we compute the 2-loop string sigma-model correction to the string energy and show that it agrees with the expression proposed by Alday and Maldacena [J. High Energy Phys. 11 (2007) 019]. We suggest that a resummation of the logarithmic l 2 ln n l terms is necessary in order to establish an interpolation to the weakly coupled gauge-theory results. In the process, we set up a general framework for the calculation of higher loop corrections to the energy of multispin string configurations. In particular, we find that in addition to the direct 2-loop term in the string energy there is a contribution from lower loop order due to a finite 'renormalization' of the relation between the parameters of the classical solution and the fixed spins, i.e., the charges of the SO(2,4)xSO(6) symmetry.
20. Post-Newtonian approximation of the maximum four-dimensional Yang-Mills gauge theory
International Nuclear Information System (INIS)
Smalley, L.L.
1982-01-01
We have calculated the post-Newtonian approximation of the maximum four-dimensional Yang-Mills theory proposed by Hsu. The theory contains torsion; however, torsion is not active at the level of the post-Newtonian approximation of the metric. Depending on the nature of the approximation, we obtain the general-relativistic values for the classical Robertson parameters (γ = β = 1), but deviations for the Nordtvedt effect and violations of post-Newtonian conservation laws. We conclude that in its present form the theory is not a viable theory of gravitation
1. Unified theory of gravitation, electromagnetism, and the Yang-Mills field
International Nuclear Information System (INIS)
Borchsenius, K.
1976-01-01
The recent modification and extension of Einstein's nonsymmetric unified field theory for gravitation and electromagnetism is generalized to include the Yang-Mills field theory. The generalization consists in assuming that the components of the linear connection and of the fundamental tensor are not ordinary c numbers but are matrices related to some unitary symmetry. As an example we consider the SU(2) case. The theory is applied to the gauge-covariant formulation of electrically and isotopically charged spin-1/2 field theories
2. Supergravity and Yang-Mills theories as generalized topological fields with constraints
International Nuclear Information System (INIS)
Ling Yi; Tung Rohsuan; Guo Hanying
2004-01-01
We present a general approach to construct a class of generalized topological field theories with constraints by means of generalized differential calculus and its application to connection theory. It turns out that not only the ordinary BF formulations of general relativity and Yang-Mills theories, but also the N=1,2 chiral supergravities can be reformulated as these constrained generalized topological field theories once the free parameters in the Lagrangian are specially chosen. We also show that the Chern-Simons action on the boundary may naturally be induced from the generalized topological action in the bulk, rather than introduced by hand
3. Pure spinors as auxiliary fields in the ten-dimensional supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Nilsson, B.E.W.
1986-01-01
A new way of introducing auxiliary fields into the ten-dimensional supersymmetric Yang-Mills theory is proposed. The auxiliary fields are commuting 'pure spinors' and constitute a non-linear realisation of the Lorentz group. This invalidates previous no-go theorems concerning the possibility of going off-shell in this theory. There seems to be a close relation between pure spinors and the concepts usually used in twistor theory. The non-Abelian theory can be constructed for all groups having pseudo-real representations. (author)
4. Integrable System and $N=2$ Supersymmetric Yang-Mills Theory
OpenAIRE
Nakatsu, T.; Takasaki, K.
1996-01-01
Comment: 6 pages,latex file with sprocl.sty, no figures, to appear in the Proceedings of the Workshop (Talk presented at the Workshop "Frontiers in Quantum Field Theory" in honor of the 60th birthday of Prof. Keiji Kikkawa, Osaka, Japan, December 1995
5. Dimensional versus lattice regularization within Luescher's Yang Mills theory
International Nuclear Information System (INIS)
Diekmann, B.; Langer, M.; Schuette, D.
1993-01-01
It is pointed out that the coefficients of Luescher's effective model space Hamiltonian, which is based upon dimensional regularization techniques, can be reproduced by applying folded diagram perturbation theory to the Kogut Susskind Hamiltonian and by performing a lattice continuum limit (keeping the volume fixed). Alternative cutoff regularizations of the Hamiltonian are in general inconsistent, the critical point beeing the correct prediction for Luescher's tadpole coefficient which is formally quadratically divergent and which has to become a well defined (negative) number. (orig.)
6. Lorentz-violating Yang-Mills theory. Discussing the Chern-Simons-like term generation
Energy Technology Data Exchange (ETDEWEB)
Santos, Tiago R.S.; Sobreiro, Rodrigo F. [UFF-Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ (Brazil)
2017-12-15
We analyze the Chern-Simons-like term generation in the CPT-odd Lorentz-violating Yang-Mills theory interacting with fermions. Moreover, we study the anomalies of this model as well as its quantum stability. The whole analysis is performed within the algebraic renormalization theory, which is independent of the renormalization scheme. In addition, all results are valid to all orders in perturbation theory. We find that the Chern-Simons-like term is not generated by radiative corrections, just like its Abelian version. Additionally, the model is also free of gauge anomalies and quantum stable. (orig.)
7. Exact Gell-Mann-Low function of supersymmetric Yang-Mills theories from instanton calculus
International Nuclear Information System (INIS)
Novikov, V.A.; Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I.
1983-01-01
The instanton contribution to the vacuum energy is supersymmetric Yang-Mills theories is considered. Using renormalizability of the theory the exact beta function for n-extended supersymmetry (n=1, 2, 4) is found. The coefficients of the beta function have a geometrical meaning - they are associated with the number of boson and fermion zero modes in the instanton field. If extra matter superfields are added the method allows one to fix the first two coefficients. A non-renormalization theorem which extends cancellation of vacuum loops to the case of the external instanton field is proved
8. Exact Gell-Mann-Low function of supersymmetric Yang-Mills theories from instanton calculus
International Nuclear Information System (INIS)
Novikov, V.A.; Shifman, M.A.; Vainshtein, A.I.; Zakharov, V.I.
1983-01-01
The instanton contribution to the vacuum energy in supersymmetric Yang-Mills theories is considered. Using the renormalizability of the theory we find the exact beta function for n-extended supersymmetry (n=1, 2, 4). The coefficients of the beta function have a geometrical meaning: they are associated with the number of boson and fermion zero modes in the instanton field. If extra matter superfields are added our method allows one to fix the first two coefficients. We prove a non-renormalization theorem which extends the cancellation of vacuum loops to the case of the external instanton field. (orig.)
9. A new quantum representation for canonical gravity and SU(2) Yang-Mills theory
International Nuclear Information System (INIS)
Loll, R.
1990-04-01
Starting from Rovelli-Smolin's infinite-dimensional graded Poisson-bracket algebra of loop variables, we propose a new way of constructing a corresponding quantum representation. After eliminating certain quadratic constraints, we 'integrate' an infinite-dimensional subalgebra of loop variables, using a formal group law expansion. With the help of techniques from the representation theory of semidirect-product groups, we find an exact quantum representation of the full classical Poisson-bracket algebra of loop variables, without any higher-order correction terms. This opens new ways of tackling the quantum dynamics for both canonical gravity and Yang-Mills theory. (orig.)
10. A new quantum representation for canonical gravity and SU(2) Yang-Mills theory
International Nuclear Information System (INIS)
Loll, R.
1991-01-01
Starting from Rovelli-Smolin's infinite-dimensional graded Poisson-bracket algebra of loop variables, we propose a new way of constructing a corresponding quantum representation. After eliminating certain quadratic constraints, we 'integrate' an infinite-dimensional subalgebra of loop variables, using a formal group law expansion. With the help of techniques from the representation theory of semidirect-product groups, we find an exact quantum representation of the full classical Poisson-bracket algebra of loop variables, without any higher-order correction terms. This opens new ways of tackling the quantum dynamics for both canonical gravity and Yang-Mills theory. (orig.)
11. Gauge theories of Yang-Mills vector fields coupled to antisymmetric tensor fields
International Nuclear Information System (INIS)
Anco, Stephen C.
2003-01-01
A non-Abelian class of massless/massive nonlinear gauge theories of Yang-Mills vector potentials coupled to Freedman-Townsend antisymmetric tensor potentials is constructed in four space-time dimensions. These theories involve an extended Freedman-Townsend-type coupling between the vector and tensor fields, and a Chern-Simons mass term with the addition of a Higgs-type coupling of the tensor fields to the vector fields in the massive case. Geometrical, field theoretic, and algebraic aspects of the theories are discussed in detail. In particular, the geometrical structure mixes and unifies features of Yang-Mills theory and Freedman-Townsend theory formulated in terms of Lie algebra valued curvatures and connections associated to the fields and nonlinear field strengths. The theories arise from a general determination of all possible geometrical nonlinear deformations of linear Abelian gauge theory for one-form fields and two-form fields with an Abelian Chern-Simons mass term in four dimensions. For this type of deformation (with typical assumptions on the allowed form considered for terms in the gauge symmetries and field equations), an explicit classification of deformation terms at first-order is obtained, and uniqueness of deformation terms at all higher orders is proven. This leads to a uniqueness result for the non-Abelian class of theories constructed here
12. Non-supersymmetric matrix strings from generalized Yang-Mills theory on arbitrary Riemann surfaces
International Nuclear Information System (INIS)
Billo, M.; D'Adda, A.; Provero, P.
2000-01-01
We quantize pure 2d Yang-Mills theory on an arbitrary Riemann surface in the gauge where the field strength is diagonal. Twisted sectors originate, as in Matrix string theory, from permutations of the eigenvalues around homotopically non-trivial loops. These sectors, that must be discarded in the usual quantization due to divergences occurring when two eigenvalues coincide, can be consistently kept if one modifies the action by introducing a coupling of the field strength to the space-time curvature. This leads to a generalized Yang-Mills theory whose action reduces to the usual one in the limit of zero curvature. After integrating over the non-diagonal components of the gauge fields, the theory becomes a free string theory (sum over unbranched coverings) with a U(1) gauge theory on the world-sheet. This is shown to be equivalent to a lattice theory with a gauge group which is the semi-direct product of S N and U(1) N . By using well known results on the statistics of coverings, the partition function on arbitrary Riemann surfaces and the kernel functions on surfaces with boundaries are calculated. Extensions to include branch points and non-abelian groups on the world-sheet are briefly commented upon
13. Reformulations of the Yang-Mills theory toward quark confinement and mass gap
Energy Technology Data Exchange (ETDEWEB)
Kondo, Kei-Ichi; Shinohara, Toru [Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522 (Japan); Kato, Seikou [Fukui National College of Technology, Sabae 916-8507 (Japan); Shibata, Akihiro [Computing Research Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 (Japan)
2016-01-22
We propose the reformulations of the SU (N) Yang-Mills theory toward quark confinement and mass gap. In fact, we have given a new framework for reformulating the SU (N) Yang-Mills theory using new field variables. This includes the preceding works given by Cho, Faddeev and Niemi, as a special case called the maximal option in our reformulations. The advantage of our reformulations is that the original non-Abelian gauge field variables can be changed into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge-independent way. Our reformulations can be combined with the SU (N) extension of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the Wilson loop operator to give a gauge-invariant definition for the magnetic monopole in the SU (N) Yang-Mills theory without the scalar field. In the so-called minimal option, especially, the restricted field is non-Abelian and involves the non-Abelian magnetic monopole with the stability group U (N− 1). This suggests the non-Abelian dual superconductivity picture for quark confinement. This should be compared with the maximal option: the restricted field is Abelian and involves only the Abelian magnetic monopoles with the stability group U(1){sup N−1}, just like the Abelian projection. We give some applications of this reformulation, e.g., the stability for the homogeneous chromomagnetic condensation of the Savvidy type, the large N treatment for deriving the dimensional transmutation and understanding the mass gap, and also the numerical simulations on a lattice which are given by Dr. Shibata in a subsequent talk.
14. Reformulations of the Yang-Mills theory toward quark confinement and mass gap
Science.gov (United States)
Kondo, Kei-Ichi; Kato, Seikou; Shibata, Akihiro; Shinohara, Toru
2016-01-01
We propose the reformulations of the SU (N) Yang-Mills theory toward quark confinement and mass gap. In fact, we have given a new framework for reformulating the SU (N) Yang-Mills theory using new field variables. This includes the preceding works given by Cho, Faddeev and Niemi, as a special case called the maximal option in our reformulations. The advantage of our reformulations is that the original non-Abelian gauge field variables can be changed into the new field variables such that one of them called the restricted field gives the dominant contribution to quark confinement in the gauge-independent way. Our reformulations can be combined with the SU (N) extension of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the Wilson loop operator to give a gauge-invariant definition for the magnetic monopole in the SU (N) Yang-Mills theory without the scalar field. In the so-called minimal option, especially, the restricted field is non-Abelian and involves the non-Abelian magnetic monopole with the stability group U (N- 1). This suggests the non-Abelian dual superconductivity picture for quark confinement. This should be compared with the maximal option: the restricted field is Abelian and involves only the Abelian magnetic monopoles with the stability group U(1)N-1, just like the Abelian projection. We give some applications of this reformulation, e.g., the stability for the homogeneous chromomagnetic condensation of the Savvidy type, the large N treatment for deriving the dimensional transmutation and understanding the mass gap, and also the numerical simulations on a lattice which are given by Dr. Shibata in a subsequent talk.
15. Unified Maxwell-Einstein and Yang-Mills-Einstein supergravity theories in five dimensions
International Nuclear Information System (INIS)
Guenaydin, Murat; Zagermann, Marco
2003-01-01
Unified N = 2 Maxwell-Einstein supergravity theories (MESGTs) are supergravity theories in which all the vector fields, including the graviphoton, transform in an irreducible representation of a simple global symmetry group of the Lagrangian. As was established long time ago, in five dimensions there exist only four unified Maxwell-Einstein supergravity theories whose target manifolds are symmetric spaces. These theories are defined by the four simple euclidean Jordan algebras of degree three. In this paper, we show that, in addition to these four unified MESGTs with symmetric target spaces, there exist three infinite families of unified MESGTs as well as another exceptional one. These novel unified MESGTs are defined by non-compact (minkowskian) Jordan algebras, and their target spaces are in general neither symmetric nor homogeneous. The members of one of these three infinite families can be gauged in such a way as to obtain an infinite family of unified N = 2 Yang-Mills-Einstein supergravity theories, in which all vector fields transform in the adjoint representation of a simple gauge group of the type SU(N,1). The corresponding gaugings in the other two infinite families lead to Yang-Mills-Einstein supergravity theories coupled to tensor multiplets. (author)
16. FIFTY YEARS OF YANG-MILLS THEORIES: A Phenomenological Point of View
Science.gov (United States)
de Rújula, Alvaro
On the occasion of the celebration of the first half-century of Yang-Mills theories, I am contributing a personal recollection of how the subject, in its early times, confronted physical reality, that is, its "phenomenology". There is nothing original in this work, except, perhaps, my own points of view. But I hope that the older practitioners of the field will find here grounds form nostalgia, or good reasons to disagree with me. Younger addicts may learn that history does not resemble at all what is reflected in current textbooks: it was orders of magnitude more fascinating.
17. Verifying the Kugo-Ojima Confinement Criterion in Landau Gauge Yang-Mills Theory
International Nuclear Information System (INIS)
Watson, Peter; Alkofer, Reinhard
2001-01-01
Expanding the Landau gauge gluon and ghost two-point functions in a power series we investigate their infrared behavior. The corresponding powers are constrained through the ghost Dyson-Schwinger equation by exploiting multiplicative renormalizability. Without recourse to any specific truncation we demonstrate that the infrared powers of the gluon and ghost propagators are uniquely related to each other. Constraints for these powers are derived, and the resulting infrared enhancement of the ghost propagator signals that the Kugo-Ojima confinement criterion is fulfilled in Landau gauge Yang-Mills theory
18. From decay to complete breaking: pulling the strings in SU(2) Yang-Mills theory.
Science.gov (United States)
Pepe, M; Wiese, U-J
2009-05-15
We study {2Q+1} strings connecting two static charges Q in (2+1)D SU(2) Yang-Mills theory. While the fundamental {2} string between two charges Q=1/2 is unbreakable, the adjoint {3} string connecting two charges Q=1 can break. When a {4} string is stretched beyond a critical length, it decays into a {2} string by gluon pair creation. When a {5} string is stretched, it first decays into a {3} string, which eventually breaks completely. The energy of the screened charges at the ends of a string is well described by a phenomenological constituent gluon model.
19. Integrable model of Yang-Mills theory with scalar field and quasi-instantons
International Nuclear Information System (INIS)
Yatsun, V.A.
1988-01-01
In the framework of Euclidean conformally invariant Yang-Mills theory with a scalar field a study is made of a Hamiltonian system with two degrees of freedom that is integrable for a definite relationship between the coupling constants. A particular solution of the Hamilton-Jacobi equation leads to first-order equations that ensure a nonself-dual solution of instanton type of the considered model. As generalization of the first-order equations a quasiself-dual equation that can be integrated by means of the 't Hooft ansatz and leads to quasiself-dual instantons - quasi-instantons - is proposed
20. Integrable model of Yang-Mills theory and quasi-instantons
International Nuclear Information System (INIS)
Yatsun, V.A.
1986-01-01
Within the framework of Euclidean conformal invariant Yang-Mills theory with a scalar field, a two-dimensional Hamiltonian system integrable for a definite relation between the coupling constants is considered. A particular solution of the Hamilton-Jacobi equation leads to a system of first-order equations providing a nonself-dual instanton-like solution of the model concerned. As a generalizationof the system, a quasi-self-duality equation is suggested which is integrated by means of the 't Hooft ansatz and results in quasi-self-dual instantons (quasi-instantons). (orig.)
1. Towards a precise determination of the topological susceptibility in the SU(3) Yang-Mills theory
CERN Document Server
Giusti, Leonardo; Petrarca, Silvano
2009-01-01
An ongoing effort to compute the topological susceptibility for the SU(3) Yang-Mills theory in the continuum limit with a precison of about 2% is reported. The susceptibility is computed by using the definition of the charge suggested by Neuberger fermions for two values of the negative mass parameter s. Finite volume and discretization effects are estimated to meet this level of precision. The large statistics required has been obtained by using PCs of the INFN-GRID. Simulations with larger lattice volumes are necessary in order to better understanding the continuum limit at small lattice spacing values.
2. Supersymmetric Yang-Mills theory on conformal supergravity backgrounds in ten dimensions
Energy Technology Data Exchange (ETDEWEB)
Medeiros, Paul de; Figueroa-O’Farrill, José [Maxwell Institute and School of Mathematics, The University of Edinburgh,James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD (United Kingdom)
2016-03-14
We consider bosonic supersymmetric backgrounds of ten-dimensional conformal supergravity. Up to local conformal isometry, we classify the maximally supersymmetric backgrounds, determine their conformal symmetry superalgebras and show how they arise as near-horizon geometries of certain half-BPS backgrounds or as a plane-wave limit thereof. We then show how to define Yang-Mills theory with rigid supersymmetry on any supersymmetric conformal supergravity background and, in particular, on the maximally supersymmetric backgrounds. We conclude by commenting on a striking resemblance between the supersymmetric backgrounds of ten-dimensional conformal supergravity and those of eleven-dimensional Poincaré supergravity.
3. Variational study of mass generation and deconfinement in Yang-Mills theory
Science.gov (United States)
Comitini, Giorgio; Siringo, Fabio
2018-03-01
A very simple variational approach to pure SU (N ) Yang-Mills theory is proposed, based on the Gaussian effective potential in a linear covariant gauge. The method provides an analytical variational argument for mass generation. The method can be improved order by order by a perturbative massive expansion around the optimal trial vacuum. At finite temperature, a weak first-order transition is found (at Tc≈250 MeV for N =3 ) where the mass scale drops discontinuously. Above the transition the optimal mass increases linearly as expected for deconfined bosons. The equation of state is found in good agreement with the lattice data.
4. A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory
International Nuclear Information System (INIS)
Badger, Simon; Mogull, Gustav; Ochirov, Alexander; O’Connell, Donal
2015-01-01
We compute the integrand of the full-colour, two-loop, five-gluon scattering amplitude in pure Yang-Mills theory with all helicities positive, using generalized unitarity cuts. Tree-level BCJ relations, satisfied by amplitudes appearing in the cuts, allow us to deduce all the necessary non-planar information for the full-colour amplitude from known planar data. We present our result in terms of irreducible numerators, with colour factors derived from the multi-peripheral colour decomposition. Finally, the leading soft divergences are checked to reproduce the expected infrared behaviour.
5. Fifty years of Yang-Mills Theories: a phenomenological point of view
CERN Document Server
De Rújula, Alvaro
2005-01-01
On the occasion of the celebration of the first half-century of Yang--Mills theories, I am contributing a personal recollection of how the subject, in its early times, confronted physical reality, that is, its "phenomenology". There is nothing original in this work, except, perhaps, my own points of view. But I hope that the older practitioners of the field will find here grounds for nostalgia, or good reasons to disagree with me. Younger addicts may learn that history does not resemble at all what is reflected in current textbooks: it was orders of magnitude more fascinating.
6. Chern-Simons theory, 2d Yang-Mills, and Lie algebra wanderers
International Nuclear Information System (INIS)
Haro, Sebastian de
2005-01-01
We work out the relation between Chern-Simons, 2d Yang-Mills on the cylinder, and Brownian motion. We show that for the unitary, orthogonal and symplectic groups, various observables in Chern-Simons theory on S 3 and lens spaces are exactly given by counting the number of paths of a Brownian particle wandering in the fundamental Weyl chamber of the corresponding Lie algebra. We construct a fermionic formulation of Chern-Simons on S 3 which allows us to identify the Brownian particles as B-model branes moving on a noncommutative two-sphere, and construct 1- and 2-matrix models to compute Brownian motion ensemble averages
7. A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Badger, Simon; Mogull, Gustav; Ochirov, Alexander [Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland (United Kingdom); O’Connell, Donal [Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland (United Kingdom); Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030 (United States)
2015-10-09
We compute the integrand of the full-colour, two-loop, five-gluon scattering amplitude in pure Yang-Mills theory with all helicities positive, using generalized unitarity cuts. Tree-level BCJ relations, satisfied by amplitudes appearing in the cuts, allow us to deduce all the necessary non-planar information for the full-colour amplitude from known planar data. We present our result in terms of irreducible numerators, with colour factors derived from the multi-peripheral colour decomposition. Finally, the leading soft divergences are checked to reproduce the expected infrared behaviour.
8. Four-loop collinear anomalous dimension in N=4 Yang-Mills theory
International Nuclear Information System (INIS)
Cachazo, Freddy; Spradlin, Marcus; Volovich, Anastasia
2007-01-01
We report a calculation in N=4 Yang-Mills of the four-loop term g (4) in the collinear anomalous dimension g(λ) which governs the universal subleading infrared structure of gluon scattering amplitudes. Using the method of obstructions to extract this quantity from the 1/ε singularity in the four-gluon iterative relation at four loops, we find g (4) =-1240.9 with an estimated numerical uncertainty of 0.02%. We also analyze the implication of our result for the strong coupling behavior of g(λ), finding support for the string theory prediction computed recently by Alday and Maldacena using AdS/CFT
9. Connection between Einstein equations, nonlinear sigma models, and self-dual Yang-Mills theory
International Nuclear Information System (INIS)
Sanchez, N.; Whiting, B.
1986-01-01
The authors analyze the connection between nonlinear sigma models self-dual Yang-Mills theory, and general relativity (self-dual and non-self-dual, with and without killing vectors), both at the level of the equations and at the level of the different type of solutions (solitons and calorons) of these theories. They give a manifestly gauge invariant formulation of the self-dual gravitational field analogous to that given by Yang for the self-dual Yang-Mills field. This formulation connects in a direct and explicit way the self-dual Yang-Mills and the general relativity equations. They give the ''R gauge'' parametrization of the self-dual gravitational field (which corresponds to modified Yang's-type and Ernst equations) and analyze the correspondence between their different types of solutions. No assumption about the existence of symmetries in the space-time is needed. For the general case (non-self-dual), they show that the Einstein equations contain an O nonlinear sigma model. This connection with the sigma model holds irrespective of the presence of symmetries in the space-time. They found a new class of solutions of Einstein equations depending on holomorphic and antiholomorphic functions and we relate some subclasses of these solutions to solutions of simpler nonlinear field equations that are well known in other branches of physics, like sigma models, SineGordon, and Liouville equations. They include gravitational plane wave solutions. They analyze the response of different accelerated quantum detector models, compare them to the case when the detectors are linterial in an ordinary Planckian gas at a given temperature, and discuss the anisotropy of the detected response for Rindler observers
10. N=4 super-Yang-Mills in LHC superspace. Part II: Non-chiral correlation functions of the stress-tensor multiplet
CERN Document Server
Chicherin, Dmitry
2017-03-09
We study the multipoint super-correlation functions of the full non-chiral stress-tensor multiplet in N=4 super-Yang-Mills theory in the Born approximation. We derive effective supergraph Feynman rules for them. Surprisingly, the Feynman rules for the non-chiral correlators differ only slightly from those for the chiral correlators. We rely on the formulation of the theory in Lorentz harmonic chiral (LHC) superspace elaborated in the twin paper \\cite{PartI}. In this approach only the chiral half of the supersymmetry is manifest. The other half is realized by nonlinear and nonlocal transformations of the LHC superfields. However, at Born level only the simple linear part of the transformations is relevant. It corresponds to effectively working in the self-dual sector of the theory. Our method is also applicable to a wider class of supermultiplets like all the half-BPS operators and the Konishi multiplet.
11. On generalized Yang-Mills theories and extensions of the standard model in Clifford (tensorial) spaces
International Nuclear Information System (INIS)
Castro, Carlos
2006-01-01
We construct the Clifford-space tensorial-gauge fields generalizations of Yang-Mills theories and the Standard Model that allows to predict the existence of new particles (bosons, fermions) and tensor-gauge fields of higher-spins in the 10 Tev regime. We proceed with a detailed discussion of the unique D 4 - D 5 - E 6 - E 7 - E 8 model of Smith based on the underlying Clifford algebraic structures in D = 8, and which furnishes all the properties of the Standard Model and Gravity in four-dimensions, at low energies. A generalization and extension of Smith's model to the full Clifford-space is presented when we write explicitly all the terms of the extended Clifford-space Lagrangian. We conclude by explaining the relevance of multiple-foldings of D = 8 dimensions related to the modulo 8 periodicity of the real Cliford algebras and display the interplay among Clifford, Division, Jordan, and Exceptional algebras, within the context of D = 26, 27, 28 dimensions, corresponding to bosonic string, M and F theory, respectively, advanced earlier by Smith. To finalize we describe explicitly how the E 8 x E 8 Yang-Mills theory can be obtained from a Gauge Theory based on the Clifford (16) group
12. Sigma-model formulation of the Yang-Mills theory on four-dimensional hypersphere
International Nuclear Information System (INIS)
Ivanov, E.A.; Krivonos, S.O.
1981-01-01
The bilocal sigma-model representation is constructed for the Yang-Mills theory in the simplest conformally flat hyperspherical spaces So(1,4)/SO(1,3), SO(2,3)/SO(1,3) and SO(5)/SO(4). Like in the case of Minkowski and Euclidean spaces, Yang-Mills potential is defined as bsub(μ)(x)=dsub(μ)sup(y)b(x,y)|y=0 , b(x,y) being a bilocal Goldstone field which takes values in the gauge group algebra and is subjected to certain covariant constraints. The minimal version of these constraints results in the ''string'' representation for b(x,y) through the P-exponential of bsub(μ)(x) along the fixed paths coinciding with geodesics. Due to the presence of closed geodesics, the contour fuctionals naturally appear in the theory, with contours being the circles with the hypersphere radius. The sigma-model representation is shown to be Weyl-covariant: its formulations indifferent conformally flat spaces are related by transformations of ysup(rho). The geometric meaning of ysup(rho) and minimal constraints is explained, and the conformal group gransformation for ysup(rho) is found [ru
13. Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond
International Nuclear Information System (INIS)
Bern, Zvi; Dixon, Lance J.; Smirnov, Vladimir A.
2005-01-01
We compute the leading-color (planar) three-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4-2ε dimensions, as a Laurent expansion about ε=0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N=4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated Ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/ε 2 pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N=4 supersymmetric Yang-Mills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD
14. Spinor analysis of the Yang-Mills theory in the Minkowski space
International Nuclear Information System (INIS)
Pervushin, V.N.; Horejsi, J.
1982-01-01
Spinorial methods are applied to the solution of self-duality equations for Yang-Mills field and the Dirac equation with an external self-dual field in the Minkowski space. Gauge group SU(2) is considered. It is shown that in the spinorial formalism an analog of the Yang construction of self-dual fields emerges naturally. Solutions of the Dirac eqUation for massless fermion with an arbitrary isospin, interacting with an external self-dual or anti-self-dual field are obtained. The external field is chosen to be the Minkowskian analog of the EUclidean Hooft Ansatz. It is shown that for the isospin 1/2 and 1 the solutions of the Dirac equation may be expressed in terms of the solutions of d'Alembert equation. The solutions obtained may be employed in the approach to gauge theories proposed recently, which is based on an analogy with the superfluidity theory; in such an approach the self-dual solutions of the Yang-Mills equations represent the vacuum
15. Canonical Yang-Mills field theory with invariant gauge-families
International Nuclear Information System (INIS)
Yokoyama, Kan-ichi
1978-01-01
A canonical Yang-Mills field theory with indefinite metric is presented on the basis of a covariant gauge formalism for quantum electrodynamics. As the first step of the formulation, a many-gauge-field problem, in which many massless Abelian-gauge fields coexist, is treated from a new standpoint. It is shown that only a single pair of a gaugeon field and its associated one can govern the gauge structure of the whole system. The result obtained is further extended to cases of non-Abelian gauge theories. Gauge parameters for respective components of the Yang-Mills fields are introduced as a group vector. There exists a q-number local gauge transformation which connects relevant fields belonging to the same invariant gauge family with one another in a manifestly covariant way. In canonical quantization, the Faddeev-Popov ghosts are introduced in order to guarantee the existence of a desirable physical subspace with positive semi-definite metric. As to treatment of the Faddeev-Popov ghosts, Kugo and Ojima's approach is adopted. Three supplementary conditions which are consistent with one another constrain the physical subspace. (author)
16. On the field/string theory approach to theta dependence in large N Yang-Mills theory
International Nuclear Information System (INIS)
1999-01-01
The theta dependence of the vacuum energy in large N Yang-Mills theory has been studied some time ago by Witten using a duality of large N gauge theories with the string theory compactified on a certain space-time. We show that within the field theory context vacuum fluctuations of the topological charge give rise to the vacuum energy consistent with the string theory computation. Furthermore, we calculate 1/N suppressed corrections to the string theory result. The reconciliation of the string and field theory approaches is based on the fact that the gauge theory instantons carry zerobrane charge in the corresponding D-brane construction of Yang-Mills theory. Given the formula for the vacuum energy we study certain aspects of stability of the false vacua of the model for different realizations of the initial conditions. The vacuum structure appears to be different depending on whether N is infinite or, alternatively, large but finite
17. Center vortex model for the infrared sector of SU(4) Yang-Mills theory: String tensions and deconfinement transition
International Nuclear Information System (INIS)
Engelhardt, M.
2006-01-01
A random vortex world-surface model for the infrared sector of SU(4) Yang-Mills theory is constructed, focusing on the confinement properties and the behavior at the deconfinement phase transition. Although the corresponding data from lattice Yang-Mills theory can be reproduced, the model requires a more complex action and considerably more tuning than the SU(2) and SU(3) cases studied previously. Its predictive capabilities are accordingly reduced. This behavior has a definite physical origin, which is elucidated in detail in the present work. As the number of colors is raised in Yang-Mills theory, the corresponding infrared effective vortex description cannot indefinitely continue to rely on dynamics determined purely by vortex world-surface characteristics; additional color structures present on the vortices begin to play a role. As evidenced by the modeling effort reported here, definite signatures of this behavior appear in the case of four colors
18. Continuum strong-coupling expansion of Yang-Mills theory: quark confinement and infra-red slavery
Energy Technology Data Exchange (ETDEWEB)
Mansfield, P. (Dept. of Mathematical Sciences, Univ. of Durham (United Kingdom))
1994-04-25
We solve Schroedinger's equation for the ground-state of four-dimensional Yang-Mills theory as an expansion in inverse powers of the coupling. Expectation values computed with the leading-order approximation are reduced to a calculation in two-dimensional Yang-Mills theory which is known to confine. Consequently the Wilson loop in the four-dimensional theory obeys an area law to leading order and the coupling becomes infinite as the mass scale goes to zero. (orig.)
19. Continuum strong-coupling expansion of Yang-Mills theory: quark confinement and infra-red slavery
Science.gov (United States)
Mansfield, Paul
1994-04-01
We solve Schrödinger's equation for the ground-state of four-dimensional Yang-Mills theory as an expansion in inverse powers of the coupling. Expectation values computed with the leading-order approximation are reduced to a calculation in two-dimensional Yang-Mills theory which is known to confine. Consequently the Wilson loop in the four-dimensional theory obeys an area law to leading order and the coupling becomes infinite as the mass scale goes to zero.
20. Yang-Mills gravity in biconformal space
International Nuclear Information System (INIS)
Anderson, Lara B; Wheeler, James T
2007-01-01
We write a gravity theory with Yang-Mills-type action using the biconformal gauging of the conformal group. We show that the resulting biconformal Yang-Mills gravity theories describe 4-dim, scale-invariant general relativity in the case of slowly changing fields. In addition, we systematically extend arbitrary 4-dim Yang-Mills theories to biconformal space, providing a new arena for studying flat-space Yang-Mills theories. By applying the biconformal extension to a 4-dim pure Yang-Mills theory with conformal symmetry, we establish a 1-1, onto mapping between a set of gravitational gauge theories and 4-dim, flat-space gauge theories
1. Geometrodynamics of gauge fields on the geometry of Yang-Mills and gravitational gauge theories
CERN Document Server
Mielke, Eckehard W
2016-01-01
This monograph aims to provide a unified, geometrical foundation of gauge theories of elementary particle physics. The underlying geometrical structure is unfolded in a coordinate-free manner via the modern mathematical notions of fibre bundles and exterior forms. Topics such as the dynamics of Yang-Mills theories, instanton solutions and topological invariants are included. By transferring these concepts to local space-time symmetries, generalizations of Einstein's theory of gravity arise in a Riemann-Cartan space with curvature and torsion. It provides the framework in which the (broken) Poincaré gauge theory, the Rainich geometrization of the Einstein-Maxwell system, and higher-dimensional, non-abelian Kaluza-Klein theories are developed. Since the discovery of the Higgs boson, concepts of spontaneous symmetry breaking in gravity have come again into focus, and, in this revised edition, these will be exposed in geometric terms. Quantizing gravity remains an open issue: formulating it as a de Sitter t...
2. The universe as a topological defect in a higher-dimensional Einstein-Yang-Mills theory
International Nuclear Information System (INIS)
Nakamura, A.; Shiraishi, K.
1989-04-01
An interpretation is suggested that a spontaneous compactification of space-time can be regarded as a topological defect in a higher-dimensional Einstein-Yang-Mills (EYM) theory. We start with D-dimensional EYM theory in our present analysis. A compactification leads to a D-2 dimensional effective action of Abelian gauge-Higgs theory. We find a 'vortex' solution in the effective theory. Our universe appears to be confined in a center of a 'vortex', which has D-4 large dimensions. In this paper we show an example with SU (2) symmetry in the original EYM theory, and the resulting solution is found to be equivalent to the 'instanton-induced compactification'. The cosmological implication is also mentioned. (author)
3. Dual computations of non-Abelian Yang-Mills theories on the lattice
International Nuclear Information System (INIS)
Cherrington, J. Wade; Khavkine, Igor; Christensen, J. Daniel
2007-01-01
In the past several decades there have been a number of proposals for computing with dual forms of non-Abelian Yang-Mills theories on the lattice. Motivated by the gauge-invariant, geometric picture offered by dual models and successful applications of duality in the U(1) case, we revisit the question of whether it is practical to perform numerical computation using non-Abelian dual models. Specifically, we consider three-dimensional SU(2) pure Yang-Mills as an accessible yet nontrivial case in which the gauge group is non-Abelian. Using methods developed recently in the context of spin foam quantum gravity, we derive an algorithm for efficiently computing the dual amplitude and describe Metropolis moves for sampling the dual ensemble. We relate our algorithms to prior work in non-Abelian dual computations of Hari Dass and his collaborators, addressing several problems that have been left open. We report results of spin expectation value computations over a range of lattice sizes and couplings that are in agreement with our conventional lattice computations. We conclude with an outlook on further development of dual methods and their application to problems of current interest
4. Non-Gaussianities in the topological charge distribution of the SU(3) Yang-Mills theory
Science.gov (United States)
Cè, Marco; Consonni, Cristian; Engel, Georg P.; Giusti, Leonardo
2015-10-01
We study the topological charge distribution of the SU(3) Yang-Mills theory with high precision in order to be able to detect deviations from Gaussianity. The computation is carried out on the lattice with high statistics Monte Carlo simulations by implementing a naive discretization of the topological charge evolved with the Yang-Mills gradient flow. This definition is far less demanding than the one suggested from Neuberger's fermions and, as shown in this paper, in the continuum limit its cumulants coincide with those of the universal definition appearing in the chiral Ward identities. Thanks to the range of lattice volumes and spacings considered, we can extrapolate the results for the second and fourth cumulant of the topological charge distribution to the continuum limit with confidence by keeping finite volume effects negligible with respect to the statistical errors. Our best results for the topological susceptibility is t02χ =6.67 (7 )×1 0-4 , where t0 is a standard reference scale, while for the ratio of the fourth cumulant over the second, we obtain R =0.233 (45 ). The latter is compatible with the expectations from the large Nc expansion, while it rules out the θ behavior of the vacuum energy predicted by the dilute instanton model. Its large distance from 1 implies that, in the ensemble of gauge configurations that dominate the path integral, the fluctuations of the topological charge are of quantum nonperturbative nature.
5. Zero value for the three-loop β function in N=4 supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Grisaru, M.; Rocek, M.; Siegel, W.
1980-01-01
This Letter describes a calculation using superfield techniques, showing that the β function is zero to three loops in N=4 supersymmetric Yang-Mills theory. This result gives further indication that the theory is likely to be finite and conformally invariant order by order in perturbation theory
6. Duality symmetry of N=4 Yang-Mills theory on T3
International Nuclear Information System (INIS)
Hacquebord, F.; Verlinde, H.
1997-01-01
We study the spectrum of BPS states in N=4 supersymmetric U(N) Yang-Mills theory. This theory has been proposed to describe M-theory on T 3 in the discrete light-cone formalism. We find that the degeneracy of irreducible BPS bound states in this model exhibits a (partially hidden) SL(5,Z) duality symmetry. Besides the electro-magnetic symmetry, this duality group also contains Nahm-like transformations that interchange the rank N of the gauge group with some of the magnetic or electric fluxes. In the M-theory interpretation, this mapping amounts to a reflection that interchanges the longitudinal direction with one of the transverse directions. (orig.)
7. High energy behavior of a six-point R-current correlator in N=4 supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Bartels, Jochen; Hentschinski, Martin; Mischler, Anna-Maria
2009-12-01
We study the high energy limit of a six-point R-current correlator in N=4 supersymmetric Yang-Mills theory for finite N c . We make use of the framework of perturbative resummation of large logarithms of the energy. More specifically, we apply the (extended) generalized leading logarithmic approximation. We find that the same conformally invariant two-to-four gluon vertex occurs as in non-supersymmetric Yang-Mills theory. As a new feature we find a direct coupling of the four-gluon t-channel state to the R-current impact factor. (orig.)
8. BRST quantization of 2d-Yang Mills theory with anomalies
International Nuclear Information System (INIS)
Kalau, Wolfgang
1992-01-01
The BRST-quantization of anomalous 2d-Yang Mills (YM) theory is discussed. Since an oscillator basis for the YM-Fock-space is used the anomaly appears already for a pure YM-system and the constraints form a Kac-Moody algebra with negative central charge. The coupling of chiral fermions is also discussed and it is found that the BRST-cohomology for systems with chiral fermions in a sufficiently large representation of the gauge group is completely equivalent to the cohomology of the finite dimensional gauge group. For pure YM theory or YM theory coupled to chiral fermions in small representations there exists an infinite number of inequivalent cohomology classes. This is discussed in some detail for the example of SU(2). (author). 15 refs
9. Wilson loop, Regge trajectory and hadron masses in a Yang-Mills theory from semiclassical strings
International Nuclear Information System (INIS)
Bigazzi, F.; Cotrone, A.L.; Martucci, L.; Pando Zayas, L.A.
2004-07-01
We compute the one-loop string corrections to the Wilson loop, glueball Regge trajectory and stringy hadron masses in the Witten model of non supersymmetric, large-N Yang-Mills theory. The classical string configurations corresponding to the above field theory objects are respectively: open straight strings, folded closed spinning strings, and strings orbiting in the internal part of the supergravity background. For the rectangular Wilson loop we show that besides the standard Luscher term, string corrections provide a rescaling of the field theory string tension. The one-loop corrections to the linear glueball Regge trajectories render them nonlinear with a positive intercept, as in the experimental soft Pomeron trajectory. Strings orbiting in the internal space predict a spectrum of hadronic-like states charged under global flavor symmetries which falls in the same universality class of other confining models. (author)
10. The anomalous dimension of the gluon-ghost mass operator in Yang-Mills theory
International Nuclear Information System (INIS)
Dudal, D.; Verschelde, H.; Lemes, V.E.R.; Sarandy, M.S.; Sobreiro, R.; Sorella, S.P.; Picariello, M.; Gracey, J.A.
2003-01-01
The local composite gluon-ghost operator (((1)/(2))A aμ A μ a +αc-bar a c a ) is analysed in the framework of the algebraic renormalization in SU(N) Yang-Mills theories in the Landau, Curci-Ferrari and maximal abelian gauges. We show, to all orders of perturbation theory, that this operator is multiplicatively renormalizable. Furthermore, its anomalous dimension is not an independent parameter of the theory, being given by a general expression valid in all these gauges. We also verify the relations we obtain for the operator anomalous dimensions by explicit 3-loop calculations in the MS-bar scheme for the Curci-Ferrari gauge
11. Supercurrent and the Adler-Bardeen theorem in coupled supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Ensign, P.; Mahanthappa, K.T.
1987-01-01
We construct the supercurrent and a supersymmetric current which satisfies the Adler-Bardeen theorem in supersymmetric Yang-Mills theory coupled to non-self-interacting chiral matter. Using the formulation recently developed by Grisaru, Milewski, and Zanon, supersymmetry and gauge invariance are maintained with supersymmetric background-field theory and regularization by dimensional reduction. We verify the finiteness of the supercurrent to one loop, and the Adler-Bardeen theorem to two loops by explicit calculations in the minimal-subtraction scheme. We then demonstrate the subtraction-scheme independence of the one-loop Adler-Bardeen anomaly and prove the existence of a subtraction scheme in which the Adler-Bardeen theorem is satisfied to all orders in perturbation theory
12. A sketch to the geometrical N=2-d=5 Yang-Mills theory over a supersymmetric group-manifold - I
International Nuclear Information System (INIS)
Borges, M.; Turin Univ.; Pio, G.
1983-03-01
This work concerns the search and the construction of a geometrical structure for a supersymmetric N=2-d=5 Yang-Mills theory on the group manifold. From criteria established throughout this paper, we build up an ansatz for the curvatures of our theory and then solve the Bianchi identities, whose solution is fundamental for the construction of the geometrical action. (author)
13. A Yang-Mills Type Gauge Theory of Gravity and the Dark Matter and Dark Energy Problems
OpenAIRE
Yang, Yi; Yeung, Wai Bong
2012-01-01
A Yang-Mills type gauge theory of gravity is shown to have a richer structure than the Einstein's General Theory of Relativity. This new structure can give an explanation of the form of the galactic rotation curves, of the amount of intergalactic gravitational lensing, and of the accelerating expansion of the Universe.
14. Pure classical SU(2) Yang-Mills theory with potentials invariant under a U(1) gauge subgroup
International Nuclear Information System (INIS)
Bacry, H.
1978-07-01
The present article is devoted to pure SU(2) classical Yang-Mills theories whose potentials are invariant under a U(1) gauge subgroup. Such potentials are shown to be associated with classical Maxwell-like fields with magnetic sources as 't Hooft's monopole is associated with the Dirac magnetic monopole. Conversely, the authors give Yang-Mills potentials corresponding to some Maxwell-like fields, in particular static magnetic fields with emphasis on those with cylindrical symmetry (including the dipole and other multipoles) and the ephemerons corresponding to an instantaneous magnetic multipole
15. Unconstrained off-shell N=3 supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Galperin, A.; Ivanov, E.; Kalitzin, S.; Ogievetsky, V.; Sokatchev, E.
1984-01-01
The harmonic superspace is used to build up an unconstrained off-shell formulation of N=3 supersymmetric Yang-Mills theory. The theory is defined in an analytic N=3 superspace having M 4 x(SU(3)/U(1)xU(1) as an even part. The basic objects are the analytic potentials which serve as gauge connections entering harmonic derivatives. The action is an integral over analytic superspace. The Lagrange density is surprisingly simple and it is gauge invariant up to total harmonic derivative. The equations of motion are integrability conditions on the internal space SU(3)/U(1)xU(1). The jumping over the ''N=3 barrier'' became possible due to the infinite number of auxiliary fields
16. Screening masses in quenched (2+1)d Yang-Mills theory: Universality from dynamics?
International Nuclear Information System (INIS)
Frigori, Rafael B.
2010-01-01
We compute the spectrum of gluonic screening-masses in the 0 ++ channel of quenched 3d Yang-Mills theory near the phase-transition. Our finite-temperature lattice simulations are performed at scaling region, using state-of-art techniques for thermalization and spectroscopy, which allows for thorough data extrapolations to thermodynamic limit. Ratios among mass-excitations with the same quantum numbers on the gauge theory, 2d Ising and λφ 4 models are compared, resulting in a nice agreement with predictions from universality. In addition, a gauge-to-scalar mapping, previously employed to fit QCD Green's functions at deep IR, is verified to dynamically describe these universal spectroscopic patterns.
17. The asymptotic spectrum of the N = 4 super-Yang-Mills spin chain
International Nuclear Information System (INIS)
Chen, H.-Y.; Dorey, Nick; Okamura, Keisuke
2007-01-01
In this paper we discuss the asymptotic spectrum of the spin chain description of planar N = 4 SUSY Yang-Mills. The states appearing in the spectrum belong to irreducible representations of the unbroken supersymmetry SU(2 vertical bar 2) x SU(2 vertical bar 2) with non-trivial extra central extensions. The elementary magnon corresponds to the bifundamental representation while boundstates of Q magnons form a certain short representation of dimension 16Q 2 . Generalising the Beisert's analysis of the Q = 1 case, we derive the exact dispersion relation for these states by purely group theoretic means
18. Lattice simulation of a center symmetric three dimensional effective theory for SU(2) Yang-Mills
International Nuclear Information System (INIS)
Smith, Dominik
2010-01-01
We present lattice simulations of a center symmetric dimensionally reduced effective field theory for SU(2) Yang Mills which employ thermal Wilson lines and three-dimensional magnetic fields as fundamental degrees of freedom. The action is composed of a gauge invariant kinetic term, spatial gauge fields and a potential for theWilson line which includes a ''fuzzy'' bag term to generate non-perturbative fluctuations between Z(2) degenerate ground states. The model is studied in the limit where the gauge fields are set to zero as well as the full model with gauge fields. We confirm that, at moderately weak coupling, the ''fuzzy'' bag term leads to eigenvalue repulsion in a finite region above the deconfining phase transition which shrinks in the extreme weak-coupling limit. A non-trivial Z(N) symmetric vacuum arises in the confined phase. The effective potential for the Polyakov loop in the theory with gauge fields is extracted from the simulations including all modes of the loop as well as for cooled configurations where the hard modes have been averaged out. The former is found to exhibit a non-analytic contribution while the latter can be described by a mean-field like ansatz with quadratic and quartic terms, plus a Vandermonde potential which depends upon the location within the phase diagram. Other results include the exact location of the phase boundary in the plane spanned by the coupling parameters, correlation lengths of several operators in the magnetic and electric sectors and the spatial string tension. We also present results from simulations of the full 4D Yang-Mills theory and attempt to make a qualitative comparison to the 3D effective theory. (orig.)
19. Lattice simulation of a center symmetric three dimensional effective theory for SU(2) Yang-Mills
Energy Technology Data Exchange (ETDEWEB)
Smith, Dominik
2010-11-17
We present lattice simulations of a center symmetric dimensionally reduced effective field theory for SU(2) Yang Mills which employ thermal Wilson lines and three-dimensional magnetic fields as fundamental degrees of freedom. The action is composed of a gauge invariant kinetic term, spatial gauge fields and a potential for theWilson line which includes a ''fuzzy'' bag term to generate non-perturbative fluctuations between Z(2) degenerate ground states. The model is studied in the limit where the gauge fields are set to zero as well as the full model with gauge fields. We confirm that, at moderately weak coupling, the ''fuzzy'' bag term leads to eigenvalue repulsion in a finite region above the deconfining phase transition which shrinks in the extreme weak-coupling limit. A non-trivial Z(N) symmetric vacuum arises in the confined phase. The effective potential for the Polyakov loop in the theory with gauge fields is extracted from the simulations including all modes of the loop as well as for cooled configurations where the hard modes have been averaged out. The former is found to exhibit a non-analytic contribution while the latter can be described by a mean-field like ansatz with quadratic and quartic terms, plus a Vandermonde potential which depends upon the location within the phase diagram. Other results include the exact location of the phase boundary in the plane spanned by the coupling parameters, correlation lengths of several operators in the magnetic and electric sectors and the spatial string tension. We also present results from simulations of the full 4D Yang-Mills theory and attempt to make a qualitative comparison to the 3D effective theory. (orig.)
20. Thermodynamics of SU(2) quantum Yang-Mills theory and CMB anomalies
Science.gov (United States)
Hofmann, Ralf
2014-04-01
A brief review of effective SU(2) Yang-Mills thermodynamics in the deconfining phase is given, including the construction of the thermal ground-state estimate in terms of an inert, adjoint scalar field φ, based on non-propagating (anti)selfdual field configurations of topological charge unity. We also discuss kinematic constraints on interacting propagating gauge fields implied by the according spatial coarse-graining, and we explain why the screening physics of an SU(2) photon is subject to an electric-magnetically dual interpretation. This argument relies on the fact that only (anti)calorons of scale parameter ρ ˜ |φ|-1 contribute to the coarse-graining required for thermal-ground-state emergence at temperature T. Thus, use of the effective gauge coupling e in the (anti)caloron action is justified, yielding the value ħ for the latter at almost all temperatures. As a consequence, the indeterministic transition of initial to final plane waves caused by an effective, pointlike vertex is fundamentally mediated in Euclidean time by a single (anti)caloron being part of the thermal ground state. Next, we elucidate how a low-frequency excess of line temperature in the Cosmic Microwave Background (CMB) determines the value of the critical temperature of the deconfining-preconfining phase transition of an SU(2) Yang-Mills theory postulated to describe photon propagation, and we describe how, starting at a redshift of about unity, SU(2) photons collectively work 3D temperature depressions into the CMB. Upon projection along a line of sight, a given depression influences the present CMB sky in a cosmologically local way, possibly explaining the large-angle anomalies confirmed recently by the Planck collaboration. Finally, six relativistic polarisations residing in the SU(2) vector modes roughly match the number of degrees of freedom in cosmic neutrinos (Planck) which would disqualify the latter as radiation. Indeed, if interpreted as single center-vortex loops in
1. Thermodynamics of SU(2 quantum Yang-Mills theory and CMB anomalies
Directory of Open Access Journals (Sweden)
Hofmann Ralf
2014-04-01
Full Text Available A brief review of effective SU(2 Yang-Mills thermodynamics in the deconfining phase is given, including the construction of the thermal ground-state estimate in terms of an inert, adjoint scalar field φ, based on non-propagating (antiselfdual field configurations of topological charge unity. We also discuss kinematic constraints on interacting propagating gauge fields implied by the according spatial coarse-graining, and we explain why the screening physics of an SU(2 photon is subject to an electric-magnetically dual interpretation. This argument relies on the fact that only (anticalorons of scale parameter ρ ∼ |φ|−1 contribute to the coarse-graining required for thermal-ground-state emergence at temperature T. Thus, use of the effective gauge coupling e in the (anticaloron action is justified, yielding the value ħ for the latter at almost all temperatures. As a consequence, the indeterministic transition of initial to final plane waves caused by an effective, pointlike vertex is fundamentally mediated in Euclidean time by a single (anticaloron being part of the thermal ground state. Next, we elucidate how a low-frequency excess of line temperature in the Cosmic Microwave Background (CMB determines the value of the critical temperature of the deconfining-preconfining phase transition of an SU(2 Yang-Mills theory postulated to describe photon propagation, and we describe how, starting at a redshift of about unity, SU(2 photons collectively work 3D temperature depressions into the CMB. Upon projection along a line of sight, a given depression influences the present CMB sky in a cosmologically local way, possibly explaining the large-angle anomalies confirmed recently by the Planck collaboration. Finally, six relativistic polarisations residing in the SU(2 vector modes roughly match the number of degrees of freedom in cosmic neutrinos (Planck which would disqualify the latter as radiation. Indeed, if interpreted as single center
2. Off-diagonal mass generation for Yang-Mills theories in the maximal Abelian gauge
International Nuclear Information System (INIS)
Dudal, D.; Verschelde, H.; Sarandy, M.S.
2007-01-01
We investigate a dynamical mass generation mechanism for the off-diagonal gluons and ghosts in SU(N) Yang-Mills theories, quantized in the maximal Abelian gauge. Such a mass can be seen as evidence for the Abelian dominance in that gauge. It originates from the condensation of a mixed gluon-ghost operator of mass dimension two, which lowers the vacuum energy. We construct an effective potential for this operator by a combined use of the local composite operators technique with algebraic renormalization and we discuss the gauge parameter independence of the results. We also show that it is possible to connect the vacuum energy, due to the mass dimension two condensate discussed here, with the non-trivial vacuum energy originating from the condensate 2 μ >, which has attracted much attention in the Landau gauge. (author)
3. Non-perturbative BRST quantization of Euclidean Yang-Mills theories in Curci-Ferrari gauges
Science.gov (United States)
Pereira, A. D.; Sobreiro, R. F.; Sorella, S. P.
2016-10-01
In this paper we address the issue of the non-perturbative quantization of Euclidean Yang-Mills theories in the Curci-Ferrari gauge. In particular, we construct a refined Gribov-Zwanziger action for this gauge, which takes into account the presence of gauge copies as well as the dynamical formation of dimension-two condensates. This action enjoys a non-perturbative BRST symmetry recently proposed in Capri et al. (Phys. Rev. D 92(4), 045039. doi: 10.1103/PhysRevD.92.045039 arXiv:1506.06995 [hep-th], 2015). Finally, we pay attention to the gluon propagator in different space-time dimensions.
4. Non-perturbative BRST quantization of Euclidean Yang-Mills theories in Curci-Ferrari gauges
International Nuclear Information System (INIS)
Pereira, A.D.; Sobreiro, R.F.; Sorella, S.P.
2016-01-01
In this paper we address the issue of the non-perturbative quantization of Euclidean Yang-Mills theories in the Curci-Ferrari gauge. In particular, we construct a refined Gribov-Zwanziger action for this gauge, which takes into account the presence of gauge copies as well as the dynamical formation of dimension-two condensates. This action enjoys a non-perturbative BRST symmetry recently proposed in Capri et al. (Phys. Rev. D 92(4), 045039. doi:10.1103/PhysRevD.92.045039. arXiv:1506.06995 [hepth], 2015). Finally, we pay attention to the gluon propagator in different space-time dimensions. (orig.)
5. Non-perturbative BRST quantization of Euclidean Yang-Mills theories in Curci-Ferrari gauges
Energy Technology Data Exchange (ETDEWEB)
Pereira, A.D. [UFF, Universidade Federal Fluminense, Instituto de Fisica, Campus da Praia Vermelha, Niteroi, RJ (Brazil); Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Potsdam (Germany); UERJ, Universidade do Estado do Rio de Janeiro, Departamento de Fisica Teorica, Rio de Janeiro (Brazil); Sobreiro, R.F. [UFF, Universidade Federal Fluminense, Instituto de Fisica, Campus da Praia Vermelha, Niteroi, RJ (Brazil); Sorella, S.P. [UERJ, Universidade do Estado do Rio de Janeiro, Departamento de Fisica Teorica, Rio de Janeiro (Brazil)
2016-10-15
In this paper we address the issue of the non-perturbative quantization of Euclidean Yang-Mills theories in the Curci-Ferrari gauge. In particular, we construct a refined Gribov-Zwanziger action for this gauge, which takes into account the presence of gauge copies as well as the dynamical formation of dimension-two condensates. This action enjoys a non-perturbative BRST symmetry recently proposed in Capri et al. (Phys. Rev. D 92(4), 045039. doi:10.1103/PhysRevD.92.045039. arXiv:1506.06995 [hepth], 2015). Finally, we pay attention to the gluon propagator in different space-time dimensions. (orig.)
6. The large N limit of the topological susceptibility of Yang-Mills gauge theory
Energy Technology Data Exchange (ETDEWEB)
Ce, Marco [Scuola Normale Superiore, Pisa (Italy); INFN, Pisa (Italy); Garcia Vera, Miguel [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Humboldt Univ. Berlin (Germany). Inst. fuer Physik; Giusti, Leonardo [Milano Bicocca Univ., Milano (Italy); INFN, Milano Bicocca (Italy); Schaefer, Stefan [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2016-10-27
We present a precise computation of the topological susceptibility χ{sub YM} of SU(N) Yang-Mills theory in the large N limit. The computation is done on the lattice, using high-statistics Monte Carlo simulations with N=3,4,5,6 and three different lattice spacings. Two major improvements make it possible to go to finer lattice spacing and larger N compared to previous works. First, the topological charge is implemented through the gradient flow definition; and second, open boundary conditions in the time direction are employed in order to avoid the freezing of the topological charge. The results allow us to extrapolate the dimensionless quantity t{sub 0}{sup 2}χ{sub YM} to the continuum and large N limits with confidence. The accuracy of the final result represents a new quality in the verification of large N scaling.
7. Periodic electromagnetic vacuum in the two-dimensional Yang-Mills theory with the Chern-Simons mass
International Nuclear Information System (INIS)
Skalozub, V.V.; Vilensky, S.A.; Zaslavsky, A.Yu.
1993-06-01
The periodic vacuum structure formed from magnetic and electric fields is derived in the two-dimensional Yang-Mills theory with the Chern-Simons term. It is shown that both the magnetic flux quantization in the fundamental sell and conductivity quantization inherent to the vacuum. Hence, the quantum Hall effect gets its natural explanation. (author). 10 refs
8. On Yang--Mills Theories with Chiral Matter at Strong Coupling
Energy Technology Data Exchange (ETDEWEB)
Shifman, M.; /Minnesota U., Theor. Phys. Inst. /Saclay, SPhT; Unsal, Mithat; /SLAC /Stanford U., Phys. Dept.
2008-08-20
Strong coupling dynamics of Yang-Mills theories with chiral fermion content remained largely elusive despite much effort over the years. In this work, we propose a dynamical framework in which we can address non-perturbative properties of chiral, non-supersymmetric gauge theories, in particular, chiral quiver theories on S{sub 1} x R{sub 3}. Double-trace deformations are used to stabilize the center-symmetric vacuum. This allows one to smoothly connect smaller(S{sub 1}) to larger(S{sub 1}) physics (R{sub 4} is the limiting case) where the double-trace deformations are switched off. In particular, occurrence of the mass gap in the gauge sector and linear confinement due to bions are analytically demonstrated. We find the pattern of the chiral symmetry realization which depends on the structure of the ring operators, a novel class of topological excitations. The deformed chiral theory, unlike the undeformed one, satisfies volume independence down to arbitrarily small volumes (a working Eguchi-Kawai reduction) in the large N limit. This equivalence, may open new perspectives on strong coupling chiral gauge theories on R{sub 4}.
9. Geometrical Lagrangian for a Supersymmetric Yang-Mills Theory on the Group Manifold
International Nuclear Information System (INIS)
Borges, M. F.
2002-01-01
Perhaps one of the main features of Einstein's General Theory of Relativity is that spacetime is not flat itself but curved. Nowadays, however, many of the unifying theories like superstrings on even alternative gravity theories such as teleparalell geometric theories assume flat spacetime for their calculations. This article, an extended account of an earlier author's contribution, it is assumed a curved group manifold as a geometrical background from which a Lagrangian for a supersymmetric N=2, d=5 Yang-Mills - SYM, N=2, d=5 - is built up. The spacetime is a hypersurface embedded in this geometrical scenario, and the geometrical action here obtained can be readily coupled to the five-dimensional supergravity action. The essential idea that underlies this work has its roots in the Einstein-Cartan formulation of gravity and in the 'group manifold approach to gravity and supergravity theories'. The group SYM, N=2, d=5, turns out to be the direct product of supergravity and a general gauge group G:G=GxSU(2,2/1)-bar
10. Three-loop renormalization of the N=1, N=2, N=4 supersymmetric Yang-Mills theories
International Nuclear Information System (INIS)
Velizhanin, V.N.
2009-01-01
We calculate the renormalization constants of the N=1, N=2, N=4 supersymmetric Yang-Mills theories in an arbitrary covariant gauge in the dimensional reduction scheme up to three loops. We have found, that the beta-functions for N=1 and N=4 SYM theories are the same from the different triple vertices. This means that the dimensional reduction scheme works correctly in these models up to third order of perturbative theory.
11. ((F, D1), D3) bound state, S-duality and noncommutative open string/Yang-Mills theory
International Nuclear Information System (INIS)
Lu, J.X.; Roy, S.; Singh, H.
2000-01-01
We study decoupling limits and S-dualities for noncommutative open string/Yang-Mills theory in a gravity setup by considering an SL(2,Z) invariant supergravity solution of the form ((F, D1), D3) bound state of type IIB string theory. This configuration can be regarded as D3-branes with both electric and magnetic fields turned on along one of the spatial directions of the brane and preserves half of the space-time supersymmetries of the string theory. Our study indicates that there exists a decoupling limit for which the resulting theory is an open string theory defined in a geometry with noncommutativity in both space-time and space-space directions. We study S-duality of this noncommutative open string (NCOS) and find that the same decoupling limit in the S-dual description gives rise to a space-space noncommutative Yang-Mills theory (NCYM). We also discuss independently the decoupling limit for NCYM in this D3 brane background. Here we find that S-duality of NCYM theory does not always give a NCOS theory. Instead, it can give an ordinary Yang-Mills with a singular metric and an infinitely large coupling. We also find that the open string coupling relation between the two S-duality related theories is modified such that S-duality of a strongly coupled open-string/Yang-Mills theory does not necessarily give a weakly coupled theory. The relevant gravity dual descriptions of NCOS/NCYM are also given. (author)
12. A Unified Field Theory of Gravity, Electromagnetism, and theA Unified Field Theory of Gravity, Electromagnetism, and the Yang-Mills Gauge Field
Directory of Open Access Journals (Sweden)
Suhendro I.
2008-01-01
Full Text Available In this work, we attempt at constructing a comprehensive four-dimensional unified field theory of gravity, electromagnetism, and the non-Abelian Yang-Mills gauge field in which the gravitational, electromagnetic, and material spin fields are unified as intrinsic geometric objects of the space-time manifold $S_4$ via the connection, with the generalized non-Abelian Yang-Mills gauge field appearing in particular as a sub-field of the geometrized electromagnetic interaction.
13. Global symmetries of Yang-Mills squared in various dimensions
Energy Technology Data Exchange (ETDEWEB)
Anastasiou, A. [Theoretical Physics, Blackett Laboratory, Imperial College London,London SW7 2AZ (United Kingdom); Borsten, L. [Theoretical Physics, Blackett Laboratory, Imperial College London,London SW7 2AZ (United Kingdom); School of Theoretical Physics, Dublin Institute for Advanced Studies,10 Burlington Road, Dublin 4 (Ireland); Hughes, M.J. [Theoretical Physics, Blackett Laboratory, Imperial College London,London SW7 2AZ (United Kingdom); Nagy, S. [Theoretical Physics, Blackett Laboratory, Imperial College London,London SW7 2AZ (United Kingdom); Department of Mathematics, Instituto Superior Técnico,Av. Rovisco Pais, 1049-001 Lisbon (Portugal)
2016-01-25
Tensoring two on-shell super Yang-Mills multiplets in dimensions D≤10 yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) D with each dimension 3≤D≤10 we obtain a formula for the supergravity U-duality G and its maximal compact subgroup H in terms of the internal global symmetry algebras of each super Yang-Mills theory. We extend our analysis to include supergravities coupled to an arbitrary number of matter multiplets by allowing for non-supersymmetric multiplets in the tensor product.
14. Hamiltonian approach to 1 + 1 dimensional Yang-Mills theory in Coulomb gauge
International Nuclear Information System (INIS)
Reinhardt, H.; Schleifenbaum, W.
2009-01-01
We study the Hamiltonian approach to 1 + 1 dimensional Yang-Mills theory in Coulomb gauge, considering both the pure Coulomb gauge and the gauge where in addition the remaining constant gauge field is restricted to the Cartan algebra. We evaluate the corresponding Faddeev-Popov determinants, resolve Gauss' law and derive the Hamiltonians, which differ in both gauges due to additional zero modes of the Faddeev-Popov kernel in the pure Coulomb gauge. By Gauss' law the zero modes of the Faddeev-Popov kernel constrain the physical wave functionals to zero colour charge states. We solve the Schroedinger equation in the pure Coulomb gauge and determine the vacuum wave functional. The gluon and ghost propagators and the static colour Coulomb potential are calculated in the first Gribov region as well as in the fundamental modular region, and Gribov copy effects are studied. We explicitly demonstrate that the Dyson-Schwinger equations do not specify the Gribov region while the propagators and vertices do depend on the Gribov region chosen. In this sense, the Dyson-Schwinger equations alone do not provide the full non-abelian quantum gauge theory, but subsidiary conditions must be required. Implications of Gribov copy effects for lattice calculations of the infrared behaviour of gauge-fixed propagators are discussed. We compute the ghost-gluon vertex and provide a sensible truncation of Dyson-Schwinger equations. Approximations of the variational approach to the 3 + 1 dimensional theory are checked by comparison to the 1 + 1 dimensional case
15. On N=8 supergravity in AdS5 and N=4 superconformal Yang-Mills theory
International Nuclear Information System (INIS)
Ferrara, S.; Zaffaroni, A.; Froensdal, C.
1998-01-01
We discuss the spectrum of states of IIB supergravity on AdS 5 x S 5 in a manifest SU(2,2/4) invariant setting. The boundary fields are described in terms of N=4 superconformal Yang-Mills theory and the proposed correspondence between supergravity in AdS 5 and superconformal invariant singleton theory at the boundary is formulated in a N=4 superfield covariant language. (orig.)
16. AdS/CFT correspondence, quasinormal modes, and thermal correlators in N=4 supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Nunez, Alvaro; Starinets, Andrei O.
2003-01-01
We use the Lorentzian AdS/CFT prescription to find the poles of the retarded thermal Green's functions of N=4 SU(N) supersymmetric Yang-Mills theory in the limit of large N and large 't Hooft coupling. In the process, we propose a natural definition for quasinormal modes in an asymptotically AdS spacetime, with boundary conditions dictated by the AdS/CFT correspondence. The corresponding frequencies determine the dispersion laws for the quasiparticle excitations in the dual finite-temperature gauge theory. Correlation functions of operators dual to massive scalar, vector and gravitational perturbations in a five-dimensional AdS-Schwarzschild background are considered. We find asymptotic formulas for quasinormal frequencies in the massive scalar and tensor cases, and an exact expression for vector perturbations. In the long-distance, low-frequency limit we recover results of the hydrodynamic approximation to thermal Yang-Mills theory
17. AdS/CFT correspondence, quasinormal modes, and thermal correlators in N=4 supersymmetric Yang-Mills theory
Science.gov (United States)
Núñez, Alvaro; Starinets, Andrei O.
2003-06-01
We use the Lorentzian AdS/CFT prescription to find the poles of the retarded thermal Green’s functions of N=4 SU(N) supersymmetric Yang-Mills theory in the limit of large N and large ’t Hooft coupling. In the process, we propose a natural definition for quasinormal modes in an asymptotically AdS spacetime, with boundary conditions dictated by the AdS/CFT correspondence. The corresponding frequencies determine the dispersion laws for the quasiparticle excitations in the dual finite-temperature gauge theory. Correlation functions of operators dual to massive scalar, vector and gravitational perturbations in a five-dimensional AdS-Schwarzschild background are considered. We find asymptotic formulas for quasinormal frequencies in the massive scalar and tensor cases, and an exact expression for vector perturbations. In the long-distance, low-frequency limit we recover results of the hydrodynamic approximation to thermal Yang-Mills theory.
18. Self-dual phase space for (3 +1 )-dimensional lattice Yang-Mills theory
Science.gov (United States)
Riello, Aldo
2018-01-01
I propose a self-dual deformation of the classical phase space of lattice Yang-Mills theory, in which both the electric and magnetic fluxes take value in the compact gauge Lie group. A local construction of the deformed phase space requires the machinery of "quasi-Hamiltonian spaces" by Alekseev et al., which is reviewed here. The results is a full-fledged finite-dimensional and gauge-invariant phase space, the self-duality properties of which are largely enhanced in (3 +1 ) spacetime dimensions. This enhancement is due to a correspondence with the moduli space of an auxiliary noncommutative flat connection living on a Riemann surface defined from the lattice itself, which in turn equips the duality between electric and magnetic fluxes with a neat geometrical interpretation in terms of a Heegaard splitting of the space manifold. Finally, I discuss the consequences of the proposed deformation on the quantization of the phase space, its quantum gravitational interpretation, as well as its relevance for the construction of (3 +1 )-dimensional topological field theories with defects.
19. Lattice Yang-Mills theory at finite densities of heavy quarks
International Nuclear Information System (INIS)
Langfeld, Kurt; Shin, Gwansoo
2000-01-01
SU(N c ) Yang-Mills theory is investigated at finite densities of N f heavy quark flavors. The calculation of the (continuum) quark determinant in the large-mass limit is performed by analytic methods and results in an effective gluonic action. This action is then subject to a lattice representation of the gluon fields and computer simulations. The approach maintains the same number of quark degrees of freedom as in the continuum formulation and a physical heavy quark limit (to be contrasted with the quenched approximation N f →0). The proper scaling towards the continuum limit is manifest. We study the partition function for given values of the chemical potential as well as the partition function which is projected onto a definite baryon number. First numerical results for an SU(2) gauge theory are presented. We briefly discuss the breaking of the color-electric string at finite densities and shed light onto the origin of the overlap problem inherent in the Glasgow approach
20. An object oriented code for simulating supersymmetric Yang-Mills theories
Science.gov (United States)
Catterall, Simon; Joseph, Anosh
2012-06-01
We present SUSY_LATTICE - a C++ program that can be used to simulate certain classes of supersymmetric Yang-Mills (SYM) theories, including the well known N=4 SYM in four dimensions, on a flat Euclidean space-time lattice. Discretization of SYM theories is an old problem in lattice field theory. It has resisted solution until recently when new ideas drawn from orbifold constructions and topological field theories have been brought to bear on the question. The result has been the creation of a new class of lattice gauge theories in which the lattice action is invariant under one or more supersymmetries. The resultant theories are local, free of doublers and also possess exact gauge-invariance. In principle they form the basis for a truly non-perturbative definition of the continuum SYM theories. In the continuum limit they reproduce versions of the SYM theories formulated in terms of twisted fields, which on a flat space-time is just a change of the field variables. In this paper, we briefly review these ideas and then go on to provide the details of the C++ code. We sketch the design of the code, with particular emphasis being placed on SYM theories with N=(2,2) in two dimensions and N=4 in three and four dimensions, making one-to-one comparisons between the essential components of the SYM theories and their corresponding counterparts appearing in the simulation code. The code may be used to compute several quantities associated with the SYM theories such as the Polyakov loop, mean energy, and the width of the scalar eigenvalue distributions. Program summaryProgram title: SUSY_LATTICE Catalogue identifier: AELS_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AELS_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 9315 No. of bytes in distributed program
1. Quantization of the Yang-Mills field and Mandelstam's theory in gauge-independent path-dependent formalism
International Nuclear Information System (INIS)
Naito, S.
1976-01-01
We derive commutation relations (CR's) between gauge-invariant quantities in the Yang-Mills field theory by applying the Peierls method. The CR's obtained are different from those given by Mandelstam in his gauge-independent, path-dependent formalism. However, our CR's are shown to give a consistently quantized field theory, while his CR's do not. In fact, there exist systematic errors in Mandelstam's treatment of the covariant Green's functions. On the other hand, if we correctly treat covariant Green's functions guided by his procedure, our CR's are shown to lead to the same Feynman rules for the Yang-Mills field as prescribed by Feynman, DeWitt, Faddeev and Popov, and Mandelstam
2. On string solutions of Bethe equations in N=4 supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Bytsko, A.G.; Shenderovich, I.E.
2007-12-01
The Bethe equations, arising in description of the spectrum of the dilatation operator for the su(2) sector of the N=4 supersymmetric Yang-Mills theory, are considered in the anti-ferromagnetic regime. These equations are deformation of those for the Heisenberg XXX magnet. It is proven that in the thermodynamic limit roots of the deformed equations group into strings. It is proven that the corresponding Yang's action is convex, which implies uniqueness of solution for centers of the strings. The state formed of strings of length (2n+1) is considered and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that non-analyticity of the left hand side of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. Whence it is concluded that the structure of the anti-ferromagnetic vacuum is determined by the behaviour of exponential corrections to string solutions in the thermodynamic limit and possibly involves strings of length 2. (orig.)
3. On string solutions of Bethe equations in N=4 supersymmetric Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Bytsko, A.G. [Rossijskaya Akademiya Nauk, St. Petersburg (Russian Federation). Inst. Matematiki]|[Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Shenderovich, I.E. [St. Petersburg State Univ. (Russian Federation). Physics Dept.
2007-12-15
The Bethe equations, arising in description of the spectrum of the dilatation operator for the su(2) sector of the N=4 supersymmetric Yang-Mills theory, are considered in the anti-ferromagnetic regime. These equations are deformation of those for the Heisenberg XXX magnet. It is proven that in the thermodynamic limit roots of the deformed equations group into strings. It is proven that the corresponding Yang's action is convex, which implies uniqueness of solution for centers of the strings. The state formed of strings of length (2n+1) is considered and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that non-analyticity of the left hand side of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. Whence it is concluded that the structure of the anti-ferromagnetic vacuum is determined by the behaviour of exponential corrections to string solutions in the thermodynamic limit and possibly involves strings of length 2. (orig.)
4. A further pathology of the Coulomb gauge in non-Abelian Yang-Mills theories
International Nuclear Information System (INIS)
Ademollo, M.; Napolitano, E.; Sciuto, S.
1978-01-01
In the first part the vacuum structure of SU(2) Yang-Mills theories in the Coulomb gauge is discussed. It is proved that the only transverse pure gauge field Asub(μ)(x) = U -1 deltasub(μ)U with U(x) → (as r→infinity) const., is the trivial one Asub(μ)(x) equivalent to 0; the features of other possible vacua with U(x) → (as r→infinity) U(theta, pli) are studied. In the second part, regular Euclidean configurations that connect a vacuum state at x 4 = -infinity to another at x 4 = +infinity are discussed. It is proved, always working in the Coulomb gauge, that the perturbative vacuum Asub(μ)(x) equivalent to 0 cannot tunnel into any other one and that regular configurations with non-vanishing Pontryagin number q cannot affect such a vacuum. Moreover, strong arguments are given to show that many-instanton configurations (mod(q)>=2) cannot be expressed at all in the Coulomb gauge, that is by a regular field Asub(μ) satisfying the transversality condition deltasub(i)Asub(i) (x, x 4 ) = 0. (Auth.)
5. Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
International Nuclear Information System (INIS)
Forrester, Peter J.; Majumdar, Satya N.; Schehr, Gregory
2011-01-01
We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=L c (N)∼√(N) in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy-Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy-Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy-Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions ('watermelons' with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy-Widom law. In addition, large deviation formulas for the maximum height are also computed.
6. Calculating the jet quenching parameter in the plasma of noncommutative Yang-Mills theory from gauge/gravity duality
Science.gov (United States)
Chakraborty, Somdeb; Roy, Shibaji
2012-02-01
A particular decoupling limit of the nonextremal (D1, D3) brane bound state system of type IIB string theory is known to give the gravity dual of space-space noncommutative Yang-Mills theory at finite temperature. We use a string probe in this background to compute the jet quenching parameter in a strongly coupled plasma of hot noncommutative Yang-Mills theory in (3+1) dimensions from gauge/gravity duality. We give expressions for the jet quenching parameter for both small and large noncommutativity. For small noncommutativity, we find that the value of the jet quenching parameter gets reduced from its commutative value. The reduction is enhanced with temperature as T7 for fixed noncommutativity and fixed ’t Hooft coupling. We also give an estimate of the correction due to noncommutativity at the present collider energies like in RHIC or in LHC and find it too small to be detected. We further generalize the results for noncommutative Yang-Mills theories in diverse dimensions.
7. Radiating black holes in Einstein-Yang-Mills theory and cosmic censorship
International Nuclear Information System (INIS)
2010-01-01
Exact nonstatic spherically symmetric black-hole solutions of the higher dimensional Einstein-Yang-Mills equations for a null dust with Yang-Mills gauge charge are obtained by employing Wu-Yang ansatz, namely, HD-EYM Vaidya solution. It is interesting to note that gravitational contribution of Yang-Mills (YM) gauge charge for this ansatz is indeed opposite (attractive rather than repulsive) that of Maxwell charge. It turns out that the gravitational collapse of null dust with YM gauge charge admits strong curvature shell focusing naked singularities violating cosmic censorship. However, there is significant shrinkage of the initial data space for a naked singularity of the HD-Vaidya collapse due to presence of YM gauge charge. The effect of YM gauge charge on structure and location of the apparent and event horizons is also discussed.
8. A non-technical introduction to confinement and N = 2 globally supersymmetric Yang-Mills gauge theories
International Nuclear Information System (INIS)
Fucito, F.
1997-01-01
The aim of this talk is to give a brief introduction to the problem of confinement in QCD and N = 2 globally supersymmetric Yang-Mills gauge theories (SYM). While avoiding technicalities as much as possible I will try to emphasize the physical ideas which lie behind the picture of confinement as a consequence of the vacua of QCD to be a dual superconductor. Finally I review the implementation of this picture in the framework of N = 2 SYM. (author)
9. The light bound states of N=1 supersymmetric SU(3) Yang-Mills theory on the lattice
Science.gov (United States)
Ali, Sajid; Bergner, Georg; Gerber, Henning; Giudice, Pietro; Montvay, Istvan; Münster, Gernot; Piemonte, Stefano; Scior, Philipp
2018-03-01
In this article we summarise our results from numerical simulations of N=1 supersymmetric Yang-Mills theory with gauge group SU(3). We use the formulation of Curci and Veneziano with clover-improved Wilson fermions. The masses of various bound states have been obtained at different values of the gluino mass and gauge coupling. Extrapolations to the limit of vanishing gluino mass indicate that the bound states form mass-degenerate supermultiplets.
10. Black holes with Yang-Mills hair
International Nuclear Information System (INIS)
Kleihaus, B.; Kunz, J.; Sood, A.; Wirschins, M.
1998-01-01
In Einstein-Maxwell theory black holes are uniquely determined by their mass, their charge and their angular momentum. This is no longer true in Einstein-Yang-Mills theory. We discuss sequences of neutral and charged SU(N) Einstein-Yang-Mills black holes, which are static spherically symmetric and asymptotically flat, and which carry Yang-Mills hair. Furthermore, in Einstein-Maxwell theory static black holes are spherically symmetric. We demonstrate that, in contrast, SU(2) Einstein-Yang-Mills theory possesses a sequence of black holes, which are static and only axially symmetric
11. Yang-Mills theory on a momentum lattice: Gauge invariance, chiral invariance, and no fermion doubling
International Nuclear Information System (INIS)
Berube, D.; Kroeger, H.; Lafrance, R.; Marleau, L.
1991-01-01
We discuss properties of a noncompact formulation of gauge theories with fermions on a momentum (k) lattice. (a) This formulation is suitable to build in Fourier acceleration in a direct way. (b) The numerical effort to compute the action (by fast Fourier transform) goes essentially like logV with the lattice volume V. (c) For the Yang-Mills theory we find that the action conserves gauge symmetry and chiral symmetry in a weak sense: On a finite lattice the action is invariant under infinitesimal transformations with compact support. Under finite transformations these symmetries are approximately conserved and they are restored on an infinite lattice and in the continuum limit. Moreover, these symmetries also hold on a finite lattice under finite transformations, if the classical fields, instead of being c-number valued, take values from a finite Galois field. (d) There is no fermion doubling. (e) For the φ 4 model we investigate the transition towards the continuum limit in lattice perturbation theory up to second order. We compute the two- and four-point functions and find local and Lorentz-invariant results. (f) In QED we compute a one-loop vacuum polarization and find in the continuum limit the standard result. (g) As a numerical application, we compute the propagator left-angle φ(k)φ(k')right-angle in the φ 4 model, investigate Euclidean invariance, and extract m R as well as Z R . Moreover we compute left-angle F μν (k)F μν (k')right-angle in the SU(2) model
12. Yang-Mills theories in axial and light-cone gauges, analytic regularization and Ward identities
International Nuclear Information System (INIS)
Lee, H.C.
1984-12-01
The application of the principles of generalization and analytic continuation to the regularization of divergent Feynman integrals is discussed. The technique, or analytic regularization, which is a generalization of dimensional regularization, is used to derive analytic representations for two classes of massless two-point integrals. The first class is based on the principal-value prescription and includes integrals encountered in quantum field theories in the ghost-free axial gauge (n.A=0), reducing in a special case to integrals in the light-cone gauge (n.A=0,n 2 =0). The second class is based on the Mandelstam prescription devised espcially for the light-cone gauge. For some light-cone gauge integrals the two representations are not equivalent. Both classes include as a subclass integrals in the Lorentz covariant 'zeta-gauges'. The representations are used to compute one-loop corrections to the self-energy and the three-vertex in Yang-Mills theories in the axial and light-cone gauges, showing that the two- and three-point Ward identities are satisfied; to illustrate that ultraviolet and infrared singularities, indistinguishable in dimensional regularization, can be separated analytically; and to show that certain tadpole integrals vanish because of an exact cancellation between ultraviolet and infrared singularities. In the axial gauge, the wavefunction and vertex renormalization constants, Z 3 and Z 1 , are identical, so that the β-function can be directly derived from Z 3 the result being the same as that computed in the covariant zeta-gauges. Preliminary results suggest that the light-cone gauge in the Mandelstam prescription, but not in the principal value prescription, has the same renormalization property of the axial gauge
13. Dilatation operator and the Super Yang-Mills duals of open strings on AdS giant gravitons
International Nuclear Information System (INIS)
Correa, Diego H.; Silva, Guillermo A.
2006-01-01
We study the one-loop anomalous dimensions of the Super Yang-Mills dual operators to open strings ending on AdS giant gravitons. AdS giant gravitons have no upper bound for their angular momentum and we represent them by the contraction of scalar fields, carrying the appropriate R-charge, with a totally symmetric tensor. We represent the open string motion along AdS directions by appending to the giant graviton operator a product of fields including covariant derivatives. We derive a bosonic lattice Hamiltonian that describes the mixing of these excited AdS giants operators under the action of the one-loop dilatation operator of N = 4 SYM. This Hamiltonian captures several intuitive differences with respect to the case of sphere giant gravitons. A semiclassical analysis of the Hamiltonian allows us to give a geometrical interpretation for the labeling used to describe the fields products appended to the AdS giant operators. It also allows us to show evidence for the existence of continuous bands in the Hamiltonian spectrum
14. Einstein-Yang-Mills from pure Yang-Mills amplitudes
Energy Technology Data Exchange (ETDEWEB)
Nandan, Dhritiman; Plefka, Jan [Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin,Zum Großen Windkanal 6, D-12489 Berlin (Germany); Schlotterer, Oliver [Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,Am Mühlenberg 1, D-14476 Potsdam (Germany); Wen, Congkao [I.N.F.N. Sezione di Roma Tor Vergata,Via della Ricerca Scientifica, 00133 Roma (Italy)
2016-10-14
We present new relations for scattering amplitudes of color ordered gluons and gravitons in Einstein-Yang-Mills theory. Tree-level amplitudes of arbitrary multiplicities and polarizations involving up to three gravitons and up to two color traces are reduced to partial amplitudes of pure Yang-Mills theory. In fact, the double-trace identities apply to Einstein-Yang-Mills extended by a dilaton and a B-field. Our results generalize recent work of Stieberger and Taylor for the single graviton case with a single color trace. As the derivation is made in the dimension-agnostic Cachazo-He-Yuan formalism, our results are valid for external bosons in any number of spacetime dimensions. Moreover, they generalize to the superamplitudes in theories with 16 supercharges.
15. Generating functional and large N limit of nonlocal 2D generalized Yang-Mills theories (nlgYM2's)
International Nuclear Information System (INIS)
2001-01-01
Using the path integral method, we calculate the partition function and the generating functional (of the field strengths) on nonlocal generalized 2D Yang-Mills theories (nlgYM 2 's), which are nonlocal in the auxiliary field. This has been considered before by Saaidi and Khorrami. Our calculations are done for general surfaces. We find a general expression for the free energy of W(φ) =φ 2k in nlgYM 2 theories at the strong coupling phase (SCP) regime (A > A c ) for large groups. In the specific φ 4 model, we show that the theory has a third order phase transition. (orig.)
16. The quantum dual string wave functional in Yang-Mills theories
International Nuclear Information System (INIS)
Gervais, J.-L.; Neveu, A.
1979-01-01
From any solution of the classical Yang-Mills equations, a string wave functional based on the Wilson loop integral is defined. Its precise definition is given by replacing the string by a finite set of N points, and taking the limit N → infinity. It is shown that this functional satisfies the Schroedinger equation of the relativistic dual string to leading order in N. The relevance of this object to the quantum problem is speculated. (Auth.)
17. Onset of chaos in the classical SU(2) Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Furusawa, Toyoaki
1988-12-28
Chaotic behaviors of color electric and magnetic fields are numerically demonstrated in the classical SU(2) Yang-Mills system in the case that the field configuration depends only on one spatial coordinate and time. We show that the homogeneous color fields evolve into the disordered one as time passes. Power spectra of the color fields are investigated and the maximum Lyapunov exponent is evaluated.
18. Finiteness of broken N=4 super Yang-Mills theory
International Nuclear Information System (INIS)
Namazie, M.A.; Salam, A.; Strathdee, J.
1982-11-01
Using a light cone gauge formulation for N=4 extended supersymmetry, it is shown that an explicit breaking of the supersymmetry by addition of mass terms does not disturb off-shell finiteness to any order provided the sum of fermion masses equals the sum of scalar masses and appropriate cubic interactions between scalars are included. (author)
19. An off-shell formulation of N=4 supersymmetric Yang-Mills theory in twistor harmonic superspace
International Nuclear Information System (INIS)
Sokatchev, E.
1989-01-01
Twistor-like harmonic variables which parametrize the coset space SO(1, 4)/SO(1, 2)xSO(2) are introduced. With their help the on-shell constraints for N=4, d=5 supersymmetric Yang-Mills theory are rewritten as conditions for flatness in the harmonic directions of superspace. A Chern-Simons off-shell action leading to those equations is proposed. There are indications that the off-shell theory might be finite, despite the fact that the on-shell one seems non-renormalizable. (orig.)
20. Numerical determination of quark potential, glueball masses, and phase structure in the N=1 supersymmetric Yang-Mills theory; Numerische Bestimmung von Quarkpotential, Glueball-Massen und Phasenstruktur in der N=1 supersymmetrischen Yang-Mills-Theorie
Energy Technology Data Exchange (ETDEWEB)
Sandbrink, Dirk
2015-01-26
One of the most promising candidates to describe the physics beyond the standard model is the so-called supersymmetry. This work was created in the context of the DESY-Muenster-Collaboration, which studies in particular the N=1 supersymmetric Yang-Mills theory (SYM). SYM is a comparatively simple theory, which is therefore well-suited to study the expected properties of a supersymmetric theory with the help of Monte Carlo simulations on the lattice. This thesis is focused on the numerical determination of the quarkpotential, the glueball masses and the phase structur of the N=1 supersymmetric Yang-Mills theory. The quarkpotential is used to calculate the Sommer scale, which in turn is needed to convert the dimensionless lattice spacing into physical units. Glueballs are hypothetical particles built out of gluons, their masses are relatively hard to determine in lattice simulations due to their pure gluonic nature. For this reason, various methods are studied to reduce the uncertainties of the mass determination. The focus lies on smearing methods and their use in variational smearing as well as on the use of different glueball operators. Lastly, a first look is taken at the phase diagram of the model at finite temperature. Various simulations have been performed at finite temperature in parallel to those performed at temperature zero to analyse the behaviour of the Polyakov loop and the gluino condensate in greater detail.
1. Two-loop ghost-antighost condensation for SU(2) Yang-Mills theories in the maximal abelian gauge
International Nuclear Information System (INIS)
Fazio, A.R.
2004-01-01
In the framework of the formalism of Cornwall et.al. for composite operators I study the ghost-antighost condensation in SU(2) Yang-Mills theories quantized in the Maximal Abelian Gauge and derive analytically a condensating effective potential at two ghost loops. I find that in this approximation the one-loop pairing ghost-antighost is not destroyed and no mass is generated if the ansatz for the propagator suggested by the tree level Hubbard-Stratonovich transformations is used
2. The topological susceptibility in the large-N limit of SU(N) Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Ce, Marco [Scuola Normale Superiore, Pisa (Italy); Istituto Nazionale di Fisica Nucleare, Pisa (Italy); Garcia Vera, Miguel [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Giusti, Leonardo [Milano-Bicocca Univ. (Italy); INFN, Milano (Italy); Schaefer, Stefan [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2016-07-15
We compute the topological susceptibility of the SU(N) Yang-Mills theory in the large-N limit with a percent level accuracy. This is achieved by measuring the gradient-flow definition of the susceptibility at three values of the lattice spacing for N=3,4,5,6. Thanks to this coverage of parameter space, we can extrapolate the results to the large-N and continuum limits with confidence. Open boundary conditions are instrumental to make simulations feasible on the finer lattices at the larger N.
3. Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem
International Nuclear Information System (INIS)
Montesinos, M.; Flores, E.
2006-01-01
The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end. (Author)
4. Quantum fields in the non-perturbative regime. Yang-Mills theory and gravity
Energy Technology Data Exchange (ETDEWEB)
Eichhorn, Astrid
2011-09-06
In this thesis we study candidates for fundamental quantum field theories, namely non-Abelian gauge theories and asymptotically safe quantum gravity. Whereas the first ones have a stronglyinteracting low-energy limit, the second one enters a non-perturbative regime at high energies. Thus, we apply a tool suited to the study of quantum field theories beyond the perturbative regime, namely the Functional Renormalisation Group. In a first part, we concentrate on the physical properties of non-Abelian gauge theories at low energies. Focussing on the vacuum properties of the theory, we present an evaluation of the full effective potential for the field strength invariant F{sub {mu}}{sub {nu}}F{sup {mu}}{sup {nu}} from non-perturbative gauge correlation functions and find a non-trivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the {beta} function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators and derive an upper bound on their scaling exponents. We then consider the theory at finite temperature and study the nature of the confinement phase transition in d = 3+1 dimensions in various non-Abelian gauge theories. For SU(N) with N= 3,..,12 and Sp(2) we find a first-order phase transition in agreement with general expectations. Moreover our study suggests that the phase transition in E(7) Yang-Mills theory also is of first order. Our studies shed light on the question which property of a gauge group determines the order of the phase transition. In a second part we consider asymptotically safe quantum gravity. Here, we focus on the Faddeev-Popov ghost sector of the theory, to study its properties in the context of an interacting UV regime. We investigate several truncations, which all lend support to the conjecture that gravity may be asymptotically safe. In a first truncation, we study the ghost anomalous dimension
5. Quantum fields in the non-perturbative regime. Yang-Mills theory and gravity
International Nuclear Information System (INIS)
Eichhorn, Astrid
2011-01-01
In this thesis we study candidates for fundamental quantum field theories, namely non-Abelian gauge theories and asymptotically safe quantum gravity. Whereas the first ones have a stronglyinteracting low-energy limit, the second one enters a non-perturbative regime at high energies. Thus, we apply a tool suited to the study of quantum field theories beyond the perturbative regime, namely the Functional Renormalisation Group. In a first part, we concentrate on the physical properties of non-Abelian gauge theories at low energies. Focussing on the vacuum properties of the theory, we present an evaluation of the full effective potential for the field strength invariant F μν F μν from non-perturbative gauge correlation functions and find a non-trivial minimum corresponding to the existence of a dimension four gluon condensate in the vacuum. We also relate the infrared asymptotic form of the β function of the running background-gauge coupling to the asymptotic behavior of Landau-gauge gluon and ghost propagators and derive an upper bound on their scaling exponents. We then consider the theory at finite temperature and study the nature of the confinement phase transition in d = 3+1 dimensions in various non-Abelian gauge theories. For SU(N) with N= 3,..,12 and Sp(2) we find a first-order phase transition in agreement with general expectations. Moreover our study suggests that the phase transition in E(7) Yang-Mills theory also is of first order. Our studies shed light on the question which property of a gauge group determines the order of the phase transition. In a second part we consider asymptotically safe quantum gravity. Here, we focus on the Faddeev-Popov ghost sector of the theory, to study its properties in the context of an interacting UV regime. We investigate several truncations, which all lend support to the conjecture that gravity may be asymptotically safe. In a first truncation, we study the ghost anomalous dimension which we find to be negative at the
6. Quark confinement: Dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang-Mills theory
Science.gov (United States)
Kondo, Kei-Ichi; Kato, Seikou; Shibata, Akihiro; Shinohara, Toru
2015-05-01
The purpose of this paper is to review the recent progress in understanding quark confinement. The emphasis of this review is placed on how to obtain a manifestly gauge-independent picture for quark confinement supporting the dual superconductivity in the Yang-Mills theory, which should be compared with the Abelian projection proposed by 't Hooft. The basic tools are novel reformulations of the Yang-Mills theory based on change of variables extending the decomposition of the SU(N) Yang-Mills field due to Cho, Duan-Ge and Faddeev-Niemi, together with the combined use of extended versions of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the SU(N) Wilson loop operator. Moreover, we give the lattice gauge theoretical versions of the reformulation of the Yang-Mills theory which enables us to perform the numerical simulations on the lattice. In fact, we present some numerical evidences for supporting the dual superconductivity for quark confinement. The numerical simulations include the derivation of the linear potential for static interquark potential, i.e., non-vanishing string tension, in which the "Abelian" dominance and magnetic monopole dominance are established, confirmation of the dual Meissner effect by measuring the chromoelectric flux tube between quark-antiquark pair, the induced magnetic-monopole current, and the type of dual superconductivity, etc. In addition, we give a direct connection between the topological configuration of the Yang-Mills field such as instantons/merons and the magnetic monopole. We show especially that magnetic monopoles in the Yang-Mills theory can be constructed in a manifestly gauge-invariant way starting from the gauge-invariant Wilson loop operator and thereby the contribution from the magnetic monopoles can be extracted from the Wilson loop in a gauge-invariant way through the non-Abelian Stokes theorem for the Wilson loop operator, which is a prerequisite for exhibiting magnetic monopole dominance for quark
7. Boson-fermion and boson-boson scattering in a Yang-Mills theory at high energy: Sixth-order perturbation theory
International Nuclear Information System (INIS)
McCoy, B.M.; Wu, T.T.
1976-01-01
Our previous study of Yang-Mills fields is extended by calculating the high-energy behavior of the boson-fermion and of the boson-boson amplitude in sixth-order perturbation theory. In the isovector and isoscalar channels of both these processes the behavior of the amplitude is the same as that found in fermion-fermion scattering
8. A novel long range spin chain and planar N=4 super Yang-Mills
International Nuclear Information System (INIS)
Beisert, N.; Dippel, V.; Staudacher, M.
2004-01-01
We probe the long-range spin chain approach to planar N=4 gauge theory at high loop order. A recently employed hyperbolic spin chain invented by Inozemtsev is suitable for the SU(2) subsector of the state space up to three loops, but ceases to exhibit the conjectured thermodynamic scaling properties at higher orders. We indicate how this may be bypassed while nevertheless preserving integrability, and suggest the corresponding all-loop asymptotic Bethe ansatz. We also propose the local part of the all-loop gauge transfer matrix, leading to conjectures for the asymptotically exact formulae for all local commuting charges. The ansatz is finally shown to be related to a standard inhomogeneous spin chain. A comparison of our ansatz to semi-classical string theory uncovers a detailed, non-perturbative agreement between the corresponding expressions for the infinite tower of local charge densities. However, the respective Bethe equations differ slightly, and we end by refining and elaborating a previously proposed possible explanation for this disagreement. (author)
9. Massive Yang-Mills fields
NARCIS (Netherlands)
Veltman, M.J.G.; Reiff, J.
1969-01-01
Two problems are studied in the paper: (i) the relation between Lagrangian and Feynman rules if the Lagrangian contains derivative couplings and/or vector meson fields and (ii) the behaviour of certain two closed loop diagrams in the perturbation theory of Yang-Mills fields. With respect to ( i ) .
10. Double-winding Wilson loops in SU(N) Yang-Mills theory - A criterion for testing the confinement models -
Science.gov (United States)
Matsudo, Ryutaro; Kondo, Kei-Ichi; Shibata, Akihiro
2018-03-01
We examine how the average of double-winding Wilson loops depends on the number of color N in the SU(N) Yang-Mills theory. In the case where the two loops C1 and C2 are identical, we derive the exact operator relation which relates the doublewinding Wilson loop operator in the fundamental representation to that in the higher dimensional representations depending on N. By taking the average of the relation, we find that the difference-of-areas law for the area law falloff recently claimed for N = 2 is excluded for N ⩾ 3, provided that the string tension obeys the Casimir scaling for the higher representations. In the case where the two loops are distinct, we argue that the area law follows a novel law (N - 3)A1/(N - 1) + A2 with A1 and A2(A1 law when (N ⩾ 3). Indeed, this behavior can be confirmed in the two-dimensional SU(N) Yang-Mills theory exactly.
11. The large N limit of N=2 super Yang-Mills, fractional instantons and infrared divergences
International Nuclear Information System (INIS)
Ferrari, Frank
2001-01-01
We investigate the large N limit of pure N=2 supersymmetric gauge theory with gauge group SU(N) by using the exact low energy effective action. Typical one-complex dimensional sections of the moduli space parametrized by a global complex mass scale v display three qualitatively different regions depending on the ratio between vertical bar v vertical bar and the dynamically generated scale Λ. At large vertical bar v vertical bar/Λ, instantons are exponentially suppressed as N→∞. When vertical bar v vertical bar ∼Λ, singularities due to massless dyons occur. They are densely distributed in rings of calculable thicknesses in the v-plane. At small vertical bar v vertical bar/Λ, instantons disintegrate into fractional instantons of charge 1/(2N). These fractional instantons give non-trivial contributions to all orders of 1/N, unlike a planar diagrams expansion which generates a series in 1/N 2 , implying the presence of open strings. We have explicitly calculated the fractional instantons series in two representative examples, including the 1/N and 1/N 2 corrections. Our most interesting finding is that the 1/N expansion breaks down at singularities on the moduli space due to severe infrared divergencies, a fact that has remarkable consequences
12. End-to-end probability for an interacting center vortex world line in Yang-Mills theory
International Nuclear Information System (INIS)
Teixeira, Bruno F.I.; Lemos, Andre L.L. de; Oxman, Luis E.
2011-01-01
Full text: The understanding of quark confinement is a very important open problem in Yang-Mills theory. In this regard, nontrivial topological defects are expected to play a relevant role to achieve a solution. Here we are interested in how to deal with these structures, relying on the Cho-Faddeev-Niemi decomposition and the possibility it offers to describe defects in terms of a local color frame. In particular, the path integral for a single center vortex is a fundamental object to handle the ensemble integration. As is well-known, in three dimensions center vortices are string-like and the associated physics is closely related with that of polymers. Using recent techniques developed in the latter context, we present in this work a detailed derivation of the equation for the end-to-end probability for a center vortex world line, including the effects of interactions. Its solution can be associated with a Green function that depends on the position and orientation at the boundaries, where monopole-like instantons are placed. In the limit of semi flexible polymers, an expansion only keeping the lower angular momenta for the final orientation leads to a reduced Green function for a complex vortex field minimally coupled to the dual Yang-Mills fields. This constitutes a key ingredient to propose an effective model for correlated monopoles, center vortices and the dual fields. (author)
13. Fuzzy knot theory interpretation of Yang-Mills instantons and Witten's 5-Brane model
International Nuclear Information System (INIS)
El Naschie, M.S.
2008-01-01
A knot theory interpretation of 'tHooft's instanton based on hyperbolic volume, crossing numbers and exceptional Lie symmetry groups is given. Subsequently it is shown that although instantons and particle-like states of Heterotic super strings may appear to be different concepts, on a very deep fuzzy level they are not
14. Total source charge and charge screening in Yang-Mills theories
International Nuclear Information System (INIS)
Campbell, W.B.; Norton, R.E.
1991-01-01
New gauge-invariant definitions for the total charge on a static Yang-Mills source are suggested which we argue are better suited for determining when true color screening has occurred. In particular, these new definitions imply that the Abelian Coulomb solution for a simple ''electric'' dipole source made up of two opposite point charges has zero total source charge and therefore no color screening. With the definition of total source charge previously suggested by other authors, such a source would have a total source charge of 2q and therefore a screening charge in the field of -2q, where q is the magnitude of the charge of either point charge. Our definitions for more general solutions are not unique because of the path dependence of the parallel transport of charges. Suggestions for removing this ambiguity are offered, but it is not known if a unique, physically meaningful definition of total source charge in fact exists
15. Running coupling from gluon and ghost propagators in the Landau gauge: Yang-Mills theories with adjoint fermions
Science.gov (United States)
Bergner, Georg; Piemonte, Stefano
2018-04-01
Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are N =1 supersymmetric Yang-Mills (SYM) theory and minimal walking technicolor, which are gauge theories coupled to one adjoint Majorana and two adjoint Dirac fermions, respectively. While confinement is a property of N =1 SYM, minimal walking technicolor is expected to be infrared conformal. We study the propagators of ghost and gluon fields in the Landau gauge to compute the running coupling in the MiniMom scheme. We analyze several different ensembles of lattice Monte Carlo simulations for the SU(2) adjoint QCD with Nf=1 /2 ,1 ,3 /2 , and 2 Dirac fermions. We show how the running of the coupling changes as the number of interacting fermions is increased towards the conformal window.
16. Analytic study of the off-diagonal mass generation for Yang-Mills theories in the maximal Abelian gauge
International Nuclear Information System (INIS)
Dudal, D.; Verschelde, H.; Gracey, J.A.; Lemes, V.E.R.; Sobreiro, R.F.; Sorella, S.P.; Sarandy, M.S.
2004-01-01
We investigate a dynamical mass generation mechanism for the off-diagonal gluons and ghosts in SU(N) Yang-Mills theories, quantized in the maximal Abelian gauge. Such a mass can be seen as evidence for the Abelian dominance in that gauge. It originates from the condensation of a mixed gluon-ghost operator of mass dimension two, which lowers the vacuum energy. We construct an effective potential for this operator by a combined use of the local composite operators technique with the algebraic renormalization and we discuss the gauge parameter independence of the results. We also show that it is possible to connect the vacuum energy, due to the mass dimension-two condensate discussed here, with the nontrivial vacuum energy originating from the condensate μ 2 >, which has attracted much attention in the Landau gauge
17. Black p-branes versus black holes in non-asymptotically flat Einstein-Yang-Mills theory
Science.gov (United States)
Habib Mazharimousavi, S.; Halilsoy, M.
2016-09-01
We present a class of non-asymptotically flat (NAF) charged black p-branes (BpB) with p-compact dimensions in higher-dimensional Einstein-Yang-Mills theory. Asymptotically the NAF structure manifests itself as an anti-de sitter spacetime. We determine the total mass/energy enclosed in a thin shell located outside the event horizon. By comparing the entropies of BpB with those of black holes in the same dimensions we derive transition criteria between the two types of black objects. Given certain conditions satisfied, our analysis shows that BpB can be considered excited states of black holes. An event horizon r+ versus charge square Q2 plot for the BpB reveals such a transition where r+ is related to the horizon radius rh of the black hole (BH) both with the common charge Q.
18. Local BRST cohomology in the antifield formalism. Pt. 2. Application to Yang-Mills theory
International Nuclear Information System (INIS)
Barnich, G.; Henneaux, M.
1995-01-01
Yang-Mills models with compact gauge group coupled to matter fields are considered. The general tools developed in a companion paper are applied to compute the local cohomology of the BRST differential s modulo the exterior spacetime derivative d for all values of the ghost number, in the space of polynomials in the fields, the ghosts, the antifields (=sources for the BRST variations) and their derivatives. New solutions to the consistency conditions sa+db=0 depending non-trivially on the antifields are exhibited. For a semi-simple gauge group, however, these new solutions arise only at ghost number two or higher. Thus at ghost number zero or one, the inclusion of the antifields does not bring in new solutions to the consistency condition sa+db=0 besides the already known ones. The analysis does not use power counting and is purely cohomological. It can be easily extended to more general actions containing higher derivatives of the curvature or Chern-Simons terms. (orig.)
19. On non-primitively divergent vertices of Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Huber, Markus Q. [Institute of Physics, University of Graz, NAWI Graz (Austria)
2017-11-15
Two correlation functions of Yang-Mills beyond the primitively divergent ones, the two-ghost-two-gluon and the four-ghost vertices, are calculated and their influence on lower vertices is examined. Their full (transverse) tensor structure is taken into account. As input, a solution of the full two-point equations - including two-loop terms - is used that respects the resummed perturbative ultraviolet behavior. A clear hierarchy is found with regard to the color structure that reduces the number of relevant dressing functions. The impact of the two-ghost-two-gluon vertex on the three-gluon vertex is negligible, which is explained by the fact that all non-small dressing functions drop out due to their color factors. Only in the ghost-gluon vertex a small net effect below 2% is seen. The four-ghost vertex is found to be extremely small in general. Since these two four-point functions do not enter into the propagator equations, these findings establish their small overall effect on lower correlation functions. (orig.)
20. Local BRST cohomology in the antifield formalism. Pt. 2. Application to Yang-Mills theory
International Nuclear Information System (INIS)
Barnich, G.; Henneaux, M.; Brandt, F.
1994-01-01
Yang-Mills models with compact gauge group coupled to matter fields are considered. The general tools developed in a companion paper are applied to compute the local cohomology of the BRST differential s modulo the exterior spacetime derivative d for all values of the ghost number, in the space of polynomials in the fields, the ghosts, the antifields (= sources for the BRST variations) and their derivatives. New solutions to the consistency conditions sa+db = 0 depending non trivially on the antifields are exhibited. For a semi-simple gauge group, however, these new solutions arise only at ghost number two or higher. Thus at ghost number zero or one, the inclusion of the antifields does not bring in new solutions to the consistency condition sa+db 0 besides the already known ones. The analysis does not use power counting and is purely cohomological. It can be easily extended to more general actions containing higher derivatives of the curvature, or Chern-Simons terms. (orig.)
1. Non-perturbative construction of 2D and 4D supersymmetric Yang-Mills theories with 8 supercharges
International Nuclear Information System (INIS)
Hanada, Masanori; Matsuura, So; Sugino, Fumihiko
2012-01-01
In this paper, we consider two-dimensional N=(4,4) supersymmetric Yang-Mills (SYM) theory and deform it by a mass parameter M with keeping all supercharges. We further add another mass parameter m in a manner to respect two of the eight supercharges and put the deformed theory on a two-dimensional square lattice, on which the two supercharges are exactly preserved. The flat directions of scalar fields are stabilized due to the mass deformations, which gives discrete minima representing fuzzy spheres. We show in the perturbation theory that the lattice continuum limit can be taken without any fine tuning. Around the trivial minimum, this lattice theory serves as a non-perturbative definition of two-dimensional N=(4,4) SYM theory. We also discuss that the same lattice theory realizes four-dimensional N=2U(k) SYM on R 2 ×(Fuzzy R 2 ) around the minimum of k-coincident fuzzy spheres.
2. N = 1 SU(2) supersymmetric Yang-Mills theory on the lattice with light dynamical Wilson gluinos
International Nuclear Information System (INIS)
Demmouche, Kamel
2009-01-01
The supersymmetric Yang-Mills (SYM) theory with one supercharge (N=1) and one additional Majorana matter-field represents the simplest model of supersymmetric gauge theory. Similarly to QCD, this model includes gauge fields, gluons, with color gauge group SU(N c ) and fermion fields, describing the gluinos. The non-perturbative dynamical features of strongly coupled supersymmetric theories are of great physical interest. For this reason, many efforts are dedicated to their formulation on the lattice. The lattice regularization provides a powerful tool to investigate non-perturbatively the phenomena occurring in SYM such as confinement and chiral symmetry breaking. In this work we perform numerical simulations of the pure SU(2) SYM theory on large lattices with small Majorana gluino masses down to about m g approx 115 MeV with lattice spacing up to a ≅0.1 fm. The gluino dynamics is simulated by the Two-Step Multi-Boson (TSMB) and the Two-Step Polynomial Hybrid Monte Carlo (TS-PHMC) algorithms. Supersymmetry (SUSY) is broken explicitly by the lattice and the Wilson term and softly by the presence of a non-vanishing gluino mass m g ≠0. However, the recovery of SUSY is expected in the infinite volume continuum limit by tuning the bare parameters to the SUSY point in the parameter space. This scenario is studied by the determination of the low-energy mass spectrum and by means of lattice SUSY Ward-Identities (WIs). (orig.)
3. Thermodynamics of SU(N) Yang-Mills theories in 2+1 dimensions II. The Deconfined phase
CERN Document Server
Caselle, Michele; Feo, Alessandra; Gliozzi, Ferdinando; Gursoy, Umut; Panero, Marco; Schafer, Andreas
2012-01-01
We present a non-perturbative study of the equation of state in the deconfined phase of Yang-Mills theories in D=2+1 dimensions. We introduce a holographic model, based on the improved holographic QCD model, from which we derive a non-trivial relation between the order of the deconfinement phase transition and the behavior of the trace of the energy-momentum tensor as a function of the temperature T. We compare the theoretical predictions of this holographic model with a new set of high-precision numerical results from lattice simulations of SU(N) theories with N=2, 3, 4, 5 and 6 colors. The latter reveal that, similarly to the D=3+1 case, the bulk equilibrium thermodynamic quantities (pressure, trace of the energy-momentum tensor, energy density and entropy density) exhibit nearly perfect proportionality to the number of gluons, and can be successfully compared with the holographic predictions in a broad range of temperatures. Finally, we also show that, again similarly to the D=3+1 case, the trace of the en...
4. Large-N limit of the non-local 2D Yang-Mills and generalized Yang-Mills theories on a cylinder
Energy Technology Data Exchange (ETDEWEB)
Saaidi, K. [Department of Physics, Tehran University (Iran); Khorrami, M. [Institute for Advanced Studies in Basic Sciences, Zanjan (Iran)
2002-04-01
The large-group behavior of the non-local YM{sub 2}'s and gYM{sub 2}'s on a cylinder or a disk is investigated. It is shown that this behavior is similar to that of the corresponding local theory, but with the area of the cylinder replaced by an effective area depending on the dominant representation. The critical areas for non-local YM{sub 2}'s on a cylinder with some special boundary conditions are also obtained. (orig.)
5. Experimentally verifiable Yang-Mills spin 2 gauge theory of gravity with group U(1) x SU(2)
International Nuclear Information System (INIS)
Peng, H.
1988-01-01
In this work, a Yang-Mills spin 2 gauge theory of gravity is proposed. Based on both the verification of the helicity 2 property of the SU(2) gauge bosons of the theory and the agreement of the theory with most observational and experimental evidence, the authors argues that the theory is truly a gravitational theory. An internal symmetry group, the eigenvalues of its generators are identical with quantum numbers, characterizes the interactions of a given class. The author demonstrates that the 4-momentum P μ of a fermion field generates the U(1) x SU(2) internal symmetry group for gravity, but not the transformation group T 4 . That particles are classified by mass and spin implies that the U(1) x SU(2), instead of the Poincare group, is a symmetry group of gravity. It is shown that the U(1) x SU(2) group represents the time displacement and rotation in ordinary space. Thereby internal space associated with gravity is identical with Minkowski spacetime, so a gauge potential of gravity carries two space-time indices. Then he verifies that the SU(2) gravitational boson has helicity 2. It is this fact, spin from internal spin, that explains alternatively why the gravitational field is the only field which is characterized by spin 2. The Physical meaning of gauge potentials of gravity is determined by comparing theory with the results of experiments, such as the Collella-Overhauser-Werner (COW) experiment and the Newtonian limit, etc. The gauge potentials this must identify with ordinary gravitational potentials
6. BPS string solutions in non-Abelian Yang-Mills theories
Energy Technology Data Exchange (ETDEWEB)
Kneipp, Marco A.C.; Brockill, Patrick [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil). Coordenacao de Teoria de Campos e Particulas]. E-mail: [email protected]; [email protected]
2001-04-01
Starting from the bosonic part of N=2 Super QCD with a 'Seiberg-Witten' N = 2 breaking mass term, we obtain string BPS conditions for arbitrary semi-simple gauge groups. We show that the vacuum structure is compatible with a symmetry breaking scheme which allows the existence of Z{sub k}-strings and which has Spin (10) {yields} SU(5) x Z{sub 2} as a particular case. We obtain BPS Z{sub k}-string solutions and show that they satisfy the same first order differential equations and string tension as the BPS string for the U(1) case. (author)
7. The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
International Nuclear Information System (INIS)
Bern, Zvi; Czakon, Michael; Dixon, Lance J.; Kosower, David A.; Smirnov, Vladimir A.
2006-01-01
We present an expression for the leading-color (planar) four-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4-2ε dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around ε = 0, and obtain analytic expressions for the poles from 1/ε 8 through 1/ε 4 . We give numerical results for the coefficients of the 1/ε 3 and 1/e 2 poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/ε 2 coefficient allows us to test a conjecture of Eden and Staudacher for the four-loop cusp (soft) anomalous dimension. We find that the conjecture is incorrect, although our numerical results suggest that a simple modification of the expression, flipping the sign of the term containing ζ 3 2 , may yield the correct answer. Our numerical value can be used, in a scheme proposed by Kotikov, Lipatov and Velizhanin, to estimate the two constants in the strong-coupling expansion of the cusp anomalous dimension that are known from string theory. The estimate works to 2.6% and 5% accuracy, providing non-trivial evidence in support of the AdS/CFT correspondence. We also use the known constants in the strong-coupling expansion as additional input to provide approximations to the cusp anomalous dimension which should be accurate to under one percent for all values of the coupling. When the evaluations of the integrals are completed through the finite terms, it will be possible to test the iterative, exponentiated structure of the finite terms in the four-loop four-point amplitude, which was uncovered earlier at two and three loops
8. Generalization of Faddeev-Popov rules in Yang-Mills theories: N = 3,4 BRST symmetries
Science.gov (United States)
Reshetnyak, Alexander
2018-01-01
The Faddeev-Popov rules for a local and Poincaré-covariant Lagrangian quantization of a gauge theory with gauge group are generalized to the case of an invariance of the respective quantum actions, S(N), with respect to N-parametric Abelian SUSY transformations with odd-valued parameters λp, p = 1,…,N and generators sp: spsq + sqsp = 0, for N = 3, 4, implying the substitution of an N-plet of ghost fields, Cp, instead of the parameter, ξ, of infinitesimal gauge transformations: ξ = Cpλ p. The total configuration spaces of fields for a quantum theory of the same classical model coincide in the N = 3 and N = 4 symmetric cases. The superspace of N = 3 SUSY irreducible representation includes, in addition to Yang-Mills fields 𝒜μ, (3 + 1) ghost odd-valued fields Cp, B̂ and 3 even-valued Bpq for p, q = 1, 2, 3. To construct the quantum action, S(3), by adding to the classical action, S0(𝒜), of an N = 3-exact gauge-fixing term (with gauge fermion), a gauge-fixing procedure requires (1 + 3 + 3 + 1) additional fields, Φ¯(3): antighost C¯, 3 even-valued Bp, 3 odd-valued B̂pq and Nakanishi-Lautrup B fields. The action of N = 3 transformations on new fields as N = 3-irreducible representation space is realized. These transformations are the N = 3 BRST symmetry transformations for the vacuum functional, Z3(0) =∫dΦ(3)dΦ¯(3)exp{(ı/ℏ)S(3)}. The space of all fields (Φ(3),Φ¯(3)) proves to be the space of an irreducible representation of the fields Φ(4) for N = 4-parametric SUSY transformations, which contains, in addition to 𝒜μ the (4 + 6 + 4 + 1) ghost-antighost, Cr = (Cp,C¯), even-valued, Brs = -Bsr = (Bpq,Bp4 = Bp), odd-valued B̂r = (B̂,B̂pq) and B fields. The quantum action is constructed by adding to S0(𝒜) an N = 4-exact gauge-fixing term with a gauge boson, F(4). The N = 4 SUSY transformations are by N = 4 BRST transformations for the vacuum functional, Z4(0) =∫dΦ(4)exp{(ı/ℏ)S(4)}. The procedures are valid for
9. Yang--Mills gauge theories and Baker--Johnson quantum electrodynamics
International Nuclear Information System (INIS)
Lemmon, J.; Mahanthappa, K.T.
1976-01-01
We show that the physical mass of a fermion in a symmetric asymptotically free non-Abelian vector gauge theory is dynamical in origin. We comment on the close analogy that exists between such a theory and the Baker--Johnson finite quantum electrodynamics. Comments are also made when there is spontaneous symmetry breaking
10. Superspace action for a 6-dimensional non-extended supersymmetric Yang-Mills theory
International Nuclear Information System (INIS)
Nilsson, B.E.W.
1980-01-01
The ordinary N = 2 extended (abelian) gauge theory is written in the framework of a non-extended theory in 6-dimensional superspace. Graded differential geometry and superfield technique is used to construct a superspace action. The x-space lagrangian arises as the term of second order in theta of the superspace lagrangian. (orig.)
11. Center vortex properties in the Laplace center gauge of SU(2) Yang-Mills theory
OpenAIRE
Langfeld, K.; Reinhardt, H.; Schafke, A.
2001-01-01
Resorting to the the Laplace center gauge (LCG) and to the Maximal-center gauge (MCG), respectively, confining vortices are defined by center projection in either case. Vortex properties are investigated in the continuum limit of SU(2) lattice gauge theory. The vortex (area) density and the density of vortex crossing points are investigated. In the case of MCG, both densities are physical quantities in the continuum limit. By contrast, in the LCG the piercing as well as the crossing points li...
12. Divergences in maximal supersymmetric Yang-Mills theories in diverse dimensions
International Nuclear Information System (INIS)
Bork, L.V.; Kazakov, D.I.; Kompaniets, M.V.; Tolkachev, D.M.; Vlasenko, D.E.
2015-01-01
The main aim of this paper is to study the scattering amplitudes in gauge field theories with maximal supersymmetry in dimensions D=6,8 and 10. We perform a systematic study of the leading ultraviolet divergences using the spinor helicity and on-shell momentum superspace framework. In D=6 the first divergences start at 3 loops and we calculate them up to 5 loops, in D=8,10 the first divergences start at 1 loop and we calculate them up to 4 loops. The leading divergences in a given order are the polynomials of Mandelstam variables. To be on the safe side, we check our analytical calculations by numerical ones applying the alpha-representation and the dedicated routines. Then we derive an analog of the RG equations for the leading pole that allows us to get the recursive relations and construct the generating procedure to obtain the polynomials at any order of perturbation theory (PT). At last, we make an attempt to sum the PT series and derive the differential equation for the infinite sum. This equation possesses a fixed point which might be stable or unstable depending on the kinematics. Some consequences of these fixed points are discussed.
13. Yang-Mills formulation of interacting strings
International Nuclear Information System (INIS)
Chan Hongmo; Tsou Sheungtsun
1988-06-01
A suggestion that the theory of interacting open bosonic string be reformulated as a generalised Yang-Mills theory is further elucidated. Moreover, a serious reservation regarding the ordering of operators in the earlier 'proof' of equivalence between the new and standard formulations is now removed. (author)
14. Einstein-Yang-Mills-Lorentz black holes
Energy Technology Data Exchange (ETDEWEB)
Cembranos, Jose A.R.; Gigante Valcarcel, Jorge [Universidad Complutense de Madrid, Departamento de Fisica Teorica I, Madrid (Spain)
2017-12-15
Different black hole solutions of the coupled Einstein-Yang-Mills equations have been well known for a long time. They have attracted much attention from mathematicians and physicists since their discovery. In this work, we analyze black holes associated with the gauge Lorentz group. In particular, we study solutions which identify the gauge connection with the spin connection. This ansatz allows one to find exact solutions to the complete system of equations. By using this procedure, we show the equivalence between the Yang-Mills-Lorentz model in curved space-time and a particular set of extended gravitational theories. (orig.)
15. Theory of motion for monopole-dipole singularities of classical Yang-Mills-Higgs fields. I. Laws of motion
International Nuclear Information System (INIS)
Drechsler, W.; Havas, P.; Rosenblum, A.
1984-01-01
In two recent papers, the general form of the laws of motion for point particles which are multipole sources of the classical coupled Yang-Mills-Higgs fields was determined by Havas, and for the special case of monopole singularities of a Yang-Mills field an iteration procedure was developed by Drechsler and Rosenblum to obtain the equations of motion of mass points, i.e., the laws of motion including the explicit form of the fields of all interacting particles. In this paper we give a detailed derivation of the laws of motion of monopole-dipole singularities of the coupled Yang-Mills-Higgs fields for point particles with mass and spin, following a procedure first applied by Mathisson and developed by Havas. To obtain the equations of motion, a systematic approximation method is developed in the following paper for the solution of the nonlinear field equations and determination of the fields entering the laws of motion found here to any given order in the coupling constant g
16. Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
International Nuclear Information System (INIS)
Chiodaroli, Marco; Günaydin, Murat; Johansson, Henrik; Roiban, Radu
2015-01-01
We expose a double-copy structure in the scattering amplitudes of the generic Jordan family of N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity theories in four and five dimensions. The Maxwell-Einstein supergravity amplitudes are obtained through the color/kinematics duality as a product of two gauge-theory factors; one originating from pure N=2 super-Yang-Mills theory and the other from the dimensional reduction of a bosonic higher-dimensional pure Yang-Mills theory. We identify a specific symplectic frame in four dimensions for which the on-shell fields and amplitudes from the double-copy construction can be identified with the ones obtained from the supergravity Lagrangian and Feynman-rule computations. The Yang-Mills/Einstein supergravity theories are obtained by gauging a compact subgroup of the isometry group of their Maxwell-Einstein counterparts. For the generic Jordan family this process is identified with the introduction of cubic scalar couplings on the bosonic gauge-theory side, which through the double copy are responsible for the non-abelian vector interactions in the supergravity theory. As a demonstration of the power of this structure, we present explicit computations at tree-level and one loop. The double-copy construction allows us to obtain compact expressions for the supergravity superamplitudes, which are naturally organized as polynomials in the gauge coupling constant.
17. Scattering amplitudes in N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity
CERN Document Server
Chiodaroli, Marco; Johansson, Henrik; Roiban, Radu
2015-01-01
We expose a double-copy structure in the scattering amplitudes of the generic Jordan family of N=2 Maxwell-Einstein and Yang-Mills/Einstein supergravity theories in four and five dimensions. The Maxwell-Einstein supergravity amplitudes are obtained through the color/kinematics duality as a product of two gauge-theory factors; one originating from pure N=2 super-Yang-Mills theory and the other from the dimensional reduction of a bosonic higher-dimensional pure Yang-Mills theory. We identify a specific symplectic frame in four dimensions for which the on-shell fields and amplitudes from the double-copy construction can be identified with the ones obtained from the supergravity Lagrangian and Feynman-rule computations. The Yang-Mills/Einstein supergravity theories are obtained by gauging a compact subgroup of the isometry group of their Maxwell-Einstein counterparts. For the generic Jordan family this process is identified with the introduction of cubic scalar couplings on the bosonic gauge-theory side, which th...
18. Zero modes and the vacuum problem: A study of scalar adjoint matter in two-dimensional Yang-Mills theory via light-cone quantization
International Nuclear Information System (INIS)
Kalloniatis, A.C.
1996-01-01
SU(2) Yang-Mills theory coupled to massive adjoint scalar matter is studied in 1+1 dimensions using discretized light-cone quantization. This theory can be obtained from pure Yang-Mills theory in 2+1 dimensions via dimensional reduction. On the light cone, the vacuum structure of this theory is encoded in the dynamical zero mode of a gluon and a constrained mode of the scalar field. The latter satisfies a linear constraint, suggesting no nontrivial vacua in the present paradigm for symmetry breaking on the light cone. I develop a diagrammatic method to solve the constraint equation. In the adiabatic approximation I compute the quantum-mechanical potential governing the dynamical gauge mode. Because of a condensation of the lowest momentum modes of the dynamical gluons, a centrifugal barrier is generated in the adiabatic potential. In the present theory, however, the barrier height appears too small to make any impact in this model. Although the theory is superrenormalizable on naive power-counting grounds, the removal of ultraviolet divergences is nontrivial when the constrained mode is taken into account. The solution of this problem is discussed. copyright 1996 The American Physical Society
19. Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, _q) and PGL(2, _q)
International Nuclear Information System (INIS)
Roche, Ph.
2016-01-01
We recall the relation between zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of zeta function representations of groups. We compute some of these functions in the case of the finite group GL(2, _q) and PGL(2, _q). We recall the table characters of these groups for any q, compute the Frobenius-Schur indicator of their irreducible representations, and give the explicit structure of their fusion rings.
20. Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem
Energy Technology Data Exchange (ETDEWEB)
Montesinos, M. [CINVESTAV-IPN, 07360 Mexico D.F. (Mexico); Flores, E. [Facultad de Fisica e Inteligencia Artificial, Universidad Veracruzana, 91000 Xalapa, Veracruz (Mexico)]. E-mail: [email protected]
2006-07-01
The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end. (Author)
1. Yang-Mills analogs of general-relativistic solutions
International Nuclear Information System (INIS)
Singlton, D.
1998-01-01
Some solutions of Yang-Mills equations, which can be found with the use of the general relativistic theory and Yang-Mills theory, are discussed. Some notes concerning possible physical sense of these solutions are made. Arguments showing that some of such solutions in the Yang-Mills theory (similar to the general relativistic ones) may be connected with the confinement phenomenon are given in particular. The motion of probe particles located into the phonon potential similar to the Schwarz-Child one is briefly discussed for this purpose [ru
2. Double-winding Wilson loops in SU(N Yang-Mills theory – A criterion for testing the confinement models –
Directory of Open Access Journals (Sweden)
Matsudo Ryutaro
2018-01-01
Full Text Available We examine how the average of double-winding Wilson loops depends on the number of color N in the SU(N Yang-Mills theory. In the case where the two loops C1 and C2 are identical, we derive the exact operator relation which relates the doublewinding Wilson loop operator in the fundamental representation to that in the higher dimensional representations depending on N. By taking the average of the relation, we find that the difference-of-areas law for the area law falloff recently claimed for N = 2 is excluded for N ⩾ 3, provided that the string tension obeys the Casimir scaling for the higher representations. In the case where the two loops are distinct, we argue that the area law follows a novel law (N − 3A1/(N − 1 + A2 with A1 and A2(A1 < A2 being the minimal areas spanned respectively by the loops C1 and C2, which is neither sum-ofareas (A1 + A2 nor difference-of-areas (A2 − A1 law when (N ⩾ 3. Indeed, this behavior can be confirmed in the two-dimensional SU(N Yang-Mills theory exactly.
3. A low-energy β-function in a finite super-Yang-Mills model with multiple mass scales
International Nuclear Information System (INIS)
Foda, O.; Helayel-Neto, J.A.
1985-01-01
We compute the one-loop contribution to the low-energy light-fermion gauge coupling in a finite supersymmetric gauge theory with two mass scales: a heavy mass that breaks an initial N=4 supersymmetry down to N=2, but respects the finiteness, and a light mass that, for simplicity, is set to zero. We find that coupling grows with the mass of the heavy intermediate states. Hence the latter do not decouple at low energies, leading to large logarithms that invalidate low-energy perturbation theory. Consequently, further manipulations are required to obtain a meaningful perturbative expansion. Enforcing decoupling through finite renormalizations, that absorb the heavy mass effects into a redefinition of the parameters of the lagrangian, introduces an arbitrary subtraction mass μ. The requirement that the S-matrix elements be independent of μ leads to a non-trivial renormalization-group equation for the low-energy theory, with a non-vanishing β-function. (orig.)
4. Low-energy. beta. -function in a finite super-Yang-Mills model with multiple mass scales
Energy Technology Data Exchange (ETDEWEB)
Foda, O.; Helayel-Neto, J.A. (International Centre for Theoretical Physics, Trieste (Italy))
1985-02-14
We compute the one-loop contribution to the low-energy light-fermion gauge coupling in a finite supersymmetric gauge theory with two mass scales: a heavy mass that breaks an initial N=4 supersymmetry down to N=2, but respects the finiteness, and a light mass that, for simplicity, is set to zero. We find that coupling grows with the mass of the heavy intermediate states. Hence the latter do not decouple at low energies, leading to large logarithms that invalidate low-energy perturbation theory. Consequently, further manipulations are required to obtain a meaningful perturbative expansion. Enforcing decoupling through finite renormalizations, that absorb the heavy mass effects into a redefinition of the parameters of the lagrangian, introduces an arbitrary subtraction mass ..mu... The requirement that the S-matrix elements be independent of ..mu.. leads to a non-trivial renormalization-group equation for the low-energy theory, with a non-vanishing ..beta..-function.
5. A low-energy β-function in a finite super-Yang-Mills model with multiple mass scales
International Nuclear Information System (INIS)
Foda, O.; Helayel-Neto, J.A.
1984-08-01
We compute the one-loop contribution to the low-energy light-fermion gauge coupling in a finite supersymmetric gauge theory with two mass scales: a heavy mass that breaks an initial N=4 supersymmetry down to N=2, but respects the finiteness, and a light mass that, for simplicity, is set to zero. We find that the coupling grows with the mass of the heavy intermediate states. Hence the latter do not decouple at low energies, leading to large logarithms that invalidate low-energy perturbation theory. Consequently, further manipulations are required to obtain a meaningful perturbative expansion. Enforcing decoupling through finite renormalizations, that absorb the heavy mass effects into a redefinition of the parameters of the Lagrangian, introduces an arbitrary subtraction mass μ. The requirement that the S-matrix elements be independent of μ leads to a non-trivial renormalization-group equation for the low-energy theory, with a non-vanishing β-function. (author)
6. Two-loop N=4 super-Yang-Mills effective action and interaction between D3-branes
International Nuclear Information System (INIS)
Buchbinder, I.L.; Petrov, A.Yu.; Tseytlin, A.A.
2002-01-01
We compute the leading low-energy term in the planar part of the 2-loop contribution to the effective action of N=4 SYM theory in 4 dimensions, assuming that the gauge group SU(N+1) is broken to SU(N)xU(1) by a constant scalar background X. While the leading 1-loop correction is the familiar c 1 F 4 /vertical bar X vertical bar 4 term, the 2-loop expression starts with c 2 F 6 /vertical bar X vertical bar 8 . The 1-loop constant c 1 is known to be equal to the coefficient of the F 4 term in the Born-Infeld action for a probe D3-brane separated by distance vertical bar X vertical bar from a large number N of coincident D3-branes. We show that the same is true also for the 2-loop constant c 2 : it matches the coefficient of the F 6 term in the D3-brane probe action. In the context of the AdS/CFT correspondence, this agreement suggests a non-renormalization of the coefficient of the F 6 term beyond two loops. Thus the result of hep-th/9706072 about the agreement between the v 6 term in the D0-brane supergravity interaction potential and the corresponding 2-loop term in the (1+0)-dimensional reduction of N=4 SYM theory has indeed a direct generalization to 1+3 dimensions, as conjectured earlier in hep-th/9709087. We also discuss the issue of gauge theory-supergravity correspondence for higher order (F 8 , etc.) terms
7. Higgs amplitudes from supersymmetric form factors Part I: $\\mathcal{N}=4$ super Yang-Mills arXiv
CERN Document Server
Brandhuber, Andreas; Penante, Brenda; Travaglini, Gabriele
In the large top-mass limit, Higgs plus multi-gluon amplitudes in QCD can be computed using an effective field theory. This approach turns the computation of such amplitudes into that of form factors of operators of increasing classical dimension. In this paper we focus on the first finite top-mass correction, arising from the operator ${\\rm Tr}(F^3)$, up to two loops and three gluons. Setting up the calculation in the maximally supersymmetric theory requires identification of an appropriate supersymmetric completion of ${\\rm Tr}(F^3)$, which we recognise as a descendant of the Konishi operator. We provide detailed computations for both this operator and the component operator ${\\rm Tr}(F^3)$, preparing the ground for the calculation in $\\mathcal{N}<4$, to be detailed in a companion paper. Our results for both operators are expressed in terms of a few universal functions of transcendental degree four and below, some of which have appeared in other contexts, hinting at universality of such quantities. An im...
8. N=4 SUSY Yang-Mills: Three loops made simple(r)
Energy Technology Data Exchange (ETDEWEB)
Dokshitzer, Yu.L. [LPTHE, Universities of Paris-VI and VII and CNRS, Paris (France); Marchesini, G. [University of Milano-Bicocca and INFN Sezione di Milano-Bicocca, Milan (Italy)]. E-mail: [email protected]
2007-03-15
We construct universal parton evolution equation that produces space- and time-like anomalous dimensions for the maximally super-symmetric N=4 Yang-Mills field theory model, and find that its kernel satisfies the Gribov-Lipatov reciprocity relation in three loops. Given a simple structure of the evolution kernel, this should help to generate the major part of multi-loop contributions to QCD anomalous dimensions, due to classical soft gluon radiation effects.
9. Dryson equations, Ward identities, and the infrared behavior of Yang-Mills theories. [Schwinger-Dyson equations, Slavnov-Taylor identities
Energy Technology Data Exchange (ETDEWEB)
Baker, M.
1979-01-01
It was shown using the Schwinger-Dyson equations and the Slavnov-Taylor identities of Yang-Mills theory that no inconsistency arises if the gluon propagator behaves like (1/p/sup 2/)/sup 2/ for small p/sup 2/. To see whether the theory actually contains such singular long range behavior, a nonperturbative closed set of equations was formulated by neglecting the transverse parts of GAMMA and GAMMA/sub 4/ in the Schwinger-Dyson equations. This simplification preserves all the symmetries of the theory and allows the possibility for a singular low-momentum behavior of the gluon propagator. The justification for neglecting GAMMA/sup (T)/ and GAMMA/sub 4//sup (T)/ is not evident but it is expected that the present study of the resulting equations will elucidate this simplification, which leads to a closed set of equations.
10. Massive Yang-Mills fields in the Kemmer's formulation
International Nuclear Information System (INIS)
Santana Cordolino, L.A. de.
1984-01-01
The Kemmer's equation, which describes the meson, is presented in the field theory formalism. Conservated, quantities are found through the Noether's identity. This formalism is used for masive Yang-Mills fields and two equations, similar to the Kemmer's equation, are obtained, although of different formates, both containing quadratic terms. In consequence two Lagrangians are defined, formally distint, for the Yang-Mills fields. The Schroedinger-like Hamiltonian is calculated for the first wave equations. This Hamiltonian presentes one spin-Yang-Mills field interaction term, PHI jk . (L.C.) [pt
11. Noncommutative Yang-Mills from equivalence of star products
International Nuclear Information System (INIS)
Jurco, B.; Schupp, P.
2000-01-01
It is shown that the transformation between ordinary and noncommutative Yang-Mills theory as formulated by Seiberg and Witten is due to the equivalence of certain star products on the D-brane world-volume. (orig.)
12. Noncommutative Yang-Mills from equivalence of star products
Energy Technology Data Exchange (ETDEWEB)
Jurco, B. [Max-Planck-Institut fuer Mathematik, Bonn (Germany); Schupp, P. [Sektion Physik, Universitaet Muenchen, Theresienstrasse 37, 80333 Muenchen (Germany)
2000-05-01
It is shown that the transformation between ordinary and noncommutative Yang-Mills theory as formulated by Seiberg and Witten is due to the equivalence of certain star products on the D-brane world-volume. (orig.)
13. An ambitwistor Yang-Mills Lagrangian
International Nuclear Information System (INIS)
Mason, L.J.; Skinner, D.
2006-01-01
We introduce a Chern-Simons Lagrangian for Yang-Mills theory as formulated on ambitwistor space via the Ward, Isenberg, Yasskin, Green, Witten construction. The Lagrangian requires the selection of a codimension-2 Cauchy-Riemann submanifold which is naturally picked out by the choice of space-time reality structure and we focus on the choice of Euclidean signature. The action is shown to give rise to a space-time action that is equivalent to the standard one, but has just cubic vertices. We identify the ambitwistor propagators and vertices and work out their corresponding expressions on space-time and momentum space. It is proposed that this formulation of Yang-Mills theory underlies the recursion relations of Britto, Cachazo, Feng and Witten and provides the generating principle for twistor diagrams for gauge theory
14. Instability of higher dimensional Yang-Mills systems
International Nuclear Information System (INIS)
Randjbar-Daemi, S.; Strathdee, J.
1983-01-01
We investigate the stability of Poincare xO(3) invariant solutions for a pure semi-simple Yang-Mills, as well as Yang-Mills coupled to gravity in 6-dimensional space-time compactified over M 4 xS 2 . In contrast to the Maxwell U(1) theory (IC-82/208) in six dimensions coupled with gravity and investigated previously, the present theory exhibits tachyonic excitations and is unstable. (author)
15. Yang--Mills vacua in Landau gauge
International Nuclear Information System (INIS)
Frampton, P.H.; Hagiwara, T.; Palmer, W.F.; Pinsky, S.S.
1977-01-01
A vacuum gauge field A/sub μ//sup a/ for Yang-Mills theory is constructed; this field is pure vacuum (A/sub μ//sup a/ = 0) at the origin, approaches at large distances, the Belavin-Polyakov-Schwartz-Tyupkin pseudo-particle, and satisfies delta/sub μ/A/sub μ//sup a/ = 0 everywhere. The net topological charge is zero, and there is a Dirac-like string terminating at the origin
16. The K-Z Equation and the Quantum-Group Difference Equation in Quantum Self-dual Yang-Mills Theory
OpenAIRE
Chau, Ling-Lie; Yamanaka, Itaru
1995-01-01
From the time-independent current $\\tcj(\\bar y,\\bar k)$ in the quantum self-dual Yang-Mills (SDYM) theory, we construct new group-valued quantum fields $\\tilde U(\\bar y,\\bar k)$ and $\\bar U^{-1}(\\bar y,\\bar k)$ which satisfy a set of exchange algebras such that fields of $\\tcj(\\bar y,\\bar k)\\sim\\tilde U(\\bar y,\\bar k)~\\partial\\bar y~\\tilde U^{-1}(\\bar y,\\bar k)$ satisfy the original time-independent current algebras. For the correlation functions of the products of the $\\tilde U(\\bar y,\\bar k... 17. More on ghost condensation in Yang-Mills theory: BCS versus Overhauser effect and the breakdown of the Nakanishi-Ojima annex SL(2,R) symmetry International Nuclear Information System (INIS) Dudal, David; Verschelde, Henri; Lemes, Vitor E.R.; Sarandy, Marcelo S.; Sorella, Silvio P.; Picariello, Marco; Vicini, Alessandro; Gracey, John A. 2003-01-01 We analyze the ghost condensates abc c b c c >, abc c-bar b c-bar c > and abc c-barbc c > in Yang-Mills theory in the Curci-Ferrari gauge. By combining the local composite operator formalism with the algebraic renormalization technique, we are able to give a simultaneous discussion of abc c b c c >, abc c-bar b c-bar c > and abc c-bar b c c >, which can be seen as playing the role of the BCS, respectively Overhauser effect in ordinary superconductivity. The Curci-Ferrari gauge exhibits a global continuous symmetry generated by the Nakanishi-Ojima (NO) algebra. This algebra includes, next to the {(anti-)BRST} transformation, a SL(2,R) subalgebra. We discuss the dynamical symmetry breaking of the NO algebra through these ghost condensates. Particular attention is paid to the Landau gauge, a special case of the Curci-Ferrari gauge. (author) 18. Combined study of the gluon and ghost condensates μ2> and abccbcc> in Euclidean SU(2) Yang-Mills theory in the Landau gauge International Nuclear Information System (INIS) Capri, M.A.L.; Lemes, V.E.R.; Sobreiro, R.F.; Sorella, S.P.; Dudal, D.; Verschelde, H.; Gracey, J.A. 2006-01-01 The ghost condensate abc c b c c > is considered together with the gluon condensate μ 2 > in SU(2) Euclidean Yang-Mills theories quantized in the Landau gauge. The vacuum polarization ceases to be transverse due to the nonvanishing condensate abc c b c c >. The gluon propagator itself remains transverse. By polarization effects, this ghost condensate induces then a splitting in the gluon mass parameter, which is dynamically generated through μ 2 >. The obtained effective masses are real when μ 2 > is included in the analysis. In the absence of μ 2 >, the already known result that the ghost condensate induces effective tachyonic masses is recovered. At the one-loop level, we find that the effective diagonal mass becomes smaller than the off-diagonal one. This might serve as an indication for some kind of Abelian dominance in the Landau gauge, similar to what happens in the maximal Abelian gauge 19. Towards the general solution of the Yang-Mills equations International Nuclear Information System (INIS) Helfer, A.D. 1985-01-01 The author presents a new non-perturbative technique for finding arbitrary self-dual solutions to the Yang-Mills equations, and of describing massless fields minimally coupled to them. The approach uses techniques of complex analysis in several variables, and is complementary to Ward's: it is expected that a combination of the two techniques will yield general, non-self-dual solutions to the Yang-Mills equations. This has been verified to first order in perturbation theory 20. Yang Mills instantons, geometrical aspects International Nuclear Information System (INIS) Stora, R. 1977-09-01 The word instanton has been coined by analogy with the word soliton. They both refer to solutions of elliptic non linear field equations with boundary conditions at infinity (of euclidean space time in the first case, euclidean space in the second case) lying on the set of classical vacua in such a way that stable topological properties emerge, susceptible to survive quantum effects, if those are small. Under this assumption, instantons are believed to be relevant to the description of tunnelling effects between classical vacua and signal some characteristics of the vacuum at the quantum level, whereas solitons should be associated with particles, i.e. discrete points in the mass spectrum. In one case the euclidean action is finite, in the other case, the energy is finite. From the mathematical point of view, the geometrical phenomena associated with the existence of solitons have forced physicists to learn rudiments of algebraic topology. The study of euclidean classical Yang Mills fields involves naturally mathematical items falling under the headings: differential geometry (fibre bundles, connections); differential topology (characteristic classes, index theory) and more recently algebraic geometry. These notes are divided as follows: a first section is devoted to a description of the physicist's views; a second section is devoted to the mathematician's vie 1. Interacting fields of arbitrary spin and N > 4 supersymmetric self-dual Yang-Mills equations International Nuclear Information System (INIS) Devchand, Ch.; Ogievetsky, V. 1996-06-01 We show that the self-dual Yang-Mills equations afford supersymmetrization to systems of equations invariant under global N-extended super-Poincare transformations for arbitrary values of N, without the limitation (N ≤ 4) applicable to standard non-self-dual Yang-Mills theories. These systems of equations provide novel classically consistent interactions for vector supermultiplets containing fields of spin up to N-2/2. The equations of motion of the component fields of spin greater than 1/2 are interacting variants of the first-order Dirac-Fierz equations for zero rest-mass fields of arbitrary spin. The interactions are governed by conserved currents which are constructed by an iterative procedure. In (arbitrarily extended) chiral superspace, the equations of motion for the (arbitrarily large) self-dual supermultiplet are shown to be completely equivalent to the set of algebraic supercurvature defining the self-dual superconnection. (author). 25 refs 2. Finite energy shifts in SU(n) supersymmetric Yang-Mills theory on T3xR at weak coupling International Nuclear Information System (INIS) Ohlsson, Fredrik 2010-01-01 We consider a perturbative treatment, in the regime of weak gauge coupling, of supersymmetric Yang-Mills theory in a space-time of the form T 3 xR with SU(n)/Z n gauge group and a nontrivial gauge bundle. More specifically, we consider the theories obtained as power series expansions around a certain class of normalizable vacua of the classical theory, corresponding to isolated points in the moduli space of flat connections, and the perturbative corrections to the free energy eigenstates and eigenvalues in the weakly interacting theory. The perturbation theory construction of the interacting Hilbert space is complicated by the divergence of the norm of the interacting states. Consequently, the free and interacting Hilbert spaces furnish unitarily inequivalent representations of the algebra of creation and annihilation operators of the quantum theory. We discuss a consistent redefinition of the Hilbert space norm to obtain the interacting Hilbert space and the properties of the interacting representation. In particular, we consider the lowest nonvanishing corrections to the free energy spectrum and discuss the crucial importance of supersymmetry for these corrections to be finite. 3. D = 4 Yang-Mills correlators from NSR strings on AdS5 x S5 International Nuclear Information System (INIS) Polyakov, D. 1999-07-01 In our previous work (hep-th/9812044) we have proposed the sigma-model action, conjectured to be the NSR analogue of superstring theory on AdS 5 x S 5 . This sigma-model is the NSR superstring action with potential term corresponding to the exotic 5-form vertex operator (branelike state). This 5-form potential plays the role of cosmological term, effectively curving the flat space-time geometry to that of AdS 5 x S 5 . In this paper we study this ansatz in more detail and provide the derivation of the correlators of the four-dimensional super Yang-Mills theory from the above mentioned sigma-model. In particular, we show that the correlation function of two dilaton vertex operators in such a model reproduces the well-known result for the two-point function in N = 4 four-dimensional super Yang-Mills theory. (author) 4. Study of the maximal Abelian gauge in SU(2) Euclidean Yang-Mills theory in the presence of the Gribov horizon International Nuclear Information System (INIS) Capri, M. A. L.; Lemes, V. E. R.; Sobreiro, R. F.; Sorella, S. P.; Thibes, R. 2006-01-01 We pursue the study of SU(2) Euclidean Yang-Mills theory in the maximal Abelian gauge by taking into account the effects of the Gribov horizon. The Gribov approximation, previously introduced in [M. A. L. Capri, V. E. R. Lemes, R. F. Sobreiro, S. P. Sorella, and R. Thibes, Phys. Rev. D 72, 085021 (2005).], is improved through the introduction of the horizon function, which is constructed under the requirements of localizability and renormalizability. By following Zwanziger's treatment of the horizon function in the Landau gauge, we prove that, when cast in local form, the horizon term of the maximal Abelian gauge leads to a quantized theory which enjoys multiplicative renormalizability, a feature which is established to all orders by means of the algebraic renormalization. Furthermore, it turns out that the horizon term is compatible with the local residual U(1) Ward identity, typical of the maximal Abelian gauge, which is easily derived. As a consequence, the nonrenormalization theorem, Z g Z A 1/2 =1, relating the renormalization factors of the gauge coupling constant Z g and of the diagonal gluon field Z A , still holds in the presence of the Gribov horizon. Finally, we notice that a generalized dimension two gluon operator can be also introduced. It is BRST invariant on-shell, a property which ensures its multiplicative renormalizability. Its anomalous dimension is not an independent parameter of the theory, being obtained from the renormalization factors of the gauge coupling constant and of the diagonal antighost field 5. YANG-MILLS FIELDS AND THE LATTICE. Energy Technology Data Exchange (ETDEWEB) CREUTZ,M. 2004-05-18 The Yang-Mills theory lies at the heart of our understanding of elementary particle interactions. For the strong nuclear forces, we must understand this theory in the strong coupling regime. The primary technique for this is the lattice. While basically an ultraviolet regulator, the lattice avoids the use of a perturbative expansion. I discuss some of the historical circumstances that drove us to this approach, which has had immense success, convincingly demonstrating quark confinement and obtaining crucial properties of the strong interactions from first principles. 6. A non-perturbative study of matter field propagators in Euclidean Yang-Mills theory in linear covariant, Curci-Ferrari and maximal Abelian gauges Energy Technology Data Exchange (ETDEWEB) Capri, M.A.L.; Fiorentini, D.; Sorella, S.P. [UERJ - Universidade do Estado do Rio de Janeiro, Departamento de Fisica Teorica, Rio de Janeiro (Brazil); Pereira, A.D. [UERJ - Universidade do Estado do Rio de Janeiro, Departamento de Fisica Teorica, Rio de Janeiro (Brazil); UFF - Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ (Brazil) 2017-08-15 In this work, we study the propagators of matter fields within the framework of the refined Gribov-Zwanziger theory, which takes into account the effects of the Gribov copies in the gauge-fixing quantization procedure of Yang-Mills theory. In full analogy with the pure gluon sector of the refined Gribov-Zwanziger action, a non-local long-range term in the inverse of the Faddeev-Popov operator is added in the matter sector. Making use of the recent BRST-invariant formulation of the Gribov-Zwanziger framework achieved in Capri et al. (Phys Rev D 92(4):045039, 2015), (Phys Rev D 94(2):025035, 2016), (Phys Rev D 93(6):065019, 2016), (arXiv:1611.10077 [hepth]), Pereira et al. (arXiv:1605.09747 [hep-th]), the propagators of scalar and quark fields in the adjoint and fundamental representations of the gauge group are worked out explicitly in the linear covariant, Curci-Ferrari and maximal Abelian gauges. Whenever lattice data are available, our results exhibit good qualitative agreement. (orig.) 7. A non-perturbative study of matter field propagators in Euclidean Yang-Mills theory in linear covariant, Curci-Ferrari and maximal Abelian gauges Science.gov (United States) Capri, M. A. L.; Fiorentini, D.; Pereira, A. D.; Sorella, S. P. 2017-08-01 In this work, we study the propagators of matter fields within the framework of the refined Gribov-Zwanziger theory, which takes into account the effects of the Gribov copies in the gauge-fixing quantization procedure of Yang-Mills theory. In full analogy with the pure gluon sector of the refined Gribov-Zwanziger action, a non-local long-range term in the inverse of the Faddeev-Popov operator is added in the matter sector. Making use of the recent BRST-invariant formulation of the Gribov-Zwanziger framework achieved in Capri et al. (Phys Rev D 92(4):045039, 2015), (Phys Rev D 94(2):025035, 2016), (Phys Rev D 93(6):065019, 2016), (arXiv:1611.10077 [hep-th]), Pereira et al. (arXiv:1605.09747 [hep-th]),the propagators of scalar and quark fields in the adjoint and fundamental representations of the gauge group are worked out explicitly in the linear covariant, Curci-Ferrari and maximal Abelian gauges. Whenever lattice data are available, our results exhibit good qualitative agreement. 8. Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff Energy Technology Data Exchange (ETDEWEB) Chankowski, Piotr H. [Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,Pasteura 5, 02-093 Warsaw (Poland); Lewandowski, Adrian [Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),Mühlenberg 1, D-14476 Potsdam (Germany); Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,Pasteura 5, 02-093 Warsaw (Poland); Meissner, Krzysztof A. [Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,Pasteura 5, 02-093 Warsaw (Poland) 2016-11-18 We perform a systematic one-loop renormalization of a general renormalizable Yang-Mills theory coupled to scalars and fermions using a regularization scheme with a smooth momentum cutoff Λ (implemented through an exponential damping factor). We construct the necessary finite counterterms restoring the BRST invariance of the effective action by analyzing the relevant Slavnov-Taylor identities. We find the relation between the renormalized parameters in our scheme and in the conventional (MS)-bar scheme which allow us to obtain the explicit two-loop renormalization group equations in our scheme from the known two-loop ones in the (MS)-bar scheme. We calculate in our scheme the divergences of two-loop vacuum graphs in the presence of a constant scalar background field which allow us to rederive the two-loop beta functions for parameters of the scalar potential. We also prove that consistent application of the proposed regularization leads to counterterms which, together with the original action, combine to a bare action expressed in terms of bare parameters. This, together with treating Λ as an intrinsic scale of a hypothetical underlying finite theory of all interactions, offers a possibility of an unconventional solution to the hierarchy problem if no intermediate scales between the electroweak scale and the Planck scale exist. 9. 5D Yang-Mills instantons from ABJM Monopoles CERN Document Server Lambert, N.; Papageorgakis, C. 2012-01-01 In the presence of a background supergravity flux, N M2-branes will expand via the Myers effect into M5-branes wrapped on a fuzzy three-sphere. In previous work the fluctuations of the M2-branes were shown to be described by the five-dimensional Yang-Mills gauge theory associated to D4-branes. We show that the ABJM prescription for eleven-dimensional momentum in terms of magnetic flux lifts to an instanton flux of the effective five-dimensional Yang-Mills theory on the sphere, giving an M-theory interpretation for these instantons. 10. Yang-Mills-Chern-Simons supergravity International Nuclear Information System (INIS) Lue, H; Pope, C N; Sezgin, E 2004-01-01 N = (1, 0) supergravity in six dimensions admits AdS 3 x S 3 as a vacuum solution. We extend our recent results presented in Lue et al (2002 Preprint hep-th/0212323), by obtaining the complete N = 4 Yang-Mills-Chern-Simons supergravity in D = 3, up to quartic fermion terms, by S 3 group manifold reduction of the six-dimensional theory. The SU(2) gauge fields have Yang-Mills kinetic terms as well as topological Chern-Simons mass terms. There is in addition a triplet of matter vectors. After diagonalization, these fields describe two triplets of topologically-massive vector fields of opposite helicities. The model also contains six scalars, described by a GL(3, R)/SO(3) sigma model. It provides the first example of a three-dimensional gauged supergravity that can be obtained by a consistent reduction of string theory or M-theory and that admits AdS 3 as a vacuum solution. There are unusual features in the reduction from six-dimensional supergravity, owing to the self-duality condition on the 3-form field. The structure of the full equations of motion in N = (1, 0) supergravity in D = 6 is also elucidated, and the role of the self-dual field strength as torsion is exhibited 11. Yang-Mills correlation functions from integrable spin chains International Nuclear Information System (INIS) Roiban, Radu; Volovich, Anastasia 2004-01-01 The relation between the dilatation operator of N = 4 Yang-Mills theory and integrable spin chains makes it possible to compute the one-loop anomalous dimensions of all operators in the theory. In this paper we show how to apply the technology of integrable spin chains to the calculation of Yang-Mills correlation functions by expressing them in terms of matrix elements of spin operators on the corresponding spin chain. We illustrate this method with several examples in the SU(2) sector described by the XXX 1/2 chain. (author) 12. Analogy between spin glasses and Yang--Mills fluids International Nuclear Information System (INIS) Holm, D.D.; Kupershmidt, B.A. 1988-01-01 A dictionary of correspondence is established between the dynamical variables for spin-glass fluid and Yang-Mills plasma. The Lie-algebraic interpretation of these variables is presented for the two theories. The noncanonical Poisson bracket for the Hamiltonian dynamics of an ideal spin glass is shown to be identical to that for the dynamics of a Yang--Mills fluid plasma, although the Hamiltonians differ for the two theories. This Poisson bracket is associated to the dual space of an infinite-dimensional Lie algebra of semidirect-product type 13. Gluing operation and form factors of local operators in N = 4 Super Yang-Mills theory Science.gov (United States) Bolshov, A. E. 2018-04-01 The gluing operation is an effective way to get form factors of both local and non-local operators starting from different representations of on-shell scattering amplitudes. In this paper it is shown how it works on the example of form factors of operators from stress-tensor operator supermultiplet in Grassmannian and spinor helicity representations. 14. On the SL(2,R) symmetry in Yang-Mills theories in the Landau, Curci-Ferrari and maximal abelian gauge International Nuclear Information System (INIS) Dudal, David; Verschelde, Henri; Rodino Lemes, Vitor Emanuel; Sarandy, Marcelo S.; Sorella, Silvio Paolo; Picariello, Marco 2002-01-01 The existence of a SL(2;R) symmetry is discussed in SU(N) Yang-Mills in the maximal abelian gauge. This symmetry, also present in the Landau and Curci-Ferrari gauge, ensures the absence of tachyons in the maximal abelian gauge. In all these gauges, SL(2;R) turns out to be dynamically broken by ghost condensates. (author) 15. Solutions of the classical SU(2) Yang-Mills theory in 2+1 dimensions with the Chern-Simons term: Ansatz building International Nuclear Information System (INIS) Teh, R. 1989-07-01 Here I would like to show a general way of writing the gauge potentials A μ α for which the SU(2) Yang-Mills equations of motion can be simplified and become solvable. A number of exact solutions can be obtained from these simplified equations of motion. (author). 14 refs 16. On the infrared behaviour of Yang-Mills Greens functions International Nuclear Information System (INIS) Olesen, P. 1976-01-01 Making certain assumptions (valid to any finite order of perturbation theory), it is shown that non-perturbatively pure Yang-Mills Greens functions are power behaved in the momenta in a limit related to the infrared limit. It is also shown that the fundamental vertices have a more singular behaviour than indicated by perturbation theory. (Auth.) 17. A connection between the Einstein and Yang-Mills equations International Nuclear Information System (INIS) Mason, L.J.; Newman, E.T. 1989-01-01 It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unified equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie-algebra as that of the volume preserving 3-dimensional diffeomorphisms). When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einstein vacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of an SO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations. (orig.) 18. Franklin Medal and Bower prize awarded to C.N. Yang. On the Yang-Mills gauge field theory International Nuclear Information System (INIS) Ma Zhongqi 1995-01-01 C.N. Yang was awarded the Benjamin Franklin Medal and 1995 Bower Prize mainly for his fundamental work on nonabelian gauge field theory. A brief introduction to this theory and its important role in the development of physics is given 19. An elementary introduction to Yang-Mills theories and to their applications to the weak and electromagnetic interactions International Nuclear Information System (INIS) Maiani, L. 1976-01-01 In these lecture notes, the author deals exclusively with a gauge theory of weak and e.m. processes. The aim is to give an elementary introduction to the subject by discussing the general underlying ideas, and the way these ideas can be put to work in a concrete theory, based on the gauge group SU(2)X(1). (Auth.) 20. Chiral anomalies and constraints on the gauge group in higher-dimensional supersymmetric Yang-Mills theories International Nuclear Information System (INIS) Townsend, P.K.; Sierra, G. 1983-01-01 Chiral anomalies for gauge theories in any even dimension are computed and the results applied to supersymmetric theories in D=6, 8 and 10. For D=8 there is an anomalous chiral U(1) invariance, just as in D=4, except for certain special groups. For D=6 and D=10 there is no anomalous chiral U(1) symmetry, but the gauge current is anomalous except for certain ''anomaly-free'' groups. For D=6 the group is thereby constrained to be one of [SU(2), SU(3), exceptional], while for D=10 it is constrained to be one of [SU(n)n 8 ]. (orig.) 1. Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, #Mathematical Double-Struck Capital F#{sub q}) and PGL(2, #Mathematical Double-Struck Capital F#{sub q}) Energy Technology Data Exchange (ETDEWEB) Roche, Ph., E-mail: [email protected] [Université Montpellier 2, CNRS, L2C, IMAG, Montpellier (France) 2016-03-15 We recall the relation between zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of zeta function representations of groups. We compute some of these functions in the case of the finite group GL(2, #Mathematical Double-Struck Capital F#{sub q}) and PGL(2, #Mathematical Double-Struck Capital F#{sub q}). We recall the table characters of these groups for any q, compute the Frobenius-Schur indicator of their irreducible representations, and give the explicit structure of their fusion rings. 2. Gauge invariant frequency splitting of the continuum Yang-Mills field International Nuclear Information System (INIS) Mitter, P.K.; Valent, G. 1977-01-01 Frequency splitting plays an important role in Wilson's theory of critical phenomena. Here the authors give a theory of gauge invariant frequency splitting of the Yang-Mills field in 4 dimensions. (Auth.) 3. The topological B-model on fattened complex manifolds and subsectors of N=4 self-dual Yang-Mills theory International Nuclear Information System (INIS) Saemann, Christian 2005-01-01 In this paper, we propose so-called fattened complex manifolds as target spaces for the topological B-model. We naturally obtain these manifolds by restricting the structure sheaf of the N=4 supertwistor space, a process, which can be understood as a fermionic dimensional reduction. Using the twistorial description of these fattened complex manifolds, we construct Penrose-Ward transforms between solutions to the holomorphic Chern-Simons equations on these spaces and bosonic subsectors of solutions to the N=4 self-dual Yang-Mills equations on C 4 or R 4 . Furthermore, we comment on Yau's theorem for these spaces. (author) 4. On N = 4 supersymmetric Yang-Mills in harmonic superspace International Nuclear Information System (INIS) Ahmed, E.; Bedding, S.; Card, C.T.; Dumbrell, M.; Nouri-Moghadam, M.; Taylor, J.G. 1985-01-01 An analysis of N=4 supersymmetric Yang-Mills theory is presented using a construction involving additional bosonic variables in the coset space SU(4)/H. No choice of H can be shown to lead to an analytic formulation of the theory. by introducing an analysis on dual planes the theory is reduced (including the reality constraint) to one involving N=2 symmetry. This approach has to be extended to include truly harmonic derivatives. For the typical case of SU(4)/SU(2)xU(1) prepotentials are introduced which solve the constraints. It has not been possible, however, to construct an action which leads to the equation of motion for the original N=4 supersymmetric Yang-Mills theory (at the linearised level). (author) 5. A new approach to the self-dual Yang-Mills equations International Nuclear Information System (INIS) Takasaki, K. 1984-01-01 Inspired by Sato's new theory for soliton equations, we find a new approach to the self-dual Yang-Mills equations. We first establish a correspondence of solutions between the self-dual Yang-Mills equations and a new system of equations with infinitely many unknown functions. It then turns out that the latter equations can be easily solved by a simple explicit procedure. This leads to an explicit description of a very broad class of solutions to the self-dual Yang-Mills equations, and also to a construction of transformations acting on these solutions. (orig.) 6. A Generalized Yang-Mills Model and Dynamical Breaking of Gauge Symmetry International Nuclear Information System (INIS) Wang Dianfu; Song Heshan 2005-01-01 A generalized Yang-Mills model, which contains, besides the vector part V μ , also a scalar part S, is constructed and the dynamical breaking of gauge symmetry in the model is also discussed. It is shown, in terms of Nambu-Jona-Lasinio (NJL) mechanism, that the gauge symmetry breaking can be realized dynamically in the generalized Yang-Mills model. The combination of the generalized Yang-Mills model and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory. 7. Fiber spaces, connections and Yang-Mills fields International Nuclear Information System (INIS) Hermann, R. 1982-01-01 From the point of view of a differential geometer, Yang-Mills Fields are connections on principal fiber bundles whose curvature satisfies certain first-order differential equations. These lectures notes assume a knowledge of the formalism of calculus on manifolds, i.e., the theory of differential forms and vector fields, and are based on the theory of connections in fiber spaces, developed primarily by E. Cartan and C. Ehresmann in the period 1920-1955. To make the material more readily accessible to someone familiar with classical physics, the emphasis will be on Maxwell electromagnetic theory, considered as a Yang-Mills with an abelian structure group. Some of the material is from Interdisciplinary Mathematics, some is new. (orig.) 8. The Yang-Mills vacuum wave functional in Coulomb gauge International Nuclear Information System (INIS) Campagnari, Davide R. 2011-01-01 Yang-Mills theories are the building blocks of today's Standard Model of elementary particle physics. Besides methods based on a discretization of space-time (lattice gauge theory), also analytic methods are feasible, either in the Lagrangian or in the Hamiltonian formulation of the theory. This thesis focuses on the Hamiltonian approach to Yang-Mills theories in Coulomb gauge. The thesis is presented in cumulative form. After an introduction into the general formulation of Yang-Mills theories, the Hamilton operator in Coulomb gauge is derived. Chap. 1 deals with the heat-kernel expansion of the Faddeev-Popov determinant. In Chapters 2 and 3, the high-energy behaviour of the theory is investigated. To this purpose, perturbative methods are applied, and the results are compared with the ones stemming from functional methods in Coulomb and Landau gauge. Chap. 4 is devoted to the variational approach. Variational ansatzes going beyond the Gaussian form for the vacuum wave functional are considered and treated using Dyson-Schwinger techniques. Equations for the higher-order variational kernels are derived and their effects are estimated. Chap. 5 presents an application of the previously obtained propagators, namely the evaluation of the topological susceptibility, which is related to the mass of the η meson. Finally, a short overview of the perturbative treatment of dynamical fermion fields is presented. 9. Loop Amplitudes in Pure Yang-Mills from Generalised Unitarity OpenAIRE Brandhuber, Andreas; McNamara, Simon; Spence, Bill; Travaglini, Gabriele 2005-01-01 We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions. 10. Loop amplitudes in pure Yang-Mills from generalised unitarity International Nuclear Information System (INIS) Brandhuber, Andreas; McNamara, Simon; Spence, Bill; Travaglini, Gabriele 2005-01-01 We show how generalised unitarity cuts in D = 4-2ε dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions 11. Loop amplitudes in pure Yang-Mills from generalised unitarity Energy Technology Data Exchange (ETDEWEB) Brandhuber, Andreas [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom); McNamara, Simon [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom); Spence, Bill [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom); Travaglini, Gabriele [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom) 2005-10-15 We show how generalised unitarity cuts in D = 4-2{epsilon} dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions. 12. Dirac equations for generalised Yang-Mills systems International Nuclear Information System (INIS) Lechtenfeld, O.; Nahm, W.; Tchrakian, D.H. 1985-06-01 We present Dirac equations in 4p dimensions for the generalised Yang-Mills (GYM) theories introduced earlier. These Dirac equations are related to the self-duality equations of the GYM and are checked to be elliptic in a 'BPST' background. In this background these Dirac equations are integrated exactly. The possibility of imposing supersymmetry in the GYM-Dirac system is investigated, with negative results. (orig.) 13. SU(5)-invariant decomposition of ten-dimensional Yang-Mills supersymmetry CERN Document Server Baulieu, Laurent 2011-01-01 The N=1,d=10 superYang-Mills action is constructed in a twisted form, using SU(5)-invariant decomposition of spinors in 10 dimensions. The action and its off-shell closed twisted scalar supersymmetry operator Q derive from a Chern-Simons term. The action can be decomposed as the sum of a term in the cohomology of Q and of a term that is Q-exact. The first term is a fermionic Chern-Simons term for a twisted component of the Majorana-Weyl gluino and it is related to the second one by a twisted vector supersymmetry with 5 parameters. The cohomology of Q and some topological observables are defined from descent equations. In this SU(5)theory is determined by only 6 supersymmetry generators, as in the twisted N=4, d=4 theory. There is a superspace with 6 twisted fermionic directions, with solvable constraints. 14. Unique specification of Yang-Mills solutions International Nuclear Information System (INIS) Campbell, W.B.; Joseph, D.W.; Morgan, T.A. 1980-01-01 Screened time-independent cylindrically-symmetric solutions of Yang-Mills equations are given which show that the source does not uniquely determine the field. However, these particular solutions suggest a natural way of uniquely specifying solutions in terms of a physical realization of a symmetry group. (orig.) 15. A class of Yang-Mills solutions International Nuclear Information System (INIS) Castillejo, L.; Kugler, M. 1980-09-01 We investigate a class of solutions of the classical SU(2) Yang-Mills equations. The symmetry of this class prescribes a natural set of gauge invariant degrees of freedom. Using these degrees of freedom we obtain a simple set of equations which enables us to find all the solutions belonging to the class under discussion. (Author) 16. Einstein constraints in the Yang-Mills form International Nuclear Information System (INIS) Ashtekar, A. 1987-01-01 It is pointed out that constraints of Einstein's theory play a powerful role in both classical and quantum theory because they generate motions in spacetime, rather than in an internal space. New variables are then introduced on the Einstein phase space in terms of which constraints simplify considerably. In particular, the use of these variables enables one to imbed the constraint surface of Einstein's theory into that of Yang-Mills. The imbedding suggests new lines of attack to a number of problems in classical and quantum gravity and provides new concepts and tools to investigate the microscopic structure of space-time geometry 17. Fate of Yang-Mills black hole in early Universe Energy Technology Data Exchange (ETDEWEB) Nakonieczny, Lukasz; Rogatko, Marek [Institute of Physics Maria Curie-Sklodowska University 20-031 Lublin, pl. Marii Curie-Sklodowskiej 1 (Poland) 2013-02-21 According to the Big Bang Theory as we go back in time the Universe becomes progressively hotter and denser. This leads us to believe that the early Universe was filled with hot plasma of elementary particles. Among many questions concerning this phase of history of the Universe there are questions of existence and fate of magnetic monopoles and primordial black holes. Static solution of Einstein-Yang-Mills system may be used as a toy model for such a black hole. Using methods of field theory we will show that its existence and regularity depend crucially on the presence of fermions around it. 18. Loop quantum corrected Einstein Yang-Mills black holes Science.gov (United States) Protter, Mason; DeBenedictis, Andrew 2018-05-01 In this paper, we study the homogeneous interiors of black holes possessing SU(2) Yang-Mills fields subject to corrections inspired by loop quantum gravity. The systems studied possess both magnetic and induced electric Yang-Mills fields. We consider the system of equations both with and without Wilson loop corrections to the Yang-Mills potential. The structure of the Yang-Mills Hamiltonian, along with the restriction to homogeneity, allows for an anomaly-free effective quantization. In particular, we study the bounce which replaces the classical singularity and the behavior of the Yang-Mills fields in the quantum corrected interior, which possesses topology R ×S2 . Beyond the bounce, the magnitude of the Yang-Mills electric field asymptotically grows monotonically. This results in an ever-expanding R sector even though the two-sphere volume is asymptotically constant. The results are similar with and without Wilson loop corrections on the Yang-Mills potential. 19. SU(2) Yang-Mills solitons in R2 gravity Science.gov (United States) Perapechka, I.; Shnir, Ya. 2018-05-01 We construct new family of spherically symmetric regular solutions of SU (2) Yang-Mills theory coupled to pure R2 gravity. The particle-like field configurations possess non-integer non-Abelian magnetic charge. A discussion of the main properties of the solutions and their differences from the usual Bartnik-McKinnon solitons in the asymptotically flat case is presented. It is shown that there is continuous family of linearly stable non-trivial solutions in which the gauge field has no nodes. 20. Polyakov lines in Yang-Mills matrix models International Nuclear Information System (INIS) Austing, Peter; Wheater, John F.; Vernizzi, Graziano 2003-01-01 We study the Polyakov line in Yang-Mills matrix models, which include the IKKT model of IIB string theory. For the gauge group SU(2) we give the exact formulae in the form of integral representations which are convenient for finding the asymptotic behaviour. For the SU(N) bosonic models we prove upper bounds which decay as a power law at large momentum p. We argue that these capture the full asymptotic behaviour. We also indicate how to extend the results to some correlation functions of Polyakov lines. (author) 1. Geometry of Yang-Mills fields International Nuclear Information System (INIS) Atiyah, M.F. 1978-01-01 In this talk I shall explain how information about classical solutions of Yang-Mills equations can be obtained, rather surprisingly, from algebraic geometry. Although direct physical interest is restricted to the case of four dimensions I shall begin by discussing the two-dimensional case. Besides preparing the ground for the four-dimensional problem this has independent mathematical (and possible physical) interest, and very complete results can be obtained. (orig.) [de 2. Massive and mass-less Yang-Mills and gravitational fields NARCIS (Netherlands) Veltman, M.J.G.; Dam, H. van 1970-01-01 Massive and mass-less Yang-Mills and gravitational fields are considered. It is found that there is a discrete difference between the zero-mass theories and the very small, but non-zero mass theories. In the case of gravitation, comparison of massive and mass-less theories with experiment, in 3. Maximally Generalized Yang-Mills Model and Dynamical Breaking of Gauge Symmetry International Nuclear Information System (INIS) Wang Dianfu; Song Heshan 2006-01-01 A maximally generalized Yang-Mills model, which contains, besides the vector part V μ , also an axial-vector part A μ , a scalar part S, a pseudoscalar part P, and a tensor part T μν , is constructed and the dynamical breaking of gauge symmetry in the model is also discussed. It is shown, in terms of the Nambu-Jona-Lasinio mechanism, that the gauge symmetry breaking can be realized dynamically in the maximally generalized Yang-Mills model. The combination of the maximally generalized Yang-Mills model and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory. 4. Color Memory: A Yang-Mills Analog of Gravitational Wave Memory Science.gov (United States) Pate, Monica; Raclariu, Ana-Maria; Strominger, Andrew 2017-12-01 A transient color flux across null infinity in classical Yang-Mills theory is considered. It is shown that a pair of test "quarks" initially in a color singlet generically acquire net color as a result of the flux. A nonlinear formula is derived for the relative color rotation of the quarks. For a weak color flux, the formula linearizes to the Fourier transform of the soft gluon theorem. This color memory effect is the Yang-Mills analog of the gravitational memory effect. 5. The Massive Yang-Mills Model and Diffractive Scattering CERN Document Server Forshaw, J R; Parrinello, C 1999-01-01 We argue that the massive Yang-Mills model of Kunimasa and Goto, Slavnov, and Cornwall, in which massive gauge vector bosons are introduced in a gauge-invariant way without resorting to the Higgs mechanism, may be useful for studying diffractive scattering of strongly interacting particles. With this motivation, we perform in this model explicit calculations of S-matrix elements between quark states, at tree level, one loop, and two loops, and discuss issues of renormalisability and unitarity. In particular, it is shown that the S-matrix element for quark scattering is renormalisable at one-loop order and is only logarithmically non-renormalisable at two loops. The discrepancies in the ultraviolet regime between the one-loop predictions of this model and those of massless QCD are discussed in detail. In addition, some of the similarities and differences between the massive Yang-Mills model and theories with a Higgs mechanism are analysed at the level of the S-matrix. As an elementary application of the model ... 6. Massive IIA string theory and Matrix theory compactification International Nuclear Information System (INIS) Lowe, David A.; Nastase, Horatiu; Ramgoolam, Sanjaye 2003-01-01 We propose a Matrix theory approach to Romans' massive Type IIA supergravity. It is obtained by applying the procedure of Matrix theory compactifications to Hull's proposal of the massive Type IIA string theory as M-theory on a twisted torus. The resulting Matrix theory is a super-Yang-Mills theory on large N three-branes with a space-dependent noncommutativity parameter, which is also independently derived by a T-duality approach. We give evidence showing that the energies of a class of physical excitations of the super-Yang-Mills theory show the correct symmetry expected from massive Type IIA string theory in a lightcone quantization 7. Asymptotic properties of spherically symmetric, regular and static solutions to Yang-Mills equations International Nuclear Information System (INIS) Cronstrom, C. 1987-01-01 In this paper the author discusses the asymptotic properties of solutions to Yang-Mills equations with the gauge group SU(2), for spherically symmetric, regular and static potentials. It is known, that the pure Yang-Mills equations cannot have nontrivial regular solutions which vanish rapidly at space infinity (socalled finite energy solutions). So, if regular solutions exist, they must have non-trivial asymptotic properties. However, if the asymptotic behaviour of the solutions is non-trivial, then the fact must be explicitly taken into account in constructing the proper action (and energy) for the theory. The elucidation of the appropriate surface correction to the Yang-Mills action (and hence the energy-momentum tensor density) is one of the main motivations behind the present study. In this paper the author restricts to the asymptotic behaviour of the static solutions. It is shown that this asymptotic behaviour is such that surface corrections (at space-infinity) are needed in order to obtain a well-defined (classical) theory. This is of relevance in formulating a quantum Yang-Mills theory 8. Renormalizability aspects of massive Yang--Mills field models International Nuclear Information System (INIS) Ktorides, C.N. 1976-01-01 We confront the problem concerning the renormalizability of massive Yang--Mills theories in which the mass term for the vector fields has been inserted by hand. Our starting Lagrangians are of a type in the past found to be nonrenormalizable. The massive Yang--Mills fields are split into transverse and longitudinal components. The latter carry all the nonrenormalizability pathologies which manifest themselves in terms of certain nonpolynomial factors involving the longtitudinal fields. The removal of the bad nonpolynomial terms (Boulware's problem) is studied within the context of the adjoint representation of the gauge group SU(2). A necessary condition for solving Boulware's problem is the introduction of extra fields. We find an explicit solution which requires the introduction of a triplet of scalar fields belonging to the adjoint representation of SU(2). We interpret the additional fields as ghost, or superfluous, fields, most probably corresponding to the ghost fields of spontaneously broken gauge theories in the R gauge. Out interpretation of the fields which combine with the longitudinal ones in order to remove the nonpolymomial factors as ghost fields is not evident in the treatment of Cornwall et al. Unlike the case of Cornwall et al., we do not just show the existence of the trnasformation which removes the undesirable terms, but also give the explicit conditions which bring about this result in the case of SU(2). A proposition relating the models under consideration to spontaneously broken gauge ones is also presented. We argue, without explicit proof, that the combination of this proposition with out main theorem corresponds to building a spontaneously broken gauge theory in the R gauge, having started from a non-Abelian theory with mass inserted by hand 9. Chaotic behavior of the lattice Yang-Mills on CUDA Directory of Open Access Journals (Sweden) Forster Richárd 2015-12-01 Full Text Available The Yang-Mills fields plays important role in the strong interaction, which describes the quark gluon plasma. The non-Abelian gauge theory provides the theoretical background understanding of this topic. The real time evolution of the classical fields is derived by the Hamiltonian for SU(2 gauge field tensor. The microcanonical equations of motion is solved on 3 dimensional lattice and chaotic dynamics was searched by the monodromy matrix. The entropy-energy relation was presented by Kolmogorov-Sinai entropy. We used block Hessenberg reduction to compute the eigenvalues of the current matrix. While the purely CPU based algorithm can handle effectively only a small amount of values, the GPUs provide enough performance to give more computing power to solve the problem. 10. Vortex-like and string-like solutions for the 2+1 dimensional SU(2) Yang-Mills theory with the Chern-Simons term International Nuclear Information System (INIS) Teh, R. 1989-07-01 Vortex-like and string-like solutions of 2+1 Dim. SU(2) YM theory with the Chern-Simons term are discussed. Two ansatze are constructed which yield respectively analytic Bessel function solutions and elliptic function solutions. The Bessel function solutions are vortex-like and tend to the same vacuum state as the Ginzburg-Landau vortex solution at large ρ. The Jacobi elliptic function solutions are string-like, have finite energy and magnetic flux concentrated along a line in the x 1 - x 2 plane. (author). 18 refs 11. The Yang-Mills gradient flow and SU(3) gauge theory with 12 massless fundamental fermions in a colour-twisted box CERN Document Server Lin, C -J David; Ramos, Alberto 2015-01-01 We perform the step-scaling investigation of the running coupling constant, using the gradient-flow scheme, in SU(3) gauge theory with twelve massless fermions in the fundamental representation. The Wilson plaquette gauge action and massless unimproved staggered fermions are used in the simulations. Our lattice data are prepared at high accuracy, such that the statistical error for the renormalised coupling, g_GF, is at the subpercentage level. To investigate the reliability of the continuum extrapolation, we employ two different lattice discretisations to obtain g_GF. For our simulation setting, the corresponding gauge-field averaging radius in the gradient flow has to be almost half of the lattice size, in order to have this extrapolation under control. We can determine the renormalisation group evolution of the coupling up to g^2_GF ~ 6, before the onset of the bulk phase structure. In this infrared regime, the running of the coupling is significantly slower than the two-loop perturbative prediction, altho... 12. Yang-Mills fields due to an infinite charge cylinder International Nuclear Information System (INIS) Campbell, W.B.; Joseph, D.W.; Morgan, T.A.; Nebraska Univ., Lincoln 1981-01-01 The problem of determining time-independent solutions of the classical Yang-Mills equations for infinitely long charge cylinders is studied. A useful expression for the total energy in the field in terms of just the sources is derived. Numerical solutions have been found in the special cases of a small charge cylinder with a magnetic field B that either lies along the axis of symmetry or encircles the axis. It is as if these two solutions were due to currents encircling the axis or parallelling it, respectively. The condition that the solutions behave well at infinity implies an exponential fall off for the fields in the azimuthal B field case and a fall off more rapid than 1/R in the axial B field case, so that in both cases the existence of a B field requires the charge on the axis to be shieled. Consequently, these solutions do not behave at infinity at all like the Maxwell solution for a charge cylinder, and they have a lower energy per unit length. They show that in Yang-Mills theories the source does not determine a unique field. A classical interpretation of this is that the field remembers how the charges were transported during the construction of the cylinder. It also suggests that a quantum mechanical version of this problem would exhibit a spontaneous symmetry breaking to a less symmetric, lower energy vacuum. These solutions exhibit a twofold degeneracy, as the magnetic field may be either left- or right-handed in the azimuthal B field case, or point along the +z or -z axis in the axial B field case. (orig.) 13. Conformally flat spaces and solutions to Yang-Mills equations International Nuclear Information System (INIS) Chaohao, G. 1980-01-01 Using the conformal invariance of Yang-Mills equations in four-dimensional manifolds, it is proved that in a simply connected space of negative constant curvature Yang-Mills equations admit solutions with any real number as their Pontryagin number. It is also shown that the space S 3 x S 1 which is the regular counterpart of the meron solution is one example of a class of solutions to Yang-Mills equations on compact manifolds that are neither self-dual nor anti-self-dual 14. Cosmological coevolution of Yang-Mills fields and perfect fluids International Nuclear Information System (INIS) Barrow, John D.; Jin, Yoshida; Maeda, Kei-ichi 2005-01-01 We study the coevolution of Yang-Mills fields and perfect fluids in Bianchi type I universes. We investigate numerically the evolution of the universe and the Yang-Mills fields during the radiation and dust eras of a universe that is almost isotropic. The Yang-Mills field undergoes small amplitude chaotic oscillations, as do the three expansion scale factors which are also displayed by the expansion scale factors of the universe. The results of the numerical simulations are interpreted analytically and compared with past studies of the cosmological evolution of magnetic fields in radiation and dust universes. We find that, whereas magnetic universes are strongly constrained by the microwave background anisotropy, Yang-Mills universes are principally constrained by primordial nucleosynthesis but the bound is comparatively weak with Ω YM rad 15. Null solution of the Yang-Mills equations International Nuclear Information System (INIS) Tafel, J. 1986-05-01 We investigate the correspondence between null solutions of the Yang-Mills equations and shearfree geodesic null congruences. We give an example of a non-Abelian null solution with twisting rays. (orig.) 16. Yang-Mills-Vlasov system in the temporal gauge International Nuclear Information System (INIS) Choquet-Bruhat, Y.; Noutchegueme, N. 1991-01-01 We prove a local in time existence theorem of a solution of the Cauchy problem for the Yang-Mills-Vlasov integrodifferential system. Such equations govern the evolution of plasmas, for instance of quarks and gluons (quagmas), where non abelian gauge fields and Yang-Mills charges replace the usual electromagnetic field and electric charge. We work with the temporal gauge and use functional spaces with appropriate weight on the momenta, but no fall off is required in the space direction [fr 17. Dynamical CP violation of the generalized Yang-Mills model International Nuclear Information System (INIS) Wang Dianfu; Chang Xiaojing; Sun Xiaoyu 2011-01-01 Starting from the generalized Yang-Mills model which contains, besides the vector part V μ , also a scalar part S and a pseudoscalar part P . It is shown, in terms of the Nambu-Jona-Lasinio (NJL) mechanism, that CP violation can be realized dynamically. The combination of the generalized Yang-Mills model and the NJL mechanism provides a new way to explain CP violation. (authors) 18. Supersymmetric Yang-Mills and supergravity amplitudes at one loop International Nuclear Information System (INIS) Hall, Anthony 2009-01-01 By applying the known expressions for super Yang-Mills (SYM) and supergravity (SUGRA) tree amplitudes, we write generating functions for the next-to-next-to-maximally helicity violating (NNMHV) box coefficients of SYM as well as the maximally helicity violating, next-to-maximally helicity violating, and NNMHV box coefficients for SUGRA. The all-multiplicity generating functions utilize covariant, on-shell superspace whereby the contribution from arbitrary external states in the supermultiplet can be extracted by Grassmann operators. In support of the relation between dual-Wilson loops and SYM scattering amplitudes at weak coupling, the SYM amplitudes are presented in a manifestly dual superconformal form. We introduce ordered box coefficients for calculating SUGRA quadruple cuts and prove that ordered coefficients generate physical cut amplitudes after summing over permutations of the external legs. The ordered box coefficients are produced by sewing ordered subamplitudes, previously used in applying on-shell recursion relations at tree level. We describe our verification of the results against the literature, and a formula for extracting the contributions from external gluons or gravitons to NNMHV superamplitudes is presented. 19. The one-loop Green's functions of dimensionally reduced gauge theories International Nuclear Information System (INIS) Ketov, S.V.; Prager, Y.S. 1988-01-01 The dimensional regularization technique as well as that by dimensional reduction is applied to the calculation of the regularized one-loop Green's functions in dsub(o)-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge group G. As a particular example, the super-symmetrically regularized one-loop Green's functions of the N=4 supersymmetric Yang-Mills model are derived. (author). 17 refs 20. AdS charged black holes in Einstein-Yang-Mills gravity's rainbow: Thermal stability and P - V criticality Science.gov (United States) Hendi, Seyed Hossein; Momennia, Mehrab 2018-02-01 Motivated by the interesting non-abelian gauge field, in this paper, we look for the analytical solutions of Yang-Mills theory in the context of gravity's rainbow. Regarding the trace of quantum gravity in black hole thermodynamics, we examine the first law of thermodynamics and also thermal stability in the canonical ensemble. We show that although the rainbow functions and Yang-Mills charge modify the solutions, the first law of thermodynamics is still valid. Based on the phenomenological similarities between the adS black holes and van der Waals liquid/gas systems, we study the critical behavior of the Yang-Mills black holes in the extended phase space thermodynamics. We also investigate the effects of various parameters on thermal instability as well as critical properties by using appropriate figures. 1. A model of unified quantum chromodynamics and Yang-Mills gravity Institute of Scientific and Technical Information of China (English) HSU Jong-Ping 2012-01-01 Based on a generalized Yang-Mills framework,gravitational and strong interactions can be unified in analogy with the unification in the clectroweak theory.By gauging T(4) × [SU(3)]color in fiat space-time,we have a unified model of chromo-gravity with a new tensor gauge field,which couples universally to all gluons,quarks and anti-quarks.The space-time translational gauge symmetry assures that all wave equations of quarks and gluons reduce to a Hamilton-Jacobi equation with the same ‘effective Riemann metric tensors' in the geometric-optics (or classical) limit.The emergence of effective metric tensors in the classical limit is essential for the unified model to agree with experiments.The unified model suggests that all gravitational,strong and electroweak interactions appear to be dictated by gauge symmetries in the generalized Yang-Mills framework. 2. A model of unified quantum chromodynamics and Yang-Mills gravity International Nuclear Information System (INIS) HSU Jongping 2012-01-01 Based on a generalized Yang-Mills framework, gravitational and strong interactions can be unified in analogy with the unification in the electroweak theory. By gauging T(4) × [SU(3)] color in flat space-time, we have a unified model of chromo-gravity with a new tensor gauge field, which couples universally to all gluons, quarks and anti-quarks. The space-time translational gauge symmetry assures that all wave equations of quarks and gluons reduce to a Hamilton-Jacobi equation with the same 'effective Riemann metric tensors’ in the geometric-optics (or classical) limit. The emergence of effective metric tensors in the classical limit is essential for the unified model to agree with experiments. The unified model suggests that all gravitational, strong and electroweak interactions appear to be dictated by gauge symmetries in the generalized Yang-Mills framework. (author) 3. Non-Abelian Yang-Mills analogue of classical electromagnetic duality International Nuclear Information System (INIS) Chan, Hong-Mo; Faridani, J.; Tsun, T.S. 1995-01-01 The classic question of non-Abelian Yang-Mills analogue to electromagnetic duality is examined here in a minimalist fashion at the strictly four-dimensional, classical field, and point charge level. A generalization of the Abelian Hodge star duality is found which, though not yet known to give dual symmetry, reproduces analogues to many dual properties of the Abelian theory. For example, there is a dual potential, but it is a two-indexed tensor T μν of the Freedman-Townsend-type. Though not itself functioning as such, T μν gives rise to a dual parallel transport A μ for the phase of the wave function of the color magnetic charge, this last being a monopole of the Yang-Mills field but a source of the dual field. The standard color (electric) charge itself is found to be a monpole of A μ . At the same time, the gauge symmetry is found doubled from say SU(N) to SU(N)xSU(N). A novel feature is that all equations of motion, including the standard Yang-Mills and Wong equations, are here derived from a ''universal'' principle, namely, the Wu-Yang criterion for monpoles, where interactions arise purely as a consequence of the topological definition of the monopole charge. The technique used is the loop space formulation of Polyakov 4. Schwinger-Dyson operator of Yang-Mills matrix models with ghosts and derivations of the graded shuffle algebra NARCIS (Netherlands) Krishnaswami, G.S. 2008-01-01 We consider large-N multi-matrix models whose action closely mimics that of Yang-Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger-Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G( ), are quadratic equations 5. Simplifying Multi-loop Integrands of Gauge Theory and Gravity Amplitudes Energy Technology Data Exchange (ETDEWEB) Bern, Z.; Carrasco, J.J.M.; Dixon, L.J.; Johansson, H.; Roiban, R. 2012-02-15 We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N = 4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude's integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N = 8 supergravity integrand, whose graph numerators are double copies (squares) of the N = 4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the duality and double-copy properties hold at loop level. The new form of the four-loop four-point supergravity amplitude makes manifest the same ultraviolet power counting as the corresponding N = 4 super-Yang-Mills amplitude. We determine the amplitude's ultraviolet pole in the critical dimension of D = 11/2, the same dimension as for N = 4 super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of N = 8 supergravity as the subleading-in-1/N{sub c}{sup 2} single-trace divergence in N = 4 super-Yang-Mills theory. 6. Euclidean self-dual Yang-Mills field configurations International Nuclear Information System (INIS) Sartori, G. 1980-01-01 The determination of a large class of regular and singular Euclidean self-dual Yang-Mills field configurations is reduced to the solution of a set of linear algebraic equations. The matrix of the coefficients is a polynomial functions of x and the rules for its construction are elementary. (author) 7. Classical Yang-Mills mechanics. Nonlinear colour oscillations International Nuclear Information System (INIS) Matinyan, S.G.; Savvidi, G.K.; Ter-Arutyunyan-Savvidi, N.G. 1981-01-01 A novel class of solutions of the classical Yang-Mills equations in the Minkowsky space which leads to nonlinear colour oscillations is studied. The system discribing these oscillations is apparently stochastic. Periodic trajectories corresponding to the solutions are found and studied and it is demonstrated that they constitute at least an enumerable set [ru 8. Gravitational matter-antimatter asymmetry and four-dimensional Yang-Mills gauge symmetry Science.gov (United States) Hsu, J. P. 1981-01-01 A formulation of gravity based on the maximum four-dimensional Yang-Mills gauge symmetry is studied. The theory predicts that the gravitational force inside matter (fermions) is different from that inside antimatter. This difference could lead to the cosmic separation of matter and antimatter in the evolution of the universe. Moreover, a new gravitational long-range spin-force between two fermions is predicted, in addition to the usual Newtonian force. The geometrical foundation of such a gravitational theory is the Riemann-Cartan geometry, in which there is a torsion. The results of the theory for weak fields are consistent with previous experiments. 9. Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields International Nuclear Information System (INIS) Cazenave, T.; Shatah, J.; Tahvildar-Zadeh, A.S. 1998-01-01 In this article we explore some of the connections between the theories of Yang-Mills fields, wave maps, and harmonic maps. It has been shown that the search for similarity solutions of wave maps leads to harmonic maps of the hyperbolic space. On the other hand, Glassey and Strauss have shown that the equations for an SO(3)-equivariant Yang-Mills connection on the Minkowski space R 3,1 with gauge group SU(2) reduce to a certain nonlinear wave equation, which we can now identify as a wave map on R 1,1 . More generally, we will here show the reduction under equivariance of a Yang-Mills system on the Minkowski space R n,1 to a wave map system on R n-2,1 in the specific case of SO(n) bundles with SO(n) symmetry. We then prove for odd n the existence of equivariant harmonic maps from the hyperbolic space H n that are smooth at the ideal boundary of H n , thus establishing the existence of similarity solutions for equivariant wave maps and Yang-Mills fields. As a consequence we show that for n ≥ 7, it is possible to have a wave map into a negatively curved target manifold that develops from smooth initial data and blows up in finite time, in sharp contrast to the elliptic case of harmonic maps. Finally we show how these singular solutions can be lifted to one dimension higher to produce singular travelling waves. (orig.) 10. On hidden symmetries of a super gauge theory and twistor string theory International Nuclear Information System (INIS) Wolf, Martin 2005-01-01 We discuss infinite-dimensional hidden symmetry algebras (and hence an infinite number of conserved nonlocal charges) of the N-extented self-dual super Yang-Mills equations for general N=4 by using the supertwistor correspondence. Furthermore, by enhancing the supertwistor space, we construct the N-extended self-dual super Yang-Mills hierarchies, which describe infinite sets of graded abelian symmetries. We also show that the open topological B-model with the enhanced supertwistor space as target manifold will describe the hierarchies. Furthermore, these hierarchies will in turn - by a supersymmetric extension of Ward's conjecture - reduce to the super hierarchies of integrable models in D<4 dimensions. (author) 11. Spinors in self-dual Yang-Mills fields in minkowski space International Nuclear Information System (INIS) Pervushin, V.N. 1981-01-01 Yang-Mills theory with infrared divergences removed by spontaneous vacuum symmetry breaking is considered. The corresponding vacuum fields are self-dual and are defined in the Minkowski space. The complete set of solutions of Dirac equations with self-dual fields, depending on certain arbitrary function, is found. Physical observables (charge, energy, spin) for the spinor fields within the self-dual vacuum are calculated and a Hermitean Hamiltonian is obtained. The physical picture corresponds to a relativistic generalization of the hadron bag model [ru 12. Non self-dual Yang-Mills fields International Nuclear Information System (INIS) Bor, G. 1991-01-01 The purpose of the thesis is to prove the existence of a new family of non self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, with SU(2) as a gauge group. The approach is that of equivalent geometry: attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry which it is proved that (1) a solution to the Yang-Mills equations exists for among them, and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by showing that the self-duality equations, linearized at a symmetric self-dual solution, cannot possess the required symmetry 13. Yang-Mills fields which are not self-dual International Nuclear Information System (INIS) Bor, G. 1992-01-01 The purpose of this paper is to prove the existence of a new family of non-self-dual finite-energy solutions to the Yang-Mills equations on Euclidean four-space, with SU(2) as a gauge group. The approach is that of 'equivariant geometry': Attention is restricted to a special class of fields, those that satisfy a certain kind of rotational symmetry, for which it is proved that (1) a solution to the Yang-Mills equations exists among them; and (2) no solution to the self-duality equations exists among them. The first assertion is proved by an application of the direct method of the calculus of variations (existence and regularity of minimizers), and the second assertion by studying the symmetry properties of the linearized-self-duality equations. The same technique yields a new family of non-self-dual solutions on the complex projective plane. (orig.) 14. Yang-Mills- SU(3) via FORM International Nuclear Information System (INIS) Costa Jorge, Patricia M. da; Peres, Patricia Duarte; Boldo, J.L. 1997-06-01 This work uses FORM software aspects for obtaining a series of formal results in the non-Abelian gauge theory, with SU(3) group. The work also studies field transformation, Lagrangian density invariance, field equations, energy distribution and the theory reparametrization in terms of fields associated to particles which are possible to be detected in accelerators 15. Theory of super LIE groups International Nuclear Information System (INIS) Prakash, M. 1985-01-01 The theory of supergravity has attracted increasing attention in the recent years as a unified theory of elementary particle interactions. The superspace formulation of the theory is highly suggestive of an underlying geometrical structure of superspace. It also incorporates the beautifully geometrical general theory of relativity. It leads us to believe that a better understanding of its geometry would result in a better understanding of the theory itself, and furthermore, that the geometry of superspace would also have physical consequences. As a first step towards that goal, we develop here a theory of super Lie groups. These are groups that have the same relation to a super Lie algebra as Lie groups have to a Lie algebra. More precisely, a super Lie group is a super-manifold and a group such that the group operations are super-analytic. The super Lie algebra of a super Lie group is related to the local properties of the group near the identity. This work develops the algebraic and super-analytical tools necessary for our theory, including proofs of a set of existence and uniqueness theorems for a class of super-differential equations 16. Monopoles and vortices in Yang-Mills plasma International Nuclear Information System (INIS) Chernodub, M. N.; Zakharov, V. I. 2009-01-01 We discuss the role of magnetic degrees of freedom in Yang-Mills plasma at temperatures above and of order of the critical temperature T c . While at zero temperature the magnetic degrees of freedom are condensed and electric degrees of freedom are confined, at the point of the phase transition both magnetic and electric degrees of freedom are released into the thermal vacuum. This phenomenon might explain the observed unusual properties of the plasma. 17. Regular behaviors in SU(2) Yang-Mills classical mechanics International Nuclear Information System (INIS) Xu Xiaoming 1997-01-01 In order to study regular behaviors in high-energy nucleon-nucleon collisions, a representation of the vector potential A i a is defined with respect to the (a,i)-dependence in the SU(2) Yang-Mills classical mechanics. Equations of the classical infrared field as well as effective potentials are derived for the elastic or inelastic collision of two plane wave in a three-mode model and the decay of an excited spherically-symmetric field 18. Wormhole instanton solution in the Einstein-Yang-Mills system International Nuclear Information System (INIS) Hosoya, Akio; Ogura, Waichi. 1989-01-01 A spherical symmetric classical solution of the Einstein and the SU(2) Yang-Mills equations is found in the four dimensional Euclidean space-time with the cosmological constant. The isospinor fermion has zero modes. Their cosmological implications are also discussed with an emphasis on the fact that wormhole instantons in general can be found not only in the sub-Planck physics but also in almost all the stages in lower energy physics. (author) 19. Space-time symmetry and quantum Yang-Mills gravity how space-time translational gauge symmetry enables the unification of gravity with other forces CERN Document Server Hsu, Jong-Ping 2013-01-01 Yang-Mills gravity is a new theory, consistent with experiments, that brings gravity back to the arena of gauge field theory and quantum mechanics in flat space-time. It provides solutions to long-standing difficulties in physics, such as the incompatibility between Einstein's principle of general coordinate invariance and modern schemes for a quantum mechanical description of nature, and Noether's 'Theorem II' which showed that the principle of general coordinate invariance in general relativity leads to the failure of the law of conservation of energy. Yang-Mills gravity in flat space-time a 20. Numerical methods for the study of a N=1 super-Yang-Mills theory with SU(2)c and SU(3)c Wilson fermions International Nuclear Information System (INIS) Ferling, Alexander 2009-01-01 A main topic of this thesis was to transfer the hybrid Monte-Carlo algorithm on a N=1 supersymmetric model. As model served the two-step multi-boson algorithm (TSMB). Beside the essential algorithm in the TSMB program further optimizations were realized. A further step was to optimize the lattice action so that discretization artefacts at finite lattice parameters were more strongly suppressed 1. The electromagnetic Dirac-Fock-Podolsky problem and symplectic properties of the Maxwell and Yang-Mills type dynamical systems International Nuclear Information System (INIS) Bogolubov, N.N. Jr.; Prykarpatsky, A.K.; Taneri, U.; Prykarpatsky, Y.A. 2009-01-01 Based on analysis of reduced geometric structures on fibered manifolds, invariant under action of a certain symmetry group, we construct the symplectic structures associated with connection forms on suitable principal fiber bundles. The application to the non-standard Hamiltonian analysis of the Maxwell and Yang-Mills type dynamical systems is presented. A symplectic reduction theory of the classical Maxwell electromagnetic field equations is formulated, the important Lorentz condition, ensuring the existence of electromagnetic waves, is naturally included into the Hamiltonian picture, thereby solving the well known Dirac, Fock and Podolsky problem. The symplectically reduced Poissonian structures and the related classical minimal interaction principle, concerning the Yang-Mills type equations, are considered. (author) 2. Gravity duals for the Coulomb branch of marginally deformed N=4 Yang-Mills CERN Document Server Hernández, R; Zoakos, D; Hernandez, Rafael; Sfetsos, Konstadinos; Zoakos, Dimitrios 2006-01-01 Supergravity backgrounds dual to a class of exactly marginal deformations of N supersymmetric Yang-Mills can be constructed through an SL(2,R) sequence of T-dualities and coordinate shifts. We apply this transformation to multicenter solutions and derive supergravity backgrounds describing the Coulomb branch of N=1 theories at strong 't Hooft coupling as marginal deformations of N=4 Yang-Mills. For concreteness we concentrate to cases with an SO(4)xSO(2) symmetry preserved by continuous distributions of D3-branes on a disc and on a three-dimensional spherical shell. We compute the expectation value of the Wilson loop operator and confirm the Coulombic behaviour of the heavy quark-antiquark potential in the conformal case. When the vev is turned on we find situations where a complete screening of the potential arises, as well as a confining regime where a linear or a logarithmic potential prevails depending on the ratio of the quark-antiquark separation to the typical vev scale. The spectra of massless excitat... 3. Constructing the tree-level Yang-Mills S-matrix using complex factorization Science.gov (United States) Schuster, Philip C.; Toro, Natalia 2009-06-01 A remarkable connection between BCFW recursion relations and constraints on the S-matrix was made by Benincasa and Cachazo in 0705.4305, who noted that mutual consistency of different BCFW constructions of four-particle amplitudes generates non-trivial (but familiar) constraints on three-particle coupling constants — these include gauge invariance, the equivalence principle, and the lack of non-trivial couplings for spins > 2. These constraints can also be derived with weaker assumptions, by demanding the existence of four-point amplitudes that factorize properly in all unitarity limits with complex momenta. From this starting point, we show that the BCFW prescription can be interpreted as an algorithm for fully constructing a tree-level S-matrix, and that complex factorization of general BCFW amplitudes follows from the factorization of four-particle amplitudes. The allowed set of BCFW deformations is identified, formulated entirely as a statement on the three-particle sector, and using only complex factorization as a guide. Consequently, our analysis based on the physical consistency of the S-matrix is entirely independent of field theory. We analyze the case of pure Yang-Mills, and outline a proof for gravity. For Yang-Mills, we also show that the well-known scaling behavior of BCFW-deformed amplitudes at large z is a simple consequence of factorization. For gravity, factorization in certain channels requires asymptotic behavior ~ 1/z2. 4. Constructing the tree-level Yang-Mills S-matrix using complex factorization International Nuclear Information System (INIS) Schuster, Philip C.; Toro, Natalia 2009-01-01 A remarkable connection between BCFW recursion relations and constraints on the S-matrix was made by Benincasa and Cachazo in 0705.4305, who noted that mutual consistency of different BCFW constructions of four-particle amplitudes generates non-trivial (but familiar) constraints on three-particle coupling constants - these include gauge invariance, the equivalence principle, and the lack of non-trivial couplings for spins > 2. These constraints can also be derived with weaker assumptions, by demanding the existence of four-point amplitudes that factorize properly in all unitarity limits with complex momenta. From this starting point, we show that the BCFW prescription can be interpreted as an algorithm for fully constructing a tree-level S-matrix, and that complex factorization of general BCFW amplitudes follows from the factorization of four-particle amplitudes. The allowed set of BCFW deformations is identified, formulated entirely as a statement on the three-particle sector, and using only complex factorization as a guide. Consequently, our analysis based on the physical consistency of the S-matrix is entirely independent of field theory. We analyze the case of pure Yang-Mills, and outline a proof for gravity. For Yang-Mills, we also show that the well-known scaling behavior of BCFW-deformed amplitudes at large z is a simple consequence of factorization. For gravity, factorization in certain channels requires asymptotic behavior ∼ 1/z 2 . 5. On self-dual Yang-Mills hierarchy International Nuclear Information System (INIS) Nakamura, Yoshimasa 1989-01-01 In this note, motivated by the Kadomtsev-Petviashvili (KP) hierarchy of integrable nonlinear evolution equations, a GL(n,C) self-dual Yang-Mills (SDYM) hierarchy is presented; it is an infinite system of SDYM equations having an infinite number of independent variables and being outside of the KP hierarchy. A relationship between the KP hierarchy and the SDYM hierarchy is discussed. It is also shown that GL(∞) SDYM equations introduced in this note are reduced to the GL(n,C) SDYM hierarchy by imposing an algebraic constraint. (orig.) 6. Supersymmetric self-dual Yang-Mills fields International Nuclear Information System (INIS) Zhao Liu 1994-01-01 A new four dimensional (4d) N = 1 supersymmetric integrable model, i.e. the supersymmetric self-dual Yang-Mills model is constructed. The equations of motion for this model are shown to be equivalent to the zero curvature condition on some superplane in the 4d superspace, the superplane being characterized by a point in the project space CP 3,4 . The linear systems are established according to this geometrical interpretation, and the effective action is also proposed in order to explain the dynamical content of the model 7. On superwistor geometry and integrability in super gauge theory International Nuclear Information System (INIS) Wolf, M. 2006-01-01 In this thesis, we report on different aspects of intgrability in supersymmetric gauge theories. Our main tool of investigation is supertwistor geometry. In the first chapter, we briefly review the basics of twistor geometry. Afterwards, we discuss self-dual super Yang-Mills (SYM) theory and some of its relatives. In particular, a detailed twistor description of self-dual SYM theory is presented. Furthermore, we introduce certain self-dual models which are, in fact, obtainable from self-dual SYM theory by a suitable reduction. Some of them can be interpreted within the context of topological field theories. To provide a twistor description of these models, we propose weighted projective superspaces as twistor space. These spaces turn out to be Calabi-Yau supermanifolds. Therefore, it is possible to write down approriate action princuples, as well. In chapter three, we then deal with the twistor formulation of a certain supersymmetric Bogomolnyi model in three space-time dimensions. The nonsupersymmetric version of this model describes static Yang-Mills-Higgs monopoles in the Prasad-Sommerfield limit. In particular, we consider a supersymmetric extension of mini-twistor space. This space is in turn a part of a certain doubble fibration. It is then possible to formulate a Chern-Simons type theory on the correspondence space of this fibration. As we explain, this theory describes partially holomorphic vector bundles. It should be noticed that the correspondence space can be equipped with a Cauchy-Riemann structure. Moreover, we formulate holomorphic BF theory on mini-supertwistor space. e then prove that the moduli spaces of all three theories are bijective. In addition, complex structure deformations on mini-supertwistor space are investigated eventually resulting in a twistor correspondence involving a supersymmetric Bogomolnyi model with massive fields. In chapter four, we review the twistor formulation of non-self-dual SYM theories. The remaining chapter is 8. Coupling of Yang-Mills to N=4, d=4 supergravity International Nuclear Information System (INIS) Bergshoeff, E.; Koh, I.G.; Sezgin, E. 1985-01-01 We couple N=4, d=4 supersymmetric Yang-Mills theory to supergravity. The scalars of the theory parametrize the coset (SO(n,6)/[SO(n)xSO(6)])xSU(1,1)/U(1)). Keeping the composite local SO(n)xSO(6)xU(1) invariance intact, we gauge an (n+6) parameter subgroup of SO(n,6) which is either (i) SU(2)xSU(2)xH(dim H=n), (ii) SO(4,1)xH(dim H=n-4) or (iii) SO(6,1)xH(dim H=n-15). In all these cases the theory has an indefinite potential. (orig.) 9. Coupling of Yang-Mills to N=4, d=4 supergravity International Nuclear Information System (INIS) Bergshoeff, E.; Koh, I.G.; Sezgin, E. 1985-01-01 We couple N=4, d=4 supersymmetric Yang-Mills theory to supergravity. The scalars of the theory parametrize the coset ((SO(n,6)/SO(n)xSO(6))x((SU(1,1)/U(1)). Keeping the composite local SO(n)xSO(6)xU(1) invariance intact, we gauge an (n+6) parameter subgroup of SO(n,6) which is either (i) SU(2)xSU(2)xH (dim H=n), (ii) SO(4,1)xH (dim H=n-4) or (iii) SO(6,1)xH (dim H=n-15). In all these cases the theory has an indefinite potential. (author) 10. Self-dual solutions to Euclidean Yang-Mills equations International Nuclear Information System (INIS) Corrigan, E. 1979-01-01 The paper provides an introduction to two approaches towards understanding the classical Yang-Mills field equations. On the one hand, the work of Atiyah and Ward showed that the self-dual equations, which are non-linear, could be regarded as a set of linear equations which turned out to be related to each other by Baecklund transformations. Fundamental to their procedure was the observation that the information carried by the vector potential could be coded into the structure of certain analytic vector bundles over a three dimensional projective space. The classification of these bundles and the subsequent recovery of the gauge field led to the infinite set of ansaetze, corresponding to the sets of linear equation mentioned already. On the other hand, Atiyah, Hitchin, Drinfeld and Manin have recently constructed, completely algebraically, the bundles of interest and indicated how the Yang-Mills potential may be obtained. Remarkably, their construction differs very little as the gauge group is changed (to any of the classical compact groups) and, uses only the elementary operations of linear algebra to yield potentials as rational functions of the spatial coordinates. (Auth.) 11. Spinning higher dimensional Einstein-Yang-Mills black holes International Nuclear Information System (INIS) Ghosh, Sushant G.; Papnoi, Uma 2014-01-01 We construct a Kerr-Newman-like spacetime starting from higher dimensional (HD) Einstein-Yang-Mills black holes via complex transformations suggested by Newman-Janis. The new metrics are a HD generalization of Kerr-Newman spacetimes which has a geometry that is precisely that of Kerr-Newman in 4D corresponding to a Yang-Mills (YM) gauge charge, but the sign of the charge term gets flipped in the HD spacetimes. It is interesting to note that the gravitational contribution of the YM gauge charge, in HD, is indeed opposite (attractive rather than repulsive) to that of the Maxwell charge. The effect of the YM gauge charge on the structure and location of static limit surface and apparent horizon is discussed. We find that static limit surfaces become less prolate with increase in dimensions and are also sensitive to the YM gauge charge, thereby affecting the shape of the ergosphere. We also analyze some thermodynamical properties of these BHs. (orig.) 12. Spinning higher dimensional Einstein-Yang-Mills black holes Energy Technology Data Exchange (ETDEWEB) Ghosh, Sushant G. [Jamia Millia Islamia, Centre for Theoretical Physics, New Delhi (India); University of Kwa-Zulu-Natal, Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, Private Bag 54001, Durban (South Africa); Papnoi, Uma [Jamia Millia Islamia, Centre for Theoretical Physics, New Delhi (India) 2014-08-15 We construct a Kerr-Newman-like spacetime starting from higher dimensional (HD) Einstein-Yang-Mills black holes via complex transformations suggested by Newman-Janis. The new metrics are a HD generalization of Kerr-Newman spacetimes which has a geometry that is precisely that of Kerr-Newman in 4D corresponding to a Yang-Mills (YM) gauge charge, but the sign of the charge term gets flipped in the HD spacetimes. It is interesting to note that the gravitational contribution of the YM gauge charge, in HD, is indeed opposite (attractive rather than repulsive) to that of the Maxwell charge. The effect of the YM gauge charge on the structure and location of static limit surface and apparent horizon is discussed. We find that static limit surfaces become less prolate with increase in dimensions and are also sensitive to the YM gauge charge, thereby affecting the shape of the ergosphere. We also analyze some thermodynamical properties of these BHs. (orig.) 13. Yang-Mills solutions and Spin(7)-instantons on cylinders over coset spaces with G2-structure International Nuclear Information System (INIS) Haupt, Alexander S. 2016-01-01 We study g-valued Yang-Mills fields on cylinders Z(G/H)=ℝ×G/H, where G/H is a compact seven-dimensional coset space with G 2 -structure, g is the Lie algebra of G, and Z(G/H) inherits a Spin(7)-structure. After imposing a general G-invariance condition, Yang-Mills theory with torsion on Z(G/H) reduces to Newtonian mechanics of a point particle moving in ℝ n under the influence of some quartic potential and possibly additional constraints. The kinematics and dynamics depends on the chosen coset space. We consider in detail three coset spaces with nearly parallel G 2 -structure and four coset spaces with SU(3)-structure. For each case, we analyze the critical points of the potential and present a range of finite-energy solutions. We also study a higher-dimensional analog of the instanton equation. Its solutions yield G-invariant Spin(7)-instanton configurations on Z(G/H), which are special cases of Yang-Mills configurations with torsion. 14. The 1+1 SU(2) Yang-Mills path integral International Nuclear Information System (INIS) Swanson, Mark S 2004-01-01 The path integral for SU(2) invariant two-dimensional Yang-Mills theory is recast in terms of the chromoelectric field strength by integrating the gauge fields from the theory. Implementing Gauss's law as a constraint in this process induces a topological term in the action that is no longer invariant under large gauge transformations. For the case that the partition function is considered over a circular spatial degree of freedom, it is shown that the effective action of the path integral is quantum mechanically WKB exact and localizes onto a set of chromoelectric zero modes satisfying antiperiodic boundary conditions. Summing over the zero modes yields a partition function that can be reexpressed using the Poisson resummation technique, allowing an easy determination of the energy spectrum, which is found to be identical to that given by other approaches 15. Nonexistence theorems for Yang-Mills fields and harmonic maps in the Schwarzschild spacetime International Nuclear Information System (INIS) Hu Hesheng 1987-01-01 The nonexistence of static solutions to pure Yang-Mills equations and nonconstant harmonic maps defined on the Schwarzschild spacetime outside the black hole (r>2M) is considered. Nonexistence theorems for pure Yang-Mills equations and harmonic maps in the region r≥5M and r≥3M are obtained, respectively. (orig.) 16. A Static Solution of Yang-Mills Equation on Anti-de Sitter Space International Nuclear Information System (INIS) Chen Li; Ren Xinan 2009-01-01 Since product metric on AdS space has played a very important role in Lorentz version of AdS/CFT correspondence, the Yang-Mills equation on AdS space with this metric is considered and a static solution is obtained in this paper, which helps to understand the AdS/CFT correspondence of Yang-Mills fields. (general) 17. Step scaling and the Yang-Mills gradient flow International Nuclear Information System (INIS) Lüscher, Martin 2014-01-01 The use of the Yang-Mills gradient flow in step-scaling studies of lattice QCD is expected to lead to results of unprecedented precision. Step scaling is usually based on the Schrödinger functional, where time ranges over an interval [0,T] and all fields satisfy Dirichlet boundary conditions at time 0 and T. In these calculations, potentially important sources of systematic errors are boundary lattice effects and the infamous topology-freezing problem. The latter is here shown to be absent if Neumann instead of Dirichlet boundary conditions are imposed on the gauge field at time 0. Moreover, the expectation values of gauge-invariant local fields at positive flow time (and of other well localized observables) that reside in the center of the space-time volume are found to be largely insensitive to the boundary lattice effects. 18. N=4 supersymmetric Yang Mills scattering amplitudes at high energies. The Regge cut contribution International Nuclear Information System (INIS) Bartels, J.; Sabio Vera, A. 2008-07-01 We further investigate, in N=4 supersymmetric Yang Mills theories, the high energy Regge behavior of six-point scattering amplitudes. In particular, for the new Regge cut contribution found in our previous paper, we compute in the leading logarithmic approximation (LLA) the energy spectrum of the BFKL equation in the color octet channel, and we calculate explicitly the two loop corrections to the discontinuities of the amplitudes for the transitions 2→4 and 3→3. We find an explicit solution of the BFKL equation for the octet channel for arbitrary momentum transfers and investigate the intercepts of the Regge singularities in this channel. As an important result we find that the universal collinear and infrared singularities of the BDS formula are not affected by this Regge-cut contribution. (orig.) 19. Towards a unified treatment of Yang-Mills and Higgs fields International Nuclear Information System (INIS) Balakrishna, B.S.; Guersey, F.; Wali, K.C. 1991-01-01 Starting from a noncommutative algebra scrA of the form scrC direct-product scrM, where scrC is the algebra of smooth functions on space-time and scrM is the algebra of nxn Hermitian matrices, we construct an exterior algebra of differential forms over scrA. We use the one-forms of this algebra to describe Yang-Mills and Higgs fields on a similar footing and construct a Lagrangian from its two-forms. We show how, in the resulting geometrical description, a Higgs potential that leads to spontaneous symmetry breaking arises naturally. We discuss the application of this formalism to the bosonic sectors of the standard electroweak theory and a grand-unified model based on SU(5)direct-product U(1) 20. Evolution of nonlinear perturbations inside Einstein-Yang-Mills black holes International Nuclear Information System (INIS) Donets, E.E.; Tentyukov, M.N.; Tsulaya, M.M. 1998-01-01 We present our results on numerical study of evolution of nonlinear perturbations inside spherically symmetric black holes in the SU(2) Einstein-Yang-Mills (EYM) theory. Recent developments demonstrate a new type of the behaviour of the metric for EYM black hole interiors; the generic metric exhibits an infinitely oscillating approach to the singularity, which is a spacelike but not of the mixmaster type. The evolution of various types of spherically symmetric perturbations, propagating from the internal vicinity of the external horizon towards the singularity is investigated in a self-consistent way using an adaptive numerical algorithm. The obtained results give strong numerical evidence in favor of nonlinear stability of the generic EYM black hole interiors. Alternatively, the EYM black hole interiors of S (schwarzschild)-type, which form only a zero measure subset in the space of all internal solutions are found to be unstable and transform to the generic type as perturbations are developed 1. Emergent gravity and noncommutative branes from Yang-Mills matrix models International Nuclear Information System (INIS) Steinacker, Harold 2009-01-01 The framework of emergent gravity arising from Yang-Mills matrix models is developed further, for general noncommutative branes embedded in R D . The effective metric on the brane turns out to have a universal form reminiscent of the open string metric, depending on the dynamical Poisson structure and the embedding metric in R D . A covariant form of the tree-level equations of motion is derived, and the Newtonian limit is discussed. This points to the necessity of branes in higher dimensions. The quantization is discussed qualitatively, which singles out the IKKT model as a prime candidate for a quantum theory of gravity coupled to matter. The Planck scale is then identified with the scale of N=4 SUSY breaking. A mechanism for avoiding the cosmological constant problem is exhibited 2. Quantisation of super Teichmueller theory International Nuclear Information System (INIS) Aghaei, Nezhla; Hamburg Univ.; Pawelkiewicz, Michal; Techner, Joerg 2015-12-01 We construct a quantisation of the Teichmueller spaces of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential. 3. Numerical methods for the study of a N=1 super-Yang-Mills theory with SU(2){sub c} and SU(3){sub c} Wilson fermions; Numerische Methoden zur Erforschung einer N=1 Super Yang-Mills-Theorie mit SU(2){sub c} und SU(3){sub c} Wilson Fermionen Energy Technology Data Exchange (ETDEWEB) Ferling, Alexander 2009-05-29 A main topic of this thesis was to transfer the hybrid Monte-Carlo algorithm on a N=1 supersymmetric model. As model served the two-step multi-boson algorithm (TSMB). Beside the essential algorithm in the TSMB program further optimizations were realized. A further step was to optimize the lattice action so that discretization artefacts at finite lattice parameters were more strongly suppressed. 4. The Hamiltonian Approach to Yang-Mills (2+1): An Update and Corrections to String Tension International Nuclear Information System (INIS) Nair, V P 2013-01-01 Yang-Mills theories in 2+1 (or 3) dimensions are interesting as nontrivial gauge theories in their own right and as effective theories of QCD at high temperatures. I shall review the basics of our Hamiltonian approach to this theory, emphasizing symmetries with a short update on its status. We will show that the calculation of the vacuum wave function for Yang-Mills theory in 2+1 dimensions is in the lowest order of a systematic expansion. Expectation values of observables can be calculated using an effective interacting chiral boson theory, which also leads to a natural expansion as a double series in the coupling constant (to be interpreted within a resummed perturbation series) and a particular kinematical factor. The calculation of the first set of corrections in this expansion shows that the string tension is modified by about −0.3% to −2.8% compared to the lowest order value. This is in good agreement with lattice estimates 5. The ultraviolet behaviour of N=4 Yang-Mills and the power counting of extended superspace International Nuclear Information System (INIS) Marcus, N.; Sagnotti, A. 1985-01-01 A two-loop calculation in the N=4 supersymmetric Yang-Mills theory is performed in various dimensions. The theory is found to be two-loop finite in six dimensions or less, but infinite in seven and nine dimensions. The six-dimensional result can be explained by a formulation of the theory in terms of N=2 superfields. The divergence in seven dimensions is naively compatible with both N=2 and N=4 superfield power counting rules, but is of a form that cannot be written as an on-shell N=4 superfield integral. The hypothesized N=4 extended superfield formalism therefore either does not exist, or at least has weaker consequences than would have been expected. This leads one to expect that four-dimensional supergravity theories diverge at three loops. Some general issues about the meaning of finiteness in nonrenormalizable theories are discussed. In particular, we discuss the use of field redefinitions, the generalization of wave function renormalization to nonrenormalizable theories, and whether counterterms should be used in calculations in finite theories. (orig.) 6. Zeta-function regularization of the quantum fluctuations around the Yang-Mills pseudoparticle International Nuclear Information System (INIS) Chadha, S.; Di Vecchia, P.; D'Adda, A.; Nicodemi, F. 1977-01-01 The hypersphere stereographic projection and the zeta-function regularization procedure are used to compute the one loop correction around the Yang-Mills pseudoparticle with scalars and fermions in an arbitrary representation of the SU(2) gauge group. (Auth.) 7. Liouville and Painleve equations and Yang--Mills strings International Nuclear Information System (INIS) Saclioglu, C.K. 1984-01-01 Stringlike solutions of the self-dual Yang--Mills equations (dimensionally reduced to R 2 ) are sought. A multistring Ansatz results in the sinh--Gordon and Liouville equations. According to a general theorem, the solutions must be either real and singular and have infinite action, or complex and nonsingular, with zero action. In the Liouville case, explicit arbitrarily separated n-string solutions of both classes are given. The magnetic flux for these solutions is found to be the Chern class of a Kaehler manifold, and it consequently assumes quantized values 4πn/e. The axisymmetric version of the sinh--Gordon is solved by the third Painleve transcendent P 3 , using the results on P 3 by Wu et al. [Phys. Rev. B 13, 316 (1976)] and McCoy et al. [J. Math. Phys. 18, 10 (1977)]. The axisymmetric case can be cast into the Ernst equation framework for the generation of further solutions. In the Appendix, the Euclideanized Ernst equation is shown to give self-dual Gibbons--Hawking gravitational instantons 8. Singular points in moduli spaces of Yang-Mills fields International Nuclear Information System (INIS) Ticciati, R. 1984-01-01 This thesis investigates the metric dependence of the moduli spaces of Yang-Mills fields of an SU(2) principal bundle P with chern number -1 over a four-dimensional, simply-connected, oriented, compact smooth manifold M with positive definite intersection form. The purpose of this investigation is to suggest that the surgery class of the moduli space of irreducible connections is, for a generic metric, a Z 2 topological invariant of the smooth structure on M. There are three main parts. The first two parts are local analysis of singular points in the moduli spaces. The last part is global. The first part shows that the set of metrics for which the moduli space of irreducible connections has only non-degenerate singularities has codimension at least one in the space of all metrics. The second part shows that, for a one-parameter family of moduli spaces in a direction transverse to the set of metrics for which the moduli spaces have singularities, passing through a non-degenerate singularity of the simplest type changes the moduli space by a cobordism. The third part shows that generic one-parameter families of metrics give rise to six-dimensional manifolds, the corresponding family of moduli spaces of irreducible connections. It is shown that when M is homeomorphic to S 4 the six-dimensional manifold is a proper cobordism, thus establishing the independence of the surgery class of the moduli space on the metric on M 9. Perturbations of the Yang-Mills field in the universe International Nuclear Information System (INIS) Zhao Wen 2009-01-01 It has been suggested that the Yang-Mills (YM) field can be a kind of candidate for the inflationary field at high energy scales or dark energy at very low energy scales, which can naturally give the equation of state -1 -2 , from which it follows that the equation of state of the YM field always goes to -1, independent of the initial conditions. By solving the first order Einstein equations and the YM field equations, we find that in the YM field inflationary models, the scale-invariant primordial perturbation power spectrum cannot be generated. Therefore, only using this kind of YM field is not enough to account for inflationary sources. However, as a kind of candidate for dark energy, the YM field has the 'sound speed' cs 2 S = -1/3 < 0, which makes the perturbation oe have a damping behavior at large scales. This provides a way to distinguish the YM field dark energy models from other kinds of models. (research papers) 10. The limit of the Yang-Mills-Higgs flow on Higgs bundles OpenAIRE Li, Jiayu; Zhang, Xi 2014-01-01 In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle$(E, H_{0})$over a compact K\\"ahler manifold$(M, \\omega )\$. We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorphic to the double dual of the graded Higgs sheaves associated to the Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle.
11. Stiefel-Skyrem-Higgs models, their classical static solutions and Yang-Mills-Higgs monopoles
International Nuclear Information System (INIS)
Dobrev, V.K.
1981-07-01
A new series of models is introduced by adding Higgs fields to the earlier proposed euclidean four-dimensional Skyrme-like models with Yang-Mills composite fields constructed from Stiefel manifold-valued fields. The classical static versions of these models are discussed. The connection with the monopole solutions of the Yang-Mills-Higgs models in the Prasad-Sommerfield limit is pointed out and the BPS monopole is reobtained as an example. (author)
12. Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic
International Nuclear Information System (INIS)
Douglas, Michael R.; Karp, Robert L.; Lukic, Sergio; Reinbacher, Rene
2007-01-01
We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P 2 . In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic
13. Moduli dynamics as a predictive tool for thermal maximally supersymmetric Yang-Mills at large N
Energy Technology Data Exchange (ETDEWEB)
Morita, Takeshi [Department of Physics, Shizuoka University,836 Ohya, Suruga-ku, Shizuoka 422-8529 (Japan); Department of Physics and Astronomy, University of Kentucky,Lexington, KY 40506 (United States); Shiba, Shotaro [Maskawa Institute for Science and Culture, Kyoto Sangyo University,Kamigamo-Motoyama, Kita-ku, Kyoto 603-8555 (Japan); Wiseman, Toby [Theoretical Physics Group, Blackett Laboratory, Imperial College,Exhibition Road, London SW7 2AZ (United Kingdom); Withers, Benjamin [Mathematical Sciences and STAG Research Centre, University of Southampton,Highfield, Southampton SO17 1BJ (United Kingdom)
2015-07-09
Maximally supersymmetric (p+1)-dimensional Yang-Mills theory at large N and finite temperature, with possibly compact spatial directions, has a rich phase structure. Strongly coupled phases may have holographic descriptions as black branes in various string duality frames, or there may be no gravity dual. In this paper we provide tools in the gauge theory which give a simple and unified picture of the various strongly coupled phases, and transitions between them. Building on our previous work we consider the effective theory describing the moduli of the gauge theory, which can be computed precisely when it is weakly coupled far out on the Coulomb branch. Whilst for perturbation theory naive extrapolation from weak coupling to strong gives little information, for this moduli theory naive extrapolation from its weakly to its strongly coupled regime appears to encode a surprising amount of information about the various strongly coupled phases. We argue it encodes not only the parametric form of thermodynamic quantities for these strongly coupled phases, but also certain transcendental factors with a geometric origin, and allows one to deduce transitions between the phases. We emphasise it also gives predictions for the behaviour of other observables in these phases.
14. Moduli dynamics as a predictive tool for thermal maximally supersymmetric Yang-Mills at large N
International Nuclear Information System (INIS)
Morita, Takeshi; Shiba, Shotaro; Wiseman, Toby; Withers, Benjamin
2015-01-01
Maximally supersymmetric (p+1)-dimensional Yang-Mills theory at large N and finite temperature, with possibly compact spatial directions, has a rich phase structure. Strongly coupled phases may have holographic descriptions as black branes in various string duality frames, or there may be no gravity dual. In this paper we provide tools in the gauge theory which give a simple and unified picture of the various strongly coupled phases, and transitions between them. Building on our previous work we consider the effective theory describing the moduli of the gauge theory, which can be computed precisely when it is weakly coupled far out on the Coulomb branch. Whilst for perturbation theory naive extrapolation from weak coupling to strong gives little information, for this moduli theory naive extrapolation from its weakly to its strongly coupled regime appears to encode a surprising amount of information about the various strongly coupled phases. We argue it encodes not only the parametric form of thermodynamic quantities for these strongly coupled phases, but also certain transcendental factors with a geometric origin, and allows one to deduce transitions between the phases. We emphasise it also gives predictions for the behaviour of other observables in these phases.
15. S-duality, deconstruction and confinement for a marginal deformation of N=4 SUSY Yang-Mills
International Nuclear Information System (INIS)
Dorey, Nick
2004-01-01
We study an exactly marginal deformation of N=4 SUSY Yang-Mills with gauge group U(N) using field theory and string theory methods. The classical theory has a Higgs branch for rational values of the deformation parameter. We argue that the quantum theory also has an S-dual confining branch which cannot be seen classically. The low-energy effective theory on these branches is a six-dimensional non-commutative gauge theory with sixteen supercharges. Confinement of magnetic and electric charges, on the Higgs and confining branches respectively, occurs due to the formation of BPS-saturated strings in the low energy theory. The results also suggest a new way of deconstructing Little String Theory as a large-N limit of a confining gauge theory in four dimensions. (author)
16. On skein relations in class S theories
International Nuclear Information System (INIS)
Tachikawa, Yuji; Watanabe, Noriaki
2015-01-01
Loop operators of a class S theory arise from networks on the corresponding Riemann surface, and their operator product expansions are given in terms of the skein relations, that we describe in detail in the case of class S theories of type A. As two applications, we explicitly determine networks corresponding to dyonic loops of N=4SU(3) super Yang-Mills, and compute the superconformal index of a nontrivial network operator of the T 3 theory.
17. Supergravity duals of supersymmetric four dimensional gauge theories
Energy Technology Data Exchange (ETDEWEB)
Bigazzi, F [Abdus Salam International Centre for Theoretical Physics, Trieste (Italy); Cotrone, A L [Centre de Physique Theorique, Ecole Polytechnique, Palaiseau Cedex (France); [INFN, Rome (Italy); Petrini, M [Centre de Physique Theorique, Ecole Polytechnique, Palaiseau (France); Zaffaroni, A [Universita di Milano-Bicocca and INFN, Milan (Italy)
2002-03-01
This article contains an overview of some recent attempts of understanding supergravity and string duals of four dimensional gauge theories using the AdS/CFT correspondence. We discuss the general philosophy underlying the various ways to realize Super Yang-Mills theories in terms of systems of branes. We then review some of the existing duals for N=2 and N=1 theories. We also discuss differences and similarities with realistic theories. (author)
18. Renormalization of Yang-Mills theory developed around an instanton
International Nuclear Information System (INIS)
Rouet, A.
1978-09-01
Faddeev-Popov like -but space time independent-ghosts have been introduced together with the usual ones to deal with the hudge Jacobian mixing the translation and dilatation zero modes contribution to the usual gauge one
19. Ice limit of Coulomb gauge Yang-Mills theory
International Nuclear Information System (INIS)
Heinzl, T.; Ilderton, A.; Langfeld, K.; Lavelle, M.; McMullan, D.
2008-01-01
In this paper we describe gauge invariant multiquark states generalizing the path integral framework developed by Parrinello, Jona-Lasinio, and Zwanziger to amend the Faddeev-Popov approach. This allows us to produce states such that, in a limit which we call the ice limit, fermions are dressed with glue exclusively from the fundamental modular region associated with Coulomb gauge. The limit can be taken analytically without difficulties, avoiding the Gribov problem. This is illustrated by an unambiguous construction of gauge invariant mesonic states for which we simulate the static quark-antiquark potential.
20. On the ground state of Yang-Mills theory
International Nuclear Information System (INIS)
Bakry, Ahmed S.; Leinweber, Derek B.; Williams, Anthony G.
2011-01-01
Highlights: → The ground state overlap for sets of meson potential trial states is measured. → Non-uniform gluonic distributions are probed via Wilson loop operator. → The locally UV-regulated flux-tube operators can optimize the ground state overlap. - Abstract: We investigate the overlap of the ground state meson potential with sets of mesonic-trial wave functions corresponding to different gluonic distributions. We probe the transverse structure of the flux tube through the creation of non-uniform smearing profiles for the string of glue connecting two color sources in Wilson loop operator. The non-uniformly UV-regulated flux-tube operators are found to optimize the overlap with the ground state and display interesting features in the ground state overlap. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9205904603004456, "perplexity": 1262.4984189378154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145443.63/warc/CC-MAIN-20200221045555-20200221075555-00322.warc.gz"} |
https://www.physicsforums.com/threads/pebble-falling-into-water-drag-force.661851/ | # Pebble falling into water; drag force
1. Jan 1, 2013
### unscientific
1. The problem statement, all variables and given/known data
The drag force experienced by the spherical pebble in water is given by: (0.5)CpwAv2
mass of pebble = 83.8g
radius of pebble = 2 cm
C = 0.7
(a) Write down the differential equation governing pebble's descent. (done)
(b) Derive an expression for terminal velocity and evaluate it. (done)
(c) The pebble is dropped 5m above surface of lake, which is also 5m deep. Show that the pebble travels about 10cm in water before reaching it's terminal velocity.
3. The attempt at a solution
Part (a)
Part (b)
Part (c)
2. Jan 1, 2013
### Staff: Mentor
In your exact analysis, the ball will never reach its terminal velocity - but it will approach it with a typical timescale, and you can determine this and show what it does not travel more than 10cm before it reaches a velocity very close to the terminal velocity.
3. Jan 1, 2013
### unscientific
Is there anything wrong with my working in part (b)?? I substituted for v2 using the differential equation..
Last edited: Jan 1, 2013
4. Jan 1, 2013
### Staff: Mentor
I don't see any result in c, and I don't understand what you are doing there.
5. Jan 1, 2013
### unscientific
sorry i meant part (b)!
6. Jan 1, 2013
### SteamKing
Staff Emeritus
1. Are you sure that C is the drag coefficient which uses the entire surface area of the pebble or just the cross-sectional area of the pebble?
2. You recognize that the acceleration of the pebble is x double dot. What about the velocity of the pebble? Can't the velocity also be represented in terms of x?
7. Jan 1, 2013
8. Jan 1, 2013
### haruspex
Sorry, misunderstood the point you were making (and for some reason it's not letting me edit the post).
9. Jan 1, 2013
### Staff: Mentor
You need to substitute x double dot = dv/dt, and solve for the velocity vs time first. Also, what happened to the buoyant force?
10. Jan 2, 2013
### SteamKing
Staff Emeritus
The point about drag coefficients is that they are usually based on the cross sectional or frontal area of the body, not the total surface area.
The OP has used (4/3)pi*r^2, which is the total area of the pebble.
11. Jan 2, 2013
### haruspex
That's not how I read it. Seems to have written the mass as (4/3)pi*r^3*density, then cancelled one r and π with the expression for cross-sectional area. Not that there was much point in doing that: the mass is given, the density is not.
12. Jan 2, 2013
### unscientific
sorry, density of pebble is 2.5g/cm3!
13. Jan 3, 2013
### unscientific
I'm more concerned about the integration part; to find the work done against resistance. Is there anything wrong with that?
14. Jan 3, 2013
### haruspex
When you lost k from your equations, you might have realised you were off course. K must feature in the answer. There are errors further along, but they're not relevant to finding a correct solution.
i don't think you can hope to integrate v2dx straight off. Work with the forces and get the differential equation for the acceleration.
15. Jan 5, 2013
### unscientific
There are 2 ways to approach this problem:
Method 1
Substitute v2 using the differential equation and integrate; which was what I did but I'm not sure what's wrong with my working...
Method 2
a) Solve the differential equation to obtain v in terms of x. show that when x = 0.1 m, v = vterm
16. Jan 5, 2013
### haruspex
As I said, that substitution lost k, and as soon as that happened you were going nowhere. At best you'd end up with tautology. As it happens, you must have made some other error, but I don't think it's interesting to figure out what.
You have $\ddot{x}=\dot{v}=g-kv^2$. You could solve that to find v in terms of t, but there is a trick to eliminate t first and solve for v in terms of x. Is that what you mean?
17. Jan 5, 2013
### Staff: Mentor
Guys,
You've left out the buoyant force on the pebble.
Chet
18. Jan 5, 2013
### haruspex
Good catch!
19. Jan 6, 2013
### unscientific
yeah. replace dv/dt by v(dv/dx) then solve the diferential equation, putting initial condition as v = √2gh.
then substitute v = vterm to find that x = 0.1 m...
20. Jan 6, 2013
### unscientific
Ignore the buoyant force on the pebble.
Similar Discussions: Pebble falling into water; drag force | {"extraction_info": {"found_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": 0.8158007264137268, "perplexity": 1510.1039526967686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1510934806465.90/warc/CC-MAIN-20171122050455-20171122070455-00382.warc.gz"} |
https://mathhelpboards.com/threads/multiplication-of-fractions-destiny-c-sweets-question-on-facebook.5106/ | # Multiplication of Fractions (Destiny C Sweet's Question on Facebook)
#### Sudharaka
##### Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Destiny C Sweet on Facebook writes:
Can someone take a look at this mixed number multiplication problem below... and explain to me how they got this answer? When you work the problem out find the product and reduce to the lowest terms...
$4\frac{3}{4}\times 9\frac{4}{8}=45\frac{1}{8}$
#### Sudharaka
##### Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Destiny C Sweet on Facebook writes:
Can someone take a look at this mixed number multiplication problem below... and explain to me how they got this answer? When you work the problem out find the product and reduce to the lowest terms...
$4\frac{3}{4}\times 9\frac{4}{8}=45\frac{1}{8}$
Hi Destiny,
First we convert the mixed fractions into improper fractions and then do the multiplication. Finally we convert the resulting improper fraction into a mixed fraction. For learn how to convert mixed fractions into improper fractions you might find >>this<< helpful.
\begin{eqnarray}
4\frac{3}{4}\times 9\frac{4}{8}&=&4\frac{3}{4}\times 9\frac{1}{2}\\
&=&\frac{19}{4}\times\frac{19}{2}\\
&=&\frac{361}{8}\\
&=&45\frac{1}{8}
\end{eqnarray} | {"extraction_info": {"found_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": 0.8925968408584595, "perplexity": 2314.0880669147473}, "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-2021-17/segments/1618038085599.55/warc/CC-MAIN-20210415125840-20210415155840-00298.warc.gz"} |
http://philpapers.org/s/Modal%20logic | ## Search results for 'Modal logic' (try it on Scholar)
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1. Tapio Korte, Ari Maunu & Tuomo Aho (2009). Modal Logic From Kant to Possible Worlds Semantics. In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.score: 93.0
This chapter begins with a discussion of Kant's theory of judgment-forms. It argues that it is not true in Kant's logic that assertoric or apodeictic judgments imply problematic ones, in the manner in which necessity and truth imply possibility in even the weakest systems of modern modal logic. The chapter then discusses theories of judgment-form after Kant, the theory of quantification, Frege's Begriffsschrift, C. I. Lewis and the beginnings of modern modal logic, the proof-theoretic approach (...)
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2. Christopher Menzel (1991). The True Modal Logic. Journal of Philosophical Logic 20 (4):331 - 374.score: 93.0
In this paper, I first trace the course of Prior's struggles with the concepts and phenomena of modality and the reasoning that led him to his own rather peculiar modal logic Q. I find myself in almost complete agreement with Prior's intuitions and the arguments that rest upon them. However, I will argue that those intuitions do not of themselves lead to Q, but that one must also accept a certain picture of what it is for a proposition (...)
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3. Robert Demolombe, Andreas Herzig & Ivan Varzinczak (2003). Regression in Modal Logic. Journal of Applied Non-Classical Logic 13 (2):165-185.score: 93.0
In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.
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4. Rohan French & Lloyd Humberstone (2009). Partial Confirmation of a Conjecture on the Boxdot Translation in Modal Logic. Australasian Journal of Logic 7:56-61.score: 93.0
The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture.
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5. Sonia Roca-Royes (2011). Essentialism Vis-à-Vis Possibilia, Modal Logic, and Necessitism. Philosophy Compass 6 (1):54-64.score: 90.0
Pace Necessitism – roughly, the view that existence is not contingent – essential properties provide necessary conditions for the existence of objects. Sufficiency properties, by contrast, provide sufficient conditions, and individual essences provide necessary and sufficient conditions. This paper explains how these kinds of properties can be used to illuminate the ontological status of merely possible objects and to construct a respectable possibilist ontology. The paper also reviews two points of interaction between essentialism and modal logic. First, we (...)
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6. Christopher Menzel (1993). Singular Propositions and Modal Logic. Philosophical Topics 21 (2):113-148.score: 90.0
According to many actualists, propositions, singular propositions in particular, are structurally complex, that is, roughly, (i) they have, in some sense, an internal structure that corresponds rather directly to the syntactic structure of the sentences that express them, and (ii) the metaphysical components, or constituents, of that structure are the semantic values — the meanings — of the corresponding syntactic components of those sentences. Given that reference is "direct", i.e., that the meaning of a name is its denotation, an apparent (...)
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7. Phillip Bricker (1989). Quantified Modal Logic and the Plural De Re. Midwest Studies in Philosophy 14 (1):372-394.score: 90.0
Modal sentences of the form "every F might be G" and "some F must be G" have a threefold ambiguity. in addition to the familiar readings "de dicto" and "de re", there is a third reading on which they are examples of the "plural de re": they attribute a modal property to the F's plurally in a way that cannot in general be reduced to an attribution of modal properties to the individual F's. The plural "de re" (...)
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8. Boudewijn de Bruin (2008). A Note on List's Modal Logic of Republican Freedom. Politics, Philosophy and Economics 7 (3):341-349.score: 90.0
In this note, I show how Christian List's modal logic of republican freedom (as published in this journal in 2006) can be extended (1) to grasp the differences between liberal freedom (noninterference) and republican freedom (non-domination) in terms of two purely logical axioms and (2) to cover a more recent definition of republican freedom in terms of arbitrary interference' that gains popularity in the literature.
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9. score: 90.0
Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises (...)
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10. score: 90.0
The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML s is not equivalent to any formula of basic modal logic (...)
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11. score: 90.0
Modal logics have in the past been used as a unifying framework for the minimality semantics used in defeasible inference, conditional logic, and belief revision. The main aim of the present paper is to add adaptive logics, a general framework for a wide range of defeasible reasoning forms developed by Diderik Batens and his co-workers, to the growing list of formalisms that can be studied with the tools and methods of contemporary modal logic. By characterising the (...)
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12. Susanne Bobzien (1986). Die Stoische Modallogik (Stoic Modal Logic). Königshausen & Neumann.score: 90.0
ABSTRACT: Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the modalities; modalities as properties of propositions; contingent propositions; the relation (...)
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13. Pierluigi Minari (2012). Infinitary Modal Logic and Generalized Kripke Semantics. Annali Del Dipartimento di Filosofia 17 (1):135-166.score: 87.0
This paper deals with the infinitary modal propositional logic Kω1, featuring countable disjunctions and conjunc- tions. It is known that the natural infinitary extension LK.
No categories
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14. Susanne Bobzien (1993). Chrysippus' Modal Logic and Its Relation to Philo and Diodorus. In K. Doering & Th Ebert (eds.), Dialektiker und Stoiker. Franz Steiner.score: 82.0
ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and (...) theorems, and to make clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)
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15. Pietro Galliani (forthcoming). The Dynamification of Modal Dependence Logic. Journal of Logic, Language and Information:1-27.score: 81.0
We examine the transitions between sets of possible worlds described by the compositional semantics of Modal Dependence Logic, and we use them as the basis for a dynamic version of this logic. We give a game theoretic semantics, a (compositional) transition semantics and a power game semantics for this new variant of modal Dependence Logic, and we prove their equivalence; and furthermore, we examine a few of the properties of this formalism and show that (...) Dependence Logic can be recovered from it by reasoning in terms of reachability. Then we show how we can generalize this approach to a very general formalism for reasoning about transformations between pointed Kripke models. (shrink)
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16. Joanna Golinska-Pilarek, Emilio Munoz Velasco & Angel Mora (2011). A New Deduction System for Deciding Validity in Modal Logic K. Logic Journal of IGPL 19 (2): 425-434.score: 78.0
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17. Joanna Golinska-Pilarek, Emilio Munoz-Velasco & Angel Mora (2012). Relational Dual Tableau Decision Procedure for Modal Logic K. Logic Journal of IGPL 20 (4):747-756.score: 78.0
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18. Peter Lohmann & Heribert Vollmer (2013). Complexity Results for Modal Dependence Logic. Studia Logica 101 (2):343-366.score: 78.0
Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p 1, . . . , p n , = (p 1, . . . , p n-1, p n ) intuitively states that the value of p n is determined by those of p 1, . . . , p n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence (...)
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19. score: 75.0
Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are (...)
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20. Angel Mora, Emilio Munoz Velasco & Joanna Golinska-Pilarek (2011). Implementing a Relational Theorem Prover for Modal Logic K. International Journal of Computer Mathematics 88 (9):1869-1884.score: 75.0
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21. score: 74.0
This landmark work provides a systematic introduction to systems of modal logic and stands as the first presentation of what have become central ideas in philosophy of language and metaphysics, from the "new theory of reference" and non-linguistic necessity and essentialism to "Kripke semantics.".
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22. score: 74.0
This long-awaited book replaces not one but both of Hughes and Cresswell's two previous classic studies of modal logic: An Introduction to Modal Logic and A Companion to Modal Logic . A New Introduction to Modal Logic has been completely rewritten by the authors to incorporate all the developments that have taken place since 1968 both in modal propositional logical and modal predicate logic, but without sacrificing the clarity of (...)
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23. score: 74.0
The Handbook of Modal Logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. The articles survey the field from a wide variety of perspectives: the underling theory is explored in depth, modern computational approaches are treated, and six major applications areas of modal logic (in Mathematics, Computer Science, Artificial Intelligence, Linguistics, Game Theory, and Philosophy) are surveyed. The book contains both (...)
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24. Brian F. Chellas (1980). Modal Logic: An Introduction. Cambridge University Press.score: 74.0
A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical (...)
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25. James W. Garson (2006). Modal Logic for Philosophers. Cambridge University Press.score: 74.0
Designed for use by philosophy students, this book provides an accessible, yet technically sound treatment of modal logic and its philosophical applications. Every effort has been made to simplify the presentation by using diagrams in place of more complex mathematical apparatus. These and other innovations provide philosophers with easy access to a rich variety of topics in modal logic, including a full coverage of quantified modal logic, non-rigid designators, definite descriptions, and the de-re de-dictio (...)
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26. Richard Patterson (1995). Aristotle's Modal Logic: Essence and Entailment in the Organon. Cambridge University Press.score: 74.0
Aristotle's Modal Logic presents a very new interpretation of Aristotle's logic by arguing that a proper understanding of the system depends on an appreciation of its connection to the metaphysics. Richard Patterson develops three striking theses in the book. First, there is a fundamental connection between Aristotle's logic of possibility and necessity, and his metaphysics, and that this connection extends far beyond the widely recognised tie to scientific demonstration and relates to the more basic distinction between (...)
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27. Patrick Blackburn (2002). Modal Logic. Cambridge University Press.score: 74.0
This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.
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28. score: 74.0
Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.
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29. Alexander Chagrov (1997). Modal Logic. Oxford University Press.score: 74.0
For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators. It starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantic and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge (...)
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30. Sally Popkorn (1994). First Steps in Modal Logic. Cambridge University Press.score: 74.0
This is a first course in propositional modal logic, suitable for mathematicians, computer scientists and philosophers. Emphasis is placed on semantic aspects, in the form of labelled transition structures, rather than on proof theory. The book covers all the basic material - propositional languages, semantics and correspondence results, proof systems and completeness results - as well as some topics not usually covered in a modal logic course. It is written from a mathematical standpoint. To help the (...)
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31. George Boolos (1979). The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press.score: 74.0
The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a (...)
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32. score: 74.0
This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.
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33. Nino B. Cocchiarella (2008). Modal Logic: An Introduction to its Syntax and Semantics. Oxford University Press.score: 66.0
In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight.
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34. Sebastian Enqvist (forthcoming). A General Lindström Theorem for Some Normal Modal Logics. Logica Universalis:1-32.score: 66.0
There are several known Lindström-style characterization results for basic modal logic. This paper proves a generic Lindström theorem that covers any normal modal logic corresponding to a class of Kripke frames definable by a set of formulas called strict universal Horn formulas. The result is a generalization of a recent characterization of modal logic with the global modality. A negative result is also proved in an appendix showing that the result cannot be strengthened to (...)
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35. Sebastian Enqvist (2009). Interrogative Belief Revision in Modal Logic. Journal of Philosophical Logic 38 (5):527 - 548.score: 63.0
The well known AGM framework for belief revision has recently been extended to include a model of the research agenda of the agent, i.e. a set of questions to which the agent wishes to find answers (Olsson & Westlund in Erkenntnis , 65 , 165–183, 2006 ). The resulting model has later come to be called interrogative belief revision . While belief revision has been studied extensively from the point of view of modal logic, so far interrogative belief (...)
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36. Hirohiko Kushida (forthcoming). The Modal Logic of Gödel Sentences. Journal of Philosophical Logic.score: 63.0
The modal logic of Gödel sentences, termed as GS , is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS . GS is also an extended system of the (...) of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005 ). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic. (shrink)
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37. Torben Braüner (2002). Modal Logic, Truth, and the Master Modality. Journal of Philosophical Logic 31 (4):359-386.score: 63.0
In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a so-called master modality. Moreover, we explore a connection between the master modality (...)
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38. Guram Bezhanishvili, Leo Esakia & David Gabelaia (2010). The Modal Logic of Stone Spaces: Diamond as Derivative. Review of Symbolic Logic 3 (1):26-40.score: 63.0
We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces.
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39. Wiebe Van Der Hoek & Maarten De Rijke (1993). Generalized Quantifiers and Modal Logic. Journal of Logic, Language and Information 2 (1).score: 63.0
We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness (...)
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40. John P. Burgess (1999). Which Modal Logic Is the Right One? Notre Dame Journal of Formal Logic 40 (1):81-93.score: 63.0
The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logic must (...)
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41. Esther Ramharter & Christian Gottschall (2011). Peirce's Search for a Graphical Modal Logic (Propositional Part). History and Philosophy of Logic 32 (2):153 - 176.score: 63.0
This paper deals with modality in Peirce's existential graphs, as expressed in his gamma and tinctured systems. We aim at showing that there were two philosophically motivated decisions of Peirce's that, in the end, hindered him from producing a modern, conclusive system of modal logic. Finally, we propose emendations and modifications to Peirce's modal graphical tinctured systems and to their underlying ideas that will produce modern modal systems.
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42. G. Aldo Antonelli & Richmond H. Thomason (2002). Representability in Second-Order Propositional Poly-Modal Logic. Journal of Symbolic Logic 67 (3):1039-1054.score: 63.0
A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.
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43. Henrik Lagerlund (2009). Avicenna and Ūsī on Modal Logic. History and Philosophy of Logic 30 (3):227-239.score: 63.0
In this article, the author studies some central concepts in Avicenna's and sī's modal logics as presented in Avicenna's Al-Ish r t wa'l Tan īh t ( Pointers and Reminders ) and in sī's commentary. In this work, Avicenna introduces some remarkable distinctions in order to interpret Aristotle's modal syllogistic in the Prior Analytics . The author outlines a new interpretation of absolute sentences as temporally indefinite sentences and argues on the basis of this that Avicenna seems to (...)
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44. M. McKeon (2005). A Defense of the Kripkean Account of Logical Truth in First-Order Modal Logic. Journal of Philosophical Logic 34 (3):305 - 326.score: 63.0
This paper responds to criticism of the Kripkean account of logical truth in first-order modal logic. The criticism, largely ignored in the literature, claims that when the box and diamond are interpreted as the logical modality operators, the Kripkean account is extensionally incorrect because it fails to reflect the fact that all sentences stating truths about what is logically possible are themselves logically necessary. I defend the Kripkean account by arguing that some true sentences about logical possibility are (...)
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45. Kosta Došen (1992). Modal Logic as Metalogic. Journal of Logic, Language and Information 1 (3):173-201.score: 63.0
The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. (...)
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46. Charles B. Cross (1993). From Worlds to Probabilities: A Probabilistic Semantics for Modal Logic. Journal of Philosophical Logic 22 (2):169 - 192.score: 63.0
I develop a probabilistic semantics for modal logic that generalizes the quantificational apparatus of Kripke models. Soundness and completeness theorems are proved for propositional M, B, S4, and S5. My semantics formalizes the idea that uncertainty about modal claims like "Possibly-A" arises from the fact that thought experiments which test the intelligibility of A may be inconclusive for a given agent. On this view, an agent who is uncertain about "Possibly-A" assigns at least as much credibility to (...)
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47. Paul Égré (2005). The Knower Paradox in the Light of Provability Interpretations of Modal Logic. Journal of Logic, Language and Information 14 (1).score: 63.0
This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in (...)
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48. Balder ten Cate (2006). Expressivity of Second Order Propositional Modal Logic. Journal of Philosophical Logic 35 (2).score: 63.0
We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic (...)
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49. Tero Tulenheimo (2009). Hybrid Logic Meets If Modal Logic. Journal of Logic, Language and Information 18 (4).score: 63.0
The hybrid logic and the independence friendly modal logic IFML are compared for their expressive powers. We introduce a logic IFML c having a non-standard syntax and a compositional semantics; in terms of this logic a syntactic fragment of IFML is singled out, denoted IFML c . (In the Appendix it is shown that the game-theoretic semantics of IFML c coincides with the compositional semantics of IFML c .) The hybrid logic is proven to (...)
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50. Balder ten Cate (2006). Expressivity of Second Order Propositional Modal Logic. Journal of Philosophical Logic 35 (2):209 - 223.score: 63.0
We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic (...)
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51. Johan Van Benthem, Patrick Girard & Olivier Roy (2009). Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences. Journal of Philosophical Logic 38 (1):83 - 125.score: 63.0
This paper presents a new modal logic for ceteris paribus preferences understood in the sense of "all other things being equal". This reading goes back to the seminal work of Von Wright in the early 1960's and has returned in computer science in the 1990' s and in more abstract "dependency logics" today. We show how it differs from ceteris paribus as "all other things being normal", which is used in contexts with preference defeaters. We provide a semantic (...)
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52. Natasha Kurtonina (1998). Categorial Inference and Modal Logic. Journal of Logic, Language and Information 7 (4):399-411.score: 63.0
This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.
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53. Mary C. MacLeod & Peter K. Schotch (2000). Remarks on the Modal Logic of Henry Bradford Smith. Journal of Philosophical Logic 29 (6):603-615.score: 63.0
H. B. Smith, Professor of Philosophy at the influential Pennsylvania School was (roughly) a contemporary of C. I. Lewis who was similarly interested in a proper account of implication. His research also led him into the study of modal logic but in a different direction than Lewis was led. His account of modal logic does not lend itself as readily as Lewis' to the received possible worlds semantics, so that the Smith approach was a (...)
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54. Maarten de Rijke (1998). A System of Dynamic Modal Logic. Journal of Philosophical Logic 27 (2):109-142.score: 63.0
In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means (...)
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55. Ronald Fagin (1994). A Quantitative Analysis of Modal Logic. Journal of Symbolic Logic 59 (1):209-252.score: 63.0
We do a quantitative analysis of modal logic. For example, for each Kripke structure M, we study the least ordinal μ such that for each state of M, the beliefs of up to level μ characterize the agents' beliefs (that is, there is only one way to extend these beliefs to higher levels). As another example, we show the equivalence of three conditions, that on the face of it look quite different, for what it means to say that (...)
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56. Eric Rosen (1997). Modal Logic Over Finite Structures. Journal of Logic, Language and Information 6 (4):427-439.score: 63.0
We investigate properties of propositional modal logic over the classof finite structures. In particular, we show that certain knownpreservation theorems remain true over this class. We prove that aclass of finite models is defined by a first-order sentence and closedunder bisimulations if and only if it is definable by a modal formula.We also prove that a class of finite models defined by a modal formulais closed under extensions if and only if it is defined by a (...)
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57. Ernst Zimmermann (2003). Elementary Definability and Completeness in General and Positive Modal Logic. Journal of Logic, Language and Information 12 (1):99-117.score: 63.0
The paper generalises Goldblatt's completeness proof for Lemmon–Scott formulas to various modal propositional logics without classical negation and without ex falso, up to positive modal logic, where conjunction and disjunction, andwhere necessity and possibility are respectively independent.Further the paper proves definability theorems for Lemmon–Scottformulas, which hold even in modal propositional languages without negation and without falsum. Both, the completeness theorem and the definability theoremmake use only of special constructions of relations,like relation products. No second order (...), no general frames are involved. (shrink)
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58. H. Kushida & M. Okada (2003). A Proof-Theoretic Study of the Correspondence of Classical Logic and Modal Logic. Journal of Symbolic Logic 68 (4):1403-1414.score: 63.0
It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the (...)
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59. Heinrich Wansing (1999). Displaying the Modal Logic of Consistency. Journal of Symbolic Logic 64 (4):1573-1590.score: 63.0
It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.
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60. Tim Fernando (1999). A Modal Logic for Non-Deterministic Discourse Processing. Journal of Logic, Language and Information 8 (4):445-468.score: 63.0
A modal logic for translating a sequence of English sentences to a sequence of logical forms is presented, characterized by Kripke models with points formed from input/output sequences, and valuations determined by entailment relations. Previous approaches based (to one degree or another) on Quantified Dynamic Logic are embeddable within it. Applications to presupposition and ambiguity are described, and decision procedures and axiomatizations supplied.
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61. Lou Goble (2000). An Incomplete Relevant Modal Logic. Journal of Philosophical Logic 29 (1):103-119.score: 63.0
The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley–Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that (...)
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62. Maarten Rijkdee (1998). A System of Dynamic Modal Logic. Journal of Philosophical Logic 27 (2):109-142.score: 63.0
In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means (...)
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63. Yde Venema (1993). Derivation Rules as Anti-Axioms in Modal Logic. Journal of Symbolic Logic 58 (3):1003-1034.score: 63.0
We discuss a negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ+ is the extension of Λ with a set of non-ξ rules, then (...)
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64. Maarten De Rijke (1998). A System of Dynamic Modal Logic. Journal of Philosophical Logic 27 (2):109 - 142.score: 63.0
In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means (...)
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65. score: 62.0
A powerful challenge to some highly influential theories, this book offers a thorough critical exposition of modal realism, the philosophical doctrine that many possible worlds exist of which our own universe is just one. Chihara challenges this claim and offers a new argument for modality without worlds.
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66. Daniel Gallin (1975). Intensional and Higher-Order Modal Logic: With Applications to Montague Semantics. American Elsevier Pub. Co..score: 62.0
CHAPTER 1. INTENSIONAL LOGIC §1. Natural Language and Intensional Logic When we speak of a theory of meaning for a natural language such as English, ...
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67. score: 62.0
INTENSIONAL LOGIC §1. Natural Language and Intensional Logic When we speak of a theory of meaning for a natural language such as English, we have in mind an ...
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68. score: 62.0
Normal propositional modal systems This first chapter has two main aims. One is to give a general account of the propositional modal systems that we shall ...
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69. Maarten Marx (1997). Multi-Dimensional Modal Logic. Kluwer Academic Publishers.score: 62.0
Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi-dimensional. (Our definition of multi ...
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70. J. C. Beall (2003). Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic. Oxford University Press.score: 62.0
Extensively classroom-tested, Possibilities and Paradox provides an accessible and carefully structured introduction to modal and many-valued logic. The authors cover the basic formal frameworks, enlivening the discussion of these different systems of logic by considering their philosophical motivations and implications. Easily accessible to students with no background in the subject, the text features innovative learning aids in each chapter, including exercises that provide hands-on experience, examples that demonstrate the application of concepts, and guides to further reading.
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71. Marcelo E. Coniglio & Newton M. Peron (2013). Modal Extensions of Sub-Classical Logics for Recovering Classical Logic. Logica Universalis 7 (1):71-86.score: 62.0
In this paper we introduce non-normal modal extensions of the sub-classical logics CLoN, CluN and CLaN, in the same way that S0.5 0 extends classical logic. The first modal system is both paraconsistent and paracomplete, while the second one is paraconsistent and the third is paracomplete. Despite being non-normal, these systems are sound and complete for a suitable Kripke semantics. We also show that these systems are appropriate for interpreting □ as “is provable in classical logic”. (...)
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72. Sara Negri (2005). Proof Analysis in Modal Logic. Journal of Philosophical Logic 34 (5-6):507 - 544.score: 61.0
A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel–Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.
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73. Rohan French (2012). Denumerably Many Post-Complete Normal Modal Logics with Propositional Constants. Notre Dame Journal of Formal Logic 53 (4):549-556.score: 61.0
We show that there are denumerably many Post-complete normal modal logics in the language which includes an additional propositional constant. This contrasts with the case when there is no such constant present, for which it is well known that there are only two such logics.
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74. score: 60.0
These lecture notes were composed while teaching a class at Stanford and studying the work of Brian Chellas (Modal Logic: An Introduction, Cambridge: Cambridge University Press, 1980), Robert Goldblatt (Logics of Time and Computation, Stanford: CSLI, 1987), George Hughes and Max Cresswell (An Introduction to Modal Logic, London: Methuen, 1968; A Companion to Modal Logic, London: Methuen, 1984), and E. J. Lemmon (An Introduction to Modal Logic, Oxford: Blackwell, 1977). The Chellas text (...)
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75. Bernard Linsky & Edward N. Zalta (1994). In Defense of the Simplest Quantified Modal Logic. Philosophical Perspectives 8:431-458.score: 60.0
The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these (...)
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76. Sten Lindström & Krister Segerberg (2007). Modal Logic and Philosophy. In Patrick Blackburn & Johan van Benthem (eds.), Handbook of Modal Logic. Elsevier.score: 60.0
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77. score: 60.0
These short notes are intended to supplement the lectures and text ntroduce some of the basic concepts of Modal Logic. The primary goal is to provide students in Philosophy 151 at Stanford University with a study guide that will complement the lectures on modal logic. There are many textbooks that you can consult for more information. The following is a list of some texts (this is not a complete list, but a pointer to books that I (...)
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78. score: 60.0
The paper presents an alternative substitutional semantics for first-order modal logic which, in contrast to traditional substitutional (or truth-value) semantics, allows for a fine-grained explanation of the semantical behavior of the terms from which atomic formulae are composed. In contrast to denotational semantics, which is inherently reference-guided, this semantics supports a non-referential conception of modal truth and does not give rise to the problems which pertain to the philosophical interpretation of objectual domains (concerning, e.g., possibilia or trans-world (...)
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79. score: 60.0
As McKinsey and Tarksi showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the "necessity" operation is modeled by taking the interior of an arbitrary subset of a topological space. in this paper the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect (...)
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80. score: 60.0
This text contains some basic facts about modal logic. For motivation, intuition and examples the reader should consult one of the standard textbooks in the field.
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81. Veikko Rantala (1982). Quantified Modal Logic: Non-Normal Worlds and Propositional Attitudes. Studia Logica 41 (1):41 - 65.score: 60.0
One way to obtain a comprehensive semantics for various systems of modal logic is to use a general notion of non-normal world. In the present article, a general notion of modal system is considered together with a semantic framework provided by such a general notion of non-normal world. Methodologically, the main purpose of this paper is to provide a logical framework for the study of various modalities, notably prepositional attitudes. Some specific systems are studied together with semantics (...)
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82. Giacomo Bonanno (2002). Modal Logic and Game Theory: Two Alternative Approaches. Risk Decision and Policy 7:309-324.score: 60.0
Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other’s rationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely modal logic, can be used to explicitly model both views.
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83. Patrick Blackburn (2001). Modal Logic as Dialogical Logic. Synthese 127 (1-2):57 - 93.score: 60.0
The title reflects my conviction that, viewed semantically,modal logic is fundamentally dialogical; this conviction is based on the key role played by the notion of bisimulation in modal model theory. But this dialogical conception of modal logic does not seem to apply to modal proof theory, which is notoriously messy. Nonetheless, by making use of ideas which trace back to Arthur Prior (notably the use of nominals, special proposition symbols which name worlds) I will (...)
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84. score: 60.0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 BASIC MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3..
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85. Dwayne Raymond (2011). Polarity and Inseparability: The Foundation of the Apodictic Portion of Aristotle's Modal Logic. History and Philosophy of Logic 31 (3):193-218.score: 60.0
Modern logicians have sought to unlock the modal secrets of Aristotle's Syllogistic by assuming a version of essentialism and treating it as a primitive within the semantics. These attempts ultimately distort Aristotle's ontology. None of these approaches make full use of tests found throughout Aristotle's corpus and ancient Greek philosophy. I base a system on Aristotle's tests for things that can never combine (polarity) and things that can never separate (inseparability). The resulting system not only reproduces Aristotle's recorded results (...)
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86. Horacio Arló-Costa & Eric Pacuit (2006). First-Order Classical Modal Logic. Studia Logica 84 (2):171 - 210.score: 60.0
The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan (...)
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87. Edward N. Zalta (1993). A Philosophical Conception of Propositional Modal Logic. Philosophical Topics 21 (2):263-281.score: 60.0
The author revises the formulation of propositional modal logic by interposing a domain of structured propositions between the modal language and the models. Interpretations of the language (i.e., ways of mapping the language into the domain of propositions) are distinguished from models of the domain of propositions (i.e., ways of assigning truth values to propositions at each world), and this contrasts with the traditional formulation. Truth and logical consequence are defined, in the first instance, as properties of, (...)
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88. score: 60.0
This paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic.
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89. John Mccarthy (1997). Modality, Si! Modal Logic, No! Studia Logica 59 (1):29-32.score: 60.0
This article is oriented toward the use of modality in artificial intelligence (AI). An agent must reason about what it or other agents know, believe, want, intend or owe. Referentially opaque modalities are needed and must be formalized correctly. Unfortunately, modal logics seem too limited for many important purposes. This article contains examples of uses of modality for which modal logic seems inadequate.I have no proof that modal logic is inadequate, so I hope modal (...)
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90. Charles G. Morgan (1973). Systems of Modal Logic for Impossible Worlds. Inquiry 16 (1-4):280 – 289.score: 60.0
The intuitive notion behind the usual semantics of most systems of modal logic is that of ?possible worlds?. Loosely speaking, an expression is necessary if and only if it holds in all possible worlds; it is possible if and only if it holds in some possible world. Of course, contradictory expressions turn out to hold in no possible worlds, and logically true expressions turn out to hold in every possible world. A method is presented for transforming standard (...) systems into systems of modal logic for impossible worlds. To each possible world there corresponds an impossible world such that an expression holds in the impossible world if and only if it does not hold in the possible world. One can then talk about such worlds quite consistently, and there seems to be no logical reason for excluding them from consideration. (shrink)
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91. Lloyd Humberstone (2007). Modal Logic for Other-World Agnostics: Neutrality and Halldén Incompleteness. Journal of Philosophical Logic 36 (1):1 - 32.score: 60.0
The logic of ‘elsewhere,’ i.e., of a sentence operator interpretable as attaching to a formula to yield a formula true at a point in a Kripke model just in case the first formula is true at all other points in the model, has been applied in settings in which the points in question represent spatial positions (explaining the use of the word ‘elsewhere’), as well as in the case in which they represent moments of time. This logic is (...)
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92. Giacomo Bonanno (2005). A Simple Modal Logic for Belief Revision. Synthese 147 (2):193 - 228.score: 60.0
We propose a modal logic based on three operators, representing intial beliefs, information and revised beliefs. Three simple axioms are used to provide a sound and complete axiomatization of the qualitative part of Bayes’ rule. Some theorems of this logic are derived concerning the interaction between current beliefs and future beliefs. Information flows and iterated revision are also discussed.
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93. Holger Sturm (2000). Elementary Classes in Basic Modal Logic. Studia Logica 64 (2):193-213.score: 60.0
Dealing with topics of definability, this paper provides some interesting insights into the expressive power of basic modal logic. After some preliminary work it presents an abstract algebraic characterization of the elementary classes of basic modal logic, that is, of the classes of models that are definable by means of (sets of) basic modal formulas. Taking that for a start, the paper further contains characterization results for modal universal classes and modal positive classes.
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94. Timothy Williamson (2010). Barcan Formulas in Second-Order Modal Logic. In Themes From Barcan Marcus. Ontos Verlag.score: 60.0
Second-order logic and modal logic are both, separately, major topics of philosophical discussion. Although both have been criticized by Quine and others, increasingly many philosophers find their strictures uncompelling, and regard both branches of logic as valuable resources for the articulation and investigation of significant issues in logical metaphysics and elsewhere. One might therefore expect some combination of the two sorts of logic to constitute a natural and more comprehensive background logic for metaphysics. So (...)
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95. G. M. Bierman & V. C. V. de Paiva (2000). On an Intuitionistic Modal Logic. Studia Logica 65 (3):383-416.score: 60.0
In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models (...)
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96. Josep Maria Font & Petr Hájek (2002). On Łukasiewicz's Four-Valued Modal Logic. Studia Logica 70 (2):157-182.score: 60.0
ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.
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97. score: 60.0
First-order modal logic, in the usual formulations, is not suf- ficiently expressive, and as a consequence problems like Frege’s morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its (...)
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98. Alberto Zanardo (1996). Branching-Time Logic with Quantification Over Branches: The Point of View of Modal Logic. Journal of Symbolic Logic 61 (1):1-39.score: 60.0
In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (...)
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99. score: 60.0
Horacio Arlo-Costa. First Order Extensions of Classical Systems of Modal Logic: The Role of Barcan Schemas. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8662017583847046, "perplexity": 2155.0227214709344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1368703682988/warc/CC-MAIN-20130516112802-00010-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/137954-how-do-i-show-limit-goes-infinity.html | # Math Help - How do I show that this limit goes to infinity?
1. ## How do I show that this limit goes to infinity?
In the limit:
lim as n--->inf 2*(n!)/4^n = 2*lim as n--->inf (n!)/4^n
how do I show that n! grows faster than 4^n?
This is not a proof question. It's just a "do it" question and I'm asking for the "how".
=infinity
Any input would be greatly appreciated!
2. Originally Posted by s3a
In the limit:
lim as n--->inf 2*(n!)/4^n = 2*lim as n--->inf (n!)/4^n
how do I show that n! grows faster than 4^n?
This is not a proof question. It's just a "do it" question and I'm asking for the "how".
$\lim_{n\rightarrow\infty} \frac{|a_{n+1}|}{|a_{n}|}$ | {"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": 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": 0.8684981465339661, "perplexity": 1628.5072223277857}, "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-2015-22/segments/1432207928078.25/warc/CC-MAIN-20150521113208-00102-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/is-it-possible-to-find-out-the-speed-of-air-given-pressure-and-size-of-exit.646204/ | # Is it possible to find out the speed of air, given pressure and size of exit
1. Oct 22, 2012
### Tugberk
For an experiment, I sent specific pressures of air strams through a tube and I was wondering if there was a way to calculate the speed of the air since I know the pressure and the radius of the tube where it left from.
2. Oct 22, 2012
### Simon Bridge
Look up "free expansion".
Though you could try a related problem - firing a slug from an air-gun.
Knowing the pressure each side of the slug, and the dimensions of the barrel, you know the muzzle-speed of the slug right?
3. Oct 22, 2012
### Staff: Mentor
If the pressure drop is not too large (say less than about 10%), you could approximate the air as being incompressible, and use the basics for the pressure drop/flow rate relationship: The pressure gradient is equal to the friction factor f times ρv2/2 times 4/D, where v is the cross section average velocity and D is the diameter, and where the friction factor is a known correlated function of the Reynolds number (unless the flow is laminar (Re<2100), in which case the friction factor is exactly 16/Re). If the flow is compressible (i.e., the pressure drop is too large), the calculation is a little more difficult, but no big problem. To find out how to do it for a compressible gas, see the oil and gas handbooks.
4. Oct 24, 2012
### Tugberk
Is the p for density, or pressure? Also, how can I multiply by 16/Re, where Re is not a number? Thank you though ^^ :)
5. Oct 24, 2012
### Staff: Mentor
The ρ is density. The Reynolds number Re is equal to the density times the velocity times the diameter divided by the viscosity. You look up the viscosity of air in any book on fluid mechanics or in a handbook, or on line. The density can be calculated from the ideal gas law.
Did you not get the connection that Re is the abbreviation for the Reynolds number?
6. Oct 24, 2012
### Tugberk
I got that Re is Reynolds number, but I'm in high school, this is the first time I've heard of Re :/, also, I don't get how this will help me because the pressure isn't used anywhere, I'm using low pressures, 1.5 Bars, 2 Bars etc., so the change in density is negligible
7. Oct 24, 2012
### Staff: Mentor
You know the pressure at the inlet of the tube and at the outlet of the tube (1 bar), correct? If the flow is laminar, then the pressure drop is related to the cross section average velocity by:
ΔP = η(32v/D2)L
where L is the tube length and η is the viscosity of air at the gas temperature.
Use this equation to solve for v (in consistent units). Then check the Reynolds number to see if the flow is really laminar.
Re = ρvD/η
where ρ is the air density at the average of the inlet and outlet pressures, evaluated using the ideal gas law.
If Re < 2100, you're done. If Re > 2100, then you have to recalculate because the flow is going to be turbulent.
For turbulent flow, the pressure drop is
ΔP = (ρv2/2) (0.079/Re)0.8(4L/D)
where Re is again given by Re = ρvD/η, but with the new velocity you are now calculating for turbulent flow being used in the equation (not the one from the laminar flow calculation)
You now need to use your algebra to solve the above turbulent ΔP equation for the velocity v in terms of ΔP. Then substitute the data into the equation to get v.
8. Oct 24, 2012
### Staff: Mentor
Why not just use a simplified form of Bernoulli's equation? I think what you are suggesting may reduce to it.
9. Oct 24, 2012
Depending on the pressure involved, it is entirely possible that none of this will work. Pretty much all instances of compressed air leaving through a hole are, in fact, compressible. That means that Bernoulli's principle does not apply and it means that things like Hagen-Poiseuille flow do not apply either. The best way to start thinking about this is to have the OP answer a few simple questions.
1) What is the pressure in your reservoir and is it just exhausting to the standard atmosphere?
2) How long is this tube through which you are sending your flow? Is it of constant cross-sectional area?
10. Oct 24, 2012
### Staff: Mentor
If doesn't. This describes the viscous pressure drop, which Bernoulli's equation does not handle. Bernoulli's equation assumes that the system is conservative.
11. Oct 24, 2012
### Staff: Mentor
I guess I made the implicit assumption that the tube is long enough (so that entrance and exit effects are negligible) and that the cross sectional area of the tube is constant. There are, of course, corrections that can be applied to Hagen_Poiseuille (or the turbulent version of Hagen Poiseuille that I gave) to account for entrance and exit effects.
The OP's comment that the pressure is nowhere higher than 1.5 of 2 bars seems to suggest the exit is a atmospheric pressure, and the inlet is at a little higher pressure, and that compressibility can be neglected as a first approximation (although a factor of 2 variation on the pressure is higher than I would like to see if compressibility is being neglected).
12. Oct 24, 2012
### Staff: Mentor
Since we know virtually nothing about the system, how can we even begin to examine losses?
13. Oct 24, 2012
Assuming his reservoir pressure is 2 bar and his exit pressure is around 1 bar, that is more than enough for compressibility to be important. Depending on geometry that could even be supersonic!
14. Oct 24, 2012
### Tugberk
The tube which the air was sent through is approximately 25 cms long. One end is closed and is where the air constantly flows in, at a constant pressure, this is unknown. A valve allows me to control how much air is put in to the 25 cm long tube. By covering the exit, I can build up pressure and read it off of a barometer.
15. Oct 24, 2012
So you cover one end and fill it, then release it?
At any rate, at a pressure ratio (p2/p01) of 0.5, your flow will be choked through the tube, meaning that you can calculate the mass flow pretty quickly and easily if you know your reservoir pressure and temperature and the area of your tube (neglecting any viscous effects). It also means the flow through the tube will be exactly Mach 1. The actual exit velocity will depend on whether the tube exhausts straight to atmosphere or has some kind expansion on the end.
When your reservoir pressure is less than about 1.89 atm, you won't reach sonic conditions in your tube and you will need a tiny bit more effort to figure out mass flow and velocity, but it is still pretty easy as long as you ignore viscosity, which may or may not be a reasonable assumption in your case. If you do make that assumption, it is just a matter of determining Mach number based on the isentropic flow assumption and working from there to get temperature, density and velocity and thus mass flow rate.
Last edited: Oct 24, 2012
16. Oct 24, 2012
### Staff: Mentor
Russ_watters is correct.... the nature of the geometry and the nature of the experiment is still very unclear.
What does this mean?: "One end is closed and is where the air constantly flows in." If the end is closed, how can air flow in?
Please provide more details of the experiment?
What is the diameter of the tube. Where are the valves located? Do you allow the air to escape slowly from the exit value (if there is an exit valve)?
So far, as best I can understand, you pressurize a cylindrical tube with air through one end while the other end is covered. The pressure reaches a certain value, and then you stop adding air. Then you suddenly open the capped end to the atmosphere and let the air escape. You want to find out what? How much air escapes when you remove the cap? How long it takes to return to atmospheric pressure? What the velocity of the air exiting the tube is as a function of time?
17. Oct 25, 2012
### Tugberk
I don't stop adding air, the air is still flowing in to the tube at the same rate. I attached a picture of the system, The little blue pipe is constantly blowing air into the tube at the same velocity/pressure. You can see the barometer and the valve right next to it, The valve allows me to control how much air I let into the main Tube. Than I cover the exit, read the barometer to see how much pressure builds up in the tube, than I let the air flow again, and I want to find out how fast the air moves when it's leaving the tube. The main blue pipe is still supplying air at this point.
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https://lexique.netmath.ca/en/irrational-equation/ | # Irrational Equation
## Irrational Equation
An equation that contains a variable under a radical sign and that can be converted into an algebraic equation.
### Example
The equation $$2x+\sqrt {x}=10$$ is an irrational equation, since it contains a variable $$x$$ under a radical sign and can be converted into the algebraic equation : $$4x^{2} – 41x + 100 = 0$$. To do so, the irrational term is isolated on either side of the equation and then both sides of the equation are squared. | {"extraction_info": {"found_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": 0.8688549995422363, "perplexity": 203.39291178968475}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585353.52/warc/CC-MAIN-20211020214358-20211021004358-00530.warc.gz"} |
http://www.maplesoft.com/support/help/Maple/view.aspx?path=worksheet/managing/OptionsDialogDisplay | Display Tab - Maple Programming Help
Home : Support : Online Help : Configure Maple : Customize the Maple System : worksheet/managing/OptionsDialogDisplay
Options Dialog - Display Tab
In the Options dialog, the Display tab allows you to change the default display settings for typesetting, plots, context menus, insert rules, and more. Click on the links below to browse to an option description.
For more information about the Options dialog and how to apply settings to current and future Maple sessions, see the Options Dialog help page.
Input Display
Control the appearance of expressions by selecting one of two Input display types: Maple notation and 2-D math notation. Changing this option changes the type of display used for any input after the current execution group.
1 From the Input display drop-down list, select Maple Notation or 2-D Math Notation. The default is 2-D Math Notation.
2 Click Apply to Session, Apply Globally, or Cancel.
Maple Notation
Use Maple Notation to display input as Maple syntax.
2-D Math Notation
Use 2-D Math Notation to display expressions in standard math notation.
Output Display
Maple provides four different formats in which to display Maple output.
1 From the Output display drop-down list, select one of Maple Notation, Character Notation, Typeset Notation, or 2-D Math Notation. The default is 2-D Math Notation.
2 Click Apply to Session, Apply Globally, or Cancel.
Future commands that are executed display output using the new setting.
Maple Notation
Use Maple Notation to display the output as Maple syntax.
Character Notation
Use Character Notation to display the output in a 2-D text-based character format. All of the special symbols are created using ASCII characters.
Typeset Notation
Typeset Notation displays the output as it would appear in typeset textbooks. In this case, you cannot edit the output.
2-D Math Notation
Use 2-D Math Notation to display standard math.
Note: In the Classic worksheet interface, you can select and edit subexpressions in the output of this format. Press Enter and the changes are displayed on a new input line, ready to execute.
Typesetting Level
The rule-based typesetting functionality is available only when Typesetting level is set to Extended. For details, see Typesetting and Typesetting Rule Assistant.
Assumed Variables
Set Maple output to indicate which variables have assumptions.
1 From the Assumed variables drop-down list, select No Annotation, Trailing Tildes, or Phrase. The default is Trailing Tildes.
2 Click Apply to Session, Apply Globally, or Cancel.
No Annotation
When No Annotation is selected, the output of the Maple command displays variables with assumptions as regular variables.
Trailing Tildes
When Trailing Tildes is selected, the output of the Maple command displays variables with assumptions by appending a tilde to them.
Phrase
When Phrase is selected, the output of the Maple command displays variables with assumptions as regular variables, but lists any variables that have assumptions.
Enter the Maple command about(variablename) to display a description of the assumption for the variable variablename.
Plot Anti-aliasing
Plot anti-aliasing is a technique for making subtle adjustments in the color of the individual pixels that make up the curve of a plot in order to make the plot appear smoother. By default, this option is Enabled.
Font Anti-aliasing
You can set font display such that characters are rendered with a smoother and rounder appearance.
1 From the Font anti-aliasing drop-down list, select from Default, Enabled, or Disabled.
2 Click Apply to Session, Apply Globally, or Cancel.
3 Exit Maple.
4 Restart Maple. The new setting is enabled.
Plot Display
Display plots in the current worksheet (inline) or in a new plot window.
1 From the Plot display drop-down list, select Inline or Window. The default is Inline.
2 Click Apply to Session, Apply Globally, or Cancel.
Note: This option does not apply to plots inserted with the Insert>Plot>2-D or 3-D menu option. These plots are displayed in the current worksheet.
Default Point Probe Mode
The point probe displays the coordinate information for points on your 2-D plots. Use this option to select the default coordinate information to display.
1 From the Default Point Probe Mode drop-down list, select one of the following options:
• None
• Cursor position
• Nearest datum
• Nearest point on line
2 Click Apply to Session or Apply Globally.
For more information on the point probe modes, see Display 2-D Plot Coordinates.
When copying task template content to another worksheet, you have the option of indicating variables. From the Show task variables on insert drop-down list, select from Always, Only on Naming Conflict, or Never. The default is Only on Naming Conflict.
With the Task content to insert option, you can copy All Content, Standard Content, or Minimal Content into the worksheet.
• All Content inserts all content present in the task template.
• Standard Content inserts all content but the Commands Used and the Description heading. This is the default setting.
• Minimal Content inserts only the commands or embedded components.
Automatically Display Legends in 2-D Plots
For 2-D plots, you can display a legend that matches each curve with a text label. To set this feature, select the Automatically display legends in 2-D plots check box. For information on customizing the text labels, see Edit a Legend.
Note: Legends vary in size, depending on the number of curves in the plot. There is a limit on the amount of space available for a plot legend. Therefore, if there are too many entries in the legend, some may not be displayed.
For information on how to display or hide the legend on a particular 2-D plot after the plot is created, see Show or Hide a Legend.
Enable Rollover Highlighting in Plots
When you roll over a plot with your mouse, the plot can be highlighted. To set this feature, select the Enable rollover highlighting in plots check box.
Use Hardware Acceleration for Plots
By default, Maple uses hardware acceleration for plots if possible, and uses a software renderer if hardware acceleration is not available.
To change the setting to always use the software renderer, clear the Use hardware acceleration for plots check box.
Always Insert New Execution Group after Executing
The Always insert new execution group after executing option controls the automatic insertion of a new execution group after Maple computes an execution group. By default, this feature is not enabled.
• If this feature is not enabled, Maple computes the execution group and then moves the cursor to the next execution group in the worksheet.
If the computed execution group is the last one in the worksheet, Maple creates a new execution group and moves the cursor to it.
• If this feature is enabled, Maple automatically inserts a new execution group after computing any execution group.
Show Equation Labels
Display equation labels in the worksheet. Each execution group generates a label that is displayed in the right-hand side of the worksheet. These labels may be referenced elsewhere in the worksheet as necessary. For instructions, see Using Equation Labels.
Note: Disabling this feature will affect the display of help pages and Maple worksheets that use equation label references.
Self-documenting context menus automatically create a record of your actions while you solve your problem. An arrow connects input and output as the result of a context menu operation and the text describing the operation is displayed above the arrow. For more information, see Self-Documenting Context Menu. | {"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": 0.8658871054649353, "perplexity": 3195.9233454322616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1476988721278.88/warc/CC-MAIN-20161020183841-00438-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/51256-explanation-delta-method-print.html | # Explanation of delta method?
Definition: For $A\subset\mathbb{R}, X_n$ are A-valued random variables and $a_n (X_n - \theta) \stackrel{D}{\rightarrow} X$ for some sequence $\{a_n\}_{n\in\mathbb{N}}$ satisfying $a_n \rightarrow \infty$ as $n \rightarrow \infty$. Then for any function $h: A \rightarrow\mathbb{R}$ that is differentiable at $\theta , a_n (h(X_n ) - h(\theta)) \stackrel{D}{\rightarrow} h'(\theta)X$ | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9196861386299133, "perplexity": 149.30005240003294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560281084.84/warc/CC-MAIN-20170116095121-00055-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/222765/writing-an-ieee-document | # Writing an IEEE document
I need to write a document by using one of the IEEE Xplore templates but I don't know how to compile it since that my TeX editors don't recognize this template [I have installed in my laptop TeX Maker and LyX].
The template I want to use is in the following link: http://www.ieee.org/conferences_events/conferences/publishing/templates.html
Can you help me please? What I have to do in order to these editors compile this kind of templates?.
Thanks for your attention and help.
• Welcome to TeX.SX! It's tedious to download templates from websites. Most probably you have to place either a `.sty` file in your directory or open the template `.tex` file and edit it. Side note: This is an English-speaking website, you don't have to put Spanish '?' marks ;-) – user31729 Jan 12 '15 at 15:43
• the template will be in compressed format (.zip or .tar.gz) uncompress it to see the .tex files – nidhin Jan 12 '15 at 15:59
• when I open the .tex document and order to compile, the editor don't recognize "this type of document". Between the files I have I don't have a .sty file but a .cls one and .tex files. I don't know where I should put them. – Jose Jan 12 '15 at 16:00
• @Jose: Place the `.cls` file in a folder your TeXMaker can find and the start a new `.tex` document with the top line `\documentclass{foo}` if your class file is named `foo.cls`. Don't ask me about input files for TeXMaker ... I don't use such editors – user31729 Jan 12 '15 at 16:06
## 4 Answers
1. Download and unpack the archive.
2. Copy the `IEEEtran.cls` class file into the same directory as your document.
3. Make the first line of your document `\documentclass{IEEEtran}`.
• (You can also load class options as described in the "How-To" file included in the archive.)
4. Compile the document the way you would normally compile a LaTeX document. (E.g., enter `pdflatex file` at the command line.)
See other questions on this site about setting up your preferred editor and using custom document classes.
If you are using a UNIX system, and plan on using similar templates more than once, here is what you can do:
1. Download IEEEtran.cls.
2. `locate article.cls` to find where you need to place `IEEEtran.cls`
3. `sudo cp IEEEtran.cls <location-of-article.cls>` to copy `IEEEtran.cls` into the folder containing `article.cls`
4. `sudo texhash` to tell TeX about the new class
5. If you are getting many error messages at this point when compiling the template, then you may need to install the texlive-fonts-recommended package: `sudo apt-get install texlive-fonts-recommended`
You should now be able to compile your template.
• Please don't fiddle with the official installation and those folders. Make your own texmf reachable instead. – percusse Feb 6 '15 at 9:59
In addition to installing an editor such as TeX Maker (I prefer TeX Studio myself), you have to set up the LaTeX production environment based on which your TeX Maker can compile your documents.
No matter what Operating System you are using, I suggest you do the following steps:
1. Install TeX Live first: https://www.tug.org/texlive/
2. Re-install a TeX editor, such as TeX Maker or TeX Studio;
3. Check whether the correct path for LaTeX compile tools are detected correctly in TeX Maker (make sure they are in the right path as installed TeX Live); e.g. http://www.xm1math.net/texmaker/doc.html#SECTION01
4. Dry-run the template to see how it works.
BTW, as a beginner, you may not want to confuse yourself by using TeX and LyX at the same time. They are quite different.
Good luck!
Try the following link, no installation, `open as template` you can write IEEE paper directly from web browser. Enjoy typing! | {"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": 0.9804964065551758, "perplexity": 2344.765690891248}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145729.69/warc/CC-MAIN-20200222211056-20200223001056-00478.warc.gz"} |
http://mathhelpforum.com/calculus/3154-trig-functions.html | # Math Help - Trig Functions?
1. ## Trig Functions?
I'm wondering if someone could help me list the Trig functions series.
For example:
Tan(x) = Sin(x)/Cos(x)
Sec(x) = 1/Cos(x)
I'm not sure if this is posted in the right section but I need these information for my Calc homeworks.
2. Originally Posted by nirva
I'm wondering if someone could help me list the Trig functions series.
For example:
Tan(x) = Sin(x)/Cos(x)
Sec(x) = 1/Cos(x)
I'm not sure if this is posted in the right section but I need these information for my Calc homeworks.
Are you sure that the examples give are what is expected?
Others are:
Cot(x)=1/Tan(x)=Cos(x)/Sin(x)
Cosec(x)=1/Sin(x).
RonL
3. Originally Posted by CaptainBlack
Are you sure that the examples give are what is expected?
Others are:
Cot(x)=1/Tan(x)=Cos(x)/Sin(x)
Cosec(x)=1/Sin(x).
RonL
Because those examples are what was given to some question like
$\int \frac {sin(x) + sec(x)} {tan(x)} dx$
Where sin(x)/tan(x) becomes sin(x)/{sin(x)/cos(x)}
What is Sec(x)/Tan(x) equal to by the way? Csc(x)?
4. yes it is equal to Csc(x)
5. Hello, nirva!
Because those examples are what was given to some question like: $\int\frac{\sin x + \sec x}{\tan x}\,dx$
Where $\frac{\sin x}{\tan x }$ becomes $\frac{\sin x}{\frac{\sin x}{\cos x}}$
What is $\frac{\sec x }{\tan x}$ equal to? . $\csc x$ ?
Yes . . . $\displaystyle{\frac{\sec x}{\tan x}\;=\;\frac{\frac{1}{\cos x}}{\frac{\sin x}{\cos x}} \;=\;\frac{1}{\cos x}\cdot\frac{\cos x}{\sin x} \;=\;\frac{1}\sin x} \;=\;\csc x }$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
By the way, I prefer to simplify the function like this:
. . . $\displaystyle{\frac{\sin x + \sec x}{\tan x} \;= \;\frac{\sin x + \frac{1}{\cos x}}{\frac{\sin x}{\cos x}} }$
Multiply top and bottom by $\cos x$ **
. . . $\frac{\cos x\left(\sin x + \frac{1}{\cos x}\right)}{\cos x\left(\frac{\sin x}{\cos x}\right)} \;= \;\frac{\sin x\cos x + 1}{\sin x}$
Then make two fractions:
. . . $\frac{\sin x\cos x}{\sin x} + \frac{1}{\sin x} \;= \;\cos x + \csc x$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
**
This is a technique used on a complex fraction,
. . a fraction with more than two "levels".
Example: $\frac{\frac{1}{3} + \frac{1}{2}}{\frac{1}{6} + \frac{1}{4}}$
Multiply top and bottom by the LCD of $all$ the denominators (12):
. . . $\frac{12\cdot\left(\frac{1}{3} + \frac{1}{2}\right)}{12\cdot\left(\frac{1}{6} + \frac{1}{4}\right)} \;= \;\frac{4 + 6}{2 + 3} \;= \;\frac{10}{5}\;=\;2$ . . . see?
6. Are you asking for the different trig identities? just wondering | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.995937168598175, "perplexity": 474.56157983571245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776431056.5/warc/CC-MAIN-20140707234031-00019-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://zbmath.org/?q=an:0757.60034&format=complete | ×
zbMATH — the first resource for mathematics
Linear rescaling of the stochastic process. (English) Zbl 0757.60034
Stochastic processes $$X$$ and $$Y$$ are considered, whose index has the form $${\mathbf t}=(t_ 1,\dots,t_ k)$$, $$t_ i>0$$, $$1\leq i\leq k$$. It is assumed that the “$${\mathbf t}$$-process” $$\alpha({\mathbf s})Y_{{\mathbf s}\cdot{\mathbf t}}+\beta({\mathbf s})$$ has finite-dimensional distributions which converge in law to those of $$X_{{\mathbf t}}$$, as $$\min_{1\leq i\leq k} s_ i$$ goes to $$\infty$$. It is proved that two cases are possible: either $$X$$ is essentially a constant process, or the following limits exist $\lim_{{\mathbf s}}[\alpha({\mathbf s})\cdot\alpha({\mathbf a}\cdot{\mathbf s})^{- 1}]=A({\mathbf a}),\quad\text{and}\quad\lim_{{\mathbf s}}[\beta({\mathbf s})- \beta({\mathbf a}\cdot{\mathbf s})\cdot\alpha({\mathbf s})\cdot\alpha({\mathbf a}\cdot{\mathbf s})^{-1}]=B({\mathbf a}).$ In the latter case, for all $${\mathbf a}$$, the processes $$X_{{\mathbf a}\cdot{\mathbf t}}$$ and $$A({\mathbf a})\cdot X_{{\mathbf t}}+B({\mathbf a})$$ have the same finite-dimensional distributions. Finally explicit expressions for the functions $$A$$ and $$B$$ are given, with, typically and respectively, multiplicative and additive forms.
MSC:
60G18 Self-similar stochastic processes 62E20 Asymptotic distribution theory in statistics 60G10 Stationary stochastic processes 60F05 Central limit and other weak theorems
Full Text: | {"extraction_info": {"found_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": 0.921116828918457, "perplexity": 343.4232812181144}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056548.77/warc/CC-MAIN-20210918154248-20210918184248-00022.warc.gz"} |
https://www.arxiv-vanity.com/papers/cond-mat/9905198/ | 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.
# Transition Temperature of the Homogeneous, Weakly Interacting Bose gas
Markus Holzmann and Werner Krauth 1 CNRS-Laboratoire de Physique Statistique
and Laboratoire Kastler-Brossel2
Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France
11krauth
22Laboratoire Kastler-Brossel is a unité de recherche de l’Ecole Normale Supérieure et de l’Université Pierre et Marie Curie, associée au CNRS.
###### Abstract
We present a Monte Carlo calculation for up to bosons in D to determine the shift of the transition temperature due to small interactions . We generate independent configurations of the ideal gas. At finite , the superfluid density changes by a certain correlation function in the limit ; the limit is taken afterwards. We argue that our result is independent of the order of limits. Detailed knowledge of the non-interacting system for finite allows us to avoid finite-size scaling assumptions.
PACS numbers: 03.75.Fi, 02.70.Lq, 05.30.Jp
Feynman [1] has provided us with a classic formula for the partition function of the canonical noninteracting Bose gas. It represents a “path-integral without paths”, as they have been integrated out. What remains is the memory of the cyclic structure of the permutations that were needed to satisfy bosonic statistics:
ZN=∑{mk}P({mk});withP({mk})=N∏k=1ρmkkmk!kmk. (1)
The partitions in eq. (1) decompose permutations of the particles into exchange cycles ( cycles of length for all with ). is a system-dependent weight for cycles of length .
In this paper we present an explicit Monte Carlo calculation for up to bosons in three dimensions, starting from eq. (1). The calculation allows us to determine unambiguously the shift in the transition temperature for weakly interacting bosons in the thermodynamic limit for an infinitesimal s-wave scattering length . This fundamental question has lead to quite a number of different and contradictory theoretical as well as computational answers (cf, e.g., [2, 3, 4]).
We will first use Eq. (1) and its generalizations to determine very detailed properties of the finite- canonical Bose gas in a box with periodic boundary conditions. We then point out that all information on the shift of for weakly interacting gases is already contained in the noninteracting system. In the linear response regime (infinitesimal interaction), it is a certain correlation function of the noninteracting system which determines the shift in . This correlation is much too complicated to be calculated directly, but we can sample it, even for very large . To do so, we generate independent bosonic configurations in the canonical ensemble. We have found a solution (based on Feynman’s formula Eq. (1)) which avoids Markov chain Monte Carlo methods. In our two-step procedure, a partition is generated with the correct probability . Then, a random boson configuration is constructed for the given partition.
We stress that all our calculations are done very close to , so that the correlation length of any macroscopic sample is much larger than the actual system size of the simulation. This condition allows us to invoke the standard finite-size scaling hypothesis[5], but also to take the limit after the limit .
A key concept in the path integral representation of bosons is that of a winding number. Consider first the density matrix of a single particle [] at inverse temperature . In a three-dimensional cubic box of length with periodic boundary conditions, with, e.g.,
ρ(x,x′,β)=∞∑wx=−∞exp[−(x−(x′+Lwx))2/2β]√2πβ. (2)
In Eq. (2), and are to be taken within the periodic box ().
It is more convenient to adopt non-periodic coordinates (), as we will do from here on. In Fig. 1, the path drawn with a thick line can equivalently be tagged by or by . This notation allows one to keep track of the topology of paths without introducing intermediate time steps , even for very small systems. With this convention, the winding number of a configuration, , is defined as
W=∑(r′i−ri)/L. (3)
The winding number in Eq. (3) is the sum of the (integer) winding numbers for each of the statistically uncorrelated cycles which comprise the configuration. The complete statistical weight of a cycle of length [cf. Eq. (1)] is given by the sum of the weights for all winding numbers :
ρk=[∞∑w=−∞ρk,w]3;ρk,w=L√2πkβexp(−L2w22kβ). (4)
Pollock and Ceperley [6] have obtained the result
ρs/ρ=L23βN, (5)
which connects the system’s superfluid density to the winding number in a rigorous fashion.
It is possible to determine the mean square winding number from Eq. (1). We first compute for a given partition
{mk}=∑mkk. (6)
Here, is the mean with respect to cycles of length , . This yields, by summation over partitions
=∑ρkkkZN−k/ZN. (7)
We have also determined the probability distribution of .
An analogous calculation formally replaces in Eq. (6), which becomes [cf. Eq. (1)]. Equation (7) is transformed into
ZN=∑ρkZN−k/N. (8)
Equation (8) allows the recursive calculation of the partition function if are known [7].
The same relation Eq. (8) allows us to identify
k=ρkZN−k/ZN (9)
as the mean number of particles in a cycle of length .
From a different point of view, the quantity determines the occupation number of single-particle energy levels for a given partition . This allows us to compute the average number of particles occupying state in the bosonic system:
=N∑k=1{e−βkϵiZN−kZN.} (10)
Eq. (10) is of crucial importance: We find that , the condensate fraction, is different from the superfluid fraction, as determined from Eqs (5) and (7) in a finite non-interacting system.
The term in Eq. (10) can be regarded as the probability of having at least particles in state . Taking the sum over all states , with the use of Eq. (9), we can connect cycle statistics with the usual occupation number representation:
k=∑iP(ni≥k). (11)
This curious result, which is of practical use in inhomogeneous systems [8], tells us that the discrete derivative of the mean cycle numbers with respect to their length is given by the probability of having particles in the same single-particle energy level.
Rescaled superfluid densities (from Eqs (7) and (5)) are plotted in Fig. 2a for as a function of the rescaled temperature , where is the critical temperature for .
A finite-size scaling ansatz, which was used in previous Monte Carlo work on the problem [3], assumes that the curves of for a weakly interacting Bose gas should intersect at the transition temperature, as they do approximately. However, the small-scale Fig, 2b clearly shows the importance of corrections to scaling (cf. [9]) already for the noninteracting gas. By continuity, the corrections to scaling for the weakly interacting Bose gas must be important, especially if the temperature shift due to interactions becomes small.
Our strategy greatly benefits from the solution Eq. (7) for the ideal gas. We compute the intersection point () for two finite systems with and particles and determine how this point is shifted under the influence of interactions (cf. Fig. 2b). Our arbitrary but fixed ratio facilitates the direct extrapolation in to .
To generate a random partition, we interpret the term in Eq. (8) as the probability to split off a cycle of length from a configuration of bosons, and to be left with a system of bosons. We can pick with probability with a simple “tower of probabilities” strategy [10]. Recursively, we can thus generate an independent random partition with great speed. The recursion stops as soon as we have split off a cycle of length from a system with particles.
To go from a random partition to a random configuration, we may treat each cycle separately. For a cycle of length , we select a winding number with probability [cf. Eq. (4)], and analogously for and . Towers of probabilities are again used. The cycle starts at a random position with , and ends at . Intermediate points are filled in with the appropriate Lévy construction [6]. We have tested our algorithm successfully against the known results (cf. Fig. 2).
We thus generate independent free boson path-integral configurations by a method very different from what is usually done in path-integral (Markov-chain) Quantum Monte Carlo calculations, but with an equivalent outcome: Any appropriate operator is sampled with the probability
0=∑P∫dR[∏iρ(ri,r′i)]O∑P∫dR[∏iρ(ri,r′i)]. (12)
Here, indicates the summation over all permutations , and is the position of the particle . In the presence of interactions, the statistical weight of each configuration is no longer given by the product of the one-particle density matrices . To lowest order in the interaction, the density matrix is exclusively modified by s-wave scattering. Likewise, only binary collisions need to be kept. This means that the correct statistical weight is given by
πa(r1,…,rN;r′1,…,r′N)=π0∏i
The contribution of collisions is to lowest order in
∏i
with
cij=(|rij|−1+|r′ij|−1)exp[|rij||r′ij|(1+cosγij)/2β].
Here, and is the angle between and . Equation (13) corresponds to the popular Path-Integral Monte Carlo “action” with the following important modifications: i) no interior time-slices are needed, ii) the interaction may be treated on the s-wave level, and iii) the interaction may be expanded in . For a consistent evaluation of the interaction with periodic boundary conditions, as schematically represented in Fig. 1, it is best to sum over all pairs shown, with the condition that be in the original simulation box (shaded in gray). As indicated by the small circles in Fig. 1, may have important contributions stemming from more than one representative of the path , especially for small systems. Of course, a cutoff procedure can be installed.
We now find for the mean-square winding number in the interacting system
a=<(1−aC)W2>0<(1−aC)>0, (15)
where we put . Expanding in , this yields
a−0=−a<(ΔW2)(ΔC)>0, (16)
where [11].
For a finite system of bosons, the shift in the superfluid density can thus be proven to be linear in
δρsρ=−XNβN1/3aρ1/3, (17)
with .
To determine quantitatively the shift of the intersection points in Fig. 2, we expand the ideal gas superfluid density around the intersection temperature of two systems with and bosons at the same density:
ρs/ρ(T)N1/3=ρs/ρ(Ts)N1/3+αN×[T−Ts]. (18)
In this formula, the linear expansion coefficients can be computed. With interactions, only is modified to linear order in . remains unchanged, as we restrict the expansion to . We find the new intersection point of the two systems to be shifted in temperature as
ΔTsTs(0):=Ts(a)−Ts(0)Ts(0)=X8N−XNα8N−αNaρ1/3. (19)
We have also computed the shift in , but found out only that it must be extremely small. We are unaware of any fundamental reason for a vanishing shift in this quantity. In Fig. 3, we plot the shift for different system sizes ranging from to vs . We have not attempted a thorough analysis of the finite-size effects, which already appear negligible for our largest systems.
We conclude that the transition temperature of the weakly interacting Bose gas increases linearly in the scattering length by an amount of
ΔTCTC=(2.3±0.25)aρ1/3. (20)
Our result Eq. (20) is almost an order of magnitude larger than what was found in a previous Monte Carlo calculation [3]. However, this calculation was restricted to very small particle numbers and it used a problematic finite-size scaling ansatz, as pointed out. The agreement of Eq. (20) with the renormalization group calculation [2] seems to be quite good.
It is very interesting to understand whether the result Eq. (20) directly applies to the current Bose-Einstein condensation experiments (cf, e.g. [12, 13]). In earlier papers [8, 14], we have pointed out the particularities of these finite systems in external potentials (cf. [15] for a general overview). Notwithstanding the differences between the two systems, a relevant parameter is for both cases , where the maximum density (at the center of the trap) at the transition point must be taken in the inhomogeneous case. The experimental value is of the order .
Within our method, we can also study finite values of the interaction, even though we no longer compute a correlation function, and also have to introduce interior time slices. Contributions beyond s-wave scattering need to be monitored, as we have in [14]. For bosons we have found agreement with the linear reponse formula Eq. (16) up to , but a decrease for the full treatment for . A detailed investigation of this question goes beyond the scope of this paper.
In conclusion, it is worth noting that we have encountered none of the difficulties which usually haunt boson calculations: We work in the canonical ensemble; therefore, the fluctuation anomaly of the grand-canonical Bose gas plays no role. The density remains automatically constant as a function of so that an expansion in is well-defined. At finite , we can furthermore prove that the shift in is linear in the interaction parameter. We have also consistently approached the weakly interacting system from the vantage point of the ideal gas. This allows us to obtain the crucial information on exactly where to do our simulation (cf. Fig. 2). Finally, our extremely powerful sampling algorithm has allowed us to partially dispel the curse of Monte Carlo simulations: limitations to small system sizes.
We thank Boris V. Svistunov for discussions. M. H. acknowledges support by the Deutscher Akademischer Austauschdienst. The FORTRAN programs used in this work are made available (from MH or WK). | {"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": 0.9061180949211121, "perplexity": 511.1566941351807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738816.7/warc/CC-MAIN-20200811150134-20200811180134-00143.warc.gz"} |
https://neurips.cc/Conferences/2021/ScheduleMultitrack?event=26351 | Timezone: »
Poster
Mao Li · Kaiqi Jiang · Xinhua Zhang
Tue Dec 07 08:30 AM -- 10:00 AM (PST) @
Probability discrepancy measure is a fundamental construct for numerous machine learning models such as weakly supervised learning and generative modeling. However, most measures overlook the fact that the distributions are not the end-product of learning, but are the basis of downstream predictor. Therefore it is important to warp the probability discrepancy measure towards the end tasks, and we hence propose a new bi-level optimization based approach so that the two distributions are compared not uniformly against the entire hypothesis space, but only with respect to the optimal predictor for the downstream end task. When applied to margin disparity discrepancy and contrastive domain discrepancy, our method significantly improves the performance in unsupervised domain adaptation, and enjoys a much more principled training process.
#### Author Information
##### Kaiqi Jiang (University of Illinois at Chicago)
I am currently a Ph.D. student concentrating on machine learning working with Professor Xinhua Zhang. My current research is domain adaptation and fairness. | {"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": 0.8354496955871582, "perplexity": 2125.6966382454184}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1679296949181.44/warc/CC-MAIN-20230330101355-20230330131355-00195.warc.gz"} |
https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy/Computational_Physics_(Chong)/09%3A_Numerical_Integration/9.05%3A_Monte_Carlo_Integration | $$\require{cancel}$$
# 9.5: Monte Carlo Integration
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The final numerical integration scheme that we'll discuss is Monte Carlo integration, and it is conceptually completely different from the previous schemes. Instead of assigning a set of discretization points (either explicitly, as in the mid-point/trapezium/Simpson's rules, or through a machine optimization procedure, as in the adaptive quadrature method), this method randomly samples the points in the integration domain. If the sampling points are independent and there is a sufficiently large number of them, the integral can be estimated by taking a weighted average of the integrand over the sampling points.
To be precise, consider a 1D integral over a domain $$x \in [a,b]$$. Let each sampling point be drawn independently from a distribution $$p(x)$$. This means that the probability of drawing sample $$x_{n}$$ in the range $$x_n \in [x, x + dx]$$ is $$p(x) dx$$. The distribution is normalized, so that
$\int_a^b p(x) \; dx = 1.$
Let us take $$N$$ samples, and evaluate the integrand at those points: this gives us a set of numbers $$\{f(x_n)\}$$. We then compute the quantity
$\mathcal{I}^{\mathrm{mc}} = \frac{1}{N} \sum_{n=0}^{N-1} \frac{f(x_n)}{p(x_n)}.$
Unlike the estimators that we have previously studied, $$\mathcal{I}^{\mathrm{mc}}$$ is a random number (because the underlying variables $$\{x_n\}$$ are all random). Crucially, its average value is equal to the desired integral:
\begin{align} \left\langle\mathcal{I}^{\mathrm{mc}}\right\rangle &= \frac{1}{N} \sum_{n=0}^{N-1} \left\langle \frac{f(x_n)}{p(x_n)}\right\rangle \\ & = \left\langle \frac{f(x_n)}{p(x_n)}\right\rangle \qquad\mathrm{for}\;\mathrm{each}\; n \\ & = \int_a^b p(x) \, \left[\frac{f(x)}{p(x)}\right]\, dx \\ & = \int_a^b f(x)\, dx \end{align}
For low-dimensional integrals, there is normally no reason to use the Monte Carlo integration method. It requires a much larger number of samples in order to reach a level of numerical accuracy comparable to the other numerical integration methods. (For 1D integrals, Monte Carlo integration typically requires millions of samples, whereas Simpson's rule only requires hundreds or thousands of discretization points.) However, Monte Carlo integration outperforms discretization-based integration schemes when the dimensionality of the integration becomes extremely large. Such integrals occur, for example, in quantum mechanical calculations involving many-body systems, where the dimensionality of the Hilbert space scales exponentially with the number of particles.
9.5: Monte Carlo Integration is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9990813732147217, "perplexity": 295.1338317862962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662561747.42/warc/CC-MAIN-20220523194013-20220523224013-00460.warc.gz"} |
http://tex.stackexchange.com/questions/28708/why-does-pagereflastpage-give-me-rather-than-page-number-of-the-last-pag | # Why does \pageref{LastPage} give me “??” rather than page number of the last page?
I am trying to use the lastpage package in order to have footers with the page number given in the form "Page x of y", where y is the page number of the last page.
This should be straightforward, but I'm getting instead: "Page x of ??"; i.e., I'm getting a pair of question marks rather than a page number.
I've distilled the problem down in order to minimize dependencies, etc. to:
\documentclass{article}
\usepackage{lastpage}
\usepackage{fancyhdr}
\pagestyle{fancy}
\cfoot{\thepage\ of \pageref{LastPage}}
\begin{document}
text
\newpage
text
\newpage
text
\end{document}
Does anyone have any idea about what I could be missing to cause \pageref{LastPage} to give me "??"?
-
Did you run it twice? – Peter Grill Sep 16 '11 at 22:14
Your code compiles OK in my system. Have you compiled it twice? – Gonzalo Medina Sep 16 '11 at 22:14
In order for TeX to know what the last page is, the document must be completed. So you need to run LaTeX twice. The first time it will write out an temporary .aux file and on the second run it will read it in and fill in the ??. When I run your document the second time it correctly displays Page 1 of 3.
@Jum Ratliff: When encountering this issue after two compiler runs (some files need three or even four compiler runs!), please look at the end of the .log file for warnings of package rerunfilecheck and for LaTeX Warning: There were undefined references. LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. – Stephen Sep 17 '11 at 11:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8589268922805786, "perplexity": 1903.4364549344816}, "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-2016-07/segments/1454701168076.20/warc/CC-MAIN-20160205193928-00251-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-5-linear-functions-5-6-parallel-and-perpendicular-lines-lesson-check-page-330/6 | ## Algebra 1
$y = 2x+4$ is a line (as an example). If another line has a slope of 2, then we would know the lines are parallel. However, if the slope of the other line is -1/2, then we would know the lines are perpendicular. | {"extraction_info": {"found_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": 0.9747254848480225, "perplexity": 188.97758836716625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1544376823228.36/warc/CC-MAIN-20181209232026-20181210013526-00047.warc.gz"} |
https://www.physicsforums.com/threads/acceleration-on-inclined-plane.809520/ | # Acceleration on inclined plane
1. Apr 19, 2015
### bongobl
Hi guys I am trying too find a formula to figure out how I would find the acceleration of a block on an incline plane IF that plane itself were accelerating in the +Y direction.
I am not even sure which frame of reference to use since the acceleration is in 2 dimensions. This hinders me from being able to draw a free body diagram
Does anybody know how to go about finding the acceleration of the block (in cartesian vector form)?
Thanks
-Alex
#### Attached Files:
• ###### inclineWithTheta.gif
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2.3 KB
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2. Apr 19, 2015
### Staff: Mentor
Is the acceleration of the plane given, or is it to be determined by the force from the block?
If it is given then just remember that acceleration is a vector. So you can just add the known acceleration of the plane to the unknown acceleration of the block relative to the plane to get the total acceleration and you can still write Newton's 2nd law. The movement of the plane shouldn't change the free body diagram at all.
If it is not given then I would use a Lagrangian approach.
3. Apr 20, 2015
### Delta²
I think it can be done in Newtonian mechanics since the problem states that the acceleration of the plane (and i hope it doesnt mean the acceleration of the c.o.m only) is only in the +y direction. Of course this means there is some sort of constraint that prevents the rotation of the plane around its c.o.m (like that the plane is moving between two vertical walls).
4. Apr 20, 2015
### Staff: Mentor
Such as if the incline were fixed to the floor of an accelerating elevator? (The simplest case.)
If so, I would use the accelerating frame of the incline, being sure to add the appropriate inertial pseudo force due to the acceleration. (Of course, you have several options.)
5. Apr 20, 2015
### rcgldr
If the inclined plane is accelerating upwards at a rate of "u" m/s^2, then let g' = g (gravitational acceleration) + u (inclined plane upwards acceleration). Solve as if the plane was not accelerating, using g' instead of g, then when completed, add the block's acceleration vector with respect to the incline plane and the upwards acceleration of the inclined plane with respect to the groung to get the total acceleration vector with respect to the ground. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9227529168128967, "perplexity": 444.4068870367805}, "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/1531676596542.97/warc/CC-MAIN-20180723145409-20180723165409-00577.warc.gz"} |
http://ab-initio.mit.edu/wiki/index.php?title=Meep_Tutorial&diff=4978&oldid=4890 | # Meep Tutorial
(Difference between revisions)
## Current revision
In this page, we'll go through a couple of simple examples that illustrate the process of computing fields, transmission/reflection spectra, and resonant modes in Meep. All of the examples here are two-dimensional calculations, simply because they are quicker than 3d computations and they illustrate most of the essential features. For more advanced functionality involving 3d computations, see the Simpetus projects page.
This tutorial uses the libctl/Scheme scripting interface to Meep, which is what we expect most users to employ most of the time. There is also a C++ interface that may give additional flexibility in some situations; that is described in the C++ tutorial.
In order to convert the HDF5 output files of Meep into images of the fields and so on, this tutorial uses our free h5utils programs. You could also use any other program, such as Matlab, that supports reading HDF5 files.
## The ctl file
The use of Meep revolves around the control file, abbreviated "ctl" and typically called something like foo.ctl (although you can use any file name you wish). The ctl file specifies the geometry you wish to study, the current sources, the outputs computed, and everything else specific to your calculation. Rather than a flat, inflexible file format, however, the ctl file is actually written in a scripting language. This means that it can be everything from a simple sequence of commands setting the geometry, etcetera, to a full-fledged program with user input, loops, and anything else that you might need.
Don't worry, though—simple things are simple (you don't need to be a Real Programmer), and even there you will appreciate the flexibility that a scripting language gives you. (e.g. you can input things in any order, without regard for whitespace, insert comments where you please, omit things when reasonable defaults are available...)
The ctl file is actually implemented on top of the libctl library, a set of utilities that are in turn built on top of the Scheme language. Thus, there are three sources of possible commands and syntax for a ctl file:
• Scheme, a powerful and beautiful programming language developed at MIT, which has a particularly simple syntax: all statements are of the form (function arguments...). We run Scheme under the GNU Guile interpreter (designed to be plugged into programs as a scripting and extension language). You don't need to know much Scheme for a basic ctl file, but it is always there if you need it; you can learn more about it from these Guile and Scheme links.
• libctl, a library that we built on top of Guile to simplify communication between Scheme and scientific computation software. libctl sets the basic tone of the interface and defines a number of useful functions (such as multi-variable optimization, numeric integration, and so on). See the libctl manual pages.
• Meep itself, which defines all the interface features that are specific to FDTD calculations. This manual is primarily focused on documenting these features.
At this point, please take a moment to leaf through the libctl tutorial to get a feel for the basic style of the interface, before we get to the Meep-specific stuff below. (If you've used MPB, all of this stuff should already be familiar, although Meep is somewhat more complex because it can perform a wider variety of computations.)
Okay, let's continue with our tutorial. The Meep program is normally invoked by running something like the following at the Unix command-line (herein denoted by the unix% prompt):
unix% meep foo.ctl >& foo.out
which reads the ctl file foo.ctl and executes it, saving the output to the file foo.out. However, if you invoke meep with no arguments, you are dropped into an interactive mode in which you can type commands and see their results immediately. If you do that now, you can paste in the commands from the tutorial as you follow it and see what they do.
## Fields in a waveguide
For our first example, let's examine the field pattern excited by a localized CW source in a waveguide— first straight, then bent. Our waveguide will have (non-dispersive) $\varepsilon=12$ and width 1. That is, we pick units of length so that the width is 1, and define everything in terms of that (see also units in meep).
### A straight waveguide
Before we define the structure, however, we have to define the computational cell. We're going to put a source at one end and watch it propagate down the waveguide in the x direction, so let's use a cell of length 16 in the x direction to give it some distance to propagate. In the y direction, we just need enough room so that the boundaries (below) don't affect the waveguide mode; let's give it a size of 8. We now specify these sizes in our ctl file via the geometry-lattice variable:
(set! geometry-lattice (make lattice (size 16 8 no-size)))
(The name geometry-lattice comes from MPB, where it can be used to define a more general periodic lattice. Although Meep supports periodic structures, it is less general than MPB in that affine grids are not supported.) set! is a Scheme command to set the value of an input variable. The last no-size parameter says that the computational cell has no size in the z direction, i.e. it is two-dimensional.
Now, we can add the waveguide. Most commonly, the structure is specified by a list of geometric objects, stored in the geometry variable. Here, we do:
(set! geometry (list
(make block (center 0 0) (size infinity 1 infinity)
(material (make dielectric (epsilon 12))))))
Dielectric function (black = high, white = air), for straight waveguide simulation.
The waveguide is specified by a block (parallelepiped) of size $\infty \times 1 \times \infty$, with ε=12, centered at (0,0) (the center of the computational cell). By default, any place where there are no objects there is air (ε=1), although this can be changed by setting the default-material variable. The resulting structure is shown at right.
Now that we have the structure, we need to specify the current sources, which is specified as a list called sources of source objects. The simplest thing is to add a point source Jz:
(set! sources (list
(make source
(src (make continuous-src (frequency 0.15)))
(component Ez)
(center -7 0))))
Here, we gave the source a frequency of 0.15, and specified a continuous-src which is just a fixed-frequency sinusoid exp( − iωt) that (by default) is turned on at t = 0. Recall that, in Meep units, frequency is specified in units of c, which is equivalent to the inverse of vacuum wavelength. Thus, 0.15 corresponds to a vacuum wavelength of about 1 / 0.15 = 6.67, or a wavelength of about 2 in the $\varepsilon=12$ material—thus, our waveguide is half a wavelength wide, which should hopefully make it single-mode. (In fact, the cutoff for single-mode behavior in this waveguide is analytically solvable, and corresponds to a frequency of 1/2√11 or roughly 0.15076.) Note also that to specify a Jz, we specify a component Ez (e.g. if we wanted a magnetic current, we would specify Hx, Hy, or Hz). The current is located at ( − 7,0), which is 1 unit to the right of the left edge of the cell—we always want to leave a little space between sources and the cell boundaries, to keep the boundary conditions from interfering with them.
Speaking of boundary conditions, we want to add absorbing boundaries around our cell. Absorbing boundaries in Meep are handled by perfectly matched layers (PML)— which aren't really a boundary condition at all, but rather a fictitious absorbing material added around the edges of the cell. To add an absorbing layer of thickness 1 around all sides of the cell, we do:
(set! pml-layers (list (make pml (thickness 1.0))))
pml-layers is a list of pml objects—you may have more than one pml object if you want PML layers only on certain sides of the cell, e.g. (make pml (thickness 1.0) (direction X) (side High)) specifies a PML layer on only the + x side. Now, we note an important point: the PML layer is inside the cell, overlapping whatever objects you have there. So, in this case our PML overlaps our waveguide, which is what we want so that it will properly absorb waveguide modes. The finite thickness of the PML is important to reduce numerical reflections; see perfectly matched layers for more information.
Meep will discretize this structure in space and time, and that is specified by a single variable, resolution, that gives the number of pixels per distance unit. We'll set this resolution to 10, which corresponds to around 67 pixels/wavelength, or around 20 pixels/wavelength in the high-dielectric material. (In general, at least 8 pixels/wavelength in the highest dielectric is a good idea.) This will give us a $160\times80$ cell.
(set! resolution 10)
Now, we are ready to run the simulation! We do this by calling the run-until function. The first argument to run-until is the time to run for, and the subsequent arguments specify fields to output (or other kinds of analyses at each time step):
(run-until 200
(at-beginning output-epsilon)
(at-end output-efield-z))
Here, we are outputting the dielectric function ε and the electric-field component Ez, but have wrapped the output functions (which would otherwise run at every time step) in at-beginning and at-end, which do just what they say. There are several other such functions to modify the output behavior—and you can, of course, write your own, and in fact you can do any computation or output you want at any time during the time evolution (and even modify the simulation while it is running).
It should complete in a few seconds. If you are running interactively, the two output files will be called eps-000000.00.h5 and ez-000200.00.h5 (notice that the file names include the time at which they were output). If we were running a tutorial.ctl file, then the outputs will be tutorial-eps-000000.00.h5 and tutorial-ez-000200.00.h5. In any case, we can now analyze and visualize these files with a wide variety of programs that support the HDF5 format, including our own h5utils, and in particular the h5topng program to convert them to PNG images.
unix% h5topng -S3 eps-000000.00.h5
This will create eps-000000.00.png, where the -S3 increases the image scale by 3 (so that it is around 450 pixels wide, in this case). In fact, precisely this command is what created the dielectric image above. Much more interesting, however, are the fields:
unix% h5topng -S3 -Zc dkbluered -a yarg -A eps-000000.00.h5 ez-000200.00.h5
Briefly, the -Zc dkbluered makes the color scale go from dark blue (negative) to white (zero) to dark red (positive), and the -a/-A options overlay the dielectric function as light gray contours. This results in the image:
Here, we see that the the source has excited the waveguide mode, but has also excited radiating fields propagating away from the waveguide. At the boundaries, the field quickly goes to zero due to the PML layers. If we look carefully (click on the image to see a larger view), we see somethinge else—the image is "speckled" towards the right side. This is because, by turning on the current abruptly at t = 0, we have excited high-frequency components (very high order modes), and we have not waited long enough for them to die away; we'll eliminate these in the next section by turning on the source more smoothly.
### A 90° bend
Now, we'll start a new simulation where we look at the fields in a bent waveguide, and we'll do a couple of other things differently as well. If you are running Meep interactively, you will want to get rid of the old structure and fields so that Meep will re-initialize them:
(reset-meep)
Then let's set up the bent waveguide, in a slightly bigger computational cell, via:
(set! geometry-lattice (make lattice (size 16 16 no-size)))
(set! geometry (list
(make block (center -2 -3.5) (size 12 1 infinity)
(material (make dielectric (epsilon 12))))
(make block (center 3.5 2) (size 1 12 infinity)
(material (make dielectric (epsilon 12))))))
(set! pml-layers (list (make pml (thickness 1.0))))
(set! resolution 10)
Bent waveguide dielectric function and coordinate system.
Note that we now have two blocks, both off-center to produce the bent waveguide structure pictured at right. As illustrated in the figure, the origin (0,0) of the coordinate system is at the center of the computational cell, with positive y being downwards in h5topng, and thus the block of size 12×1 is centered at ( − 2, − 3.5). Also shown in green is the source plane at x = − 7 (see below).
We also need to shift our source to y = − 3.5 so that it is still inside the waveguide. While we're at it, we'll make a couple of other changes. First, a point source does not couple very efficiently to the waveguide mode, so we'll expand this into a line source the same width as the waveguide by adding a size property to the source (Meep also has an eigenmode source feature which can be used here and is covered in a separate tutorial). Second, instead of turning the source on suddenly at t = 0 (which excites many other frequencies because of the discontinuity), we will ramp it on slowly (technically, Meep uses a tanh turn-on function) over a time proportional to the width of 20 time units (a little over three periods). Finally, just for variety, we'll specify the (vacuum) wavelength instead of the frequency; again, we'll use a wavelength such that the waveguide is half a wavelength wide.
(set! sources (list
(make source
(src (make continuous-src
(wavelength (* 2 (sqrt 12))) (width 20)))
(component Ez)
(center -7 -3.5) (size 0 1))))
Finally, we'll run the simulation. Instead of running output-efield-z only at the end of the simulation, however, we'll run it at every 0.6 time units (about 10 times per period) via (at-every 0.6 output-efield-z). By itself, this would output a separate file for every different output time, but instead we'll use another feature of Meep to output to a single three-dimensional HDF5 file, where the third dimension is time:
(run-until 200
(at-beginning output-epsilon)
(to-appended "ez" (at-every 0.6 output-efield-z)))
Here, "ez" determines the name of the output file, which will be called ez.h5 if you are running interactively or will be prefixed with the name of the file name for a ctl file (e.g. tutorial-ez.h5 for tutorial.ctl). If we run h5ls on this file (a standard utility, included with HDF5, that lists the contents of the HDF5 file), we get:
unix% h5ls ez.h5
ez Dataset {161, 161, 330/Inf}
That is, the file contains a single dataset ez that is a 162×162×330 array, where the last dimension is time. (This is rather a large file, 69MB; later, we'll see ways to reduce this size if we only want images.) Now, we have a number of choices of how to output the fields. To output a single time slice, we can use the same h5topng command as before, but with an additional -t option to specify the time index: e.g. h5topng -t 229 will output the last time slice, similar to before. Instead, let's create an animation of the fields as a function of time. First, we have to create images for all of the time slices:
unix% h5topng -t 0:329 -R -Zc dkbluered -a yarg -A eps-000000.00.h5 ez.h5
This is similar to the command before, with two new options: -t 0:329 outputs images for all time indices from 0 to 329, i.e. all of the times, and the the -R flag tells h5topng to use a consistent color scale for every image (instead of scaling each image independently). Then, we have to convert these images into an animation in some format. For this, we'll use the free ImageMagick convert program (although there is other software that will do the trick as well).
unix% convert ez.t*.png ez.gif
Here, we are using an animated GIF format for the output, which is not the most efficient animation format (e.g. ez.mpg, for MPEG format, would be better), but it is unfortunately the only format supported by this Wiki software. This results in the following animation :
x by time slice of bent waveguide (vertical = time).
It is clear that the transmission around the bend is rather low for this frequency and structure—both large reflection and large radiation loss are clearly visible. Moreover, since we operating are just barely below the cutoff for single-mode behavior, we are able to excite a second leaky mode after the waveguide bend, whose second-order mode pattern (superimposed with the fundamental mode) is apparent in the animation. At right, we show a field snapshot from a simulation with a larger cell along the y direction, in which you can see that the second-order leaky mode decays away, leaving us with the fundamental mode propagating downward.
Instead of doing an animation, another interesting possibility is to make an image from a $x \times t$ slice. Here is the y = − 3.5 slice, which gives us an image of the fields in the first waveguide branch as a function of time.
unix% h5topng -0y -35 -Zc dkbluered ez.h5
Here, the -0y -35 specifies the y = − 3.5 slice, where we have multiplied by 10 (our resolution) to get the pixel coordinate.
#### Output tips and tricks
Above, we outputted the full 2d data slice at every 0.6 time units, resulting in a 69MB file. This is not too bad by today's standards, but you can imagine how big the output file would get if we were doing a 3d simulation, or even a larger 2d simulation—one can easily generate gigabytes of files, which is not only wasteful but is also slow. Instead, it is possible to output more efficiently if you know what you want to look at.
To create the movie above, all we really need are the images corresponding to each time. Images can be stored much more efficiently than raw arrays of numbers—to exploit this fact, Meep allows you to output PNG images instead of HDF5 files. In particular, instead of output-efield-z as above, we can use (output-png Ez "-Zc dkbluered"), where Ez is the component to output and the "-Zc dkbluered" are options for h5topng (which is the program that is actually used to create the image files). That is:
(run-until 200 (at-every 0.6 (output-png Ez "-Zc bluered")))
will output a PNG file file every 0.6 time units, which can then be combined with convert as above to create a movie. The movie will be similar to the one before, but not identical because of how the color scale is determined. Before, we used the -R option to make h5topng use a uniform color scale for all images, based on the minimum/maximum field values over all time steps. That is not possible, here, because we output an image before knowing the field values at future time steps. Thus, what output-png does is to set its color scale based on the minimum/maximum field values from all past times—therefore, the color scale will slowly "ramp up" as the source turns on.
The above command outputs zillions of .png files, and it is somewhat annoying to have them clutter up our directory. Instead, we can use the following command before run-until:
(use-output-directory)
This will put all of the output files (.h5, .png, etcetera) into a newly-created subdirectory, called by default filename-out/ if our ctl file is filename.ctl.
What if we want to output an $x \times t$ slice, as above? To do this, we only really wanted the values at y = − 3.5, and therefore we can exploit another powerful Meep output feature—Meep allows us to output only a subset of the computational cell. This is done using the in-volume function, which (like at-every and to-appended) is another function that modifies the behavior of other output functions. In particular, we can do:
(run-until 200
(to-appended "ez-slice"
(at-every 0.6
(in-volume (volume (center 0 -3.5) (size 16 0))
output-efield-z))))
The first argument to in-volume is a volume, specified by (volume (center ...) (size ...)), which applies to all of the nested output functions. (Note that to-appended, at-every, and in-volume are cumulative regardless of what order you put them in.) This creates the output file ez-slice.h5 which contains a dataset of size 162×330 corresponding to the desired $x \times t$ slice.
## Transmission spectrum around a waveguide bend
Above, we computed the field patterns for light propagating around a waveguide bend. While this is pretty, the results are not quantitatively satisfying. We'd like to know exactly how much power makes it around the bend, how much is reflected, and how much is radiated away. How can we do this?
The basic principles were described in the Meep introduction; please re-read that section if you have forgotten. Basically, we'll tell Meep to keep track of the fields and their Fourier transforms in a certain region, and from this compute the flux of electromagnetic energy as a function of ω. Moreover, we'll get an entire spectrum of the transmission in a single run, by Fourier-transforming the response to a short pulse. However, in order to normalize the transmission (to get transmission as a fraction of incident power), we'll have to do two runs, one with and one without a bend.
This control file will be more complicated than before, so you'll definitely want it as a separate file rather than typing it interactively. See the bend-flux.ctl file included with Meep in its examples/ directory.
Above, we hard-coded all of the parameters like the cell size, the waveguide width, etcetera. For serious work, however, this is inefficient—we often want to explore many different values of such parameters. For example, we may want to change the size of the cell, so we'll define it as:
(define-param sx 16) ; size of cell in X direction
(define-param sy 32) ; size of cell in Y direction
(set! geometry-lattice (make lattice (size sx sy no-size)))
(Notice that a semicolon ";" begins a comment, which is ignored by Meep.) define-param is a libctl feature to define variables that can be overridden from the command line. We could now do meep sx=17 tut-wvg-bend-trans.ctl to change the X size to 17, without editing the ctl file, for example. We'll also define a couple of parameters to set the width of the waveguide and the "padding" between it and the edge of the computational cell:
(define-param pad 4) ; padding distance between waveguide and cell edge
(define-param w 1) ; width of waveguide
In order to define the waveguide positions, etcetera, we will now have to use arithmetic. For example, the y center of the horizontal waveguide will be given by -0.5 * (sy - w - 2*pad). At least, that is what the expression would look like in C; in Scheme, the syntax for 1 + 2 is (+ 1 2), and so on, so we will define the vertical and horizontal waveguide centers as:
(define wvg-ycen (* -0.5 (- sy w (* 2 pad)))) ; y center of horiz. wvg
(define wvg-xcen (* 0.5 (- sx w (* 2 pad)))) ; x center of vert. wvg
Now, we have to make the geometry, as before. This time, however, we really want two geometries: the bend, and also a straight waveguide for normalization. We could do this with two separate ctl files, but that is annoying. Instead, we'll define a parameter no-bend? which is true for the straight-waveguide case and false for the bend.
(define-param no-bend? false) ; if true, have straight waveguide, not bend
Now, we define the geometry via two cases, with an if statement—the Scheme syntax is (if predicate? if-true if-false).
(set! geometry
(if no-bend?
(list
(make block
(center 0 wvg-ycen)
(size infinity w infinity)
(material (make dielectric (epsilon 12)))))
(list
(make block
(size (- sx pad) w infinity)
(material (make dielectric (epsilon 12))))
(make block
(size w (- sy pad) infinity)
(material (make dielectric (epsilon 12)))))))
Thus, if no-bend? is true we make a single block for a straight waveguide, and otherwise we make two blocks for a bent waveguide.
The source is now a gaussian-src instead of a continuous-src, parameterized by a center frequency and a frequency width (the width of the Gaussian spectrum), which we'll define via define-param as usual.
(define-param fcen 0.15) ; pulse center frequency
(define-param df 0.1) ; pulse width (in frequency)
(set! sources (list
(make source
(src (make gaussian-src (frequency fcen) (fwidth df)))
(component Ez)
(center (+ 1 (* -0.5 sx)) wvg-ycen)
(size 0 w))))
Notice how we're using our parameters like wvg-ycen and w: if we change the dimensions, everything will now shift automatically. The boundary conditions and resolution are set as before, except that now we'll use set-param! so that we can override the resolution from the command line.:
(set! pml-layers (list (make pml (thickness 1.0))))
(set-param! resolution 10)
Finally, we have to specify where we want Meep to compute the flux spectra, and at what frequencies. (This must be done after specifying the geometry, sources, resolution, etcetera, because all of the field parameters are initialized when flux planes are created.)
(define-param nfreq 100) ; number of frequencies at which to compute flux
(define trans ; transmitted flux
(if no-bend?
(make flux-region
(center (- (/ sx 2) 1.5) wvg-ycen) (size 0 (* w 2)))
(make flux-region
(center wvg-xcen (- (/ sy 2) 1.5)) (size (* w 2) 0)))))
(define refl ; reflected flux
(make flux-region
(center (+ (* -0.5 sx) 1.5) wvg-ycen) (size 0 (* w 2)))))
We compute the fluxes through a line segment twice the width of the waveguide, located at the beginning or end of the waveguide. (Note that the flux lines are separated by 1 from the boundary of the cell, so that they do not lie within the absorbing PML regions.) Again, there are two cases: the transmitted flux is either computed at the right or the bottom of the computational cell, depending on whether the waveguide is straight or bent.
Here, the fluxes will be computed for 100 (nfreq) frequencies centered on fcen, from fcen-df/2 to fcen+df/2. That is, we only compute fluxes for frequencies within our pulse bandwidth. This is important because, to far outside the pulse bandwidth, the spectral power is so low that numerical errors make the computed fluxes useless.
Now, as described in the Meep introduction, computing reflection spectra is a bit tricky because we need to separate the incident and reflected fields. We do this in Meep by saving the Fourier-transformed fields from the normalization run (no-bend?=true), and loading them, negated, before the other runs. The latter subtracts the Fourier-transformed incident fields from the Fourier transforms of the scattered fields; logically, we might subtract these after the run, but it turns out to be more convenient to subtract the incident fields first and then accumulate the Fourier transform. All of this is accomplished with two commands, save-flux (after the normalization run) and load-minus-flux (before the other runs). We can call them as follows:
(if (not no-bend?) (load-minus-flux "refl-flux" refl))
(run-sources+ 500 (at-beginning output-epsilon))
(if no-bend? (save-flux "refl-flux" refl))
This uses a file called refl-flux.h5, or actually bend-flux-refl-flux.h5 (the ctl file name is used as a prefix) to store/load the Fourier transformed fields in the flux planes. The (run-sources+ 500) runs the simulation until the Gaussian source has turned off (which is done automatically once it has decayed for a few standard deviations), plus an additional 500 time units.
Why do we keep running after the source has turned off? Because we must give the pulse time to propagate completely across the cell. Moreover, the time required is a bit tricky to predict when you have complex structures, because there might be resonant phenomena that allow the source to bounce around for a long time. Therefore, it is convenient to specify the run time in a different way: instead of using a fixed time, we require that the | Ez | 2 at the end of the waveguide must have decayed by a given amount (e.g. 1/1000) from its peak value. We can do this via:
(run-sources+
(stop-when-fields-decayed 50 Ez
(if no-bend?
(vector3 (- (/ sx 2) 1.5) wvg-ycen)
(vector3 wvg-xcen (- (/ sy 2) 1.5)))
1e-3))
stop-when-fields-decayed takes four arguments: (stop-when-fields-decayed dT component pt decay-by). What it does is, after the sources have turned off, it keeps running for an additional dT time units every time the given |component|2 at the given point has not decayed by at least decay-by from its peak value for all times within the previous dT. In this case, dT=50, the component is Ez, the point is at the center of the flux plane at the end of the waveguide, and decay-by=0.001. So, it keeps running for an additional 50 time units until the square amplitude has decayed by 1/1000 from its peak: this should be sufficient to ensure that the Fourier transforms have converged.
Finally, we have to output the flux values:
(display-fluxes trans refl)
This prints a series of outputs like:
flux1:, 0.1, 7.91772317108475e-7, -3.16449591437196e-7
flux1:, 0.101010101010101, 1.18410865137737e-6, -4.85527604203706e-7
flux1:, 0.102020202020202, 1.77218779386503e-6, -7.37944901819701e-7
flux1:, 0.103030303030303, 2.63090852112034e-6, -1.11118350510327e-6
flux1:, ...
This is comma-delimited data, which can easily be imported into any spreadsheet or plotting program (e.g. Matlab): the first column is the frequency, the second is the transmitted power, and the third is the reflected power.
Now, we need to run the simulation twice, once with no-bend?=true and once with no-bend?=false (the default):
unix% meep no-bend?=true bend-flux.ctl | tee bend0.out
unix% meep bend-flux.ctl | tee bend.out
(The tee command is a useful Unix command that saves the output to a file and displays it on the screen, so that we can see what is going on as it runs.) Then, we should pull out the flux1 lines into a separate file to import them into our plotting program:
unix% grep flux1: bend0.out > bend0.dat
unix% grep flux1: bend.out > bend.dat
Now, we import them to Matlab (using its dlmread command), and plot the results:
What are we plotting here? The transmission is the transmitted flux (second column of bend.dat) divided by the incident flux (second column of bend0.dat), to give us the fraction of power transmitted. The reflection is the reflected flux (third column of bend.dat) divided by the incident flux (second column of bend0.dat); we also have to multiply by − 1 because all fluxes in Meep are computed in the positive-coordinate direction by default, and we want the flux in the x direction. Finally, the loss is simply 1 - transmission - reflection.
We should also check whether our data is converged, by increasing the resolution and cell size and seeing by how much the numbers change. In this case, we'll just try doubling the cell size:
unix% meep sx=32 sy=64 no-bend?=true bend-flux.ctl |tee bend0-big.out
unix% meep sx=32 sy=64 bend-flux.ctl |tee bend-big.out
Again, we must run both simulations in order to get the normalization right. The results are included in the plot above as dotted lines—you can see that the numbers have changed slightly for transmission and loss, probably stemming from interference between light radiated directly from the source and light propagating around the waveguide. To be really confident, we should probably run the simulation again with an even bigger cell, but we'll call it enough for this tutorial.
## Modes of a ring resonator
As described in the Meep introduction, another common task for FDTD simulation is to find the resonant modes—frequencies and decay rates—of some electromagnetic cavity structure. (You might want to read that introduction again to recall the basic computational strategy.) Here, we will show how this works for perhaps the simplest example of a dielectric cavity: a ring resonator, which is simply a waveguide bent into a circle. (This can be also found in the examples/ring.ctl file included with Meep.) In fact, since this structure has cylindrical symmetry, we can simulate it much more efficiently by using cylindrical coordinates, but for illustration here we'll just use an ordinary 2d simulation.
As before, we'll define some parameters to describe the geometry, so that we can easily change the structure:
(define-param n 3.4) ; index of waveguide
(define-param w 1) ; width of waveguide
(define-param r 1) ; inner radius of ring
(define-param dpml 2) ; thickness of PML
(define sxy (* 2 (+ r w pad dpml))) ; cell size
(set! geometry-lattice (make lattice (size sxy sxy no-size)))
How do we make a circular waveguide? So far, we've only seen block objects, but Meep also lets you specify cylinders, spheres, ellipsoids, and cones, as well as user-specified dielectric functions. In this case, we'll use two cylinder objects, one inside the other:
(set! geometry (list
(make cylinder (center 0 0) (height infinity)
(radius (+ r w)) (material (make dielectric (index n))))
(make cylinder (center 0 0) (height infinity)
(set! pml-layers (list (make pml (thickness dpml))))
(set-param! resolution 10)
Later objects in the geometry list take precedence over (lie "on top of") earlier objects, so the second air ($\varepsilon=1$) cylinder cuts a circular hole out of the larger cylinder, leaving a ring of width w.
Now, we don't know the frequency of the mode(s) ahead of time, so we'll just hit the structure with a broad Gaussian pulse to excite all of the (TM polarized) modes in a chosen bandwidth:
(define-param fcen 0.15) ; pulse center frequency
(define-param df 0.1) ; pulse width (in frequency)
(set! sources (list
(make source
(src (make gaussian-src (frequency fcen) (fwidth df)))
(component Ez) (center (+ r 0.1) 0))))
Finally, we are ready to run the simulation. The basic idea is to run until the sources are finished, and then to run for some additional period of time. In that additional period, we'll perform some signal-processing on the fields at some point with harminv to identify the frequencies and decay rates of the modes that were excited:
(run-sources+ 300
(at-beginning output-epsilon)
(after-sources (harminv Ez (vector3 (+ r 0.1)) fcen df)))
The signal-processing is performed by the harminv function, which takes four arguments: the field component (here Ez) and position (here (r + 0.1,0)) to analyze, and a frequency range given by a center frequency and bandwidth (here, the same as the source pulse). Note that we wrap harminv in (after-sources ...), since we only want to analyze the frequencies in the source-free system (the presence of a source will distort the analysis). At the end of the run, harminv prints a series of lines (beginning with harminv0:, to make it easy to grep for) listing the frequencies it found:
harminv0:, frequency, imag. freq., Q, |amp|, amplitude, error
harminv0:, 0.118101575043663, -7.31885828253851e-4, 80.683059081382, 0.00341388964904578, -0.00305022905294175-0.00153321402956404i, 1.02581433904604e-5
harminv0:, 0.147162555528154, -2.32636643253225e-4, 316.29272471914, 0.0286457663908165, 0.0193127882016469-0.0211564681361413i, 7.32532621851082e-7
harminv0:, 0.175246750722663, -5.22349801171605e-5, 1677.48461212767, 0.00721133215656089, -8.12770506086109e-4-0.00716538314235085i, 1.82066436470489e-7
There are six columns (in addition to the label), comma-delimited for easy import into other programs. The meaning of these columns is as follows. Harminv analyzes the fields f(t) at the given point, and expresses this as a sum of modes (in the specified bandwidth):
$f(t) = \sum_n a_n e^{-i\omega_n t}$
for complex amplitudes an and complex frequencies ωn. The six columns relate to these quantities. The first column is the real part of ωn, expressed in our usual c units, and the second column is the imaginary part—a negative imaginary part corresponds to an exponential decay. This decay rate, for a cavity, is more often expressed as a dimensionless "lifetime" Q, defined by:
$Q = \frac{\mathrm{Re}\,\omega}{-2 \mathrm{Im}\,\omega} .$
(Q is the number of optical periods for the energy to decay by exp( − 2π), and 1 / Q is the fractional bandwidth at half-maximum of the resonance peak in Fourier domain.) This Q is the third column of the output. The fourth and fifth columns are the absolute value | an | and complex amplitudes an. The last column is a crude measure of the error in the frequency (both real and imaginary)...if the error is much larger than the imaginary part, for example, then you can't trust the Q to be accurate. Note: this error is only the uncertainty in the signal processing, and tells you nothing about the errors from finite resolution, finite cell size, and so on!
An interesting question is how long should we run the simulation, after the sources are turned off, in order to analyze the frequencies. With traditional Fourier analysis, the time would be proportional to the frequency resolution required, but with harminv the time is much shorter. Here, for example, there are three modes. The last has a Q of 1677, which means that the mode decays for about 2000 periods or about 2000/0.175 = 104 time units. We have only analyzed it for about 300 time units, however, and the estimated uncertainty in the frequency is 10 − 7 (with an actual error of about 10 − 6, from below)! In general, you need to increase the run time to get more accuracy, and to find very high Q values, but not by much—in our own work, we have successfully found Q = 109 modes by analyzing only 200 periods.
In this case, we found three modes in the specified bandwith, at frequencies of 0.118, 0.147, and 0.175, with corresponding Q values of 81, 316, and 1677. (As was shown by Marcatilli in 1969, the Q of a ring resonator increases exponentially with the product of ω and ring radius.) Now, suppose that we want to actually see the field patterns of these modes. No problem: we just re-run the simulation with a narrow-band source around each mode and output the field at the end.
In particular, to output the field at the end we might add an (at-end output-efield-z) argument to our run-sources+ function, but this is problematic: we might be unlucky and output at a time when the Ez field is almost zero (i.e. when all of the energy is in the magnetic field), in which case the picture will be deceptive. Instead, at the end of the run we'll output 20 field snapshots over a whole period 1/fcen by appending the command:
(run-until (/ 1 fcen) (at-every (/ 1 fcen 20) output-efield-z))
Now, we can get our modes just by running e.g.:
unix% meep fcen=0.118 df=0.01 ring.ctl
After each one of these commands, we'll convert the fields into PNG images and thence into an animated GIF (as with the bend movie, above), via:
unix% h5topng -RZc dkbluered -C ring-eps-000000.00.h5 ring-ez-*.h5
unix% convert ring-ez-*.png ring-ez-0.118.gif
The resulting animations for (from left to right) 0.118, 0.147, and 0.175, are below, in which you can clearly see the radiating fields that produce the losses:
(Each of these modes is, of course, doubly-degenerate according to the representations of the $C_{\infty\mathrm{v}}$ symmetry group. The other mode is simply a slight rotation of this mode to make it odd through the x axis, whereas we excited only the even modes due to our source symmetry. Equivalently, one can form clockwise and counter-clockwise propagating modes by taking linear combinations of the even/odd modes, corresponding an angular φ dependence $e^{\pm i m\phi}$ for m = 3, 4, and 5 in this case.)
You may have noticed, by the way, that when you run with the narrow-bandwidth source, harminv gives you slightly different frequency and Q estimates, with a much smaller error estimate—this is not too strange, since by exciting a single mode you generate a cleaner signal that can be analyzed more accurately. For example, the narrow-bandwidth source for the ω = 0.175 mode gives:
harminv0:, 0.175247426698716, -5.20844416909221e-5, 1682.33949533974, 0.185515412838043, 0.127625313330642-0.13463932485617i, 7.35320734698267e-12
which differs by about 0.000001 (10 − 6) from the earlier estimate; the difference in Q is, of course, larger because a small absolute error in ω gives a larger relative error in the small imaginary frequency.
### Exploiting symmetry
In this case, because we have a mirror symmetry plane (the x axis) that preserves both the structure and the sources, we can exploit this mirror symmetry to speed up the computation. (See also exploiting symmetry in Meep.) In particular, everything about the input file is the same except that we add a single line, right after we specify the sources:
(set! symmetries (list (make mirror-sym (direction Y))))
This tells Meep to exploit a mirror-symmetry plane through the origin perpendicular to the y direction. Meep does not check whether your system really has this symmetry—you should only specify symmetries that really preserve your structure and your sources.
Everything else about your simulation is the same: you can still get the fields at any point, the output file still covers the whole ring, and the harminv outputs are exactly the same. Internally, however, Meep is only doing computations with half of the structure, and the simulation is around twice as fast (YMMV).
In general, the symmetry of the sources may require some phase. For example, if our source was in the y direction instead of the z direction, then the source would be odd under mirror flips through the x axis. We would specify this by (make mirror-sym (direction Y) (phase -1)). See the Meep reference for more symmetry possibilities.
In this case, we actually have a lot more symmetry that we could potentially exploit, if we are willing to restrict the symmetry of our source/fields to a particular angular momentum (i.e. angular dependence eimφ). See also: Ring resonator in cylindrical coordinates for how to solve for modes of this cylindrical geometry much more efficiently.
## More examples
The examples above suffice to illustrate the most basic features of Meep. However, there are many more advanced features that have not been demonstrated here. So, we hope to add, over time, a sequence of examples that exhibit more complicated structures and computational techniques. The ones we have so far are listed below.
If you have a good example you would like to share, feel free to register for a user name and log in, and then add a link to a new page for your example below. (Please prefix the link with "Meep Tutorial/". Simple examples illustrating a specific concept are preferred.)
## Editors and ctl
It is useful to have emacs use its scheme-mode for editing ctl files, so that hitting tab indents nicely, and so on. emacs does this automatically for files ending with ".scm"; to do it for files ending with ".ctl" as well, add the following lines to your ~/.emacs file:
(push '("\\.ctl\\'" . scheme-mode) auto-mode-alist)
or if your emacs version is 24.3 or earlier and you have other ".ctl" files which are not Scheme:
(if (assoc "\\.ctl" auto-mode-alist)
nil
(Incidentally, emacs scripts are written in "elisp," a language closely related to Scheme.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 12, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8151353597640991, "perplexity": 1597.8922767151594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945082.84/warc/CC-MAIN-20180421071203-20180421091203-00517.warc.gz"} |
http://math.stackexchange.com/questions/237838/fitting-curve-for-newtons-cooling-law-data-programatically?answertab=active | # Fitting curve for Newton's cooling law data programatically?
The data are for the model $T(t) = T_{s} - (T_{s}-T_{0})e^{-\alpha t}$, where $T_0$ is the temperature measured at time 0, and $T_{s}$ is the temperature at time $t=\infty$, or the environment temperature. $T_{s}$ and $\alpha$ are parameters to be determined.
How can I fit my data against this model? I'm trying to solve $T_{s}$ by $T_{s}=(T_{0}T_{2}-T_{1}^{2})/(T_{0}+T_{2}-2T_{1})$, where $T_{1}$ and $T_{2}$ are measurements in time $\Delta t$ and $2\Delta t$, respectively.
However, the results are varying a lot through the whole data set.
Shall I try gradient descent for the parameters?
-
If you use gradient descent, what cost function are you going to minimize? – littleO Nov 15 '12 at 9:32
I think $T_s$ was actually supposed to be the temperature at time $t = \infty$ (at steady state). Perhaps there was an incorrect edit. – littleO Nov 15 '12 at 9:39
Yes, t=∞, indeed. The environment is thought to be a source whose capacity is big enough to keep its temperature stable. – ZhangChn Nov 15 '12 at 11:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8938294649124146, "perplexity": 439.6164098270197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830903.34/warc/CC-MAIN-20140820021350-00119-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://www.numericalmethod.com/javadoc/suanshu/com/numericalmethod/suanshu/analysis/integration/univariate/riemann/substitution/package-summary.html | Package com.numericalmethod.suanshu.analysis.integration.univariate.riemann.substitution
• Interface Summary
Interface Description
SubstitutionRule
A substitution rule specifies $$x(t)$$ and $$\frac{\mathrm{d} x}{\mathrm{d} t}$$.
• Class Summary
Class Description
DoubleExponential
This transformation speeds up the convergence of the Trapezoidal Rule exponentially.
DoubleExponential4HalfRealLine
This transformation is good for the region $$(0, +\infty)$$.
DoubleExponential4RealLine
This transformation is good for the region $$(-\infty, +\infty)$$.
Exponential
This transformation is good for when the lower limit is finite, the upper limit is infinite, and the integrand falls off exponentially.
InvertingVariable
This is the inverting-variable transformation.
MixedRule
The mixed rule is good for functions that fall off rapidly at infinity, e.g., $$e^{x^2}$$ or $$e^x$$ The integral region is $$(0, +\infty)$$.
NoChangeOfVariable
This is a dummy substitution rule that does not change any variable.
PowerLawSingularity
This transformation is good for an integral which diverges at one of the end points.
StandardInterval
This transformation is for mapping integral region from [a, b] to [-1, 1].
• Enum Summary
Enum Description
PowerLawSingularity.PowerLawSingularityType
the type of end point divergence | {"extraction_info": {"found_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": 0.9693424701690674, "perplexity": 1748.5036973863403}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039748901.87/warc/CC-MAIN-20181121133036-20181121155036-00346.warc.gz"} |
https://secgroup.dais.unive.it/teaching/cryptography/polyalphabetic-ciphers/ | # Polyalphabetic ciphers
We have seen that monoalphabetic ciphers are prone to statistical attacks, since they preserve the statistical structure of the plaintext. To overcome this issue, it is important that the same plain symbol is not always mapped to the same encrypted symbol. When this happens the cipher is called polyalphabetic.
### Vigenére cipher
This simple polyalphabetic cipher works on “blocks” of m letters with a key of length m. In fact, a key is also a block of m letter. Formally, ${\cal P} = {\cal C} = {\cal K} = \mathbb{Z}_{26}^m$, where $\mathbb{Z}_{26}^m$ is $\mathbb{Z}_{26}\times\mathbb{Z}_{26}\times\ldots\times\mathbb{Z}_{26}$, m times. Encryption and decryption are defined as follows:
$\sf \begin{array}{rcl} E_{k_1, \ldots, k_m}(x_1, \ldots, x_m) & = & (x_1 + k_1, \ldots, x_m + k_m) \mbox{ mod } 26 \\ D_{k_1, \ldots, k_m}(y_1, \ldots, y_m) & = & (y_1 - k_1, \ldots, y_m - k_m) \mbox{ mod } 26 \end{array}$
For example, the sentence “THISISAVERYSECRETMESSAGE” encrypted under key “FLUTE” is computed as follows:
THISISAVERYSECRETMESSAGE +
FLUTEFLUTEFLUTEFLUTEFLUT =
YSCLMXLPXVDDYVVJEGXWXLAX
The plaintext is split into block of length 5 and the key FLUTE is repeated as necessary and used to encrypt each block.
The number of possible keys is $26^m$, i.e., all the possible sequence of letters of length m. For m big enough this prevents brute force attacks. Note that the number of keys grows exponentially with respect to the length. You can also notice, from the example, that one letter is not always mapped to the same one (unless the are at a distance that is multiple of m). For example the fist two letters “I” are encrypted as “C” and “M”, respectively. While the “S” in position 6 and the last one are both encrypted as “X” using the “F” of “FLUTE”. They are, in fact, at distance 15 which is a multiple of 5.
#### Security of the cipher
Even if Vigenére cipher hides the statistic structure of the plaintext better than monoalphabetic ciphers, it still preserve most of it. There are two famous methods to break this cipher. The first is due to Friedrick Kasiski (1863) and the second to Wolfe Friedman (1920). We illustrate the latter since it is more suitable to be mechanized. Both are based on the following steps:
1. Recover the length m of the key
2. Recover the key
The Friedman method uses statistical measures. In particular for step 1 we consider the index of coincidence:
$\displaystyle I_c(x)=\frac{\sum_{i=1}^{26} f_i(f_i -1)}{n(n-1)}\approx\sum_{i=1}^{26}p_i^2$
where $f_i$ is the frequency of i-th letter in a text of length $n$, i.e., the number of times it occurs in such a text, and $p_i = \frac{f_i}{n}$ is the probability of letter i-th. Intuitively, this measure gives the probability that two letters, chosen at random from a text, are the same.
Notice that the value of the index is minimum, with value 1/26 ≈ 0.038, for texts composed of letters chosen with uniform probability 1/26 while it is maximum, with value 1, for texts composed of just a single letter repeated n times. It is, in fact, a measure of how non-uniformely letters are distributed in a text. So each natural language has a characteristic index of coincidence. For English the value is approximatively 0.065
1) Recover the length m of the key. The idea is to find the length by brute-forcing, following this algorithm:
m = 1
LIMIT=0.06 # this is to check that ICs are above 0.06 and thus close to 0.065
found = False
sub = subciphers() # takes the m subciphertexts sub[m] obtained by selecting one letter every m
found = True
for i in range(0,m): # compute the Ic of all subtexts
if Ic(sub[i]) < LIMIT:
# if one of the Ic is not as expected try to increase length
found = False
m += 1
break
# survived the check, all Ic's are above LIMIT
output(m)
For example, for length 2 we get the two subciphertexts composed of only the letters in odd positions and even positions, respectively. We require that the index of coincidence of all the subtexts is close to the characteristic index of the plaintext language. Typically, the bigger is the index the higher is the probability that the frequencies of the letters are close to the one of the plaintext language. It is thus enough to choose a lower bound such as 0.06 and check that the ICs are above that.
2) Recover the key. We find the relative shift between two subciphers by using the mutual index of coincidence
$\displaystyle MI_c(x,x')=\frac{\sum_{i=1}^{26} f_i f'_i }{n n'}=\sum_{i=1}^{26}p_i p'_i$
representing the probability that two letters taken from two texts x and x' are the same. The idea is to shift one subcipher until the mutual index of coincidence with the first subcipher becomes close to the one of the plaintext language. When this happens, we know that the applied shift is the relative shift between the two subciphers and, consequently, between the corresponding letters of the key. This is encoded in the following algorithm. In fact, what we do, is to select the relative shift that maximizes the mutual index of coincidence.
key = [] # empty list
for i in range(0,m): # for any letter of the key
k = 0 # current relative shift
mick = 0 # maximum index so far (we start with 0)
for j in range(0,26): # for any possible relative shift
# compute the mutual index of coincidence between the first subcipher
# sub[0] and the i-th subcipher shifted by j
mic = MIc(sub[0], shift(j,sub[i]))
if mic > mick: # if it is the biggest so far
k = j # we remember the relative shift
mick = mic # ... and the new maximum
key.append(k) # we append to the list the shift we have found
In the list key we obtain the list of relative shifts. For example key = [0,4,6,3,9] means that the second letter of the key is equal to the first plus 4 while the third is the first plus 6 and so on. The final step is to try all the possible 26 first letter of the key, giving 26 possible keys. (This step could be avoided computing the MIc with a reference text written in the plaintext language.)
Challenge. Can you write a program that breaks the following ciphertext?
WTTNRAVWSKAMFFEVREBKZXMKLCLANMOZSWDXKOHKTQDMCDVWOIIUHXZWXIYVNRYSWNUZAFCVZEUMIKKU
ZZURVXKZEFAEOICEQEKBQAETAXDQVZIWWWEBAZCHJAEGFEJYAZLMOMOLFRYYFVAZESRLZHXK | {"extraction_info": {"found_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": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.897058367729187, "perplexity": 809.8636482236375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1627046153521.1/warc/CC-MAIN-20210728025548-20210728055548-00284.warc.gz"} |
https://en.wikipedia.org/wiki/Talk:Random_walk | # Talk:Random walk
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## Probabilistic interpretation
Please, would you be so kind to give an example of a random walk under this heading, for example, the geometric distribution. It would especially be helpful if you could show how the probability distribution $P(N = n) = p(1-p)^{(n-1)}$ for $n = 1, 2, 3, ...$ could be derived in the framework of a random walk as well as the central moments and, if possible, the maximum likelihood estimator for $p$. — Preceding unsigned comment added by Ad van der Ven (talkcontribs) 15:27, 17 June 2011 (UTC)
## Higher Dimensions
The article says: "Will the drunkard ever get back to his home from the bar? It turns out that he will" Wouldn't "Will the drunkard always get back to his home from the bar? It turns out that he will" be more accurate? For me "ever" implies any probabilty > 0 (which is trivially true) whereas "always" would more accurately describe "p = 1". --flatfish89 (talk) 17:46, 19 October 2008 (UTC)
As I understand it, with infinite time (as implied by "ever"), any probability > 0 would mean that the drunkard will get home - saying that he always gets home is redundant. --Aseld talk 15:48, 7 February 2009 (UTC)
## RMS Distance (1D)
Actually, the average distance after n steps is of the order of $\sqrt{n}$ for practically any definition of average. So the original formulation was quite correct. I made some "compromise" with Miguel, but thinking about it, maybe this sentence can be just removed? What do you say?
You're probably right, except that the root-mean-square is *exactly* sqrt n. I wouldn't remove it, it is one of those intuitive statements that are useful to know. Note that I did not replace "average" by "root-mean-square", but rather supplemented the nontechnical statement with a technical one. — Miguel 18:17, 16 Jul 2004 (UTC)
Can someone show a proof that the rms is exactly sqrt n? This is not a Poisson distribution, it's a random walk - see e.g. http://mathworld.wolfram.com/RandomWalk1-Dimensional.html and i seem to have a simple proof that it's wrong:
The (absolute) distance x after n steps can be anywhere in the range from 0 to n inclusive. Taking the definition from root-mean-square, the only way that xrms can equal sqrt n is if xi = n for every i. This is clearly wrong, since xi should range from 0 to n (with probabilities related to the binomial distribution). Therefore xrms must be strictly less than sqrt n. QED.
So i've modified the rms statement. Please show that my proof is wrong and the sqrt n is right if you wish to reinstate the exactly claim. Boud 13:38, 29 Apr 2005 (UTC)
Your proof is wrong. The mean squared distance is n2, not n, ,if xi = n for every i, giving an RMS distance of n, not sqrt(n).
Sqrt(n) is indeed the correct RMS for a symmetrical random walk, and is always less than or equal to n. Jheald 17:05, 5 February 2007 (UTC)
I can't provide a correct proof (that's what I'm looking for), but I can show you that your proof is invalid. It was a simple oversight: when you sum over the values, you square them first. If every value is n, then the RMS is n, not sqrt n. Luqui 05:21, August 15, 2005 (UTC)
Surely the sqrt(2/pi) factor for the direct mean (as opposed to the rms) is only for the 1 dimensional walk (see the above referenced Mathworld 1-d walk article)? If so, this should definately be pointed out in this article. The proof that the rms is exactly sqrt(n) in 2d (and 3d) is easy to show...see the link in the Mathworld article to the 2d random walk. I have never seen a value for the 'direct' mean in the 2d walk...I assume it's probably next to impossible to calculate, hence why people only worry about the rms, but I assume it wouldn't be sqrt(2/pi). ScottRShannon 05:45, 5 October 2005 (UTC)
In 1 dimension, starting from 0, for n steps there are 2^n possible paths p each with some ending point E(p). The sum over p of (E(p))^2 is exactly n*2^n (easy proof by induction). The mean value of (E(p))^2 is therefore n and the rms is therefore sqrt(n).
Sketch of inductive proof: Clear if n=0. Suppose true for some fixed n. In going from n to n+1, each summand x^2 is replaced by two summands (x-1)^2 + (x+1)^2 = 2x^2+ 2. The sum S of all 2^n terms therefore goes to 2S + 2*2^n. Since S = n*2^n by hypothesis, the new sum is n*2^(n+1) + 2^(n+1) = (n+1)*2^(n+1), as desired. —Preceding unsigned comment added by [[User:{{{1}}}|{{{1}}}]] ([[User talk:{{{1}}}|talk]] • [[Special:Contributions/{{{1}}}|contribs]])
A useful thing to note in the 1D case is that you must move on every step.
So the number of moves to the left mL is simply n - mR; and x = mR - mL = 2 mR - n.
But mR has a binomial distribution, with mean n p and variance n p q. So the mean square distance, <x2> is given by
<x2> = <(2 mR - n)2>
= 4 <mR2> - 4 <mRn> + <n2>
= 4 (<mR>2 + var(mR)) - 4 <mRn> + n2
= 4 n2p2 + 4 n p q - 4 n2p + n2
= 4 n p q + n2(2 p - 1)2
= 4 n p q + n2(p - q)2 (ie: the square of the mean, plus the variance)
= n if p = q = 1/2.
-- Jheald 17:05, 5 February 2007 (UTC)
## Stricto Sensu
Well, I didn't really see the point here. A graph is a graphic representation of something and there is no "strict sense" in which it has to be one way or the other. The time axis is just as important as the space axis. Could you please clarify? Gadykozma 05:17, 24 Jul 2004 (UTC)
Well, the walk takes place in space, so it is only "upwards" or "downwards". It gets difficult for beginners to grasp this if they only see the planar representation. But I may be overly cautios and punctillious. Pfortuny 10:14, 24 Jul 2004 (UTC)
## Cities are Infinite?
I don't agree that cities are infinite. Maybe the word can be changed to indicate that cities are vast, but not infinite. "Imagine now a drunkard walking around in the city. The city is infinite and completely ordered, and at every corner he chooses one of the four possible routes (including the one he came from) with equal probability."
it doesn't say cities are infinite, it says "imagine a city [which is] infinite". This is is mathematics. --Taejo | Talk 15:31, 11 November 2005 (UTC)
## clarification and consistency of figure
I just added a sentence at the end of the first paragraph of the relation to brownian motion section.
"Brownian motion is the scaling limit of random walk in dimension 1."
Whereas the brownian motion article says:
"The term Brownian motion (in honor of the botanist Robert Brown) refers to either
1. The physical phenomenon that minute particles immersed in a fluid move about randomly; or
2. The mathematical models used to describe those random movements."
It's only 2. that is the scaling limit for a random walk. 1. is just a particular type of random movement (follwing the langevin equation).
Right; I've made this more precise by changing "Brownian motion" to "Wiener process" in that section. David-Sarah Hopwood (talk) 17:16, 12 February 2009 (UTC)
Then I have another consistency problem, with the picture that shows "steps" of a brownian motion (it's in both articles). If a brownian motion has steps, then it's a random walk, it's not the scaling limit. If we are speaking of brownian motion as a scaling limit, then there can't be any discernable steps (i.e. the steps are infinitesimal).ThorinMuglindir 13:45, 27 October 2005 (UTC)
OK now I understand the picture... Well, the caption in the random walk article should be the same as in the brownian motion article. It is rather important to mention that each step is distributed according to a normal distribution, because that's what makes it "brownian motion" (the scaling limit of the random walk).ThorinMuglindir 14:32, 27 October 2005 (UTC)
## error/precision in average distance
For a mere (uncorrelated) random walk, if the steps are constant and equal to 1 unit then for the distance from the starting point (net displacement): - the rms is equal to sqrt(n) in both 1 and 2 dimensions (the expected net squared displacement is equal to n) - the average distance asymptotes to sqrt(2n/pi) in 1 dimension but to sqrt(pi*n/4) in 2 dimensions (as consequences of the distribution of a khi law with 1 or 2 dof) - the expressions for correlated random walks are much more complex
[email protected]
## Langton's Ant
Should Langton's Ant be mentioned (and linked to) as another example of a random walker, or is that diverting too far from the main point? 213.106.64.203 17:09, 20 January 2006 (UTC) Buckjack
There doesn't seem to be any randomness in Langton's ant.--GrafZahl 07:41, 21 January 2006 (UTC)
## Non-encyclopedic POV
This article tells the reader to imagine things and asks rhetorical questions. This is not written from an encyclopedic POV, and these parts should be replaced or deleted. Steevven1 (Talk) (Contribs) (Gallery) 13:40, 5 February 2007 (UTC)
The language used here is fairly standard fare for a mathematics discussion. The "Imagine..." is setting up an analogy to the problem, and the question "Will the drunkard ever get back to his home from the bar?" then asked is not rhetorical, it is also analogy to the equivalent hypothesis, "A random walk from point A will eventually reach point B". While it may seem "non-encyclopedic" to someone who does not frequently read lay-discussions of mathematics, this is probably the clearest way to illustrate this concept. Dolohov 19:57, 12 March 2007 (UTC)
Dolohov, please see Wikipedia:Manual of Style (mathematics) especially the section about writing style in mathematics articles. While it is pretty standard for the mathematics community, it is inappropriate and unnecessary in an encyclopedia. Use of the "royal we" should be avoided whenever possible. Cliff (talk) 06:02, 5 April 2011 (UTC)
Er... do you realize you replied to a comment more than 4 years old? I think it may be time to archive some things on this page. ~Amatulić (talk) 06:26, 5 April 2011 (UTC)
Of course I do. This is my attempt to revive discussion on this point. It makes no sense to create a new section to discuss a topic that still (after 4 years!) Has not been fixed. It's important to see old discussion on a topic. Cliff (talk) 15:57, 10 April 2011 (UTC)
## Reorganization
I added a picture and sorted the sections with the more elementary on top. This makes the article a bit non-rigorous (with examples and pictures before the definition) but should make for a better read I hope. Oleg Alexandrov (talk) 03:19, 18 April 2007 (UTC)
## A random walk will always return to the origin regardless of the number of dimensions.
What is wrong with this explanation, assuming that "In a one dimensional random walk every point (including the origin) is crossed an infinite number of times in an infinite amount of time." is true?
In a one dimensional random walk every point (including the origin) is crossed an infinite number of times in an infinite amount of time. If we add a second dimension the line representing the point in the original 1 dimension will be crossed an infinite number of times. Since both dimensions are independent of each other there will be an infinite number of occurrences of both dimensions crossing the origin at the same time. This analogy can be extended to any number of dimensions where the third dimension will cross the plane, representing the original 2 dimensions, an infinite number of times.
From this I would conclude that a random walk will always return to the point of origin regardless of the number of dimensions if an infinite amount of time is allowed. In other words there always exists the chance that a random walk will take the shortest possible path to it's origin at any point in time and space. For example after n steps the random walker could be at coordinate (0,0,n) with probability $p=\left({1\over 6}\right)^n$ thus there exists with equal probability that the random walker would take the shortest path to the origin from the farthest possible distance. How can their exist a chance that it would never(p=0) return to the origin if their always exists(p>0) a chance that it could return to the origin in any finite number of steps from any conceivable (Manhatten) distance from the origin. --ANONYMOUS COWARD0xC0DE 03:46, 2 June 2007 (UTC)
What mathematicans are usually concerned with is if the random walk visits a given point infinitely many times, not just one time. In particular, a simple symmetric random walk on the d-dimensional lattice doesn't visit any point infinitely often for dimension greater than two. The section on higher dimension needs this distinction to be brought up and clarified. Filam3nt 22:46, 3 June 2007 (UTC)
Lets generalize to 'd' dimensions travelling 'n' units straight out and back to the origin $p=\left({1\over 2d}\right)^{2n}$. This means their exists a chance for a random walk to return to the origin once in any number of dimensions. Lets say k times: $p=\left(\left({1\over 2d}\right)^{2n}\right)^k$ and $\lim_{k\rightarrow\infty}{\left(\left({1\over 2d}\right)^{2n}\right)^k}=0$, but this also applies to the 2-D case so their must be something I am missing. In addition [1] never mentions a random walk returing to the origin an infinite number of times. --ANONYMOUS COWARD0xC0DE 03:46, 29 June 2007 (UTC)
Nobody's claiming that the probability of returning to the origin is zero. The claim is that in 3 or more dimensions, the probability is less than one. The flaw in your "take each dimension separately" argument is that, although the path will almost certainly cross the x-y plane infinitely often, there's no reason to suppose that any of those crossings will coincide with crossing the y-z plane and the z-x plane. 65.57.245.11 05:26, 10 September 2007 (UTC)
You're right, the crossing coincidence is not certain, just almost certain. However, I don't see why the chance of hitting an arbitrary bound with a 1D walk is anything but almost certain either, which is why I call into question the statement "A simple random walk on $\mathbb Z$ will cross every point an infinite number of times." 80.203.160.34 (talk) 14:09, 9 October 2015 (UTC)
I had the same difficulty understanding this so I had to read a book describing the proof. I ended up conceptualizing it as similar in nature to the issue of an escape velocity, although there is a force pulling the two objects together, they never meet. In the same way although there is always a certain non-zero probability that the random walk will return to the origin the probability does not sum to one even after considering an infinite sum in the same way that an infinitely long integral of the force over time applied to an object at it's escape velocity will only, manage to cancel out the momentum, not reverse it (in order to meet). So just think of the third dimension as the "probabilistic" escape velocity of a random walker. I think such an analogy works quite well. 71.147.50.115 (talk) 21:15, 25 March 2010 (UTC)
I just realized that the analogy works quite well in another regard. In three dimensions the inverse square law holds for forces. So the force at a distance 'd' is proportional to 1/d^2. Since $\int_{1}^{\infty} \frac{1}{x^2} \, dx=1$ is finite it only takes a finite amount of kinetic energy to achieve an escape velocity. Whereas if the forces applied only in two dimensions the force of gravity would be proportional to 1/d, and since $\int_{1}^{\infty} \frac{1}{x} \, dx=\infty$ is infinite it would take an infinite amount of kinetic energy to achieve an escape velocity (which is impossible, therefore the probability of returning is always one in less than or equal to two dimensions). 71.147.50.115 (talk) 21:59, 25 March 2010 (UTC)
## Drunk Sailor
Rather than drunkard's walk, I remember the term "drunk sailor's walk" - or is this only used in German? —Preceding unsigned comment added by 84.136.218.223 (talk) 14:58, 27 September 2007 (UTC)
## Some clarification on the a and b formula in the one dimensional case
Would someone please clarify the sentence leading up to the b/(a+b) formula. a and b are fixed positions. The current usage of "steps" could refer either to steps taken or a fixed number of steps from the origin. It would probably read better like this: "The expected number of steps until a one dimensional random walk goes up to position b or down to position -a is ab. The probability that the random walk will go up to position b before going down to position a is ..." —Preceding unsigned comment added by Speculator mike (talkcontribs) 21:45, 28 September 2007 (UTC)
I tried to derive this property... Ouch ! Could somebody give me little hints of derivation ? Sincerely yours, Damien. — Preceding unsigned comment added by Lamina-le-sédentaire (talkcontribs) 09:15, 12 October 2011 (UTC)
## Clarification of the general definition
Does the "move distribution" for a generic random walk have to be independent of the current position? If so (or if not so) could this be specified somewhere in the introduction. If so then presumably the sample space has to follow certain symmetry conditions which depend on the "move distribution." For example you couldn't have a Gaussian random walk on the interval [0,1] because of the boundaries. —Preceding unsigned comment added by 129.31.244.53 (talk) 20:01, 16 October 2010 (UTC)
## Redid intro and defition
I revised/expanded the intro somewhat and totally changed the definition. Hopefully the intro covers the topic more broadly. As for the definition, it formerly said:
This is far too narrow since RW's can be correlated, biased and have any step-length distribution. However, there is some inconsistency in the literature about definitions (for example: is a random walk a kind of Markov process? or are some Markov processes a limited form of a random walk?) and I do not presume that the definition/notation I provide is canonical. Best, Eliezg (talk) 19:57, 14 January 2008 (UTC)
## The general case random walk
There does need to be some clarification of terminology etc., if the article is to be extended (or split) to deal with more general types of random walks. For example, Feller Vol 2 p 192, use ordinary random walk for the case where steps are of sizes -1,+1 only (with possibly unequal probabilities), but simple random walk for the same case on p213, simple binomial random walk on p393 and binomial random walk on p395, while Feller Vol 1 p363 uses generalized random walk for cases where steps may be of any integer size. In addition, Feller Vol 2 p190 uses general random walk for cases where the step size may have any (1-dimensional) distribution. In all cases the steps are independent across times. Melcombe (talk) 16:41, 11 June 2008 (UTC)
Indeed, there is no perfect consistency in the terminology in textbooks. But that's not a big deal. The article currently is ordered by increasing generality. There is some point to be made that for this particular subject the more general setup of simple random walks on graphs is a better starting point than the specific case of simple random walk on ${\mathbb Z}$. But I don't think that this is sufficiently compelling to warrant a rewrite. Oded (talk) 21:24, 15 June 2008 (UTC)
## Formal Definition
The article could do with a mathematical definition somewhere, perhaps after the introduction? The descriptive definition is good, but it plunges into examples without a formal statement of what a random walk is. Wjastle (talk) 16:17, 16 August 2010 (UTC)
Even the Mathematica web site has a similar definition: http://mathworld.wolfram.com/RandomWalk.html
However, here's a mathematical one:
For the sum $S_n=x_1+x_2+x_3+\dots +x_n$, where $\{x\}_{k=1}^\infty$ is a sequence of independent discrete random variables, the sequence of sums $\{S\}_{n=1}^\infty$ is known as a random walk.
Paraphrased from http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter12.pdf
I'm not sure that such a definition adds any clarity to the article. I wouldn't object if someone adds it, though. ~Amatulić (talk) 17:36, 16 August 2010 (UTC)
## Gaussian random walk - Is this really correct?
>For steps distributed according to any distribution with a finite variance (not necessarily just a normal distribution), the root mean squared expected translation distance after n steps is sigma x sqrt(n)
This doesn't seem right. Should it no be "any symmetric distribution" (3rd moment = 0)? —Preceding unsigned comment added by 156.109.18.2 (talk) 11:14, 24 August 2010 (UTC)
The statement is correct. The distance from the origin is the sum of a number of step. The variance of the distance is the sum of the variances of the step. The "root mean squared translation distance" is just the square root of the variance. A slightly odd terminology, which doesn't need the "expected" where it was. But there is no need to assume symmetry, just the existence of the variance of the distribution of steps. Melcombe (talk) 13:58, 24 August 2010 (UTC)
Thank you very much for your reply. I am not familiar with the term "root mean squared translation distance", but the formula seems to say it is sqrt(E(S_n^2)), where S_n is the position after n steps.
Could you clarify my mistake: if I step +1 with probability 1 then sigma is 0 and S_n=n so sqrt(E(S_n^2))=n, which does not seem to match the formula
Also, when you say "root mean squared translation distance" is just the square root of the variance, do you mean the variance of S_n? Again this does not appear to match the formula unless E(S_n) = 0?
Thanks
OK, for that formula a zero mean is required for the step size, and I have put that in. (A zero mean is not the same as symmetry.) I guess some more general formulae could be put in to cover the more general case, but the obvious way to do it would not match in well with the rest of the article. Melcombe (talk) 16:59, 24 August 2010 (UTC)
I originally wrote that section. Thanks for fixing it up. ~Amatulić (talk) 17:09, 24 August 2010 (UTC)
Thanks for the reply. I think this formula is interesting because it says that when the mean is 0 you can expect stay reasonably close to the starting point (within sqrt(n)) no matter what the distribution. In the general case you can be all over the place so the corresponding formula would probably not say very much —Preceding unsigned comment added by 156.109.18.2 (talk) 17:39, 24 August 2010 (UTC)
But, in the more general case, you will be within a smallish distance (within sqrt(n)) of a known location (that depends on the mean step size) that drifts away from zero at a fixed rate, which is actually quite simple. Melcombe (talk) 08:46, 31 August 2010 (UTC)
## Clarification of the general definition
Does the "move distribution" for a generic random walk have to be independent of the current position? If so (or if not so) could this be specified somewhere in the introduction. If so then presumably the sample space has to follow certain symmetry conditions which depend on the "move distribution." For example you couldn't have a Gaussian random walk on the interval [0,1] because of the boundaries. 129.31.244.53 (talk) 20:02, 16 October 2010 (UTC)
## Bounded random walk
I don't see anything about bounded random walks here. That seems a shame. A walk on a finite graph is finite, OK, but what about the continuous case? e.g. random walk in a space with one boundary (e.g. position can't go below zero). --mcld (talk) 15:32, 19 February 2011 (UTC)
## How does this passage square with the transience in 3d ???
The section Random walk on graphs begins as follows:
"Assume now that our city is no longer a perfect square grid. When our drunkard reaches a certain junction he picks between the various available roads with equal probability. Thus, if the junction has seven exits the drunkard will go to each one with probability one seventh. This is a random walk on a graph. Will our drunkard reach his home? It turns out that under rather mild conditions, the answer is still yes. For example, if the lengths of all the blocks are between a and b (where a and b are any two finite positive numbers), then the drunkard will, almost surely, reach his home. Notice that we do not assume that the graph is planar, i.e. the city may contain tunnels and bridges."
Can someone please explain how this is consistent with the fact that the symmetric random walk in 3D (i.e., on the 1-skeleton of the cubical tiling) is known to be transient?
Certainly all "blocks" have equal lengths. The statement "Notice that we do not assume that the graph is planar" seems to allow a grid like this 3D one. What properties of the graph are assumed???
If this is not what is intended here, perhaps the above-quoted statement could be modified to make clear just why such grids as the 3D one (and for that matter all the corresponding nD ones, n ≥ 3) do not qualify. Thanks.Daqu (talk) 23:50, 5 March 2011 (UTC)
This section is extremely confusing and could really do with some rewriting. Or decent citations. I'm really interested in this subject and frustrated that all the sources I can find are hopelessly impenetrable; Someone should really fix this section up! 152.179.216.94 (talk) 15:25, 17 April 2014 (UTC)
## I want to submit a couple of Random Walk .gif's
I created an account so I could submit my Random Walk animated gifs but now it tells me I must be Autoconfirmed. Can I just give these to someone to put on the page? I think they're perfect ... small and actually a Random Walk, unlike the Brownian motion simulated by Random Walk video. — Preceding unsigned comment added by DrDInfinity (talkcontribs) 00:59, 2 July 2011 (UTC)
## Video not working
I saw Firefox 13.0.1 cannot play the video, it previews, but when I click play, nothing happens. Now3d (talk) 12:22, 6 July 2012 (UTC) | {"extraction_info": {"found_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": 15, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8635328412055969, "perplexity": 732.9862886927692}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1443737951049.91/warc/CC-MAIN-20151001221911-00231-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/covariant-derivative-along-a-horizontal-lift-in-an-associated-vector-bundle.609714/ | # Covariant derivative along a horizontal lift in an associated vector bundle
1. May 29, 2012
### n.evans
I am trying to familiarize myself with the use of fibre bundles and associated bundles but am having some problems actually making calculations. I would like to show that the covariant derivative along the horizontal lift of a curve in the base space vanishes (which should be just a matter of employing definitions correctly I think):
Consider a principal fibre bundle $(E, \pi, M)$ with a structure group $G$ associated to a vector bundle $(E_F, \pi_F, M)$, where $F$ is a vector space. Let $\alpha(t): [a, b] \rightarrow M$ and $\alpha^{\uparrow}_F(t): M \rightarrow E_F$ be the horizontal lift of $\alpha$ in the associated bundle. Let $\Psi(x): M \rightarrow E_F$, with $x \in M$ be a section of the associated bundle, such that $\Psi(\alpha(t)) = \alpha^{\uparrow}_F(t)$
Show that the covariant derivative $\nabla_{\alpha}\Psi$ evaluated along $\alpha(t)$ vanishes.
I know that the covariant derivative can be written as
$\nabla_{\mu}\Psi(x) = \partial_{\mu}\Psi(x) + A_{\mu}(x)\Psi(x)$
but I cannot work out how to use the relation between $\Psi$ and $\alpha^{\uparrow}_F(t)$ to show that it vanishes (if indeed it should). | {"extraction_info": {"found_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": 0.9897503852844238, "perplexity": 87.39523541913586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1534221210105.8/warc/CC-MAIN-20180815122304-20180815142304-00346.warc.gz"} |
https://pure.au.dk/portal/en/[email protected] | # Department of Engineering
## Johannes Liljenhjerte
PhD Student
### Profile
Project title: Extended reach interventions
Project description:
When very long pipes/rods are subjected to a significant compressive load, instabilities can occur in the form of buckling. A phenomenon that can be observed by placing a long slender ruler vertically on a table and then press down its top edge until it starts bending.
This project is focusing on the oil and gas industry where kilometers long coiled tubings (CT) are injected into oil wells (of length up to 12km) for various operations such as removing scales or releasing chemicals. A build-up of compressive forces form due to friction between the CT and well casing. This build-up of compressive forces can initiate different buckling modes along the length of the tubes that eventually end up in a helical shape on the inside of the well. When this happens, friction between the tube and well increases to an extent where the tube locks up inside the well and thus can reach no further.
In this project, a novel solution to reduce friction and provide increased structural integrity for the tube is proposed. The desired properties and viability of the solution is investigated computationally and later experimentally before introducing it in the field.
Instabilities in pipes/rods appear across various length scales and industries, thus the investigation can find its usefulness in other contexts as well, for example when inserting catheters into the body, jamming nanorods into confined channels or DNA packing inside viral capsules.
Supervisor: Assoc. Prof. Jens Vinge Nygaard
• ## Participation Certificate
Prize: Prizes, scholarships, distinctions
ID: 129318628 | {"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": 0.8487055897712708, "perplexity": 2194.160448836886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585371876625.96/warc/CC-MAIN-20200409185507-20200409220007-00298.warc.gz"} |
http://www.nucleonica.net/wiki/index.php/Help:Gamma_Spectrum_Generator | # Help:Gamma Spectrum Generator
Create html file for this document
## INTRODUCTION
Nowadays the γ-spectrometry is used on a daily basis in different basic and applied fields of nuclear science and technology. A variety of instruments and measurement techniques, involving γ-spectrometry measurements, are employed routinely by nuclear and radio-chemists, health physicists, nuclear facility operators, radiation protection staff, safeguards inspectors, border police, customs and low-enforcement officers. Needs for education and training in these areas are high and, obviously, they will be increasing in the future as new challenges, such as strengthening nuclear safeguards and security, nuclear terrorism prevention and implementation of new standards in radiation safety and protection, arise.
To address these growing demands in education & training, an interactive web-assessible simulation tool, Gamma Spectrum Generator, has been developed and integrated into the NUCLEONICA nuclear science and data portal. The simulator presents an efficient visual teaching aid that is especially useful in training facilities, which have restrictions on the use of radioactive substances, or when sources of special interest (e.g. spent fuel, enriched U, weapon grade Pu or other highly radiotoxic materials) are not readily available.
This document describes the underlying mathematical approach and user interface of the Gamma Spectrum Generator. It also contains the results of the experimental validation of the simulator and some examples of its application.
## BASICS OF THE GAMMA-SPECTROSCOPY
Gamma-spectroscopy is an extremely important nuclear and radioanalytical method. Most radioactive sources produce γ-rays of various energies and intensities. When these emissions are collected and analyzed with a γ-spectroscopy system, a gamma energy spectrum can be produced. A detailed analysis of this spectrum is typically used to determine the identity and quantity of γ-emitters present in the source. The γ-spectrum is characteristic of the γ-emitting nuclides contained in the source. The equipment used in γ-spectroscopy includes an energy sensitive radiation detector, a pulse sorter (multichannel analyzer), and associated amplifiers and data readout devices. The most common detectors include sodium iodide (NaI) scintillation counter and High Purity Germanium detectors (HPGe).
For more details on the γ-spectroscopy method and system components see the following references in Wikipedia
## GAMMA-SPECTRUM SIMULATION APPROACH
### Basic assumptions
The γ-ray spectrum is modeled assuming that:
• a point-like γ-source is measured, i.e. γ-ray self-attenuation is negligibly small,
• the detector pulse pile-up effect due to both true and random coincidence summing is negligible,
• the losses of counts due to the spectrometer dead time are negligible, and
• a multi-channel analyzer with absolute linearity is used.
Based on these assumptions, the number of counts F(N) in the N-th channel of a simulated spectrum is calculated as
$F(N)=\sum_{i=1}^{N_\gamma} N_i\,\int_{\Delta N}^{\Delta(N+1)}R(E,E_i)\,dE$,
where Nγ - the number of γ-rays with different energies emitted by a source, Ei - the energy of the i-th γ-ray, keV, Ni - the total number of i-th γ-rays emitted during the spectrum measurement, Δ - the spectrum channel width in energy units, keV/channel, R(E,Ei) - the detector response for a γ-ray with energy Ei.
The number of γ-rays emitted during the spectrum measurement is calculated from the number of disintegrations of particular nuclide, which is evaluated using the Nucleonica Decay Engine. The γ-ray energies and emission probabilities are taken from the Nucleonica Database, which is currently based on the evaluated nuclear data library JEFF3.1-RDD.
### Detector response model
Figure 3.1. Illustration to the origin of the detector response components: solid and dotted photon trajectories give rise to the P(E,Ei) and S(E,Ei) components of the detector response, respectively.
The detector response is presented in the form
$R(E,E_i)=\epsilon_{p,tot} (E_i) \left \{ \, P(E,E_i) \, + \, k_s \, S(E,E_i) \, \right \}.$
Here εp,tot(Ei) is the total detection efficiency for γ-rays with energy Ei.
The function P(E,Ei) represents the main part of the detector response resulting from the detection of primary γ-rays, which are transported directly from a source to the detector sensitive volume (see solid lines in Fig.3.1). It is important to note that only those primary γ-rays, which upon arrival to the detector sensitive volume immediately undergo interaction and leave there at least part of their energy, are considered to contribute to this component of the detector response and to the total detection efficiency εp,tot.
The function S(E,Ei) presents additional contribution to the detector response and to the total detection efficiency, which is resulted from the detection of secondary γ-rays (shown by the dotted lines in Fig.3.1). These secondary γ-rays are produced in the interactions of the primary photons with materials surrounding the detector sensitive volume, such as the materials of the detector construction elements, measurement setup components, experimental room objects etc. The dimensionless factor ks determines the magnitude of this additional contribution.
#### Total detection efficiency (primary gammas)
Figure 3.2. Illustration to the total efficiency εp,tot calculation.
The detection of a primary γ-ray can be considered in terms of the following three probabilities:
• the probability εd for a γ-ray to be emitted in the direction of the detector sensitive volume,
• the probability εa for a γ-ray to escape absorption in all absorbing layers on its pathway from the source to the detector sensitive volume, and
• the probability εi for a γ-ray to interact with the detector sensitive volume, so that a signal at the detector output is produced (this porbability is ofter refered as the intrinsic efficiency of a detector).
For a collimated or pencil photon beam the total detection efficiency can be represented simply as a product of these three probabilities, i.e. εp,tot = εd × εa × εi, whereas for uncollimated sources and broad photon beams an integartion over the detector solid angle is required.
In the currently implemented approach the total detection efficiency εp,tot is calculated assuming that a point-like isotropic source located on the axis of a cylindrically symmetric sensitive volume (detector crystal) and separated from it by a number of absorbing layers, as shown in Fig.3.2. An opening at the rear side of the crystal represents a rear contact feature of the conventional coaxial HPGe detectors. The total detection efficiency calcualtion procedure implements the following formula
$\epsilon_{p,tot}(E_i)=\frac{1}{2}\,\sum_{n=0}^{3} \, \int_{t_{n}}^{t_{n+1}}\exp \left ( -\textstyle \sum_m \mu_{a,m}(E_i)\,d_{a,m}/t \right ) \, \left \{ 1- \exp (-\mu_{s}(E_i)\,x_n(t)) \right \} \,dt\,,$
where μa,m - the total photon attenuation coefficient for the material of the m-th absorbing layer, da,m - the thickness of the m-th absorbing layer, μs - the total photon attenuation coefficient for the material of the detector sensitive volume, xn(t) - the length of the photon pathway in the detector sensitive volume for a given value of the polar angle cosine t = cosθ. The formulas for the limits of the polar angle cosine intervals tn and the respective lengths of the photon pathways xn(t) are presented in Table 3.1. In the Table, z is the source-to-detector sensitive volume distance and z0 = R l / (R-r) - L, where R, L, l and r are the dimensions of the sensitive volume defined in Fig.3.2.
Table 3.1. The polar angle cosine intervals and lengths of the photon pathways used in the total detection efficiency calculation.
$n$ $t_n$ $x_n(t)$
$0$ $\frac{z}{\sqrt{z^2+R^2}}$ $\frac{R}{\sqrt{1-t^2}}-\frac{z}{t}$
$1,\, z \geqslant z_0$ $\frac{z+L}{\sqrt{(z+L)^2+R^2}}$ $\frac{L}{t}$
$1,\, z < z_0$ $\frac{z+L-l}{\sqrt{(z+L-l)^2+r^2}}$ $\frac{L-l}{t}+\frac{R-r}{\sqrt{1-t^2}}$
$2,\, z \geqslant z_0$ $\frac{z+L-l}{\sqrt{(z+L-l)^2+r^2}}$ $\frac{z+2L-l}{t}-\frac{r}{\sqrt{1-t^2}}$
$2,\, z < z_0$ $\frac{z+L}{\sqrt{(z+L)^2+R^2}}$ $\frac{z+2L-l}{t}-\frac{r}{\sqrt{1-t^2}}$
$3$ $\frac{z+L}{\sqrt{(z+L)^2+r^2}}$ $\frac{L-l}{t}$
$4$ $1$ $-$
The integrals in the above formula are computed numerically with relative precision 10-9 using optimized recursive adaptive quadratures [1]. The total photon attenuation coefficients μa,m and μs for a given photon energy are computed by the interpolation of tabulated reference photon attenuation data, taken from the Evaluated Photon Data Library EPDL-97 [2]. It must be noted that, because of its highly peaked forward angular distribution and absence of any energy deposition, the coherent scattering is not included into the total photon attenuation coefficients μa,m and μs.
#### Primary photon contribution
The function P(E,Ei) consists of several components representing the main features of a typical detector response shown in Fig.3.3. These features are the Full Energy Peak (FEP), X-ray Escape Peaks (XEP), Single Escape Peak (SEP), Double Escape Peak (DEP), and Compton continuum.
Figure 3.3. Components of the detector response R(E,Ei) for the energy of incident photons Ei = Eγ.
Mathematically the primary photon contribution is presented by the formula
$P(E,E_i)=\omega_{fep,i} \, F_{fep}(E,E_{fep,i}) \, + \, \omega_{sep,i} \, F_{sep}(E,E_{sep,i}) \, + \, \omega_{dep,i} \, F_{dep}(E,E_{dep,i}) \, +$
$+ \, \textstyle \sum_j \omega_{xep,ij} \, F_{xep}(E,E_{xep,ij}) \, + \, \left \{ \, 1 - \omega_{fep,i} - \omega_{sep,i} - \omega_{dep,i} - \textstyle \sum_j \omega_{xep,ij} \, \right \} \, F_{cp}(E,E_i).$
Here, ωfep,i = εfep,i / εtot,i, ωsep,i = εsep,i / εtot,i, ωdep,i = εdep,i / εtot,i, ωxep,ij = εxep,ij / εtot,i – ratios of the FEP, SEP, DEP and XEP efficiencies to the total detection efficiency, respectively. The index i indicates that the ratios must be evaluated for the incident photon energy Ei. The index j denotes that the corresponding value is related to the escape of the j-th characteristic X-ray from the detector sensitive volume.
The probability density distribution functions Ffep, Fsep, Fdep and Fxep describe profiles of the FEP, SEP, DEP and XEP, respectively. The peak centroids are calculated as: Efep,i = Ei, Esep,i = Ei - m0c2, Edep,i = Ei - 2m0c2 and Exep,ij = Ei - Ex,j, where m0c2 = 511 keV is the electron rest mass and Ex,j is the energy of the j-th characteristic X-ray of the constituent chemical elements of the detector sensitive volume. The probability density distribution function Fcp describes the profile of the continuum created in the primary photon interactions, which result in the partial transfer of the photon energy to the detector sensitive volume. Since all profiles are normalized to unity, so as the primary photon contribution
$\int_0^{\infty} P(E,E_i) \, dE = 1.$
#### Secondary photon contribution
The function S(E,Ei) includes contributions from two sources
$S(E,E_i)=\omega_{ann,i}\,F_{ann}(E,m_0c^2)\,+\,\omega_{cs,i}\,F_{cs}(E,E_i).$
The first term describes a γ-ray peak at 511 keV (see Fig.3.3), which is due to the full absorption of 511 keV photons from the annihilation of positrons. These positrons are created in the interactions of high-energy photons (Eγ > 2 m0c2) and supposed to annihilate outside the detector sensitive volume. The ωann is the annihilation peak efficiency εann to total detection efficiency εp,tot ratio. The probability density distribution function Fann describes the profile of the 511 keV peak.
The second term in the formula represents a part of the continuum associated with the detection of the secondary photons (see Fig.3.3). The ωcs gives the magnitude of this contribution relative to the total detection efficiency εp,tot. The probability density distribution function Fcs describes the profile of this continuum.
Since both annihilation and continuum profiles are normalized to unity, the normalization of the secondary photon distribution function S(E,Ei) is
$\int_0^{\infty} S(E,E_i) \, dE = \omega_{ann,i} \, + \, \omega_{cs,i}.$
#### Total detection efficiency (all gammas)
Taking into account the above normalizations of the primary and secondary photon distribution functions, one can easily obtain the following formula for the total detection efficiency
$\epsilon_{tot}(E_i)=\int_{0}^{\infty}R(E,E_i)\,dE\,=\,\epsilon_{p,tot}(E_i)\, \left \{ 1 \,+\, k_s\, (\omega_{ann,i} \,+\, \omega_{cs,i}) \right \}.$
Here both the primary and secondary photon contributions are included. Using the factor ks, one can scale the secondary photon contribution, thus, adapting the detector response model to different measurement conditions, i.e. different detector environment.
#### Peak profiles
The shapes of the full energy γ-peaks are simulated using the Gaussian distribution function
$F_{fep}(E,E_{fep})=G(E;E_{fep},\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}\,\exp{\left ( - \frac{(E-E_{fep})^2}{2 \,\sigma^2} \right )}.$
The energy dependent standard deviation σ is calculated from the energy resolution properties of a detector, described by the function Δ1/2(E) - the Full Width at Half Maximum (FWHM) of a peak, using the relation
$\sigma(E_{fep})=\frac{\Delta_{1/2}(E_{fep})}{2\sqrt{2\ln{2}}}\approx 0.4246609\,\Delta_{1/2}(E_{fep}).$
The energy dependence of the FWHM is treated differently for NaI and HPGe detectors. Particularly, for NaI detectors it is taken as a linear function of the photon energy
$\Delta_{1/2}(E})=a\,+\,b\,E,$
whereas for HPGe detectors it is considered to be propotional to the square root of the photon energy
$\Delta_{1/2}(E})=a\,+\,b\,\sqrt{E}.$
The coefficients a and b in these formulas are calculated based on the user-specified FWHM values for the 122 keV ( 57Co ) and 1332.5 keV ( 60Co ) full energy γ-peaks.
The SEP, DEP and 511 keV annihilation peak profiles are also modeled using the above Gaussian distribution. The only modification concerns the single escape and 511 keV peaks, whose widths include additional contribution from the Doppler broadening, i.e.
$\sigma(E_{sep,511})=\sqrt{\frac{\Delta_{1/2}^2(E_{sep,511})}{8\ln{2}} \,+\, \sigma_{Doppler}^2}.$
The conventional estimate for the Doppler shift of the annihilation photon energies σDoppler ≈ 1 keV [3] is used.
The shapes of the full energy X-ray peaks and X-ray escape peaks are modeled using the Voigt profile, defined as a convolution of the Gaussian and Cauchy-Lorentz distributions
$F_{fep,xep}(E,E_{fep,xep})=\int_{-\infty}^{+\infty}G(E';E_{fep,xep},\sigma) \, L(E-E';\Gamma) \, dE'.$
Here L(E-E';Γ) is the Lorentzian profile
$L(E-E',\Gamma)=\frac{\Gamma}{2\pi}\,\frac{1}{(E-E')^2+\Gamma^2/4}$
with Γ being the full natural width at the half maximum of an X-ray line. The standard deviation σ of the Gaussian distribution represents the detector energy resolution properties and is calculated like in the case of the full energy γ-peaks (see above). A rapid algorithm [4] is employed to calculate the Voigt profile function. The natural widths of the X-ray lines are calculated based on the widths of the atomic levels involved in the respective intra-atomic transitions, taken from the Evaluated Atomic Data Library [5].
#### Continuum profiles
The primary and secondary photon continuum profiles Fcp and Fcs are obtained by convoluting the respective physical continuum profiles Hcp and Hcs with the Gaussian distribution function
$F_{cp,cs}(E,E_i)=\int_{0}^{+\infty} H_{cp,cs}(E';E_i) \, G(E';E,\sigma) \, dE'.$
Here σ represents the energy resolution properties of a detector and is calculated based on the FWHM function Δ1/2, like in the case of the Gaussian full energy γ-peak profile (see above).
The models of the physical profiles are shown in Fig.3.4 and 3.5. They are represented as piecewise continuous functions with the continuity intervals defined based on the physical considerations, briefly explained in the Figures.
Figure 3.4. Physical continuum profile Hcp associated with the detection of primary gammas. The vertical dotted lines shows physically grounded devision of the full energy range into the intervals of the function continuity.
The continuum described by the profile Hcp is mostly formed by the Compton scattering of the primary gammas. For the primary gamma-rays with energies > 2 m0c2, an additional contribution from the Compton scattering events of annihilation gammas created within the detector sensitive volume is present.
Figure 3.5. Physical continuum profile Hcs associated with the detection of secondary gammas. The vertical dotted lines shows physically grounded devision of the full energy range into the intervals of the function continuity.
The continuum Hcs represents the contribution from the secondary photons, which, after being scattered in the detector environment, strike the detector sensitive volume, thus, forming a so-called backscatter peak. Additional contribution to this continuum component appears for photon energies > 2 m0c2. It represents the Compton continuum associated with the detection of the 511 keV annihilation photons created in the materials surrounding the detector sensitive volume.
On each of the continuity intervals the physical profiles are represented by the 8-th order polynomial function.
### Detector response database
In the course of modeling, an extensive detector response database is used for evaluating contributions of the spectrum components. The database has been created using a specially developed and validated Monte Carlo program. It contains a large set of the peak-to-total and continuum-to-total efficiency ratios as well as the parameterized continuum profiles, calculated on grids of the detector crystal dimensions, γ-ray energies and source-to-detector distances. To calculate the efficiency ratios and continuum profiles for arbitrary measurement setup and photon energy, a set of interpolation techniques described in [6] is applied. More details on the detector response database are given in the sub-sections which follow.
#### Scope of the database
The database contains tabulated response data arrays, which were pre-calculated for the generalized models of NaI and HPGe detectors shown in Fig.3.6.
Figure 3.6. Models of NaI and HPGe detectors used in the detector response database calculations (dimensions in mm).
The both models imply a point-like γ-source located on the axis of a cylindrically symmetric crystal (the detector sensitive volume), which is encompassed by a number of fixed standard construction elements (absorbing layers). For the NaI crystal (ρ = 3.667 g/cm3) these fixed elements include:
• the 0.5 mm Al detector end cap (ρ = 2.7 g/cm3),
• the 5 mm crystal packaging made of foam plastic (ρ = 0.2 g/cm3),
• the 0.5 mm Al crystal casing, and
• the 0.5 mm MgO reflector (ρ = 2.0 g/cm3).
The fixed construction elements encompassing the HPGe crystal (ρ = 5.323 g/cm3) include:
• the 1.5 mm Al detector end cap (ρ = 2.7 g/cm3),
• the 0.7 mm inactive Ge at the front and sides of the crystal (ρ = 5.323 g/cm3), and
• the 1.0 mm Al crystal holder (ρ = 2.7 g/cm3).
In both models there is the 30 mm thick Al disk, which simulates the presence of all other detector construction elements (e.g. the photomultiplier tube, cooling system, electronics etc.) behind the detector sensitive volume.
The response data have been tabulated on the grids of the detector sensitive volume dimensions (L and D in Fig.3.6), source-to-detector distance (X in Fig.3.6) and photon energy. The sensitive volume dimension grid has 10 mm step and spans 20 mm to 120 mm ranges for both length and diameter (in total 121 grid points). For HPGe detectors it corresponds to the relative efficiency range from 1.5% to more than 300%, thus, encompassing the practical range of existing high-energy photon detectors. The γ-ray energy grid consists of 61 points, which are equally spaced in logarithmic scale from 10 keV to 10 MeV, covering the useful energy ranges of radionuclide decay and prompt activation gammas. There are only 4 points in the source-to-detector distance grid, specifically 0 cm, 1 cm, 5 cm and 25 cm, which is due to a weak dependence of the response profile properties on this parameter.
Thus, in total, there are: 2 (detector types) × 121 (sensitive volume dimensions) × 61 (photon energies) × 4 (source-to-detector distances) = 59048 records in the database. Each record contains information about:
• the peak-to-total efficiency ratios for FEP, SEP, DEP, XEP and 511 keV peaks,
• the total number and energies of the XEP peaks (8 peaks per each constituting chemical element in the detector sensitive volume, assosiated with the emission of Kα1,2,3 and Kβ1,2,3,4,5 X-rays),
• the cumulative peak-to-total efficiency ratio for XEP peaks,
• the standard uncertainties for all peak-to-total and continuum-to-total ratios,
• the coefficients of the 8-th order polynomial parameterization for each continuity interval of the primary and secondary photon continuum profiles.
#### Detector response generator
The database has been filled up with the data using a specially developed Monte Carlo based Detector Response Generator program DRGen (Version 1.3). Its main features are: (a) the full control over the statistical uncertainties of the target values, i.e. uncertainties of the efficiency ratios and continuum components, (b) the separation of the primary and secondary continuum components in accordance to the definition shown in Fig.3.1, and (c) the automatic processing of the simulated data and storage of the target values in the database. Screen captures of the program main form are shown in Fig.3.7 and Fig.3.8.
Figure 3.7. Screen capture of the DRGen program main form: response calculation running for the 1 MeV incident photons.
Figure 3.8. Screen capture of the DRGen program main form: response calculation running for the 9 MeV incident photons.
For each datapoint in the detector response grid, the DRGen starts with Nmin trials and then proceeds by steps of 100 thousand trials. After the first set of trials and each subsequent step, the simulated data are processed and the statistical uncertainties of the target values are checked against the predefined uncertainty thresholds
• 1% for the peak-to-total and continuum-to-total efficiency ratios,
• 5% for the average uncertainties in each continuity interval of the primary and secondary components of the continuum.
The calculation continues until the uncertainties of all target values are below the respective thresholds or the maximum number of trials Nmax is reached. The minimum and maximum numbers of trials are chosen different in different energy intervals of the incident photons, particularly
• Nmin = 106 and Nmax = 107 for 10 keV ≤ Eγ < 40 keV,
• Nmin = 5 · 106 and Nmax = 1.5 · 107 for 40 keV ≤ Eγ < 1 MeV, and
• Nmin = 107 and Nmax = 2 · 107 for 1 MeV ≤ Eγ < 10 MeV.
The transport of photons through the detector geometry is performed by modeling the main types of interaction - coherent (Rayleigh) scattering, photoelectric effect, pair production effect and incoherent (Compton) scattering. Photon interaction cross-sections are calculated based on the EPDL-97 library [2]. The Compton and Rayleigh scatterings are simulated using GEANT Low-Energy Compton Scattering package GLECS 3.36 [7], which includes atomic electron binding effects (Doppler broadening). In photoelectric interactions the emission of characteristic X-rays of K- and L-series is modeled.
Because of the large number of the detector response grid datapoints and huge computational overhead associated with the electron transport, DRGen uses a simplified approach for tracking electrons, which is beleived, however, to be sufficient for typical dimensions of the γ-detector sensitive volumes and γ-ray energy range of interest. The energy deposited by an electron is assumed to be proportional to a part of its projected range, which entirely lies in the detector sensitive volume. This part of the range is evaluated along the straight line projected in the direction of the electron emission. The bremsstrahlung photons are sampled uniformly along the electron pathway. The photon numbers and energies are sampled using the Thick Target Bremsstrahlung (TTB) approximation. The transport of positrons is modeled using similar approach. The point of the positron annihilation is sampled from the uniform distribution along its pathway. The positron annihilation at rest is assumed by modeling the emission of two 511 keV photons in opposite directions.
To futher enhance the performance of the calculations, the directional cosines of the source photons are sampled only within the solid angle subtended by the detector. For the low-energy photons (Eγ < 40 keV), which suffer severely from the absorption, an additional variance reduction technique is applied. These photons are transported directely to the detector sensitive volume with appropriate reduction of their weight: w = exp(-Σμx), where μ = μtot - μcoh is the total photon attenuation coefficient with the coherent scattering excluded, x is the absorbing layer thickness, and the sum is over all absorbing layers separating source and detector sensitive volume along the photon pathway.
The calculation performance is also optimized through a set of the energy cut-offs / thresholds, which are
• 1 keV and 10 keV - the cut-off energies for photons and electrons respectively, below which the local deposition of their energy is assumed,
• 40 keV - the lower incident photon energy, for which the continuum is calculated,
• 1200 keV - the maximum energy of the incident photons, for which the XEP peak-to-total efficiency ratio is calculated, and
• 1200 keV - the lower threshold for the SEP, DEP and 511 keV peak-to-total efficiency ratio calculation.
#### Uncertainty of the data
The minimum, maximum and average statistical uncertainties of the data in the detector response database are summarized in Table 3.2 and Table 3.3 for NaI and HPGe detectors respectively. The uncertainties are presented for different energy regions of incident photons.
The mean uncertainties of the FEP-to-total efficiency ratio ωfep do not exceed 0.3% in the whole energy range. The individual values of the ratio are well below 0.6% for photon energies up to 3 MeV and for all crystal dimensions and detector types. For higher energies and small detector sensitive volumes (where the contribution of the FEP becomes not so important) the uncertainty of the ratio can reach about 4%.
The individual uncertainties of the cumulative XEP-to-total efficiency ratio ωxep in the low-energy interval are below 0.32% and 1.1% for the NaI and HPGe detectors respectively. The individual uncertainties of the SEP and DEP peak-to-total ratios for energies greater than 3 MeV are below 1.3% and 0.7% respectively. The mean uncertainties of the ratios do not exceed 1% in the full energy range.
The 511 keV peak-to-total ratios have greater uncertainties with the mean values in the higher energy region within 1.0-1.5% and extreme values up to 3.4%. This still can be considered as appropriate since this peak is not usually used in the quantitative γ-spectrometry.
The uncertainties of the continuum-to-total ratios are well below 0.5% in the energy intervals, where the continuum contribution is expected to be significant.
Table 3.2. The relative statistical uncertainties of the NaI detector response data. The values shown are 1σ uncertainties in percent. Cases where the contribution of a particular detector response component can be significant are marked with the grey background.
Energy range Value $\omega_{fep,i}$ $\textstyle\sum_j\omega_{xep,ij}$ $\omega_{sep,i}$ $\omega_{dep,i}$ $\omega_{ann,i}$ $\omega_{cp,i}$ $\omega_{cs,i}$
10 keV - 100 keV min
max
mean
0.017
0.22
0.081
0.10
0.32
0.16
0.29
0.39
0.33
0.12
0.17
0.14
100 keV - 1 MeV min
max
mean
0.028
0.24
0.056
0.24
2.33
0.80
0.049
0.50
0.17
0.097
0.32
0.17
1 MeV - 3 MeV min
max
mean
0.034
0.54
0.11
0.99
2.19
1.27
0.14
3.84
0.56
0.24
2.00
0.63
0.97
17.1
2.82
0.041
0.11
0.055
0.087
0.21
0.13
3 MeV - 10 MeV min
max
mean
0.063
4.18
0.29
0.11
1.25
0.23
0.20
0.69
0.27
0.65
3.36
1.35
0.042
0.110
0.061
0.12
0.26
0.17
Table 3.3. The relative statistical uncertainties of the HPGe detector response data. The values shown are 1σ uncertainties in percent. Cases where the contribution of a particular detector response component can be significant are marked with the grey background.
Energy range Value $\omega_{fep,i}$ $\textstyle\sum_j\omega_{xep,ij}$ $\omega_{sep,i}$ $\omega_{dep,i}$ $\omega_{ann,i}$ $\omega_{cp,i}$ $\omega_{cs,i}$
10 keV - 100 keV min
max
mean
0.027
0.40
0.16
0.61
1.05
0.79
0.18
1.50
0.55
0.10
0.49
0.20
100 keV - 1 MeV min
max
mean
0.022
0.31
0.054
0.89
14.4
3.30
0.043
0.19
0.077
0.078
0.21
0.11
1 MeV - 3 MeV min
max
mean
0.034
0.58
0.12
4.33
12.8
6.20
0.23
6.27
0.81
0.30
3.36
0.92
0.81
13.0
2.05
0.037
0.094
0.051
0.081
0.19
0.12
3 MeV - 10 MeV min
max
mean
0.070
3.25
0.30
0.14
1.20
0.29
0.24
0.63
0.36
0.66
2.60
1.04
0.050
0.100
0.061
0.11
0.20
0.15
#### Accuracy of the data
The accuracy of the data has been studied by comparing the DRGen results with the reference data obtained with help of the MCNP-4C [8] general purpose Monte Carlo N-Particle transport code. The MCNP calculations were performed with the detailed electron transport (mode p e) and realistic bremsstahlung creation (phys:e bnum=1) models. The reference geometries included a point-like γ-source at 25 cm distance from the bare cylindrical Ge crystals with dimensions 2 × 2 cm, 6 × 6 cm, and 12 × 12 cm. The incident photon energies considered were 1 MeV, 3 MeV, and 10 MeV. The results of the calculations are presented in Table 3.4.
Table 3.4. The reference efficiency data obtained with the use of the MCNP code. The numbers in paranthesis indicate the statistical uncertainties at 1σ level. For absolute values, multiply the respective figures by 10-6.
Crystal dimensions, cm E, MeV εtot εfep εsep εdep
2 × 2 1
3
10
168.11 (0.77)
115.69 (0.16)
105.61 (0.18)
13.91 (0.23)
3.066 (0.032)
0.131 (0.007)
1.018 (0.019)
0.823 (0.018)
5.944 (0.055)
4.428 (0.068)
6 × 6 1
3
10
2555.7 (2.0)
2018.7 (1.6)
1889.8 (2.6)
763.0 (2.0)
303.6 (0.9)
60.4 (0.7)
65.0 (0.4)
102.4 (0.9)
37.4 (0.3)
53.2 (0.6)
12 × 12 1
3
10
11101 (7)
9657 (6)
9261 (13)
5725 (9)
3308 (6)
1327 (8)
381.5 (2.1)
817.2 (6.4)
56.8 (0.8)
112.9 (2.4)
For the sake of comparison, the MCNP calculations were also performed using two simplified approaches
• MCNP-1: without electron transport and without bremsstrahlung photon creation (mode p, phys:p ides=1),
• MCNP-2: without electron transport but with bremsstrahlung photon creation modeling using TTB approximation (mode p, phys:p ides=0).
The relative deviations of the DRGen, MCNP-1 and MCNP-2 results from the reference values are summarized in Table 3.5. The data show that all approaches give perfectly accurate estimations for the total detection efficiency. However, the accuracy of the peak efficiencies is strognly influenced by the underlying physical approximations. Particularly, disregarding the electron transport and utilization of an inadequate bremsstrahlung creation model result in the overestimation of the peak efficiencies, which are about a few percent for 1 MeV photons, a few tens percent for 3 MeV photons, and several hundereds percent for 10 MeV photons.
As for the DRGen accuracies, they are perfectly below 1% for the FEP efficiencies at 1 MeV, and FEP and SEP efficiencies of the medium and large sensitive volume detectors at 3 MeV. For the smallest crystal and 3 MeV photons the accuracy of the FEP efficiency is about 4.5%, which is also quite good if one takes into account that 3 MeV is far beyond the typical practical energy range of the detectors with such small crystals. For 10 MeV photons the peak efficiencies are biased to higer values not more than by 25%, which is still very good if compared to the respective deviations of the MCNP-1 and MCNP-2 values.
Table 3.5. The relative deviations (%) of the MCNP-1, MCNP-2 and DRGen results from the reference efficiency data. Cases when the deviations are significant with the confidence probability p > 0.99 are marked with red.
E, MeV Crystal dimensions, cm Value εtot εfep εsep εdep
1 2 × 2 MCNP-1
MCNP-2
DRGen
0.19
0.23
0.00
9.15
6.06
0.25
1 6 × 6 MCNP-1
MCNP-2
DRGen
0.00
0.00
-0.18
1.79
0.74
-0.60
1 12 × 12 MCNP-1
MCNP-2
DRGen
-0.04
-0.03
-0.28
0.86
0.43
-0.57
3 2 × 2 MCNP-1
MCNP-2
DRGen
0.06
0.22
0.18
50.09
17.68
-4.46
9.65
9.31
-0.47
29.64
15.83
3.81
3 6 × 6 MCNP-1
MCNP-2
DRGen
0.02
0.01
-0.08
17.64
4.74
0.00
8.76
2.95
0.07
16.79
10.11
2.93
3 12 × 12 MCNP-1
MCNP-2
DRGen
-0.02
0.00
-0.29
8.30
2.16
-0.12
7.19
3.68
0.85
17.90
11.77
4.36
10 2 × 2 MCNP-1
MCNP-2
DRGen
0.16
0.20
0.18
1026.15
176.81
2.70
507.09
101.68
19.31
603.66
129.55
24.52
10 6 × 6 MCNP-1
MCNP-2
DRGen
0.25
0.26
0.17
182.49
29.20
17.21
183.90
39.06
22.54
237.70
55.16
22.32
10 12 × 12 MCNP-1
MCNP-2
DRGen
0.03
0.01
-0.25
76.17
14.42
13.29
107.44
27.62
15.86
144.18
29.76
-0.51
The computed continuum distributions are compared in Fig.3.9-3.11. One can see that, in contrast to the simplified MCNP-1 and MCNP-2 approaches, the DRGen distributions quite accurately reproduce the reference continuum spectra for all crystal dimensions and photon energies tested.
Figure 3.9. Comparison of the continuum distributions simulated for 2 × 2 cm bare Ge crystal.
Figure 3.10. Comparison of the continuum distributions simulated for 6 × 6 cm bare Ge crystal.
Figure 3.11. Comparison of the continuum distributions simulated for 12 × 12 cm bare Ge crystal.
## EXPERIMENTAL VALIDATION
### Coaxial 60% High Purity Germanium Detector
#### Experimental
The experimental setup consisted of the GC6020 coaxial HPGe detector with 61.8% relative efficiency and 2110 TRP preamplifier. Detector signal processing and spectrum aquisition were carried out by high-throughput electronics consisting of a DSP9660 digital signal processor and an AIM556 interface module. The energy resolution of the system was FWHM = 1.75 keV at 1.33 MeV and FWHM = 0.95 keV at 122 keV.
The measurements were performed with 137Cs, 60Co and 152Eu thin standard spectrometry γ-ray sources. The activities of the sources, as provided in the manufacturer's certificates, were known with 3% relative cumulative uncertainty at 3σ-level. The sources were located on the detector axis and measured twice, at 5 cm and 17 cm from the detector end cap. The background spectrum was measured in a separate run and appropriately subtracted from the source spectra.
The spectrum processing was performed by Genie-2000 spectroscopic software, inhanced by the interactive peak fiiting option. The experimental efficiencies at 5 cm distance were corrected for the cascade summing effect using the Monte Carlo approach described in [9].
#### Detector model parameters
The detector crystal dimensions as well as the thickness of inactive layers were taken from the detector certificate provided and used without any modifications: crystal diameter – 74 mm, crystal height – 53 mm, inactive germanium at the front and sides of the crystal – 0.7 mm, Al end cap – 1.5 mm, end cap to crystal distance – 5 mm. The rear contact dimensions of the crystal (diameter - 10 mm, length - 36 mm) were obtained from the detector manufacturer.
#### Results
Fig.4.1 shows calculated and measured FEP efficiencies for 5 cm and 17 cm distances between source and detector end cap. Open circles represent data points not corrected for cascade summing. One may see that the generator provides an excellent description of the experimental data. Small discrepancies between modelled and measured efficiencies can be caused by the uncertainties of the Ge inactive layer thickness and detector sensitive volume dimensions, which not always correspond to the certified values.
Fig.4.2-4.4 compare the simulated and experimental detector responses. All responses were generated with the value kS = 2.0 for the secondary photon contribution scaling coefficient. One can see that the model spectra reproduce the experimental data quite well in both shape and absolute values. The observed differences are due to the contribution of the secondary radiation (backscatter peak), the shape of which is highly sensitive to the peculiarities of the actual measurement geometry.
Figure 4.1. Comparison of calculated (curves) and experimental (circles) FEP efficiencies for a point source at 5 cm (upper curve) and 17 cm (lower curve) distance from the detector end cap.
Figure 4.2. Comparison of the calculated (curve) and experimental (circles) detector responses for a 60% HPGe detector and point 137Cs source located at 17 cm distance from the detector end cap.
Figure 4.3. Comparison of the calculated (curve) and experimental (circles) detector responses for a 60% HPGe detector and point 152Eu source located at 17 cm distance from the detector end cap.
Figure 4.4. Comparison of the calculated (curve) and experimental (circles) detector responses for a 60% HPGe detector and point 60Co source located at 17 cm distance from the detector end cap.
## PROGRAM INTERFACE DESCRIPTION
### Getting started
The Gamma Spectrum Generator is an interactive web-accessible application which can be used to simulate the gamma spectrum of radioactive substances. The simulator presents an efficient visual teaching aid that is especially useful in training facilities which have restrictions on the use of radioactive substances, or when sources of special interest (e.g. spent fuel, enriched U, weapon grade Pu or other highly radiotoxic materials) are not readily available.
The Gamma Spectrum Generator module interface is shown in Fig.5.1. The basic geometric arrangement of the source, filters and detector is shown schematically.The lines shown indicate some of the paths of photons which lead to a contribution in the detector. Associated with source, filters and detector are a number of input boxes in which one can specify the source and its strength, the filter materials, the source detector distance, type of detector etc.
The module is setup such that the user can run the program immediately through a simple “one-click” calculation with default parameters. To do this, press the “Start” button shown in Fig.5.1. Thereafter, the lower half of the page will immediately show the “Calculation results” tab, which displays the resulting simulated spectrum with default view settings as shown in Fig.5.2.
Figure 5.1. Gamma Spectrum Generator starting page
Figure 5.2. Calculation results of the “one-click” calculation with defaults setting
This “one-click” calculation simulates the spectrum for the 10 MBq 60Co γ-source, as specified by the default settings of the “Element”, “Mass” and “Quantity” controls below the nuclide’s image (see Fig.5.1).
The activity of the source is related exactly to the measurement starting point in time as indicated by the default setting of the “Reference point” dropdown box to the right of the “Quantity” controls.
The “Measurement time” controls at the top of the “Measurement setup” tab is defaulted to 1000 seconds. This is normally a sufficient time interval for detecting nuclide’s signatures in the real conditions.
The detector setup in the “Current configuration” dropdown box in Fig.5.1 is defaulted to an unshielded NaI detector with crystal dimensions 3" x 3" (76.2 x 76.2 mm) positioned at 25 cm distance from the source.
The graphical presentation of the results is defaulted to the spectral distribution of intensity of counts (count rate in counts per second) at the start of the measurement (see “Data displayed” dropdown box in Fig.5.2).
See the quick visual guide in Section 5.2 below for a brief overview of additional calculation settings and results presentation options available. Section 5.3 provides a detailed description of the user interface.
### Quick visual guide
Figure 5.3. Start view of the Gamma Spectrum Generator page. Controls for selecting nuclide / mixture and setting up the measurement geometry.
Figure 5.4. Measurement geometry setup controls.
Figure 5.5. Calculation options available.
Figure 5.6. Controls available on the "Calculation results" tab.
Figure 5.7. Customizing appearance of the detection efficiency graph.
### Detailed description
#### Choosing nuclide or mixture
By default the page is opened in the Nuclide Selection Mode. To select a nuclide, use "Element" and "Mass" dropdown lists (shown in Fig.5.8) for specifiying the chemical element and mass number of the isotope of interest. In the "Mass" list the stable isotopes are marked with symbol "s" (e.g. "27 s" indicates stable isotope of Al) and the isomers are marked with symbols "m", "n", and "p", indicating the first, second and third isomeric states of the nuclide respectively (e.g., "26 m" corresponds to the first isomeric state of Al-26).
Figure 5.8. Nuclide selection controls.
Figure 5.9. Nuclide selected.
A nuclide of interest can be chosen also from Nucleonica’s Help:Nuclide Explorer page, which can be reached by clicking on the corresponding link provided - the Nuclide Explorer logo box to the right of the "Mass" dropdown list (see Fig.5.8).
Once a source nuclide is selected, the box at the top left of the page (see Fig.5.9) shows an image with its basic properties (decay modes, half-live, existing isomers) indicated. The nuclide’s name to the right of the image provides a link to a part of the Nucleonica Wiki, which describes in detail properties of the chemical element.
To switch to the Mixture Selection Mode, click on the "Nuclide Mixtures Selector" link to the right of the Explorer logo box (see Fig.5.8). This will enable the "Nuclide Mixtures" dropdown list shown in Fig.5.10, which contains a set of the user-specified and pre-defined Nuclide mixtures. When the required mixture is selected, its name appears below the page title (see Fig.5.11) providing a link to the Nucleonica's Nuclide mixtures editor. Clicking on this link redirects to the Nuclide mixtures editor page, which is opened with the current mixture selected.
Figure 5.10. Mixture selection controls.
Figure 5.11. Nuclide mixture selected.
Alternatively, one can start from the selection/creation/editing of a required mixture on the Nucleonica's Nuclide mixtures page. When the mixture is prepared, one can switch to the Gamma Spectrum Generator using the corresponding link provided on the Nuclide mixtures page.
To return to the Nuclide Selection Mode, click on the "Nuclide Selector" link to the right of the "Nuclide Mixtures" dropdown list (see Fig.5.10).
#### Specifying nuclide/mixture quantity
The quantity of a nuclide or a mixture in the source can be specified using the controls shown in Fig.5.12. First, specify the quantity type using the "Quantity" dropdown list and then enter the corresponding numeric value in the edit box to the right of it. The possible quantity types are
• the activity either in Bequerrels or Curries,
• the mass in grams (only for a single nuclide), and
• the number of atoms (only for a single nuclide).
It is important to note, that switching between different types of the nuclide/mixture quantity does not convert the numeric value into the corresponding measurement unit automatically.
Figure 5.12. Nuclide quantity selection controls.
Figure 5.13. Specifying "cooling" time interval.
The nuclide/mixture quantity can be specified either at the moment of its production/certification or at the spectrum measurement starting point of time. This is defined by the selection of an appropriate item in the "Reference point" dropdown list (see Fig.5.13). Note, that the list becomes enabled only if the corresponding option in the "Options" tab, which enables the decay calculations, is selected (see section 5.3.7).
By default, the reference point of time is the "Measurement start". If the "Nuclide creation" is chosen instead, then the "Cooling time" controls for specifying the duration of the source "cooling" time interval become available (see Fig.5.13). Use the respective dropdown list to select a measurement unit for the source "cooling" time interval. The options available are "sec", "min", "hour", "day" and "year". The length of the interval must be entered in the corresponding edit box to the right. Note, that the length value is not converted automatically when a new measurement unit for the "cooling" interval is selected.
#### Specifying the measurement time interval
Figure 5.14. Measurement time controls.
The respective controls (the dropdown list and edit box shown in Fig.5.14) are provided at the top of the "Measurement setup" tab. Use the dropdown list to select an appropriate measurement unit for the time interval. The available options are shown in the figure. Specify the value of the interval in the edit box to the right.
#### Selecting and managing detector configurations
The detector configuration controls shown in Fig.5.15 are available at the top of the "Measurement setup" tab.
Figure 5.15. Detector configuration controls.
The "Current configuration" dropdown list shows available measurement setups, which include 6 default configurations with NaI and HPGe detectors. The default configurations are marked with "(default)" at the end and they appear always on the top of the list. Parameters of these pre-defined configurations are shown in Table 5.1.
Table 5.1. Parameters of the pre-defined (default) measurement setups available in Nucleonica's Gamma Spectrum Generator.
Configuration identification Crystal
L × D, mm
Rear contact
L × D, mm
Source-to-detector, mm Input window, mm Crystal packaging, mm
NaI, L × D = 3 in × 3 in 76.2 × 76.2 None 250.0 Al, 0.5 None
NaI, L × D = 1 in × 2 in 25.4 × 50.8 None 250.0 Al, 0.5 Plastic, 5.0
LEGe, 20 mm × 28 cm2, 0.5 Be 20 × 59.7 None 25.0 Be, 0.5 Vacuum, 2.5
HPGe, coaxial, p-type, rel. eff. 150% 95.0 × 82.0 60.0 × 10.0 250.0 Al, 0.5 Vacuum, 3.0
HPGe, coaxial, p-type, rel. eff. 50% 70.0 × 59.0 45.0 × 10.0 250.0 Al, 0.5 Vacuum, 3.0
BEGe, 30 mm × 50 cm2, rel. eff. 30% 30.0 × 79.8 None 50.0 Be, 0.5 Vacuum, 3.0
Table 5.1. continued
Configuration identification Inactive layer, mm Number of channels Conversion, keV/channel FWHM at 122 keV, keV FWHM at 1332 keV, keV
NaI, L × D = 3 in × 3 in MgO, 0.5 2048 1.0 18 90
NaI, L × D = 1 in × 2 in MgO, 0.5 512 3.0 20 140
LEGe, 20 mm × 28 cm2, 0.5 Be Ge, 0.0003 4096 0.07 0.75 1.64
HPGe, coaxial, p-type, rel. eff. 150% Ge, 0.8 8128 0.30 1.3 2.3
HPGe, coaxial, p-type, rel. eff. 50% Ge, 0.5 8128 0.30 0.8 1.8
BEGe, 30 mm × 50 cm2, rel. eff. 30% Ge, 0.0003 8128 0.30 0.75 2.2
The last entry "<...Edit...>" in the configurations list enables all measurement setup controls on the "Measurement setup" tab, so that new user-specific configurations can be created.
Hint: To start editing a new measurement setup based on an existing configuration, first, choose the configuration, which you would like to have as the basis, and then switch to the edit mode by selecting "<...Edit...>".
Using the "Save as" button, the new configuration can be saved in one's Nucleonica user account. Fig.5.16 shows the "Configuration save" dialog, which appears after the "Save as" button is pressed. Here the user is prompted to enter a unique (!) identification (decription) string for his new measurement setup.
Figure 5.16. Detector configuration save dialog.
After the new configuration is saved, it appears in the list of the available measurement setups after the default configurations.
An outdated user-specific configuration can be deleted using the "Delete" button (see Fig.5.15). After pressing the button the user will be prompted to confirm the deletion. Note, that both "Save as" and "Delete" buttons are disabled when one of the default configurations is selected.
#### Editing the measurement setup
The edit mode is turned on when the last entry ("<...Edit...>") in the "Current configuration" dropdown list on the "Measurement setup" tab is selected (see previous section). This enables the controls on the underlying measurement setup drawing, whereby one can configure his own γ-spectrometer.
By default the drawing displays the basic configurable parameters of the measurement setup shown in Fig.5.17 and 5.18 for NaI and HPGe detectors respectively. These basic parameters include the length and diameter of the detector crystal, the length and diameter of the rear contact of the crystal (available only for HPGe crystals), and source-to-detector distance. One can switch between models in Fig.5.17 and 5.18 by selecting the "NaI" or "HPGe" item in the "Crystal" dropdown list.
Use the "Dimensions in" dropdown list at the left top of the drawing to select an appropriate measurement unit ("mm", "cm" or "inch") for the dimensions entered in the respective text edit controls. Note that all dimensions are recalculated automatically when a new unit is chosen.
Figure 5.17. Basic configurable parameters of the measurement setup with a NaI detector.
Figure 5.18. Basic configurable parameters of the measurement setup with a HPGe detector.
More measurement setup controls appear (see Fig.5.19) when "Show more settings" checkbox at the bottom right of the measurement setup drawing is selected. These additional controls include dropdown lists with associated text edit boxes for specifying the material and thickness of
• the detector input window made of Al, Be or Mylar,
• the detector crystal packaging made of foam plastic or polyethylene in the case of NaI crystals and should be taken as void (vacuum) in the case of HPGe crystals,
• the inactive Ge layer in the front of the HPGe crystal or MgO reflector in the case of NaI detector, and
• additional absorbing filters between source and detector made of Al, Cu, Fe, Pb, Sn, or polyethylene.
Figure 5.19. Advanced parameters of the measurement setup.
The filters can be used to reduce unwanted contribution to the simulated spectrum from the intense low-energy γ-radiation. One can also use them to simulate γ-spectra from containerized or/and self-attenuating sources.
To add a filter to the measurement setup, first, choose appropriate filter material in the "Filter" dropdown box, then specify the filter thickness in the related edit text box and, finally, click the button "Add filter layer". The filter will appear in the table. Up to 6 non-redundant filter layers can be specified this way.
To remove a filter layer from the configuration, select the layer in the table by clicking somewhere within the respective row. The selection will be marked with the grey background and the button "Remove filter layer" will be enabled (see Fig.5.19). Pressing the button will remove the selected filter from the measurement setup.
• the dropdown list "Number of channels in the spectrum accumulated" for selecting the number of channels in the spectrum,
• the text edit box "Channel-to-energy conversion factor, keV/channel" for specifying the channel-to-energy conversion coefficient, i.e. Analog-to-Digital Converter (ADC) conversion gain, and
• two text edit boxes "Energy resolution (FWHM) at 122 keV, keV" and "Energy resolution (FWHM) at 1332 keV, keV" for specifying the energy resolution properties of the spectrometer, i.e. FWHMs at 122 keV and 1332.5 keV.
The available options and valid ranges of the measurement configuration parameters are summarized in Table 5.2. The properties of the materials used in the measurement setup definition are given in Table 5.3.
Table 5.2. Available options and valid ranges of the measurement configuration parameters.
Parameter Control name Control type Options / Range
Detector crystal type "Crystal" dropdown list "NaI", "HPGe"
Dimension measurement units "Dimensions in" dropdown list "mm", "cm", "inch"
Crystal length "Crystal length" text edit box 20 ÷ 120 mm
Crystal diameter "Crystal diameter" text edit box 20 ÷ 120 mm
Crystal rear contact length "Contact length" text edit box 0 ÷ "Crystal length"
Crystal rear contact diameter "Contact diameter" text edit box 0 ÷ 20 mm
Source-to-detector distance "Source to Detector distance" text edit box 0 ÷ 10300 mm
Input window material "Input window" dropdown list Al, Be, Mylar
Input window thickness "Input window" text edit box 0 ÷ 10 mm
Filter layer material "Filter" dropdown list Al, Cu, Pb, Fe, Sn, Polyethylene
Filter layer thickness "Filter" text edit box 0 ÷ 1000 mm
Crystal packaging material "Crystal packaging" dropdown list Vacuum, Foam plastic, Polyethylene
Crystal packaging thickness "Crystal packaging" text edit box 0 ÷ 20 mm
Inactive layer / Reflector material "Inactive layer / Reflector" dropdown list Ge, MgO
Inactive layer / Reflector thickness "Inactive layer / Reflector" text edit box 0 ÷ 5 mm
Number of channels "Number of channels in the spectrum accumulated" dropdown list 28, 29, 210, 211, 212 and 213
ADC's conversion gain "Channel-to-energy conversion factor, keV/channel" text edit box 0.1 ÷ 10 keV/channel
Energy resolution at 122 keV "Energy resolution (FWHM) in keV at 122 keV" text edit box 0.01 keV ÷ "FWHM at 1332 keV"
Energy resolution at 1332.5 keV "Energy resolution (FWHM) in keV at 1332 keV" text edit box "FWHM at 122 keV" ÷ 200 keV
Table 5.3. Properties of the materials used in the γ-spectrometer setup.
Material Chemical formula Density, g/cm3
Aluminium Al 2.70
Beryllium Be 1.85
Maylar C5H4O2 1.39
Foam plastic C2H4 0.20
Polyethylene C2H4 0.935
Germanium Ge 5.323
Magnesium Oxide MgO 2.00
Copper Cu 8.96
Iron Fe 7.87
Tin Sn 7.31
#### Starting calculations
The calculations can be started either in an on-line or background mode by pressing the respective button, i.e. "Start" or "Start in background", to the right of the "Measurement time" setup controls on the "Measurement setup" tab (see Fig.5.20).
Figure 5.20. Press "Start" or "Start in background" button to launch calculations.
If background mode is chosen, a notification will be sent via email and a respective alert will be raised in user’s Nucleonica account, once the task has been completed.
#### Selecting calculation options
Settings on the “Options” tab (see Fig.5.21) provide additional control over the calculation result output and spectrum simulation. These options include
• the "Display detector efficiency curves" checkbox that enables/disables the efficiency graph on the "Calculation results" tab,
• the "Consider decay transformations..." checkbox that enables/disables decay calculations, which will/will not allow a parent nuclide to decay and daugther nuclides to accumulate during the source "cooling" and spectrum measurement time intervals. When the box is checked the additional checkbox "Include gamma-rays of daugther nuclides" and text edit box "Decay Engine's accuracy factor" appear, as shown in Fig.5.21
• if the "Include gamma-rays of daugther nuclides" checkbox is disabled then the contribution to the simulated γ-spectrum from daugther nuclides accumulated during the source "cooling" and measurement time intervals is suppressed
• use the "Decay Engine's accuracy factor" text edit box to enter the value of the accuracy factor used by Nucleonica's Decay Engine in decay calculations
• the "Consider effects of backscatter radiation" checkbox enables/disables the contribution of the secondary photons to the full spectrum. If the box is checked, an additional text box "Backscatter peak normalization factor" appears, which allows specifying the value of the secondary photon contribution scaling coefficient ks (see section 3.2 for more details).
Figure 5.21. Additional settings for the simulation and output control.
Note: All settings on the "Options" tab are disabled by default !
#### Viewing simulated spectrum
The standard output, which appears in the "Calculation results" tab after the calculations have finished, is shown in Fig.5.22. It consists of a spectrum graph with a variaty of associated controls, which allow to switch between different types of the spectral information plotted and adapt appearance of the graph according to one’s needs and requirements.
Important: All new control settings come into force only after the button "Update spectrum graph" is pressed.
Different types of the spectral data can be displayed by selecting respective items in the "Data displayed" dropdown list, in particular
• "Count rate at start" and "Count rate at end" show spectral distributions of the detector count rate (i.e. spectral intensities) at the measurement starting and ending points of time respectively. The datapoints plotted present the intensity of counts in counts per second (cps) in respective spectrum channels,
• "Theoretical number of counts" shows the true mean numbers of counts in the respective channels, which one would obtain if performed averaging on the infinitely large number of spectra measured in absolutely the same conditions (general totality of spectra), and
• "Statistical number of counts" shows one of the possible realizations from the general totality of spectral distributions, i.e. the spectrum which is usually obtained in γ-spectroscopic measurements. The random number of counts in the spectrum channels are obtained from the Poisson distribution, whose mean values are taken from the respective datapoints in the "Theoretical number of counts" spectrum.
Note: If calculation is repeated with exactly the same settings, the only piece of information, which may differ, is the "Statistical number of counts" spectrum and values, which have been derived from it (e.g. count rates, total number of counts etc.).
Hint: It may happen that the statistical spectrum will not contain any significant number of counts when a source with low activity or at large distance from the detector is considered. In this case switching to the "Theoretical number of counts" spectrum is recommended.
Figure 5.22. The γ-spectrum graph with related controls on the "Calculation results" tab.
By default, the full spectrum (i.e. spectrum that is cumulative on the contributing nuclides and different spectrum components) is plotted on the graph as a function of the channel number. The checkboxes "Energy scale", "Spectrum continuum" and "Contribution of scattered photons" beneath the graph allow to convert the channel numbers on the X-axis to the energy units as well as to separate the peaks from the primary and secondary photon continuum contributions. This is demonstrated in Fig.5.23.
Figure 5.23. Use "Energy scale", "Spectrum continuum" and "Contribution of scattered photons" checkboxes to enable energy scale on the X-axis and to visualize different parts of the spectrum continuum.
The contributions of selected nuclides to the full spectrum can be visualized by checking boxes in the last column of the respective rows in the Table below the graph (see Fig.5.24). The other columns in the Table present the count rates at start and end of the measurement and the number of spectrum counts associated with individual nuclides. The last row in the Table shows cumulative count rates and number of counts in the full spectrum.
Note: All decay daugthers appear in separate rows in the Table below the graph, when a single nuclide source is considered. In the case of nuclide mixtures, only starting nuclides appear in the Table. The quotation marks around the starting nuclide name indicate that the figures shown include also contributions from all decay products of this nuclide.
Figure 5.24. Use checkboxes in the table below the graph to visualize contributions of individual nuclides to the full spectrum.
A panel of additional graph controls (see Fig.5.25) becomes available if the "More graph options" checkbox is selected. For each axis on the graph the panel provides similar blocks of controls, which include
• the "Scale" block for selecting appropriate data ranges to be displayed,
• the "Ticks" block for enabling/disabling axis ticks and tick lables,
• the "Tick steps" block for specifying appropriate values for the major and minor tick intervals, and
• the "Grid lines" block for enabling/disabling the major and minor grid lines, which are superimposed on the spectrum image.
The Y-axis can be turned to the logarithmic scale using the "Log" checkbox provided on the respective "Scale" block.
Note: The automatic mode is enabled by default for the "Scale" and "Tick steps" settings.
Figure 5.25. Use additional spectrum graph options to select a spectrum region of interest and to format the graph axises according to your preferences.
#### Viewing efficiency graph
The detection efficiency graph appears below the spectrum graph in the "Calculation results" tab, if the corresponding option in the "Options" tab is activated (see Section 5.3.7). The graph displays the total detection efficiency for the current spectrometer setup, as well as FEP, SEP, DEP and XEP efficiencies as functions of the incident photon energy (see Fig.5.26).
Figure 5.26. The efficiency graph with related controls.
Use "Efficiency displayed" checkbox controls below the graph to enable/disable plotting particular efficiency curves in the graph. A panel with additional graph options appears when the "More graph options" checkbox is selected. The panel appearance, content and function of controls are the same as shown in Fig.5.25. The only difference is the additional "Log" checkbox in the X-axis "Scale" block, which allows to enable/disable the logarithmic scale on the photon energy.
Both spectrum and efficiency graphs can be downloaded to user's computer in PNG or BMP formats. To do this, click the right mouse button when cursor is somewhere within the graph region and select "Save as" option from the context menue, which appears (see Fig.5.27). Then you will be prompted to specify the type of the graphical file and the destination folder and after that downloading begins.
Figure 5.27. Saving spectrum graph using the respective option in the context menue.
Use other options available in the context menue to print the image, or send it via e-mail, or see its properties etc.
A detailed report, containing the complete collection of spectral and efficiency numerical data, can be generated and downloaded as a text or Excel spreadsheet file. The corresponding links are provided in the "View/Save results in Text or Excel format" string located just above the right corner of the spectrum graph (see Fig.5.22-24).
The information in the files is structured as follows
• the "Parameters" block/sheet (see Fig.5.28) contains the full set of the settings, including the spectrometer parameters, source specification and calculation options, which have been used in the calculation,
Figure 5.28. An example of the "Parameters" sheet in the Excel file.
• the "Nuclides" block/sheet (see Fig.5.29) contains the cumulative and nuclide-specific information on the calculated activities, number of decays, count rates and number spectrum counts (theoretical and statistical) at the measurement starting and ending points of time,
Figure 5.29. An example of the "Nuclides" sheet in the Excel file.
• the "X- and Gamma-rays" block/sheet (see Fig.5.30) contains the properties of the γ- and X-rays emitted by the source with associated . For each γ- and X-ray the table indicates the ancestor and emitter nuclides, the photon energy and emission rates at the measurement start and end, the total number of photons emitted during the spectrum measurement, the total and FEP efficiencies, the peak and background number of counts, and the Minimal Detectable Activity (MDA(0)) of the ancestor nuclide at the measurement starting point of time. The MDA(0) values are calculated according to the approach described in [10],
Figure 5.30. An example of the "X- and Gamma-rays" sheet in the Excel file.
• the "Efficiency" block/sheet (see Fig.5.31) contains the efficiency data for the current measurement setup. The total and peak efficiencies are tabulated on the energy grid, which consists of 61 points spaced uniformely on the logarithmic scale in the energy range from 10 keV to 10 MeV. Additional pairs of energy points (one pair per each absorbing layer between source and detector crystal) are added to the energy grid to reproduce accurately the step-shaped behaviour of the efficiency curves near the K-edges,
Figure 5.31. An example of the "Efficiency" sheet in the Excel file.
• the "Nuclide's name" blocks/sheets (one per each nuclide in the source) contain the complete set of the spectral information related to induvidual nuclides (see Fig.5.32). This information includes the count rate spectral distributions at the start and at the end of the measurement time interval, as well as the theoretical and statistical distributions of the numbers of counts in the spectrum. For each type of the distributions the primary and secondary continuum contributions as well as the total spectrum are shown,
Figure 5.32. An example of the Excel sheet with the nuclide specific spectral information.
• the "Full spectrum" block/sheet presents the full spectrum information, i.e. the sum of the respective spectral distributions over all nuclides in the source. The data are structured in the same way as it is done in the nuclide's specific datasheets, shown in Fig.5.32.
## APPLICATION EXAMPLES
This section contains results of different case studies performed with the application of the Gamma Spectrum Generator. All graphs were created using the generator’s tools and downloaded directly from its web-page.
### Unshielded Co-60 and Eu-152 and 3" × 3" NaI
Fig.6.1 and 6.2 show γ-spectra simulated for the 100 kBq 60Co and 100 kBq 152Eu sources and NaI (3" × 3") detector. The detector setup includes 0.5 mm MgO reflector and 0.5 mm Al casing. The energy resolution of the detector is assumed to be FWHM = 18 keV at 122 keV and FWHM = 90 keV at 1332 keV. The sources are positioned at 25 cm distance from the detector and measured for 1000 s. The graphs demonstrate a powerful feature of the generator, which allows to visualize peak and continuum components of the spectrum. In addition, a backscatter peak contribution is shown as a separate continuum component in the graphs. The calculated total and peak efficiencies of the detector setup are shown in Fig.6.3.
Figure 6.1. γ-spectrum simulated for 60Co 100 kBq source and NaI (3" × 3") detector.
Figure 6.2. γ-spectrum simulated for 152Eu 100 kBq source and NaI (3" × 3") detector.
Figure 6.3. Total and peak detection efficiencies calculated for NaI (3" × 3") detector and point-like source at 25 cm distance.
### Spectra from thin NatU sample
The spectra were simulated for a 1 g U sample and three different measurement setups
• NaI detector (crystal dimensions - 76.2 mm × 76.2 mm (3" × 3"), crystal casing - 0.5 mm Al, MgO reflector - 0.5 mm), energy resolution - FWHM = 18 keV at 122 keV and FWHM = 90 keV at 1332.5 keV, number of channels - 2048, shielding - 1 mm Sn, channel-to-energy conversion - 1.0 keV/channel, source-to-detector distance - 250 mm,
• 30% BEGe detector (crystal length and diameter - 30 mm × 79.8 mm, crystal to end cap - 3 mm, inactive germanium - 0.3 um, input window - 0.5 mm Be), energy resolution - FWHM = 750 eV at 122 keV and FWHM = 2.2 keV at 1332.5 keV, shielding - 1 mm Sn, number of channels - 8192, channel-to-energy conversion - 0.3 keV/channel, source-to-detector distance - 50 mm, and
• LEGe detector (crystal length - 20 mm, active area - 2800 mm2, crystal to end cap - 2.5 mm, inactive germanium - 0.3 um, input window - 0.5 mm Be), energy resolution - FWHM = 750 eV at 122 keV and FWHM = 1.64 keV at 1332.5 keV, shielding - 0.5 mm Sn, number of channels - 4096, channel-to-energy conversion - 0.07 keV/channel, source-to-detector distance - 25 mm.
It was assumed that U had been separated from the ore 2 years before the measurement and had natural abundances of 234U, 235U and 238U at the date of the separation. The spectra include contributions from all decay products, which have been accumulated since this date. To obtain better statistics of counts, the spectrum measurement time was 105 s in all cases. The spectra are shown in Fig.6.4-6.9. Figures 6.4, 6.6 and 6.8 present spectrum peak and continuum components. Figures 6.5, 6.7 and 6.9 demostrate contrinbutions of different uranium isotopes to the full spectrum.
Figure 6.4. γ-spectrum simulated for 1 g natual uranium sample and NaI (3" × 3") detector. Spectrum peak and continuum components are shown.
Figure 6.5. γ-spectrum simulated for 1 g natual uranium sample and NaI (3" × 3") detector. Nuclide contrtibutions to the full spectrum are shown.
Figure 6.6. γ-spectrum simulated for 1 g natual uranium sample and BEGe detector. Spectrum peak and continuum components are shown.
Figure 6.7. γ-spectrum simulated for 1 g natual uranium sample and BEGe detector. Nuclide contrtibutions to the full spectrum are shown.
Figure 6.8. γ-spectrum simulated for 1 g natual uranium sample and LEGe detector. Spectrum peak and continuum components are shown.
Figure 6.9. γ-spectrum simulated for 1 g natual uranium sample and LEGe detector. Nuclide contrtibutions to the full spectrum are shown.
### Spectra from shielded TRU waste
Fig.6.10 and Fig.6.11 show low- and high-resolution γ-spectra for a 5.25 TBq source, which represents actinides extracted from a 1 kg sample of 6-year-aged PWR spent fuel. The isotopic composition of the fuel was calculated using NUCLEONICA’s webKORIGEN, assuming 4.2% for the original enrichment and 50 GWd/t for the final burnup of the fuel.
The low-resolution spectrum corresponds to a NaI detector with crystal dimensions - 76.2 mm × 76.2 mm (3" × 3"), crystal casing - 0.5 mm Al, MgO reflector - 0.5 mm, energy resolution - FWHM = 18 keV at 122 keV and FWHM = 90 keV at 1332.5 keV, number of channels - 2048, and channel-to-energy conversion coefficient - 1.0 keV/channel.
The high-resolution spectrum was simulated assuming 30% BEGe detector with the following parameters: crystal length and diameter - 30 mm × 79.8 mm, crystal to end cap - 3 mm, inactive germanium - 0.3 um, input window - 0.5 mm Be, energy resolution - FWHM = 750 eV at 122 keV and FWHM = 2.2 keV at 1332.5 keV, number of channels - 8192, channel-to-energy conversion coefficient - 0.3 keV/channel.
In both cases the source was assumed to be shielded with a 5 mm Pb filter and located at 25 cm distance from the detectors. The measurement time is 1000 s. From the spectra shown one can see easily the advantages of the high-resolution γ-spectrometry when accurate characterization of the sample is required.
Figure 6.10. γ-spectrum simulated for actinides in 1 kg PWR spent fuel shielded with 5 mm Pb and NaI (3" × 3") detector. Nuclide contrtibutions to the full spectrum are shown.
Figure 6.11. γ-spectrum simulated for actinides in 1 kg PWR spent fuel shielded with 5 mm Pb and BEGe (30% rel. eff.) detector. Nuclide contrtibutions to the full spectrum are shown.
### Simulation of a HPGe Detector with the GSG Pro
In this section we show how a HPGe detector can be simulated using the GSG Pro. To enter the detector edit mode select "Edit" from the "Current configuration" drop-down list. The main detector settings can then be changed directly in the schematic detector setup. Additional parameters can be set by activating the "Show more settings" check box. Once the settings have been set, the detector configuration should be given a name and save for future use.
Figure 6.12. Simulation of a HPGe detector using the GSG Pro: Detector Settings
Detector settings:
-Source to detector distance --> 111 mm
-Crystal --> HpGe
-Crystal length --> 80.4 mm
-Contact length --> 50 mm
-Contact diameter --> 10.0 mm
-Crystal diameter --> 59.1 mm
-Absorbing filter layers --> Aluminium 1 mm
-Input window --> Aluminum 0 mm
-Crystal packaging --> Vacuum 3.0 mm
-Inactive layer/reflector --> Germanium 0.0003 mm
-Number of spectrum channels --> 4096
-Channel to energy conversion factor --> 0.3
-Energy resolution at 122 keV--> 0.8
-Energy resolution at 1332 keV --> 1.95
Once the new detector parameters have been specified, the user is ready to make a simulation. As a first step the newly defined detector is selected from the "Current configuration" drop-down list. Thereafter the nuclide Ba137m (1 MBq) is selected. The final results of the simulation are shown in fig. 6.13
Figure 6.13. Simulation results for Ba137m with the new detector settings
## INFORMATION BROCHURE
Gamma Spectrum Generator
## PUBLICATIONS
N. Stefanakis, A Comparison of a Measurement using the Canberra BEGe Detector and a Simulation using Nucleonica, presented at the Nucleonica training course in April 2013. Link to pdf
A. Berlizov et al.,Fast and Accurate Approach to -Spectrum Modelling: A Validation Study with a Shielded / Unshielded Voluminous Uranium Sample, Applied Radiation and Isotopes 68 (2010) 1822–1831. http://dx.doi.org/10.1016/j.apradiso.2010.03.019
V. Kleinrath, a Study of Gamma Interference Scenarios for Nuclear Security Purposes, Measurements and Modelling using the Nucleonica Tools, JRC-ITU-TN-2010/21.
A. Berlizov, et al., A Collection of Reference HPGe Gamma-Spectra for Shielded / Unshielded Radionuclide Sources and Special Nuclear Materials, Technical Note JRC-ITU-TN-2009/25, 2009.
A.N. Berlizov and R. Dreher, Web-accessible γ-spectrum simulator with on-line Monte Carlo for voluminous and shielded γ-sources: First results of experimental validation. http://dx.doi.org/10.1016/j.nima.2009.07.010
A. Berlizov, Introduction to the Gamma Spectrum Generator, 2011
J. Magill, A. Berlizov, R. Dreher, Web-based Education & Training for Illicit Trafficking and Consequence Management Associated with Nuclear and Radiological Terrorism, NATO Advanced Research Workshop: Threat Detection, Response and Consequence Management Associated with Nuclear and Madiological Terrorism, November 17 - 21, 2008, Brussels, Belgium.
Full paper
## Experimental Gamma-Ray Spectrum Catalogues
Online experimental gamma spectrum catalogue
Heath Catalogue (NaI)
Heath Catalogue (HPGe)
## REFERENCES
1. A. Berlizov, A. Zhmudsky, The recursive adaptive quadrature in MS Fortran-77, Preprint LANL phyics/9905035, 1999
2. D.E. Cullen, J.H. Hubbell, L. Kissel, EPDL97: the Evaluated Photon Data Library, ‘97 Version, UCRL-50400, Vol. 6, Rev. 5, Lawrence Livermore National Laboratory, Livermore, CA, September 1997
3. I. Prochazka, Materials Structure, 8, No.2 (2001) 55.
4. R.J. Wells, Rapid Approximation to the Voigt/Faddeeva Function and its Derivatives, JQSRT, 62 (1999) 29-48.
5. EADL: Evaluated Atomic Data Library
6. A.N. Berlizov et al., NAAPRO Detector Model, a Versatile and Efficient Approach to γ-ray Spectrum Simulation, Nucl. Instrum. Meth. Phys. Res. A, 562 (2006) 245-253.
7. R.M. Kippen, The GEANT low energy Compton scattering (GLECS) package for use in simulating advanced Compton telescopes, New Astronomy Reviews, 48 (2004) 221-225.
8. J.F. Briesmeister. MCNP – a general Monte Carlo N-particle transport code. Los Alamos National Laboratory Report, 1997, LA-12625-M.
9. A.N. Berlizov, V.V. Tryshyn, A Monte Carlo Approach to True-Coincidence Summing Correction Factor Calculation for Gamma-Ray Spectrometry Applications, Journ. Radioanal. Nucl. Chem., 264 (2005) 169-174.
10. A.N. Berlizov, Limits of Detection in the Presence of the Correlated Background, Journ. Radioanal. Nucl. Chem., 271 (2007) 363-370. | {"extraction_info": {"found_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": 55, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8422214388847351, "perplexity": 3652.182746465666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657131304.74/warc/CC-MAIN-20140914011211-00333-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://byjus.com/question-answer/with-increasing-quantum-number-the-energy-difference-between-adjacent-energy-levels-in-atomsdecreasesincreasesremains-constantdecreases-for/ | Question
# With increasing quantum number, the energy difference between adjacent energy levels in atoms
A
decreases
B
increases
C
remains constant
D
decreases for low Z and increases for high Z
Solution
## The correct option is A decreasesAs $$n$$ increases, energy difference between adjacent energy levels decreases.Physics
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http://en.wikipedia.org/wiki/Fundamental_diagram | # Fundamental diagram of traffic flow
(Redirected from Fundamental diagram)
The fundamental diagram of traffic flow is a diagram that gives a relation between the traffic flux (vehicles/hour) and the traffic density (vehicles/km). A macroscopic traffic model involving traffic flux, traffic density and velocity forms the basis of the fundamental diagram. It can be used to predict the capability of a road system, or its behaviour when applying inflow regulation or speed limits.
Fundamental Diagram of traffic flow
## Basic statements
• There is a connection between traffic density and vehicle velocity: The more vehicles are on a road, the slower their velocity will be.
• To prevent congestion and to keep traffic flow stable, the number of vehicles entering the control zone has to be smaller or equal to the number of vehicles leaving the zone in the same time.
• At a critical traffic density and a corresponding critical velocity the state of flow will change from stable to unstable.
• If one of the vehicles brakes in unstable flow regime the flow will collapse.
The primary tool for graphically displaying information in the study traffic flow is the fundamental diagram. Fundamental diagrams consist of 3 different graphs: flow-density, speed-flow, and speed-density. The graphs are two dimensional graphs. All the graphs are related by the equation “flow = speed * density”; this equation is the essential equation in traffic flow. The fundamental diagrams were derived by the plotting of field data points and giving these data points a best fit curve. With the fundamental diagrams researchers can explore the relationship between speed, flow, and density of traffic.
### Speed-Density
The speed-density relationship is linear with a negative slope; therefore, as the density increases the speed of the roadway decreases. The line crosses the speed axis, y, at the freeflow speed, and the line crosses the density axis, x, at the jam density. Here the speed approaches freeflow speed as the density approaches zero. As the density increases, the speed of the vehicles on the roadway decreases. The speed reaches zero when the density equals the jam density.
### Flow-Density
In the study of traffic flow theory, the flow-density diagram is used to determine the traffic state of a roadway. Currently, there are two types of flow density graphs. The first is the parabolic shaped flow-density curve, and the second is the triangular shaped flow-density curve. Academia views the triangular shaped flow-density curve as more the accurate representation of real world events. The triangular shaped curve consists of two vectors. The first vector is the freeflow side of the curve. This vector is created by placing the freeflow velocity vector of a roadway at the origin of the flow-density graph. The second vector is the congested branch, which is created by placing the vector of the shock wave speed at zero flow and jam density. The congested branch has a negative slope, which implies that the higher the density on the congested branch the lower the flow; therefore, even though there are more cars on the road, the number of cars passing a single point is less than if there were fewer cars on the road. The intersection of freeflow and congested vectors is the apex of the curve and is considered the capacity of the roadway, which is the traffic condition at which the maximum number of vehicles can pass by a point in a given time period. The flow and capacity at which this point occurs is the optimum flow and optimum density, respectively. The flow density diagram is used to give the traffic condition of a roadway. With the traffic conditions, time-space diagrams can be created to give travel time, delay, and queue lengths of a road segment.
### Speed-Flow
Speed flow diagrams are used to determine the speed at which the optimum flow occurs. There are currently two shapes of the speed-flow curve. The speed-flow curve also consists of two branches, the freeflow and congested branches. The diagram is not a function, allowing the flow variable to exist at two different speeds. The flow variable existing at two different speeds occurs when the speed is higher and the density is lower or when the speed is lower and the density is higher, which allows for the same flow rate. In the first speed-flow diagram, the freeflow branch is a horizontal line, which shows that the roadway is at freeflow speed until the optimum flow is reached. Once the optimum flow is reached, the diagram switches to the congested branch, which is a parabolic shape. The second speed flow diagram is a parabola. The parabola suggests that the only time there is freeflow speed is when the density approaches zero; it also suggests that as the flow increases the speed decreases. This parabolic graph also contains an optimum flow. The optimum flow also divides the freeflow and congested branches on the parabolic graph.
## Macroscopic Fundamental Diagram
A macroscopic fundamental diagram (MFD) is type of traffic flow fundamental diagram that relates space-mean flow, density and speed of an entire network with n number of links as shown in Figure 1. The MFD thus represents the capacity, $\mu(n)$, of the network in terms of vehicle density with $\mu_1$ being the maximum capacity of the network and $\eta$ being the jam density of the network. The maximum capacity or “sweet spot” of the network is the region at the peak of the MFD function.
Figure 1: Sample Traffic Flow Macroscopic Fundamental Diagram
Figure 2: Space-time Diagram for the ith link in a traffic flow network
### Flow
The space-mean flow, $\bar q$, across all the links of a given network can be expressed by:
$\bar q = \frac{\sum_{k=1}^n d_i(B)}{nTL}$, where B is the area in the time-space diagram shown in Figure 2.
### Density
The space-mean density, $\bar k$, across all the links of a given network can be expressed by:
$\bar k = \frac{\sum_{k=1}^n t_i(B)}{nTL}$, where B is the area in the time-space diagram shown in Figure 2.
### Speed
The space-mean speed, $\bar v$, across all the links of a given network can be expressed by:
$\bar v = \frac{\bar q}{\bar k}$, where B is the area in the space-time diagram shown in Figure 2.
### Average Travel Time
The MFD function can expressed in terms of the number of vehicles in the network such that:
$n=\bar k \sum_{k=1}^n l_i = \bar k L$ where $L$ represents the total lane miles of the network.
Let $d$ be the average distance driven by a user in the network. The average travel time ($\tau$) is:
$\tau = \frac{d}{\bar v} = \frac{nd}{MFD(n)L}$
### Application of the MFD
In 2008, the traffic flow data of the city street network of Yokohama, Japan was collected using 500 fixed sensors and 140 mobile sensors. The study[1] revealed that city sectors with approximate area of 10 km^2 are expected to have well-defined MFD functions. However, the observed MFD does not produce the full MFD function in the congested region of higher densities. Most beneficially though, the MFD function of a city network was shown to be independent of the traffic demand. Thus, through the continuous collection of traffic flow data the MFD for urban neighborhoods and cities can be obtained and used to for analysis and traffic engineering purposes. These MFD functions can aid agencies in improving network accessibility and help to reduce congestion by monitoring the number of vehicles in the network. In turn, using congestion pricing, perimeter control an other various traffic control methods, agencies can maintain optimum network performance at the "sweet spot" peak capacity. Agencies can also use the MFD to estimate average trip times for public information and engineering purposes. | {"extraction_info": {"found_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": 14, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9177901744842529, "perplexity": 723.8126190773064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345758904/warc/CC-MAIN-20131218054918-00096-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://fearlessmath.net/mod/page/view.php?id=116 | ## Video and Examples
Finding the volume of cylinders.
Volume of cylinders In the earlier lesson, you learned that the volume of the rectangular prism is found by multiplying the area of the base by the height of the prism. Mathematicians commonly write this as V = B • h Finding the volume of a cylinder is the exact same formula! Step 1: Find the area of the base Step 2: Multiply it by the height In other words: Volume = (Area of base) • height Volume = (Area of a circle) • height Volume = (π • r • r) • h
Example 1 This cylinder has a radius of 4 cm and a height of 10 cm. Find its volume. V = B • h V = (π • r • r) • h V = (3.14 • 4 • 4) • 10 V = 502.4 cm^3
Example 2 This cylinder has a diameter of 6 cm and a height of 9 cm. What is its volume? First we need to find the radius, so we divide the diameter by 2. Radius = 6 ÷ 2 = 3 cm V = B • h V = (π • r • r) • h V = (3.14 • 3 • 3) • 9 V = 254.34 cm^3
Try this GeoGebra applet:
Move the hint slider to 0 to hide the hints and solution. Create a cylinder and then find the volume. Use the slider to check your answer.
# Self-Check
Question 1 Find the volume of a cylinder with a radius of 7 cm and a height of 11 cm. [show answer] V = B • h V = (π • r • r) • h V = (3.14 • 7 • 7) • 11 V = 1692.46 cm^3
Question 2 Find the volume of a cylinder with a radius of 3 cm and a height of 8 cm. [show answer] V = B • h V = (π • r • r) • h V = (3.14 • 3 • 3) • 8 V = 226.08 cm^3
Question 3 Find the volume of a cylinder with a diameter of 10 cm and a height of 10 cm. [show answer] V = B • h V = (π • r • r) • h V = (3.14 • 5 • 5) • 10 V = 785 cm^3 | {"extraction_info": {"found_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": 0.8605411648750305, "perplexity": 664.1060414668852}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1631780057524.58/warc/CC-MAIN-20210924110455-20210924140455-00576.warc.gz"} |
http://mathhelpforum.com/algebra/115742-make-perfect-square.html | Math Help - Make a perfect square
1. Make a perfect square
I need to make 8x^2+36x a perfect square, which number would i add to it to do so?(this is for parametric sequences in calc 2, but i just need to know the number i use to make that part a perfect square)
2. The idea here is that you want to write an equation such that $a^2x^2 + bx + c^2 = (ax + c)^2$, so you have to come up with a way to solve for c and $c^2$. In this case you want to know $c^2$.One thing we know about squaring an polynomial of the form $(ax + c)$ is that b from above can be given by $b = 2ac$. Therefore in your case you can solve for c and square it to get the value you need, i.e. solve for c in the following and then square the result
$36 = 2\sqrt{8}c\Rightarrow36^2 = 4\cdot 8 \cdot c^2$
here for the given problem if we add 41 it will make a perfect square
8x2+36x+41
4. Originally Posted by upender
here for the given problem if we add 41 it will make a perfect square
8x2+36x+41
For that equation $b^2-4ac=-16$ so it will be ok in $\mathbb{C}$ but I think the OP wants a real solution
I'd use the discriminant to find c. For it to be a perfect square we know that $b^2-4ac=0$
$36^2 = 4 \times 8 \times c$
$c = \frac{36^2}{32} = 40.5$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8592751026153564, "perplexity": 171.43597262598}, "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/1410657135558.82/warc/CC-MAIN-20140914011215-00104-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://spie.org/Publications/Proceedings/Paper/10.1117/12.2278085 | Share Email Print
### Proceedings Paper
Modelling of propagation and scintillation of a laser beam through atmospheric turbulence
Author(s): Fedor V. Shugaev; Ludmila S. Shtemenko; Olga I. Dokukina; Oxana A. Nikolaeva; Natalia A. Suhareva; Dmitri Y. Cherkasov
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PDF \$17.00 \$21.00
Paper Abstract
The investigation was fulfilled on the basis of the Navier-Stokes equations for viscous heat-conducting gas. The Helmholtz decomposition of the velocity field into a potential part and a solenoidal one was used. We considered initial vorticity to be small. So the results refer only to weak turbulence. The solution has been represented in the form of power series over the initial vorticity, the coefficients being multiple integrals. In such a manner the system of the Navier- Stokes equations was reduced to a parabolic system with constant coefficients at high derivatives. The first terms of the series are the main ones that determine the properties of acoustic radiation at small vorticity. We modelled turbulence with the aid of an ensemble of vortical structures (vortical rings). Two problems have been considered : (i) density oscillations (and therefore the oscillations of the refractive index) in the case of a single vortex ring; (ii) oscillations in the case of an ensemble of vortex rings (ten in number). We considered vortex rings with helicity, too. The calculations were fulfilled for a wide range of vortex sizes (radii from 0.1 mm to several cm). As shown, density oscillations arise. High-frequency oscillations are modulated by a low-frequency signal. The value of the high frequency remains constant during the whole process excluding its final stage. The amplitude of the low-frequency oscillations grows with time as compared to the high-frequency ones. The low frequency lies within the spectrum of atmospheric turbulent fluctuations, if the radius of the vortex ring is equal to several cm. The value of the high frequency oscillations corresponds satisfactorily to experimental data. The results of the calculations may be used for the modelling of the Gaussian beam propagation through turbulence (including beam distortion, scintillation, beam wandering). A method is set forth which describes the propagation of non-paraxial beams. The method admits generalization to the case of inhomogeneous medium.
Paper Details
Date Published: 29 September 2017
PDF: 14 pages
Proc. SPIE 10425, Optics in Atmospheric Propagation and Adaptive Systems XX, 104250N (29 September 2017); doi: 10.1117/12.2278085
Show Author Affiliations
Fedor V. Shugaev, M.V. Lomonosov Moscow State Univ. (Russian Federation)
Ludmila S. Shtemenko, M.V. Lomonosov Moscow State Univ. (Russian Federation)
Olga I. Dokukina, M.V. Lomonosov Moscow State Univ. (Russian Federation)
Oxana A. Nikolaeva, M.V. Lomonosov Moscow State Univ. (Russian Federation)
Natalia A. Suhareva, M.V. Lomonosov Moscow State Univ. (Russian Federation)
Dmitri Y. Cherkasov, M.V. Lomonosov Moscow State Univ. (Russian Federation)
Published in SPIE Proceedings Vol. 10425:
Optics in Atmospheric Propagation and Adaptive Systems XX
Karin U. Stein; Szymon Gladysz, Editor(s) | {"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": 0.8177860379219055, "perplexity": 2911.4487734140107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1575540543252.46/warc/CC-MAIN-20191212102302-20191212130302-00020.warc.gz"} |
https://www.physicsforums.com/threads/pressure-on-organisms.140495/ | # Pressure on organisms
1. Oct 29, 2006
### mikefitz
The maximum pressure most organisms can survive is about 1000 times atmospheric pressure. Only small, simple organisms such as tadpoles and bacteria can survive such high pressures. What then is the maximum depth at which these organisms can live under the sea (assuming that the density of seawater is 1026 kg/m3)?
Ptotal = 10^5 (1000)= 100000000 Pa
100000000 Pa = (1026)(9.81)(d)
d= 9935m= 9.94km?
This is wrong but I am unsure why...
2. Oct 29, 2006
### mikefitz
any ideas??:tongue2: thanks!
3. Oct 31, 2006
### mikefitz
ahh, i'm still workin on this one, here is my latest try:
1000 atm=101325000 Pa
101325000 = 101325 + 1026*9.81*d
d=909.64m
=.909km
this is still wrong but i've no idea why? Thanks
4. Oct 31, 2006
### NateTG
Hmm... that equation you came up with looks like
$$10^8\approx 10^5+10^4d$$
so it should work out to roughly
$$d \approx 10^4$$
but you come up with
$$d \approx 10^3$$
I'd guess that your first answer was correct modulo some approximation or rounding errors.
Last edited: Oct 31, 2006 | {"extraction_info": {"found_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": 0.9351741075515747, "perplexity": 4932.150220097672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583660020.5/warc/CC-MAIN-20190118090507-20190118112507-00393.warc.gz"} |
https://www.hydrol-earth-syst-sci.net/22/5967/2018/hess-22-5967-2018.html | Journal cover Journal topic
Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
Journal topic
Hydrol. Earth Syst. Sci., 22, 5967-5985, 2018
https://doi.org/10.5194/hess-22-5967-2018
Hydrol. Earth Syst. Sci., 22, 5967-5985, 2018
https://doi.org/10.5194/hess-22-5967-2018
Research article 22 Nov 2018
Research article | 22 Nov 2018
Inundation mapping based on reach-scale effective geometry
Inundation mapping based on reach-scale effective geometry
Cédric Rebolho1, Vazken Andréassian1, and Nicolas Le Moine2 Cédric Rebolho et al.
• 1Irstea, UR HYCAR, 1 Rue Pierre-Gilles de Gennes, 92160 Antony, France
• 2Sorbonne Universités, UPMC Univ Paris 06, CNRS, EPHE, UMR 7619 Metis, 4 Place Jussieu, 75005 Paris, France
Abstract
The production of spatially accurate representations of potential inundation is often limited by the lack of available data as well as model complexity. We present in this paper a new approach for rapid inundation mapping, MHYST, which is well adapted for data-scarce areas; it combines hydraulic geometry concepts for channels and DEM data for floodplains. Its originality lies in the fact that it does not work at the cross section scale but computes effective geometrical properties to describe the reach scale. Combining reach-scale geometrical properties with 1-D steady-state flow equations, MHYST computes a topographically coherent relation between the “height above nearest drainage” and streamflow. This relation can then be used on a past or future event to produce inundation maps. The MHYST approach is tested here on an extreme flood event that occurred in France in May–June 2016. The results indicate that it has a tendency to slightly underestimate inundation extents, although efficiency criteria values are clearly encouraging. The spatial distribution of model performance is discussed and it shows that the model can perform very well on most reaches, but has difficulties modelling the more complex, urbanised reaches. MHYST should not be seen as a rival to detailed inundation studies, but as a first approximation able to rapidly provide inundation maps in data-scarce areas.
1 Introduction
Floods are a recurring phenomenon in France: in September 2014, intense rainfall affected the south of the country, leading to several deaths and about EUR 0.6 billion worth of damage. The following year, in October, about 20 people died in the south-east due to massive flooding, which caused a loss of EUR 0.5 billion. Then, in June 2016, large-scale flooding occurred over the Seine and Loire catchments, mainly affecting their tributaries and resulting in four deaths at a cost of EUR 1.4 billion. These are only examples which underline the value of flood inundation mapping to anticipate the impact of such events. Public authorities and insurance companies are showing a growing interest in the field of rapid inundation modelling, and for the development of simple methods, that would work for any river with easily available data.
Flood hazard assessment usually combines rainfall observations or simulations, a hydrological model, streamflow simulations or observations, and an inundation model in order to generate inundation extents, height maps and sometimes other information (e.g. velocities). Traditionally, flood inundation models are derived from the shallow water equations (SWEs) in one or two dimensions (the so-called hydraulic models), with various simplifications that have proved to give satisfying results. For instance, the Regional Flood Model (RFM), probably one of the most comprehensive approaches published so far, is made of four parts : a daily distributed rainfall–runoff model, a 1-D hydraulic model for channel routing, a 2-D hydraulic model for floodplain mapping and a flood loss estimation model. Its application on the Mulde catchment in Germany showed mixed results concerning inundation extents, correctly predicting only 50 % of the flooded area for the August 2002 event. This underestimation was explained by dike breaches that were not accounted for within the model. A lack of observed data did not allow validation on other events.
Not all hydraulic models need to have this degree of complexity. It is indeed possible to neglect specific parts of the SWEs depending on the situation. Usually, 2-D models use the complete Saint-Venant equations while 1-D models often disregard one or several terms, leading, for instance, to the diffusive wave or kinematic wave approximations (Moussa and Cheviron2015). Some methods choose to couple 1-D and 2-D models, the former for streamflow routing and the latter for overbank flow . Despite the accuracy of such models, studies often try to further simplify them because of the large computing time to simulate small areas and the lack of precise data required to run these models.
LISFLOOD-FP , a hydraulic model developed to simulate floodplain inundation, was used in several studies . The model offers different possibilities: using 2-D equations or 1-D equations decoupled on a 2-D grid with kinematic, diffusive or inertial approximations . published a comparison between different models with gradually increasing complexities (1-D, 1-D on 2-D grid and 2-D) and, surprisingly, showed that the 1-D model had a better ability to reproduce the two events that were used in validation. The subsequent analysis concluded that the reach studied was relatively narrow and could easily be modelled using simple methods, and the authors argued that the other models would be more appropriate for more complex reaches.
However, these examples concern relatively small and well-instrumented reaches and assessing flood hazard at a larger scale may require different approaches. applied LISFLOOD-ACC, an inertial version of LISFLOOD-FP with decoupled 1-D equations on a 100 m resolution grid over Europe in order to map flood hazards for a 100-year return period, assuming a constant return period along the reaches. Broadly speaking, the model splits rivers into small reaches, to apply the hydraulic models independently and to merge simulated maps together, but only for rivers with a catchment larger than 500 km2. The model was then validated against regional and national hazard maps for six catchments in Germany and the United Kingdom and showed a general over-prediction. Another variation of LISFLOOD-FP for large-scale flood inundation modelling was introduced by , including a new sub-grid representation of channel networks for improved model accuracy .
developed an approach aimed at the forecasting context, in order to cope with excessive computing times. The solution chosen was to run a simple 1-D hydraulic model during a “pre-analysis phase” and create a catalogue of inundation extents corresponding to various return periods. These maps are then used, in a forecasting context, to give an estimate of the level of flooding, depending on the forecast discharge.
The lack of precise data (especially for channel cross sections) and the computing time required by numerical methods for solving the SWE motivated the development of potentially alternative methods, mostly based on DEM analysis. For instance, the rapid flood spreading method (Gouldby et al.2008) was chosen to divide floodplains into impact zones of different elevations in order to explore the effects of dike breaches using a spilling algorithm based on water depth. Other methods derive inundation maps from topographic information only: one can cite EXZECO , which introduces elevation noise in the DEM in order to create a single map of “maximum flow accumulation” that can be seen as a potential inundation area, and HAND (“height above nearest drainage”), a descriptor originally used for terrain classification , which has recently been adapted to static flood inundation mapping and is increasingly used to produce flood maps (Afshari et al.2018; McGrath et al.2018; Speckhann et al.2018). HAND calculates the difference between river cells' elevation and that of the connected floodplain cells, thus giving relative height information which can be compared to observed flood depths and the corresponding inundation extent.
MHYST, the method presented in this paper, is a simplified approach developed with the aim of rapidly producing inundation maps in data-scarce areas. It combines (i) concepts of hydraulic geometry to characterise channel geometry and (ii) DEM-derived relative elevations to characterise the floodplain; it does not work at the cross section scale but computes effective geometrical properties representative of the reach scale. Combining reach-scale geometrical properties with simplified steady-state hydraulic laws allows one to rapidly generate flood inundation maps while ensuring reach-scale coherence. After describing the method and the calibration dataset, MHYST is compared against the inundation extent observed for the major event that occurred in May–June 2016 in France. The last section discusses the spatial distribution of performance and the impact of uncertainties on the results obtained.
2 MHYST: a simplified steady-state hydraulic approach
The MHYST model stands for Modélisation HYdraulique simplifiée en écoulement STationnaire, i.e. Simplified Steady-state Hydraulic Modelling. It is a flood inundation model which aims to map inundation extents at the reach scale. Where classic hydraulic models use cross sections, this method is based on an effective geometry representative of each river reach. Since no detailed geometric data were available to describe the shape and roughness of the channel river bed for this study, a sub-grid representation of the channel was derived from hydraulic geometry relationships linking the drainage area with bankfull width and height . When discharge exceeds bankfull capacity, the model computes a reach-scale relation between streamflow and the HAND defined by . This relation can finally be used to assess which height corresponds to the given streamflow, and thus to derive the corresponding inundation map.
2.1 Processing of DEM: from elevations to height above nearest drainage
The initial step consists of processing the digital elevation model (DEM) in order (i) to obtain a flowing drainage direction map, (ii) to identify the subcatchments (corresponding to the river reaches), and (iii) to compute the height above nearest drainage (HAND) in each subcatchment. This initial processing is the basis of the floodplain analysis in MHYST. To compute the drainage direction map, we used the D8 method from the Flow Direction function provided by ArcGIS 10.3. It computes the drainage direction by calculating the steepest slope from the eight possible directions for a given cell.
Figure 1 shows the procedure used to compute HAND values: for a given floodplain cell, it is the difference between its elevation and that of the closest river cell in terms of drainage direction. For instance, the cell of elevation 25 at the top of the figure is linked to (i.e. flows towards) the most upstream red river cell which has an elevation of 18: thus, its HAND value is $\mathrm{25}-\mathrm{18}=\mathrm{7}$. This relative height has been used as a proxy for inundation height by various studies . To derive an inundation map from HAND values, we must define a threshold height HT: the flooded area corresponds to all the cells whose HAND value is strictly lower than HT .
Figure 1Processing of DEM and calculation of the HAND value for a hypothetical catchment.
2.2 Model description
MHYST is mostly based on a DEM and its derivatives (drainage map and drainage areas) and on the hydraulic equations describing a steady uniform flow at the reach scale. This means that for a given time step (day in this case), at a given reach, we make the approximation that the flow is constant over time and space (this is obviously a strong simplification that we will discuss later). Table 1 sums up the variables used in the following equations as well as their respective units and interpretations. Table 2 describes the two free parameters of the model. The following equations show the path to build a reach-scale relation between HT and the streamflow Q by calculating, with hydraulic formulas, the discharge value corresponding to a given HT. Once this relation is known, the model can easily simulate a hydrological event by inverting the relation, and by searching for the HT which corresponds to the given Q (Fig. 2).
Figure 2Representation of the model structure: the reach-scale geometry is derived from hydraulic geometry relationships and DEM data and is then used to compute a relation between the threshold height HT and the discharge Q. L is a fixed characteristic of the reach.
Other variables can be directly calculated from the DEM (Fig. 4): for a given threshold height HT at a reach of length L (L is a fixed parameter of the model), V(HT) is the sum of volumes above all flooded pixels and S(HT) is the area occupied by the flooded cells. A(HT), in Eq. (1), is the average cross section area of the flooded reach and it depends on V(HT) and on the bankfull cross section area of the channel (${A}_{\mathrm{b}}={h}_{\mathrm{b}}\cdot {W}_{\mathrm{b}}$, Fig. 3). This variable can also be defined as the sum of the channel cross section area $\left({A}_{\mathrm{ch}}={A}_{\mathrm{b}}+{H}_{\mathrm{T}}\cdot {W}_{\mathrm{b}}\right)$ and the floodplain cross section area $\left({A}_{\mathrm{fp}}=A\left({H}_{\mathrm{T}}\right)-{A}_{\mathrm{ch}}\right)$. B(HT), in Eq. (2), is the average surface width of the flooded reach, defined similarly from S(HT) and L.
$\begin{array}{}\text{(1)}& & A\left({H}_{\mathrm{T}}\right)=\frac{\mathrm{1}}{L}\cdot V\left({H}_{\mathrm{T}}\right)+{A}_{\mathrm{b}}={A}_{\mathrm{ch}}+{A}_{\mathrm{fp}}\text{(2)}& & B\left({H}_{\mathrm{T}}\right)=\frac{\mathrm{1}}{L}\cdot S\left({H}_{\mathrm{T}}\right)\end{array}$
Figure 3Typical cross section segmentation, with the cross section area of the channel (Ach), that of the floodplains (Afp) and the bankfull cross section area (Ab) which is calculated from the average bankfull height (hb) and width (Wb) computed from downstream hydraulic geometry relationships.
Figure 4Representation of the reach-scale geometry derived from HAND and the DEM. A(HT) and B(HT) are derived from V(HT) and S(HT) respectively (Eqs. 1 and 2).
The only unknown variables in these equations are sub-grid parameters hb (bankfull water level) and Wb (bankfull width), i.e. the bankfull geometry, which cannot be obtained from usual DEMs and are only available from detailed surveys for a small number of rivers. Indeed, in situ bathymetric data are quite scarce and red lasers cannot penetrate the water surface more than a few centimetres, which means that the real elevation of the river bed is mostly not correctly represented in the DEMs. This is why we chose to use downstream hydraulic geometry equations to estimate these geometric parameters, assuming a rectangular channel, the size of which depends on the upstream drainage area (Eqs. 3 and 4). To assess the coefficients α and β, we used satellite images from the French platform Géoportail in order to link observed bankfull widths and drainage areas. The values found for the Loing catchment are α=0.053 and β=0.822. The other coefficients, δ and ω, were taken from a study by , which attempted to regionalise these parameters in the US. We used the general values found for the whole set of catchments: δ=0.27 and ω=0.21. Although these values probably add uncertainties in the model, they are an accessible way to assess bankfull channel geometry and could still be improved by local bankfull studies when available.
$\begin{array}{}\text{(3)}& & {W}_{\mathrm{b}}=\mathit{\alpha }\cdot {A}_{\mathrm{D}}^{\mathit{\beta }}\text{(4)}& & {h}_{\mathrm{b}}=\mathit{\delta }\cdot {A}_{\mathrm{D}}^{\mathit{\omega }}\end{array}$
The fundamental equations of the MHYST model come from an experimental study by which defines the DEBORD formulation as in Eqs. (5) to (7). Building on the Manning–Strickler formula, these authors proposed an empirical parameterisation of turbulent momentum exchange between the channel and the floodplain. This formulation expresses the conveyance capacity depending on channel-related and floodplain-related variables. The coefficient C takes into account the interaction of flows between the fast-flowing channel and the slow-flowing floodplain, as well as the corresponding head losses.
$\begin{array}{ll}& {D}_{e}={K}_{\mathrm{ch}}\cdot C\cdot {A}_{\mathrm{ch}}\cdot {R}_{\mathrm{ch}}^{\mathrm{2}/\mathrm{3}}+{K}_{\mathrm{fp}}\\ \text{(5)}& & \cdot \sqrt{{A}_{\mathrm{fp}}^{\mathrm{2}}+{A}_{\mathrm{ch}}\cdot {A}_{\mathrm{fp}}\cdot \left(\mathrm{1}-{C}^{\mathrm{2}}\right)}\cdot {R}_{\mathrm{fp}}^{\mathrm{2}/\mathrm{3}}\text{(6)}& & C=\left\{\begin{array}{rlr}& {C}_{\mathrm{0}}=\mathrm{0.9}\cdot {\left(\frac{{K}_{\mathrm{fp}}}{{K}_{\mathrm{ch}}}\right)}^{\mathrm{1}/\mathrm{6}}& \text{if}\phantom{\rule{0.25em}{0ex}}r=\frac{{R}_{\mathrm{fp}}}{{R}_{\mathrm{ch}}}>\mathrm{0.3}\\ & \frac{\mathrm{1}-{C}_{\mathrm{0}}}{\mathrm{2}}\cdot \mathrm{cos}\left(\frac{\mathit{\pi }\cdot r}{\mathrm{0.3}}\right)+\frac{\mathrm{1}+{C}_{\mathrm{0}}}{\mathrm{2}}& \text{if}\phantom{\rule{0.25em}{0ex}}\mathrm{0}\le r\le \mathrm{0.3}\end{array}\right\\text{(7)}& & Q={D}_{e}\cdot \sqrt{{I}_{f}}\end{array}$
The streamflow Q is finally defined from the conveyance capacity and the channel slope, since we hypothesise a uniform flow. Rch and Rfp can easily be calculated from the assumed reach geometry (Eqs. 8 and 9), which only leaves the Strickler coefficients as unknown variables.
$\begin{array}{}\text{(8)}& & {R}_{\mathrm{ch}}=\frac{{A}_{\mathrm{ch}}}{{W}_{\mathrm{b}}+\mathrm{2}\cdot {h}_{\mathrm{b}}}\text{(9)}& & {R}_{\mathrm{fp}}=\frac{{A}_{\mathrm{fp}}}{B\left({H}_{\mathrm{T}}\right)-{W}_{\mathrm{b}}}\end{array}$
Table 1Names, units and interpretations of the variables used in the geometric and hydraulic equations of the MHYST model.
Table 2Names, units and interpretations of the free parameters of MHYST's structure.
The two Strickler coefficients add 2 degrees of freedom, and Kch is additionally used to calculate the bankfull flow from the Manning–Strickler formula (Eq. 10).
$\begin{array}{}\text{(10)}& {Q}_{\mathrm{b}}={K}_{\mathrm{ch}}\cdot {\left(\frac{{W}_{\mathrm{b}}\cdot {h}_{\mathrm{b}}}{{L}_{\mathrm{b}}+\mathrm{2}\cdot {h}_{\mathrm{b}}}\right)}^{\mathrm{2}/\mathrm{3}}\cdot \sqrt{{I}_{f}}\cdot {h}_{\mathrm{b}}\cdot {W}_{\mathrm{b}}\end{array}$
Here, we sum up the procedure, which operates at the reach scale:
1. For a given threshold height HT, we use the DEBORD formulation to calculate the corresponding discharge Q.
2. By repeating the operation for all possible HT, we obtain a reach-specific table matching values of HT and Q.
3. When working on an event where only Q is known, when it is greater than Qb (which means that the river overflowed), the model looks for the corresponding HT value in the table, by calculating a linear interpolation between two values if necessary, and then assigns to each cell in the subcatchment a flooded height ${h}_{\mathrm{flood}}=\mathrm{max}\left(\mathrm{0};{H}_{\mathrm{T}}-{\mathrm{HAND}}_{\text{cell}}\right)$.
Although this method and that of were developed independently, they share a lot of similarities, both using HAND to derive a reach-scale geometry which is used as input for a simplified hydraulic model. However, in addition to HAND, MHYST uses downstream hydraulic geometry relationships to evaluate a sub-grid representation of the channel geometry. The hydraulic model is also different: use the Manning–Strickler formula, while MHYST computes streamflow values from the DEBORD formulation.
2.3 Boundary conditions
MHYST can work with either simulated or observed flows. In this paper, observed data from 12 measurement stations of the French HYDRO database were used to create an observed distributed streamflow map by interpolating flows based on drainage area (Eq. 11) for river pixels between outlets:
$\begin{array}{}\text{(11)}& Q={Q}_{\mathrm{up}}+\frac{{A}_{\mathrm{D}}-{A}_{\mathrm{D},\mathrm{up}}}{{A}_{\mathrm{D},\mathrm{down}}-{A}_{\mathrm{D},\mathrm{up}}}×\left({Q}_{\mathrm{down}}-{Q}_{\mathrm{up}}\right),\end{array}$
where Q and AD are the streamflow and drainage area of any river cell between two outlets, Qup, Qdown, AD,up and AD,down are the direct upstream and downstream outlet discharges and drainage areas. This way, streamflow is coherently interpolated over the network, and then averaged at the reach scale.
3 Material
3.1 Generic data
In this study, we used a 5 m resolution DEM with a vertical resolution of 0.01 m covering the Loing catchment (Fig. 5) from IGN (the French national institute for geographic information), which was filled and corrected to avoid depressions and to allow a strict coherence of flow directions, meaning that every pixel flows to the sea. Drainage directions and areas were derived from this DEM and used as model inputs along with elevations. The adaptations and modifications of the DEM were conducted using ESRI ArcGIS 10.3.
Figure 55 m depressionless DEM used in this study. Elevations go from 45 to 390 m. Corrections have been applied so that each pixel flows to the sea.
Daily observed discharges were obtained from the French HYDRO database and the stations were used to delineate the hydrological network over the catchment. Calibration data for the Loing catchment were obtained from the activation EMSN028 of the Copernicus Emergency Management Service (© 2016 European Union). The original Copernicus study covered a small part of the River Seine and half of the Loing catchment (Fig. 6). However, since the study area and the defined river network were smaller, we cropped the inundation extent to match the study area (Fig. 6). These calibration data are post-processed observed data, meaning that the original maps came from satellite observations but they were then modified to build a more homogeneous inundation extent, i.e. nearby areas whose elevations were below the observed flood level were added to the inundation extent and merged with all the others. The maximum flood extent was then validated by the European Service against reported flood damage and hydrological measurements (SERTIT2016).
Figure 6Maximum flood extent for the May–June 2016 event over the Loing catchment produced by the Copernicus Emergency Management Service.
3.2 Event of May–June 2016
Following an extremely wet month of May (namely the wettest on record for many stations), a heavy rainfall event started on 30 May 2016 over the centre of France, affecting the Upper and Middle Seine basin and the Middle Loire basin. This episode lasted until 6 June and, combined with highly saturated soils due to a series of preceding minor events, led to major flood inundations. Over this period, overall precipitation reached 180 mm in Paris and Orléans, while in some tributaries, such as the River Loing, peak flows largely exceeded those of the record 1910 flood event (Fig. 7). The flood resulted in 4 deaths, 24 people injured and EUR 1.4 billion worth of damage. A total of 1148 cities were declared to be in a state of natural disaster and insurance companies received about 182 000 claims (CCR2016).
Since calibration data were available for June 2016 event, we chose to use our model to simulate this episode and compare the results with observations. We conducted this study over the River Loing, tributary to the River Seine, with a catchment covering 3900 km2, a mean elevation of 148 m and a mean slope of 0.03 m m−1. This catchment was heavily impacted by the flood event and contains a significant proportion of the inundated area, making it a suitable area to carry out the study. Streamflow data were interpolated from measurements, so no hydrological model is involved in this paper.
Figure 7Daily hydrograph of the River Loing at Épisy (3900 km2) during the event of June 2016. Overall precipitation reached 130 mm. The peak discharge was the largest ever observed on the catchment and reached about 500 m3 s−1.
4 Results
4.1 Calibration procedure
To assess the model's performance, we used several criteria based on the contingency table in Fig. 8. These scores are presented in detail by and are defined as a ratio between members of the table where n1 is the number of hits, i.e. the number of flooded cells correctly forecast; n4 is the number of pixels correctly forecast as dry; n2 is the number of false alarms; and n3 the number of observed flooded cells missed by the model. Table 3 summarises the formulas and the interpretations of each score used in this study.
The POD (probability of detection), which is also called Correct or M1 , calculates the percentage of observed inundated pixels intersected by the simulation map. Its main drawback is that it does not take into account the false alarms and thus it can give good results for a clearly overestimating inundation extent. On the contrary, the FAR (false alarm ratio) or M2 computes the proportion of cells wrongly flooded by the model. But similarly, if the model does not flood anything, the FAR can reach its optimal value. The critical success index (CSI), also known as Fit, F index or FAI , is a criterion which tries to give an overall performance of the simulation by calculating the percentage of correctly flooded cells above the total number of flooded cells (observed and simulated). In this way, the score is penalised by the over- and underestimation. However, this criterion does not specify if the model is over- or underestimating the observed extent. This is why we also looked at the BIAS, which computes the ratio between the number of simulated and observed flooded cells. If it is above 1, the model overestimates, and if it is below 1, it underestimates. However, a value of 1 does not equal a perfect simulation since there may be a balance between the misses and the false alarms.
These ratios are particularly reliable if they are used to compare simulations and exhaustive observations. This is almost the case with Copernicus calibration data, which represent a “maximum flood extent”. However, MHYST outputs are dated, which is not the case for the observed map. This is why all daily simulated inundation extents were merged into one maximum simulated extent, meaning that we did not try to validate the temporal dynamic of the flood, but only aimed to assess its largest area. Thus, the preceding scores will only evaluate MHYST's ability to reproduce the maximum flood extent.
Figure 8Contingency table gathering the different scenarios encountered during calibration (the numbers refer to pixels).
Table 3Table of forecast scores used to assess the performance of a flood simulation. All criteria are based on the contingency table (Fig. 8) and reflect one characteristic of the model. Taken together, they provide a comprehensive analysis of the model's behaviour.
4.2 Parameterisation
Figure 9Forecast scores obtained by the model on the River Loing versus Copernicus data for all the parameter values tested, (a) BIAS contour lines and (b) CSI contour lines for various values of Kch and Kfp.
MHYST has two free parameters (Table 2): Kch (the Strickler roughness coefficient for the channel) and Kfp (the Strickler roughness coefficient for the floodplains). Preliminary studies showed that, for the Loing catchment, a length of 1000 m was a good trade-off between accuracy and computation time; consequently L was fixed at 1000 m in the rest of this study. Kch and Kfp values were tested in the range [0.1;30] in order to explore a wide range of possibilities (121 combinations were tested).
To help make a decision on the optimal parametrisation of the model, we used the following graphs, on which each (Kch, Kfp) couple is characterised by one overall value:
• two contour plots (Fig. 9) showing the impact of Kch and Kfp for the two main scores (BIAS and CSI);
• a Pareto plot (Fig. 10a) showing the role played by the Kfp parameter in balancing the POD and the FAR;
• a Pareto plot (Fig. 10b) showing that the CSI identifies the best compromises between the POD and the FAR.
Last, to be able to analyse the variability of results between reaches (we have a total of 90 reaches affected by the inundation), we also computed the CSI and BIAS reach by reach, and produced two cumulative distribution plots showing these results (Fig. 11). We found the following:
• The fit criteria are very sensitive to the Kfp value and much less to the Kch value (Fig. 9): this should not be a surprise given that we deal with the maximum flood extents for calibration, where Kch only plays a minor role. Remember also that (i) we are modelling a very extreme event (T>500 years) with substantial overflowing, and (ii) that we are working with a channel geometry derived from hydraulic geometry relationships. All this contributes to making the estimation of the channel roughness coefficient more difficult.
• The CSI clearly shows an optimal zone around Kfp=5 and ${K}_{\mathrm{ch}}\le \mathrm{10}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$. The best CSI values (greater than 0.66) correspond to combinations where $\mathrm{0.1}\le {K}_{\mathrm{ch}}\le \mathrm{10}$ with Kfp=5 or $\mathrm{1}\le {K}_{\mathrm{ch}}\le \mathrm{10}$ with Kfp=4. Given the equifinality, a good way to choose a combination in this range could be to use the most physical one, which, in this case, would be Kch=10 and ${K}_{\mathrm{fp}}=\mathrm{5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$. Indeed, over the catchment, floodplains mainly consist of 44 % non-irrigated arable land, 17 % broad-leaved forest and 10 % pastures with corresponding roughness coefficients reported in the literature of 8, 2 and $\mathrm{4}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$, respectively .
• Another way to confirm the validity of this choice (Kch=10 and ${K}_{\mathrm{fp}}=\mathrm{5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$) is to look at how this parametrisation behaves at the reach scale. Given a total of 90 reaches, we can compute the CSI and BIAS criteria for each of them and draw a distribution (Fig. 11): we observe that the “optimal” distribution is unbiased and that it represents a solution among the best available for each percentile, we can thus trust this parametrisation as a relatively “all-terrain” one for the Loing catchment.
Last, Fig. 10 provides a good illustration of how parameter sets interact with the FAR, POD and CSI criteria: choosing from the parameter sets with the best CSI makes it possible to find a compromise between a high POD and a low FAR.
Figure 10Pareto diagram for two forecast scores, POD and FAR. 1-FAR is used so that each criterion evolves in the same way, (a) distribution of Kfp values and (b) distribution of CSI values according to POD and 1-FAR.
Figure 11Cumulative frequency of CSI and BIAS values for all combinations of parameters and for the 90 affected reaches. Green lines correspond to the best combinations identified in Fig. 9 while the red line refers to the physical parametrisation. The other parameters are displayed in grey.
4.3 Model behaviour
Figures 12 to 14 provide a further illustration with a colour-coded classification of each reach depending on its CSI and BIAS value. A total of 11 regions are highlighted and numbered because of their poor performance. The reasons of why MHYST was not able to reproduce the inundation extent in these regions are explained below.
Figure 12Reach-scale performance of (a) BIAS and (b) CSI for the physical combination of parameters, Kch=10 and ${K}_{\mathrm{fp}}=\mathrm{5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$, for the downstream part of the catchment. Criteria values have been categorised as follows: excellent (dark green), good (green), average (orange) and poor (red). The black lines delineate the reaches. Locations 1 to 4 correspond to areas where the model struggles to reproduce the observation (orange and/or red zones).
1. For the downstream-most part of the Loing (Fig. 12), the reaches are red or orange because this area is only partially covered by the observation, which stops just after the confluence with the small tributary.
2. The small tributary (Fig. 12) is mainly red or orange for various reasons: downstream, at the confluence, the DEM is full of small high-elevation zones (not corrected in the DEM) which the model cannot reach, thus degrading the simulation. Along the tributary, the reason can be either the observed discharge values which seem small compared to the rest of the catchment or simply the effective geometry defined by the model, which does not correspond to the actual one. Finally, the upstream part of the tributary is not covered by the observation, which stops in the middle of what MHYST simulated. However, the study zone defined by the Copernicus Emergency Management Service goes further, so we cannot know whether it was not flooded or whether the service did not map this part because it was too insignificant.
3. The orange part in the middle of the BIAS map (Fig. 12) is due to the railway tracks which act like a wall in the DEM, preventing the model from reaching the other side (from east to west), where a small tributary, which looks like a partly subterranean urban stream, overflowed in its open air part.
4. Finally, the red and orange zones in the south of the presented map (Fig. 12) correspond to a part of the river where the Loing man-made waterway plays a major role, running parallel with the main river. This configuration is difficult for MHYST because we only consider the main river, defined by the DEM, with an effective reach-scale geometry and we cannot take into account such specificities, which would require a 2-D hydraulic model.
5. The area identified (Fig. 13) shows a slight underestimation leading to a moderate CSI. This issue can be explained by a motorway which is represented in the DEM by a more elevated area. This motorway separates the reach into two parts linked by artificial openings made by the producers of the DEM. This and the Loing waterway and another road act as dikes that prevent the model from reaching a further part of the reach. The parameterisation of the model is not suitable to address this difficulty.
6. Similarly to the previous area (Fig. 13), a railway crosses the DEM from north to south with only one opening for the water. Given the parameterisation of the model, it is not possible to go over the railway to flood the missed area.
7. In that case (Fig. 13), the model clearly overestimates the flood. The water fills a depression which looks like a tributary but is only a thalweg. Once more, the parameterisation of the model does not provide an adequate representation of this reach.
8. In this area (Fig. 14), MHYST underestimates the inundation extent due to a road that works like a dike. However, with another parameterisation, the model would be able to provide enough water to go over the road.
9. In the western part of the upstream area (Fig. 14), MHYST overestimates the flood because it is a relatively flat zone. The exceeding water, still due to the parameterisation, is thus spread over the area.
10. This area (Fig. 14) is special because the overestimation of MHYST is due to a non-continuous observation map, creating large parts of reaches that are observed to be dry. However, since MHYST works at the reach scale, it necessarily floods the whole river reach. Moreover, one tributary, the Solin, is not defined in the hydrographic network used by the model, because no observed discharges were available, whereas it appears in the observed map, leading to an underestimation of the flooded area.
11. The most upstream part of the simulated area (Fig. 14) suffers from an excess of water and a non-continuous observation, leading to similar effects. Moreover, several elevated roads appear in the DEM and force the model to flood the area using artificial openings across the roads.
Figure 13Reach-scale performance of (a) BIAS and (b) CSI for the physical combination of parameters, Kch=10 and ${K}_{\mathrm{fp}}=\mathrm{5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$, for the centre part of the catchment. Criteria values have been categorised as follows: excellent (dark green), good (green), average (orange) and poor (red). The black lines delineate the reaches. Locations 5 to 7 correspond to areas where the model struggles to reproduce the observation (orange and/or red zones).
Figure 14Reach-scale performance of (a) BIAS and (b) CSI for the physical combination of parameters, Kch=10 and ${K}_{\mathrm{fp}}=\mathrm{5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$, for the upstream part of the catchment. Criteria values have been categorised as follows: excellent (dark green), good (green), average (orange) and poor (red). The black lines delineate the reaches. Locations 8 to 11 correspond to areas where the model struggles to reproduce the observation (orange and/or red zones).
In order to complete our interpretation of MHYST behaviour, we conducted two sensitivity analyses, one with the Morris method (Morris1991) and the other with the Sobol method (Sobol2001). We chose to assess the effect of six potential parameters, Kch, Kfp, α, β, δ and ω, that may play a major role in the computation of HTQ relationships. In both analyses, we found that ω, which parameterises the regionalisation of bankfull heights, has the most substantial effect on the performance and that, surprisingly, Kfp has no influence at all. As a matter of fact, when we conducted the Sobol analysis with fixed hydraulic geometry parameters, we showed that Kfp is considerably more influential than Kch. We concluded that the previous results were due to the fact that these sensitivity analyses explore the parameter space in detail, and even with reasonable boundaries, they can reach values that may not be consistent with the characteristics of the catchment studied.
4.4 Influence of the DEM resolution
It is possible to assess the sensitivity to the DEM in two ways: first by aggregating our DEM from 5 m to various resolutions (10, 25, 50 and 100 m) and then by changing the source of the DEM. Figure 15 provides the CSI scores obtained by the model while changing the resolution. It shows that the resolution has relatively little effect on the optimal value, which varies between 0.65 and 0.69. However, the position of this optimal, i.e. the combination of parameters (Kch and Kfp) leading to it, changes. We can also see that for some resolutions, such as 25 or 50 m, the equifinality zone is much smaller than the one for the 100 m resolution, for example. If we also look at the “physical” set of parameters we previously identified (Kch=10 and ${K}_{\mathrm{fp}}=\mathrm{5}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{1}/\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$), we can see that the CSI reached by the model for this combination varies between the resolutions. Nevertheless, the result still seems satisfying, so it could be used as a “default” parameterisation, for instance for ungauged catchments. But this should be tested on other catchments with observed data to lead to a more comprehensive conclusion.
Figure 15CSI scores obtained by the model on the River Loing versus Copernicus data for all the parameter values tested and for various resolutions of the DEM, aggregated from the 5 m resolution DEM: (a) 10 m, (b) 25 m, (c) 50 m and (d) 100 m.
Before using the RGE 5 m DEM from IGN, we tried to use the 25 m EU-DEM from the European Environment Agency, and it showed poorer results, because it was not precise enough. Figure 16 shows the evolution of CSI for the same combinations of parameters as before. We see that the best combinations of parameters only lead to a 0.53 maximal CSI, which is more than 10 points below what we can obtain with the RGE DEM. There is also strictly no connection between the best values of BIAS and those of CSI, the latter being obtained for a clear overestimation of the flood extent (BIAS∼1.5). These results are due to the lack of precision of the EU-DEM, which does not distinguish the channel from the floodplain, leading to a 2 km wide channel in some parts of the river.
Figure 16(a) BIAS and (b) CSI scores obtained by the model on the River Loing versus Copernicus data for all the parameter values tested and for another source of data: EU-DEM.
5 Conclusions and outlooks
The objective of this paper was to present and validate a simple hydraulic model for rapid inundation mapping in data-scarce areas. MHYST is based on DEM analyses and simple hydraulic equations, creating a reach-scale relation between the average discharge and the average “height above nearest drainage” which can then be used to simulate any event, past or future, as long as streamflow information (observed or simulated) is available. This model was calibrated against an observed exceptional flood which occurred in 2016 on the Loing River near Paris and showed results that are certainly not perfect, but from our point of view and for our objectives quite encouraging. Furthermore, we compared our methodology with the traditional HAND approach, using a single threshold height of 4 m (measured height at the outlet) for the whole catchment. The simple HAND model reached CSI=0.49, BIAS=1.55, POD=0.84 and FAR=0.46. It is clearly penalised by the overestimation (almost 50 % of false alarms), which is not surprising according to other studies .
The simple structure of MHYST allows it to be used almost anywhere with few data and only two parameters. The model can, however, be used in first approximation, when a lack of time and data restrains the use of a more complex method.
For the sake of honesty, we would like to specify the theoretical limits of the MHYST approach:
• The model equations were solved by using the hypothesis of a reach-scale steady uniform flow (probably one of the most simplifying assumptions one can make). This simplification is probably too extreme for highly complex situations, especially in the presence of dikes and bridges. Indeed, on the one hand, the DEM resolution is too coarse to precisely take into account hydraulic structures, and on the other hand, the DEBORD formulation is not sufficient to describe the interaction between the flow and these structures.
• The DEM is a critical part of the model, because geometrical relationships and variables are directly related to the shape and distribution of elevations. Another DEM was actually tested as model input and showed much poorer results.
• Moreover, since the channel geometry was unknown, hydraulic geometry equations were used to assess bankfull height and width, with fixed parameters from another study in the case of height, which may not be the optimum for this catchment, adding its share of uncertainty.
• Finally, there is at this point no continuity equation between reaches, since the calculations were made for each reach separately. Uncertainties may therefore be higher in areas around connection points between reaches, especially if it is a confluence of rivers. One way to address this issue could be to add a continuity equation between the reaches, which might increase the overall coherence of the flood. However, at this point of the development of the model, we have not included this specificity.
Thus, the maps produced by MHYST should be seen as a maximum extent of the flood which can be used as a first and rapid estimation. To further test this approach, we consider that attention should first be given to the following: assessing the impact of the DEM choice, resolution and quality; testing the approach on a range of (less extreme) events and catchments, to better assess the range and stability of its parameters and performance; and improving the treatment of possible discontinuities between reaches.
Data availability
Data availability.
The IGN DEM cannot be freely downloaded. Copernicus Emergency Management Service data and the corresponding report can be downloaded at http://emergency.copernicus.eu/mapping/list-of-components/EMSN028 (last access: 20 November 2018). French observed discharges can be downloaded at http://hydro.eaufrance.fr/indexd.php (last access: 20 November 2018).
Appendix A: Sensitivity analysis
In order to assess the sensitivity of the model to its main parameters (Kfp, Kch, α, β, ω, δ), we conducted two sensitivity analyses, using different but complementary well-known methods: Morris (Morris1991) and Sobol (Sobol2001).
A1 Morris method
The Morris method (Morris1991) provides a qualification of the effect a parameter can have on the outputs. It is a OAT (one-at-a-time) methodology, which means that the effect of a parameter is measured by changing its value by adding ±Δ without modifying the other parameters and by comparing the outputs. In order to provide a relevant analysis, we generated 160 sets of parameters, using the Latin hypercube sampling method, which acts as starting points from where the Morris method can assess the significance of parameters by changing their values one-at-a-time. Thus, more than 1000 simulations are needed to conduct the analysis. By using the 5 m resolution DEM we used in this paper, this study would take several days, if not weeks, to complete. But since we showed that the performance of MHYST did not really change with the resolution, we chose to use a coarser version of our DEM, which was aggregated at a 50 m resolution, by simply averaging the elevations, allowing us to complete this sensitivity analysis in only a few hours. For each permutation and for each parameter, Di, the difference in CSI divided by the computing step, is calculated. The results in terms of means and standard deviations are presented in Fig. A1. The analysis shows that the model is very sensitive to changes of ω, the exponent in the calculation of the regionalised bankfull width (Wb). The most surprising part of the analysis is the fact that Kfp has little or no effect on the model, while Kch has a moderate effect. This is contradicted by Fig. 9, which clearly shows that for a given value of Kfp, the CSI value varies only slightly for a Kch between 0.1 and 20. Kfp is, contrary to what the Morris analysis shows, a significant parameter of the model, particularly in a major overflowing event such as the one studied here, where the channel only represents a fraction of the water.
Figure A1Results of the Morris method applied to MHYST with a 50 m resolution DEM on the Loing catchment for the six parameters (Kfp, Kch, α, β, ω, δ).
The problem might be that despite the use of a Latin hypercube sampling method, the “good” values of the parameters never meet, i.e. when ω has a sensible value, Kfp has not and vice versa. And of course, if the ω value does not coherently represent the channel, the model is not able to conduct a correct simulation (i.e. little or no flooding), leading to little or no influence of the Kfp parameter.
Moreover, the issue with sensitivity analyses such as the Morris method is that the results can be very different depending on the catchment or the event modelled. Indeed, if the water is concentrated in the channel part for a very steep catchment, a very flat one will on the contrary rely on the floodplains, and so the parameterisation of the model will add more value to Kch or Kfp. Thus, the conclusions one can make by interpreting one analysis of an example do not necessarily reflect the global behaviour of the model.
A2 Sobol method
The Sobol method (Sobol2001) is a variance-based sensitivity analysis which aims to compute the fraction of the variance that can be attributed to each parameter. For this study, 2×500 sets of parameters were randomly chosen with a Latin hypercube sampling method, thus creating two 500×6 matrices, XA and XB. Each column of XA has sequentially been substituted by a column of XB, corresponding to one of the six parameters, leading to six other matrices. In order to limit the computation time, the interaction of several parameters (i.e. substituting two or more columns of XA by those of XB) has not been assessed. Indeed, MHYST has been launched with the 4000 sets of parameters, with a resolution of 50 m, which takes longer than the Morris method that only needed about a thousand simulations. The first-order Sobol indices Si, which indicate the contribution of one parameter to the total variance, and the total-effect indices Si, which calculate the total contribution of one parameter to the variance, including the possible interactions between parameters, have been computed. Then, with a bootstrap re-sampling method, the distributions of Si and Si have been assessed, allowing several characteristics such as the bias to be computed, the standard deviation and the confidence intervals.
The results of this analysis are presented in Table A1 for Si and Table A2 for Si. The first-order indices confirm parts of what was concluded from the Morris analysis, interpreting ω as the most influential parameter, Kch and α as moderately influential and Kfp as not influential, despite the observations we made in the article when we calibrated the parameters. The total-effect indices complete the analysis and confirm the conclusions we made with the Morris method, adding β to the list of influential parameters.
Table A1Sobol first-order indices for the six parameters of MHYST. Confidence interval is denoted as conf. int. here.
The distributions of Si and Si show that the values calculated are not biased, but the 95 % confidence interval is rather large, which means that in some cases, the interpretation may differ. This might explain why when we set values for all downstream hydraulic geometry equations parameters (α, β, δ, ω) from regionalised studies or observations, Kfp has a greater influence which is not highlighted by the sensitivity analyses. These methodologies (Morris, Sobol) indeed explore the parameter space in detail, and even with reasonable boundaries, they can reach values that may not be consistent with the characteristics of the catchment studied. Another limitation is the fact that these analyses are only valid for this particular example (the Loing catchment and the event of May–June 2016). They should ideally be used with a larger set of catchments and events to be reliably trusted.
Table A2Sobol total-effect index for the six parameters of MHYST. Confidence interval is denoted as conf. int. here.
In order to understand why Morris and Sobol give, contrary to our initial expectation, so little importance to Kfp, we conducted a quick Sobol analysis with fixed hydraulic geometry parameters, i.e. we considered the α, β, δ and ω values used in the original study and only made Kch and Kfp vary. This time, the results confirm what we observed: ${S}_{{K}_{\mathrm{ch}}}=\mathrm{0.15}$ and ${S}_{{K}_{\mathrm{fp}}}=\mathrm{0.85}$, which means that Kfp is a major parameter in our situation, and that Kch has a smaller role.
The hydraulic geometry parameters are clearly important, but if they are fixed to legitimate values estimated by observations or tables of regionalised values, their impact becomes minor in front of the Strickler coefficients.
Author contributions
Author contributions.
The model presented in this paper was developed and analysed by CR during his PhD work. He also wrote the paper, which was corrected by VA and NLM.
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Acknowledgements
Acknowledgements.
The first author was funded by a grant from the AXA Research Fund. Thanks are extended to Rafal Zielinksi, who helped us access data from the Copernicus Emergency Management Service. We would also like to thank the AXA Global P&C research team for their advice and our discussions on the development of simple conceptual inundation models. The MHYST model was developed using R (R Core Team, 2015) and GFortran, Gnu compiler collection (gcc) Version 4.9.2.
Edited by: Roger Moussa
Reviewed by: Renata Romanowicz and one anonymous referee
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McGrath, H., Bourgon, J.-F., Proulx-Bourque, J.-S., Nastev, M., and Abo El Ezz, A.: A comparison of simplified conceptual models for rapid web-based flood inundation mapping, Nat. Hazards, 93, 905–920, https://doi.org/10.1007/s11069-018-3331-y, 2018. a
Morales-Hernández, M., Petaccia, G., Brufau, P., and García-Navarro, P.: Conservative 1D–2D coupled numerical strategies applied to river flooding: The Tiber (Rome), Appl. Math. Model., 40, 2087–2105, https://doi.org/10.1016/j.apm.2015.08.016, 2016. a
Morris, M. D.: Factorial Sampling Plans for Preliminary Computational Experiments, Technometrics, 33, 161–174, https://doi.org/10.2307/1269043, 1991. a, b, c
Moussa, R. and Cheviron, B.: Modeling of Floods – State of the Art and Research Challenges, in: Rivers – Physical, Fluvial and Environmental Processes, edited by: Rowiński, P. and Radecki-Pawlik, A., 169–192, Springer International Publishing, Cham, https://doi.org/10.1007/978-3-319-17719-9_7, 2015. a
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Nobre, A. D., Cuartas, L. A., Momo, M. R., Severo, D. L., Pinheiro, A., and Nobre, C. A.: HAND contour: a new proxy predictor of inundation extent: Mapping Flood Hazard Potential Using Topography, Hydrol. Process., 30, 320–333, https://doi.org/10.1002/hyp.10581, 2016. a, b, c, d
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Zheng, X., Tarboton, D. G., Maidment, D. R., Liu, Y. Y., and Passalacqua, P.: River channel geometry and rating curve estimation using height above the nearest drainage, J. Am. Water Resour. As., 54, 785–806, https://doi.org/10.1111/1752-1688.12661, 2018. a, b | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 23, "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": 0.8103287816047668, "perplexity": 2426.0849120316198}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998473.44/warc/CC-MAIN-20190617103006-20190617125006-00328.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/1-visible-light-wavelength-480-nm-passes-two-slitswith-separation-024-mm-interference-patt-q841357 | 1. When visible light of wavelength 480 nm passes through two slitswith a separation of 0.24 mm, the interference pattern in Figure(a) is observed. The pattern is viewed on a screen located 1.55 mfrom the slits. What is the distance between the two verticaldashed lines in Figure (a) above?
2. When green light passes through a double slitarrangement in air, the interference pattern in Figure (a) above isobserved on a distant screen. It is possible to alter theinterference pattern to look like Figure (b) by changing specificparameters. Which of the following changes would result inobtaining the pattern shown in Figure (b)?
1. change the light from green to blue
2. move the entire apparatus, green light and all, into avacuum environment where there is no air at all
3. increase the intensity of the incidentlight
4. decrease the frequency of the light
5. increase the slit spacing
i can't get either of these. The answer for #1 is numericaland the answers for #2 are true or false. Any help isappreciated. | {"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": 0.8110200762748718, "perplexity": 1112.9288804230061}, "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-2015-11/segments/1424936462548.53/warc/CC-MAIN-20150226074102-00322-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://epjc.epj.org/articles/epjc/abs/2003/08/100520175/100520175.html | 2020 Impact factor 4.590
Particles and Fields
Eur. Phys. J. C 28, 175-201 (2003)
DOI: 10.1140/epjc/s2003-01163-y
## Measurement of high- Q2e-p neutral current cross sections at HERA and the extraction of xF3
The ZEUS collaboration
(Received: 2 August 2002 / Revised version: 30 January 2003 / Published online: 24 March 2003 )
Abstract
Cross sections for e-p neutral current deep inelastic scattering have been measured at a centre-of-mass energy of 318 GeV using an integrated luminosity of 15.9 pb -1 collected with the ZEUS detector at HERA. Results on the double-differential cross-sec tion in the range GeV 2 and 0.0037 < x < 0.75, as well as the single-differential cross-sec tions , and for Q2 > 200 GeV 2, are presented. To study the effect of Z-boson exchange, has also been measured for GeV 2. The structure function xF3 has been extracted by combining the e-p results presented here with the recent ZEUS measurements of e+p neutral current deep inelastic scattering. All results agree well with the predictions of the Standard Model.
© Società Italiana di Fisica, Springer-Verlag 2003 | {"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": 0.9530389904975891, "perplexity": 3033.911349477213}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584567.81/warc/CC-MAIN-20211016105157-20211016135157-00433.warc.gz"} |
https://atmos.ut.ee/en/publication/isi-000419134200001/ | # Influence of temperature on the molecular composition of ions and charged clusters during pure biogenic nucleation
### Abstract
It was recently shown by the CERN CLOUD experiment that biogenic highly oxygenated molecules (HOMs) form particles under atmospheric conditions in the absence of sulfuric acid, where ions enhance the nucleation rate by 1-2 orders of magnitude. The biogenic HOMs were produced from ozonolysis of alpha-pinene at 5 degrees C. Here we extend this study to compare the molecular composition of positive and negative HOM clusters measured with atmospheric pressure interface time-of-flight mass spectrometers (APi-TOFs), at three different temperatures (25, 5 and -25 degrees C). Most negative HOM clusters include a nitrate (NO3-) ion, and the spectra are similar to those seen in the nighttime boreal forest. On the other hand, most positive HOM clusters include an ammonium (NH4+) 4) ion, and the spectra are characterized by mass bands that differ in their molecular weight by similar to 20 C atoms, corresponding to HOM dimers. At lower temperatures the average oxygen to carbon (O : C) ratio of the HOM clusters decreases for both polarities, reflecting an overall reduction of HOM formation with decreasing temperature. This indicates a decrease in the rate of autoxidation with temperature due to a rather high activation energy as has previously been determined by quantum chemical calculations. Furthermore, at the lowest temperature (-25 degrees C), the presence of C-30 clusters shows that HOM monomers start to contribute to the nucleation of positive clusters. These experimental findings are supported by quantum chemical calculations of the binding energies of representative neutral and charged clusters.
Type
Publication
ATMOSPHERIC CHEMISTRY AND PHYSICS | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9247081279754639, "perplexity": 2119.3268239611207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1620243989030.87/warc/CC-MAIN-20210510033850-20210510063850-00033.warc.gz"} |
https://www.physicsforums.com/threads/linear-combination-and-orthogonality.837876/ | # Linear combination and orthogonality
1. Oct 15, 2015
### JulienB
1. The problem statement, all variables and given/known data
Given the vectors a = (5,2,-1), b = (3,2,1), c = (1,2,3), b' = (1,1,0), c' = (3,-3,-2)
We assume that the vector a is a linear combination of the vectors b and c and b' and c' respectively, so that:
a = xb + yc = x'b' + y'c'
a) Determine the factors x and y through construction of different scalar products from the given equation.
b) Repeat the calculation for the vectors b' and c' to obtain x' and y'. What do you notice?
c) How can you verify the conditions given at the beginning?
2. Relevant equations
Anything about basic vector operations I guess.
3. The attempt at a solution
I think I was able to solve a) and b), but my question is that I have not been using scalar product, but instead a simple system of linear equations:
a) a = xb + yc = (3x,2x,x) + (y,2y,3y) = (3x + y, 2x + 2y, x + 3y)
From there, I determine x and y using the fact that I know the components of the vector a:
3x + y = 5
2x + 2y = 2
x = 2 and y = -1
b) a = x'b' + y'c' = (x',x',0) + (3y',-3y',-2y') = (x' + 3y', x' - 3y', -2y')
Using the same method as a), I obtain:
x' = 7/3 and y' = 1/2
Not only did I not use scalar product (or did I? since multiplying a vector by a constant is considered a scalar product, right?), but I also notice nothing really :)
J.
2. Oct 15, 2015
### Staff: Mentor
Scalar product AKA dot product.
No. Multiplying a vector by a scalar is considered scalar multiplication. Scalar product implies multiplying two vectors to get a scalar.
This problem is an extension of the work you did in your other thread. Part a is to find constants x and y so that a is a linear combination of b and c. IOW, so that a = xb + yc. xb is the vector projection of a in the direction of b, yc is the vector projection of a in the direction of c. Part b is the same idea, except using two different vectors.
If you do this problem without the use of projections and such, your instructor will almost certainly take off some or all of the points.
3. Oct 15, 2015
### JulienB
Aah yes, I guess I can figure that out now. Thanks a lot again!
J.
4. Oct 16, 2015
### JulienB
I keep having a wrong result, can someone tell me where I make a mistake? Thank you very much in advance.
a = xb + yc
xb = ((a.b)/|b|).(b/|b|) = ((a.b)/|b|2).b = (9/7)b
So x = 9/7
yc = ((a.c)/|c|).(c/|c|) = ((a.c)/|c|2).c = (3/7)c
So y = 3/7
However, when I compute the results I found back into xb + yc to check the result, it comes out wrong and is no more equal to a.
5. Oct 16, 2015
### JulienB
What is strange is that it works perfectly in the second part of the problem:
a = x'b' + y'c'
x'b' = ((a.b')/|b'|).(b'/|b'|) = ((a.b')/|b'|2).b' = (7/2)b'
So x' = 9/7
y'c' = ((a.c')/|c'|).(c'/|c'|) = ((a.c')/|c'|2).c' = (1/2)c'
So y' = 3/7
Check: (7/2)b' + (1/2)c' = (7/2,7/2,0) + (3/2,-3/2,-2/2) = (5,2,-1) = a
Did I make (repeatedly!) a calculus error in the first part, or is there something else I didn't quite get? (The last question of b) says "What do you notice?")
6. Oct 16, 2015
### Ray Vickson
The vectors $\vec{b}'$ and $\vec{c}'$ are orthogonal, while $\vec{b}$ and $\vec{c}$ are not.
7. Oct 17, 2015
### JulienB
What I don't get, is how the dot product is even relevant for a) ? With a simple system of linear equations, I come very quickly to the result x = 2 and y = -1, but I can't seem to reach that in any way with the projection of a over b, probably because b and c and indeed not perpendicular.
8. Oct 17, 2015
### HallsofIvy
That's right- the dot product is not relevant in (a). The point is that because b' and c' are orthogonal, it is much easier to find the coefficients to write another vector as a linear combination of them.
If they were orthonormal it would be even easier.
9. Oct 17, 2015
### Staff: Mentor
It's relevant because its use is a requirement of the problem.
Is that a good enough reason? Even more important, I believe this problem is setting the stage for a technique to be presented later, the Gram-Schmidt process, which is used to construct an orthogonal basis from a set of basis vectors.
Part a of this problem is to find scalars x and y so that a = xb + yc, with the vectors a, b, and c given.
I believe that the intent of parts a and b is to get you to use the dot product (AKA scalar product) to come up with equations so you can find the constants in each part.
If you dot both sides of the equation above with b, you get another equation. If you dot both sides of the equation above with c, you get another equation. The two equations in x and y can be solved for these variables.
Last edited: Oct 17, 2015
10. Oct 17, 2015
### JulienB
"a) Determine the factors x and y through construction of different scalar products from the given equation.
b) Repeat the calculation for the vectors b' and c' to obtain x' and y'. What do you notice?"
As Mark said earlier, I don't think I can go through a) without dot product and simply ignore the "repeat the calculation" of b).
11. Oct 17, 2015
### Staff: Mentor
See my post #9, which talks about how the dot product is intended to be used (I believe).
12. Oct 17, 2015
### JulienB
Thank you Mark, every post you send definitely helps me a lot. The problem was solved using the technique you mentioned in post #9.
13. Oct 17, 2015
### Staff: Mentor
I'm glad to be able to help! | {"extraction_info": {"found_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": 0.88188636302948, "perplexity": 649.5075757763035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1521257646178.24/warc/CC-MAIN-20180318224057-20180319004057-00239.warc.gz"} |
https://www.knowpia.com/knowpedia/Galactic_halo | BREAKING NEWS
Galactic halo
Summary
A galactic halo is an extended, roughly spherical component of a galaxy which extends beyond the main, visible component.[1] Several distinct components of galaxies comprise the halo:[2][3]
The distinction between the halo and the main body of the galaxy is clearest in spiral galaxies, where the spherical shape of the halo contrasts with the flat disc. In an elliptical galaxy, there is no sharp transition between the other components of the galaxy and the halo.
A halo can be studied by observing its effect on the passage of light from distant bright objects like quasars that are in line of sight beyond the galaxy in question.[4]
Components of the galactic halo
Stellar halo
The stellar halo is a nearly spherical population of field stars and globular clusters. It surrounds most disk galaxies as well as some elliptical galaxies of type cD. A low amount (about one percent) of a galaxy's stellar mass resides in the stellar halo, meaning its luminosity is much lower than other components of the galaxy.
The Milky Way's stellar halo contains globular clusters, RR Lyrae stars with low metal content, and subdwarfs. Stars in our stellar halo tend to be old (most are greater than 12 billion years old) and metal-poor, but there are also halo star clusters with observed metal content similar to disk stars. The halo stars of the Milky Way have an observed radial velocity dispersion of about 200 km/s and a low average velocity of rotation of about 50 km/s.[5] Star formation in the stellar halo of the Milky Way ceased long ago.[6]
Galactic corona
A galactic corona is a distribution of gas extending far away from the center of the galaxy. It can be detected by the distinct emission spectrum it gives off, showing the presence of HI gas (H one, 21 cm microwave line) and other features detectable by X-ray spectroscopy.[7]
Dark matter halo
The dark matter halo is a theorized distribution of dark matter which extends throughout the galaxy extending far beyond its visible components. The mass of the dark matter halo is far greater than the mass of the other components of the galaxy. Its existence is hypothesized in order to account for the gravitational potential that determines the dynamics of bodies within galaxies. The nature of dark matter halos is an important area in current research in cosmology, in particular its relation to galactic formation and evolution.[8]
The Navarro–Frenk–White profile is a widely accepted density profile of the dark matter halo determined through numerical simulations.[9] It represents the mass density of the dark matter halo as a function of ${\displaystyle r}$ , the distance from the galactic center:
${\displaystyle \rho (r)={\frac {\rho _{\text{crit}}\delta _{c}}{(r/r_{s})(1+r/r_{s})^{2}}}}$
where ${\displaystyle r_{s}}$ is a characteristic radius for the model, ${\displaystyle \rho _{\text{crit}}=3H^{2}/8\pi G}$ is the critical density (with ${\displaystyle H}$ being the Hubble constant), and ${\displaystyle \delta _{c}}$ is a dimensionless constant. The invisible halo component cannot extend with this density profile indefinitely, however; this would lead to a diverging integral when calculating mass. It does, however, provide a finite gravitational potential for all ${\displaystyle r}$ . Most measurements that can be made are relatively insensitive to the outer halo's mass distribution. This is a consequence of Newton's laws, which state that if the shape of the halo is spheroidal or elliptical there will be no net gravitational effect from halo mass a distance ${\displaystyle r}$ from the galactic center on an object that is closer to the galactic center than ${\displaystyle r}$ . The only dynamical variable related to the extent of the halo that can be constrained is the escape velocity: the fastest-moving stellar objects still gravitationally bound to the Galaxy can give a lower bound on the mass profile of the outer edges of the dark halo.[10]
Formation of galactic halos
The formation of stellar halos occurs naturally in a cold dark matter model of the universe in which the evolution of systems such as halos occurs from the bottom-up, meaning the large scale structure of galaxies is formed starting with small objects. Halos, which are composed of both baryonic and dark matter, form by merging with each other. Evidence suggests that the formation of galactic halos may also be due to the effects of increased gravity and the presence of primordial black holes.[11] The gas from halo mergers goes toward the formation of the central galactic components, while stars and dark matter remain in the galactic halo.[12]
On the other hand, the halo of the Milky Way Galaxy is thought to derive from the Gaia Sausage.
• Disc galaxy – Type of galactic form
• Galactic bulge – Tightly packed group of stars within a larger formation
• Galactic corona – Hot, ionised, gaseous component in the Galactic halo
• Galactic coordinate system – Celestial coordinate system in spherical coordinates, with the Sun as its center
• Galaxy formation and evolution – From a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time
• Spiral arm – Regions of stars that extend from the center of spiral and barred spiral galaxies
References
1. ^ "OpenStax Astronomy". OpenStax.
2. ^ Helmi, Amina (June 2008). "The stellar halo of the Galaxy". The Astronomy and Astrophysics Review. 15 (3): 145–188. arXiv:0804.0019. Bibcode:2008A&ARv..15..145H. doi:10.1007/s00159-008-0009-6. ISSN 0935-4956. S2CID 2137586.
3. ^ Maoz, Dan (2016). Astrophysics in a Nutshell. Princeton University Press. ISBN 978-0-691-16479-3.
4. ^ August 2020, Meghan Bartels 31 (31 August 2020). "The Andromeda galaxy's halo is even more massive than scientists expected, Hubble telescope reveals". Space.com. Retrieved 2020-09-01.
5. ^ Setti, Giancarlo (30 September 1975). Structure and Evolution of Galaxies. D. Reidel Publishing Company. ISBN 978-90-277-0325-5.
6. ^ Jones, Mark H. (2015). An Introduction to Galaxies and Cosmology Second Edition. Cambridge University Press. ISBN 978-1-107-49261-5.
7. ^ Lesch, Harold (1997). The Physics of Galactic Halos.
8. ^ Taylor, James E. (2011). "Dark Matter Halos from the Inside Out". Advances in Astronomy. 2011: 604898. arXiv:1008.4103. Bibcode:2011AdAst2011E...6T. doi:10.1155/2011/604898. ISSN 1687-7969.
9. ^ Navarro, Julio F.; Frenk, Carlos S.; White, Simon D. M. (May 1996). "The Structure of Cold Dark Matter Halos". The Astrophysical Journal. 462: 563–575. arXiv:astro-ph/9508025. Bibcode:1996ApJ...462..563N. doi:10.1086/177173. ISSN 0004-637X. S2CID 119007675.
10. ^ Binney and Tremaine (1987). Galactic Dynamics. Princeton University Press.
11. ^ Worsley, Andrew (October 2018). "Advances in Black Hole Physics and Dark Matter Modelling of the Galactic Halo".
12. ^ Zolotov, Adi; Willman, Beth; Brooks, Alyson M.; Governato, Fabio; Brook, Chris B.; Hogg, David W.; Quinn, Tom; Stinson, Greg (2009-09-10). "The Dual Origin of Stellar Halos". The Astrophysical Journal. 702 (2): 1058–1067. arXiv:0904.3333. Bibcode:2009ApJ...702.1058Z. doi:10.1088/0004-637X/702/2/1058. ISSN 0004-637X. S2CID 16591772. | {"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": 0.9697458744049072, "perplexity": 1722.333818940479}, "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-2023-14/segments/1679296950363.89/warc/CC-MAIN-20230401221921-20230402011921-00488.warc.gz"} |
http://link.springer.com/article/10.1007/s10509-012-1181-8 | Astrophysics and Space Science
, Volume 342, Issue 1, pp 155–228
# Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests
• Kazuharu Bamba
• Salvatore Capozziello
• Shin’ichi Nojiri
• Sergei D. Odintsov
Invited Review
DOI: 10.1007/s10509-012-1181-8
Bamba, K., Capozziello, S., Nojiri, S. et al. Astrophys Space Sci (2012) 342: 155. doi:10.1007/s10509-012-1181-8
## Abstract
We review different dark energy cosmologies. In particular, we present the ΛCDM cosmology, Little Rip and Pseudo-Rip universes, the phantom and quintessence cosmologies with Type I, II, III and IV finite-time future singularities and non-singular dark energy universes. In the first part, we explain the ΛCDM model and well-established observational tests which constrain the current cosmic acceleration. After that, we investigate the dark fluid universe where a fluid has quite general equation of state (EoS) [including inhomogeneous or imperfect EoS]. All the above dark energy cosmologies for different fluids are explicitly realized, and their properties are also explored. It is shown that all the above dark energy universes may mimic the ΛCDM model currently, consistent with the recent observational data. Furthermore, special attention is paid to the equivalence of different dark energy models. We consider single and multiple scalar field theories, tachyon scalar theory and holographic dark energy as models for current acceleration with the features of quintessence/phantom cosmology, and demonstrate their equivalence to the corresponding fluid descriptions. In the second part, we study another equivalent class of dark energy models which includes F(R) gravity as well as F(R) Hořava-Lifshitz gravity and the teleparallel f(T) gravity. The cosmology of such models representing the ΛCDM-like universe or the accelerating expansion with the quintessence/phantom nature is described. Finally, we approach the problem of testing dark energy and alternative gravity models to general relativity by cosmography. We show that degeneration among parameters can be removed by accurate data analysis of large data samples and also present the examples.
### Keywords
Modified theories of gravityDark energyCosmology
## Authors and Affiliations
• Kazuharu Bamba
• 1
• Salvatore Capozziello
• 2
• 3
• Shin’ichi Nojiri
• 1
• 4
• Sergei D. Odintsov
• 5
• 6
• 7
• 8
1. 1.Kobayashi-Maskawa Institute for the Origin of Particles and the UniverseNagoya UniversityNagoyaJapan
2. 2.Dipartimento di Scienze FisicheUniversità di Napoli “Federico II”NapoliItaly
3. 3.INFN Sez. di NapoliCompl. Univ. di Monte S. AngeloNapoliItaly
4. 4.Department of PhysicsNagoya UniversityNagoyaJapan
5. 5.Instituciò Catalana de Recerca i Estudis Avançats (ICREA)BarcelonaSpain
6. 6.Instituto de Ciencias del Espacio (CSIC) and Institut d Estudis Espacials de Catalunya (IEEC-CSIC)Facultat de CienciesBellaterra (Barcelona)Spain
7. 7.Tomsk State Pedagogical UniversityTomskRussia
8. 8.Eurasian International Center for Theoretical Physics and Department of General & Theoretical PhysicsEurasian National UniversityAstanaKazakhstan | {"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": 0.8909526467323303, "perplexity": 4809.37028950416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1474738661775.4/warc/CC-MAIN-20160924173741-00255-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://search.datacite.org/works/10.4230/LIPICS.FSTTCS.2010.96 | The effect of girth on the kernelization complexity of Connected Dominating Set
Neeldhara Misra, Geevarghese Philip, Venkatesh Raman & Saket Saurabh
In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In... | {"extraction_info": {"found_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": 0.870441734790802, "perplexity": 164.57091034334456}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828134.97/warc/CC-MAIN-20171024033919-20171024053919-00233.warc.gz"} |
http://www.bh7lsw.cn/archives/1079 | Fundamentals of the MiniWhip antenna
Pieter-Tjerk de Boer, PA3FWM [email protected]
(This is an adapted version of an article that I wrote for the Dutch amateur radio magazine Electron, January 2014.)
MiniWhip天线基本原理
（这篇文章刊登在荷兰业余无线电杂志《电子》2014年1月刊，略有删减。）
2017年11月
A well-known active antenna for the LF, MF and HF bands is the “MiniWhip” designed by PA0RDT; see [1]. Many ideas and misunderstandings have been voiced about how this antenna works. In this article, I hope to shed some light on this, by using some elementary theory.
PA0RDT大神设计的MiniWhip是一款众所周知的长中短波接收有源天线，关于它的工作方式，江湖上有许多误解。在这篇小文章里，我将用一些电磁学理论解释一下它的工作原理。
1、MiniWhip概况
2、无线电波电场的规律
3、支撑杆为绝缘体的情况
4、无线电波的极化方式
5、振子用小金属板好还是用鞭状导线好
6、方向性
7、小尺寸天线的方向性
8、MiniWhip的方向性
9、结论
10、参考文献
关于本文作者及其WebSDR
1The MiniWhip
The figure shows a sketch of a typical MiniWhip setup. It consists of a mast of a few meters high, ideally in the open field, with on top of it a small metal plate and an amplifier, together in a plastic enclosure (the actual MiniWhip). A coaxial cable runs from the MiniWhip down along the mast to a receiver. For now we assume the mast is conductive and grounded, but we will later see what happens if this is not the case. The amplifier is a voltage follower with a very high input impedance in order not to load the metal plate, and a low output impedance to be able to deliver sufficient power to the 50 ohm coaxial cable; see [1,2,3]. The idea is that the metal plate “measures” the electrical field at its location, and sends the result via the coax to the receiver.
1MiniWhip概况
MiniWhip天线由一块小金属板（译注：小金属板即天线振子）和放大器构成，密封在塑料防水盒里，由同轴电缆引入室内的接收机。
2Principle and electric field
Let’s make the following assumptions: the mast’s height is small compared to the wavelength, and the signal to be received is vertically polarized. These are reasonable assumptions: the MiniWhip is often touted as an antenna for LF and MF (so with wavelengths of hundreds of meters), and those signals are predominantly vertically polarized (due to the influence of the relatively nearby conducting earth). At higher HF frequencies, these assumptions become less realistic, depending on the mast’s height.
2、无线电波电场的规律
Such a vertically polarized radio signal produces vertical fieldlines in the area around the antenna, and as a consquence the so-called equipotential surfaces (surfaces on which the potential, i.e. the voltage w.r.t. ground, is the same everywhere). The metal plate in the MiniWhip will be at the same potential as the equipotential surface crossing it.
However, the amplifer in the MiniWhip is not just connected to the metal plate, but also to the grouned mast. To be more precise: the amplifier measurs the potential difference between the plate and the mast, buffers this, and applies this same potential difference between the shield and the center conductor of the coaxial cable. This is crucial: the signal that ends up in the receiver, is the potential difference between the plate and the mast.
How large is this potential difference? The simplest reasoning says that the plate has the same potential as the field a few meters above the ground (the height of the mast), and the mast itself is at ground potential (because its lower end is grounded). However, this is a simplification. If the entire mast is at ground potential, the higher equipotential surfaces cannot cross it, and thus must be distorted.
This figure shows what the distorted surfaces look like (calculated by having my computer solve the relevant Maxwell equation). The black line at the bottom denotes the earth. On it, there is a (rather thick, cylindrical) mast, and a metal block floats above it, both also shown in black. The metal block is the MiniWhip’s metal plate. The red lines are the equipotential surfaces, or rather, cuts through them. Each of these lines corresponds to a potential, expressed in volts: the voltage of this equipotential surface w.r.t. ground. The earth and the mast itself are at ground potential, say 0 volts. The lowest red line could then e.g. be at 1 μV, the next at 2 μV, and so on.
Far away from the mast, the equipotential lines/surfaces are almost horizontal, as one would expect for a vertically polarized electric field. Around the mast, they are distorted, because the entire mast is at ground potential. And also around the metal block above it, the lines are distorted, because the potential on a conductor is the same everywhere. But actually, the distortion is not too bad; at the metal block the potential is hardly different than far away from the mast at the same height. Further calculations show that the distortion decreases as the mast becomes thinner.
3Isolated mast
What if the mast is not conductive? The amplifier will still measure the potential difference between the plate and the “ground” of the amplifier circuit. If the mast is not conductive, then the only thing connected to the circuit ground is the shield of the coaxial cable. In that case, the potential difference will be measured between the plate and whatever that cable shield goes to. If the cable shield is solidly connected to ground somewhere further on, it will work just as well as with a grounded mast. But if the shield is not grounded, goes into the shack and connects there to a “dirty” ground (e.g., the mains safety ground), well, then all noise on that dirty ground will contribute to the potential difference at the amplifier input, and thus end up in the receiver. Hence the importance of good grounding.
3、支撑杆为绝缘体的情况
It might be an idea to replace the coaxial cable by a glass fiber. That would eliminate all noise coming into the amplifier via the coaxial cable. But without any conductive connection to the outside, the entire circuit will be at the same potential, so the received signal will not cause a potential difference that can be passed on to the receiver. As a consequence, nothing will be received. PA0RDT recently tried this in practice and reported about it on the RSGB-LF mailing list: indeed, he received nothing.
4Polarization
Another interesting experiment by PA0RDT was to not put the antenna on top of a vertical pole in the garden, but on a horizontal pole out of a window, with the coaxial cable also attached to that horizontal pole. He did this in such a way that the metal plate in either case ended up at the same spot, and noticed that the reception of a vertically polarized MF signal was equally strong. At first sight, this suggests that the antenna is not polarized: the reception is the same, even though the entire setup has been rotated from vertical to horizontal.
4、无线电波的极化方式
PA0RDT做了另一个有趣的研究，不把MiniWhip天线垂直安置，而是伸出窗外，水平放置，同样用同轴电缆引进室内的接收机。他发现接收效果跟垂直放置的时候一样。乍看起来，这个天线似乎并不在乎极化方式，它横竖怎么摆放都可以。
However, this conclusion is not correct. The amplifier still measures the potential difference between the plate and the pole (if conductive and grounded) or the coax shield (which presumably is grounded somewhere, possibly via the mains). So, the potential difference is still measured between the plate (which is again at the same spot) and the ground (which also has not changed), so it is to be expected that the resulting signal is the same. Whether the ground connection goes down vertically, or takes a partially horizontal detour, does not matter, as long as the detour is short compared to the wavelength.
5Plate or whip
Most active electric field antennas do not, like the MiniWhip, use a metal plate, but a whip of about a meter long. This makes no essential difference for the operation. Such a whip will, if it is short compared to the wavelength, assume the average potential of its surroundings, in this case the potential about half a meter above the top of the mast. This half meter extra height will hardly affect the potential difference to ground.
5、振子用小金属板好还是用鞭状导线好
However, there is another important difference, namely the capacitance of the plate or whip. A whip has a capacitance of almost 10 pF per meter of length, slightly dependent on it thickness. A circular metal plate has a capacitance of about 0.35 pF per cm diameter (proportional to the diameter, not to the area, as one might expect). I haven’t found a formula for a rectangular plate, but the shape should not matter too much, so a typical MiniWhip has about 2 pF of plate capacitance. That capacitance is important, because together with the amplifier input capacitance it forms a capacitive voltage divider. If the plate or whip’s capacitance is smaller, less voltage remains when the amplifier is connected.
6Directivity
Before we can say anything about the directivity of an antenna, it’s good to have a closer look at what is or determines the “direction” of a radio signal. The figure shows a vertically polarized transmitting antenna, and the electric and magnetic field lines produced by this antenna at a large distance (the so-called far field). We see that the electric fieldlines are vertical, not surprisingly since the electric field is caused by e.g. the top half of the dipole being charged positively and the bottom half negatively (or the other way around, half a period later). We also see that the magnetic field lines are horizontal, forming a large circle around the antenna; this is also to be expected, since we know that magnetic field lines form circles around a current-carrying wire.
6、方向性
The figure also shows the so-called Poynting vector. It is named after the English physicist J.H. Poynting, and points in the direction in which the wave propagates. Mathematically it is given by the so-called outer product of the electric and magnetic field vectors. Its direction can be determined by turning one’s lefthand such that it catches the magnetic field lines into its palm, and the fingers are aligned with the electric fieldlines; then the thumb indicates the direction of the Poynting vector.
How can an antenna be more sensitive to signals from one direction than from another? If the antenna could directly sense the Poynting vector, it would be easy, since this vector indicates directly the propagation direction. Unfortunately however, antennas do not respond to the Poynting vector but only to the electric and/or magnetic field.
The first way in which an antenna can have directional sensitivity, is by measuring the electric or magnatic field at several places and “comparing” the phase of the signal at those places. This happens e.g. in a Yagi antenna: a signal that arrives from straight ahead, reaches the first director earlier than the dipole. However, for small antennas this principle doesn’t work: if the antenna is small compared to the wavelength, the signal arrives almost simultaneously everywhere in the antenna, and thus does not give a significant phase difference.
7Directivity of small antennas
For a small antenna to be directionally sensitive, the only possibility is to use the direction of the electric and magnetic field lines themselves. Unfortunately, they do not always reveal the direction from which the signal is coming.
7、小尺寸天线的方向性
Consider the vertically polarized field in the previous figure. The electrical field lines at the place of the receiver are vertical, regardless of whether the transmitter is left or right, front or rear. Thus we cannot conclude from the electric field from which direction the signal is coming. (Well, we can conclude that the signal is coming in horizontally rather than steeply from up in the sky. But usually that’s not so interesting.)
In contrast, the magnetic field lines do say something about the direction. If e.g. the transmitter is west of us, then the magnetic field lines are in a north-south direction. If the transmitter is north of us, the magnetic field lines run east/west. But this is not unambiguous: if the transmitter would be south of us, the magnetic field lines would also be east/west. In other words: in case of a vertically polarized signal, the magnetic field lines tell us from which direction the signal is coming, albeit with a 180 degree uncertainty. And this is of course well known from portable mediumwave radios with a built-in ferrite rod antenna: such an antenna is directionally sensitive, but if one rotates it by 180 degrees, reception doesn’t change.
Fox hunters / ARDF participants on 80 meters also use ferrite antennas that respond to the direction of the magnetic field lines. To resolve the 180 degree ambiguity, these receivers often have an additional “sense antenna”: a whip that responds to the electric field. As noted before, that field says nothing about the direction of the signal, but it can resolve the 180-degree ambiguity of the magnetic fieldlines: depending on the direction, the electric signal is in phase or 180 degrees out of phase with the magnetic signal.
A nice application of these principles is DF6NM’s directional longwave receiver [4]. He uses two magnetic antennas under 90 degree angles to determine the direction of the signal, and an electric antenna to resolve the 180-degree ambiguity. He uses this data to produce a waterfall diagram in which the color indicates the direction.
80米波段的测向机也用了一根磁棒来确定测向信号的磁场方向，为了解决存在的180度不确定性，测向接收机通常加了根鞭状天线来感应电场方向。如前所述，电场线并不能告诉我们信号的方向，但它能在依靠磁棒天线确定大体方向之后帮助我们解决那180度不确定方向的问题：判断电场方向与磁场线是否同相位即可。DF6NM设计的一种测向机有两根互相垂直的磁棒，外加一根测电场的鞭状天线，可以直接在图形上指示出信号方向。
All considerations so far were about vertically polarized signals. For horizontally polarized signals the situation is reversed: the electric field lines run horizontal and reveal the direction, while the magnetic field is vertical and says nothing about the direction.
8Directivity of the MiniWhip
And what about the MiniWhip? We have already seen that it responds to the electric component of a vertically polarized signal, and that it is small compared to the wavelength. Then only one conclusion is possible: it is not directionally sensitive.
8MiniWhip的方向性
However, the MiniWhip does have a dip straight up: it will not respond to signals coming from straight above. Such signals have both their electric and magnetic field lines horizontal, and then there is nothing left for this antenna to respond to. This is quite noticeable with my WebSDR at the University of Twente. Occasionally Dutch users complain that its antenna is bad because they cannot hear Dutch stations on 80 m too well; but those signals bounce off the ionosphere almost vertically.
Some people have suggested that in order to receive signals coming from straight above, one should mount a horizontal plate on top of the MiniWhip’s vertical plate. This is not going to work: the MiniWhip would still measure the potential difference between the plate and the ground, and for signals from straight up that difference is 0.
9Conclusions
What can we conclude from all of this theory?
• The MiniWhip is vertically polarized.
• Grounding is important: if the antenna is only grounded in the shack, via the coax cable, much noise can be picked up. B.t.w., that grounding does not need to be galvanic: a large piece of metal, even if not connected directly to the earth, may have enough capacity to serve as ground.
• The strength of the received signal is directly proportional to the height of the antenna above ground, as long as this is small w.r.t. the wavelength.
• Whether the mast is conductive or not hardly matters for the reception. However, if the mast is conductive, the antenna’s plate must of course not be mounted besides but above the mast.
• The antenna is omni-directional, except for a dip straight up.
• The orientation or shape of the metal plate do not matter; b.t.w., this is also true for the whip in case of whip-based active antennas.
9、结论
1、MiniWhip是垂直极化的。
2、接地非常重要。如果天线只是通过同轴电缆接到了电台的地，会引入噪音。但“地”并非一定要能流通的地线，不与大地连接的虚拟地也是可以的，只要金属部分足够大，以满足放大器输入端输入电容的需求。
3、接收到的信号强度与天线离地面的高度成正比，但高度要比波长小。
4、支撑杆是否金属无关紧要。但如果支撑杆是金属的，那么MiniWhip天线要放置在支撑杆顶上方适当位置，而不能放置在杆身旁边。
5、MiniWhip天线接收是全向的，除了头顶方向之外。
6、小金属板的形状和摆放方向是无关紧要的，即使用鞭状振子也如此。
10References
10、参考文献
[1] De pa0rdt-Mini-Whip, een actieve ontvangantenne voor 10 kHz tot 20 MHz, PA0RDT, Electron 5/2006.
[3] Technische notities van PA3FWM, Electron 3/2010.
[4] http://df6nm.de/ColourDF/ColourDF.htm
[5] AN-32 FET Circuit Applications, Texas Instruments.
[6] Active Reception Antennas, Observations, Calculations and Experiments; Detlef Burchard, VHF Communications 2/96 (and UKW-Berichte 4/94).
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https://www.maplesoft.com/support/help/errors/view.aspx?path=alias | alias - Maple Help
alias
define an abbreviation or denotation
Calling Sequence alias(e1, e2, ..., eN)
Parameters
e1, e2, ..., eN - zero or more equations
Description
• Mathematics has many special notations and abbreviations. Typically, these notations are found in written statements such as Let J denote the Bessel function of the first kind'' or Let alpha denote a root of the polynomial x^3-2''. The alias facility allows you to state abbreviations for the longer unique names that Maple uses and, more generally, to give names to arbitrary expressions.
• To define F as an alias for fibonacci, use alias(F=fibonacci). To redefine F as an alias for hypergeom, use alias(F=hypergeom). To remove this alias for F, use alias(F=F).
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• The arguments to alias are equations. When alias is called, the equations are evaluated from left to right, but are not subject to existing aliases. Therefore, you cannot define one alias in terms of another. Next, the aliases are defined. Finally, a sequence of all existing aliases is returned as the function value.
• An alias may be defined for any Maple object except a string or numerical constant. You may assign to a variable by assigning to its alias. Parameters and local variables are not affected by aliases.
• The alias command is thread-safe as of Maple 15.
Examples
Define an alias for the binomial function.
> $\mathrm{alias}\left(C=\mathrm{binomial}\right)$
${C}$ (1)
> $C\left(4,2\right)$
${6}$ (2)
> $C\left(n,m\right)$
${C}{}\left({n}{,}{m}\right)$ (3)
Redefine C as an alias for cat.
> $\mathrm{alias}\left(C=\mathrm{cat}\right)$
${C}$ (4)
> $C\left("C","is no longer an alias for the binomial function."\right)$
${"C is no longer an alias for the binomial function."}$ (5)
Remove the alias for C.
> $\mathrm{alias}\left(C=C\right)$
> $C\left(4,2\right)$
${C}{}\left({4}{,}{2}\right)$ (6)
You can define more than one alias in one line.
> $\mathrm{alias}\left(F=F\left(x\right),\mathrm{Fx}=\mathrm{diff}\left(F\left(x\right),x\right)\right)$
${F}{,}{\mathrm{Fx}}$ (7)
> $\mathrm{diff}\left(F,x\right)$
${\mathrm{Fx}}$ (8)
> $\mathrm{diff}\left(\mathrm{Fx},x\right)$
$\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}$ (9)
> $\mathrm{has}\left(\mathrm{Fx},x\right)$
${\mathrm{true}}$ (10)
You cannot define one alias in terms of another.
> $\mathrm{alias}\left(\mathrm{Fxx}=\mathrm{diff}\left(\mathrm{Fx},x\right)\right)$
> $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{2}-2\right)\right)$
${F}{,}{\mathrm{Fx}}{,}{\mathrm{\alpha }}$ (11)
> $\mathrm{factor}\left({x}^{4}-4,\mathrm{\alpha }\right)$
${-}\left({{x}}^{{2}}{+}{2}\right){}\left({-}{x}{+}{\mathrm{\alpha }}\right){}\left({x}{+}{\mathrm{\alpha }}\right)$ (12)
> $\mathrm{alias}\left(f=g\right)$
${F}{,}{\mathrm{Fx}}{,}{\mathrm{\alpha }}{,}{f}$ (13)
> $f≔\mathrm{sin}$
${f}{≔}{\mathrm{sin}}$ (14)
> $g\left(\mathrm{\pi }\right)$
${0}$ (15) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 32, "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": 0.9170149564743042, "perplexity": 2636.5467186139144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320306301.52/warc/CC-MAIN-20220128152530-20220128182530-00494.warc.gz"} |
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Tags Alfven waves , Birkeland currents , hannes alfven , Kristian Birkeland
16th November 2011, 01:19 PM #5001 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by GeeMack Since the burden of proof in this thread is the responsibility of the...... Clinger has been *HANDWAVING AWAY* for over a FULL YEAR now about his "vacuum" contraption that would demonstrate "magnetic reconnection". This is HIS CLAIM, not mine, and HIS BURDEN of proof, not mine. He's also the one claiming that B lines have a beginning and ending. His burden of proof, not mine. No monopoles, no beginnings or endings of B lines. Has he got a monopole up his sleeve or what? Last edited by Michael Mozina; 16th November 2011 at 01:20 PM.
16th November 2011, 01:31 PM #5002 GeeMack Banned Join Date: Aug 2007 Posts: 7,237 Originally Posted by Michael Mozina Clinger has been *HANDWAVING AWAY* for over a FULL YEAR now about his "vacuum" contraption that would demonstrate "magnetic reconnection". This is HIS CLAIM, not mine, and HIS BURDEN of proof, not mine. W.D.Clinger has done a pretty good job so far of making his explanation simple and clear to those who have the ability to understand it. Complaining about not understanding it does not make the explanations any less correct or valid. Quote: He's also the one claiming that B lines have a beginning and ending. His burden of proof, not mine. No monopoles, no beginnings or endings of B lines. Has he got a monopole up his sleeve or what? Since this is the electric Sun thread, the burden of proof in this thread is the responsibility of the electric Sun proponents. The correct and honest way to proceed is to support that conjecture rather than demanding that other people explain the physics supporting the contemporary consensus position.
16th November 2011, 01:41 PM #5003 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by GeeMack W.D.Clinger has done a pretty good job so far of making his explanation simple and clear to those who have the ability to understand it. Sure, and as long as you don't mind the fact that he's GROSSLY violating the laws of physics, praying to Origin the NULL, the beginning and ending of B lines, it's fine! When he, you, RC, and PS are ready to admit that a NULL point is *NOT* the beginning or the ending of any B line or collection of B lines, and B lines are continuous, without a beginning or an end, please let me know. Until then, IMO you're all clueless. Last edited by Michael Mozina; 16th November 2011 at 01:44 PM.
16th November 2011, 01:47 PM #5004 GeeMack Banned Join Date: Aug 2007 Posts: 7,237 Originally Posted by Michael Mozina Sure, and as long as you don't mind the fact that he's GROSSLY violating the laws of physics, it's fine! When he, you, RC, and PS are ready to admit that a NULL point is *NOT* the beginning or the ending of any B line or collection of B lines, and B lines are continuous, without a beginning or an end, please let me know. Until then, IMO you're all clueless. Not understanding W.D.Clinger's explanation is not a valid criticism of it. That would be an argument from ignorance. This is the electric Sun thread. The burden of proof here is the responsibility of the electric Sun proponents. The honest way to proceed is to support that conjecture rather than demanding that other people explain the physics of the contemporary consensus position.
16th November 2011, 01:54 PM #5005 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by GeeMack Not understanding W.D.Clinger's explanation is not a valid criticism of it. Oh, I understand (and have explained) the fact that his belief in Origin the beginor and endor of B lines violates the laws of physics, because a NULL is not a beginning or an ending of a B line. Any FRESHMAN can understand a FRESHMAN mistake. Not one of you four even UNDERSTANDS his mistake. How pitiful.
16th November 2011, 02:09 PM #5006 Reality Check Penultimate Amazing Join Date: Mar 2008 Location: New Zealand Posts: 18,384 Originally Posted by Michael Mozina I have HONESTLY seen high school students *CORRECTLY* verbally explain this experiment in a vacuum WITHOUT claiming that any B lines "begin" or "end" in the NULL. HOLY MOTHER OF PHYSICS! HONESTLY - given your ignorance of physics you have no way to tell if these mythical high school students *CORRECTLY* verbally explained this experiment. MM: The definition of magnetic field lines = no lines at a neutral point Michael Mozina's ignorance of high school science (the right hand rule) Michael Mozina's delusion that permeability is inductance! Michael Mozina's delusion about "*RECONNECTIONS* per unit length"). HONESTLY any high school students that verbally explain this freshman-level experiment in a vacuum withut mentioning that the B field lines cannot exists at the neutral point are *INCORRECTLY* verbally explaining this experiment in a vacuum. This would not be surprising because they would not have the knowledge needed for this. They will have the intelligence to understand the explanation:The neutral point has B=0. Magnetic field lines have a density that is proportional to the magnetic field strength. If B=0 then the density of magnetic field lines is zero. Thus there are no magnetic field lines at the neutral point. Therefore any magnetic field line that crosses a neutral point must break, i.e. end before it and start after it. MM: The definition of magnetic field lines = no lines at a neutral point HOLY MOTHER OF PHYSICS! __________________ NASA Finds Direct Proof of Dark Matter (another observation) (and Abell 520) Electric comets still do not exist!
16th November 2011, 02:29 PM #5007 Reality Check Penultimate Amazing Join Date: Mar 2008 Location: New Zealand Posts: 18,384 Originally Posted by Michael Mozina No, you are wrong: http://en.wikipedia.org/wiki/Magneti...ines_never_end No, you remain deluded:MM: The definition of magnetic field lines = no lines at a neutral point The Wikipedia article is wrong or at least not clearly stating its restriction to magnetic fields without a neutral point and field lines that cross that neutral point. __________________ NASA Finds Direct Proof of Dark Matter (another observation) (and Abell 520) Electric comets still do not exist!
16th November 2011, 02:36 PM #5008 Reality Check Penultimate Amazing Join Date: Mar 2008 Location: New Zealand Posts: 18,384 Originally Posted by Michael Mozina OMG. ... OMG. I cannot believe that you are writing inane rants! And lying about me still: I am not an "EU hater". I am a pitier of the ignorance, inability/unwillingness to learn and delusions displayed by you. As for the electric universe - I also pity anyone ignorant enough to think that it is a valid scientific theory. __________________ NASA Finds Direct Proof of Dark Matter (another observation) (and Abell 520) Electric comets still do not exist!
16th November 2011, 03:25 PM #5009 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by Reality Check OMG. I cannot believe that you are writing inane rants! And lying about me still: I am not an "EU hater". I am a pitier of the ignorance, inability/unwillingness to learn and delusions displayed by you. As for the electric universe - I also pity anyone ignorant enough to think that it is a valid scientific theory. It's absolutely amazing to me what you've learned about EU theory *WITHOUT* ever picking up, let alone reading a single book on the subject. (Sorry Belz, I just couldn't resist. ) Last edited by Michael Mozina; 16th November 2011 at 03:28 PM.
16th November 2011, 03:49 PM #5010 GeeMack Banned Join Date: Aug 2007 Posts: 7,237 Originally Posted by GeeMack This is the electric Sun thread. The burden of proof here is the responsibility of the electric Sun proponents. The honest way to proceed is to support that conjecture rather than demanding that other people explain the physics of the contemporary consensus position. And the follow-up... Originally Posted by Michael Mozina Oh, I [...] Nope. Empty. Originally Posted by Michael Mozina It's [...] Nope, nothing here. Lots of words keep appearing but they barely even relate to an electric Sun conjecture much less actually provide any support for it. Apparently I was correct many, many pages ago when I noted that it's unsupportable.
16th November 2011, 03:54 PM #5011 Almo Masterblazer Join Date: Aug 2005 Location: Montreal, Quebec Posts: 6,821 Originally Posted by Michael Mozina Look folks, it's very simple. Until Clinger gives up his "religion" in the beginning and ending of B lines *IN A VACUUM*, he's not moving on to part 5, it's that simple. He's demonstrated very clearly with his experiment in part 4 how this works. __________________ Almo! My Blog "No society ever collapsed because the poor had too much." — LeftySergeant "It may be that there is no body really at rest, to which the places and motions of others may be referred." –Issac Newton in the Principia
16th November 2011, 04:02 PM #5012 Reality Check Penultimate Amazing Join Date: Mar 2008 Location: New Zealand Posts: 18,384 Originally Posted by Michael Mozina It's absolutely amazing to me what you've learned about EU theory *WITHOUT* ever picking up, let alone reading a single book on the subject. It's absolutely not amazing to me given your displayed ignorance of physics that you have not learned that EU theory does not have any validity ! It is off topic because your electric sun fantasy has little to do with EU theory but there are plenty of problems with the EU solar fantasy, e.g. that the currents they demand to happenare not detected and would make the Sun explode! __________________ NASA Finds Direct Proof of Dark Matter (another observation) (and Abell 520) Electric comets still do not exist!
16th November 2011, 04:25 PM #5013 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by Almo He's demonstrated very clearly with his experiment in part 4 how this works. It evidently "works" by violating the laws of physics. http://en.wikipedia.org/wiki/Magneti...ines_never_end Unless he has a monopole in his pocket, his "beginning and ending" B line claims are coming to an end very soon.
16th November 2011, 04:31 PM #5014 W.D.Clinger Illuminator Join Date: Oct 2009 Posts: 3,134 MM explains his confusion, part 6 Originally Posted by Michael Mozina Since I don't have a clue how to proceed now, I'll make it simple. The instant you give up your PURE BLIND FAITH that Origin the great NULL is the "beginning" or the "ending" of any B line, you may proceed to part 5, and not one second before then. You WILL NOT violate the basic laws of physics in part 4. To the best of my knowledge, I have never violated any fundamental laws of physics. If I have violated any laws of physics, then they shouldn't be laws of physics. It should be impossible to violate a law of physics. What Michael Mozina means, of course, is that he thinks the experiment I've been recommending to him since last December somehow violates the laws of physics, or that some part of my simple derivation of magnetic reconnection involves calculations that are inconsistent with the laws of physics. Note well, however, that Michael Mozina has been unable to identify any law of physics that is inconsistent with my derivation. He has tried to suggest that my calculations are inconsistent with Gauss's law for magnetism, but it's absolutely trivial to prove that all of the magnetic fields I have described satisfy Gauss's law for magnetism. (Michael Mozina is unable to confirm that fact for himself, because he doesn't bark math.) It is also easy to prove that all magnetic field lines of B4 that run along the diagonals begin or end at a neutral point. (Michael Mozina is unable to confirm that fact for himself, because he doesn't bark math.) Since he doesn't bark math, Michael Mozina can only argue from the authority of Wikipedia and other sources that have dumbed down the science for an audience that doesn't bark math: Originally Posted by Michael Mozina Originally Posted by Reality Check MM, you are wrong. No, you are wrong: http://en.wikipedia.org/wiki/Magneti...ines_never_end Quote: B-field lines never end Main article: Gauss's law for magnetism Field lines are a useful way to represent any vector field and often reveal sophisticated properties of fields quite simply. One important property of the B-field revealed this way is that magnetic B field lines neither start nor end (mathematically, B is a solenoidal vector field); a field line either extends to infinity or wraps around to form a closed curve.[nb 8] To date no exception to this rule has been found. (See magnetic monopole below.) Unless you pull a magic monopole out of a hat during this conversation, you're wrong. In this case, Wikipedia is wrong (but Wikipedia's more mathematical statement, in parentheses, is correct; there's an important lesson there). One of the good things that will come out of this discussion is that someone will eventually repair the multiple Wikipedia articles that repeat the almost-but-not-quite-true myth to which Michael Mozina is so desperately clinging. Another benefit of this discussion is that it has answered one of Michael Mozina's perennial questions: Why is his inability/refusal to bark math relevant? Answer: Because Michael Mozina doesn't bark math, he can't discuss the actual science. When his authorities' mistakes are identified and refuted, Michael Mozina can't answer (or even understand) the criticism. Originally Posted by W.D.Clinger Repeatedly shouting his dumbed-down "physics for poets" over-simplifications of Gauss's law for magnetism will not improve anyone's understanding of magnetic reconnection, including his.
16th November 2011, 10:49 PM #5019 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by W.D.Clinger Thank you. I have a lot of grading to do and an exam to finish this week, so the final part will probably have to wait until the weekend. Don't even bother or think about starting the plasma switcheroo part of your presentation until you've lost your religion in "Origin the Null, the great beginor and endor of every magnetic line in the B field vacuum universe." You're Kludging the vacuum part of your own experiment to the point of pure absurdity, starting with the fact that you believe that NULL point is a "beginning" and "ending" of a magnetic line, and monopoles are unrelated to the B field "beginning and ending" rule that you're KLUDGING TO HELL. Last edited by Michael Mozina; 16th November 2011 at 10:51 PM.
16th November 2011, 11:28 PM #5021 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Clinger, your ENTIRE argument is based upon a misconception you have about NULL points in lines being the "beginning" or the "ending" of the line. The NULL point is NOT a "beginning" of that line. It's just a NULL point in the CONTINUOUS line. No B lines "begin" nor end, not EVER. They are "created" in full continuum form by OBJECTS, specifically MOVING CHARGE PARTICLES. They exist only as a FULL CONTINUUM a complete *FIELD*, without beginning, without ending, and without discrete lines as you imagine them to be in your oversimplified little 2D viewpoint. If and when you ever get around to ACCEPTING Guass's law of magnetism, let me know. At the moment you're in pure denial of empirical physics. B lines do not begin or end at their null point Clinger. Give it up already! Last edited by Michael Mozina; 16th November 2011 at 11:32 PM.
17th November 2011, 12:17 AM #5022 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Will this help clear up your confusion? http://en.wikipedia.org/wiki/Field_line Quote: For example, Gauss's law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. (They can also potentially form closed loops, or extend to or from infinity). A gravitational field has no sources, it has sinks at masses, and it has neither elsewhere, gravitational field lines come from infinity and end at masses. A magnetic field has no sources or sinks (Gauss's law for magnetism), so its field lines have no start or end: they can only form closed loops, or extend to infinity in both directions. Last edited by Michael Mozina; 17th November 2011 at 12:21 AM.
17th November 2011, 12:30 AM #5023 W.D.Clinger Illuminator Join Date: Oct 2009 Posts: 3,134 MM explains his confusion, part 7 Originally Posted by Michael Mozina You are in fact violating basic physics, starting with Guass's law of magnetism because no monopoles exist. ...snip... Quote: What Michael Mozina means, of course, is that he thinks the experiment I've been recommending to him since last December somehow violates the laws of physics, It does. It absolutely, positively violates Guass's law of magnetism. You don't have any monopoles hiding in your pocket do you? Quote: Note well, however, that Michael Mozina has been unable to identify any law of physics that is inconsistent with my derivation. Bull. Quote: He has tried to suggest that my calculations are inconsistent with Gauss's law for magnetism, Yes, because you keep prattling on and on about how Origin is the beginning and the and ending of the EM universe! Quote: but it's absolutely trivial to prove that all of the magnetic fields I have described satisfy Gauss's law for magnetism. (Michael Mozina is unable to confirm that fact for himself, because he doesn't bark math.) All your "math" demonstrates is that 0+0=0 and magnetic fields exist in a quadrapole experiment. Michael Mozina thinks B4 violates Gauss's law for magnetism. That's easy enough to check. Away from the rods themselves, \begin{align*} \hbox{{\bf B}}_4 &= \hbox{{\bf B}}_E + \hbox{{\bf B}}_W + \hbox{{\bf B}}_N + \hbox{{\bf B}}_S \\ &= \frac{\mu_0}{2 \pi} \frac{I}{(x-1)^2+(y-0)^2} \left( - (y-0) \, \hbox{{\bf e}}_x + (x-1) \, \hbox{{\bf e}}_y \right) \\ &+ \frac{\mu_0}{2 \pi} \frac{I}{(x+1)^2+(y-0)^2} \left( - (y-0) \, \hbox{{\bf e}}_x + (x+1) \, \hbox{{\bf e}}_y \right) \\ &- \frac{\mu_0}{2 \pi} \frac{I}{(x-0)^2+(y-1)^2} \left( - (y-1) \, \hbox{{\bf e}}_x + (x-0) \, \hbox{{\bf e}}_y \right) \\ &- \frac{\mu_0}{2 \pi} \frac{I}{(x-0)^2+(y+1)^2} \left( - (y+1) \, \hbox{{\bf e}}_x + (x-0) \, \hbox{{\bf e}}_y \right) \end{align*}so \begin{align*} \hbox{{\bf B}}_4 &= \frac{\mu_0 I}{2 \pi} \left( \frac{-y}{(x-1)^2 + y^2} + \frac{-y}{(x+1)^2 + y^2} + \frac{y-1}{x^2 + (y-1)^2} + \frac{y+1}{x^2 + (y+1)^2} \right) \hbox{{\bf e}}_x \\ &+ \frac{\mu_0 I}{2 \pi} \left( \frac{x-1}{(x-1)^2 + y^2} + \frac{x+1}{(x+1)^2 + y^2} - \frac{x}{x^2 + (y-1)^2} - \frac{x}{x^2 + (y+1)^2} \right) \hbox{{\bf e}}_y \end{align*}so \begin{align*} \hbox{{\bf div}} \; \hbox{{\bf B}}_4 &= \frac{\mu_0 I}{2 \pi} \left( \frac{2(x-1)y}{((x-1)^2 + y^2)^2} + \frac{2(x+1)y}{((x+1)^2 + y^2)^2} + \frac{-2x(y-1)}{(x^2 + (y-1)^2)^2} + \frac{-2x(y+1)}{(x^2 + (y+1)^2)^2} \right) \\ &+ \frac{\mu_0 I}{2 \pi} \left( \frac{-2(x-1)y}{((x-1)^2 + y^2)^2} + \frac{-2(x+1)y}{((x+1)^2 + y^2)^2} + \frac{2x(y-1)}{(x^2 + (y-1)^2)^2} + \frac{2x(y+1)}{(x^2 + (y+1)^2)^2} \right) \\ &= 0 \end{align*} so Gauss's law for magnetism holds. Michael Mozina's wrong again.
17th November 2011, 12:34 AM #5024 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by W.D.Clinger so Gauss's law for magnetism holds. It only "holds" when you ACCEPT that B lines have no beginning or an ending. In the field line vernacular, it has no sources or sinks and it forms FULL CIRCLES, without a beginning and without an ending! You can't start them and stop them *WITHOUT A MONOPOLE!* Do you have one in your back pocket by any chance?
17th November 2011, 12:36 AM #5025 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by W.D.Clinger so..... 0+0=0 and literally NOTHING began or ended or reconnected at X!
17th November 2011, 12:37 AM #5026 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 You handwave around math formulas that you DON'T EVEN UNDERSTAND!
17th November 2011, 12:42 AM #5027 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Will this help clear up your confusion? http://en.wikipedia.org/wiki/Field_line Quote: For example, Gauss's law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. (They can also potentially form closed loops, or extend to or from infinity). A gravitational field has no sources, it has sinks at masses, and it has neither elsewhere, gravitational field lines come from infinity and end at masses. A magnetic field has no sources or sinks (Gauss's law for magnetism), so its field lines have no start or end: they can only form closed loops, or extend to infinity in both directions.
17th November 2011, 01:04 AM #5028 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 You can't turn "Origin the NULL point in the vacuum" into a B SOURCE or a B SINK Clinger!
17th November 2011, 03:47 AM #5029 Argumemnon World Maker Join Date: Oct 2005 Location: In the thick of things Posts: 53,708 Originally Posted by Michael Mozina It's absolutely amazing to me what you've learned about EU theory *WITHOUT* ever picking up, let alone reading a single book on the subject. (Sorry Belz, I just couldn't resist. ) You can never resist. You're an ironymeteroholic. __________________ "History remembers kings, not soldiers. Tomorrow we'll batter down the gates of Troy. I'll build monuments to victory on every island of Greece. I'll carve 'Agamemnon' in the stone. _My_ name will last through the ages. Your name is written in sand for the waves to wash away."
17th November 2011, 08:01 AM #5030 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by Belz... You can never resist. You're an ironymeteroholic. Unfortunately, you may be right about that.
17th November 2011, 08:25 AM #5031 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by Belz... You can never resist. You're an ironymeteroholic. http://www.solarmonitor.org/full_dis...ter&indexnum=1 http://sdo.gsfc.nasa.gov/assets/img/..._1024_0131.mpg FYI Belz, there are some pretty powerful electrical discharges taking place in the southern hemisphere near the active region 11351 at the moment. That region is just starting to rotate to a position that faces Earth. Some high energy discharges from that active region could end spewing mass in our direction. It's worth keeping an eye on that particular active region IMO. When the "magnetic flux tubes" release their energy, they can release the whole circuit energy of the magnetic flux tube all at once. That is the primary process that generates flares and CME's from the E orientation of plasma physics. The iron in each flux tube is already being heavily ionized by the powerful current running through the tube, even before the discharge process takes place. Last edited by Michael Mozina; 17th November 2011 at 08:26 AM.
17th November 2011, 09:34 AM #5032 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Well Clinger, I'm running out of creative ideas on how I might reach you, and get you to abide by the laws of physics, specifically Gauss's law of magnetism. I've even tried speaking to you in your native mathematical tongue ("vector field geek speak") about sources and sinks and about your BLATANT misuse of vector field equations at X. I'm really running out of creative ideas now on how I can explain your error to you. There is NO NEED, nor any purpose in evoking B field line reconnection processes at X which would REQUIRE the existence of monopoles. This part of your vacuum experiment can *EASILY* be verbally explained in terms of magnetic flux changes in a vacuum. There's absolutely no need, nor any purpose in your violation of Gauss's law. The fact that your invisible friend Origin is a NULL is not a mathematical proof that Origin is a source or sink of B field lines. You're KLUDGING basic physics and *IGNORING* the laws of physics. As soon as you stop that ridiculous behavior, I'll be happy to let you move on to part five. As long as you remain in steadfast prayer to the mythical monopole creator, I have no way to even speak PHYSICS with you. Last edited by Michael Mozina; 17th November 2011 at 09:38 AM.
17th November 2011, 10:00 AM #5033 Argumemnon World Maker Join Date: Oct 2005 Location: In the thick of things Posts: 53,708 Originally Posted by Michael Mozina http://www.solarmonitor.org/full_dis...ter&indexnum=1 http://sdo.gsfc.nasa.gov/assets/img/..._1024_0131.mpg FYI Belz, there are some pretty powerful electrical discharges taking place in the southern hemisphere near the active region 11351 at the moment. That region is just starting to rotate to a position that faces Earth. Some high energy discharges from that active region could end spewing mass in our direction. It's worth keeping an eye on that particular active region IMO. When the "magnetic flux tubes" release their energy, they can release the whole circuit energy of the magnetic flux tube all at once. That is the primary process that generates flares and CME's from the E orientation of plasma physics. The iron in each flux tube is already being heavily ionized by the powerful current running through the tube, even before the discharge process takes place. What's the source of your obsession with me, I wonder ? __________________ "History remembers kings, not soldiers. Tomorrow we'll batter down the gates of Troy. I'll build monuments to victory on every island of Greece. I'll carve 'Agamemnon' in the stone. _My_ name will last through the ages. Your name is written in sand for the waves to wash away."
17th November 2011, 10:17 AM #5034 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by Belz... What's the source of your obsession with me, I wonder ? I guess I just "assumed" that since you got involved in this particular thread, of all your choices of various threads on this board, that you expected to be 'educated' in electric sun theory. You've asked me for evidence as well. I was simply discussing the topic with you. If you don't want me to do that, let me know. I try to respond to EVERYONE that posts here, not just you Belz.
17th November 2011, 11:59 AM #5035 GeeMack Banned Join Date: Aug 2007 Posts: 7,237 Originally Posted by Michael Mozina FYI Belz, there are some pretty powerful electrical discharges taking place in the southern hemisphere near the active region 11351 at the moment. This is an unsupported assertion.
17th November 2011, 12:02 PM #5036 Reality Check Penultimate Amazing Join Date: Mar 2008 Location: New Zealand Posts: 18,384 Originally Posted by Michael Mozina You can't turn "Origin the NULL point in the vacuum" into a B SOURCE or a B SINK Clinger! Usual giibberish and ignorance, Michael Mozina. The definition of field lines means any neutral point at any place in the universe (not only your obsession with the origin) breaks the field lines that cross it: MM: The definition of magnetic field lines = no lines at a neutral point Is the definition of magnetic field lines yet another thing that you are ignorant of? It is idiotic to rely on an Wikipedia article while ignoring all of the scientific literature on magnetic reconnection which states that the field lines break. For example, from Michael Mozina's delusions about Somov's 'Reconnection in a Vacuum' section Quote: Cosmic plasma physics By Boris V. Somov Quote: Chapter 4. Motion of a Particle in a Field 4.4.2 Reconnection in a Vacuum. ... This process is realized as follows: Two field lines approach the X-point, merge there, forming a separatrix, and then they reconnect forming a field line which encloses both currents. Such a process us termed reconnection of field lines or magnetic reconenction. A2 is that last reconnect field line. Magnetic reconnection is of fundamental importance for the nature of many non-stationary phenomena in cosmic plasma. We shall discuss the physics of this process more fully in chapters 16 to 22. Suffice it to say that reconnection is inevitable associated with electric field generation. The field is the inductive one, since [equation 4.65] where A is the vector potential of magnetic field, [equation 4.66] In the above example, the electric field is directed along the z axis. It is clear if that if dt is the characteristic time of the reconnection process shown in Figure 4.17 then according to (4.65) [equation 4.67] the last equality will be justified n Section 9.2 Reconnection in vacuum is a real physical process: magnetic field lines move to the X-type neutral point and reconnect in it as well as | the electric field is induced and can accelerate a charge particle or | particles in the vicinity of the neutral point. (emphasis added) __________________ NASA Finds Direct Proof of Dark Matter (another observation) (and Abell 520) Electric comets still do not exist!
17th November 2011, 12:06 PM #5037 Reality Check Penultimate Amazing Join Date: Mar 2008 Location: New Zealand Posts: 18,384 Originally Posted by Michael Mozina FYI Belz, there are some pretty powerful electrical discharges taking place in the southern hemisphere near the active region 11351 at the moment. FYI Michael Mozina: You are remain deluded about thisMichael Mozina's delusion about electrical discharges in plasma . There are some pretty powerful magnetic reconnections taking place in the southern hemisphere near the active region 11351 at the moment. __________________ NASA Finds Direct Proof of Dark Matter (another observation) (and Abell 520) Electric comets still do not exist!
17th November 2011, 12:15 PM #5038 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by GeeMack This is an unsupported assertion. No, it's not. Dungey supported it. You can't grab hold of Dungey's work and then ignore his claims about ELECTRICAL DISCHARGES. I won't let you do that.
17th November 2011, 12:18 PM #5039 Michael Mozina Banned Join Date: Feb 2009 Posts: 9,361 Originally Posted by Reality Check Usual giibberish and ignorance, Michael Mozina. No, your statements are utter gibberish and based upon pure ignorance including a KLUDGED quote mine from a book and author you've never actually read. Whereas Clinger's actual math skills might end up being his personal salvation, I don't think you even know what a source or sink might be in terms of vector fields and vector calculus. You're WAY out of your league on every level. All you can do is spew hate and remain in pure denial of Dungey's explanation of an ELECTRICAL DISCHARGE in a plasma. Round and round you go on that "hate-go-round".
17th November 2011, 12:35 PM #5040 GeeMack Banned Join Date: Aug 2007 Posts: 7,237 Originally Posted by Michael Mozina Dungey's explanation of [...] ... the energy released in solar flares and the heating of the corona was a theory which he pioneered and aptly nicknamed "magnetic reconnection".
International Skeptics Forum
Bookmarks Digg del.icio.us StumbleUpon Google Reddit | {"extraction_info": {"found_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": 3, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8626305460929871, "perplexity": 2913.557990417601}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1474738660882.14/warc/CC-MAIN-20160924173740-00146-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://scoop.eduncle.com/special-theory-of-relativity-1 | IIT JAM Follow
September 14, 2020 6:06 pm 30 pts
Does the forces of gravity change in accordance with Special Theory of Relativity if an object is moving with considerable high speed?
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see, gravitational forces depends upon mass of the two objects between which it's operating and the distance between the two bodies. At considerable high speeds mass varies 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": 0.8640008568763733, "perplexity": 1035.6905711808633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988850.21/warc/CC-MAIN-20210508061546-20210508091546-00561.warc.gz"} |
https://digital.lib.washington.edu/researchworks/handle/1773/4892/browse?rpp=20&etal=-1&type=title&starts_with=D&order=ASC&sort_by=1 | Now showing items 5-24 of 28
• #### Drivers of Turbulence in the Neutral Interstellar Medium of Dwarf Galaxies
(2013-07-23)
The cause of HI velocity dispersions in the interstellar medium (ISM) of galaxies is often attributed to star formation, but recent evidence has shown these two quantities are not connected in regions of low star formation. ...
• #### The Effect of Star-Planet Interactions on Planetary Climate
The goal of the work presented here is to explore the unique interactions between a host star, an orbiting planet, and additional planets in a stellar system, and to develop and test methods that include both radiative and ...
• #### The Effects of Refraction and Forward Scattering on Exoplanet Transit Transmission Spectroscopy
Transit transmission spectroscopy may provide the first glimpse at the atmosphere of an Earth-like exoplanet, and therefore the first chance to detect habitability markers and biosignatures, or signs of life. Within the ...
• #### The formation and evolution of the large magellanic cloud from selected clusters and star fields
(1998)
We have obtained deep Hubble Space Telescope color-magnitude diagrams of fields centered on the six old LMC globular clusters NGC 1754, NGC 1835, WGC 1898, NGC 1916, NGC 2005, and NGC 2019. The data have been carefully ...
• #### A Ground-Based Search for Transit Timing Variations from the Apache Point Observatory
(2013-02-25)
The following dissertation presents work done as part of a ground-based transit follow-up program designed to look for transit timing variations in well-known transiting systems called, the Apache Point Survey of Transit ...
• #### HI selected galaxies in the Sloan Digital Sky Survey
(2005)
We present the results from a study of HI selected galaxies that fall within both the HI Parkes All Sky Survey (HIPASS) and the Sloan Digital Sky Survey (SDSS). By comparing the optical properties derived from the SDSS ...
• #### A hyperbolic tetrad approach to numerical relativity
(2003)
An in-depth numerical study using 3 + 1 formulations of the Einstein equations for 1D colliding plane waves reveals factors which increase accuracy and stability: using "mixed" variables, hyperbolicity, satisfying the ...
• #### Karhunen-Loeve Analysis for Weak Gravitational Lensing
(2013-02-25)
In the past decade, weak gravitational lensing has become an important tool in the study of the universe at the largest scale, giving insights into the distribution of dark matter, the expansion of the universe, and the ...
• #### The Milky Way in SDSS and in N-body Models
(2013-11-14)
In this thesis, I present a comparison between recent Sloan Digital Sky Survey (SDSS) Galactic studies and N-body simulations to aid in the interpretation of observed structural features. I investigate the origin of the ...
• #### Modeling the Extragalactic Epoch of Reionization Foreground
The Epoch of Reionization represents a largely unexplored yet fundamental chapter of the early universe. During this period, spanning several hundred million years, the first stars and galaxies formed and the Hydrogen-dominated ...
• #### Observational Constraints on Models of Rapidly Evolving Luminous Stars
(2014-02-24)
Resolved stellar populations in galaxies are excellent laboratories for testing our understanding of galaxy formation, integrated colors and luminosities, supernova progenitor masses, and energy input from stellar feedback. ...
• #### The optical counterparts of the luminous x-ray binary stars in globular clusters
(1998)
Ten percent of our Galaxy's luminous (LX≳1036 ergs-1 ) X-ray point sources are located in globular clusters (GCs), but globular clusters contribute a much smaller fraction of normal stars to the Galaxy. X-ray bursts ...
• #### The Optical Variability of Quasars as Seen by SDSS
(2012-09-13)
I provide a quantitative analysis of the database of 3.5 million photometric measurements for 80,000 spectroscopically confirmed quasars recently assembled by the Sloan Digital Sky Survey (SDSS). This database is an excellent ...
• #### Physical conditions in giant extragalactic H II regions
(1984)
Giant Extragalactic H II Regions (GEHRs) in nearby galaxies have been observed at both radio and optical wavelengths. Narrow band optical imaging, large and small aperture optical spectrophotometry, radio continuum imaging ...
• #### Properties of Stellar Clusters and their Relation to Molecular Gas in the Andromeda Galaxy
The apparent age and mass of a stellar cluster can be strongly affected by stochastic sampling of the stellar initial mass function, when inferred from the integrated color of low mass clusters (\$\lesssim\$ \$10^4\$ \$M_{\odot}\$). ...
• #### Self-Interacting Dark Matter in Cosmological Simulations
Self-Interacting Dark Matter is a cosmologically consistent alternative theory to Cold Dark Matter that solves problems of the Cold Dark Matter model on small scales. Our N-body simulations demonstrate that Self-Interacting ...
• #### Simulating and Characterizing the Pale Blue Dot
(2013-02-25)
The goal of this work is to develop and validate a comprehensive model of Earth's disk-integrated spectrum. Earth is our only example of a habitable planet, or a planet capable of maintaining liquid water on its surface. ...
• #### Spots and Flares: Stellar Activity in the Time Domain Era
Time domain photometric surveys for large numbers of stars have ushered in a new era of statistical studies of astrophysics. This new parameter space allows us to observe how stars behave and change on a human timescale, ...
• #### Star Cluster Formation Efficiency in the Andromeda Galaxy
This work revolutionizes the study of star clusters in the Local Group galaxy Andromeda (Messier 31) using high spatial resolution, multi-wavelength imaging from the Hubble Space Telescope obtained as part of the Panchromatic ...
• #### Star Formation in N-Body + SPH Simulations
(2014-02-24)
The primary focus of my thesis work is to study star formation using a series of high resolution cosmological N-body Simulations. Specifically, I have studied the total stellar-to-halo mass ratio as a function of halo mass ... | {"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": 0.8173732757568359, "perplexity": 2564.4413530664815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257836397.31/warc/CC-MAIN-20160723071036-00251-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.computer.org/csdl/trans/tp/1982/02/04767228-abs.html | Issue No. 02 - February (1982 vol. 4)
ISSN: 0162-8828
pp: 208-215
Richard W. Hall , Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 15261.
ABSTRACT
This correspondence defines approaches for the efficient generation of a spiral-like search pattern within bounded rectangularly tessellated regions. The defined spiral-like search pattern grows outward from a given source in a two-dimensional space, thus tending to minimize search time in many sequential tracking tasks. Efficient spiral generation is achieved by minimizing the number of operations required for interaction with boundaries. Algorithms are developed for both rectangular search regions and for arbitrary convex search regions.
INDEX TERMS
CITATION
Richard W. Hall, "Efficient Spiral Search in Bounded Spaces", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 4, no. , pp. 208-215, February 1982, doi:10.1109/TPAMI.1982.4767228 | {"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": 0.8350875973701477, "perplexity": 3175.8524838484636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560280308.24/warc/CC-MAIN-20170116095120-00044-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/212564 | Infoscience
Journal article
# Kinetic model for calcium sulfate alpha-hemihydrate produced hydrothermally from gypsum formed by flue gas desulfurization: 3
Modeling of the kinetics of the synthesis process for calcium sulfate alpha-hemihydrate from gypsum formed by flue gas desulfurization (FGD) is important to produce high-performance products with minimal costs and production cycles under hydrothermal conditions. In this study, a model was established by horizontally translating the obtained crystal size distribution (CSD) to the CSD of the stable phase during the transformation process. A simple method was used to obtain the nucleation and growth rates. A nonlinear optimization algorithm method was employed to determine the kinetic parameters. The model can be successfully used to analyze the transformation kinetics of FGD gypsum to alpha-hemihydrate in an isothermal batch crystallizer. The results showed that the transformation temperature and stirring speed exhibit a significant influence on the crystal growth and nucleation rates of alpha-hemihydrate, thus altering the transformation time and CSD of the final products. The characteristics obtained by the proposed model can potentially be used in the production of alpha-hemihydrate. | {"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": 0.909758985042572, "perplexity": 2292.7691627152903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463610342.84/warc/CC-MAIN-20170528162023-20170528182023-00041.warc.gz"} |
https://openclimb.io/practice/p3/q5/ | How many functions $g$ can be defined from set $A = \{0,1, \cdots, 2^n - 1, 2^n\}$ to set $B = \{0, \cdots, n\}$ such that $g(2^x) = x$ for all $x$ in $B?$
Without any restriction, for each input $x,$ how many possible output values $g(x)$ are there?
How do we count the number of possible functions between two sets when we know the sets’ cardinality?
How many output values could an input of the form $2^x$ take if $x$ is in $B?$
How many output values could an input of a different form take?
To calculate the number of functions we count how many possible outputs there are for each input. In this case we also have a restriction. For each input of the form $2^x,$ where $x$ is in $B,$ the output value is fixed to $x;$ that is, every function we define must include these relationships, so the number of possible functions we can define is given only by the other inputs. This leaves only $2^n-n$ “free” inputs, and for each of these the output could be any of the $n+1$ elements in $B.$ Therefore, the number of functions that can be defined is $(n+1)^{2^{n} - n}.$ | {"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": 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": 0.9455072283744812, "perplexity": 106.83606509599412}, "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-2020-45/segments/1603107906872.85/warc/CC-MAIN-20201030003928-20201030033928-00605.warc.gz"} |
https://de.maplesoft.com/support/help/errors/view.aspx?path=combinat/cartprod | cartprod - Maple Help
combinat
cartprod
iterate over a list of lists or sets
Calling Sequence cartprod(LL)
Parameters
LL - list of sets or lists of anything
Description
• The command cartprod is special iterating function. It allows one to iterate over the Cartesian product of a list of lists or sets of values, as illustrated in the example below. It returns a table with two entries finished and nextvalue.
• The nextvalue entry is a function. When called repeatedly, it iterates through the values in the cartesian product of the list of lists or sets LL.
• The finished entry is either true or false and indicates whether the iteration is complete. If the finished flag is true, the nextvalue function can be called to get the next value in the cartesian product. If false, the sequence is finished.
• The command with(combinat,cartprod) allows the use of the abbreviated form of this command.
Examples
> $\mathrm{with}\left(\mathrm{combinat},\mathrm{cartprod}\right)$
$\left[{\mathrm{cartprod}}\right]$ (1)
> $T≔\mathrm{cartprod}\left(\left[\left[1,2,3\right],\left\{a,b\right\}\right]\right):$
> $\mathbf{while}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{not}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}T\left[\mathrm{finished}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}T\left[\mathrm{nextvalue}\right]\left(\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}$
$\left[{1}{,}{a}\right]$
$\left[{1}{,}{b}\right]$
$\left[{2}{,}{a}\right]$
$\left[{2}{,}{b}\right]$
$\left[{3}{,}{a}\right]$
$\left[{3}{,}{b}\right]$ (2)
> | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "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": 0.9811220765113831, "perplexity": 1078.6183317916443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305277.88/warc/CC-MAIN-20220127163150-20220127193150-00193.warc.gz"} |
http://link.springer.com/chapter/10.1007%2F11846802_40 | Lecture Notes in Computer Science Volume 4192, 2006, pp 275-284
# Can MPI Be Used for Persistent Parallel Services?
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## Abstract
MPI is routinely used for writing parallel applications, but it is not commonly used for writing long-running parallel services, such as parallel file systems or job schedulers. Nonetheless, MPI does have many features that are potentially useful for writing such software. Using the PVFS2 parallel file system as a motivating example, we studied the needs of software that provide persistent parallel services and evaluated whether MPI is a good match for those needs. We also ran experiments to determine the gaps between what the MPI Standard enables and what MPI implementations currently support. The results of our study indicate that MPI can enable persistent parallel systems to be developed with less effort and can provide high performance, but MPI implementations will need to provide better support for certain features. We also describe an area where additions to the MPI Standard would be useful. | {"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": 0.8380242586135864, "perplexity": 1304.5311704685807}, "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/1429246641054.14/warc/CC-MAIN-20150417045721-00220-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://vhdi.lasalsese.it/exponential-decay-examples-in-real-life.html | Exponential Decay Examples In Real Life
vertical), something we never actually see in real life. The range (co-domain) is all positive real numbers. Improve your math knowledge with free questions in "Exponential growth and decay: word problems" and thousands of other math skills. Same thing goes for figuring out how many people will be living on the earth in ten years or something. Oct 26, 2018 - Explore joshua barriga's board "exponential growth" on Pinterest. The examples highlight the manipulation of indices (exponents) and the index laws. • The structure of an exponential equation, and how these equations can be used to understand real-world situations. It is used to represent exponential growth, which has uses in virtually all scientific disciplines and is also prominent in finance. This provides students with varied learning styles a chance to participate in and understand the concept of exponential growth and decay. This is important since the rate of decay cannot change. determine the boundaries and appropriate scale when graphing an exponential function. Example 2: Find the best fit exponential smoothing approximation to the data Example 1, using the MAE measure of accuracy. The decay could be faster because of horizontal drilling. • For 0< a < 1, the function y =ax is decreasing, and the situation is one of exponential decay. Real World Uses. If your data modeled an exponential function, use the following steps: Decide what the starting value A o is. Solving Exponential and Logarithmic Equations In section 3. The dentist’s office B. The original population. An exponential function is a function with the general form y=a∙ b x , a≠0, with b>0, and b≠1. !The!value!!!of!the!car!decreases!by!16%!each!year. It seems to me that the increased cooldown at high heat levels that exponential decay gives would actually stimulate high heat alpha strikes rather t. real life examples - logarithmic (0, 1) and (1, base) (1, 0) and (Base, 1) population growth. f x b ± ax c, b c y ax Section 3. Then every year after that, the population has decreased by 3% as a result of heavy pollution. If a car costs $15000 and you get a loan for it, are you really only going to pay$15000? Using the scenario of investing $100 at 8% interest per year, the students complete the first task of finding the amount in the account after 4 years ( Math Practice 8 ). For both functions, the y-intercept is 3, the asymptote is y = 0, the domain is all real numbers, and the range is y > 0. The parent form of the exponential function appears in the form: !!=!! where !, known as the base is a fixed number and ! is any real number. Click on the transfer function in the table below to jump to that example. I will add an example of how to do this in the next release of the Real Statistics software. 718) y = ex −4 3 y exponential. Write functions in continuous exponential growth form 5. r is the growth rate when r>0 or decay rate when r 0, in percent. The line y = 0 (the x-axis) is a horizontal asymptote. Please feel free to comment with any ideas you have for improvement or any questions you have. We're at the typical "logarithms in the real world" example: Richter scale and Decibel. EXPONENTIAL DECAY Exponential!DecayFormula:!!! a=_____! r=_____! t!=_____!! EXAMPLE! 1. So, the process of cooling of a kettle after the heat is off is a good example of an exponential decay. 1,2,4,8,16,32,64). In AQA's sample assessment materials (Question 23 in Higher Paper 3 ) students are shown a graph representing the depth of water in a container over time. 25 x (Where a = 25,000 and b = 0. Write an exponential decay model to represent the real-life problem. 2 Graph Exponential decay functions No graphing calculators!! EXPONENTIAL DECAY A function of the form y =a⋅bx where a > 0 and the base b is between 0 and 1. The half life of iodine-131 is 8. 5 is negative, the function an exponential growth function. The graphs of the basic functions y = ex and y = e−x are shown. If A 0 is the initial amount, then the amount at time t is given by A = A 0 1 − r t, where r is called the decay rate, 0 < r="">< 1,and="" 1 − r ="" is="" called="" the="" decay="">. 7 Applications of Exponential Equations Examples of reallife situations involving exponential equations carbon dating richter scale population growth population decay cooling curve. However, it is the second equation that clearly shows that the backbone grows faster than the skull. The half-life of cesium-137 is 30 years. 459 Example 1: Since 2005, the amount of money spent at restaurants in the United States. This animated presentation provides four examples of the use of exponential functions to model real-life practical situations. Putting money in a savings account 2. determine the boundaries and appropriate scale when graphing an exponential function. Write an exponential decay model to represent the real-life problem. To evaluate, substitute a number in for x and find y. Exponential Models Some real-life quantities increase or decrease by a fixed percent each year (or some other time period). If you click on the link in each column labelled "New" it will take you to a page I have recently written that demonstrates the construction of the Bode plot for an arbitrary transfer function. • The range is y > 0 if a > 0 and y < 0 if a < 0. A radioactive substance has a half-life of one week. Real world algebraic expressions worksheet pdf. Solving problems with exponential growth. This article presents you with the definition and some examples of exponential distribution, as well as with the exponential distribution formula and an example of applying it in real life. ANSWER exponential function 0 1 y 2 x – 2 – 1 0. This is important since the rate of decay cannot change. Exponential Growth. To solve real-life problems, such as finding the. For those studying for their GCSEs, it would be appropriate to explore radioactive decay theory and how this forms the basis of carbon dating, including topics such as half-lives and what radioactivity is. The value of a car decreases exponentially exponential decay! Don't Let Your Car Own You Too many people today view their car as their status symbol. If b is greater than 1, the function continuously increases in value as x increases. 6 g=1158 people Exponential Function Decay: Exponential function decay d=c(p)^t where, c-Number at initial p. The exponential decay of the substance is a time-dependent decline and a prime example of exponential decay. Because a = 3 is positive and b. To see this, define. radioactive. One specific example of exponential decay is purified kerosene, used for jet fuel. Example 3 Sketch the graph of $$g\left( x \right) = 5{{\bf{e}}^{1 - x}} - 4$$. The example we will talk about here is radioactive growth and decay, but examples from other fields include the recovery of a muscle after some exertion, and the filling of a cistern. Part 2: View and comment on the work of at least 2 other students. 401,828 repetition cases have been included in the graph. Write an exponential decay model to represent the real-life problem. See full list on mathinsight. As an example let us assume we have a $100$ pounds of a substance with a half-life of $5$ years. Exponential Function Formula An exponential equation is an expression where both sides can be presented in the form of same based and it can be solved with the help of a property. Choice 1:$16,190. Exponential functions follow all the rules of functions. Till the rate of change is constant and not the amount of change, you are looking at exponential growth or decay. 9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). So, the process of cooling of a kettle after the heat is off is a good example of an exponential decay. vaniercollege. Exponential functions have the form: f(x) = b^x where b is the base and x is the exponent (or power). 2 Graph Exponential decay functions No graphing calculators!! EXPONENTIAL DECAY A function of the form y =a⋅bx where a > 0 and the base b is between 0 and 1. 6 g=1158 people Exponential Function Decay: Exponential function decay d=c(p)^t where, c-Number at initial p. This sort of equation represents what we call "exponential growth" or "exponential decay. This is the second lesson in a three-lesson series about isotopes, radioactive decay, and the nucleus. See full list on shelovesmath. In AQA's sample assessment materials (Question 23 in Higher Paper 3 ) students are shown a graph representing the depth of water in a container over time. The general formula is A(f) = A(i)[decay factor]^t where A(f) is the final amount, A(i) is the initial amount, decay factor is a number less than one (for example if the amount decreases at 10% per year, the decay factor is 0. 436 Chapter 3 Exponential and Logarithmic Functions 140. Examples of exponential decay are radioactive decay and population decrease. The amount y of such a quantity after t years can be modeled by one of these equations. The value of in the exponential decay model determines the. This is an example of exponential decay. Description: With the Half-Life Laboratory, students gain a better understanding of radioactive dating and half-lives. Within the Frayer model, students will explore:• Equations• Real- world Examples• Tables• GraphsI used the information provided to compare and contrast growth a. Exponential decay formula. Example 1:. One example of an exponential function in real life would be interest in a bank. Alpha Decay. radioactive. f x b ± ax c, b c y ax Section 3. The current population is 100,000; what will it be in 100 years?1363. 8b Use the properties of exponents to interpret expressions for exponential functions. com Exponential growth is the increase in number or size at a constantly growing rate. We say that they have a limited range. In the exponential model y=a⋅bx, what does the a value represent? BE SPECIFIC. real life examples - logarithmic (0, 1) and (1, base) (1, 0) and (Base, 1) population growth. The initial amount will earn interest according to a set rate, usually compounded after a set amount of time. Exponential Decay. In carbon dating,we use the fact that all liv-ing organisms contain two kinds of carbon,carbon 12 (a stable carbon) and carbon 14 (a radioactive carbon with a half-life of 5600 years). The second tab has students write the equation from a table. Example 4: The population growth of a city can be modeled exponentially with a constant of k = 0. exponential growth functions in the form 2. Alpha Decay. In 1899, Ernest Rutherford wrote the following words: "These experiments show that the uranium radiation is complex and that there are present at least two distinct types of radiation - one that is very readily absorbed, which will be termed for convenience the alpha-radiation, and the other of more penetrative character which will be termed the beta-radiation. #67 - Use exponential growth and decay functions to model and solve real-life problems. The simple answer is: there is no difference. Move the constant to the other side situations of exponential growth and decay – Exponential growth – growth that occurs rapidly • Money in a bank – Exponential decay – decay that occurs rapidly • Half-life of radioactive materials Solve real-world problems involving exponential growth. Logically, it cannot exist in nature as a universal law of nature because it is impossible for a population to keep growing forever without hitting a limit to growth. Solve Real-World Problems Involving Exponential Decay Exponential decay problems appear in several application problems. Exponential Growth Model Exponential Decay Model Note that a is the initial amount and r is the percent increase or decrease written as a. If you need help, go to the worked-out Examples on pages 466 through 468. Derivatives of exponential functions involve the natural logarithm function, which itself is an important limit in Calculus, as well as the initial exponential function. Case 1: 0 < a < 1, Exponential Decay. EXAMPLE 3 A Differential Equation with Initial Condition Solve y˜ = 3y, y(0) = 2. The half-life of Radioactive decay (also known as For example, gamma decay was and there are no known natural limits to how brief or long a decay half-life for radioactive Real World Application Americium-243 undergoes alpha decay with a half-life of 7,370 years. New! Exponential Growth Word Problems. Graphing transformations of exponential functions. True exponential behavior requires a trend towards an infinite gradient (i. In the example above, we gave the formula for the mass of a radioactive substance to be M = 100 × t g. If a person deposits £100 into an account which gets 3% interest a month then the balance each month would be (assuming the money is untouched):. It is used to represent exponential growth, which has uses in virtually all scientific disciplines and is also prominent in finance. As liquid oil depletes, society is switching to mining tar sands. 9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Within the Frayer model, students will explore:• Equations• Real- world Examples• Tables• GraphsI used the information provided to compare and contrast growth a. Exponential equations come in two forms. In order for b2 > 4mk the damping constant b must be relatively large. Example 3: Radioactive Decay. EX #3:A slow economy caused a company’s annual revenues to drop from $530, 000 in 2008 to$386,000 in 2010. Here is a simple example and how it is so powerful. Logarithmic functions are very helpful when working with phenomena that have a very wide range of values, because they allow you to keep the values you actually work with in a smaller range. For example, you will have to decide where you will bank. Exponential Functions - Explanation and Examples Using Graphs and Tables Exponential Function - Practice Problems with Solutions Real-Life Examples of Exponential Growth and Decay. !The graph of !!=! will always contain the point (0,1). The half life of iodine-131 is 8. Exponential decay formula. 298 Chapter 6 Exponential and Logarithmic Functions Solving a Real-Life Problem The value of a car y (in thousands of dollars) can be approximated by the model y = 25(0. • Exponential decay also occurs with functions of the form y =a−x, where a > 1. 2 Exponential Decay - Algebra 1 Common Core Exponential decay refers to an amount of substance decreasing exponentially. The half-life of each mRNA was determined by using the decay law t 1/2 = [(x 2 − x 1)/log 2 (y 1 /y 2)], with x 1 and x 2 being the time points the samples were taken (x 1 always = 0, x 2 = 10, 15, or 20 min) and y 1 and y 2 being the signal intensities at time points x 1 and x 2, respectively. Notes: x y 5 7 −24 exponential growth (0, 1) (1, 2. Exponential growth and decay by percentage. Many real life data sets follow an exponential pattern, including population growth and decline, environmental concentrations (Ott, 1995) and—oddly—even the amount of revenue collected yearly by the IRS (Larson & Falvo, 2012, p. Then find the population in 5. Let's consider exponential decay and exponential growth by inspecting their respective general shapes of their graphical representations. Writing Strategies 5. Choose a data point (A, t) and plug it into A = A o e kt; Divide both sides by A o, take the natural log of both sides, then solve for the constant k. com Exponential growth is the increase in number or size at a constantly growing rate. (0, 1) (1, b) f(x) =bx. Exponential function, in mathematics, a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Determine the value of the car after 30 months. Exponential decay and exponential growth are used in carbon dating and other real-life applications. Vocabulary Strategies 3. Exponential Decay Exponential growth functions are often used to model population growth. key points - exponential. The base b determines the rate of growth or decay: If 0 b 1 , the function decays as x increases. Exponential Growth & Decay 06/01/09 Bitsy Griffin PH 8. You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day. The first tab provides a place to write and explain the formula along with an example of exponential decay in the form of an equation, table, and graph. Example 2: Find the best fit exponential smoothing approximation to the data Example 1, using the MAE measure of accuracy. The graph shows the general shape of an exponential decay function. Exponential Decay and Half Life Many harmful materials, especially radioactive waste, take a very long time to break down to safe levels in the environment. 2 Graph Exponential decay functions No graphing calculators!! EXPONENTIAL DECAY A function of the form y =a⋅bx where a > 0 and the base b is between 0 and 1. EXAMPLE 3 A Differential Equation with Initial Condition Solve y˜ = 3y, y(0) = 2. 10 Exponential Decay Instead of increasing, it is decreasing. of Equation & Graph of Exponential Decay Function. For instance, in Exercise 70 on page 228, an exponential function is used to model the atmospheric pressure at different altitudes. 0433 percent of the radium decays away each year, or 433 parts per million per year. SOLUTION Here, k = 3 and P 0 = 2. Exponential Models Exponential Growth Model: !=#1+23 3 Exponential Decay Model: !=#1−2 *** IMPORTANT NOTE! When using a percentage “r” is always written in decimal form! *** Example 2: Solving a Real-Life Problem The value of a car y (In thousands of dollars) can be approximated by the model !=250. Before look at the problems, if you like to learn about exponential growth and decay, You can also visit the following web pages on different stuff in math. 01)(12t), y = (1. (y = k^{-x}\) graphs decrease in value. Estimate the value after 2 years. 2 Exponential Growth and Decay. Problems such as this arise naturally when we deal with exponential growth and decay. Answer: The domain of an exponential function of this form is all real numbers. Finding the Inverse of an Exponential Function I will go over three examples in this tutorial showing how to determine algebraically the inverse of an exponential function. For example, radium-226 has a half-life of 1,602 years, an mean lifetime of (1,602)/ln2 = 2,311 years, and a decay rate of 1/(2,311) = 0. Exponential Decay Certain materials, such as radioactive substances, decrease with time, rather than increase, with the rate of decrease proportional to the amount. Choose a data point (A, t) and plug it into A = A o e kt; Divide both sides by A o, take the natural log of both sides, then solve for the constant k. This example is more about the evaluation process for exponential functions than the graphing process. The exponential decay is a model in which the exponential function plays a key role and is one very useful model that fits many real life application theories. If you need help, go to the worked-out Examples on pages 466 through 468. For example, suppose that the population of a city was 100,000 in 1980. A capacitor stores charge, and the voltage V across the capacitor is proportional to the charge q stored, given by the relationship V = q/C, where C is called the capacitance. This foldable provides an organized introduction to exponential decay. In order to simplify any exponential expression, we must first identify a common base in the expression and then use our rules for exponents as necessary. Therefore, the half-life of this medication, given this constant, is approximately 5 hours, based on using this model for exponential decay. To find: The time taken by the sample of radium-221 to decay 95 %. Exponential Decay Function: An exponential decay function is an exponential function that decreases. Putting money in a savings account. Figure: Cumulative forgetting curve for learning material of mixed complexity, and mixed stability. A growth factor greater than 1 gives exponential growth and a growth factor between 0 and 1 gives exponential decay. Improve your math knowledge with free questions in "Exponential growth and decay: word problems" and thousands of other math skills. EXPONENTIAL DECAY Exponential!DecayFormula:!!! a=_____! r=_____! t!=_____!! EXAMPLE! 1. Students are able to visualize and model what is meant by the half-life of a reaction. Explanation:. In 1899, Ernest Rutherford wrote the following words: "These experiments show that the uranium radiation is complex and that there are present at least two distinct types of radiation - one that is very readily absorbed, which will be termed for convenience the alpha-radiation, and the other of more penetrative character which will be termed the beta-radiation. 2/ A rare coin is bought at an auction in 1998 for $500. The dentist’s office B. determine half-life as a form of exponential decay graph an exponential function constructed from a table, sequence or a situation. Then find the population in 5. We need a process for solving exponential equations. To evaluate, substitute a number in for x and find y. 436 Chapter 3 Exponential and Logarithmic Functions 140. With exponential growth or decay, quantities grow or decay at a rate directly proportional to their size. For instance, to find how much of an initial 10 grams of isotope with a half-life of 1599 years is left after 500 years, substitute this information into the model Using the value of found above and 10, the amount left is. R=70% on the right edge of the graph). And after 57,300 years, due to the property of exponential decay, there will still be. Growth: Money, Populations, Antiques, speeds of computers Decay: Diseases, half-life of elements Exponential growth/decay can be modeled by the equations below. Figure 1: The damped oscillation for example 1. Example 1 - Relevance and application of exponential functions in real life situations A group of 1000 people increase by 5% in an hour near to accident place. does not change linearly with time but follows a curve. This question is testing one's ability to recognize real life situations that have a exponential growth or decay over a certain interval and how to deal with them in function form. For b > 1, f(x) is increasing -- its graph rises to the right. The half-life of cesium-137 is 30 years. An exponential smoothing over an already smoothed time series is called double-exponential smoothing. Without introducing a factor to suppress it, exponential growth is an infectious disease doctor's. 314 exponential decay, p. Objective: In this lesson you learned how to use exponential growth models, exponential decay models, Gaussian models, logistic growth models, and logarithmic models to solve real-life problems. A capacitor stores charge, and the voltage V across the capacitor is proportional to the charge q stored, given by the relationship V = q/C, where C is called the capacitance. 3 to find the time. See full list on studiousguy. And it is on Earth. Determine the value of the car after 30 months. As mentioned above, there are a number of fields that use the exponential decay (and growth) formula to determine results of consistent business transactions, purchases, and exchanges as well as politicians and anthropologists who study population trends like voting and consumer fads. radioactive. Exponential growth is a specific way in which an amount of some quantity can increase over time. The graphs of the basic functions y = ex and y = e−x are shown. Now that we have defined bx for every positive number b and every real number x, we can define an appropriate function. In the function: y = a(b)x, a is the y-intercept and b is the base that determines the direction of the graph and the steepness. Explain the importance of e 4. Damping of oscillating system. An example of exponential growth is the rapid population growth rate of bacteria. If we use the symbol to denote the half life of a process, and to represent the amount of a substance initially present then. #68 - Use Gaussian functions to model and solve real-life problems. All radioactive substances have a specific half-life, which is the time required for half of the radioactive substance to decay. Exponential decay is a particular form of a very rapid decrease in some quantity. of Equation & Graph of Exponential Decay Function. The exponential function is used when the quantity grows or decrease at the rate of its current value which can be found by the exponent calculator. Example 3: Radioactive Decay. Critics of the simple exponential growth model (growth at a constant rate of exponential growth) are quite right to dismiss it as having no real-life meaning. This should occur every few sessions. In other words, it will take you 7. As an example let us assume we have a $100$ pounds of a substance with a half-life of $5$ years. classify an exponential function as a growth or decay. As liquid oil depletes, society is switching to mining tar sands. The exponential decay model is a a aaaaaa a a a a. Putting money in a savings account. 5)x is an exponential decay function; as x increases, y decreases. and gamma decay with reactions written out. Exponential growth is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. For instance, to find how much of an initial 10 grams of 226Ra isotope with a half-life of 1599 years is left after 500 years, substitute this information into the model y –= ae bt. One extremely important thing to notice is that in this case the roots. Just as exponential growth, there is also Exponential decay. Another application of the exponential function is exponential decay, which occurs in radioactive decay and the absorption of light. Exponential function, in mathematics, a relation of the form y = a x, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. This equation for r will allow us to find the rate of decay whenever we are given the half-life h. Finance: Future. This article presents you with the definition and some examples of exponential distribution, as well as with the exponential distribution formula and an example of applying it in real life. Slowing the rate of new cases requires dramatic measures and is the key to preventing healthcare systems from becoming overwhelmed. Goal 3: Using Exponential Decay Models When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by this equation:. Exponential growth and decay by percentage. And after 57,300 years, due to the property of exponential decay, there will still be. Such negative growth is described by exponential functions, very much like exponential growth except for a negative sign in the exponent. Figure: Cumulative forgetting curve for learning material of mixed complexity, and mixed stability. If you continue browsing the site, you agree to the use of cookies on this website. One example models the average amount spent(to the nearest dollar) by a person at a shopping mall after x hours and is The base of the. This is important since the rate of decay cannot change. You are responsible for these equations. In the following example, the graph of is used to graph functions of the form where and are any real numbers. Big Ideas: Exponential decay occurs when a quantity is decreasing at a constant multiplicative rate. Write an exponential function to model this situation. notebook March 25, 2013. The "half life" is how long it takes for a value to halve with exponential decay. 2 Exponential Decay - Algebra 1 Common Core Exponential decay refers to an amount of substance decreasing exponentially. ƒ(x) = 3e2x is an exponential growth function, since 2 > 0. For example, radium-226 has a half-life of 1,602 years, an mean lifetime of (1,602)/ln2 = 2,311 years, and a decay rate of 1/(2,311) = 0. Online Resources 6. The line y = 0 (the x-axis) is a horizontal asymptote. Growth factor between 0 and 1, it is considered exponential decay. Exponential growth is so powerful not because it's necessarily fast, but because it's relentless. EXAMPLES State whether the function represents exponential growth or exponential decay: 8. Pupils move quantities of rice to a chessboard and calculate the amount of rice for each day. " Other examples of exponential functions include: $$y=3^x$$$$f(x)=4. y = 100 x 1. For those studying for their GCSEs, it would be appropriate to explore radioactive decay theory and how this forms the basis of carbon dating, including topics such as half-lives and what radioactivity is. Examples of how to use “exponential function” in a sentence from the Cambridge Dictionary Labs. • Exponential decay also occurs with functions of the form y =a−x, where a > 1. When such a function describes a “real life” situation, we say that the situation is one of exponential growth. In addition to the alpha parameter for controlling smoothing factor for the level, an additional smoothing factor is added to control the decay of the influence of the change in trend called beta ( b ). Estimate the value after 2 years. This time we minimize the value of MAE (cell J21 in Figure 3) by changing the value in cell H21 subject to the constraint that H21 <= 1. Exponential Growth and Decay Exponential growth refers to an amount of substance increasing exponentially. Half-Life : Paper, M&M’s, Pennies, or Puzzle Pieces. Use exponential growth and decay to model real-life situations - B Contact Us If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. = Initial temperature difference at time t=0. The amount y of such a quantity after t years can be modeled by one of these equations. 97) t, y = (1. Vocabulary Strategies 3. A Look Ahead (More Zombies) We covered the basics on Zombies, Exponents, and a “simple” intro to Exponential Growth, but we have yet to tie those 3 concepts together in a way that will make you truly scared to go outside and risk being chased down. g(x) = 3eº2x is an exponential decay function, since º2 < 0. The graph is obtained by superposition of 400 forgetting curves normalized for the decay constant of 0. 2) Describe examples of real-life applications that use exponential growth and decay functions. Same thing goes for figuring out how many people will be living on the earth in ten years or something. This question is testing one's ability to recognize real life situations that have a exponential growth or decay over a certain interval and how to deal with them in function form. Find real & complex roots of higher degree polynomial equations using the factor theorem, remainder theorem, rational root theorem & fundamental theorem of algebra, incorporating complex & radical conjugates. If your data modeled an exponential function, use the following steps: Decide what the starting value A o is. You will find activities, lessons, projects, notes and assessments. Reading Strategies 4. All the major topics of exponential functions are covered including growth and decay, graphing, tables, equations, compound interest and real-life examples. The first one yields more money. 8b Use the properties of exponents to interpret expressions for exponential functions. Example 4: The population growth of a city can be modeled exponentially with a constant of k = 0. For the second decay mode, you add another exponential term to the model. Exponential Functions. For example, if aquantitygrows at 10% per year, then it will take 72/10 or 7. The "half life" is how long it takes for a value to halve with exponential decay. Exponential growth is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. 436 Chapter 3 Exponential and Logarithmic Functions 140. 1,2,4,8,16,32,64). Write an exponential decay model giving the television's value y (in dollars) after t years. Problems such as this arise naturally when we deal with exponential growth and decay. Year 1981 or 1 year after:. Below is an interactive demonstration of the population growth of a species of rabbits whose population grows at 200% each year and demonstrates the power of exponential population growth. Finding an exponential function given its graph. Exponential decay also happens, for example radioactive decay and the absorption of light. Angelie Abiera 18,750 views. decayof radioactive isotopes. Solving Exponential Growth Problems using Differential Equations. Objective: In this lesson you learned how to use exponential growth models, exponential decay models, Gaussian models, logistic growth models, and logarithmic models to solve real-life problems. Please feel free to comment with any ideas you have for improvement or any questions you have. While simple exponential smoothing requires stationary condition, the double-exponential smoothing can capture linear trends, and triple-exponential. Finance: Future. Growth: Money, Populations, Antiques, speeds of computers Decay: Diseases, half-life of elements Exponential growth/decay can be modeled by the equations below. of an exponential decay function. The information found can help predict what the half-life of a radioactive material is or what the population will be for a city or colony in the future. Graph the model. 097g of carbon in the sample. For b > 1, f(x) is increasing -- its graph rises to the right. of Equation & Graph of Exponential Decay Function. 7 Applications Involving Exponential Functions December 12, 2016 Ex. These types of functions adhere to certain properties: 1. This is important since the rate of decay cannot change. Example 2: Simplify ( ) ( ) Goal: 1. Example of Exponential Growth. For example, identify percent rate of change in functions, and classify them as representing exponential growth or decay. In real-life situations we use x as time and try to find out how things change exponentially over time. For the second decay mode, you add another exponential term to the model. = Temperature difference between soup and water in sink at time t. Example 3: Radioactive Decay. Let's consider exponential decay and exponential growth by inspecting their respective general shapes of their graphical representations. We're at the typical "logarithms in the real world" example: Richter scale and Decibel. In this section we describe an exponential decay model for the concentration of a drug in a patient's body. Example: A new television costs$1200. an exponential decay function involves the expression a b^x, where a>0 and 0 1, while exponential decay functions have b < 1. All the major topics of exponential functions are covered including growth and decay, graphing, tables, equations, compound interest and real-life examples. A function of the form y = aerx is called a natural base exponential function. Exponential+Growthand+DecayWord+Problems+!! 8. The second tab has students write the equation from a table. Another application of the exponential function is exponential decay, which occurs in radioactive decay and the absorption of light. Exponential function Suppose b is a positive number, with b 6= 1. In other words, 0. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0. A great example of exponential growth is the interest earned on a savings account over time. Every radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive decay. In this definition, $$a$$ is known as the coefficient, $$b$$ is called the base, and $$x$$ is the exponent. tick GRADE Your avatar Add comment Radioactive Decay Avatar of Caroline Huang Caroline Huang 1yr Radioactive Decay Radioactive decay in chemistry is the exponential process of radioactive elements emitting mass as they change forms. • Visually the graph can help you understand a problem better. • Many real life applications involve exponential functions. 01 12t, y = (1. SWBAT use the formula for exponential growth and decay to predict future values in real-life situations. The equation is y equals 2 raised to the x power. However, because they also make up their own unique family, they have their own subset of rules. Solve real-life problems involving exponential growth and decay. Use exponential growth and decay to model real-life situations Contact Us If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. We usually see Exponential Growth and Decay problems relating to populations, bacteria, temperature, and so on, usually as a function of time. Exponential equations come in two forms. 1 Exponential Growth 465 Graph exponential growth functions. I give a real life example like car payments. An Example of an exponential function: Many real life situations model exponential functions. Logarithmic models are y = a + b ln x and y = a + b log10x. of an exponential function I can apply my knowledge of growth and decay to real life situations. classify an exponential function as a growth or decay. The examples highlight the manipulation of indices (exponents) and the index laws. Exponential decay model Substitute 15,000 for and 0. I think this is a good problem to think about because it is so useful in all sorts of areas. Carbon-14, for example, has a half-life of approximately 6,000 years. This decrease is radioactive decay The half-life is defined as the time for m a sample of radioactive material. We explain Exponential Decay in the Real World with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. Examples of exponential growth include contagious diseases for which a cure is unavailable, and biological populations whose growth is uninhibited by predation, environmental factors, and so on. 3) exponential decay and 4% 4) exponential decay and 96% Example 5: The fish population of Lake Collins is decreasing at a rate of 4% per year. Jan 15, 2020 - This board is exclusively for exponential function activities. Objectives: Estimate half life by analyzing graph Devise an equation to represent graphical data Use technology to visually study exponential functions Students will use the decay model and import the data from Netlogo to Fathom. Alpha Decay. An exponential smoothing over an already smoothed time series is called double-exponential smoothing. In one case, it is possible to get the same base on each side of the equation. The half-life of each mRNA was determined by using the decay law t 1/2 = [(x 2 − x 1)/log 2 (y 1 /y 2)], with x 1 and x 2 being the time points the samples were taken (x 1 always = 0, x 2 = 10, 15, or 20 min) and y 1 and y 2 being the signal intensities at time points x 1 and x 2, respectively. Exponential growth and decay by a factor. Why you should learn it Exponential functions can be used to model and solve real-life problems. Exponential Functions - Explanation and Examples Using Graphs and Tables Exponential Function - Practice Problems with Solutions Real-Life Examples of Exponential Growth and Decay. The most common examples of 'graphs showing real-life situations in geometry' are those that model water flow. Specifically, my question is about commonly used statistical distributions (normal - beta- gamma etc. Introduction (Page 420) The exponential growth model is a a aa aa a a a a a. C is the initial value of y, and k is the proportionality constant. Pupils move quantities of rice to a chessboard and calculate the amount of rice for each day. 000433 per year. We say that they have a limited range. Theta decay doesn't depend on the in the moneyness. Several examples of the construction of Bode Plots are included in this file. Part 1: How are exponential growth and decay present in the real world? Give at least 2 examples for exponential growth and 2 examples of exponential decay. Build new functions from existing functions. Exponential growth functions are used in real-life situations involving compound interest. Because a = 3 is positive and b. The value of the television decreases by 21% each year. Probably the most well known example of exponential decay in the real world involves the half-life of radioactive substances. Lesson idea. Here is a simple example and how it is so powerful. Solving problems with exponential growth. 5%!per!year. The value of a car decreases exponentially exponential decay! Don't Let Your Car Own You Too many people today view their car as their status symbol. Writing Strategies 5. The topics typically used are population, radioactive elements, credit card accounts, etc. Exponential decay and exponential growth are used in carbon dating and other real-life applications. Exponential decay occurs when a quantity decreases by the same proportion r in each time period t. A Geiger counter detects 40 counts per second from a sample of iodine-131. Exponential growth is so powerful not because it's necessarily fast, but because it's relentless. 05 t, where t is given in years. Examples of exponential growth include contagious diseases for which a cure is unavailable, and biological populations whose growth is uninhibited by predation, environmental factors, and so on. Specifically, my question is about commonly used statistical distributions (normal - beta- gamma etc. 8b Use the properties of exponents to interpret expressions for exponential functions. 6 g=1158 people Exponential Function Decay: Exponential function decay d=c(p)^t where, c-Number at initial p. If your data modeled an exponential function, use the following steps: Decide what the starting value A o is. In real life there is no reason for the decay rate to match the growth rate. Finance: Compound interest. Then graph the function. Exponential Functions - Explanation and Examples Using Graphs and Tables Exponential Function - Practice Problems with Solutions Real-Life Examples of Exponential Growth and Decay. Exponential functions help define things like population, bacterial growth, and virus spread. Although, interest earned is expressed as an annual rate, the interest is usually compounded more frequently than once per year. The range (co-domain) is all positive real numbers. r is the growth rate when r>0 or decay rate when r 0, in percent. 25 x (Where a = 25,000 and b = 0. But before you take a look at the worked examples, I suggest that you review the suggested steps below first in order to have a good grasp of … Inverse of Exponential Function Read More ». Enter the value of x in the exponential function calculator, it automatically calculates e x. Reading Strategies 4. 02)t, y = (0. This time we minimize the value of MAE (cell J21 in Figure 3) by changing the value in cell H21 subject to the constraint that H21 <= 1. Same thing goes for figuring out how many people will be living on the earth in ten years or something. Graph the model. • The range is y > 0 if a > 0 and y < 0 if a < 0. This is because any number raised to the 0 power will always be 1. Finding the Inverse of an Exponential Function I will go over three examples in this tutorial showing how to determine algebraically the inverse of an exponential function. Choose a data point (A, t) and plug it into A = A o e kt; Divide both sides by A o, take the natural log of both sides, then solve for the constant k. Write the prediction equation. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Example 3 Sketch the graph of $$g\left( x \right) = 5{{\bf{e}}^{1 - x}} - 4$$. Population: The population of the popular town of Smithville in 2003 was estimated to be 35,000 people with an annual rate of …. 003567, which corresponds with recall of 70% at 100% of the presented time span (i. One example of an exponential function in real life would be interest in a bank. • The range is y > 0 if a > 0 and y < 0 if a < 0. Introduction (Page 238) The exponential growth model is aaa a aa a a a a a. While simple exponential smoothing requires stationary condition, the double-exponential smoothing can capture linear trends, and triple-exponential. The exponential decay model is Exercises for Example 1 1. The graph shows the general shape of an exponential decay function. Carbon-14, for example, has a half-life of approximately 6,000 years. A 70 delta call and a 30 delta call have very close theta decay at any given moment. Exponential decay is a particular form of a very rapid decrease in some quantity. Exponential and Logarithmic Models What You’ll Learn: #66 - Recognize the five most common types of models involving exponential or logarithmic functions. This is an example of exponential decay. 05 t, where t is given in years. Move the constant to the other side situations of exponential growth and decay – Exponential growth – growth that occurs rapidly • Money in a bank – Exponential decay – decay that occurs rapidly • Half-life of radioactive materials Solve real-world problems involving exponential growth. 7 Exponential Equations Applications. Write an exponential decay model for the value of the car. The idea is to put events which can vary drastically (earthquakes) on a single scale with a small range (typically 1 to 10). 9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Use exponential growth and decay to model real-life situations Contact Us If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. You!buy!anew!car!for!$24,000. reteach exponential functions growth and decay Media Publishing eBook, ePub, Kindle PDF View ID 8464301e2 Mar 08, 2020 By Robin Cook we have exponential decay in many applications the input variable x denotes time but lets apply the. We need a process for solving exponential equations. 1 Exponential Growth 465 Graph exponential growth functions. Because a = 3 is positive and b. For example, radium-226 has a half-life of 1,602 years, an mean lifetime of (1,602)/ln2 = 2,311 years, and a decay rate of 1/(2,311) = 0. This is an example of an exponential graph. Oct 26, 2018 - Explore joshua barriga's board "exponential growth" on Pinterest. Exponential Growth & Decay 06/01/09 Bitsy Griffin PH 8. Use the model to estimate the value after 2 years. If you continue browsing the site, you agree to the use of cookies on this website. The examples highlight the manipulation of indices (exponents) and the index laws. Consequently, the students are able to experience how quickly exponential growth and decay occurs as the number of M&Ms they are having to count, collect, shake, and dump on their desk grows or shrinks rapidly. In 1899, Ernest Rutherford wrote the following words: "These experiments show that the uranium radiation is complex and that there are present at least two distinct types of radiation - one that is very readily absorbed, which will be termed for convenience the alpha-radiation, and the other of more penetrative character which will be termed the beta-radiation. y = 3 x (Where a = 1 and b = 3) 2. Examples of newton’s law in everyday life based on second law of motion: A truck which carries less mass will have a bigger acceleration that a truck which carries more mass When we push a small and a big table, the small table will have a bigger acceleration so that the smaller table will get to the destination faster. The most famous application of exponential decay has to do with the behavior of radioactive materials. !Find! the!valueof!theinvestment!after!30yr. For example, identify percent rate of change in functions such as y = (1. We assume that the drug is administered intravenously, so that the concentration of the drug in the bloodstream jumps almost immediately to its highest level. • Solve more complex exponential and logarithmic equations. The dentist’s office B. key points - exponential. In real life there is no reason for the decay rate to match the growth rate. Exponential Growth and Decay Exponential decay refers to an amount of substance decreasing exponentially. These types of functions adhere to certain properties: 1. Just like PageRank, each 1-point increase is a 10x improvement in power. r is the growth rate when r>0 or decay rate when r 0, in percent. Use exponential models to solve real-life problems. Let T be the time (in days) between hits. In AQA's sample assessment materials (Question 23 in Higher Paper 3 ) students are shown a graph representing the depth of water in a container over time. Logically, it cannot exist in nature as a universal law of nature because it is impossible for a population to keep growing forever without hitting a limit to growth. This plot shows decay for decay constant (λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5. One specific example of exponential decay is purified kerosene, used for jet fuel. 3 to find the time. This Differential Equations Representing Growth and Decay: Rice Legend Interactive is suitable for 11th - Higher Ed. f x b ± ax c, b c y ax Section 3. Exponential Growth Exponential Decay ! P=p o 1+r ( ) t/n! P=p o 1"r ( ). If we ask the question, when is the mass equal to say 30g, then we need to solve t = 0. Improve your math knowledge with free questions in "Exponential growth and decay: word problems" and thousands of other math skills. The second tab has students write the equation from a table. Choose an appropriate model for data. As liquid oil depletes, society is switching to mining tar sands. Both are used in science, for very different reasons. Explore real phenomena related to exponential & logarithmic functions including half-life & doubling time MM3A3a. Exponential Models Some real-life quantities increase or decrease by a fixed percent each year (or some other time period). If two decay modes exist, then you must use the two-term exponential model. To evaluate, substitute a number in for x and find y. based on your observation, list out 4 points on the characteristics of logarithmic or exponential functions and their graphical representation. The range is y > O. If a car costs$15000 and you get a loan for it, are you really only going to pay $15000? Using the scenario of investing$100 at 8% interest per year, the students complete the first task of finding the amount in the account after 4 years ( Math Practice 8 ). GUIDED PRACTICE for Examples 1 and 2 2. Find real & complex roots of higher degree polynomial equations using the factor theorem, remainder theorem, rational root theorem & fundamental theorem of algebra, incorporating complex & radical conjugates. 9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Let T be the time (in days) between hits. Any idiot knows that the Malthusian Growth Model doesn't apply to real-life populations (W. Use exponential growth and decay to model real-life situations - B Contact Us If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. Online Resources 6. How many people will be in the crowd after 3 hour? Given: c=1000 p =1+r =1+0. "The Complete Idiot's Guide To Calculus"): "Truth be told, there are not a lot of natural cases in which exponential growth is exhibited. A quantity undergoing exponential decay. Objective: In this lesson you learned how to use exponential growth models, exponential decay models, Gaussian models, logistic growth models, and logarithmic models to solve real-life problems. Notes: x y 5 7 −24 exponential growth (0, 1) (1, 2. 05 t =3 hour Solution: Exponential function growth g=c(p)^t Substitute the given data in the formulae g=1000(1. The simple answer is: there is no difference. The closest we’ve come is the Financial results folk seem to want, in the form of never ending growth, profitability, real estate prices etc – our busts confirm the finite resources at our disposal. A resistor dissipates electrical energy, and the voltage V across it is. In probability for example, polynomial decay and exponential decay are the two regimes generally discussed for the tail behaviour of distributions of random variables, and all sorts of things are qualitatively different in the two cases. Exponential Decay Exponential growth functions are often used to model population growth. Use the model to estimate the value after 2 years. For example, suppose that the population of a city was 100,000 in 1980. The 'radioactive dice' experiment is a commonly used classroom analogue to model the decay of radioactive nuclei. Explain the importance of e 4. Exponential Decay: The population of a town is decreasing at a rate 0 1% er year. EXAMPLE 3 A Differential Equation with Initial Condition Solve y˜ = 3y, y(0) = 2. The first tab provides a place to write and explain the formula along with an example of exponential decay in the form of an equation, table, and graph. 02) t, y = (0. 2 years to doubleinvalue. 7 Applications Involving Exponential Functions December 12, 2016 Ex. "The Complete Idiot's Guide To Calculus"): "Truth be told, there are not a lot of natural cases in which exponential growth is exhibited. ) Smaller values of b lead to faster rates of decay. Logarithmic and exponential functions can be used to model real-world situations. The topics typically used are population, radioactive elements, credit card accounts, etc. Since exponential growth is truly a universal model, we’ll start with something that is intuitive for most people – money in your bank account. Exponential and Logarithmic Models What You’ll Learn: #66 - Recognize the five most common types of models involving exponential or logarithmic functions. EXPONENTIAL DECAY Exponential!DecayFormula:!!! a=_____! r=_____! t!=_____!! EXAMPLE! 1. Theta is the decay of extrinsic value. 2 years to double yourmoney if you put it into an account that pays 10% interest. It seems to me that the increased cooldown at high heat levels that exponential decay gives would actually stimulate high heat alpha strikes rather t. Critics of the simple exponential growth model (growth at a constant rate of exponential growth) are quite right to dismiss it as having no real-life meaning. These types of functions adhere to certain properties: 1. Case 1: 0 < a < 1, Exponential Decay. Examples of Real-Life Arithmetic Sequences; 9 Exponential Functions Activities That Are A Must! Real-life Examples of Solids of Revolution and Cross-Sections. Exponential decay occurs when a quantity decreases by the same proportion r in each time period t. Growth factor > 1, it is considered exponential growth. SOLUTION Here, k = 3 and P 0 = 2. Case (ii) Overdamping (distinct real roots) If b2 > 4mk then the term under the square root is positive and the char acteristic roots are real and distinct. This article presents you with the definition and some examples of exponential distribution, as well as with the exponential distribution formula and an example of applying it in real life. Slowing the rate of new cases requires dramatic measures and is the key to preventing healthcare systems from becoming overwhelmed. Graph the model. Exponential Decay: The population of a town is decreasing at a rate 0 1% er year. Determine whether this model is an exponential growth or exponential decay, and which equation can be used to find the population in 2008? 1) exponential growth ; y = 1250(0. Half-Life : Paper, M&M’s, Pennies, or Puzzle Pieces. Let’s look at examples of these exponential functions at work. Critics of the simple exponential growth model (growth at a constant rate of exponential growth) are quite right to dismiss it as having no real-life meaning. Big Ideas: Exponential decay occurs when a quantity is decreasing at a constant multiplicative rate. Enter the value of x in the exponential function calculator, it automatically calculates e x. Compound growth is a term usually used in finance to describe exponential growth in interest or dividends. The half life of a substance or a decaying material (or population) is the amount of time it takes for 50% of the original amount of substance (or material or population) to decay. Phosphorus has a half-life of 14 days. An example of decay is the depreciation of the value of a car, and the radioactive decay of isotopes. The kerosene is purified by removing pollutants, using a clay filter. notebook March 25, 2013. Exponential decay is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. If you need help, go to the worked-out Examples on pages 466 through 468. For example, you will have to decide where you will bank. 8b Use the properties of exponents to interpret expressions for exponential functions. In particular then, the half life of a radioactive element is the time required for half of it to decay (i. In the exponential model y=a⋅bx, what does the a value represent? BE SPECIFIC. 1 Exponential Growth 465 Graph exponential growth functions. Some examples of this are money growing in a bank account by a certain percentage every year or the population of a city growing by a certain percentage every year. The decay could be slower because we tax to conserve or successfully invest in technologies. Examples of Applications of Exponential Functions We have seen in past courses that exponential functions are used to represent growth and decay. This is because any number raised to the 0 power will always be 1. 1b Video Notes Assignment pg 304-305, 3-43 ODD Omit 25, Don't Draw Graphs. Goal 3: Using Exponential Decay Models When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by this equation:. tick GRADE Your avatar Add comment Radioactive Decay Avatar of Caroline Huang Caroline Huang 1yr Radioactive Decay Radioactive decay in chemistry is the exponential process of radioactive elements emitting mass as they change forms. The domain off(x) = blis all real numbers. Write functions in continuous exponential growth form 5. So, we must modify the exponential growth function for compound interest problems. Exponential Growth & Decay 06/01/09 Bitsy Griffin PH 8. Notes/Examples What You Will Learn Use and identify exponential growth and decay functions. 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https://hal-ujm.archives-ouvertes.fr/ujm-00628784 | # Multiscale modeling of light absorption in tissues: limitations of classical homogenization approach.
* Auteur correspondant
Abstract : In biophotonics, the light absorption in a tissue is usually modeled by the Helmholtz equation with two constant parameters, the scattering coefficient and the absorption coefficient. This classic approximation of "haemoglobin diluted everywhere" (constant absorption coefficient) corresponds to the classical homogenization approach. The paper discusses the limitations of this approach. The scattering coefficient is supposed to be constant (equal to one) while the absorption coefficient is equal to zero everywhere except for a periodic set of thin parallel strips simulating the blood vessels, where it is a large parameter ω. The problem contains two other parameters which are small: ε, the ratio of the distance between the axes of vessels to the characteristic macroscopic size, and δ, the ratio of the thickness of thin vessels and the period. We construct asymptotic expansion in two cases: ε --> 0, ω --> ∞, δ --> 0, ωδ --> ∞, ε2ωδ --> 0 and ε --> 0, ω --> ∞, δ --> 0, ε2ωδ --> ∞, and and prove that in the first case the classical homogenization (averaging) of the differential equation is true while in the second case it is wrong. This result may be applied in the biomedical optics, for instance, in the modeling of the skin and cosmetics.
Type de document :
Article dans une revue
PLoS ONE, Public Library of Science, 2010, 5 (12), pp.e14350. 〈10.1371/journal.pone.0014350〉
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https://hal-ujm.archives-ouvertes.fr/ujm-00628784
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Dernière modification le : jeudi 11 janvier 2018 - 06:20:35
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journal.pone.0014350.pdf
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### Citation
Stéphane Mottin, Grigory Panasenko, S Sivaji Ganesh. Multiscale modeling of light absorption in tissues: limitations of classical homogenization approach.. PLoS ONE, Public Library of Science, 2010, 5 (12), pp.e14350. 〈10.1371/journal.pone.0014350〉. 〈ujm-00628784〉
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https://zbmath.org/?q=an:0862.32021 | ×
# zbMATH — the first resource for mathematics
Cohomology of the complement of a free divisor. (English) Zbl 0862.32021
The authors give conditions on a divisor on a smooth complex manifold, which assure that the cohomology of its complement can be calculated by the logarithmic de Rham complex.
##### MSC:
32S20 Global theory of complex singularities; cohomological properties 32S25 Complex surface and hypersurface singularities 14F40 de Rham cohomology and algebraic geometry 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory
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##### References:
[1] E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 279 – 284. [2] Egbert Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol$$^{\prime}$$d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, 1973, pp. 21 – 44. Lecture Notes in Math., Vol. 317 (French). [3] Robert Ephraim, Isosingular loci and the Cartesian product structure of complex analytic singularities, Trans. Amer. Math. Soc. 241 (1978), 357 – 371. · Zbl 0395.32006 [4] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95 – 103. · Zbl 0145.17602 [5] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. · Zbl 0237.14008 [6] E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series, vol. 77, Cambridge University Press, Cambridge, 1984. · Zbl 0552.14002 [7] John N. Mather, Stability of \?^{\infty } mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 279 – 308. John N. Mather, Stability of \?^{\infty } mappings. IV. Classification of stable germs by \?-algebras, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 223 – 248. John N. Mather, Stability of \?^{\infty } mappings. V. Transversality, Advances in Math. 4 (1970), 301 – 336 (1970). · Zbl 0207.54303 [8] John N. Mather, Stability of \?^{\infty } mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 279 – 308. John N. Mather, Stability of \?^{\infty } mappings. IV. Classification of stable germs by \?-algebras, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 223 – 248. John N. Mather, Stability of \?^{\infty } mappings. V. Transversality, Advances in Math. 4 (1970), 301 – 336 (1970). · Zbl 0207.54303 [9] Bernd Wegner, Decktransformationen transnormaler Mannigfaltigkeiten, Manuscripta Math. 4 (1971), 179 – 199 (German, with English summary). · Zbl 0205.51603 [10] D. G. Northcott, Injective envelopes and inverse polynomials, J. London Math. Soc. (2) 8 (1974), 290 – 296. · Zbl 0284.13012 [11] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. · Zbl 0757.55001 [12] Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265 – 291. · Zbl 0496.32007 [13] C. T. C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), no. 6, 481 – 539. · Zbl 0451.58009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | {"extraction_info": {"found_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": 0.83304762840271, "perplexity": 1212.1052907862406}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060677.55/warc/CC-MAIN-20210928092646-20210928122646-00065.warc.gz"} |
http://physics.stackexchange.com/questions/32635/when-is-the-stationary-phase-approximation-exact?answertab=votes | # when is the stationary phase approximation exact?
I am thinking about some topological field theories, and I am wondering when one can say that the stationary phase approximation (ie. a sum of the first-order variations about each vacuum) is exact.
I am looking perhaps for conditions on how the space of vacua is embedded into the space of all field configurations. I suspect that when the action is a Morse function (and I suppose the space of field configurations is finite dimensional) that the exactness of the stationary phase approximation implies some very strict topological constraints on the configuration space... torsion-free and so on.
Anyone have a good reference or some wisdom?
Also I'd like to dedicate this question to the memory of theoreticalphysics.stackexchange
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In general, the situation where the stationary phase approximation is exact is described by the Duistermaat Heckman theorem, which states (not in its most general form) that if $M$ is a compact symplectic manifold and $H$ is a Hamiltonian generationg a torus action on $M$, then for the "partition" function
$Z = \int_M e^{it H} d_L(M)$
the stationary phase approximation is exact ($d_L(M)$ is the Liouville measure) and the integral can be computed by summing the contributions from the extrema of $H$ (fixed points of the torus action).
An equivalent characterization of the hamiltonian $H$ is that it is a perfect Morse function.
Two very known examples are the Gaussian integral and the spin partition function in a magnetic field where the classical and the quantum partition functions are exactly the same.
This theorem was applied and generalized to more complicated situations (e.g., when the fixed points are not isolated), to path integrals of certain theories (coherent state path integrals), loop spaces and to topological field theories.
Further reserach of the Duistermaat-Heckman theorem and its generalizations led to the discovery of a general phenomenon leading to this type of exactness, now called "equivariant localization".
Please see the following review article"Equivariant Localization of Path Integrals" by Richard J. Szabo, where numerous applications are described.
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+1, thanks for the wonderful review (and for preventing me from posting a totally wrong answer). – Ron Maimon Jul 23 '12 at 7:54
The Duistermaat Heckman theorem is restricted to finitley many degrees of freedom. How would it apply to a topological field theory? – Arnold Neumaier Jul 23 '12 at 13:40
@Arnold. The generalization of the Duistermaat-Heckman theorem to field theories is only to the physics level of rigor. Actually, the field theory examples for which this formalism was applied constitute of finite dimensional reduced phase spaces. For example the Chern-Simons theory whose reduced phase space is the moduli space of flat connections, or the coherent state path integral which is reduced to the lowest landau level by supersymmetry. I think that the first one who in roduced this generalization is Antti Niemi, please see for example Arxiv: hep-th/9301059 (He has also earlier works). – David Bar Moshe Jul 23 '12 at 14:22
Thanks for your answer, David! That review article looks like a good reference for me. – Ryan Thorngren Jul 23 '12 at 16:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9707129597663879, "perplexity": 385.86435277656017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860125175.9/warc/CC-MAIN-20160428161525-00127-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/179266 | Infoscience
Journal article
# Mechanism of the hydrogen/platinum(111) fuel cell
We discuss our recently proposed mechanism for the electro-oxidation/reduction on Pt(111) surfaces (J. Electroanal. Chem. 2002, 537, 7) in the presence of sulfuric acid. The bisulfate ion has a large dipole moment and is strongly adsorbed on the positive electrode. Due to the large field gradients, the oxygen atoms of the adsorbed water molecules (and the dipoles) point down and bind to the on-top positions of the platinum substrate. As the electrode becomes more negative, the field gradient changes direction, and the water dipoles gradually reverse their orientation. At a certain critical value of the orientational parameter (which depends also on the bisulfate surface concentration), a two-dimensional honeycomb array of hydrogen bonded water molecules is formed. This is a new form of solid water, a true two-dimensional "ice". For these negative potentials, the stable structure has one of the hydrogen atoms of the water pointing down, and this means that it is adsorbed by the hollow site of the Pt lattice. To satisfy the stoichiometry of the hydrogen bonds, we need to adsorb one-third of the surface sites of H+ ions. The following reversible reaction occurs: (H5O2+)(3) + 6e(-) reversible arrow 6H + (H3O3-)(3). For the (111) surface of platinum and because of the geometrical matchup (the Pt-Pt distance is 2.77 Angstrom, and the water diameter is 2.76 Angstrom) this reaction occurs as a first-order transition, visible in the voltammogram as a sharp peak. From the [H+] concentration dependence of this sharp spike, we get an effective charge of 1.02 +/- 0.02 for the adsorbed moiety. High-accuracy quantum calculations on a five-layer platinum metal slab show that this compound is stable in the absence of bisulfate ions. The quantum calculations show also that the hydrogen atoms in the hollow positions are neutralized. Since there are two-thirds of the Pt sites in the hollow positions, our model gives a natural explanation to the well-known fact that the hydrogen yield is 2/3 on this surface. We have revised our theory to shift the turning point of the water molecules to the transition potential where the HER honeycomb phase is formed. The turning point is in general agreement with the recent laserinduced measurements of the potential of zero charge.
#### Reference
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https://www.physicsforums.com/threads/what-is-location.35614/ | # What is location?
1. Jul 18, 2004
### Simons
what is location??
I have a question, hopefully someone can provide some insight on it:
what, fundamentally, is location?
there are only a handfull of really "unique" aspects of matter at this time, with many physicists trying to simplify them even further. But it seems that we have mass, electromagnetic charge, velocity, and many "lesser" properties of subatomic particles like charm, top/bottom, etc. One way of looking at this question is to see location, too, as a property of matter. But what defines this property? Obviously, whatever the "value" of this property is for various particles, must influence how the various forces produced by these particles affect each other (for example, the earth will not have the same gravitational effects on a particle on the moon, or a particle in andromeda galaxy). And if we were able to fundamentally understand this property, and how it was represented subatomically, could we devise a way of manipulating it for perhaps teleportation (?)
Can you help with the solution or looking for help too?
Draft saved Draft deleted
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http://slideplayer.com/slide/225865/ | # The Beginning of the Quantum Physics
## Presentation on theme: "The Beginning of the Quantum Physics"— Presentation transcript:
The Beginning of the Quantum Physics
Beginning of the Quantum Physics
Some “Problems” with Classical Physics Vastly different values of electrical resistance Temperature Dependence of Resistivity of metals Blackbody Radiation Photoelectric effect Discrete Emission Lines of Atoms Constancy of speed of light
Blackbody Radiation: Solids heated to very high temperatures emit visible light (glow) Incandescent Lamps (tungsten filament) The color changes with temperature At high temperatures emission color is whitish, at lower temperatures color is more reddish, and finally disappear Radiation is still present, but “invisible” Can be detected as heat Heaters; Night Vision Goggles
Experiment: Focus the sun’s rays or direct a parabolic mirror with a heating spiral onto combustible material the material will flare up and burn Materials absorb as well as emit radiation
Blackbody Radiation All object at finite temperatures radiate electromagnetic waves (emit radiation) Objects emit a spectrum of radiation depending on their temperature and composition From classical point of view, thermal radiation originates from accelerated charged particles in the atoms near surface of the object
A blackbody is an object that absorbs all radiation incident upon it Its emission is universal, i.e. independent of the nature of the object Blackbodies radiate, but do not reflect and so are black Blackbody Radiation is EM radiation emitted by blackbodies
Blackbody Radiation There are no absolutely blackbodies in nature – this is idealization But some objects closely mimic blackbodies: Carbon black or Soot (reflection is <<1%) The closest objects to the ideal blackbody is a cavity with small hole (and the universe shortly after the big bang) Entering radiation has little chance of escaping, and mostly absorbed by the walls. Thus the hole does not reflect incident radiation and behaves like an ideal absorber, and “looks black”
Kirchoff's Law of Thermal Radiation (1859)
absorptivity αλ is the ratio of the energy absorbed by the wall to the energy incident on the wall, for a particular wavelength. The emissivity of the wall ελ is defined as the ratio of emitted energy to the amount that would be radiated if the wall were a perfect black body at that wavelength. At thermal equilibrium, the emissivity of a body (or surface) equals its absorptivity αλ = ελ If this equality were not obeyed, an object could never reach thermal equilibrium. It would either be heating up or cooling down. For a blackbody ελ = 1 Therefore, to keep your frank warm or your ice cream cold at a baseball game, wrap it in aluminum foil What color should integrated circuits be to keep them cool?
Emission is continuous The total emitted energy increases with temperature, and represents, the total intensity (Itotal) – the energy per unit time per unit area or power per unit area – of the blackbody emission at given temperature, T. It is given by the Stefan-Boltzmann Law σ = 5.670×10-8 W/m2-K4 To get the emission power, multiply Intensity Itotal by area A
Blackbody Radiation The maximum shifts to shorter wavelengths with increasing temperature the color of heated body changes from red to orange to yellow-white with increasing temperature 5780 K is the temperature of the Sun
Blackbody Radiation The wavelength of maximum intensity per unit wavelength is defined by the Wien’s Displacement Law: b = 2.898×10-3 m/K is a constant For, T ~ 6000 K,
Blackbody Radiation Laws: Classical Physics View
Average energy of a harmonic oscillator is <E> Intensity of EM radiation emitted by classical harmonic oscillators at wavelength λ per unit wavelength: Or per unit frequency ν:
Blackbody Radiation Laws: Classical Physics View
In classical physics, the energy of an oscillator is continuous, so the average is calculated as: is the Boltzmann distribution
This gives the Rayleigh-Jeans Law Agrees well with experiment long wavelength (low frequency) region Predicts infinite intensity at very short wavelengths (higher frequencies) – “Ultraviolet Catastrophe” Predicts diverging total emission by black bodies No “fixes” could be found using classical physics
Max Planck postulated that
Planck’s Hypothesis Max Planck postulated that A system undergoing simple harmonic motion with frequency ν can only have energies where n = 1, 2, 3,… and h is Planck’s constant h = 6.63×10-34 J-s
E is a quantum of energy
Planck’s Theory E is a quantum of energy For = 3kHz
Planck’s Theory As before: Now energy levels are discrete, So
Sum to obtain average energy:
Blackbody Radiation c is the speed of light, kB is Boltzmann’s constant, h is Planck’s constant, and T is the temperature
Planck’s Theory
Planck’s Theory
High Frequency - h >> kT
At room temperature, 300 K, kT= 1/40 eV At = 1 m: At 300 K:
Plank’s curve Stefan-Boltzmann Law IBB T4 IBB = T4 Stefan-Boltzmann constant =5.67×10-8 J/m2K4 More generally: I = T4 is the emissivity Wien's Displacement Law peak T = 2.898×10-3 m K At T = 5778 K: peak = 5.015×10-7 m = 5,015 A λmax
Best test of law – it is experiment agreement is better than 1/100,000
Energy Balance of Electromagnetic Radiation 50% of energy emitted from the sun in visible range Appears as white light above the atmosphere, peaked Appears as yellow to red light due to Rayleigh scattering by the atmosphere Earth radiates infrared electromagnetic (EM) radiation Glass Prism White light is made of a range of wave lengths
Step 4: Calculate energy emitted by Earth
Earth emits terrestrial long wave IR radiation Assume Earth emits as a blackbody. Calculate energy emission per unit time (Watts) Blackbody Radiation 29 Notice color change as turn up power on light bulb.
Assume Earth characterized by single T
Result depends only weakly on albedo What is wrong – we have neglected the influence of the atm.
Greenhouse Effect Visible light passes through atmosphere and warms planet’s surface Atmosphere absorbs infrared light from surface, trapping heat Why is it cooler on a mountain tops than in the valley? 31
Albedo and Atmosphere Affect Planet Temperature
32
Venus has 90 atm. of pressure
Consider a spherical cow
Sequel - Consider a cylindrical cow Two layers- two levels of atm. Lower level emits and absorbed in second layer Putting in the empirical formulas. (IR gets out by reradiating into free windows as well as by diffusion) Get 16C
Einstein’s Photon Interpretation of Blackbody Radiation
EM Modes: Two sine waves traveling in opposite directions create a standing wave For EM radiation reflecting off a perfect metal, the reflected amplitude equals the incident amplitude and the phases differ by rad E = 0 at the wall For allowed modes between two walls separated by a: sin(kx) = 0 at x = 0, a This can only occur when, ka = n, or k = n/a, n = 1,2,3… In terms of the wavelength, k = 2/ = n/a, or /2 = a/n This is for 1D, for 2D, a standing wave is proportional to: For 3D a standing wave is proportional to:
Density of EM Modes, 1 May represent allowed wave vectors k by points on a unit lattice in a 3D abstract number space k = 2/. But f = c, so f = c/ = c/[(/2))(2)] = c/[(1/k)((2)]=ck/2 f is proportional to k = n /a in 1D and can generalize to higher dimensions: where, n is the distance in abstract number space from the origin (0,0,0) To the point (n1,n2 n3)
Density of EM Modes, 2 The number of modes between f and (f+df) is the number of points in number space with radii between n and (n+dn) in which n1, n2, n3,> 0, which is 1/8 of the total number of points in a shell with inner radius n and outer radius (n+dn), multiplied by 2, for a total factor of 1/4 The first factor arises because modes with positive and negative n correspond to the same modes The second factor arises because there are two modes with perpendicular polarization (directions of oscillation of E) for each value of f Since the density of points in number space is 1 (one point per unit volume), the number of modes between f and (f+df) is the number of points dN in number space in the positive octant of a shell with inner radius n and outer radius (n+dn) multiplied by 2 dN = 2 dV', where dV‘ = where dV‘ is the relevant volume in numbr space The volume of a complete shell is the area of the shell multiplied by its thickness, 4 n2dn The number of modes with associated radii in number space between n and (n+dn) is, therefore, dN = 2 dV‘ = (2)(1/8)4 n2dn = n2dn
Density of EM Modes, 3 The density of modes is the number of modes per unit frequency: This may be expressed in terms of f once n and dn/df are so expressed This is density of modes in a volume a3 For a unit volume, the density of states is:
Modes Density How many EM modes per unit frequency are there in a cubic cavity with sides a = 10 cm at a wavelength of = 1 micron = 10-6 m? f = c, f = c/ = 3x108/10-6 = 3x1014
Blackbody Radiation Einstein argued that the intensity of black body radiation I(f), reflects the state of thermal equilibrium of the radiation field The energy density (energy per unit volume per unit frequency) within the black body is: , where is the average energy of a mode of EM radiation at frequency f and temperature T The intensity is given by: Since (a) only ½ the flux is directed out of the black body and (b) the average component of the velocity of light In a direction normal to the surface is ½
Blackbody Radiation But hf = e and , as before So | {"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": 0.9193353056907654, "perplexity": 1366.5262236459428}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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-20170924190521-20170924210521-00116.warc.gz"} |
https://www.physicsforums.com/threads/astrophysics-orbital-dynamics-duration-of-a-transit-of-venus.418048/ | # Astrophysics: orbital dynamics - duration of a transit of Venus
1. Jul 24, 2010
### joriarty
Edit: perhaps this might belong in one of the Physics help forums? Oops. Mods, please move if so!
1. The problem statement, all variables and given/known data
Show that a transit of Venus across the Sun’s disk lasts at most about 8 hours. The synodic period of Venus is 584 days and its orbital radius is 0.723 A.U. The Sun’s angular diameter is 32′. Assume that the orbits of Venus and the Earth are coplanar and circular. Do not use any other numerical data. (Hint: find the angular velocity of Venus relative to the Earth-Sun line and as seen from the Earth. The greatest duration of transit is when Venus passes through a diameter of the solar disk.)
2. Relevant equations
ωVenus - ωEarth = ωsynodic
3. The attempt at a solution
I understand the concept of synodic period, I'm just unsure how to tackle this problem. The transit must occur at inferior conjunction. I know the synodic period, which I can use to calculate the synodic angular velocity as ~ 0.02568 degrees per hour. This gives a maximum transit duration of nearly 21 hours! That's not right.
How can I convert this synodic angular velocity (which is with respect to a reference frame that is co-rotating with the Earth-Sun line) to an angular velocity seen by an observer on Earth?
Please don't be too specific with your hints as this is coursework, and I do want to figure things out for myself. I just need a nudge in the right direction :)
Thanks!
2. Jul 24, 2010
### Filip Larsen
Try think of this as a purely circular geometric problem. Near inferior conjunction the Earth moves as if it rotates 360 degrees around Venus in 584 days at a distance of 1-0.723 AU. Now think about how much angle the 32' seen from Earth over 1 AU corresponds to when seen from Venus at 0.723 AU and then think about how fast this angle is covered when doing 360/584 deg/day.
Last edited: Jul 24, 2010
3. Jul 24, 2010
### joriarty
Got it now! Thank you :). Just needed to think in terms of Earth's frame of reference - I was trying to solve it from a "top down" solar system model.
Similar Discussions: Astrophysics: orbital dynamics - duration of a transit of Venus | {"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": 0.9379090070724487, "perplexity": 889.8127798062668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948588294.67/warc/CC-MAIN-20171216162441-20171216184441-00606.warc.gz"} |
http://math.stackexchange.com/questions/33506/design-function-that-is-maximized-at-finite-value | # Design function that is maximized at finite value
Given two functions $f_1(m) = a^m$ and $f_2(m) = b^m$, how to design another function $f(m)=g(f_1(m), f_2(m))$ such that $f(m)$ is maximized at some finite value $m=m_o$ (with $m_o$ not equal to $0$ or $\infty$). Or prove there is no such function $g(\cdot,\cdot)$.
For instance, $g(x,y) = \frac{x}{y}$ does not meet the requirement, since if $a\le{}b$, $m_o=0$. If $a\ge{}b$, $m_o=\infty$.
-
Would $\frac{(a/b)^m}{(c+(a/b)^m)^{2n}}$ where $c > 0$ and $n$ is an integer work? – Guess who it is. Apr 17 '11 at 18:09
Would $g(x,y)=-|\log(kx)\log(ky)|$ work? – Isaac Apr 17 '11 at 18:12
Can't you just cheat? Let $m_0$ be the value you wish to maximize at, define $g(f_1(m_0),f_2(m_0))=1$ and for all other values define $g$ to be 0. – JSchlather Apr 17 '11 at 18:21
Just take $g(x,y)$ to be a function of two variables that attains a maximum at $(x,y) = (f_1(m_0), f_2(m_0))$. For example, $g(x,y) = - (x - f_1(m_0))^2 - (y - f_2(m_0))^2$ will work.
There are many examples. One idea is to note that the maximum of $$H_1(m)=\frac{1}{\frac{a^m}{a}+\frac{a}{a^m}}=\frac{a\cdot a^{m}}{a^2+(a^{m})^2}$$ is reached at $m=1$. So if we let $$f(x,y)=\frac{ax}{a^2+x^2}+\frac{by}{b^2+y^2}$$ then $f(a^m, b^m)$ will reach its maximum at $m=1$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9685664176940918, "perplexity": 168.67265372504153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1438042987034.19/warc/CC-MAIN-20150728002307-00118-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://codedump.io/share/Jc1JHHcEm8Hh/1/shell-command-in-vba-execution | user5619709 - 1 year ago 114
Bash Question
# Shell Command in VBA Execution
I'm trying to run two lines of commands using shell. I haven't been able to find a good source on how to actually execute shell in VBA. So far, I have been able to figure out how to open a specific directory.
``````Sub shellCMD()
Shell ("cmd.exe /k CD\Users\n808037\Desktop\OTHER")
End Sub
``````
This will at least lead me to the directory where I need to go. However, now that I've gotten to the directory I need to go, I need to execute a command after. That is
copy *.csv merged.csv
How do I do this in shell?
Each `Shell` call runs in its own process, so you can't run separate commands by calling `Shell` consecutively. Generally, you'd want to either run multiple commands as a batch file or script if you were going to do a lot of processing. In this case, just specify the full path for `copy`. There's no need to change the working directory at all:
``````Shell "cmd.exe /k copy C:\Users\n808037\Desktop\OTHER\*.csv C:\Users\n808037\Desktop\OTHER\merged.csv" | {"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": 0.8614792227745056, "perplexity": 1415.3775018143767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1508187822480.15/warc/CC-MAIN-20171017181947-20171017201947-00349.warc.gz"} |
https://www.physicsforums.com/threads/question-from-saunders-maclane-and-birkhoff-algebra-book.295472/ | # Question from Saunders Maclane and Birkhoff Algebra book.
1. Feb 26, 2009
### MathematicalPhysicist
If S is a subset of G with G finite, while f:G->H is a morphism with kernel N, prove that:
[G:S]=[f(G):f(S)][N:SnN] where 'n' stands for intersection.(The question is on page 78 exercise 10).
Now as far I can tell [G:S] is the number of right or left cosets of S in G, if the morphism was monomorphism it would be equal [f(G):f(S)], now [N:(SnN)] is the number of cosets m(SnN) where m is in N, I must find a way to show that in order to count the number of gS's for each f(g)f(S) I need to count the number of preimages, but f(gS)=f(g)f(S), I don't know how to procceed.
Any hints?
Can you offer guidance or do you also need help?
Similar Discussions: Question from Saunders Maclane and Birkhoff Algebra book. | {"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": 0.8139761686325073, "perplexity": 1193.7159687480018}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103891.56/warc/CC-MAIN-20170817170613-20170817190613-00468.warc.gz"} |
https://energyeducation.ca/wiki/index.php?title=Coefficient_of_performance | # Coefficient of performance
Figure 1: The coefficient of performance of a system to be cooled (refrigerator, air conditioning) is the ratio of heat out of the system to the work put into the system.[1]
Coefficient of performance (K) is a number that describes the effectiveness of refrigerators or air conditioners, by comparing the heat that is dispelled from it to the work that had to be done to do so. It is similar to the thermal efficiency of heat engines, as they both relate what benefits you are getting to what you had to pay.[2] The coefficient of performance is given by the equation:
where:[2]
• is the heat dispelled from the refrigerator
• is the work input to the system
• is the coefficient of performance
A better refrigerator will require less work to remove a given amount of heat, thus having a larger coefficient of performance. From the equation, it is clear that if no work was input to the system () the coefficient would equal infinity. This would constitute a perfect refrigerator, which is forbidden by the Second law of thermodynamics. Hence, .[2]
## Maximum Coefficient of Performance
Just like there is a thermal Carnot efficiency, there is also a Carnot coefficient of performance, which describes the maximum K value for a refrigerator. This is given by the equation:
where:[2]
• is the temperature of the environment to which the heat is dispelled to
• is the temperature of the space to cool off
• is the maximum coefficient of performance
## References
1. Created internally by a member of the Energy Education team.
2. R. D. Knight, "Heat Engines and Refrigerators" in Physics for Scientists and Engineers: A Strategic Approach, 3nd ed. San Francisco, U.S.A.: Pearson Addison-Wesley, 2008, ch.19, sec.2, pp. 532-533 and 544-545
## Authors and Editors
Bethel Afework, Jordan Hanania, Kailyn Stenhouse, Jason Donev
Last updated: May 18, 2018
Get Citation | {"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": 0.8764933347702026, "perplexity": 1020.8807985934777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247484928.52/warc/CC-MAIN-20190218094308-20190218120308-00294.warc.gz"} |
https://thealevelbiologist.co.uk/surface-area-to-volume-ratio/ | Select Page
# 🐁 Surface area to volume ratio (Edexcel)
Organisms exchange substances and heat with their environment all the time, and this possibility is crucial to survival. The specific way in which this is achieved is very tightly related to the shape and structure of the specific organism, as well as its environment. For example, unicellular organisms are so small that molecules such as oxygen and water can readily diffuse in and out via the membrane, due to the short diffusion pathway. Could this be achieved by a human, or even a bee? No – they are simply too big.
Two properties are important to consider here: the volume of an organism, and the surface area of an organism. The volume is what determines the amount of substances which need exchanging, while the surface area determines the amount which can be exchanged.
Key principle: as the size of an organism increases, the surface area to volume ratio decreases.
That might seem hard to really understand. Why use a ratio in the first place? Well, the ratio shows the relationship between surface area and volume… | {"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": 0.8724724650382996, "perplexity": 469.18852824262524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400232211.54/warc/CC-MAIN-20200926004805-20200926034805-00074.warc.gz"} |
https://www.groundai.com/project/impacts-of-gravitational-wave-standard-siren-observation-of-the-einstein-telescope-on-weighing-neutrinos-in-cosmology/ | Impacts of gravitational-wave standard siren observation of the Einstein Telescope on weighing neutrinos in cosmology
# Impacts of gravitational-wave standard siren observation of the Einstein Telescope on weighing neutrinos in cosmology
Ling-Feng Wang Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China Xuan-Neng Zhang Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China Jing-Fei Zhang Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China Xin Zhang111Corresponding author Department of Physics, College of Sciences, Northeastern University, Shenyang 110004, China Center for High Energy Physics, Peking University, Beijing 100080, China
###### Abstract
We investigate the impacts of the gravitational-wave (GW) standard siren observation of the Einstein Telescope (ET) on constraining the total neutrino mass. We simulate 1000 GW events that would be observed by the ET in its 10-year observation by taking the standard CDM cosmology as a fiducial model. We combine the simulated GW data with other cosmological observations including cosmic microwave background (CMB), baryon acoustic oscillations (BAO), and type Ia supernovae (SN). We consider three mass hierarchy cases for the neutrino mass, i.e., normal hierarchy (NH), inverted hierarchy (IH), and degenerate hierarchy (DH). Using Planck+BAO+SN, we obtain eV for the NH case, eV for the IH case, and eV for the DH case. After considering the GW data, i.e., using Planck+BAO+SN+GW, the constraint results become eV for the NH case, eV for the IH case, and eV for the DH case. We find that the GW data can help reduce the upper limits of by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. In addition, we find that the GW data can also help break the degeneracies between and other parameters. We show that the GW data of the ET could greatly improve the constraint accuracies of cosmological parameters.
## I Introduction
On 17 August 2017, the signal of a gravitational wave (GW) produced by the merger of a binary neutron-star system (BNS) was detected for the first time [1], and the electromagnetic (EM) signals generated by the same transient source were also observed subsequently, which indicates that the age of gravitational-wave multi-messenger astronomy is coming. The measurement of GWs from the merger of a binary compact-object system involves the information of absolute luminosity distance of the transient source [2], and thus if we can simultaneously accurately measure the GW and EM signals from the same merger event of a BNS or a binary system consisting of a neutron star (NS) and a black hole (BH), then we are able to establish a true luminosity distance–redshift () relation. Therefore, the GW observations can serve as a cosmic “standard siren”, which can be developed to be a new cosmological probe if we can accurately observe a large number of merger events of this class.
The current mainstream cosmological probes include the measurements of cosmic microwave background (CMB) anisotropies (temperature and polarization), baryon acoustic oscillations (BAO), type Ia supernovae (SNIa), and the Hubble constant, etc. In addition, there are also some observations for the growth history of large-scale structure (LSS), such as the shear measurement of the weak gravitational lensing, the galaxy clusters number counts in light of Sunyaev-Zeldovich (SZ) effect, and the CMB lensing measurement, etc. When using these observational data to make cosmological parameter estimation, some problems occur including mainly: (i) there are degeneracies between some parameters, and in some of which the correlations are rather strong, and (ii) there are apparent tensions between some observations. The GW standard siren observations have some peculiar advantages in breaking the parameter degeneracies, owing to the fact that the GW observations can directly measure the true luminosity distances, but the SNIa observations actually can only measure the ratios of luminosity distances at different redshifts, but not the true luminosity distances. In addition, compared to the standard candle provided by the SNIa observations [3, 4, 5], which needs to cross-calibrate the distance indicators on different scales, the GW observations allow us to directly measure the luminosity distances up to higher redshifts [6]. To obtain the information of redshifts, one needs to detect the EM counterparts of the GW sources. In fact, there are some forthcoming large facility observation programs, such as Large Synoptic Survey Telescope (LSST) [7], Square Kilometer Array (SKA) [8], and Extremely Large Telescope (ELT) [9], which can detect the EM counterparts by optically identifying the host galaxies. In this way, in the near future, the relation would be obtained by the GW standard siren measurements, and we could use this powerful tool to explore the expansion history of the universe.
Recently, the related issues have been discussed by some authors [10, 11, 12, 13, 15, 16, 17, 14, 18, 19] (see also Ref. [20] for a recent review). For example, in Ref. [10], Cai and Yang estimated the ability of using GW data to constrain cosmological parameters. They considered to use the GW detector under planning, the Einstein Telescope (ET), to simulate the GW data, which is a third-generation ground-based detector of GWs [21]. ET is ten times more sensitive than the current advanced ground-based detectors and it covers the frequency range of Hz. According to their results [10], the errors of cosmological parameters can be constrained to be and when using 1000 GW events, whose sensitivity is comparable to that of the Planck data [22]. From their study, we can see that the GW data can indeed be used to improve the constraint accuracies of parameters. In Ref. [23], we can see how the parameter degeneracies are broken by the GW observations in an efficient way. In this work, we investigate the issue of measuring the neutrino mass in light of cosmological observations and we will discuss what role the GW observations of the ET will play as a new cosmological probe in this study.
Since the phenomenon of neutrino oscillation was discovered, which proved that the neutrino masses are not zero [24], the determination of neutrino masses has been an important issue in the field of particle physics. Due to the fact that neutrino oscillation experiments are only sensitive to the squared mass differences between the neutrino mass eigenstates, it is a great challenge to determine the absolute masses of neutrinos by particle physics experiments. Although the solar and atmospheric neutrino oscillation experiments can give two squared mass differences between the mass eigenstates: eV and eV, we cannot determine whether the third neutrino is heaviest or lightest. Therefore, these measurements can only give two possible mass orders, i.e., the normal hierarchy (NH) with and the inverted hierarchy (IH) with . If the total mass of neutrinos () can be measured, then the absolute masses of neutrinos could be solved by combining the total mass and the two squared mass differences.
Massive neutrinos play an significant role in the evolution of the universe, and thus they leave distinct signatures on CMB and LSS at different epochs of the evolution of the universe. These signatures can actually be extracted from the cosmological observations, from which the total mass of neutrinos can be effectively constrained. In recent years, the combinations of various cosmological observations have been providing more and more tight constraint limits for the total mass of neutrinos. For the latest progresses on this aspect, see e.g. Refs. [25, 26, 27, 28, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].
In fact, in a recent paper [44], the constraints on the total neutrino mass have been discussed by considering the inclusion of the actual observation of GW and EM emission produced by the merger of BNS, GW170817. In Ref. [44], the authors considered a 12-parameter extended cosmological model that contains the dark-energy equation-of-state parameter and , the spatial curvature , the total neutrino mass , the effective number of relativistic species , and the running of the scalar spectral index , besides the 6 base parameters, and they made a comparison for the constraint results of Planck and Planck+GW170817. For such a 12-parameter model, using only Planck data gives eV, and the combination of Planck+GW170817 gives eV, showing that the inclusion of GW170817 leads to an about 30% improvement for the upper limit of , compared to the result obtained from the Planck data alone. However, it is clearly known that the 12-parameter model actually cannot be well constrained by the current observations. In the present work, we wish to scrutinize the standard cosmology, i.e., a 7-parameter cold dark matter (CDM) model that contains the 6 base parameters plus . We aim to see how the future GW observations help improve the constraints on the total neutrino mass in a 7-parameter CDM cosmology.
In this study, our main focus is on the ability of the GW observations of ET to constrain the neutrino mass. We follow Ref. [10] to generate the catalogue of GW events, from which we can get the corresponding relation. According to the influences of instruments and weak lensing, we can estimate the uncertainty on the measurement of . We simulate 1,000 GW events that could be observed by the ET in its 10-year observation. Then, we combine these simulated GW data with other current observations to constrain the total mass of neutrinos, , by using the Markov-chain Monte Carlo (MCMC) [45] approach.
The paper is organized as follows. In Sec. II, we describe the method to generate simulated GW events and get the luminosity distances with the simulated measurement errors. In addition, the fiducial model and the data processing method are also briefly introduced. In Sec. III, we report the results of this work. In Sec. IV, we make a conclusion for this work.
## Ii Methods of simulating data and constraining parameters
### ii.1 Method of simulating data
The first step for generating GW data is to simulate the redshift distribution of the sources. Following Refs. [10, 16], the distribution takes the form
P(z)∝4πd2C(z)R(z)H(z)(1+z), (1)
where is the comoving distance at redshift and denotes the time evolution of the burst rate and takes the form [46, 47, 10]
R(z)=⎧⎪⎨⎪⎩1+2z,z≤1,34(5−z),1
According to the prediction of the Advanced LIGO-Virgo network, we take the ratio between BHNS (the binary system of a BH and a NS) and BNS events to be 0.03. For the mass distributions of NS and BH in the simulation, we randomly sample the mass of NS in the interval and the mass of BH in the interval (here is the solar mass), as the same as in Ref. [10].
After the distribution of the sources is known, the second step is to generate the catalogue of the GW sources through the fiducial model, i.e., the CDM model. If we consider a flat Friedmann-Robertson-Walker universe, the Hubble parameter for the CDM cosmology can be written as
H(z)2 =H20[(1−Ωm−Ωr)+Ωm(1+z)3+ Ωr(1+z)4], (3)
where and represent the fractional energy densities of matter and radiation, respectively. In the late universe, can be neglected, and , where , , and represent the fractional energy densities of baryons, cold dark matter, and neutrinos, respectively. Note that can be expressed as
Ων=∑mν94h2eV, (4)
where is the reduced Hubble constant (the Hubble constant ). Since the radiation component can be ignored in the late universe, we set here.
The luminosity distance can be calculated by
dL=(1+z)H0∫z0dz′E(z′), (5)
where . According to the redshift distribution of the GW sources, we can then use Eq. (5) to generate a catalogue of the GW sources. That is to say, the relation can be obtained for every GW event in a CDM cosmology.
The next step is to get the error of of the GW source, which is denoted by in this paper. We first need to generate the waveform of GWs. Because the GW amplitude depends on the luminosity distance , we can get the information of and from the amplitude of waveform.
Following Ref. [10], the strain in GW interferometers can be written as
h(t)=F+(θ,ϕ,ψ)h+(t)+F×(θ,ϕ,ψ)h×(t), (6)
where is the polarization angle and (,) are angles describing the location of the source relative to the detector. Here the antenna pattern functions of the ET (i.e. and ) are [16]
F(1)+(θ,ϕ,ψ)= √32[12(1+cos2(θ))cos(2ϕ)cos(2ψ) −cos(θ)sin(2ϕ)sin(2ψ)], F(1)×(θ,ϕ,ψ)= √32[12(1+cos2(θ))cos(2ϕ)sin(2ψ) +cos(θ)sin(2ϕ)cos(2ψ)]. (7)
For the other two interferometers, their antenna pattern functions can be derived from above equations because the interferometers have with each other.
Following Refs. [16, 17], we can compute the Fourier transform of the time domain waveform ,
H(f)=Af−7/6exp[i(2πft0−π/4+2ψ(f/2)−φ(2.0))], (8)
where is the Fourier amplitude that is given by
A= 1dL√F2+(1+cos2(ι))2+4F2×cos2(ι) ×√5π/96π−7/6M5/6c, (9)
where is the “chirp mass”, is the total mass of coalescing binary with component masses and , and is the symmetric mass ratio. Note that all the masses here refer to the observed mass rather than the intrinsic mass. The observed mass is larger than the intrinsic mass by a factor of , i.e., . is the angle of inclination of the binary’s orbital angular momentum with the line of sight. Due to the fact that the short gamma ray bursts (SGRBs) are expected to be strongly beamed, the coincidence observations of SGRBs imply that the binaries should be orientated nearly face on (i.e., ) and the maximal inclination is about . In fact, averaging the Fisher matrix over the inclination and the polarization with the constraint is roughly the same as taking in the simulation [17]. Thus, when we simulate the GW source we can take . But, when we estimate the practical uncertainty of the measurement of , the impacts of the uncertainty of inclination should be taken into account. Actually, the consideration of the maximal effect of the inclination (between and ) on the signal-to-noise (SNR) leads to a factor of 2 [see Eq. (13)]. The definitions of other parameters and the values of the parameters can be found in Ref. [10].
After knowing the waveform of GWs, we can then calculate the SNR. A GW detection is confirmed if it produces a combined SNR of at least 8 in ET [21, 48] (see also Refs. [10, 16, 11, 14]). The combined SNR for the network of three independent interferometers is
ρ= ⎷3∑i=1(ρ(i))2, (10)
where , with the inner product being defined as
⟨a,b⟩=4∫fupperflower~a(f)~b∗(f)+~a∗(f)~b(f)2dfSh(f), (11)
where a “” above a function denotes the Fourier transform of the function and is the one-side noise power spectral density. In this work, we take of the ET to be the same as in Ref. [16].
Using the Fisher information matrix, we can estimate the instrumental error on the measurement of , which can be written as
(12)
Because we only focus on the parameter in the waveform, we find that due to . Considering the effect from the inclination angle , we add a factor 2 in front of the error, so the error is written as
σinstdL≃2dLρ. (13)
Following Ref. [10], we set the additional error = , which represents the error from weak lensing. Thus, the total error on the measurement of can be expressed as
σdL =√(σinstdL)2+(σlensdL)2 =√(2dLρ)2+(0.05zdL)2. (14)
Using the method described above, we can generate the catalogue of the GW events with their , , and . According to the results in Ref. [10], we know that it requires more than 1000 GW events to match the Planck sensitivity, and so we simulate 1000 GW events that are expected to be detected by the ET in its 10-year observation.
### ii.2 Method of constraining parameters
In order to constrain cosmological parameters, we use the MCMC method to infer the posterior probability distributions of parameters. To measure the total neutrino mass in light of cosmological observations, we set to be a free parameter in the cosmological fit.
In this work, we add the simulated GW data in the combined cosmological data. For the GW standard siren measurement with simulated data points, we can write its as
χ2GW=N∑i=1⎡⎣¯diL−dL(¯zi;→Ω)¯σidL⎤⎦2, (15)
where , , and are the th redshift, luminosity distance, and error of luminosity distance of the simulated GW data, and represents the set of cosmological parameters.
To show the constraining capability of the simulated GW data, we consider two data combinations for comparison in this work: (i) Planck+BAO+SN and (ii) Planck+BAO+SN+GW. For the CMB data, we use the Planck 2015 temperature and polarization data. For the BAO data, we use the measurements of the six-degree-field galaxy (6dFGS) at [49], the SDSS main galaxy sample (MGS) at [50], the baryon oscillation spectroscopic survey (BOSS) LOWZ at [51], and the BOSS CMASS at [51]. For the SN data, we use the “joint light-curve analysis” (JLA) sample [52]. For the simulated GW data, we consider 1,000 GW events that could be observed by the ET in its 10-year observation.
For the neutrino mass measurement in this work, we consider three mass hierarchy cases, i.e., the normal hierarchy (NH), the inverted hierarchy (IH), and the degenerate hierarchy (DH). For the details of this aspect, see Refs. [29, 30].
## Iii Results and discussion
Our constraint results for the neutrino mass and other cosmological parameters are shown in Table 1 and Figs. 14. Note that in this work we have considered three mass hierarchy cases for massive neutrinos and we have used two data combinations to make the analysis.
For convenience, we also use “Data1” to represent the Planck+BAO+SN data combination, and use “Data2” to represent the Planck+BAO+SN+GW data combination; e.g., see Table 1. In Table 1, we show the best-fit results with the CL uncertainties for the cosmological parameters, but owing to the fact that the neutrino mass cannot be well constrained, we only give the CL upper limits for the neutrino mass . In addition, the derived parameters and are also listed in this table.
In Fig. 1, we show the one-dimensional posterior distributions of using the two data combinations, for the three mass hierarchy cases. In Figs. 24, we show the two-dimensional posterior distribution contours ( and CL) in the , , and planes, for the three mass hierarchy cases, also using the two data combinations. In these figures, the blue lines and contours represent the results from the Planck+BAO+SN data, and the red lines and contours represent the results from the Planck+BAO+SN+GW data.
First, we discuss the effect of the simulated GW data of the ET on constraining the total neutrino mass. From the one-dimensional posterior distributions of in Fig. 1, we find that, when the GW data are considered, the constraints on in all the three mass hierarchy cases become tighter, i.e., smaller values of the upper limits of are obtained. The detailed constraint results have been given in Table 1. Using the data combination Planck+BAO+SN, we obtain: eV for the NH case, eV for the IH case, and eV for the DH case. After adding the GW data of the ET, namely when using the data combination Planck+BAO+SN+GW, the constraint results become: eV for the NH case, eV for the IH case, and eV for the DH case. We find that the GW data help reduce the upper limits of by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. Obviously, the GW data can indeed effectively improve the constraints on the neutrino mass.
Next, we discuss how the GW data help improve the constraint accuracies for other cosmological parameters. The constraint results of the derived parameters and are also listed in Table 1. Using the data combination of Planck+BAO+SN, we obtain: and for the NH case, and for the IH case, and and for the DH case. After considering the GW data of the ET, i.e., using the data combination of Planck+BAO+SN+GW, we obtain: and for the NH case, and for the IH case, and and for the DH case. Comparing the results from the two data combinations, we find that the accuracy of is increased by about and the accuracy of is increased by about when the GW data of the ET are considered in the cosmological fit. This indicates that the GW data of the ET can significantly improve the constraint accuracies of cosmological parameters.
We also display the two-dimensional posterior distribution contours in Figs. 24. From the blue contours (Planck+BAO+SN) in the and planes, we can see that is anti-correlated with and is positively correlated with . From the red contours (Planck+BAO+SN+GW) in the and planes, we find that, when the GW data are considered, the parameter space is greatly shrunk in each plane and the constraints on and become much tighter. Moreover, after adding the GW data, there is no obvious correlation between and other cosmological parameter. This indicates that the degeneracies between parameters including the neutrino mass existing in the cosmological EM observations can be effectively broken by the GW observations. This is because when we use the Planck data to do the cosmological fit, the parameter combinations must be constrained to a constant (the observed angular size of acoustic scale ), which will cause the degeneracies between parameters. Also, neither BAO or SN can truly measure the cosmological distances ( or ). But the GW data contain the absolute distance information at low redshifts relative to the CMB data, so they can help to break the degeneracies between and other parameters. Hence, the upper limits on the total neutrino mass are also reduced.
To briefly summarize, our results show that the GW data can indeed improve the constraints on the total neutrino mass. When the GW data of the ET are considered, tighter bounds on the total neutrino mass could be obtained. Also, after considering the GW data, the constraints on the derived parameters and become much tighter. In addition, the GW data can help break the degeneracies between and other parameters.
## Iv Conclusion
In this paper, we investigated the constraint capability of the GW observation of the ET on the total neutrino mass . We constrained the total neutrino mass in the CDM cosmology by using the simulated GW data of ET in combination with other cosmological data including CMB, BAO and SN. For the three-generation neutrinos, we considered the cases of normal hierarchy, inverted hierarchy, and degenerate hierarchy. For the GW data, we considered the CDM model as our fiducial model to simulate 1000 GW events that could be detected by the ET in its 10-year observation. In order to show the effect of the GW data, we used two data combinations including Planck+BAO+SN and Planck+BAO+SN+GW to constrain cosmological parameters.
Through comparing the constraint results from the two data combinations, we find that the GW data can indeed effectively improve the constraints on the total neutrino mass. Using Planck+BAO+SN, we obtain eV for the NH case, eV for the IH case, and eV for the DH case. After considering the GW data, i.e., using Planck+BAO+SN+GW, the constraint results become eV for the NH case, eV for the IH case, and eV for the DH case. It is found that the GW data can help reduce the upper limits of by 13.7%, 7.5%, and 10.3% for the NH, IH, and DH cases, respectively. For the derived parameters and , the GW data can significantly improve the constraint accuracies of them. The accuracy of is increased by about and the accuracy of is increased by about , when considering the GW data in the cosmological fit. In addition, the GW data can help to break the degeneracies between and other parameters.
###### Acknowledgements.
We would like to thank Zhou-Jian Cao, Tao Yang, Jue Zhang, and Wen Zhao for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grants No. 11690021 and No. 11522540), the Top-Notch Young Talents Program of China, and the Provincial Department of Education of Liaoning (Grant No. L2012087).
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http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion | # Kepler's laws of planetary motion
For a more historical approach, see in particular the articles Astronomia nova and Epitome Astronomiae Copernicanae.
In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.
Figure 1: Illustration of Kepler's three laws with two planetary orbits.
(1) The orbits are ellipses, with focal points ƒ1 and ƒ2 for the first planet and ƒ1 and ƒ3 for the second planet. The Sun is placed in focal point ƒ1.
(2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2.
(3) The total orbit times for planet 1 and planet 2 have a ratio a13/2 : a23/2.
1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1]
3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
Most planetary orbits are almost circles, and careful observation and calculation is required in order to establish that they are actually ellipses. Calculations of the orbit of the planet Mars first indicated to Johannes Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, also have elliptical orbits.
Kepler's work improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.[2]
Isaac Newton showed in 1687 that relationships like Kepler's would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation.
Kepler's laws are part of the foundation of modern astronomy and physics.[3]
## Comparison to Copernicus
Kepler's laws improve the model of Copernicus. If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agrees with Copernicus:
1. The planetary orbit is a circle
2. The Sun at the center of the orbit
3. The speed of the planet in the orbit is constant
4. The square of the sidereal period is proportional to the cube of the distance from the Sun.
The eccentricities of the orbits of the planets known to Copernicus and Kepler are small, so the rules above give good approximations of planetary motion, but Kepler's laws fit observations even better.
Kepler's corrections are not at all obvious:
1. The planetary orbit is not a circle, but an ellipse
2. The Sun is not at the center but at a focal point of the elliptical orbit
3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant.
4. The square of the sidereal period is proportional to the cube of the mean between the maximum and minimum distances from the Sun.
The nonzero eccentricity of the orbit of the earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the sun parallel to the equator of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately
$\varepsilon\approx\frac \pi 4 \frac {186-179}{186+179}\approx 0.015,$
which is close to the correct value (0.016710219). (See Earth's orbit). The calculation is correct when the perihelion, the date that the Earth is closest to the Sun, is on a solstice. The current perihelion, near January 4, is fairly close to the solstice on December 21 or 22.
## Nomenclature
It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws".[4][5] The Biographical Encyclopedia of Astronomers in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande.[6] It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler (1804) that made up the set of three laws, by adding in the third.[7] Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.[5][8]
Further, the current usage of "Kepler's Second Law" is something of a misnomer. Kepler had two versions of it, related in a qualitative sense, the "distance law" and the "area law". The "area law" is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.[9]
## History
Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.[10][2][11] Kepler's third law was published in 1619.[12][2]
Kepler in 1621 and Godefroy Wendelin in 1643 noted that Kepler's third law applies to the four brightest moons of Jupiter.[Nb 1] The second law, in the "area law" form, was contested by Nicolaus Mercator in a book from 1664; but by 1670 he was publishing in its favour in Philosophical Transactions, and as the century proceeded it became more widely accepted.[13] The reception in Germany changed noticeably between 1688, the year in which Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.[14]
Newton is credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.[15]
## Formulary
The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.
### First law
The orbit of every planet is an ellipse with the Sun at one of the two foci.
Figure 2: Kepler's first law placing the Sun at the focus of an elliptical orbit
Figure 4: Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = rmin and for θ = 180°, r = rmax.
Mathematically, an ellipse can be represented by the formula:
$r=\frac{p}{1+\varepsilon\, \cos\theta},$
where p is the semi-latus rectum, and ε is the eccentricity of the ellipse, and r is the distance from the Sun to the planet, and θ is the angle to the planet's current position from its closest approach, as seen from the Sun. So (rθ) are polar coordinates.
For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the sun at the centre.
At θ = 0°, perihelion, the distance is minimum
$r_\min=\frac{p}{1+\varepsilon}.$
At θ = 90° and at θ = 270°, the distance is equal to the semi-latus rectum.
At θ = 180°, aphelion, the distance is maximum
$r_\max=\frac{p}{1-\varepsilon}.$
The semi-major axis a is the arithmetic mean between rmin and rmax:
$\,r_\max - a=a-r_\min$
$a=\frac{p}{1-\varepsilon^2}.$
The semi-minor axis b is the geometric mean between rmin and rmax:
$\frac{r_\max} b =\frac b{r_\min}$
$b=\frac p{\sqrt{1-\varepsilon^2}}.$
The semi-latus rectum p is the harmonic mean between rmin and rmax:
$\frac{1}{r_\min}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_\max}$
$pa=r_\max r_\min=b^2\,.$
The eccentricity ε is the coefficient of variation between rmin and rmax:
$\varepsilon=\frac{r_\max-r_\min}{r_\max+r_\min}.$
The area of the ellipse is
$A=\pi a b\,.$
The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = π r2.
### Second law
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1]
The same blue area is swept out in a given time. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity
The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation where the planet travels faster when close to the sun, then slower when far from the sun. Kepler's second law states that the blue sector has constant area.
In a small time $dt\,$ the planet sweeps out a small triangle having base line $r\,$ and height $r \, d\theta$ and area $dA=\tfrac 1 2\cdot r\cdot r d\theta$ and so the constant areal velocity is $\frac{dA}{dt}=\tfrac{1}{2}r^2 \frac{d\theta}{dt}.$
The area enclosed by the elliptical orbit is $\pi ab.\,$ So the period $P\,$ satisfies
$P\cdot \tfrac 12r^2 \frac{d\theta}{dt}=\pi a b$
and the mean motion of the planet around the Sun
$n = 2\pi/P$
satisfies
$r^2 \, d\theta = a b n \,dt .$
### Third law
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
This captures the relationship between the distance of planets from the Sun, and their orbital periods.
For a brief biography of Kepler and discussion of his third law, see: NASA: Stargaze
Kepler enunciated in 1619 [12] this third law in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation.[16] So it was known as the harmonic law.[17]
Mathematically, the law says that the expression
$P^2/a^3$
has the same value for all the planets in the solar system. Here P is the time taken for a planet to complete an orbit round the sun, and a is the mean value between the maximum and minimum distances between the planet and sun.
The modern formulation, with the constant evaluated, reads as:
$\frac{T^2}{r^3} = \frac{4 \pi^2}{GM}$
where
• T is the orbital period of the orbiting body,
• M is the mass of the star,
• G is the universal gravitational constant and
• r is the radius, i.e. the semi-major axis of the ellipse.
In the full formulation under Newton's laws of motion, M should be replaced by
$M+m,$
where m is the mass of the orbiting body. Consequently, the proportionality constant is not truly the same for each planet. Nevertheless, given that m is so small relative to M for planets in our solar system, the approximation is good in the original setting.
## Planetary acceleration
A sudden sunward velocity change is applied to a planet. Then the areas of the triangles defined by the path of the planet for fixed time intervals will be equal. (Click on image for a detailed description.)
Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second law.
1. The direction of the acceleration is towards the Sun.
2. The magnitude of the acceleration is in inverse proportion to the square of the distance from the Sun.
This suggests that the Sun may be the physical cause of the acceleration of planets.
Newton defined the force on a planet to be the product of its mass and the acceleration. (See Newton's laws of motion). So:
1. Every planet is attracted towards the Sun.
2. The force on a planet is in direct proportion to the mass of the planet and in inverse proportion to the square of the distance from the Sun.
Here the Sun plays an unsymmetrical part, which is unjustified. So he assumed Newton's law of universal gravitation:
1. All bodies in the solar system attract one another.
2. The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.
As the planets have small masses compared to that of the Sun, the orbits conform to Kepler's laws approximately. Newton's model improves upon Kepler's model and fits actual observations more accurately. (See two-body problem).
A deviation in the motion of a planet from Kepler's laws due to the gravity of other planets is called a perturbation.
Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.
### Acceleration vector
From the heliocentric point of view consider the vector to the planet $\mathbf{r} = r \hat{\mathbf{r}}$ where $r$ is the distance to the planet and the direction $\hat {\mathbf{r}}$ is a unit vector. When the planet moves the directions change:
$\frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}} = \dot\theta \hat{\boldsymbol\theta},\qquad \frac{d\hat{\boldsymbol\theta}}{dt}=\dot{\hat{\boldsymbol\theta}} = -\dot\theta \hat{\mathbf{r}}$
where $\hat{\boldsymbol\theta}$ is the unit vector orthogonal to $\hat{\mathbf{r}}$ and pointing in the direction of rotation, and $\theta$ is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.
So differentiating the position vector twice to obtain the velocity and the acceleration vectors:
$\dot{\mathbf{r}} =\dot{r} \hat{\mathbf{r}} + r \dot{\hat{\mathbf{r}}} =\dot{r} \hat{\mathbf{r}} + r \dot{\theta} \hat{\boldsymbol{\theta}},$
$\ddot{\mathbf{r}} = (\ddot{r} \hat{\mathbf{r}} +\dot{r} \dot{\hat{\mathbf{r}}} ) + (\dot{r}\dot{\theta} \hat{\boldsymbol{\theta}} + r\ddot{\theta} \hat{\boldsymbol{\theta}} + r\dot{\theta} \dot{\hat{\boldsymbol{\theta}}}) = (\ddot{r} - r\dot{\theta}^2) \hat{\mathbf{r}} + (r\ddot{\theta} + 2\dot{r} \dot{\theta}) \hat{\boldsymbol{\theta}}.$
So
$\ddot{\mathbf{r}} = a_r \hat{\boldsymbol{r}}+a_\theta\hat{\boldsymbol{\theta}}$
$a_r=\ddot{r} - r\dot{\theta}^2$
and the transversal acceleration is
$a_\theta=r\ddot{\theta} + 2\dot{r} \dot{\theta}.$
### The inverse square law
Kepler's laws say that
$r^2\dot \theta = nab$
is constant.
The transversal acceleration $a_\theta$ is zero:
$\frac{d (r^2 \dot \theta)}{dt} = r (2 \dot r \dot \theta + r \ddot \theta ) = r a_\theta = 0.$
So the acceleration of a planet obeying Kepler's laws is directed towards the sun.
The radial acceleration $a_r$ is
$a_r = \ddot r - r \dot \theta^2= \ddot r - r \left(\frac{nab}{r^2} \right)^2= \ddot r -\frac{n^2a^2b^2}{r^3}.$
Kepler's first law states that the orbit is described by the equation:
$\frac{p}{r} = 1+ \varepsilon \cos\theta.$
Differentiating with respect to time
$-\frac{p\dot r}{r^2} = -\varepsilon \sin \theta \,\dot \theta$
or
$p\dot r = nab\,\varepsilon\sin \theta.$
Differentiating once more
$p\ddot r =nab \varepsilon \cos \theta \,\dot \theta =nab \varepsilon \cos \theta \,\frac{nab}{r^2} =\frac{n^2a^2b^2}{r^2}\varepsilon \cos \theta .$
The radial acceleration $a_r$ satisfies
$p a_r = \frac{n^2 a^2b^2}{r^2}\varepsilon \cos \theta - p\frac{n^2 a^2b^2}{r^3} = \frac{n^2a^2b^2}{r^2}\left(\varepsilon \cos \theta - \frac{p}{r}\right).$
Substituting the equation of the ellipse gives
$p a_r = \frac{n^2a^2b^2}{r^2}\left(\frac p r - 1 - \frac p r\right)= -\frac{n^2a^2}{r^2}b^2.$
The relation $b^2=pa$ gives the simple final result
$a_r=-\frac{n^2a^3}{r^2}.$
This means that the acceleration vector $\mathbf{\ddot r}$ of any planet obeying Kepler's first and second law satisfies the inverse square law
$\mathbf{\ddot r} = - \frac{\alpha}{r^2}\hat{\mathbf{r}}$
where
$\alpha = n^2 a^3\,$
is a constant, and $\hat{\mathbf r}$ is the unit vector pointing from the Sun towards the planet, and $r\,$ is the distance between the planet and the Sun.
According to Kepler's third law, $\alpha$ has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire solar system.
The inverse square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line. See Kepler orbit.
### Newton's law of gravitation
By Newton's second law, the gravitational force that acts on the planet is:
$\mathbf{F} = m_\text{Planet} \mathbf{\ddot r} = - m_\text{Planet} \alpha r^{-2} \hat{\mathbf{r}}$
where $m_\text{Planet}$ is the mass of the planet and $\alpha$ has the same value for all planets in the solar system. According to Newton's third Law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, $m_\text{Sun}$. So
$\alpha = Gm_\text{Sun}$
where $G$ is the gravitational constant.
The acceleration of solar system body number i is, according to Newton's laws:
$\mathbf{\ddot r_i} = G\sum_{j\ne i} m_j r_{ij}^{-2} \hat{\mathbf{r}}_{ij}$
where $m_j$ is the mass of body j, $r_{ij}$ is the distance between body i and body j, $\hat{\mathbf{r}}_{ij}$ is the unit vector from body i towards body j, and the vector summation is over all bodies in the world, besides i itself.
In the special case where there are only two bodies in the world, Earth and Sun, the acceleration becomes
$\mathbf{\ddot r}_\text{Earth} = G m_\text{Sun} r_{\text{Earth},\text{Sun}}^{-2} \hat{\mathbf{r}}_{\text{Earth},\text{Sun}}$
which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.
If the two bodies in the world are Moon and Earth the acceleration of the Moon becomes
$\mathbf{\ddot r}_\text{Moon} = G m_\text{Earth} r_{\text{Moon},\text{Earth}}^{-2} \hat{\mathbf{r}}_{\text{Moon},\text{Earth}}$
So in this approximation the Moon moves around the Earth according to Kepler's laws.
In the three-body case the accelerations are
$\mathbf{\ddot r}_\text{Sun} = G m_\text{Earth} r_{\text{Sun},\text{Earth}}^{-2} \hat{\mathbf{r}}_{\text{Sun},\text{Earth}} + G m_\text{Moon} r_{\text{Sun},\text{Moon}}^{-2} \hat{\mathbf{r}}_{\text{Sun},\text{Moon}}$
$\mathbf{\ddot r}_\text{Earth} = G m_\text{Sun} r_{\text{Earth},\text{Sun}}^{-2} \hat{\mathbf{r}}_{\text{Earth},\text{Sun}} + G m_\text{Moon} r_{\text{Earth},\text{Moon}}^{-2} \hat{\mathbf{r}}_{\text{Earth},\text{Moon}}$
$\mathbf{\ddot r}_\text{Moon} = G m_\text{Sun} r_{\text{Moon},\text{Sun}}^{-2} \hat{\mathbf{r}}_{\text{Moon},\text{Sun}}+ G m_\text{Earth} r_{\text{Moon},\text{Earth}}^{-2} \hat{\mathbf{r}}_{\text{Moon},\text{Earth}}$
These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations. See Lunar theory.
## Position as a function of time
Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.
The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, is the following four steps:
1. Compute the mean anomaly M = nt where n is the mean motion.
$n\cdot P=2\pi$ radians where P is the period.
2. Compute the eccentric anomaly E by solving Kepler's equation:
$\ M=E-\varepsilon \sin E$
3. Compute the true anomaly θ by the equation:
$(1-\varepsilon) \tan^2\frac \theta 2 = (1+\varepsilon) \tan^2\frac E 2$
4. Compute the heliocentric distance
$r=a(1-\varepsilon \cos E).$
The important special case of circular orbit, ε = 0, gives θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.
The proof of this procedure is shown below.
### Mean anomaly, M
FIgure 5: Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled S and the planet P. The auxiliary circle is an aid to calculation. Line xd is perpendicular to the base and through the planet P. The shaded sectors are arranged to have equal areas by positioning of point y.
The Keplerian problem assumes an elliptical orbit and the four points:
s the Sun (at one focus of ellipse);
z the perihelion
c the center of the ellipse
p the planet
and
$\ a=|cz|,$ distance between center and perihelion, the semimajor axis,
$\ \varepsilon={|cs|\over a},$ the eccentricity,
$\ b=a\sqrt{1-\varepsilon^2},$ the semiminor axis,
$\ r=|sp| ,$ the distance between Sun and planet.
$\theta=\angle zsp,$ the direction to the planet as seen from the Sun, the true anomaly.
The problem is to compute the polar coordinates (r,θ) of the planet from the time since periheliont.
It is solved in steps. Kepler considered the circle with the major axis as a diameter, and
$\ x,$ the projection of the planet to the auxiliary circle
$\ y,$ the point on the circle such that the sector areas |zcy| and |zsx| are equal,
$M=\angle zcy,$ the mean anomaly.
The sector areas are related by $|zsp|=\frac b a \cdot|zsx|.$
The circular sector area $\ |zcy| = \frac{a^2 M}2.$
The area swept since perihelion,
$|zsp|=\frac b a \cdot|zsx|=\frac b a \cdot|zcy|=\frac b a\cdot\frac{a^2 M}2 = \frac {a b M}{2},$
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
$M=n t, \,$
where n is the mean motion.
### Eccentric anomaly, E
When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ = f(M) is, however, not elementary.[18] Kepler's solution is to use
$E=\angle zcx$, x as seen from the centre, the eccentric anomaly
as an intermediate variable, and first compute E as a function of M by solving Kepler's equation below, and then compute the true anomaly θ from the eccentric anomaly E. Here are the details.
$\ |zcy|=|zsx|=|zcx|-|scx|$
$\frac{a^2 M}2=\frac{a^2 E}2-\frac {a\varepsilon\cdot a\sin E}2$
Division by a2/2 gives Kepler's equation
$M=E-\varepsilon\cdot\sin E.$
This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used.
Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.
### True anomaly, θ
Note from the figure that
$\overrightarrow{cd}=\overrightarrow{cs}+\overrightarrow{sd}$
so that
$a\cdot\cos E=a\cdot\varepsilon+r\cdot\cos \theta.$
Dividing by $a$ and inserting from Kepler's first law
$\ \frac r a =\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}$
to get
$\cos E =\varepsilon+\frac{1-\varepsilon^2}{1+\varepsilon\cdot\cos \theta}\cdot\cos \theta$$=\frac{\varepsilon\cdot(1+\varepsilon\cdot\cos \theta)+(1-\varepsilon^2)\cdot\cos \theta}{1+\varepsilon\cdot\cos \theta}$$=\frac{\varepsilon +\cos \theta}{1+\varepsilon\cdot\cos \theta}.$
The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.
A computationally more convenient form follows by substituting into the trigonometric identity:
$\tan^2\frac{x}{2}=\frac{1-\cos x}{1+\cos x}.$
Get
$\tan^2\frac{E}{2} =\frac{1-\cos E}{1+\cos E}$$=\frac{1-\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}{1+\frac{\varepsilon+\cos \theta}{1+\varepsilon\cdot\cos \theta}}$$=\frac{(1+\varepsilon\cdot\cos \theta)-(\varepsilon+\cos \theta)}{(1+\varepsilon\cdot\cos \theta)+(\varepsilon+\cos \theta)}$$=\frac{1-\varepsilon}{1+\varepsilon}\cdot\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\varepsilon}{1+\varepsilon}\cdot\tan^2\frac{\theta}{2}.$
Multiplying by 1+ε gives the result
$(1-\varepsilon)\cdot\tan^2\frac \theta 2 = (1+\varepsilon)\cdot\tan^2\frac E 2$
This is the third step in the connection between time and position in the orbit.
### Distance, r
The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:
$\ r\cdot(1+\varepsilon\cdot\cos \theta)=a\cdot(1-\varepsilon^2)$
Using the relation above between θ and E the final equation for the distance r is:
$\ r=a\cdot(1-\varepsilon\cdot\cos E).$
## Notes
1. ^ Godefroy Wendelin wrote a letter to Giovanni Battista Riccioli about the relationship between the distances of the Jovian moons from Jupiter and the periods of their orbits, showing that the periods and distances conformed to Kepler's third law. See: Joanne Baptista Riccioli, Almagestum novum … (Bologna (Bononia), (Italy): Victor Benati, 1651), volume 1, page 492 Scholia III. In the margin beside the relevant paragraph is printed: Vendelini ingeniosa speculatio circa motus & intervalla satellitum Jovis. (The clever Wendelin's speculation about the movement and distances of Jupiter's satellites.)
In 1621, Johannes Kepler had noted that Jupiter's moons obey (approximately) his third law in his Epitome Astronomiae Copernicanae [Epitome of Copernican Astronomy] (Linz (“Lentiis ad Danubium“), (Austria): Johann Planck, 1622), book 4, part 2, page 554.
## References
1. ^ a b Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.
2. ^ a b c Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. ISBN 0-8135-2908-5. Retrieved December 27, 2009.
3. ^ See also G. E. Smith, "Newton's Philosophiae Naturalis Principia Mathematica", especially the section Historical context ... in The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.).
4. ^ Voltaire, Eléments de la philosophie de Newton [Elements of Newton's Philosophy] (London, England: 1738). See, for example:
• From p. 162: "Par une des grandes loix de Kepler, toute Planete décrit des aires égales en temp égaux : par une autre loi non moins sûre, chaque Planete fait sa révolution autour du Soleil en telle sort, que si, sa moyenne distance au Soleil est 10. prenez le cube de ce nombre, ce qui sera 1000., & le tems de la révolution de cette Planete autour du Soleil sera proportionné à la racine quarrée de ce nombre 1000." (By one of the great laws of Kepler, each planet describes equal areas in equal times ; by another law no less certain, each planet makes its revolution around the sun in such a way that if its mean distance from the sun is 10, take the cube of that number, which will be 1000, and the time of the revolution of that planet around the sun will be proportional to the square root of that number 1000.)
• From p. 205: "Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ... " (It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun … )
5. ^ a b Wilson, Curtis (May 1994). "Kepler's Laws, So-Called". HAD News (Washington, DC: Historical Astronomy Division, American Astronomical Society) (31): 1–2. Retrieved December 27, 2009.
6. ^ De la Lande, Astronomie, vol. 1 (Paris, France: Desaint & Saillant, 1764). See, for example:
• From page 390: " … mais suivant la fameuse loi de Kepler, qui sera expliquée dans le Livre suivant (892), le rapport des temps périodiques est toujours plus grand que celui des distances, une planete cinq fois plus éloignée du soleil, emploie à faire sa révolution douze fois plus de temps ou environ; … " ( … but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (paragraph 892), the ratio of the periods is always greater than that of the distances [so that, for example,] a planet five times farther from the sun, requires about twelve times or so more time to make its revolution [around the sun]; … )
• From page 429: "Les Quarrés des Temps périodiques sont comme les Cubes des Distances. 892. La plus fameuse loi du mouvement des planetes découverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient à les parcourir; … " (The squares of the periods are as the cubes of the distances. 892. The most famous law of the movement of the planets discovered by Kepler is that of the relation that exists between the sizes of their orbits and the times that the [planets] require to traverse them; … )
• From page 430: "Les Aires sont proportionnelles au Temps. 895. Cette loi générale du mouvement des planetes devenue si importante dans l'Astronomie, sçavior, que les aires sont proportionnelles au temps, est encore une des découvertes de Kepler; … " (Areas are proportional to times. 895. This general law of the movement of the planets [which has] become so important in astronomy to know, [namely] that areas are proportional to times, is one of Kepler's discoveries; … )
• From page 435: "On a appellé cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce célebre Inventeur; … " (One called this law of areas proportional to times (the law of Kepler) as well as that of paragraph 892, by the name of that celebrated inventor; … )
7. ^ Robert Small, An account of the astronomical discoveries of Kepler (London, England: J Mawman, 1804), pp. 298–299.
8. ^ Robert Small, An account of the astronomical discoveries of Kepler (London, England: J. Mawman, 1804).
9. ^ Bruce Stephenson (1994). Kepler's Physical Astronomy. Princeton University Press. p. 170. ISBN 0-691-03652-7.
10. ^ In his Astronomia nova, Kepler presented only a proof that Mars' orbit is elliptical. Evidence that the other known planets' orbits are elliptical was presented only in 1621.
See: Johannes Kepler, Astronomia nova … (1609), p. 285. After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical. From the top of page 285: "Ergo ellipsis est Planetæ iter; … " (Thus, an ellipse is the planet's [i.e., Mars'] path; … ) Later on the same page: " … ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; … " ( … as will be revealed in the next chapter: where it will also then be proved that any figure of the planet's orbit must be relinquished, except a perfect ellipse; … ) And then: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ) The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289–290.
Kepler stated that all planets travel in elliptical orbits having the Sun at one focus in: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III. De Figura Orbitæ (III. On the figure [i.e., shape] of orbits), pages 658-665. From p. 658: "Ellipsin fieri orbitam planetæ … " (Of an ellipse is made a planet's orbit … ). From p. 659: " … Sole (Foco altero huius ellipsis) … " ( … the Sun (the other focus of this ellipse) … ).
11. ^ In his Astronomia nova ... (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome of 1621. Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".
• His "distance law" is presented in: "Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." (Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, Astronomia nova … (1609), pp. 165–167. On page 167, Kepler states: " … , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." ( … , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.) That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times. However, as Kepler presented it, his argument is accurate only for circles, not ellipses.
• His "area law" is presented in: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ), Protheorema XIV and XV, pp. 291–295. On the top p. 294, it reads: "Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it. Therefore, the area that is swept out by a radius from the Sun to Mars as Mars moves along an arc of its elliptical orbit is proportional to the time that Mars requires to move along that arc. Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.
In 1621, Kepler restated his second law for any planet: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, page 668. From page 668: "Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales: accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem." (It has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun.) That is, a planet's speed along its orbit is inversely proportional to its distance from the Sun. (The remainder of the paragraph makes clear that Kepler was referring to what is now called angular velocity.)
12. ^ a b Johannes Kepler, Harmonices Mundi [The Harmony of the World] (Linz, (Austria): Johann Planck, 1619), book 5, chapter 3, p. 189. From the bottom of p. 189: "Sed res est certissima exactissimaque quod proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis mediarum distantiarum, … " (But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialternate proportion [i.e., the ratio of 3:2] of their mean distances, … ")
An English translation of Kepler's Harmonices Mundi is available as: Johannes Kepler with E.J. Aiton, A.M. Duncan, and J.V. Field, trans., The Harmony of the World (Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especially p. 411.
13. ^ Wilbur Applebaum (13 June 2000). Encyclopedia of the Scientific Revolution: From Copernicus to Newton. Routledge. p. 603. ISBN 978-1-135-58255-5.
14. ^ Roy Porter (25 September 1992). The Scientific Revolution in National Context. Cambridge University Press. p. 102. ISBN 978-0-521-39699-8.
15. ^ Victor Guillemin; Shlomo Sternberg (2006). Variations on a Theme by Kepler. American Mathematical Soc. p. 5. ISBN 978-0-8218-4184-6.
16. ^ Burtt, Edwin. The Metaphysical Foundations of Modern Physical Science. p. 52.
17. ^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 45. ISBN 0-8135-2908-5.
18. ^ MÜLLER, M (1995). "EQUATION OF TIME -- PROBLEM IN ASTRONOMY". Acta Physica Polonica A. Retrieved 23 February 2013.
## Bibliography
• A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J. L. (1971) [1966]. "Dynamics, 2nd ed". New York: John Wiley. ISBN 0-471-59601-9..
• Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0-521-57597-4
• V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0-387-96890-3 | {"extraction_info": {"found_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": 124, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9347970485687256, "perplexity": 1820.1384512464863}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131298015.2/warc/CC-MAIN-20150323172138-00186-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.physicsoverflow.org/16221/what-neatest-way-describe-non-autonomous-lagrangian-system | # What is the neatest way to describe a “non-autonomous” (lagrangian) system?
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The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$ if the constraint is nice enough (i.e. if $0$ is a regular value of $\Phi$), can be described as a $k$-dimensional submanifold of $\mathbb R ^{3n}$, and this clearly has some advantages. Those systems are called “autonomous” (and if someone could throw some light on the terminology I'd also be grate).
Now, suppose that $\Phi$ depends on time, i.e. the constraints are:$$\Phi(x,t)=0.$$ In this case it isn't immediately obvious to me what would be the most natural way to describe the configuration space. For example, one might define $g^t(x)=\Phi(x,t)$, suppose that $0\in \mathbb R ^{3n-k}$ is still a regular value of $g^t$ and define the configuration space at time $t$ as $$M^t=(g^t)^ {-1}(0),$$ and describe the position of the system at time $t$ with $k+1$ parameters $(q_1,\dots,q_k,t)$. This works good if, for example, the manifolds $M^t$ are essentially the same: for example, if $M^t\subset \mathbb R ^3$ is a ring that rotates about the $z$ axis, the manifold is simply $S^1$. But is this description always possible? I mean, $M^t$ could possibly change in time so that the coordinates $(q_1,\dots,q_k,t)$ don't mean a thing for some $t$.
Another conceivable way, I suppose, would be to consider the ($k+1$-dimensional) manifold: $$M=\Phi ^{-1}(0)\subset \mathbb R ^{3n+1}\ni (x_1,...,x_n,t).$$ For example, in this post, I sketched a proof of the Noether's theorem for non-autonomous systems following a similar idea. The main problem that occurs to me is that, in this way, we make the statement of propositions like, for example, d'Alembert principle more complicated, because we can no more consider virtual displacements as tangent vectors to the configuration space.
So, to summarize: given a constraint of the form $\Phi(x,t)=0$, what is the most natural way to describe the configuration space of such a system?
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user pppqqq
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Regardless of the form of whichever (holonomic) constraints you may have, non-autonomous systems are most naturally understood from a field theoretical viewpoint. More precisely, one should understand Lagrangian mechanics as Lagrangian field theory in $0+1$ dimensions (that is, the space-time manifold is just the real time line). There, coordinates over each time instant constitute the fibers (which could well be manifolds instead of vector spaces) of a fiber bundle over the real line, and the pairing of positions and velocities are the fibers of the corresponding first-order jet bundle of this fiber bundle. Histories, on their turn, are then understood as field configurations. Hence, the Noether theorem for non-autonomous systems is just the Noether theorem for classical field theory. One advantage of this viewpoint is that the apparent distinction in the description of autonomous and non-autonomous systems disappears.
Holonomic constraints can still be understood, as in the autonomous case, as conditions on initial data which are preserved under the dynamics. In other words, there should be an one-to-one correspondence between solutions of the equations of motion and initial data at a certain instant of time (= "initial-data surface") satisfying such constraints. In particular, there can be no "topology change" among fibers, in compliance with the fiber bundle picture. Notice as well that, since the base manifold is 1-dimensional, all fiber bundles over it are trivial, so the total space of the bundle is always of the form (position space) $\times$ (time). With these considerations in mind, one concludes that holonomic constraints of the kind $\Phi(x,t)=0$ should specify a sub-bundle of the above fiber bundle. More generally, if the constraints involve the veolcities as well, one should instead consider sub-bundles of the first-order jet bundle.
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
answered Feb 8, 2014 by (580 points)
Hi Pedro, I'm sorry but I know nothing about field theory, and I can understand little from the second part of your answer. Could you possibly try to state it in simpler terms? From what I get, you are saying that the world line of the system always lies in a $k+1$-dimensional manifold of the form $\mathbb R \times M$, where $k=$ number of constraints? Also, by saying that “there should be a 1-1 correspondece...“, do you mean the uniqueness of the solution of the equations of motion? I'm sorry, but my “vocabulary” is a bit limited. Thank you.
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user pppqqq
Indeed, all possible worldlines live in a manifold of the kind $\mathbb{R}\times M$, where $M$ can be thought of as the "position manifold". However, this includes both worldlines which are solutions of the equations of motion and those which are not. If there were no constraints in the dynamics, the dimension $k$ of $M$ would be the number of degrees of freedom of the system. Once you have constraints as above, you have to count the number of physical degrees of freedom, which is $k$ minus the number of independent constraints. This is the dimension of the so-called constraint manifold.
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
As for the "1-1 correspondence" I've mentioned, you got it right, it refers to uniqueness of the solutions of the equations of motion for given initial data.
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
(continued from my first comment) The constraint manifold is just the submanifold of $M$ where the "physical" (i.e. constrained) trajectories live. One can see this viewpoint as a "field theoretical" one because each worldline can be thought of as a "field configuration" over the time line. The value of the field configuration at a given time $t$ is just the position of the trajectory at that time.
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register. | {"extraction_info": {"found_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": 0.8818063735961914, "perplexity": 329.6288521896708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1652662604794.68/warc/CC-MAIN-20220526100301-20220526130301-00391.warc.gz"} |
https://interhacker.wordpress.com/2016/01/02/algorithms-notes-chapter-8/ | # Algorithms Notes – Chapter 8
xkcd/761
Depth-First Search
One thing that is important to understand is that the Breadth-First Search algorithm generates a tree as it searches a graph (for a proof of this, refer to Lemma 22.6 in Chapter 22 of Introduction to Algorithms 3rd Edition by Cormen et al.). This is called a Breadth-First Tree. A Depth-First Search, on the other hand, does not generate a single tree. Instead, it generates several trees (called depth-first trees) and creates a depth-first forest.
Check this animation out to get a general understand of how a depth-first search works.
Here is the pseudocode for the Depth-First Search algorithm:
```
DFS(G)
for each vertex u ϵ G.V
u.color = WHITE
u.π = NIL
time = 0
for each vertex u ϵ G.V
if u.color = WHITE
DFS-VISIT(G, u)
DFS-VISIT(G, u)
time = time + 1 // white vertex discovered
u.d = time
u.color = GRAY
for each v ϵ G.Adj[u] // explore edge (u, v)
if v.color == WHITE
v.π = u
DFS-VISIT(G, v)
u.color = BLACK // u is finished. Mark it black.
time = time + 1
u.f = time
```
Notation
Most of the notation is very similar to that used in the Breadth-First Search pseudocode with one key difference. Let $v$ be a vertex in $G$. The property $v.d$ does not store the distance from the source, unlike in BFS. Instead it, along with the new property $v.f$ store time-stamps.
• $v.d$ stores the time when the vertex $v$ is discovered by the algorithm
• $v.f$ stores the time when the algorithm finishes scanning the adjacency list of $v$ and sets its color to black.
• The global variable $time$ is used for the above described time-stamping.
The following two GIFs provide a good comparison between a BFS and a DFS. | {"extraction_info": {"found_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": 9, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8001118898391724, "perplexity": 448.59034299856785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818695726.80/warc/CC-MAIN-20170926122822-20170926142822-00232.warc.gz"} |
https://www.ncbi.nlm.nih.gov/pubmed/8410105?dopt=Abstract | Format
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J Clin Epidemiol. 1993 Oct;46(10):1203-11.
# The design of prospective epidemiological studies: more subjects or better measurements?
### Author information
1
Academic Department of Genito-Urinary Medicine, University College and Middlesex School of Medicine, London.
### Abstract
Prospective epidemiological studies which seek to relate potential risk factors to the risk of disease are subject to appreciable biases which are often unrecognized. The inability to precisely measure subjects' true values of the risk factors under consideration tends to result in bias towards unity in the univariate relative risks associated with them--the more imprecisely a risk factor is measured, the greater the bias. When correlated risk factors are measured with different degrees of imprecision the adjusted relative risk associated with them can be biased towards or away from unity. When designing a new prospective study cost considerations usually limit the total number of subject-evaluations that are available. The usual design approach is to maximize the study size and evaluate each subject on one occasion only. An alternative approach involves recruitment of a smaller number of subjects so that each can be evaluated on more than one occasion, thus resulting in a more precise measure of subjects' risk factor values and hence less bias in the relative risk estimates. In this paper we use a simulation approach to show that under conditions that prevail for most major prospective epidemiological studies the latter approach is actually more likely to produce accurate relative risk estimates. This emphasizes the importance of bias due to exposure measurement imprecision and suggests that attempts to anticipate and control it be given at least as high a priority as that given to sample size assessment in the design of epidemiological studies.
### Comment in
PMID:
8410105
DOI:
10.1016/0895-4356(93)90120-p
[Indexed for MEDLINE] | {"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": 0.803769052028656, "perplexity": 1305.863152123148}, "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-2019-47/segments/1573496668594.81/warc/CC-MAIN-20191115065903-20191115093903-00520.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/212823-hypothesis-tests-regression-coefficients.html | # Thread: Hypothesis tests for regression coefficients
1. ## Hypothesis tests for regression coefficients
Hey,
doing a question on hypothesis testing for regression coefficients and a formula I have to work out the t value is t = (β - b)/root( σ2(X'X)-1jj) for my beta I have 0.04399 and for sigma squared I have 0.0215 and for (X'X)-1 I have ( 0.28553 -0.0058 0.00713; -0.0058 0.00013 -0.00019; 0.00713 -0.00019 0.00047) the only thing I'm struggling with is understanding how to calculate (σ2 * (X'X)33 ) don't really know how to calculate the matrix bit really, all I know is it's the diagonal element. Any help appreciated thanks!
Nathan
2. ## Re: Hypothesis tests for regression coefficients
Hey NathanBUK.
Just calculate the transpose of the matrix X (which is X') and then multiply this by X and you should get a covariance estimate matrix which you can use for inference. | {"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": 0.9607821106910706, "perplexity": 1233.2194989551977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1526794863811.3/warc/CC-MAIN-20180520224904-20180521004904-00475.warc.gz"} |
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